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Preface Man kann einen jeden Begriff, einen jeden Titel, darunter viele Erkenntnisse geh¨ oren, einen logischen Ort nennen. Immanuel Kant [258, p. B 324]
This book’s title subject, The Topos of Music, has been chosen to communicate a double message: First, the Greek word “topos” (τ ´ oπoς = location, site) alludes to the logical and transcendental location of the concept of music in the sense of Aristotle’s [20, 592] and Kant’s [258, p. B 324] topic. This view deals with the question of where music is situated as a concept— and hence with the underlying ontological problem: What is the type of being and existence of music? The second message is a more technical understanding insofar as the system of musical signs can be associated with the mathematical theory of topoi, which realizes a powerful synthesis of geometric and logical theories. It laid the foundation of a thorough geometrization of logic and has been successful in central issues of algebraic geometry (Grothendieck, Deligne), independence proofs and intuitionistic logic (Cohen, Lawvere, Kripke). But this second message is intimately entwined with the first since the present concept framework of the musical sign system is technically based on topos theory, so the topos of music receives its topos-theoretic foundation. In this perspective, the double message of the book’s title in fact condenses to a unified intention: to unite philosophical insight with mathematical explicitness. According to Birkh¨ auser’s initial plan in 1996, this book was first conceived as an English translation of my former book Geometrie der T¨ one [340], since the German original had suffered from its restricted access to the international public. However, the scientific progress since 1989, when it was written, has been considerable in theory and technology. We have known new subjects, such as the denotator concept framework, performance theory, and new software platforms for composition, analysis, and performance, such as RUBATOr or OpenMusic. Modeling concepts via the denotator approach in fact results from an intense collaboration of mathematicians and computer scientists in the object-oriented programming paradigm and supported by several international research grants. v
vi Also, the scientific acceptance of mathematical music theory has grown since its beginnings in the late 1970s. As the first acceptance of mathematical music theory was testified to by von Karajan’s legendary Ostersymposium “Musik und Mathematik” in 1984 in Salzburg [190], so is the significantly improved present status of acceptance testified to by the Fourth Diderot Forum on Mathematics and Music [365] in Paris, Vienna, and Lisbon 1999, which was organized by the European Mathematical Society. The corresponding extension of collaborative efforts in particular entail the inclusion of works by other research groups in this book, such as the “American Set Theory”, the Swedish school of performance research at Stockholm’s KTH, or the research on computer-aided composition at the IRCAM in Paris. Therefore, as a result of these revised conditions, The Topos of Music appears as a vastly extended English update of the original work. The extension is visibly traced in the following parts which are new with respect to [340]: Part II exposes the theory of denotators and forms, part V introduces the topological theories of rhythms and motives, part VIII introduces the structure theory of performance, part IX deals with the expressive semantics of performance in the language of performance operators and stemmata (genealogical trees of successively refined performance), part X is devoted to the description of the RUBATOr software platform for representation, analysis, composition, and performance, part XI presents a statistical analysis of musical analysis, part XII concludes the subject of performance with an inverse performance theory, in fact a first formalization of the problem of music criticism. This does however not mean that the other parts are just translations of the German text. Considerable progress has been made in most fields, except the last part XIV which reproduces the status quo in [340]. In particular, the local and global theories have been thoroughly functorialized and thereby introduce an ontological depth and variability of concepts, techniques, and results, which by far transcend the semiotically naive geometric approach in [340]. The present theory is as different from the traditional geometric conceptualization as is Grothendieck’s topos theoretic algebraic geometry from classical algebraic geometry in the spirit of Segre, van der Waerden, or Zariski. Beyond this topos-theoretic generalization, the denotator language also introduces a fairly exceptional technique of circular concept constructions. This more precisely is rooted in Finsler’s pioneering work in foundations of set theory [153], a thread which has been rediscovered in modern theoretical computer sciences [4]. The present state of denotator theory rightly could be termed a Galois theory of concepts in the sense that circular definitions of concepts play the role of conceptual equations (corresponding to algebraic equations in algebraic Galois theory), the solutions of which are concepts instead of algebraic numbers. Accordingly, the mathematical apparatus has been vastly extended, not only in the field of topos theory and its intuitionistic logic, but also with regard to general and algebraic topology, ordinary and partial differential equations, P´olya theory, statistics, multiaffine algebra and functorial algebraic geometry. It is mandatory that these technicalities had to be placed in a more elaborate semiotic perspective. However, this book does not cover the full range of music semiotics, for which the reader is referred to [361]. Of course, such an extension on the technical level has consequences for the readability of the theory. In view of the present volume of over 1300 pages, we could however not even make the attempt to approach a non-technical presentation. This subject is left to subsequent efforts. The critical reader may put the question whether music is really that complex. The answer is yes, and the reason is straightforward: We cannot pretend that Bach, Haydn, Mozart, or Beethoven, just to name some of the most prominent
vii composers, are outstanding geniuses and have elaborated masterworks of eternal value, without trying to understand such singular creations with adequate tools, and this means: of adequate depth and power. After all, understanding God’s ‘composition’, the material universe, cannot be approached without the most sophisticated tools as they have been elaborated in physics, chemistry, and molecular biology. So who is recommended to read this book? A first category of readers is evidently the working scientist in the fields of mathematical music theory, the soft- and hardware engineer in music informatics, but also the mathematician who is interested in new applications from the above fields of pure mathematics. A second category are those theoretical mathematicians or computer scientists interested in the Galois theory of concepts; they may discover interesting unsolved problems. A third category of potential readers are all those who really want to get an idea of what music is about, of how one may conceptualize and turn into language the “ineffable” in music for the common language. Those who insist on the dogma that precision and beauty contradict each other, and that mathematics only produces tautologies and therefore must fail when aiming at substantial knowledge, should not read such a book. Despite the technical character of The Topos of Music, there are at least four different approaches to its reading. To begin with, one may read it as a philosophical text, concentrating on the qualitative passages, surfing over technical portions and leaving those paragraphs to others. One may also take the book as a dictionary for computational musicology, including its concept framework and the lists of musical objects and processes (such as modulation degrees, contrapuntal steps) in the appendices. Observe however, that not all existing important lists have been included. For example, the list of all-interval series and the list of self-addressed chords are omitted, the reader may find these lists in other publications. Thirdly, the working scientist will have to read the full-fledged technicalities. And last, but not least, one may take the book as a source for ideas of how to go on with the whole subject of music. The GPL (General Public License1 ) software sources in the appended CD-ROM may support further development. The prerequisites to a more in-depth reading of this book are these. Generally speaking, a good acquaintance with formal reasoning as mathematics (including formal logic) preconizes, is a conditio sine qua non. As to musicology and music theory, the familiarity with elementary concepts, like chords, motives, rhythm, and also musical notation, as well as a real interest in understanding music and not simply (ab)using it, are recommended. For the more computeroriented passages, familiarity with the paradigm of object-oriented programming is profitable. We have not included the appendix on mathematical basics because it should help the reader get familiar with mathematics, but as an orientation in fields where the specialized mathematician possibly needs a specification of concepts and notation. The appendix was also included to expose the spectrum of mathematics which is needed to tackle the formal problems of computational musicology. It is by no means an overkill of mathematization: We have even omitted some non-trivial fields, such as statistics or Lambda calculus, for which we have to apologize. There are different supporting instances to facilitate orientation in this book. To begin with, the table of contents and an extensive subject and name index may help find one’s keywords. Further, following the list of contents, a leitfaden (on page xxix) is included for a generic navigation. Each chapter and section is headed by a summary that offers a first orientation 1A
legal matter file is contained in the book’s CD-ROM, see page xxx.
viii about specific contents. Finally, the book is also available as a file ToposOfMusic.pdf with bookmarks and active cross-references in the appended CD-ROM (see page xxx for its contents). This version is also attractive because the figures’ colors are visible only in this version. In order to obtain a consistent first reading, we recommend chapters 1 to 5, and then appendix A: Common Parameter Spaces (appendix B is not mandatory here, though it gives a good and not so technical overview of auditory physiology). After that, the reader may go on with chapter 6 on denotators and then follow the outline of the leitfaden (see page xxix). This book could not have been realized without the engaged support of nineteen collaborators and contributors. Above all, my PhD students Stefan G¨oller and Stefan M¨ uller at the MultiMedia Laboratory of the Department of Information Technology at the University of Zurich have collaborated in the production of this book on the levels of the LATEX installation, the final production of hundreds of figures, and the contributions sections 20.2 through 20.5 (G¨oller) and sections 46.3 through 46.3.6.2 (M¨ uller). My special gratitude goes to their truly collaborative spirit. Contributions to this book have been delivered by (in alphabetic order): By Carlos Agon, and G´erard Assayag (both IRCAM) with their precious Lambda-calculus-oriented presentation of the object-oriented programming principles in the composition software OpenMusic described in chapter 51, Moreno Andreatta (IRCAM) with an elucidating discourse on the American Set Theory in section 11.5.2 and section 16.3, Jan Beran (Universit¨at Konstanz) with his contribution to the compositional strategies in his original composition [49] in section 11.5.1.1, as well as with his inspiring work on statistics as reported in chapters 43 and 44, Chantal Buteau (Universit¨at and ETH Z¨ urich) with her detailed review of chapter 22, Roberto Ferretti (ETH urich) with his progressive contributions to the algebraic geometry of inverse performance Z¨ theory in sections 39.8 and 46.2, Anja Fleischer (Technische Universit¨at Berlin) with her short but critical preliminaries in chapter 23, Harald Fripertinger (Universit¨at Graz) with his ‘killer’ formulas concerning enumeration of finite local and global compositions in sections 11.4, 16.2.2 and appendix C.3.6, J¨ org Garbers (Technische Universit¨at Berlin) with his portation of the RUBATOr application to Mac OS X, as documented in the screenshots in chapters 40, 41, Werner Hemmert (Infineon) with a very up-to-date presentation of room acoustics in section A.1.1.1 and auditory physiology in appendix B.1 (we would have loved to include more of his knowledge), Michael Leyton (DIMACS, Rutgers University) with a formidable cover figure entitled “Dark Theory”, a beautiful subtitle to this book, as well as with innumerable discussions around time and its reduction to symmetries as presented in chapter 47, Emilio Lluis Puebla (UNAM, Mexico City) with his unique and engaged promotion and dissipation of mathematical music theory on the American continent, especially also in the preparation and critical review of this book, Mariana Montiel Hernandez (UNAM, Mexico City) with her critical review of the theory of circular forms and denotators in section 6.5 and appendix G.2.2.1, Thomas Noll (Technische Universit¨ at Berlin) with his substantial contributions to the functorial theory of compositions, and for his revolutionary rebuilding of Riemann’s harmony and its relations to counterpoint, Joachim Stange-Elbe (Universit¨at Osnabr¨ uck) with a very clear and innovative description of his outstanding RUBATOr performance of Bach’s contrapunctus III in the Art of Fugue in sections 42.2 through 42.4.3, Hans Straub with his adventurous extensions of classical cadence theory in section 26.2.2 and his classification of four-element motives in appendix M.4, and, last but not least, Oliver Zahorka (Out Media Design), my former collaborator and chief programmer of the NeXT RUBATOr application, which has contributed so much to the
ix success of the Z¨ urich school of performance theory. To all of them, I owe my deepest gratitude and recognition for their sweat and tears. My sincere acknowledgments go to Alexander Grothendieck, whose encouraging letters and, no doubt, awe inspiring revolution in mathematical thinking has given me so much in isolated phases of this enterprise. My acknowledgments also go to my engaged mentor Peter Stucki, director of the MultiMedia Laboratory of the Department of Information Technology at the University of Zurich; without his support, this book would have seen its birthday years later, if ever. My thanks also go to my brother Silvio, who once again (he did it already for my first book [328]) supported the final review efforts by an ideal environment in his villa in Vulpera. My thanks also go to the unbureaucratic management of the book’s production by Birkh¨auser’s lector Thomas Hempfling and the very patient copy editor Edwin Beschler. All these beautiful supports would have failed without my wife Christina’s infinite understanding and vital environment—if this book is a trace of humanity, it is also, and strongly, hers.
Vulpera, June 2002
Guerino Mazzola
Contents I
Introduction and Orientation
1
1 What is Music About? 1.1 Fundamental Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Fundamental Scientific Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Topography 2.1 Layers of Reality . . . . . . . . . . . . . . . . 2.1.1 Physical Reality . . . . . . . . . . . . 2.1.2 Mental Reality . . . . . . . . . . . . . 2.1.3 Psychological Reality . . . . . . . . . . 2.2 Molino’s Communication Stream . . . . . . . 2.2.1 Creator and Poietic Level . . . . . . . 2.2.2 Work and Neutral Level . . . . . . . . 2.2.3 Listener and Esthesic Level . . . . . . 2.3 Semiosis . . . . . . . . . . . . . . . . . . . . . 2.3.1 Expressions . . . . . . . . . . . . . . . 2.3.2 Content . . . . . . . . . . . . . . . . . 2.3.3 The Process of Signification . . . . . . 2.3.4 A Short Overview of Music Semiotics 2.4 The Cube of Local Topography . . . . . . . . 2.5 Topographical Navigation . . . . . . . . . . .
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3 Musical Ontology 23 3.1 Where is Music? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Depth and Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4 Models and Experiments in Musicology 4.1 Interior and Exterior Nature . . . . . . . . . . 4.2 What Is a Musicological Experiment? . . . . 4.3 Questions—Experiments of the Mind . . . . . 4.4 New Scientific Paradigms and Collaboratories xi
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29 32 33 34 35
xii
II
CONTENTS
Navigation on Concept Spaces
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5 Navigation 5.1 Music in the EncycloSpace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Receptive Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Productive Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39 40 44 45
6 Denotators 6.1 Universal Concept Formats . . . . . . . . . . . . . . 6.1.1 First Naive Approach To Denotators . . . . . 6.1.2 Interpretations and Comments . . . . . . . . 6.1.3 Ordering Denotators and ‘Concept Leafing’ . 6.2 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Variable Addresses . . . . . . . . . . . . . . . 6.2.2 Formal Definition . . . . . . . . . . . . . . . . 6.2.3 Discussion of the Form Typology . . . . . . . 6.3 Denotators . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Formal Definition of a Denotator . . . . . . . 6.4 Anchoring Forms in Modules . . . . . . . . . . . . . 6.4.1 First Examples and Comments on Modules in 6.5 Regular and Circular Forms . . . . . . . . . . . . . . 6.6 Regular Denotators . . . . . . . . . . . . . . . . . . . 6.7 Circular Denotators . . . . . . . . . . . . . . . . . . 6.8 Ordering on Forms and Denotators . . . . . . . . . . 6.8.1 Concretizations and Applications . . . . . . . 6.9 Concept Surgery and Denotator Semantics . . . . . .
47 48 50 55 58 61 61 63 66 67 67 69 70 76 79 85 89 93 99
III
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Local Theory
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103
7 Local Compositions 7.1 The Objects of Local Theory . . . . . . 7.2 First Local Music Objects . . . . . . . . 7.2.1 Chords and Scales . . . . . . . . 7.2.2 Local Meters and Local Rhythms 7.2.3 Motives . . . . . . . . . . . . . . 7.3 Functorial Local Compositions . . . . . 7.4 First Elements of Local Theory . . . . . 7.5 Alterations Are Tangents . . . . . . . . 7.5.1 The Theorem of Mason–Mazzola
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105 106 108 109 114 118 121 122 127 129
8 Symmetries and Morphisms 8.1 Symmetries in Music . . . . . . . . 8.1.1 Elementary Examples . . . 8.2 Morphisms of Local Compositions 8.3 Categories of Local Compositions .
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135 137 139 154 158
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CONTENTS 8.3.1 8.3.2 8.3.3 8.3.4 8.3.5
xiii Commenting the Concatenation Principle . . . Embedding and Addressed Adjointness . . . . Universal Constructions on Local Compositions The Address Question . . . . . . . . . . . . . . Categories of Commutative Local Compositions
9 Yoneda Perspectives 9.1 Morphisms Are Points . . . . . . . . . . . . . . . 9.2 Yoneda’s Fundamental Lemma . . . . . . . . . . 9.3 The Yoneda Philosophy . . . . . . . . . . . . . . 9.4 Understanding Fine and Other Arts . . . . . . . 9.4.1 Painting and Music . . . . . . . . . . . . . 9.4.2 The Art of Object-Oriented Programming 10 Paradigmatic Classification 10.1 Paradigmata in Musicology, Linguistics, and 10.2 Transformation . . . . . . . . . . . . . . . . 10.3 Similarity . . . . . . . . . . . . . . . . . . . 10.4 Fuzzy Concepts in the Humanities . . . . .
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161 163 166 169 171
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175 178 181 184 185 185 188
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Mathematics . . . . . . . . . . . . . . . . . . . . . . . .
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11 Orbits 11.1 Gestalt and Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Framework for Local Classification . . . . . . . . . . . . . . . . . . . . . . 11.3 Orbits of Elementary Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Classification Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 The Local Classification Theorem . . . . . . . . . . . . . . . . . . . . . 11.3.3 The Finite Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.4 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.5 Chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.6 Empirical Harmonic Vocabularies . . . . . . . . . . . . . . . . . . . . . . 11.3.7 Self-addressed Chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.8 Motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Enumeration Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 P´ olya and de Bruijn Theory . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Big Science for Big Numbers . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Group-theoretical Methods in Composition and Theory . . . . . . . . . . . . . 11.5.1 Aspects of Serialism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 The American Tradition . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Esthetic Implications of Classification . . . . . . . . . . . . . . . . . . . . . . . 11.6.1 Jakobson’s Poetic Function . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 Motivic Analysis: Schubert/Stolberg “Lied auf dem Wasser zu singen...” 11.6.3 Composition: Mazzola/Baudelaire “La mort des artistes” . . . . . . . . 11.7 Mathematical Reflections on Historicity in Music . . . . . . . . . . . . . . . . . 11.7.1 Jean-Jacques Nattiez’ Paradigmatic Theme . . . . . . . . . . . . . . . . 11.7.2 Groups as a Parameter of Historicity . . . . . . . . . . . . . . . . . . . .
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203 203 204 205 205 207 216 217 219 221 225 228 231 232 238 241 243 247 258 259 262 268 271 272 272
xiv
CONTENTS
12 Topological Specialization 12.1 What Ehrenfels Neglected . . . . . . . . . . . . . . . . . 12.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Metrical Comparison . . . . . . . . . . . . . . . . 12.2.2 Specialization Morphisms of Local Compositions 12.3 The Problem of Sound Classification . . . . . . . . . . . 12.3.1 Topographic Determinants of Sound Descriptions 12.3.2 Varieties of Sounds . . . . . . . . . . . . . . . . . 12.3.3 Semiotics of Sound Classification . . . . . . . . . 12.4 Making the Vague Precise . . . . . . . . . . . . . . . . .
IV
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Global Theory
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275 276 277 279 281 284 284 291 294 295
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13 Global Compositions 13.1 The Local-Global Dichotomy in Music . . . . . . . . . . . . . . . . 13.1.1 Musical and Mathematical Manifolds . . . . . . . . . . . . . 13.2 What Are Global Compositions? . . . . . . . . . . . . . . . . . . . 13.2.1 The Nerve of an Objective Global Composition . . . . . . . 13.3 Functorial Global Compositions . . . . . . . . . . . . . . . . . . . . 13.4 Interpretations and the Vocabulary of Global Concepts . . . . . . . 13.4.1 Iterated Interpretations . . . . . . . . . . . . . . . . . . . . 13.4.2 The Pitch Domain: Chains of Thirds, Ecclesiastical Modes, Quaternary Degrees . . . . . . . . . . . . . . . . . . . . . . 13.4.3 Interpreting Time: Global Meters and Rhythms . . . . . . . 13.4.4 Motivic Interpretations: Melodies and Themes . . . . . . .
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. 318 . 326 . 331
14 Global Perspectives 14.1 Musical Motivation . . . . . . . . . . . . . . . . . 14.2 Global Morphisms . . . . . . . . . . . . . . . . . 14.3 Local Domains . . . . . . . . . . . . . . . . . . . 14.4 Nerves . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Simplicial Weights . . . . . . . . . . . . . . . . . 14.6 Categories of Commutative Global Compositions
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333 333 334 341 343 345 347
15 Global Classification 15.1 Module Complexes . . . . . . . . . . . . . . . . . 15.1.1 Global Affine Functions . . . . . . . . . . 15.1.2 Bilinear and Exterior Forms . . . . . . . . 15.1.3 Deviation: Compositions vs. “Molecules” 15.2 The Resolution of a Global Composition . . . . . 15.2.1 Global Standard Compositions . . . . . . 15.2.2 Compositions from Module Complexes . . 15.3 Orbits of Module Complexes Are Classifying . . 15.3.1 Combinatorial Group Actions . . . . . . .
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349 350 350 353 355 356 356 358 363 364
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299 300 307 308 310 314 316 317
CONTENTS
xv
15.3.2 Classifying Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 16 Classifying Interpretations 16.1 Characterization of Interpretable Compositions . . . . . . . . 16.1.1 Automorphism Groups of Interpretable Compositions 16.1.2 A Cohomological Criterion . . . . . . . . . . . . . . . 16.2 Global Enumeration Theory . . . . . . . . . . . . . . . . . . . 16.2.1 Tesselation . . . . . . . . . . . . . . . . . . . . . . . . 16.2.2 Mosaics . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.3 Classifying Rational Rhythms and Canons . . . . . . . 16.3 Global American Set Theory . . . . . . . . . . . . . . . . . . 16.4 Interpretable “Molecules” . . . . . . . . . . . . . . . . . . . .
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369 370 372 374 376 376 378 380 382 385
17 Esthetics and Classification 387 17.1 Understanding by Resolution: An Illustrative Example . . . . . . . . . . . . . . . 387 17.2 Var`ese’s Program and Yoneda’s Lemma . . . . . . . . . . . . . . . . . . . . . . . 392 18 Predicates 18.1 What Is the Case: The Existence Problem . . . . . . . 18.1.1 Merging Systematic and Historical Musicology 18.2 Textual and Paratextual Semiosis . . . . . . . . . . . . 18.2.1 Textual and Paratextual Signification . . . . . 18.3 Textuality . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.1 The Category of Denotators . . . . . . . . . . . 18.3.2 Textual Semiosis . . . . . . . . . . . . . . . . . 18.3.3 Atomic Predicates . . . . . . . . . . . . . . . . 18.3.4 Logical and Geometric Motivation . . . . . . . 18.4 Paratextuality . . . . . . . . . . . . . . . . . . . . . . .
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397 397 398 400 401 402 402 406 412 419 424
19 Topoi of Music 19.1 The Grothendieck Topology . . . . 19.1.1 Cohomology . . . . . . . . . 19.1.2 Marginalia on Presheaves . 19.2 The Topos of Music: An Overview
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427 427 430 434 435
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439 439 442 442 443 444 445 446 446 448 448
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20 Visualization Principles 20.1 Problems . . . . . . . . . . . . . . . 20.2 Folding Dimensions . . . . . . . . . . 20.2.1 R2 → R . . . . . . . . . . . . 20.2.2 Rn → R . . . . . . . . . . . . 20.2.3 An Explicit Construction of µ 20.3 Folding Denotators . . . . . . . . . . 20.3.1 Folding Limits . . . . . . . . 20.3.2 Folding Colimits . . . . . . . 20.3.3 Folding Powersets . . . . . . 20.3.4 Folding Circular Denotators .
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xvi
CONTENTS 20.4 Compound Parametrized Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 20.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
V
Topologies for Rhythm and Motives
453
21 Metrics and Rhythmics 21.1 Review of Riemann and Jackendoff–Lerdahl Theories 21.1.1 Riemann’s Weights . . . . . . . . . . . . . . . 21.1.2 Jackendoff–Lerdahl: Intrinsic Versus Extrinsic 21.2 Topologies of Global Meters and Associated Weights 21.3 Macro-Events in the Time Domain . . . . . . . . . .
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455 455 456 457 459 461
22 Motif Gestalts 22.1 Motivic Interpretation . . . . . . . . . . . 22.2 Shape Types . . . . . . . . . . . . . . . . 22.2.1 Examples of Shape Types . . . . . 22.3 Metrical Similarity . . . . . . . . . . . . . 22.3.1 Examples of Distance Functions . 22.4 Paradigmatic Groups . . . . . . . . . . . . 22.4.1 Examples of Paradigmatic Groups 22.5 Pseudo-metrics on Orbits . . . . . . . . . 22.6 Topologies on Gestalts . . . . . . . . . . . 22.6.1 The Inheritance Property . . . . . 22.6.2 Cognitive Aspects of Inheritance . 22.6.3 Epsilon Topologies . . . . . . . . . 22.7 First Properties of the Epsilon Topologies 22.7.1 Toroidal Topologies . . . . . . . . 22.8 Rudolph Reti’s Motivic Analysis Revisited 22.8.1 Review of Concepts . . . . . . . . 22.8.2 Reconstruction . . . . . . . . . . . 22.9 Motivic Weights . . . . . . . . . . . . . .
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465 466 468 469 472 472 473 475 477 479 479 481 482 484 487 490 491 493 496
VI
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Harmony
499
23 Critical Preliminaries 501 23.1 Hugo Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 23.2 Paul Hindemith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 23.3 Heinrich Schenker and Friedrich Salzer . . . . . . . . . . . . . . . . . . . . . . . . 503 24 Harmonic Topology 24.1 Chord Perspectives . . . . . . . . 24.1.1 Euler Perspectives . . . . 24.1.2 12-tempered Perspectives 24.1.3 Enharmonic Projection .
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505 506 506 512 514
CONTENTS 24.2 Chord 24.2.1 24.2.2 24.2.3 24.2.4
xvii Topologies . . . . . . . . Extension and Intension Extension and Intension Faithful Addresses . . . The Saturation Sheaf .
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518 518 520 523 526
25 Harmonic Semantics 25.1 Harmonic Signs—Overview . . . . . . . . . . . . . . . . 25.2 Degree Theory . . . . . . . . . . . . . . . . . . . . . . . 25.2.1 Chains of Thirds . . . . . . . . . . . . . . . . . . 25.2.2 American Jazz Theory . . . . . . . . . . . . . . . 25.2.3 Hans Straub: General Degrees in General Scales 25.3 Function Theory . . . . . . . . . . . . . . . . . . . . . . 25.3.1 Canonical Morphemes for European Harmony . . 25.3.2 Riemann Matrices . . . . . . . . . . . . . . . . . 25.3.3 Chains of Thirds . . . . . . . . . . . . . . . . . . 25.3.4 Tonal Functions from Absorbing Addresses . . .
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529 530 532 532 534 537 538 540 543 545 546
26 Cadence 26.1 Making the Concept Precise . . . . . . . . . . . . . . . . . . . 26.2 Classical Cadences Relating to 12-tempered Intonation . . . . 26.2.1 Cadences in Triadic Interpretations of Diatonic Scales 26.2.2 Cadences in More General Interpretations . . . . . . . 26.3 Cadences in Self-addressed Tonalities of Morphology . . . . . 26.4 Self-addressed Cadences by Symmetries and Morphisms . . . 26.5 Cadences for Just Intonation . . . . . . . . . . . . . . . . . . 26.5.1 Tonalities in Third-Fifth Intonation . . . . . . . . . . 26.5.2 Tonalities in Pythagorean Intonation . . . . . . . . . .
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551 552 553 553 555 556 558 560 560 561
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563 564 565 565 568 571 574 576 581 586 586 587 590 591
27 Modulation 27.1 Modeling Modulation by Particle Interaction . . . . . . . . 27.1.1 Models and the Anthropic Principle . . . . . . . . . 27.1.2 Classical Motivation and Heuristics . . . . . . . . . . 27.1.3 The General Background . . . . . . . . . . . . . . . 27.1.4 The Well-Tempered Case . . . . . . . . . . . . . . . 27.1.5 Reconstructing the Diatonic Scale from Modulation 27.1.6 The Case of Just Tuning . . . . . . . . . . . . . . . . 27.1.7 Quantized Modulations and Modulation Domains for 27.2 Harmonic Tension . . . . . . . . . . . . . . . . . . . . . . . 27.2.1 The Riemann Algebra . . . . . . . . . . . . . . . . . 27.2.2 Weights on the Riemann Algebra . . . . . . . . . . . 27.2.3 Harmonic Tensions from Classical Harmony? . . . . 27.2.4 Optimizing Harmonic Paths . . . . . . . . . . . . . .
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xviii
CONTENTS
28 Applications 28.1 First Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.1.1 Johann Sebastian Bach: Choral from “Himmelfahrtsoratorium” 28.1.2 Wolfgang Amadeus Mozart: “Zauberfl¨ote”, Choir of Priests . . 28.1.3 Claude Debussy: “Pr´eludes”, Livre 1, No.4 . . . . . . . . . . . 28.2 Modulation in Beethoven’s Sonata op.106, 1st Movement . . . . . . . . 28.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.2 The Fundamental Theses of Erwin Ratz and Jrgen Uhde . . . . 28.2.3 Overview of the Modulation Structure . . . . . . . . . . . . . . 28.2.4 Modulation B[ G via e−3 in W . . . . . . . . . . . . . . . . 28.2.5 Modulation G E[ via Ug in W . . . . . . . . . . . . . . . . . 28.2.6 Modulation E[ D/b from W to W ∗ . . . . . . . . . . . . . . B via Ud/d] = Ug] /a within W ∗ . . . . . . 28.2.7 Modulation D/b 28.2.8 Modulation B B[ from W ∗ to W . . . . . . . . . . . . . . . 28.2.9 Modulation B[ G[ via Ub[ within W . . . . . . . . . . . . . 28.2.10 Modulation G[ G via Ua[ /a within W . . . . . . . . . . . . . 28.2.11 Modulation G B[ via e3 within W . . . . . . . . . . . . . . . 28.3 Rhythmical Modulation in “Synthesis” . . . . . . . . . . . . . . . . . . 28.3.1 Rhythmic Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3.2 Composition for Percussion Ensemble . . . . . . . . . . . . . .
VII
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Counterpoint
29 Melodic Variation by Arrows 29.1 Arrows and Alterations . . . . . . . . . . . 29.2 The Contrapuntal Interval Concept . . . . . 29.3 The Algebra of Intervals . . . . . . . . . . . 29.3.1 The Third Torus . . . . . . . . . . . 29.4 Musical Interpretation of the Interval Ring 29.5 Self-addressed Arrows . . . . . . . . . . . . 29.6 Change of Orientation . . . . . . . . . . . .
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593 594 595 598 600 603 603 605 607 608 608 608 609 609 610 610 610 610 611 613
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617 617 619 620 620 622 625 626
30 Interval Dichotomies as a Contrast 30.1 Dichotomies and Polarity . . . . . . . . . . . . . . . . 30.2 The Consonance and Dissonance Dichotomy . . . . . . 30.2.1 Fux and Riemann Consonances Are Isomorphic 30.2.2 Induced Polarities . . . . . . . . . . . . . . . . 30.2.3 Empirical Evidence for the Polarity Function . 30.2.4 Music and the Hippocampal Gate Function . .
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31 Modeling Counterpoint by Local Symmetries 31.1 Deformations of the Strong Dichotomies . . . . . . . . . . . . . . . . . . . . . . 31.2 Contrapuntal Symmetries Are Local . . . . . . . . . . . . . . . . . . . . . . . . 31.3 The Counterpoint Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
645 . 645 . 647 . 649
CONTENTS
xix
31.3.1 31.3.2 31.3.3 31.3.4
Some Preliminary Calculations . . . . . . . . . . . . . . . . . . . . . . . Two Lemmata on Cardinalities of Intersections . . . . . . . . . . . . . . An Algorithm for Exhibiting the Contrapuntal Symmetries . . . . . . . Transfer of the Counterpoint Rules to General Representatives of Strong Dichotomies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 The Classical Case: Consonances and Dissonances . . . . . . . . . . . . . . . . 31.4.1 Discussion of the Counterpoint Theorem in the Light of Reduced Strict Style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4.2 The Major Dichotomy—A Cultural Antipode? . . . . . . . . . . . . . .
VIII
Structure Theory of Performance
. 649 . 651 . 651 . 655 . 655 . 656 . 657
661
32 Local and Global Performance Transformations 32.1 Performance as a Reality Switch . . . . . . . . . . . . . . . . 32.2 Why Do We Need Infinite Performance of the Same Piece? . 32.3 Local Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3.1 The Coherence of Local Performance Transformations 32.3.2 Differential Morphisms of Local Compositions . . . . . 32.4 Global Structure . . . . . . . . . . . . . . . . . . . . . . . . . 32.4.1 Modeling Performance Syntax . . . . . . . . . . . . . . 32.4.2 The Formal Setup . . . . . . . . . . . . . . . . . . . . 32.4.3 Performance qua Interpretation of Interpretation . . .
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663 665 666 667 667 668 672 674 675 679
33 Performance Fields 33.1 Classics: Tempo, Intonation, and Dynamics . . . . . . 33.1.1 Tempo . . . . . . . . . . . . . . . . . . . . . . . 33.1.2 Intonation . . . . . . . . . . . . . . . . . . . . . 33.1.3 Dynamics . . . . . . . . . . . . . . . . . . . . . 33.2 Genesis of the General Formalism . . . . . . . . . . . . 33.2.1 The Question of Articulation . . . . . . . . . . 33.2.2 The Formalism of Performance Fields . . . . . 33.3 What Performance Fields Signify . . . . . . . . . . . . 33.3.1 Th.W. Adorno, W. Benjamin, and D. Raffman 33.3.2 Towards Composition of Performance . . . . .
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681 681 681 683 685 686 687 689 690 691 693
34 Initial Sets and Initial Performances 34.1 Taking off with a Shifter . . . . . . . 34.2 Anchoring Onset . . . . . . . . . . . 34.3 The Concert Pitch . . . . . . . . . . 34.4 Dynamical Anchors . . . . . . . . . . 34.5 Initializing Articulation . . . . . . . 34.6 Hit Point Theory . . . . . . . . . . . 34.6.1 Distances . . . . . . . . . . . 34.6.2 Flow Interpolation . . . . . .
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695 696 697 699 701 701 703 704 706
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xx
CONTENTS
35 Hierarchies and Performance Scores 35.1 Performance Cells . . . . . . . . . . . 35.2 The Category of Performance Cells . . 35.3 Hierarchies . . . . . . . . . . . . . . . 35.3.1 Operations on Hierarchies . . . 35.3.2 Classification Issues . . . . . . 35.3.3 Example: The Piano and Violin 35.4 Local Performance Scores . . . . . . . 35.5 Global Performance Scores . . . . . . 35.5.1 Instrumental Fibers . . . . . .
IX
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Expressive Semantics
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711 711 713 714 718 718 722 723 728 728
731
36 Taxonomy of Expressive Performance 36.1 Feelings: Emotional Semantics . . . . . 36.2 Motion: Gestural Semantics . . . . . . 36.3 Understanding: Rational Semantics . . 36.4 Cross-semantical Relations . . . . . . .
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733 734 737 741 745
37 Performance Grammars 37.1 Rule-based Grammars . . . . . . 37.1.1 The KTH School . . . . . 37.1.2 Neil P. McAgnus Todd . . 37.1.3 The Zurich School . . . . 37.2 Remarks on Learning Grammars
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747 748 749 751 752 753
38 Stemma Theory 38.1 Motivation from Practising and Rehearsing . . . . . . . . . . . . . 38.1.1 Does Reproducibility of Performances Help Understanding? 38.2 Tempo Curves Are Inadequate . . . . . . . . . . . . . . . . . . . . 38.3 The Stemma Concept . . . . . . . . . . . . . . . . . . . . . . . . . 38.3.1 The General Setup of Matrilineal Sexual Propagation . . . 38.3.2 The Primary Mother—Taking Off . . . . . . . . . . . . . . 38.3.3 Mono- and Polygamy—Local and Global Actions . . . . . . 38.3.4 Family Life—Cross-Correlations . . . . . . . . . . . . . . .
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755 756 757 758 762 763 765 769 771
39 Operator Theory 39.1 Why Weights? . . . . . . . . . 39.1.1 Discrete and Continuous 39.1.2 Weight Recombination . 39.2 Primavista Weights . . . . . . . 39.2.1 Dynamics . . . . . . . . 39.2.2 Agogics . . . . . . . . . 39.2.3 Tuning and Intonation . 39.2.4 Articulation . . . . . . .
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773 774 775 776 777 777 780 782 783
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CONTENTS
xxi
39.2.5 Ornaments . . . . . . . . . . . . 39.3 Analytical Weights . . . . . . . . . . . . 39.4 Taxonomy of Operators . . . . . . . . . 39.4.1 Splitting Operators . . . . . . . . 39.4.2 Symbolic Operators . . . . . . . 39.4.3 Physical Operators . . . . . . . . 39.4.4 Field Operators . . . . . . . . . . 39.5 Tempo Operator . . . . . . . . . . . . . 39.6 Scalar Operator . . . . . . . . . . . . . . 39.7 The Theory of Basis-Pianola Operators 39.7.1 Basis Specialization . . . . . . . 39.7.2 Pianola Specialization . . . . . . 39.8 Locally Linear Grammars . . . . . . . .
X
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RUBATOr
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783 785 787 788 789 791 792 793 794 795 797 801 801
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40 Architecture 807 40.1 The Overall Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 40.2 Frame and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 41 The 41.1 41.2 41.3 41.4 41.5
RUBETTEr Family MetroRUBETTEr . . . . MeloRUBETTEr . . . . . HarmoRUBETTEr . . . . PerformanceRUBETTEr PrimavistaRUBETTEr .
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42 Performance Experiments 42.1 A Preliminary Experiment: Robert Schumann’s 42.2 Full Experiment: J.S. Bach’s “Kunst der Fuge” 42.3 Analysis . . . . . . . . . . . . . . . . . . . . . . 42.3.1 Metric Analysis . . . . . . . . . . . . . . 42.3.2 Motif Analysis . . . . . . . . . . . . . . 42.3.3 Omission of Harmonic Analysis . . . . . 42.4 Stemma Constructions . . . . . . . . . . . . . . 42.4.1 Performance Setup . . . . . . . . . . . . 42.4.2 Instrumental Setup . . . . . . . . . . . . 42.4.3 Global Discussion . . . . . . . . . . . .
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813 814 816 819 824 831
“Kuriose Geschichte” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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833 833 834 835 835 839 841 841 842 849 850
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Statistics of Analysis and Performance
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853
43 Analysis of Analysis 855 43.1 Hierarchical Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 43.1.1 General Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855
xxii
CONTENTS
43.1.2 Hierarchical Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857 43.1.3 Hierarchical Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 858 43.2 Comparing Analyses of Bach, Schumann, and Webern . . . . . . . . . . . . . . . 860 44 Differential Operators and Regression 44.0.1 Analytical Data . . . . . . . . . . . . . . 44.1 The Beran Operator . . . . . . . . . . . . . . . 44.1.1 The Concept . . . . . . . . . . . . . . . 44.1.2 The Formalism . . . . . . . . . . . . . . 44.2 The Method of Regression Analysis . . . . . . . 44.2.1 The Full Model . . . . . . . . . . . . . . 44.2.2 Step Forward Selection . . . . . . . . . . 44.3 The Results of Regression Analysis . . . . . . . 44.3.1 Relations between Tempo and Analysis 44.3.2 Complex Relationships . . . . . . . . . . 44.3.3 Commonalities and Diversities . . . . . 44.3.4 Overview of Statistical Results . . . . .
XII
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Inverse Performance Theory
XIII
Operationalization of Poiesis
871 873 874 874 877 880 880 881 881 882 883 884 897
903
45 Principles of Music Critique 45.1 Boiling down Infinity—Is Feuilletonism Inevitable? . . . . . . . . . . . . . . . . 45.2 “Political Correctness” in Performance—Reviewing Gould . . . . . . . . . . . . 45.3 Transversal Ethnomusicology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Critical Fibers 46.1 The Stemma Model of Critique . . . . . . . . . . . . . . . . . . . . . 46.2 Fibers for Locally Linear Grammars . . . . . . . . . . . . . . . . . . 46.3 Algorithmic Extraction of Performance Fields . . . . . . . . . . . . . 46.3.1 The Infinitesimal View on Expression . . . . . . . . . . . . . 46.3.2 Real-time Processing of Expressive Performance . . . . . . . 46.3.3 Score–Performance Matching . . . . . . . . . . . . . . . . . . 46.3.4 Performance Field Calculation . . . . . . . . . . . . . . . . . 46.3.5 Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . 46.3.6 The EspressoRUBETTEr : An Interactive Tool for Expression 46.4 Local Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46.4.1 Comparing Argerich and Horowitz . . . . . . . . . . . . . . .
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905 . 905 . 906 . 909
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911 911 912 916 916 917 918 919 921 922 925 927
931
47 Unfolding Geometry and Logic in Time 933 47.1 Performance of Logic and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 934 47.2 Constructing Time from Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 935 47.3 Discourse and Insight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937
CONTENTS
xxiii
48 Local and Global Strategies in Composition 48.1 Local Paradigmatic Instances . . . . . . . . . . 48.1.1 Transformations . . . . . . . . . . . . . 48.1.2 Variations . . . . . . . . . . . . . . . . . 48.2 Global Poetical Syntax . . . . . . . . . . . . . . 48.2.1 Roman Jakobson’s Horizontal Function 48.2.2 Roland Posner’s Vertical Function . . . 48.3 Structure and Process . . . . . . . . . . . . . .
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939 940 940 941 941 942 942 943
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945 945 948 949 952
50 Case Study I:“Synthesis” by Guerino Mazzola 50.1 The Overall Organization . . . . . . . . . . . . . . . . . . . . . 50.1.1 The Material: 26 Classes of Three-Element Motives . . . 50.1.2 Principles of the Four Movements and Instrumentation . 50.2 1st Movement: Sonata Form . . . . . . . . . . . . . . . . . . . . 50.3 2nd Movement: Variations . . . . . . . . . . . . . . . . . . . . . 50.4 3rd Movement: Scherzo . . . . . . . . . . . . . . . . . . . . . . . 50.5 4th Movement: Fractal Syntax . . . . . . . . . . . . . . . . . . .
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955 956 956 956 958 959 963 964
51 Object-Oriented Programming in OpenMusic 51.1 Object-Oriented Language . . . . . . . . . . . . 51.1.1 Patches . . . . . . . . . . . . . . . . . . 51.1.2 Objects . . . . . . . . . . . . . . . . . . 51.1.3 Classes . . . . . . . . . . . . . . . . . . 51.1.4 Methods . . . . . . . . . . . . . . . . . . 51.1.5 Generic Functions . . . . . . . . . . . . 51.1.6 Message Passing . . . . . . . . . . . . . 51.1.7 Inheritance . . . . . . . . . . . . . . . . 51.1.8 Boxes and Evaluation . . . . . . . . . . 51.1.9 Instantiation . . . . . . . . . . . . . . . 51.2 Musical Object Framework . . . . . . . . . . . 51.2.1 Internal Representation . . . . . . . . . 51.2.2 Interface . . . . . . . . . . . . . . . . . . 51.3 Maquettes: Objects in Time . . . . . . . . . . . 51.4 Meta-object Protocol . . . . . . . . . . . . . . . 51.4.1 Reification of Temporal Boxes . . . . . . 51.5 A Musical Example . . . . . . . . . . . . . . . .
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967 968 969 969 970 970 971 971 971 972 973 973 973 975 978 982 984 986
49 The 49.1 49.2 49.3 49.4
Paradigmatic Discourse on prestor The prestor Functional Scheme . . . . . Modular Affine Transformations . . . . Ornaments and Variations . . . . . . . . Problems of Abstraction . . . . . . . . .
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xxiv
CONTENTS
XIV
String Quartet Theory
991
52 Historical and Theoretical Prerequisites 52.1 History . . . . . . . . . . . . . . . . . . . . . . 52.2 Theory of the String Quartet Following Ludwig 52.2.1 Four Part Texture . . . . . . . . . . . . 52.2.2 The Topos of Conversation Among Four 52.2.3 The Family of Violins . . . . . . . . . .
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993 994 994 995 996 997
53 Estimation of Resolution Parameters 999 53.1 Parameter Spaces for Violins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000 53.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003 54 The 54.1 54.2 54.3
XV
Case of Counterpoint Counterpoint . . . . . . Harmony . . . . . . . . Effective Selection . . .
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Harmony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix: Sound
A Common Parameter Spaces A.1 Physical Spaces . . . . . . . . . . . . A.1.1 Neutral Data . . . . . . . . . A.1.2 Sound Analysis and Synthesis A.2 Mathematical and Symbolic Spaces . A.2.1 Onset and Duration . . . . . A.2.2 Amplitude and Crescendo . . A.2.3 Frequency and Glissando . .
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1013 . 1013 . 1014 . 1018 . 1028 . 1028 . 1029 . 1031
B Auditory Physiology and Psychology B.1 Physiology: From the Auricle to Heschl’s Gyri . . . . . . . . . B.1.1 Outer Ear . . . . . . . . . . . . . . . . . . . . . . . . . B.1.2 Middle Ear . . . . . . . . . . . . . . . . . . . . . . . . B.1.3 Inner Ear (Cochlea) . . . . . . . . . . . . . . . . . . . B.1.4 Cochlear Hydrodynamics: The Travelling Wave . . . . B.1.5 Active Amplification of the Traveling Wave Motion . . B.1.6 Neural Processing . . . . . . . . . . . . . . . . . . . . B.2 Discriminating Tones: Werner Meyer-Eppler’s Valence Theory B.3 Aspects of Consonance and Dissonance . . . . . . . . . . . . . B.3.1 Euler’s Gradus Function . . . . . . . . . . . . . . . . . B.3.2 von Helmholtz’ Beat Model . . . . . . . . . . . . . . . B.3.3 Psychometric Investigations by Plomp and Levelt . . . B.3.4 Counterpoint . . . . . . . . . . . . . . . . . . . . . . . B.3.5 Consonance and Dissonance: A Conceptual Field . . .
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1035 . 1036 . 1036 . 1037 . 1037 . 1041 . 1042 . 1044 . 1046 . 1049 . 1049 . 1051 . 1052 . 1052 . 1053
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CONTENTS
XVI
xxv
Appendix: Mathematical Basics
1055
C Sets, Relations, Monoids, Groups C.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . C.1.1 Examples of Sets . . . . . . . . . . . . . . C.2 Relations . . . . . . . . . . . . . . . . . . . . . . C.2.1 Universal Constructions . . . . . . . . . . C.2.2 Graphs and Quivers . . . . . . . . . . . . C.2.3 Monoids . . . . . . . . . . . . . . . . . . . C.3 Groups . . . . . . . . . . . . . . . . . . . . . . . . C.3.1 Homomorphisms of Groups . . . . . . . . C.3.2 Direct, Semi-direct, and Wreath Products C.3.3 Sylow Theorems on p-groups . . . . . . . C.3.4 Classification of Groups . . . . . . . . . . C.3.5 General Affine Groups . . . . . . . . . . . C.3.6 Permutation Groups . . . . . . . . . . . .
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D Rings and Algebras D.1 Basic Definitions and Constructions . . . . . D.1.1 Universal Constructions . . . . . . . . D.2 Prime Factorization . . . . . . . . . . . . . . D.3 Euclidean Algorithm . . . . . . . . . . . . . . D.4 Approximation of Real Numbers by Fractions D.5 Some Special Issues . . . . . . . . . . . . . . . D.5.1 Integers, Rationals, and Real Numbers
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E Modules, Linear, and Affine Transformations E.1 Modules and Linear Transformations . . . . . . . . . . . . . E.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . E.2 Module Classification . . . . . . . . . . . . . . . . . . . . . . E.2.1 Dimension . . . . . . . . . . . . . . . . . . . . . . . . E.2.2 Endomorphisms on Dual Numbers . . . . . . . . . . E.2.3 Semi-Simple Modules . . . . . . . . . . . . . . . . . E.2.4 Jacobson Radical and Socle . . . . . . . . . . . . . . E.2.5 Theorem of Krull–Remak–Schmidt . . . . . . . . . . E.3 Categories of Modules and Affine Transformations . . . . . E.3.1 Direct Sums . . . . . . . . . . . . . . . . . . . . . . . E.3.2 Affine Forms and Tensors . . . . . . . . . . . . . . . E.3.3 Biaffine Maps . . . . . . . . . . . . . . . . . . . . . . E.3.4 Symmetries of the Affine Plane . . . . . . . . . . . . E.3.5 Symmetries on Z2 . . . . . . . . . . . . . . . . . . . E.3.6 Symmetries on Zn . . . . . . . . . . . . . . . . . . . E.3.7 Complements on the Module of a Local Composition E.3.8 Fiber Products and Fiber Sums in Mod . . . . . . . E.4 Complements of Commutative Algebra . . . . . . . . . . . .
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xxvi
CONTENTS E.4.1 E.4.2 E.4.3 E.4.4
Localization . . . . Projective Modules Injective Modules . Lie Algebras . . .
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F Algebraic Geometry F.1 Locally Ringed Spaces . . . . . . . . . . . . . . . F.2 Spectra of Commutative Rings . . . . . . . . . . F.2.1 Sober Spaces . . . . . . . . . . . . . . . . F.3 Schemes and Functors . . . . . . . . . . . . . . . F.4 Algebraic and Geometric Structures on Schemes F.4.1 The Zariski Tangent Space . . . . . . . . F.5 Grassmannians . . . . . . . . . . . . . . . . . . . F.6 Quotients . . . . . . . . . . . . . . . . . . . . . .
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G Categories, Topoi, and Logic G.1 Categories Instead of Sets . . . . . . . . . . . . . . . . . . . . . G.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . G.1.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . G.1.3 Natural Transformations . . . . . . . . . . . . . . . . . . G.2 The Yoneda Lemma . . . . . . . . . . . . . . . . . . . . . . . . G.2.1 Universal Constructions: Adjoints, Limits, and Colimits G.2.2 Limit and Colimit Characterizations . . . . . . . . . . . G.3 Topoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.3.1 Subobject Classifiers . . . . . . . . . . . . . . . . . . . . G.3.2 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . G.3.3 Definition of Topoi . . . . . . . . . . . . . . . . . . . . . G.4 Grothendieck Topologies . . . . . . . . . . . . . . . . . . . . . . G.4.1 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . G.5 Formal Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.5.1 Propositional Calculus . . . . . . . . . . . . . . . . . . . G.5.2 Predicate Logic . . . . . . . . . . . . . . . . . . . . . . . G.5.3 A Formal Setup for Consistent Domains of Forms . . . .
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H Complements on General and Algebraic Topology H.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . H.1.1 General . . . . . . . . . . . . . . . . . . . . . . H.1.2 The Category of Topological Spaces . . . . . . H.1.3 Uniform Spaces . . . . . . . . . . . . . . . . . . H.1.4 Special Issues . . . . . . . . . . . . . . . . . . . H.2 Algebraic Topology . . . . . . . . . . . . . . . . . . . . H.2.1 Simplicial Complexes . . . . . . . . . . . . . . . H.2.2 Geometric Realization of a Simplicial Complex H.2.3 Contiguity . . . . . . . . . . . . . . . . . . . . . H.3 Simplicial Coefficient Systems . . . . . . . . . . . . . .
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CONTENTS
xxvii
H.3.1 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1150 I
Complements on Calculus I.1 Abstract on Calculus . . . . . . . . . . . . . . . . I.1.1 Norms and Metrics . . . . . . . . . . . . . I.1.2 Completeness . . . . . . . . . . . . . . . . I.1.3 Differentiation . . . . . . . . . . . . . . . I.2 Ordinary Differential Equations (ODEs) . . . . . I.2.1 The Fundamental Theorem: Local Case . I.2.2 The Fundamental Theorem: Global Case I.2.3 Flows and Differential Equations . . . . . I.2.4 Vector Fields and Derivations . . . . . . . I.3 Partial Differential Equations . . . . . . . . . . .
XVII
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Appendix: Tables
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J Euler’s Gradus Function
1165
K Just and Well-Tempered Tuning
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L Chord and Third Chain Classes 1169 L.1 Chord Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1169 L.2 Third Chain Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175 M Two, Three, and Four Tone Motif Classes M.1 Two Tone Motifs in OnP iM od12,12 . . . . . M.2 Two Tone Motifs in OnP iM od5,12 . . . . . M.3 Three Tone Motifs in OnP iM od12,12 . . . . M.4 Four Tone Motifs in OnP iM od12,12 . . . . . M.5 Three Tone Motifs in OnP iM od5,12 . . . .
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N Well-Tempered and Just Modulation Steps 1197 N.1 12-Tempered Modulation Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197 N.1.1 Scale Orbits and Number of Quantized Modulations . . . . . . . . . . . . 1197 N.1.2 Quanta and Pivots for the Modulations Between Diatonic Major Scales (No.38.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199 N.1.3 Quanta and Pivots for the Modulations Between Melodic Minor Scales (No.47.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1200 N.1.4 Quanta and Pivots for the Modulations Between Harmonic Minor Scales (No.54.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202 N.1.5 Examples of 12-Tempered Modulations for all Fourth Relations . . . . . . 1203 N.2 2-3-5-Just Modulation Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203 N.2.1 Modulation Steps between Just Major Scales . . . . . . . . . . . . . . . . 1203 N.2.2 Modulation Steps between Natural Minor Scales . . . . . . . . . . . . . . 1204 N.2.3 Modulation Steps From Natural Minor to Major Scales . . . . . . . . . . 1205
xxviii
CONTENTS N.2.4 N.2.5 N.2.6 N.2.7
Modulation Steps From Major to Natural Minor Scales Modulation Steps Between Harmonic Minor Scales . . . Modulation Steps Between Melodic Minor Scales . . . . General Modulation Behaviour for 32 Alterated Scales .
O Counterpoint Steps O.1 Contrapuntal Symmetries . O.1.1 Class Nr. 64 . . . . . O.1.2 Class Nr. 68 . . . . . O.1.3 Class Nr. 71 . . . . . O.1.4 Class Nr. 75 . . . . . O.1.5 Class Nr. 78 . . . . . O.1.6 Class Nr. 82 . . . . . O.2 Permitted Successors for the
XVIII
References
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Bibliography
1223
Index
1255
I. Introduction and Orientation
XI. Statistics of Analysis and Performance
X. RUBATO®
VII. Counterpoint
XIV. String Quartet Theory
VI. Harmony
XIII.Operationalization of Poiesis
XII. Inverse Performance Theory
IX. Expressive Semantics
VIII. Structure Theory of Performance
IV. Global Theory
III. Local Theory
II. Navigation on Concept Spaces
V. Topologies for Rhythm and Motives
CD-ROM
XVIII. References Bibliography Index
XVII. Appendix: Tables
XVI. Appendix: Mathematical Basics
XV. Appendix: Sound
CONTENTS xxix
Leitfaden
xxx
ToM_CD
CONTENTS
ToposOfMusic.pdf legal
GPL.pdf README.txt
software
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documentation
hbdisk README.txt
examples music
czerny_chopin mystery_child synthesis
audio_files presto_files
programs
prestino.prg presto README.txt
sources
README.txt source1 source2
rubato
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rubato README.txt
source nextrubato
documentation source
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contrapunctus_III
audio_files stemmata
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alptraeumerei takefive
Part I
Introduction and Orientation
1
Chapter 1
What is Music About? Musik ist das ganze Leben. Rudolf Wille [578] Summary. This chapter describes the overall extension of music-related activities in the spacetime of human existence. The basic scope of the systematization as described in this book is declared. Our selection of corresponding fundamental scientific domains is not meant as a qualitative judgment over other scientific or artistic domains; it merely names the scientific pillars of the musical realm. –Σ– This book deals with the topos, the very concept of music. However, music appears in a wide range of human realms. It is a universal phenomenon of symbolic and physical, formal and emotional, individual and social, systematic and historic presence. Therefore, developing a concept of music should not get off with a definition ex machina, but offer propaedeutic orientation tools in order to make the reader understand why certain conceptual mechanisms or definitions are built. The need for such a support reveals a strong distinction from chemistry, physics or other natural sciences. The point is not that these sciences are dispensed from the fundamental question of what they are about. Rather the characteristic difference to musicology—and other humanities—is that natural sciences offer a fast access to effective activities, to the “working scientist” paradigm: Realization of one’s scientific status by doing science. “Doing” musicology is much less easy. The historical overhead in musicology is an index of the effort requested to get off the ground, to navigate within a safe concept framework. Later in this book we shall discuss some of the reasons for this deficiency, but for the time being, we just notice that musicology still lacks a stable concept framework, see [361] for a detailed discussion of the “unscientific status of musicology”. It is one of the main goals of this work to develop such a framework. In fact—as a result of a lengthy concept development—an explicit concept framework for musicology will only be achieved in section 19.2. We shall, however, not deal with the full reality of music, as it appears in psychological, physiological, social, religious and political contexts. For a semiotically motivated systematization of music which comprises these aspects, refer to [361]. 3
4
CHAPTER 1. WHAT IS MUSIC ABOUT?
In this vein we want to start the preliminary orientation by presenting the fundamental activities related to music, and then their foundation upon established scientific research fields.
1.1
Fundamental Activities
Summary. The musical realm distributes among four types of artistic or scientific action and reflection: production, reception, documentation, and communication. These activities testify to a universally ramified presence of music in culture. –Σ–
Perception
Communication
Production
Documentation Figure 1.1: The four fundamental activities in music are visualized as sides of a tetrahedron. The following classification of music-related activities probably applies to many cultural fields. We recognize that developing a consciousness of their presence in music sheds a particular light on this subject. This is due to the fact that music traditionally relies on a strong connection between artistic facticity and intellectual reflection. In most sciences, and even arts, such an intertwinement is an exotic aspect, but musicology has to deal with it and this makes the case a very special one. The four activities are as follows: production, reception, documentation, and communication. For the sake of coherent representation, and because of the discussion in section 1.2, we represent these activities as sides of a tetrahedron, see figure 1.1. These concepts are understood as follows: → Production refers to whatever is recognized as being on the side of the ‘making’ of music, and this on any level or reality. For example, we can make a sound on an instrument, or write or let a computer software ‘write’ a composition in a mental reality, or make an expressive performance. But we cannot ‘make’ an analysis in this sense, since this kind of activity receives a given musical body. → Reception refers to whatever is recognized as being on the side of ‘taking’ something from music on any level or reality. For example, we may hear a sound or let a machine decompose
1.1. FUNDAMENTAL ACTIVITIES
5
it into its partials, or make an analysis of a musical score—by hand or supported by a computer. But we may not perform a composition in the sense of a creative restatement. Whatever is on the side of a production of some new musical entity is understood as being on the productive side. → Documentation deals with whatever is related to bringing musical facts into more or less permanent sign systems. Musical notation, data base systems for musical objects, tradition of musical texts, the question of how and on which media ethnomusical performances should be stored, all this is subject to music documentation. → Communication is related to whatever happens when the three preceding activities are put into relation with each other. Production leads to documents, documents are retrieved and interpreted by instrumentalists or musicologists, live production of musical sounds by improvising musicians is perceived by other musicians of the band, then answered, and so on. Communication deals with the omnidirectional processes between production, documentation, and reception. We do not contend that these four fundamental activities represent the only possible classification scheme, but they are broad enough to evidence the immense variety of perspectives when dealing with music. And we insist on the fact that all these activities are of comparable relevance. Let us make this clear. Since music is an art where time plays an omnipresent role, the making of music, its physical production in performance cannot be separated from the product as a preliminary task; it is substantial. The same holds for reception. On the other hand, documentation is not a necessary evil. It evokes the fundamental question of identity of an artistic work: Can we say that Beethoven’s Fifth is that determined text? Or is performance intrinsically part of the Fifth’s identity? Trying to understand music under permanent abstraction from any one of these aspects cannot succeed. And, above all, they are activities and not passive contemplation, a basic fact that we shall discuss in depth in chapter 5. It is further remarkable that the synopsis of these equally important music activities draws a picture of perfect encyclopedic universality. In the “Encyclop´edie” of Denis Diderot and Jean le Rond d’Alembert, we have three basic conceptual coordinates [14, 25] which go back to the global geography of science as designed by Francis Bacon in [34]: Reason/Philosophy, Documentation/History, and Imagination/Arts. If we compare these entities to the music activities, they correspond as follows: Imagination/Arts is the counterpart of production; the arts express a creative activity of making a work from one’s imagination. Reason/Philosophy is the counterpart of reception; it deals with what in German is better conceived as “Vernunft”, which means to perceive, receive, understand, observe something which is presented to your intellect. Documentation/History is verbally the same as in our list. What we have not inserted so far is the activity of communication. Now, the genuine encyclopedic concern is communication of the global orbit of human knowledge. Without this driving motor, reason, imagination, and documentation are worthless abstracta. Summarizing, we can state that the main concerns in the classical encyclopedia of the Age of Enlightenment are congruent to the four basic music activities. This is not only a comfortable raison d’ˆetre for music as a cultural phenomenon, it also has deep consequences for the construction of a viable concept framework. Such a body must comprise a variety of
6
CHAPTER 1. WHAT IS MUSIC ABOUT?
ontological1 perspectives and cannot refrain from contents in favor of formal virtuosity. The system of predicates to be developed in chapter 18 will account for this specification.
1.2
Fundamental Scientific Domains
Summary. The fundamental scientific domains relating to the activities described in the preceding section include semiotics, physics, mathematics, and psychology. This is explicated and justified—in particular with regard to the only apparant elimination of the historical perspective which dominates traditional musicology. The relation between the two lists of activities and domains is discussed. –Σ– To begin with, we have to make clear what “fundamental scientific domains” should signify. We may view the collection of sciences as being a hierarchy of disciplines with its particular relations being defined by a substantial dependency. For instance, chemistry is dominated by physics, but not vice versa. And every natural science is dominated by mathematics. Many of the hierarchical relations are quite problematic, blurred or otherwise unclear by the very definition of basics of particular disciplines; we have to accept such general uncertainties and will proceed in full consciousness of the provisional state of the art of epistemological classification. Now, if one has to locate a special research subject, such as music, within the hierarchy of disciplines, one is looking for a minimal set, a kind of ‘basis’ of disciplines which are necessary to cover the subject. In this sense, we argue that four sciences: semiotics, physics, mathematics, and psychology, are such a ‘basis’. Let us explain this selection. We have to show that the four disciplines are sufficient to cover the field of music and that no one of these sciences is superfluous in that it either is not involved at all, or that it is only involved via a proper sub-discipline. At this point, we cannot circumvent a provisional description of what music is about, though this is only a first approximation in order to delimitate the scientific overall extension. Definition 1 (First provisional version) Music is a system of signs composed of complex forms which may be represented by physical sounds, and which in this way mediate between mental and psychic contents. To deal with such a system, semiotics are naturally evoked. To describe these forms, mathematics is the adequate language. To represent these forms on the physical level, physics are indispensable. And to understand the psychic contents, psychology is the predestined science. Is this selection a “basic” one as claimed above? There are several critical points which we should clarify. First, the historic dimension of music seems to be neglected. However, semiotics has a strong competence in diachronic study of a sign system. Music history is prominently the history of a system of significant signs (much like diachronic linguistics), and here, diachronic semiotics is the adequate research field, whereas general history would be too generic in view of the given subject. Musical forms are not only mediated via physical representation. They have a strong properly mental (“symbolic”) existence, and this evokes mathematics as a directly 1 Ontology
is the philosophical discipline dealing with the concept of “being”.
1.2. FUNDAMENTAL SCIENTIFIC DOMAINS
7
involved research aspect. Physics has a prominent role in music, and this not only with respect to sound production (classical acoustics and physics of musical instruments), but also in computer science which is concerned with data representation and processing in music—be it on the generic level of software or on the level of advanced digital sound synthesis and analysis. Hence, we do “absorb” computer science and sound engineering within physics, together with the mathematical formalisms in theoretical computer science. Finally, social aspects of music are viewed as objects of social psychology on one, and socio-semiotics on the other hand. Also pedagogical problems (music education) are subsumed under the label of psychology since it is the human psyche which is educated. It is an extremely important point to recognize the relative autonomy of these four research fields in their descriptive force. This means that we do not, and never will hope or hypothesize reductionism whatsoever. There is no interest to view psychological reality as a surface of neurophysiological depth structure. In this sense, our selection is not pragmatic but ontological. We come back to this theme in chapter 2. To complete the picture, we should try to fix the relative positions of the four basic sciences with respect to the four fundamental activities of production, reception, documentation, and communication. It is evident that activity is more or less related to some of the four scientific disciplines. If we “label” each vertex of the tetrahedron shown in figure 1.1 by a basic science, it turns out that every triangular activity side is “spanned” by three of four vertices. If we pay attention to the obvious semantics of this configuration, we have to place the labels in such a way that each activity is delimited by the three most relevant sciences for its execution. We propose the labeling drawn in figure 1.2 and leave it to the reader to judge our selection.
Perception
Production
Psychology Communication
Semiotics
Physics Mathematics Documentation Figure 1.2: The four fundamental activities, resp. their counterparts of basic sciences in music, are visualized as sides, resp. vertices, of a tetrahedron. Let us simply give one example: Documentation is concerned with signs, their formal representation, and the physical storage techniques. The psychological aspect is less substantial,
8
CHAPTER 1. WHAT IS MUSIC ABOUT?
whence documentation is “spanned” by the respective vertices. We do, however, not overdress this elegant configuration as a “magic” tetrahedron, the visualization merely serves as a concise and relatively well positioned synopsis of the overall situation, where the question “What is music about?” can be initiated. And it is also a very practical orientation scheme for designing future music research policy and strategy.
Chapter 2
Topography Die Linien des Lebens sind verschieden Wie Wege sind, und wie der Berge Gr¨ anzen. Was wir hier sind, kann dort ein Gott erg¨ anzen Mit Harmonien und ewigem Lohn und Frieden. Friedrich H¨olderlin: Die Linien des Lebens. . . (1843) [469] Summary. This chapter deals with an ontological orientation in the subject of music. It was already contended in chapter 1 that music is communication, has meaning and mediates on the physical level between its mental and psychological levels. Such an orientation is topographic in nature since it offers a number of ontological “dimensions” and “coordinates” to profile musicological discourse and helps avoiding misplaced or blurred arguments. The topography involves three mutually independent dimensions: communication, reality, and semiosis. The local nature of this orientation scheme is discussed. –Σ– Whoever is dealing with formal, logical, mathematical or computational methods in music does not necessarily have to be an expert in music philosophy. Nonetheless, these methods require a sufficiently refined orientation within the complex ontology of music in order to avoid erroneous interpretations of results obtained by use of these exact methods. It is well known that the precision of mathematical results, together with a poor knowledge about the delicate ontology of music may provoke a dogmatism for which mathematics is unjustly made responsible. On the other hand, the extensive concept framework which we want to set up cannot even be approached without a firm pointer to musical ontology. In the following “topography of music” we shall sketch an ontological model without any dogmatic or unmodifiable character, though constantly available as a powerful coordinate system to locate the problems within the organism of music. In view of the generic (and first provisional) delimitation of the concept of music (definition 1), it is proposed to set up from the very beginning a “three-dimensional” ontology, i.e. an ontological system which tells rather where the concept of music subsists than what its being 9
10
CHAPTER 2. TOPOGRAPHY
is like. Let us shortly deviate on this philosophical line to justify the somewhat particular approach. To be clear, we are not concerned with music as such (a Kantian “Ding an sich”) but with music that we localize and conceive within our knowledge system. Following Immanuel Kant [258, p. B 324], a concept is a local place, a topos in the knowledge space. Ontology then questions the place of the concept, the space-coordination of the music topos. We do, in other words, stress three ontological coordinates as an intrinsic unity pointing at the concept of music. These coordinates are • REALITY • COMMUNICATION • SEMIOSIS1 We shall hereafter (sections 2.1, 2.2, 2.3) introduce them in detail. To understand music as a whole, you have to specify simultaneously its levels of reality, its semiotic character, and its communicative extension. Being a fact of music means having these three perspectives or ontological coordinates. Omitting any one of these determinants is an abstraction (though not an aberration) from the full ontology. It might be argued that this topographical reduction of ontology simply delegates the difficult question to the ontological coordinates: instead of asking for one ontological specification, we have to examine three partial specifications. The argument is correct, but it is also true that these partials are simpler, more elementary. Dealing with semiosis without being concerned with its level of reality (and vice versa) is a huge advantage. Giving the ontology a topographical “turnaround” simplifies the problem. Whether the overall ontological question is settled by this procedure is not answered here. Let us instead begin with a closer inspection of the three proposed coordinates—we shall come back to this subject after that in section 2.5.
2.1
Layers of Reality
Summary. The dimension of musical reality involves physical, psychological, and mental layers. It is not question of reducing one of these realities to the others: Either of them has an autonomous existence which can at most be transformed into others, but not eliminated. The great majority of reality-specific phenomena cannot even be translated into external layers without substantial deficiencies. –Σ– Music takes place on a wide range of realities. They may be grouped into physical, mental and psychological levels. Differentiation of realities is crucial for avoiding widespread misunderstandings about the nature of musical facts. A representative example of this problem is Fourier’s theorem, roughly stating that every periodic function is a unique sum of sinoidal components. Its a priori status is a mental one, a theorem of pure mathematics. In musical acoustics it is often claimed that—according to Fourier’s theorem—a sound “is” composed of “pure” sinoidal partials. However, there is no 1 Semiosis
refers to the fact that music deals with a system of signs.
2.1. LAYERS OF REALITY
11
physical law to support this claim. Without a specific link to physics, Fourier’s statement is just one of an infinity of mathematically equivalent orthonormal decompositions based on “pure” functions of completely general character, see [127, ch. VI]. To give the claim a physical status, it would be necessary to refer to a concrete dynamical system, such as the cochlea of the inner ear (see appendix, section B.1), which is physically sensitive to the first seven partials in Fourier’s sense. Methodologically, there is no reason nor is it ontologically possible to reduce one reality to others. The problem is rather to describe the transformation rules from the manifestation of a phenomenon in one reality to its correspondences within the others. To be clear, a neurophysiological transformation (“explanation”) of a psychological phenomenon does not, however, conserve the psychological ontology of the phenomenon. The specific phenomenon within the psychological topos corresponds to another phenomenon within the physiological topos. But ontologically, the phenomena do not collapse. We now give an overview of the three fundamental topoi of reality and their specific characters.
2.1.1
Physical Reality
Music is essentially manifested as an acoustical phenomenon, made by means of special instruments and perceived by humans. Nonetheless, its acoustical characteristics are less—if at all—condensed within a unique physical sound quality than in the physical input-output systems for sound processing. The use of physical sounds is a function of very special devices for synthesis and analysis. This is reflected in the fact that to this date, there is no generally accepted classification method for musical sounds, see section 12.3 for a thorough discussion. This is not due to missing synthesis or analysis methods and techniques. Besides classical analog sound synthesis methods as they are realized on musical instruments, there are various digital sound synthesis methods, see appendix A.1.2. The problem is rather that classification of musical sounds is arbitrary without reference to their semantic potential, see section 12.3.3. Physical reality of music is only relevant as an interface between ‘expressive’ and ‘impressive’ dynamical systems. All physical sounds are equally natural, be they produced by a live violin performance, a computer driven synthesizer via loudspeakers, or by the tape patchwork of musique concr`ete. On the other hand, the central receptive system for music is the human auditory system: Peripherical and inner ear, auditory nerve, its path through multiple relays stations of the brain stem, the neo- and archicortical centers for auditory processing and memory, such as Heschl’s gyrus and the hippocampal formation, see appendix B. This extremely complex physiological system is far from being understood. Even though some insights into the dynamics of the cochlear subsystem do exist, it is not known which analysis of the musical sounds takes place on the higher cortical levels. It is not even clear how the elementary pitch property of an ordinary tone is recognized [72]. This means that on the cognitive level, human sound analysis is not yet understood. Therefore, reference to particular sound representation models is good for synthesis options, and for speculative models of cognitive science [292], but not as a firm reference to human sound processing.
12
2.1.2
CHAPTER 2. TOPOGRAPHY
Mental Reality
Just like mathematical, logical and poetical constructions, musical creations germinate as autonomous mental entities. It is a common misunderstanding that musical notation is an awkward form to designate physical sound entities. Being a trace of intrinsically human activity, the phenomenological surface of music is linked to mental schemes which we call scores: oral or written text frames of extra-physical specification. Scores are mental guidelines to an ensemble of musical objects. They reflect the fact that music is composed as well as analyzed on a purely mental level. Obviously, scores do point at physical realization, but only as a projection of a mental stratum into physical reality. A fact of harmony or counterpoint is an abstractum much the same as an ideal triangle in geometry. In this sense, playing a chord on a piano corresponds to drawing a triangle on a sheet of paper.
2.1.3
Psychological Reality
Besides its physical manifestation and its mental framework, music fundamentally expresses emotional states of its creators and emotionally affects its listeners. This was already known to Pythagoreans [541] and defined as a central issue of music by Ren´e Descartes [126], see [120]. Such an emotional reality of music is neither subordinate nor abusive, on the contrary: to music lovers the emotional response is a dominant aspect, individually, and socially. It is not by chance that song titles such as “It don’t mean a thing if it ain’t got that swing” de facto reduce musical meaning to an emotional category, the feeling of swing. Like other realities of music, the psychological dimension cannot and needn’t be reduced to others, it is a manifestation of an autonomous, irreducible ontology.
2.2
Molino’s Communication Stream
Summary. Communication is the second dimension of our topography. Following Jean Molino, it splits into a three-part stream starting from the poietic instance of the creator of a work of art, traversing the mediating neutral datum of the virtual work, and ending on the receptive side of the listener’s esthesic perception. –Σ– According to Jean Molino [377], we describe the tripartite communicative character of music, see figure 2.1. Molino’s scheme is an abstraction to the essentials; they may be complex and interlocked in concrete situations, see section 2.5. Its structure is comparable to Oskar B¨atschmann’s “grosses abstrakt-reales Bezugssystem der Auslegung” which he proposed as a hermeneutical framework for the analytical discourse in fine arts [43]. Like Molino’s scheme, B¨atschmann’s construction issued from the insight that a dispassionate discourse on arts, be it “parler peinture” or “parler musique”, cannot succeed without considering their communicative structure. Molino’s scheme partitions the communication process into three instances: creator, work, and listener. To these, three types of analytical discourse correspond: poiesis, neutral level, and esthesis. These concepts are technical terms which have to be explicated in the following. In
2.2. MOLINO’S COMMUNICATION STREAM
Creator
Work
Production
Poiesis
Neutral Niveau
13
Reception
Listener
Esthesis
Figure 2.1: The three-part scheme of music communication following Molino. particular, Molino does not prejudice such a thing as causality between creator and listener by means of the work, or receptive passivity of the listener towards work and creator.2
2.2.1
Creator and Poietic Level
To begin with, a creator instance subsumes all factors which are essential and sufficient for the production of a musical work. This level describes the “sender” instance of the “message”, classically realized by the composer. The corresponding analytical discourse of the work’s poiesis has been introduced by Etienne Gilson [183]. According to the Greek etymology (πoιειν = to make), “poiesis” relates to the one who makes the work of art. Poiesis is concerned with the individual condition of the creator as a result of its history of development as well as with the role of the broader socio-cultural frame of the work. In specific cultural contexts or in a more refined discourse about art production, the creator instance may as well be the musician or the performing artist. In jazz, for example, the improvisational aspect is a genuine making of the music, and the performance of a classical piece of music is a creational act. Poiesis is a sharp analytical instrument. Already the very definition of a sound by means of its defining parameters, such as pitch, color, etc. bears a strong poietical flavor—quite contrary to the naive and widespread belief. For example pitch (logarithm of frequency) is no invariant of the sound, it usually results from the sound construction as a product of a periodic vibration of a determined frequency and of an envelope (see appendix A.1.2.1). Neither frequency of the periodic vibration nor the envelope are uniquely determined by their product. And, worse, not even the frequency of an ideal periodic vibration is uniquely determined from the vibration as such. In fact, Fourier’s theorem only states uniqueness of its coefficients if the fundamental period is given. But the latter is not automatically inscripted in a vibration. Such parameters are part of poiesis: the way of making the sound, not the “neutral” sound object. This shows that poiesis is not restricted to psychological aspects of “how to fabricate a composition”, it can relate to completely “objective” physical or mathematical facts. 2 We should like to relate these concepts to the activities of production, reception, documentation, and communication, as described in section 1.1. Activities are instances of what here is denoted by “communication”. What we call “poietic level” corresponds to “production activity”, “reception” is on the side of “esthesic level”, “documentation” relates to “neutral level”, and the activity of communication corresponds to the transition processes between all the three levels in Molino’s system. We have nonetheless maintained these concept frameworks since Molino’s technical terminology does not reflect the aspect of overall activity in music.
14
2.2.2
CHAPTER 2. TOPOGRAPHY
Work and Neutral Level
The work is the turning point of musical communication. It subsumes whatever determines directly the production of sound events within a specific context. Its identification depends upon the contract of creator (sender) and listener (receiver) on the common object of consideration. Musicology has difficulties with the work concept since its determinants seem to be of fuzzy nature. This relates to the dramatic difference between music and fine arts. As a matter of fact, a painting is determined down to its humidity and temperature—at least from the moment it has gained ‘eternal’ appreciation and allocation in a professional museum. In music, the definition of a work is left with ample variants. Decisions about including sound color or tuning prescriptions in Bach’s “Wohltemperiertes Klavier” do in any case redefine a new and different work. And above that, such a decision transcends the strict definition and establishes a valuation of the work in the light of historical parameters of Bach’s creativeness. In this context, a score may be viewed as being the organizational scheme of a work. In fact, physical realization of music is always based upon mental schemes which we call scores: oral or written text frames of extra-physical specification. Apart from classical western traditions where this fact is evident, the score concept is also adequate for describing traditions which are more towards ethnomusicological antipodes. First example: In the music of Noh theater [269], there are different score instances, e.g. for vocal utai music denoted in melodic units (fushi) to the right of texts, or for the hayashi notation systems for flutes and drums. Second example: The improvisational culture of jazz—which in its making only marginally relates to traditional western scores—is based on the concept of the interior score (“partition int´erieure” [487]). This means that, even for free jazz improvisers, there is an interior reference system of lexical character, together with a selection code which guides performance. The fundamental fact behind the basic role of the score concept for music is that human organization in a complex time-space of acoustical and gestural nature cannot be executed without an interpersonal spiritual orientation common to the responsible participants. Following Molino, the analytical discourse which—independently of the selection of the tools—is strictly oriented towards the given work is termed neutral level. In particular, the historical justification for a work’s delimitation is not part of the neutral level. Molino’s concept has only attracted controversies since it—faultily—seems to imply the preference of a particular analytical tool/method to the exclusion of other possible approaches. But such a thing as the “unique ideal interpretation” is precisely the famous unicorn of hermeneutics—a superfluous illusion. A momentous analytical perspective is just one among many possible accounts of the work, and neutral analysis is the variety of all these perspectives. Such a work may be the divine creation of a medieval tone system, a work of computer music which is explicated as far as its acoustical details, or the beat of waves on the open waters of the sea, no matter: Neutral level means propaedeutic analytical precision work preceding any valuation whatsoever. Evidently, exact methods will play an eminent role in realizing such a task.
2.2.3
Listener and Esthesic Level
The listener is the instance which perceives and interprets a given work. It is subject to the same type of determinants as the creator. Coordinates of a listener will vary from case to case and
2.2. MOLINO’S COMMUNICATION STREAM
15
coincide only punctually with each other or with the creator’s coordinates. The characteristic difference between listener(s) and creator is the time arrow towards the common work. Whereas the creator produces a work, the listener perceives and interprets an already existing work. From the moment of a work’s existence, the creator’s existence is fixed whereas the number of the listeners’ existences grows as a function of those who are engaged in the work. According to a proposal of Paul Val´ery, [537] the analytical discourse on the listener is termed esthesis, following the Greek term αισθησισ (perception), in order to avoid confusion with esthetics, the study of beauty. Esthesis deals with perceptive valuation of a work by the listener. This valuation of the work’s attributes is realized as a function of the listener’s individually variable position. This interpretative valuation from a determined perspective is not less active than the creator’s activity in the creation of the work. And this not only regarding the artistic performance, but quite generally. Like poiesis, esthesis is integral part of the analytical elements constituting the existence of a work. The distinction between poietical and esthesic levels is fundamental for a sound conceptualization in musicology. Only in very rare cases can we expect the conservation of results from a poietical discourse within the esthesic point of view, and vice versa. Nonetheless, we observe again and again the faulty trial to implant historically and systematically clearcut poietical concepts, such as “pitch”, into an esthesic discourse. The famous paradigmatic justification of such a malfunction of analysis is known under the dictum that “the ear is the highest musical instance”. On one hand, such a dogmatic phantom as “the ear” does not exist, is incompatible with the multiplicity of listening cultures. On the other, the ear fails as a metonymy of cortical music processing: there is a huge transformation process between the auditory cortex and the ear’s cochlear system. Poietics cannot be taken as the ideal access to a music work. There is no thing such as a causal relation between creator and work. The latter is a result which may be generated by very different processes. Within the communication process, the creator above all differs from the listener by his/her position in time. This fact becomes manifest when we imagine the work’s production process as a fictional unfolding in a retrograde movie. This visualizes a well-known insight from fine arts theories that the artist/creator is the first observer. In this way, retrograde poiesis is embedded in esthesis. And vice versa, the esthesic discourse may be restated as a manifold of variants of imaginary retrograde poieses. Example 1 The problem of symmetries in music is a good illustration of the communicationsensitive aspect. A classical conflict concerning the role of retrograde in music arises from the observation that this construction “cannot be heard and thus is a problematic feature”. Communication coordinates make this discussion more transparent: The retrograde construction as a poietic technique is a common compositional tool. It fits into the toolkit of contrapuntal constructs for organizing the compositional corpus. On the other hand, the esthesic perspective of the retrograde is concerned with the question whether and how clearly such a construct can be decoded by the listener. This latter question is a completely different topic and cannot be identified with the former. More precisely, the role of the retrograde as an organizational instance is not a function of its perceptibility as an isolated structure. The psychological question of whether a retrograde can be perceived is rather this: “Can a retrograde be distinguished from random?” Finally, retrograde structures may be recognized as objective facts within the neutral level of the score without being either constructed be the composer or consciously perceived by
16
CHAPTER 2. TOPOGRAPHY
the listener. Summarizing, the communicative coordinates help localizing and thereby making more precise the musicological discourse. 2.2.3.1
The Problem of Identity
From the preceding communication-theoretic considerations, the identity of a work is triply stratified. To begin with, we envisage a work that is given before any analysis is performed. We call this data its abstract identity. This first identity is then enriched to yield the neutral identity as a result of the multiplicity of neutral, valuation-free analyses. This identity is the basis for a variety of esthesic—in particular: retrograde poietic—valuations. The esthesic identity is built from all these esthesic valuations and their mutual relations. Accordingly, a work is only identified when the infinite process of esthesic valuation, built upon abstract and neutral identification, is completed. This yields at least one justification for an incessant performance practice in classical music, but see section 32.2 for more details on performance and identification.
2.3
Semiosis
Summary. Independently on what is communicated, and on which level of reality this takes place, music intrinsically involves complex signification processes. The generic setup of semiotics understands a sign as being a tripartite object. It consists of a significant expression, inducing the signification act of translation, and thereby producing the expression’s content, the significate. We give a short review of dichotomies of structuralist semiotics. Musical semiosis reveals a complex concatenation of meta- and connotation-layers in the sense of Louis Hjelmslev. Rather than being absent, musical meaning is distributed over a sequence of semiotic subsystems. –Σ– By use of a highly developed textuality of musical notation as well as by the very intention of musical expression, music is structured as a complex system of signs. This is not only a marginal aspect: Music is one of the most developed non-linguistic systems of signs. We shall first present the elementary sign character of musical objects (subsections 2.3.1, 2.3.2, and 2.3.3), and then (2.3.4) review the overall semiotic perspective of music.
2.3.1
Expressions
Already the earliest medieval music notation is motivated by the very nature of the graphical neumes: etymologically as well as substantially they are gestural hints pointing at movements in pitch and rhythm. This coincides with the latin etymology of sign: signare = to point at, give a hint. Neumes are aliquid pro aliquo, they express something in the sense of semiotics. Besides and beyond musical notation, music is often viewed as an expression of emotions, spiritual contents or gestural units. In any case, music has a phenomenological surface that is organized in a spatio-temporal syntax. Albeit more complex than linguistic syntax, the musical syntax shares some of its characteristics, see section 2.3.4.
2.3. SEMIOSIS
2.3.2
17
Content
According to the famous dictum of Eduard Hanslick [206], we learn that: “Der Inhalt der Musik sind t¨onend bewegte Formen.” This evidences that the notated complex of musical graphems is not the content but points at some kind of sounding content: they mean something. Hanslick’s characterization is a minimal semantic setup but some kind of content can be identified. The remarkable aspect of Hanslick’s approach is that it associates musical content with mathematical content. As a matter of fact, a triangle is a mathematical object that essentially reduces to form. But mathematicians do associate it with a content, usually with a platonic entity addressed by abstract symbols or by drawings with precise quantifications.
2.3.3
The Process of Signification
Signification is the most important instance for the realization of a sign. It is responsible for the transformation from the expressive surface to the hidden meaning. The fact that performance is such a central issue of music gives a strong proof of the qualified presence of signification in music. Without the laborious effort of turning significant expressions into the reality of their meaning, music is not what it is meant to be. For musical signs, this semiotic process bears a highly differentiated structure which is sensitive to Ferdinand de Saussure’s dichotomies [41, 471], we review these items in the following section 2.3.4.
2.3.4
A Short Overview of Music Semiotics
Semiotics of music is a complex subject which cannot be dealt with in this context. For a full account, we refer to [361]. This does, however, not dispense us from a short overview of this subject since there is no hope to understand music without a minimum of semiotical prerequisites. We shall describe music semiotics from the perspective of structuralist semiology as it has been sketched by Roland Barthes [41] as a generalization of the linguistic theory of Ferdinand de Saussure [471]. We pursue this project according to Andr´e Martinet’s principle of relevance [318]. It states that selection of materials and methods has to be justified by relevant criteria in the sense of points of view towards the actual object—music in our case. Let us recall that Barthes’ position does not mean that music is interpreted as being a type of language. On the contrary, we shall repeatedly stress significant differences between the music system and linguistics. To say that a system is semiotic means that it is articulate according to the fundamental stratification of signs into significant, signification, and significate layers (Saussure’s signifiant/signification/signifi´e). This stratification articulates the process of production of meaning (the signification), which points from the expressive surface of the mediator (the significant), to the underlying meaning (the significate). Slightly deviating from Saussure’s original approach, the existentiality of the significate is that of a concept, i.e. it is part of a “lexical body”, of a system of sayable things (λεκτ oν in the stoic tradition, in contrast to the “real” things or to the psychological imagination, the latter being Saussure’s preferred significates). Like every existing semiotic system, music is embedded in history. If we investigate the system’s state at a determined moment, we deal with synchronic analysis. Of course, the concept of a moment really means a short duration wherein the system remains fixed. If, instead, the system’s objects are observed in their temporal development, we deal with diachronic analysis.
18
CHAPTER 2. TOPOGRAPHY
For example, the etymology of the concept of consonance is a diachronic subject, whereas looking at semantics of the augmented triad in Franz Liszt’s “Faust” symphony points at a synchronic analysis. Evidently, sychronicity and diachronicity are not strictly separable since—just as with physics—the space-time continuum of music is indecomposable. The special class of shifter or deictic signs is very important in music and is characterized as follows: Their significate transcends the lexical reality and penetrates the unsayable existence of the system’s user. Whereas, for example, the concept of a sonata form is lexical, the emotional significate of music can neither be “lexicalized” by decree, nor verbalized whatsoever—contrary to numerous flopped attempts. In linguistics, shifters (such as “I”, “here”, “now”) are exceptional signs. In music deixis, such as emotional signification, is the standard situation. But this does not mean that the lexical part of the system is marginal. On the contrary, a rich deixis cannot succeed without a rich lexical support. It is evidently a difficult business to “hit” the unsayable by use of an arsenal of lexical approximation pointers. As a function of the type of signification process, Saussure distinguishes between arbitrary and motivated signification. Signification is arbitrary if it is defined by pure convention. For example, the signification “highten a pitch by a semitone!” of the sharp # is pure convention, whereas the signification of a tremolated kettledrum beat towards the significate “thunder” is motivated as onomatopoeia. Notice that the arbitrariness of musical notation is a major source of the elitarian music culture; fortunately, a huge number of semiotically well-motivated music software has started to break this artificial obstacle to the world of music. A standard means for diminishing arbitrariness is the use of system immanent logic or of instances exterior to the system which are at disposal as given evidences. The latter is the case if our system lives in the context of other systems (e.g. software environments) which may help to operationalize it. A famous example of system immanent logic is Roman Jakobson’s poetical function. It enables production of significates by means of “projection of the paradigmatic axis into the syntagmatic axis” (see below and sections 11.6.1, 48.2.1). As a whole, the system has sign character but it is composed from single signs. If two signs are adjacent in space or time they are called contiguous. Contiguity is “in praesentia”. It is a fact of concrete nature, no further abstraction is needed. A chain or network3 of signs in successive contiguity is called syntagm, in linguistics also: syntagmatix axis. A progression of degrees or a contrapuntal interval sequence are syntagmata in time; a texture of four parts (soprano, alto, tenor, bass) or a discantus which is set against a cantus firmus voice, these are syntagmata in pitch space. As a consequence, syntagmatic articulation of music looks quite different from linguistics. Whereas linguistic syntagm is one-dimensional (in time), the music syntagm extends simultaneously in several dimensions. If two signs are similar then they are called in apposition. (Martinet uses the term “opposition” in linguistics but in music this binary term is too narrow.) Similarity may regard any of the three layers—significant, signification, or significate. The collection of all signs which are in apposition to a given one define the paradigm of that sign. Apposition is a relation “in absentia”. It requires a high degree of abstraction, we come back to this important difference to the syntagmatic contiguity in section 49.4. For example, tones which are perceived with loudness equal to that of a reference tone define the paradigm of loudness of this tone. Two such tones do not necessarily yield the same loudness perception (see appendix B.2)! 3 This extension transcends the usual definition, we allow a multiplicity of simultaneous concatenations, not only one chain such as the case in ordinary language.
2.4. THE CUBE OF LOCAL TOPOGRAPHY
19
According to Louis Hjelmslev [226], a given semiotic system can appear as a sign-theoretic instance (significant, signification, or significate) of another such system. If it appears as a significant layer, we speak about connotation. The significant system is the termed denotative level, where as the superior system is termed connotative level. If, conversely, a system occupies the significate layer of another system, the superior system is termed metalevel or metasystem, whereas the inferior system represents the objectlevel or objectsystem. Generalizing Hjelmslev’s approach, one should also envisage the case where a system appears as the signification layer of its supersystem. In general, semiotic systems are complex interlockings of connotative, metasystemic, and signification layers. For example, the double articulation in language: grapheme ⇒ acoustical significate ⇒ conceptual significate can also be observed in music. Here, in Hanslick’s terminology, we have the double articulation Note ⇒ Ton ⇒ t¨onend bewegte Form. On the other hand, grammar of language and harmony condivide their metasystemic character [70, ?]. For an overview of the entire music system articulation in the sense of Hjelmslev, see [361]. The last of the famous Saussurean dichotomies—it is also decisive for distinguishing music from language—concerns language against speech (Saussure: langage/parole, Barthes: structure/proc`es). This means the contrast between general reglementation and concrete syntagms and paradigms as far as dialects and revolutionary poetical invention. In order to avoid confusion with the linguistic context, we shall adopt the dichotomy of (music) norm /(music) process (corresponding to language/speech). This dichotomy shows a strong difference to the linguistic dichotomy in so far as such a thing as a music norm is neither a synchronic nor a diachronic constant. And emotional significates of music by definition forbid any normative fixation, they are music processes, “sound speech”(“Klangrede”) as they used to call it. On the other hand, the musical denotation level in classical score notation is extremely normative, and as such a source for elite formation whose control is often confused with musicality. The analphabet differs from the score illiterate in so far as production of musical meaning has very little to do with score notation whereas competence in linguistic articulation is intimately related to lexical competence. The Hjelmslev articulation of music [361] shows that musical meaning is not concentrated on a semiotic spot, it is on the contrary distributed among several system layers of denotation and connotation. In other words, musical meaning is the result of a complex development of signs, it grows from simple categories of signification to more and more all-embracing and even transcendental categories as they appear in highly spiritual musical expression.
2.4
The Cube of Local Topography
Summary. Putting together the three dimensions of musical topography, the topographic cube is defined. While each of the described dimensions splits into three “coordinate” values, the cube involves 33 = 27 distinguished topographic locations. Their role in guiding a sophisticated discourse on music is discussed and exemplified. However, this cube’s orientation is of local nature, it does not provide a global point of view. –Σ– After presenting the ontological dimensions of reality, communication, and semiosis, we may view these contributions as coordinate axes of a more complete musical ontology. This
20
CHAPTER 2. TOPOGRAPHY
means that an ontological localization can be interpreted as a point in a three-dimensional cube spanned by the axes of reality, communication, and semiosis, each of them being articulated in three “values”: • reality: physical–psychological–mental. • communication: creator–work–listener • semiosis: significant–signification–significate This suggests a geometric representation of the ontological variety as a “topographic cube” of musical ontology, see figure 2.2. To describe the ontological position of a musical object, we have to specify its three coordinates in reality, communication, and semiosis. Each coordinate may take one of three values as listed above. This produces a set of 33 = 27 possible topographic locations. Of course, it is not necessary that a general object be localized at a single location, it may as well occupy any subset of the cube. So, the 27 points are just the elementary positions from which more complex ontological situations are composed. Significate Semiosis
Signification Significant Mental
Reality
Psychic
Physical Creator Work Listener Communication
Figure 2.2: The cube of musical ontology. To illustrate this topography, we present a classical text of music critique. It is the review of a concert by Franz Liszt written by Ludwig Rellstab in Berlin’s Vossischen Zeitung , cited following Knapp [265, p.86]: Unter dieser Erweckung der vorteilhaftesten Eindr¨ ucke setzte er sich an das Instrument. Jetzt wird ein neuer Geist in ihm lebendig. Er lebt die Musikst¨ ucke in sich, die er vortr¨ agt. W¨ ahrend er mit erstaunensw¨ urdigster Gewalt der Mechanik eigentlich alles leistet, was bisher von irgend jemand einzelnem bezwungen worden
2.5. TOPOGRAPHICAL NAVIGATION
21
ist, und außerdem noch ein ganzes F¨ ullhorn neuer Erfindungen, v¨ ollig ungekannter Effekte und mechanischer Kombinationen vor uns aussch¨ uttet, so daß die aufs h¨ ochste gespannte Erwartung und Forderung sich weit u ¨berfl¨ ugelt sieht: bleibt doch der eigent¨ umliche Geist, den er diesen Formen einhaucht, das bei weitem anziehendere, anregendere und fesselndere Element. Diese geistige Bedeutsamkeit seines Kunstwerkes pr¨ agt sich aber auf das lebendigste in seiner Pers¨ onlichkeit aus. Die Affekte seines Spiels werden zu Affekten seiner leidenschaftlich aufget¨ urmten Seele und finden in seiner Physiognomie und Haltung den treuesten Spiegel. Seine k¨ unstlerische Leistung wird zugleich eine Tatsche des Inneren, sie bleibt nicht getrennt von ihm, sondern wirkt in dem m¨ achtigen B¨ undnis mit dem Geist, der sie erzeugt. Rellstab addresses all three levels of reality. Words such as “Instrument”, “Gewalt der Mechanik”, “mechanische Kombination”, “Haltung”, and “Physiognomie” refer to physical reality.—Expressions such as “Affekt seines Spiels”, “Affekte der leidenschaftlich aufget¨ urmten Seele”, “anregend”, “fesselnd”, and “lebt die Musik in sich” refer to psychological reality.—The mental level is addressed in expressions such as “neuer Geist wird lebendig”, “Geist wird den Formen eingehaucht”, “geistige Bedeutsamkeit”, “B¨ undnis mit dem Geist”, and “Tatsache des Inneren”. The communicative dimension specifies in its poietical coordinate with expressions such as “tr¨agt vor”, “haucht Geist ein”, Geist “erzeugt” k¨ unstlerische Leistung”, “Erfindung (F¨ ullhorn)”.—The neutral level is addressed in passages such as “Musikst¨ ucke vortragen”, “Geist den Formen einhauchen”, and “Kunstwerk”. The esthesic coordinate is addressed by allusions like “wecket Eindr¨ ucke”, “vor uns aussch¨ uttet”, “Erwartung (der H¨orer)”, “Forderung”, “anregend”, and “fesselnd”. The semiotic system is instantiated when Rellstab talks about the significant coordinate in expressions such as “Musikst¨ uck”, “Kunstwerk”, “Form”, “Pers¨onlichkeit als Gef¨aß f¨ ur die Bedeutung”, “Physiognomie”, and “Haltung”. (Observe that “Haltung”, for instance, simultaneously addresses physical reality and the semiotic significant.)—Signification is traced in expressions such as “lebt die Musikst¨ ucke”, “geistige Bedeutsamkeit”, “einhauchen”, and “in dem m¨achtigen B¨ undnis mit dem Geist”.—Finally, significates are addressed in “Leidenschaft”, ¨ “Geist”, “Affekt”, and “Uberfl¨ ugelung der Erwartung und Forderung”.
2.5
Topographical Navigation
Summary. The topographic cube offers a local and recursive orientation. Hence the corresponding discursive navigation receives a ramified path structure: A priori, each topographic location may open or participate in another topographic variety. The local and recursive nature of such a ramified navigation is described and exemplified. –Σ– Observe that the topographic specification of a fact of music is a local resp. recursive one in the following sense which one may visualize as a conceptual zoom-in effect: • Parts of a sign may be entire signs of their own right. For example, the significate of a sign in a metasystem—by definition—is an entire sign of the objectsystem.
22
CHAPTER 2. TOPOGRAPHY • Also, in the communication chain, the performing artist is a creator for the auditory, but he/she includes an entire communication process, starting from the composer, and being communicated through the score. • Third, regarding levels of reality, an acoustic sound is essentially a physical entity, but its description refers to mental instances, such as real numbers for parameter values.
In other words, the topographic cube yields a local conceptual orientation, and by regression, the topographical description of a fact of music may open a complex tree of ramifications, each knot being loaded by a localization within the cube. Supposedly, there is no consistent ontology without such a self-referential regression. In particular, it is not necessary to introduce such a thing as a “topographic metacube” for the description of metamusical facts (e.g. harmony syntactics) since a metalanguage precisely means recursiveness on the level of the significate.
Chapter 3
Musical Ontology Musik ist ohne Begriffe. Hans Heinrich Eggebrecht [101, p.192] Summary. This chapter introduces the difficult subject of musical ontology. Such a discussion is substantial for a reconciliation of traditional musicology and innovative perspectives in cognitive and computational musicology. –Σ– It is a common argument of traditional musicologists against cognitive and computational methods that the full extent of musical being cannot be grasped by these methods. It is claimed that the very depth and transcendence of music are beyond any analytical and quantitative reasonment. We do not condivide such a credo for the following reasons. Above all, cognitive and computational methods are neither restricted to quantification, nor to simple analysis. Rather modern mathematics and logic is an issue of conceptual explicitness; simple numerical quantification is a very special subject of old-style mathematics that has been overridden by structure theories since the development of set theory, modern algebra, topology, and geometric logic in the first half of the twentieth century. Further, semiotics of general sign systems has proven that formal explicitness on the levels of significants and signification processes does not prevent significates from important added value of depth and transcendence. In other words, depth and transcendence are not missed by computational and cognitive approaches. Rather we can observe an increasing distribution of aspects of depth and transcendence among the conceptual topography. This provokes a crucial question of ontology: Can the problem of whatness be tackled by a differentiated discussion of the problem of whereness? At this point, we should make more precise the ontological problem. In fact, it is not a question of specifying the music’s whatness as a Kantian “Ding an sich”, but of how we conceive music: “What is the concept of music?” But a concept is—again in Kant’s words [258]—a topos, a logical place. That is, asking for the concept of music involves a topos in a particular ‘concept space’. Recall [417] that platonic ideas live in an ontological fundus, the hyperouranios topos, which is an explicit topographic site: Relative position and hierarchical location of its instances serve as basements for the whatness of ideas. 23
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CHAPTER 3. MUSICAL ONTOLOGY
It is therefore a legitimate procedure to open the ontological discourse by a view of the conceptual topography of music.
3.1
Where is Music?
Summary. The difficult localization of musical existence is due to three factors: (1) the interweaved usage of topographically distinct locations in the classical ontological discourse; (2) the tendency of classical musicologists to override topographical complexity by all-embracing breviloquent approaches in order to uphold the discourse; (3) the genuinely related activities of thinking and playing—music is one of the best operationalized fields of human cognition. We discuss and exemplify these factors. –Σ– A closer look at musicological terminology reveals that many concepts are topographically blurred since their breviloquent function inhibits differentiation. For example, the concept pairing of consonance and dissonance is of complex reality (see Appendix B.3). It relates to psychic categories, to mental compositional principles, and to physical reality of hearing and cortical sound processing. Questioning this concept pairing without topographical differentiation (of reality coordinates in our case) yields a confusion of meanings and creates bunches of pseudoproblems (see loc. cit.). Historically, one can understand such a difficulty as a capitulation in view of exorbitant complexity, but systematically, it is necessary to distribute a concept over its topographical specializations. This procedure at first sight destroys comfortable unity and simplicity, but science is not about comfort, it is about truth. One may then, after establishing a differentiated and adequate terminology, return to unity by conceptual constructions which include complexity rather than repress it. A most dramatic breviloquent repression strategy within traditional musicology was proposed by Hans Heinrich Eggebrecht on various occasions [134, 135]. Centered around the fundamental question, where to localize music ontologically, he concludes that music resides in the ”I”, a kind of super-breviloquent subject icon. This conclusion is deduced from a discussion of where music could be found elsewhere: in the work, in the composer, etc. Eggebrecht rightly observes that music is not exclusively in the work, nor is it exclusively in the composer, etc. Whence he draws the erroneous conclusion that only a kind of abstract ego may comprise the totality of music. It cannot be discussed in detail, how radical a declaration of scientific bankruptcy such a consequence does represent. But it is evident that the foundation of musical ontology in a whatever abstract subject opens any issue of subjectivity and prevents scientific discourse whenever it suits corresponding needs. A musicological discourse which is founded in the subject can go on forever, but it is for nothing. A third difficulty for locating musical ontology is a fundamental, but difficult relation between thinking and making music. From the very beginning of European music theory in the Pythagorean tradition, playing an instrument (the monochord in those early days) was a method to gain evidence of the transcendent truth of music. It was contended that a metaphysical tetractys symbol was the core of musical ontology, and by playing the monochord, we could access that transcendent being [330, 541]. This approach was not just a comfortable way to understand music, it was the unique solution to gain understanding. In our days, this is still a
3.2. DEPTH AND COMPLEXITY
25
strong belief among musicologists, as the following proposition by Helga de la Motte [121, p.232] shows: “Musikalisches Denken ist grunds¨ atzlich Denken in Musik.” Not only is it admitted that ‘thinking music by examples’ is useful, no, it is fundamentally so! This is, of course, in the line of Eggebrecht’s belief that music is without concepts. Thinking music is thinking in music, beyond words, ineffable, to rephrase it with the words of Diana Raffman [432]. On the one hand, this problematic stems from a poor localization on the semiotic axis. It is not recognized that signs of non-verbal type—mathematical formalisms, for example—could effectively resolve simple verbal ineffability. Sticking to common language, the dominant part of traditional musicological language, is the wrong strategy. No non-trivial insight in modern physics could be obtained via common language—not to speak of modern mathematics. We shall see in part VIII on performance that performance nuances, one of Raffman’s advanced arguments for ineffability of music, can be perfectly formulated in the mathematical language of vector fields. In other words, thinking music by its direct invocation is first of all the proof of a defective language. The claim that something is definitely ineffable presupposes knowledge of any possible language, an absurdity. And it is wrong to believe that music is a special issue of science in that it deals with objects that point at non-mental strata of reality. Psychology, for example, deals with emotions, physics deals with elementary particles. All these objects share aspects that transcend human conceptualization. But we may very well conceive them in a system of knowledge, model their behavior with remarkable success for our cognition. Music is not less and not more accessible than physics. But we have to establish a sophisticated system of signs in order to grasp the significates, common language is not the tool for a musical concept space. On the other hand, music is substantially tied to its communicative dimension: Unlike physics, music was always pronouncedly distributed over poiesis, neutral niveau, and esthesis. However, this aspect is completely disjoint from the question of ineffability. It simply stresses the need for a communicative differentiation when dealing with music. Forgetting about the making of music, boiling it down to an object independently of its making and perceiving is a basic error, just as it is an error to make physical experiments without quantum mechanical uncertainty and complementarity. A famous defective argument which stems from the confusion of communicative coordinates is the judgment of contrapuntal symmetries in music. For instance, the retrograde movement has a completely different existentiality as a compositional (poietic) tool, than when discussed as a (esthesic) fact of perception, we have already discussed this issue in example 1 of section 2.2.3. Summarizing, we can state that most of the musical objects cannot be treated properly outside their topographically specified ontological modality.
3.2
Depth and Complexity
Summary. The ramified discourse based on orientation via local topographic cubes is described as a tool for generating controlled complexity in understanding music. The fundamental pointer character of this tree structure is compared to pretended depth in traditional approaches. The latter is identified as a rhetoric chain of external reference pointers. –Σ– Both, traditional musicology and mathematics, use the word “depth”, however in significantly different ways. To begin with, mathematicians speak of a “deep theorem” or a “deep
26
CHAPTER 3. MUSICAL ONTOLOGY
theory” to indicate that it marks a result of a complex development, and that this result gives a compact, pregnant insight into the addressed complexity. For example, Fermat’s last theorem1 is a deep result since it involves an incredible complexity of mathematical structures, but it is a very compact, in fact elementary, statement. In this sense depth involves complexity, but rather in an encapsulated form; its surface shows a simple appearance. Semiotically speaking, mathematical depth deals with simple significants that are built from complex, viz ramified and recursive signification to connotation and/or metasystem levels. What, then, is the difference to depth in musicology? Strictly speaking, it is pretended encapsulation without the license to access contents. Let us make this clear in the following example. Eggebrecht, an authority of traditionalist German musicology, has stated that “music is without concepts” (see the catchword heading this chapter [101, p.192]). This is a prototypical deep statement since the reader is supposed to know that Eggebrecht is a scientific authority who has written a large number of books and papers on music. In fact, the citation is one of seven final theses in a book entitled “Was ist Musik?” which Eggebrecht and the other German pope of traditionalist musicology, Carl Dahlhaus, have written to discuss the ‘ultimate and deepest’ questions about music. At first sight, Eggebrecht’s statement provokes a contradiction: How can somebody write hundreds of pages about music and simultaneously claim that conceptualization must fail? The solution is depth: We have to recognize that Eggebrecht has gone an incredibly long and audacious way in search of the ‘real thing’, but alas! has to report that—ultimately— the real thing is beyond conceptual comprehension. We, his humble pupils and admirers have to accept that music is an unreachable secret and He, The Master, does rightly encapsulate a deep insight. This capsule can—however—not be opened, the encapsulation remains locked, and this is mandatory for two reasons. First, by the pronounced authority, He, The Master has experienced the horrific failure of concepts and we, the pupils, should just accept and admire this report from hell. Second, the capsule’s key has been lost, no trace down to the very proof of the pretended conceptual failure is at hand. The text [101, p.192] does not give any hint, it simply states the verdict. Such a type of “depth” is completely standard in musicology. You give an encapsulated statement and prevent any access to the (presumably) hidden complexity. Whereas in mathematics, the access to complexity is possible and its realization eventually yields insight, the musicological encapsulation sticks to a pointer to insight, however, the pointer points into the void. You cannot obtain a follow-up, information flow breaks down, and verification of the claim is not offered on the base of explicit complexity. Rather is the dark path to the hidden (pretended) rationale ornamented by metaphors. In our case, Eggebrecht terminates by stigmatizing music with mysterious weirdness, thus turning depth into mystery—and science into fabulation. As a basic rhetoric device, this kind of void pointer has been used in one of the most ambitious handbooks of traditional musicology [103]. The discourse of important parts of that handbook relies on void pointers to never ending chains of implicit (!) external references. Fortunately, musicology is not doomed to void pointer depth. If applied with all its ramifications, the topographic cube (see section 2.4 and figure ??) offers an effective tool of complexity which can help hammer out depth in the mathematical sense, i.e. without void encapsulation. In section 2.4, we have already 1 This theorem is one of the most famous and hardest results in the history of mathematics. It was proven by Andrew Wiles in 1994 and states that any non-trivial integer solution of the equation xn + y n = z n implies n = 1 or n = 2.
3.2. DEPTH AND COMPLEXITY
27
Figure 3.1: The topographic cube of musical ontology has local character. A concept of music (represented by balls in this figure) may open different localizations with successively refined concept analysis. introduced the local character of ontological topography. It means that the decision to attribute to a given situation specific topographic coordinates is no obstruction to other coordinates in a successive step of refined analysis, see figure 3.1. But it does not mean that this recursion from one local perspective to the next tends to more elementary and eventually “atomic” ontology. Ontological atomism is neither desired nor even possible as a foundation of knowledge: Being has no elements, only perspectives. The cube of topography is a local coordinate system for such perspectives, it offers orientation, but no roots. Example 2 On the string of communication, the performing artist is on the esthesic coordinate with respect to the composer. He or she receives the work and analyzes it as a mental entity. This complex identity of the performer is then encapsulated and taken as a new communication unity of the concrete performance. In this role, the performer stays on the poietic side and produces a work of physical reality, namely the sounding performance. This latter is an expression of the artist’s reflection and will now transmit the message to the listeners of a concert hall, say. In this description, the listener is still encapsulated as an esthesic instance, but we should open it with respect to what is going on when understanding a performance of a work of art. In this perspective—the inner dialog of the listener with his or her consciousness—the acoustical performance is again transformed into a mental structure and interpreted along the personal lines of musical insight. The result is a new type of work of mental and/or psychic reality which the listener will communicate internally or externally in a further processing of the musical experience. Etc.– We have just sketched some en- and decapsulation processes on the communicative dimension, but the reader can easily extend this flip-flop of perspectives on any other topographic
28
CHAPTER 3. MUSICAL ONTOLOGY
dimension. It becomes clear now that opening ontological perspectives is by no means an approach to ontological roots, but it is a clarification and gives ample orientation which void pointers definitely do refuse. It is one of the main tasks of part II to develop a profound and detailed navigation formalism through music, a formalism which provides encapsulation and reliable pointers to hidden complexity at the same time. But a word of caution is imperative: Our task cannot be attained without a considerable amount of technical machinery—nihil ex nihilo.
Chapter 4
Models and Experiments in Musicology Plato’s pessimistic picture of empirical observation caused him to deny the validity of physical models and was largely responsible for the eclipse of empiricism for 2,000 years. Peter Wegner [563] Summary. This chapter introduces the paradigm of experimental humanities. Such a perspective is basic to all computational, resp. effective, methods in musicology. We discuss the parallel to the epochal Galilean change from speculative to experimental natural sciences. –Σ– In the final chapter of this first introductory part, it is necessary to review the overall epistemology of musicology since the preceding ontological topography has questioned traditional standpoints in a measure that does not allow of uncritical takeover of epistemological fundamentals. Above all, it is the requirement of free access to (encapsuled) complexity that creates a boundless analytical attitude: It is no longer possible to ‘cultivate’ private, inaccessible regions of knowledge. This is not only due to the fact that music belongs to humans and no longer appears as a revelation of divinities of whatever flavor. It is also due to the massively improved arsenal of information and communication technology where the knowledge space becomes a concretely accessible site, and any lack of precise or explicit information is immediately blamed. This situation enforces a new, fundamentally interactive understanding of human knowledge. The paradigm of contemplative science which was essentially promoted by religious constraints [363, 477] has to be abandoned, knowledge is constantly updated and extended. The global knowledge space has become an ocean which is in permanent metamorphosis, on which we navigate and experiment in a spirit of dynamical space-time [362]. 29
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CHAPTER 4. MODELS AND EXPERIMENTS IN MUSICOLOGY
We shall make these points more explicit in the following sections. It is useful for the understanding of new paradigms in musicology, however, to have a closer look at the development of modern experimental science which prepared the Galilean revolution. In this preliminary discussion, we refer to Isabelle Stengers’ article [485, p.395]. The case of Galileo Galilei is important for musicology since it is intimately related to a common problem: definition of musical tempo and physical velocity. Galileo was concerned with the definition of instantaneous velocity. Recall that, for historical reasons, he could not yet refer to calculus and define velocity as derivative ds/dt of space s(t) as a function of time t. He had to develop the very concept of instantaneous velocity since the traditional concept of uniformly accelerated motion, as it was understood by medieval scientists such as Nicholas Oresme, could only view it as a succession of portions of uniform motion, changing discontinuously their constant velocity after a short duration. Galileo’s concern was not just another speculative concept but resulted from intense observation of the movement of physical bodies on an inclined plane. His proposals (dated around 1604) rather aimed at measuring (by impact) local velocity in a determined moment of time.
Figure 4.1: Galileo’s method concentrated on observation and experimental interaction with nature. This detail of a fresco by Giuseppe Bezzuoli (Museo di Storia della Scienza, Florence) shows the famous experiment of a ball running down an inclined plane (right) and the traditional consultation of Aristotelian works (left). In other words, Galileo’s approach was essentially built upon observation and measurement, it did not rely on abstract speculative reflections. The turning point here is precisely the passage from speculative encapsulation to explicit accessibility by ‘doing’ science. Measuring the body’s mechanical impact invokes a momentous property of that body. Though Galileo could
31 not make this property explicit on mathematical terms, he succeeded in discovering a physical one-to-one correspondence to instantaneous velocity. The interesting point in this procedure is a fundamentally operational method: thinking by doing. In fact, if one views Galileo’s twohundred-page notices written around 1604, it becomes clear that he was constantly dialoguing with his experiments, tables, number lists, and diagrams—and not with Aristotle’s authoritative writings—in order to construct a homogeneous concept space of physical movement, see figure 4.1. We have insisted on this episode since the musicological analog of velocity, musical tempo, shows an astonishingly parallel concept history—though nearly five hundred years delayed from physics.
Figure 4.2: The figure shows a typical musicological ‘digitized’ version of the tempo concept, as used by Hermann Gottschewski [110]. The rectangular regions define regions of constant tempo (visualized here by reciprocal duration values). Since no instantaneous tempo concept is given, the author obtains different averaged tempo regions, according to the respective time window. In traditional European musicology [110, p.317], we encounter a concept of tempo which corresponds to Oresme’s setting: Tempo is the quotient of musical duration (measured in quarters) and physical duration (measured in minutes), this yields the classical M¨alzel metronome m.m. in quarters per minute (see section 33.1 for a detailed discussion of tempo). It is essential that this concept does not include instantaneous tempo in the sense of a derivative, traditional musicologists deny the musical relevance of such a concept, see figure 4.2. This refutation stems from the completely speculative handling of tempo in traditional musicology. Many of the tempo-related concepts, such as fermatas, ritardandi, and accelerandi, are encapsulated locked objects, and theorists would not know how to give a workable description of such tempo phenomena. The opening and explication of these void pointers are left to musicians and programmers of music software. Instead of learning from Galileo’s experimental observations theorists insist on a discrete tempo fantasm a la Oresme—despite the rich language of modern mathematics. This parallels physics to musicology in a basic issue. Precisely as Galileo, modern working
32
CHAPTER 4. MODELS AND EXPERIMENTS IN MUSICOLOGY
musicians and engineers are at the cutting edge of explicitness and leave irrelevant speculations to feuilletonism where they are well-placed. We learn from this double experimental revolution that it is not banausic artisanship that enforces opening of concept capsules, but the mere desire to redeem conceptual pointers, and to drive out at last void rhetorics. Galileo’s revolution was the answer to pretended depth of rhetoric discourse. Just as medieval medicine was traditionally split into speculative academics (without license to practice!) and artisanal surgery (practised by charlatans), physics was split into mechanical engineering and speculative philosophy. Scientists such as physicist Galileo and the French surgean Ambroise Par´e had to ‘dissect’ the concepts and—hitherto tabooed—human bodies to explicate complexity [463]. We shall now take a closer look at the epistemological implications of this revolution.
4.1
Interior and Exterior Nature
Summary. Reasons for the categorical distinction between interior and exterior nature. Can human thought be an experimental field? –Σ– Experimental sciences are classically related to empirical exterior nature. According to a traditional opinion condivided by Immanuel Kant [258], experiments are understood as interrogations of a passive witness. Nature is passive and will answer to the hearing in an objective way, i.e. independently of the interrogator’s subject. These two characteristics, passivity and objective response, have since been recognized as erroneous idealizations. Physical experiments are substantially interactive processes, this was evidenced by uncertainty and complementarity principles of quantum mechanics. But physical nature is also responsive on a more conceptual level. Each response to an experiment may alter the theoretical position and the concepts which drive the experiments. This is exactly Galileo’s method, there are no platonic ideas which we have to retrace from empirical reality, the concepts are in incessant metamorphosis and mutate as a result of experimental cognition. As with genetics, only the fittest concepts can survive the experimental struggle. In what respect is then inner nature so fundamentally different to exterior nature? It is said that inner human nature is subjective and that there is, consequently, no such a thing as objective facts. But the argument of subjectivity is misleading. It is definitely not true that we have a more indepth control of inner nature because of subjectivity. The inner human nature is a vast field of instances which we do not any better access than exterior objects. Subjectivity is only a minor part of the inner nature. Let us give two examples: In mathematics, it is admitted that the Peano axioms for number theory1 are trivial creations of the human mind. But the impression vanishes immediately when abording such a simple question as the famous and still unsolved Goldbach conjecture2 : Can every even natural number be written as the sum of two prime numbers? This makes evident that the complexity of 1 (1) The number 0 (zero) is a natural number; (2) every natural number a has a unique successor a+ among the natural numbers; (3) if a+ = b+ , then a = b; (4) 0 is not a successor; (5) if a property φ for natural numbers is such that 1) φ(0) and 2) φ(a) implies φ(a+ ), for every a, then the property holds for all natural numbers. 2 Conjecture established by German mathematician Christian Goldbach, 1690-1764.
4.2. WHAT IS A MUSICOLOGICAL EXPERIMENT?
33
our mental creations is not more under control than physical objects. We believe that the Peano axioms are under control. But in their immediate conceptual vicinity, there are statements which do escape our mental power. In musical composition, we typically deal with the creation of a score. But it is an illusion to believe that the richness of such a creation can be controlled by the composer. His or her contribution is but a germ of an incredible complexity that implies combinatorial processes, interpretation strategies, semiotic stratification and so on. In other words, a composition is quite the same as a piece of exterior nature, the part of subjectivity vanishes in front of an exorbitant structural variety. So, what is the difference between exterior and interior nature? It relies on the more apparant, though not fundamental, autonomy of exterior ‘natural’ phenomena. But a second view reveals an inner nature—especially on the level of man-made universes such as mathematics and music—which is not less complex and uncontrollable than exterior nature. As a piece of uncontrolled nature, Bach’s “Art of the Fugue” is not fundamentally different from a supernova out in interstellar space.
4.2
What Is a Musicological Experiment?
Summary. The Pythagorean tradition and the general concept of a music instrument. Candidates for experimental layers. Testing the congruence of a scientific model with corresponding material, such as a composition, performance, or analysis. –Σ– We know from section 3.1 that Pythagorean tradition offers a type of “thinking by doing”. Can we view this paradigm as an archaic version of Galileo’s experimental approach? We could if it provided us with a conceptual laboratory where concepts could be fabricated while making music. But this fails, playing the monochord was nothing more and nothing less than a passive verification of already fixed ideas, in fact the metaphysical tetractys. It was a kind of doing on the ground of prefabricated thoughts. Nonetheless, the insistence on the aspect of activity in thinking music has survived all speculative assaults from tetractys to Keplerian harmony. The shape in which we still encounter this method has of course changed. No longer can it be found under the auspices of divinities, it has been saved by composers, the experimental scientists of music. This is somewhat exaggerated since composers often condivide the status of medieval charlatans. But in their best performance, such as with Pierre Boulez, Giorgy Ligeti, or Iannis Xenakis, they really appear as prophets of a future type of experimental musicologists. What happens is that these forerunners are left alone by a speculative void-pointed musicology. Already did Arnold Sch¨ onberg quit the theoretical background in view of the speculative overhead that could not cope with requirements of effective conceptualization in harmony3 . The so-called “emancipation of dissonance” is among others the emancipation of experimental musicology from speculative rhetorics. It is not by chance that Boulez demanded to blow up all opera houses. 3 This does not contradict his book on harmony [478] which is much less a warmed-up catechism than a critical review of failures of harmony. However, Sch¨ onberg’s treatise on harmony did rather stress than solve those problems.
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CHAPTER 4. MODELS AND EXPERIMENTS IN MUSICOLOGY
So what is a musicological experiment? It must clearly include both, thinking and playing, reflection and action. But in the form of mutual inspection, i.e. it should neither be a passive sonification of given speculations nor should it reduce to speculation without musical realization. So the decisive criterion is a bidirectional dialog between a flexible conceptualization and an intelligent sonic realization. Such a dialog can only succeed if the transformation channels between thinking and playing are explicit, operational and highly efficient. These requirements are necessary, but not sufficient to conceive such a thing as a musicological experiment. We are still missing the adaptation of the crucial concept of experience to the context of humanities. It is by no means clear that such a category can subsist within inner nature. This will be dealt with in the next section.
4.3
Questions—Experiments of the Mind
Summary. Experience in the humanities. Separating speculative questions from musicological experiments. Necessary and sufficient conditions for questions to be accessible on the level of the experimental setup. Improvement of scientific conceptualization by constraints from experimental options. –Σ– The naive view of experience in natural sciences evokes a bunch of facts in the space-time of exterior nature. But this is not what an experimental setup is dealing with; already in its very data processing are the ‘naked facts’ codified in numbers, figures, and all kinds of symbols. What is accessible to scientific reflection is already a body of information in a well-structured concept space. Measurement of physical events includes a transformation into data streams within a mental knowledge space. Experience then boils down to a complex navigation process along this data stream, and under use of statistical orientation tools and theoretical hypotheses to be verified or falsified. When put together with the foregoing approach to the experimental setup, this yields a concept of a dialog between a set of data that are embedded in a concept space, and a set of analytical tools living in the same concept or knowledge space. In this perspective, the question underlying the experiment appears as a search mission towards a statement which fits the theoretical orientation defined by hypotheses and evaluation tools. In other words, the experiment turns out to be a search mission while navigating on a concept space, the navigation being driven by two ‘dialog partners’: the data stream and the hypotheses and evaluation tools. But such a generic restatement of the concept of an experiment has erased any specific reference to exterior nature. We just deal with navigation on a data stream issued from a specific research object and a bunch of analytical orientation devices. This entire process is located on a (mental) knowledge space. Under this perspective, an experiment may naturally occur in the context of humanities. Navigation may thus be restated as interactive interpretation of a given human work, be it music, painting, literature or whatever can be subsumed under the general structure of a verbal or nonverbal semiotic ‘text’. Navigation on such a text crystallizes in form of a search for structural coherence between the variety of the text’s syntax and semantics on one side, and corresponding hypotheses and (text) theories as a background of orientation on the other. Observe that the
4.4. NEW SCIENTIFIC PARADIGMS AND COLLABORATORIES
35
concept of empirical experience has been absorbed by the concept of navigation on a stream in a concept or knowledge space. The parameter of external physical time is no longer relevant, navigation unfolds on an ‘abstract’ space, and time is replaced by an abstract ‘curve parameter’ of the navigation trajectory. We should stress that the experimental dialog is strongly interactive in that there is no priority of theories over ‘data streams’. A theory may completely mutate under the influence of new directions in artistic creativity. And conversely, the inner nature is no “nature morte”, it is in constant renewal. This being the case, it is important to review the role of speculative void pointer concepts under the experimental paradigm. In the situation of an explicit dialog between given data and theoretical instances, void pointer concepts are immediately ruled out. For example, an encapsulated locked concept of dominant, tonic and subdominant in function harmony cannot resolve any task of harmonic analysis for a concrete score since it asks for harmonic function values of any possible chord that may theoretically appear in the given score. In cases where the function value(s) cannot be determined, the data stream cannot be navigated and the experiment breaks down. Galilei could not work with the medieval velocity chimaeras since they evidently provided no navigation through the attentive observation of balls running down the inclined plane. Experiments are excellent exterminators of empty pointers: Their dialog immediately lays bare deficiencies hidden in locked concept pointers. The experiment, if viewed as a question of the mind, not as an autistic reflection on locked concepts, rather as a dialoguing navigation, can help us understand what happens when we build concepts and theories.
4.4
New Scientific Paradigms and Collaboratories
Summary. Mutual influence of communicative networks and conceptual precision. Doing musicological experiments on a distributed laboratory. Redefining scientific competence from the communicative point of view. –Σ– From the preceding discussion it is straightforward that musicology (and humanities in general) in its experimental profile cannot subsist in the private ambience of classical humanities. The experimental navigation dialog is demanding for all partners, and the very nature of dialog enforces data exchange on powerful channels. Socially restated, such a type of science has to be realized in a strongly collaborative style. On the level of institutions, this demands the structure of collaboratories4 , as introduced by the US Department of Energy. Just as physicists could not survive in isolated research units, musicologists have to initiate intense communication within research in order to succeed. We insist on communication while doing research, not merely in noble and non-binding conference small talk. Collaboration is meant as a working style, not as a title for informal politeness. Moreover, the communicative network of collaborative science enforces a radical conceptual precision. This is not only about the formal constraints when dealing with computer programs, it 4 The term “collaboratory” was coined by Bill Wulf in 1993 and is merged from “collaboration” and “laboratory”, see [363] for further information.
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CHAPTER 4. MODELS AND EXPERIMENTS IN MUSICOLOGY
is really about communicating as a working method. Learning by doing science can only succeed on the basis of unlimited access to encapsulated complexity. Evidently, scientific competence must then be rebuilt on the fundament of conceptual communicativeness while private regions of knowledge will lose their historically grown honorability. We should keep in mind that the entire navigation and orientation metaphor in the preceding discussion of the knowledge or concept space is not merely general knowledge science slang, but more concretely an adequate expression of the topographical (in fact: topological) setting which was developed in chapter 2. Navigation happens to move along (local) coordinates of topographic cubes, and we shall learn in part II that the general metaphor of a concept space can be realized in form of a veritable geometric space.
Part II
Navigation on Concept Spaces
37
Chapter 5
Navigation Verlassen wir das strenge Labyrinth der zementierten Begriffe, und ergehen wir uns ungezwungen in improvisierten Architekturen, und dies selbst dann, wenn wir reden. Pierre Boulez [60, II, p.78] Summary. Well-conceived information produces knowledge. But conceiving means building concepts, and this amounts to having well-structured access modes. This enforces the development of concept-oriented access modalities to given information, including associated concept spaces. Music is an excellent field to exemplify such knowledge spaces. It has become virtually impossible to navigate through music information without developing powerful concept formats. –Σ– We initiate this part by a reflection on conceptual navigation since the concept spaces which will be defined in the subsequent chapters are strongly motivated by universal orientation paradigms while surfing on encyclopedias of inhomogeneous music-related information and knowledge. More precisely, observe that knowledge involves two components: information and its mental organization. Whereas boiling down knowledge to mere information would mean getting drowned in a sea of amorphous substance, abstraction from information towards pure mental organization would mean getting stuck on Georg Wilhelm Friedrich Hegel’s germs of logic, i.e. the zero state of philosophy as exposed in the first pages of [214]. Both reductions cannot cope with what knowledge deals with, viz building concepts of something. Building concepts then amounts to conceiving this data, organizing the access to information and doing this by use of well-defined access modes. Knowledge has an object, and it reaches that object via its concept. Conceiving means being able to navigate in a conceptual coordinate system to attend that something. In other words, pure information is substantial, but it is uncontrollable without a conceptual form, a coordinate system where we may place and retrieve substance. This is precisely why so-called digital information has nothing to do with knowledge whatsoever, and why therefore the digital paradigm is completely irrelevant to the yoga of electronic age: Digital substance, reduced to the BIT = {0, 1} of two substance values, 0 and 1, or OFF 39
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CHAPTER 5. NAVIGATION
and ON, the name does not matter, is just a minimal substance set, but it cannot per se be responsible for any kind of knowledge. In fact, a digital record is worthless without being organized in a concept form, i.e. as encoded knowledge. ‘Understanding’, i.e., decoding information is needed in order to control and manage its knowledge potential. The digital age is not centered around “Bits and Bytes” but around their accessibility and handling, in short: data navigation. From the above we learn that a powerful concept system for any field of knowledge must provide us with a thorough navigation method that works on an extensive concept space. And from the preceding discussion regarding depth and complexity (section 3.2) we deduce that navigation must have access to any encapsulated concept, and permission to navigate on any possible “concept path”. It is essential for a successful concept navigation methodology in music to realize a highly interactive and non-authoritative, autonomous orientation and decision strategy. Rather than ending in obscure concept mist conceptual navigation should lay bare semantical lacunae, and this means developing a language for the open-ended navigation if ever this is the state of the art. In no case is it any longer acceptable to be navigated by dogmatic magisterial guides. This is what the Boulez citation heading this chapter suggests: To create a conceptual environment that allows quasi-improvisational and free-will driven discourse on music. Boulez must have felt that “cemented concept labyrinths” no longer cope with the requirements of advanced musicology, be it on the creative, or be it on the reflexive side. Now, the term “improvisational” in Boulez’ statement is somewhat misleading. It suggests an antagonist to the score-dominated way of making music. Like improvisation it should be interactive—this is to be retained. But it is much more and much more important than a way of making music. The key word is “ cemented”. Refraining from cemented concept architectures means that the concept architecture has become dynamic, ever-changing and soft. Concrete is a rigid material, hard and insensitive to whatever. Boulez’ vision is that of a flexible concept environment, rather than improvisational it should be termed dynamic, i.e. interactive and adapting to changing demands. The problem is how to realize such a dynamic concept architecture without loosing rigor and reliability. We shall propose such an architecture in chapter 6. For the time being, let us concentrate on the generic navigation paradigm as suggested by Boulez and stressed in the ongoing transformation from energy to information society. In particular we should reconsider navigation as an interactive interpenetration of knowledge agents and data bases. This latter idea turns out to be one of the deepest concerns of music in the making of our own world.
5.1
Music in the EncycloSpace
Summary. We discuss the navigation problem on music data in relation to the general concept of an encyclopedic knowledge space, the EncycloSpace. The latter is introduced as an upgrading of the classical concept of an encyclopedia, as conceived by the French encyclopedists Denis Diderot and Jean le Rond d’Alembert. The upgrading is characterized by three changes: (1) the static cosmos is replaced by a dynamically developing data organism in space-time; (2) the passive “speculum mundi”, i.e. the purely receptive view, is replaced by an interactive instrumental relation to the data organism; (3) the textually oriented alphabetic ordering is generalized to a universal navigational orientation. Consequently, encyclopedic navigation splits
5.1. MUSIC IN THE ENCYCLOSPACE
41
into a receptive and a productive variant. –Σ– In [363], a general definition of an encyclopedic knowledge space, called EncycloSpace, was given. Let us shortly recall that definition and then explain the concept’s genealogy and ingredients: Definition 2 EncycloSpace is the topological corpus of global human knowledge which evolves dynamically in a virtual space-time, is coupled to human knowledge production in an interactive and ontological way, and allows of unrestricted navigation according to universal orientation within a hypermedially represented concept space. According to Sylvain Auroux’ analysis of Denis Diderot’s Encyclop´edie [25], the encyclopedic principles of the Age of Enlightenment can be viewed as a combination of the completeness of a dictionary, the unity of philosophical synopsis, and the discoursivity of a mathematically inspired ordered representation (the alphabetic ordering of words), see figure 5.1.
?
Und so kam es, dass man sich eines Tages entsinnte, dass alles so, wie es da stand für mmer so sein würde und dies ohne je etwas daran ändern zu können
A, B,
C
Figure 5.1: The principles of classical encyclopedia: completeness (complete circumference of the circle), unity (the circle’s perfect shape), and discoursivity (alphabetical discourse along the circle’s line). In a critical review of these encyclopedic characteristics, the Encyclop´edie is recognized as being a faithful realization of the medieval idea of a “speculum mundi”: “L’encyclop´edie assume par sa fixit´e relative le rˆ ole d’un miroir du monde naturel, tel qu’il peu s’offrir aux sujets connaissants.” This insight specifies the attribute of completeness, symbolized by the
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CHAPTER 5. NAVIGATION
orbit, as a fixed cosmos. This is a traditional static world view, similar to the photography of a stationary system. However, such a snapshot cannot claim to represent a world where knowledge is rapidly and incessantly growing. We have to accept that the purely spatial coordinates of a virtual cosmos (in Diderot’s words: “connaissances ´eparses sur la surface de la terre”) must be completed by the time coordinate. We have to deal with a dynamic universe of knowledge embedded in virtual space-time, and subjected to laws of coupled synchronic and diachronic nature (see section 2.3.4 for these semiotic concepts). The development of knowledge is not an accumulation of essentially immutable and isolated time-slices. See figure 5.2 for the following discussion.
Instrument
Virtual Space-Time
Orientation
Figure 5.2: The EncycloSpace characteristics: Universal orientation in a dynamic knowledge space-time which is interactively coupled to human knowledge production. Further, the change of this dynamic system evidently does not happen by an autonomous activity: It is a result of a constant and substantial interaction of humanity with the corpus of knowledge. Much like political and civilization dynamics or brute continental shift, the encyclopedic body is an open system which is incessantly reshaped in synergy with human knowledge production. Consequently, the metaphor of a “speculum”, i.e. a passive vision of given things, is not adequate. We do not only look at prefixed things, we do also define and redefine them, add new knowledge and relativize old-fashioned approaches. One could then replace the metaphor of a “speculum” by that of an “instrumentum”. This latter is perfectly adequate to the computer as a bidirectionally active interface between humans and knowledge bases. This aspect is substantial since it also questions the very nature of knowledge. Concepts are no eternal entities who live out in a platonic sky (hyperouranios topos). They rather represent operational units with an ontology that is rightly defined by the very accessibility of conceptual components.
5.1. MUSIC IN THE ENCYCLOSPACE
43
(In a more radical perspective, it could be argued that concepts are operators.) In this vein, understanding the concept of “house”, for example, boils down to the way you activate its components (roof, walls, windows, etc.) when asked to prove your understanding (to yourself or to others, there is no difference). Finally, the order of representation of knowledge within the Encyclop´edie is twofold: On the very surface, it is the alphabetic ordering dictated by the traditional textual representation. More in-depth, it is the geographic orientation given by the Encyclop´edie’s cross-reference system [25, p.321 f]. Whereas the former is strictly linear, the latter is rather a partial relation of “pointers”, far from being an ordering in the technical sense. This is also due to the fact that the concepts are not cast in a formal language and hence, beyond the alphabetic ordering no intrinsic organization is visible on the representational level. This point is not just a marginal note concerning formal aspects of knowledge. Rather do we envisage the in-depth question of the structure of the space where concepts live. We know from Plato’s allegory of the cave [417], from Aristotle’s “Topic” [20] and from Immanuel Kant’s comment on Aristotle’s ”Topic” in “Kritik der reinen Vernunft” [258], that the spatial metaphor for the ontology of concepts is crucial for any effective discussion of the system of concepts. In this sense, ontology of concepts is essentially topology: a study of the topoi where concepts subsist. In this theoretical sense, alphabetic navigation is not sufficient, conceptual navigation must provide tools and paradigms which transcend the textual alphabetism and include generic principles of answering to primordial discursive questions: Where do I come from? Where am I? In what direction do I proceed? From a more operative and practical point of view, the alphabetic ordering inherent in the linguistic text paradigm must be completed by orderings which are genuinely related to non-textual data formats, such as geometric spaces, or spaces of set collections (see chapter 6 for precise definitions). Such more general data types emerge in a natural way in hypermedia documents including visual and acoustic instances, such as pictures, diagrams, sounds, music or dance scores, for example. Navigation orderings have to cope with hypermedia orientation paradigms for practical and for theoretical reasons: There is no reason why the completely arbitrary textual reduction should and could grasp the intrinsic ordering of concepts and thoughts. To be clear, the brute reduction to the definitely textual binary code encompasses virtually every possible information record. But it is a different business to understand information—this is definitely not in the reach of the binary code. “Wor¨ uber man nicht sprechen kann, dar¨ uber muss man schweigen.” This killer sentence which terminates Ludwig Wittgenstein’s “tractatus logico-philosophicus” [580] collapses in the age of hypermedia discourse. It should be replaced by a recommendation of visual/geometric, acoustic/musical, or haptic/gestural alternatives to textual dead ends. Summarizing, we have reviewed the concept of Diderot’s Encyclop´edie and updated its three attributes to meet the needs of a dynamic, active, and universally oriented knowledge society. Thus, we may baptize this update by the name of EncycloSpace, and determine the concept as explained in the above definition 2. Though “navigation” stems from Latin “ navigare”, and this one from “navis” plus “agere”, which means giving motion to a ship, the present meaning of “navigation” is restricted to the passive steering of a vehicle which is already moved by some motor. However, navigation on the EncycloSpace must regain the original activity, since EncycloSpace has been determined as an interactively handled body of knowledge, a body which is not built, altered and developed by mysterious forces but by our genuine agitation upon the object. Therefore, the original
44
CHAPTER 5. NAVIGATION
etymology of “navigation” is restituted: We shall distinguish between receptive and productive navigation. Receptive navigation means moving around within the body of knowledge without changing it. In contrast, productive navigation does change the body of knowledge. We will discuss these two modalities in the following sections.
5.2
Receptive Navigation
Summary. Receptive navigation leaves the EncycloSpace unaltered. It deals with universal orderings to find and represent knowledge for optimized understanding. Since musical data are of very inhomogeneous and general types, universal navigation concepts are essential in innovative musicology. –Σ– Receptive navigation is the common situation in classical encyclopedias. One is looking for a bunch of information in a more or less well-determined, and immutable field of knowledge. The main requirement in this situation is an optimal orientation environment. In case of a simple name search, the alphabetic ordering is adequate: We just leaf through the linear alphabetic ordering of words. But this does not cover less trivial search problems. For instance, if we are looking for all chords in the score of Schumann’s “Tr¨aumerei”, say, the alphabetic orientation crashes. Surfing through the set of chords in a determined score is a new situation, it requires ordering principles for chords as special types of note sets. The EncycloSpace features should comprise a representation of all chords built from the 463 notes of the “Tr¨aumerei”. They should then be able to order these chords according to universal ordering principles since we cannot (1) redefine from scratch orderings among EncycloSpace objects and (2) build individual orientation frames for each particular configuration. Given such a universal ordering, every chord could be retrieved according to corresponding intuitive orientation paradigms. We recognize that inspecting such an ordered chord list is only the last link of an extended chain of encylopedic objects which finally leads to the required chords. We would typically start on the level of composer names, then—once “Robert Schumann” has been found—descend to piano works, then to “Kinderszenen”, and finally to “Tr¨aumerei”. At this point, we will have to surf through different types of musical score signs. Once arrived on the level of notes, representing ordered lists of chords requires non-alphabetic ordering principles. In fact, piano notes are points in a space of four dimensions, pitch, onset, duration, and loudness. Such a space requires different approaches to ordering—and chords do so a fortiori. Universal orderings among EncycloSpace objects is a central feature for receptive navigation because it yields orientation and thusly prepares the field for recruitment of added knowledge which is to be created from productive navigation. Whatever are the searched for objects, it turns out to be crucial for valid receptive navigation to be provided with universal ordering principles: orientation must be generic, and generically representable on the visual interface.—This will be an important feature of the denotator system to be introduced in chapter 6.
5.3. PRODUCTIVE NAVIGATION
5.3
45
Productive Navigation
Summary. Productive navigation interacts with the EncycloSpace such that the latter is enriched by a surplus value of added knowledge. The EncycloSpace is a repertoire as well as a laboratory. Productive navigation is a central issue in extending musicological and musical knowledge. –Σ– In the above example of an ordered list of chords, we supposed the existence of such a list on the EncycloSpace. However, it is not probable that all such lists of chords drawn from scores of the piano music repertoire are incorporated in the EncycloSpace. More probably most of what is required by an interested user will not be allocated. This can be the case on very different levels of knowledge. In the simplest level, it can be a numerical information we want to obtain, e.g.: “How many quarter notes are contained in Thelonious Monk’s original tune “Blue Monk”?” This information usually is not part of the knowledge base, but it can easily be calculated. Searching for the answer is a trivial extension of knowledge, but it differs from receptive navigation: Navigation which is defined by the question and terminates upon its answer produces a (tiny) extension of the EncycloSpace. On a less trivial level, the calculation of complex data, such as the number of (1) chords in the “Tr¨aumerei” which (2) contain more than three notes, which (3) are dominant in F major, and which (4) contain a diminished third, can require too much calculation time to allow forgetting about the result. In this case, we try to save the result and access it as soon as new navigation trips ask for this information. On a still more invasive level, it may turn out that we have built new concepts, such as a special type of musical motives which meet non-standard conditions. Suppose that such a concept is named “Non-Standard-Motif”. We do not want to be excluded from navigation because this concept is not yet part of the EncycloSpace’s vocabulary! In other words, Boulez’ requirement of an “improvisational” discourse/navigation must include free vocabulary extension and then the operation upon the extended vocabulary. Summarizing, we see that productive navigation responds to the demand for dynamic navigation, interacting with the body of knowledge, and incessantly producing new data to be added to the given EncycloSpace. Evidently, such an extended functionality adds to the original repertory character of the EncycloSpace the character of a laboratory where questions are transformed into mental experiments as introduced in chapter 4. Besides the technical problem of how to realize such a functionality without losing control, we are also confronted with the level of accessibility of extended EncycloSpace contents by third parties. But this is not a question of fundamental ordering, rather technology should be concerned with the social aspect of navigation-induced knowledge extension. Boulez’ idea of an improvisational concept navigation provokes fundamental questions of communication which are well known from music when improvisation started to penetrate the music market and musicology as a science. In fact, most jazz productions essentially rely on sound tracks which override pure score or songbook data. The introduction of sound files as a major trace of and reference to the musical creativity is responsible for the very existence of the present jazz culture. Though technologically low level, the mere possibility to access ad infinitum an improvised piece of music initiated a discourse about music which was never
46
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possible before. And it enforced the relativization of classical scores as an identification of the musical work. On the level of conceptual navigation, we experience the same effect: Classical ‘scores of knowledge’, i.e. hard-coded knowledge data bases in books and encyclopedias, are being confronted with dynamic records of ever moving knowledge streams. ‘Improvisation’ within knowledge can now be traced and integrated in the global stream of knowledge production. No need to stress the fundamental transformation of knowledge sociology under such perspectives. We would like to insist on this background throughout the present book since it exerts an incredible impact on how we recognize important from irrelevant aspects of formal conceptualization in future musicology.
Chapter 6
Denotators C’est peut-ˆetre dans la fa¸con de se repr´esenter l’esprit qu’on pourrait d´echiffrer l’esprit. Paul Val´ery [538, III] Summary. This chapter introduces the universal data format of denotators to describe musical objects. Denotators generalize the structures of local compositions in mathematical music theory as well as the data model used in the RUBATOr software. They do, however, not include deeper semantic layers which are related to physical, psychic, social or religious meaning. Their semiotic structure represents an elaborate denotative baseline and resides in a purely mental level usually attributed to mathematical objects. –Σ– This chapter introduces the formal framework of the entire book. We believe that the preceding chapters have given a huge motivation to elaborate a formally rigorous concept framework. Therefore, we shall no longer comment on the general purpose of this issue. Let us instead describe the genealogy of the denotator concept. After the publication of the German precursor [340] of this volume, and in connection with the development of the composition software prestor [338], it was believed that the concept framework of mathematical music theory was sufficiently generic to handle formal issues in computational systematic musicology. This framework was built upon so-called local and global compositions. However, the theoretical and program design groundwork for the analysis and performance platform RUBATOr taught the developers that these concepts had several severe drawbacks: 1. They were centered around a special type of mathematical structures (essentially finite point sets in particular parameter spaces) which could not comprise less theoretical objects, such as to include attributes with character strings for names, or other non-numerical data. 2. Also were they ‘hard-coded’ in the sense that introduction of new concept types was not possible. In particular it was not possible to define concepts by recursion to already given concepts. 47
48
CHAPTER 6. DENOTATORS 3. The mathematical flavor was not compatible with programming requirements for data bases. 4. The naming policy did not allow clear distinction between objects and names (in a strictly semiotic sense). 5. The so-called functorial point of view was not taken seriously enough. Intuitively, this means that we had to stick to one fixed ‘ontological’ point of view. 6. The objects were not well distinguished from the spaces they live in. This hindered recombination of object selections—which now is possible, and meets the requirements of productive navigation.
The denotator concept framework will eliminate these defects. But it will also require a rather complex concept scheme. We shall develop the details in several steps, starting with a more down-to-earth version of the denotator framework (section 6.1), and then extending to the universally valid generic setup. Denotators are conceived according to the general criteria of EncycloSpace navigation, so observe that we do not follow mathematical tradition but navigation methodology, including ‘mathematical navigation’ as a special case. Also we should take care of abstraction from any deeper semantic loading with denotators. The reader should postpone semantic issues to chapter 18, the present discussion is uniquely concerned with the universal denotation formalism. Notwithstanding their formal power, denotators are radically pre-semantic, at least regarding to ‘real’ meaning of physical, psychic, social or religious character.
6.1
Universal Concept Formats
Summary. Discussion of three principles of universal data format typology: unity, completeness, and discursivity. These are realized by a typology which is recursive, of extensive ramification, and open to unrestricted modular data recombination. The denotator concept is modeled following the Aristotelian pairing of substance and form or of the geometric paradigm of point and space, respectively. Each of these components: substance and form are arranged according to the structure of signs. We discuss the fundamental pointer nature of points as basic concepts of axiomatized semiotics. –Σ– From the discussion of encyclopedic characteristics: unity, completeness, and discoursivity, we deduce three denotator attributes which cope with these characteristics. encyclopedic characteristic unity completeness discoursivity
=⇒ =⇒ =⇒ =⇒
denotator attribute recursive typology extensive ramification modes order and recombination
6.1. UNIVERSAL CONCEPT FORMATS
49
The idea is that unity in creating concepts is achieved by means of recursive constructions. One builds new denotators from old ones by systematic reuse. Completeness must be guaranteed by extensive ramification modalities when stepping down along recursive paths. Each ramification mode ensures a particular way of stepping downwards in recursivity. Finally, discoursivity is met by means of free construction and recombination of denotators and by universal orderings among these objects. Unlike with usual data base management systems [551] there is no fixed ensemble of possible denotator constructs. Figure 6.1 illustrates the denotator ‘flow chart’.
type
ramification mode ramification knot
types given by recursion
type
type
type
Figure 6.1: Denotators are designed to cope with universal encyclopedic characteristics, such as unity, completeness, and discoursivity. This flow chart visualizes the corresponding denotator attributes. A type refers to other types in a recursive way through different ramification modes as indicated in the ramification knot. Construction of new and recombination of given denotators is non-restrictive. The system of denotators can be ordered in a universal way.
50
6.1.1
CHAPTER 6. DENOTATORS
First Naive Approach To Denotators
Summary. A first approach to denotators is given without requiring further mathematical background. –Σ– The universality of our claim implies that we cannot just set up a generic space and take the objects as being “points” in such a space. Rather is it necessary that each point carries its space with it like a snail carries its shell. Referring to Aristotle’s description of real things as a combination of substance and form, we view a denotator as being a substance-point within its form-space, see Figure 6.2.
+
substance-point
=
form-space
real thing
Figure 6.2: Like a real thing, a denotator is a substance-point, together with a proper formspace. The naive definition1 of a denotator is split into a recursive definition of the form-space, and then, based upon this concept, the substance-points are defined. The recursive structure of a form is a triple, consisting of Form-Name, Type, and Coordinator, denoted in this way: F orm-N ame → T ype(Coordinator).
(6.1)
The Form-Name is any character string, such as ′Note′2 . By definition, the type is either compound or simple. Compound types are ramification types are recursive start Q modes, simple ` points. There are four3 compound types: Product , coproduct , powerset {}, and synonymy Syn. For products and coproducts, the coordinator is a finite sequence F1 ,..., Fn of forms. For powerset and synonymy4 , the coordinator is one form F . By definition, four5 simple types are possible: STRING, BOOLE, INTEGER, FLOAT. Their coordinators look as follows: 1 In the mathematically rigorous framework of form semiotics as described in appendix G.5.3, the naive approach is related to the topos Sets of sets. 2 The special ′...′quotation marks reduce the word to a character string, i.e. the sequence of characters without any further meaning. 3 These are just the ones which can be easily understood within this naive introduction. Later, we shall generalize the types, but no fundamentally different ideas will be added. 4 In the rigorous framework, synonymy is superfluous and may be mimicked by a product with one single factor. 5 These simple domains will be vastly extended, the present selection is made with regard to practical and programming-oriented usage.
6.1. UNIVERSAL CONCEPT FORMATS
51
• STRING: The set CHR of character strings6 . • BOOLE: The set BIT = {0, 1}. • INTEGER: The set Z = {0, ±1, ±2, ±3, . . .} of integers. • FLOAT: The set R of (floating point) decimal numbers7 , such as 234.0157, −25.19022879, etc. Since coordinators are uniquely determined for simple forms, we shall omit the coordinator bracket in these cases, i.e. INTEGER(Z) simplifies to INTEGER, etc. The general naming policy for forms is that two different forms should bear different names. This enables us to identify a form with its name (the name is a “key” in the terminology of database theory [551]). For example, if we have the product form ′Piano-Note′ →
Y (Onset, Pitch, Loudness, Duration),
(6.2)
with coordinators Onset, Pitch, Loudness, and Duration, then we want to give it the shorthand name Piano-Note. In this sense, we often adopt the bracketless denotation Y Piano-Note → (Onset, Pitch, Loudness, Duration) (6.3) if no confusion is likely8 . With this naming convention in mind we introduce the following simple forms as a first set of ad hoc examples for this ‘naive’ introduction: • Onset→FLOAT, • Pitch→INTEGER, • Loudness→STRING, • Duration→FLOAT. A rigorous expression for such a form would read ′Onset′→ SIMPLE(R), for example, and the form itself would be called Onset. If we deal with complex compound forms and wish to represent them in a more graphical way, the symbolism can be shown in a vertical arrangement, as illustrated for the Piano-Note form in Figure 6.3. Observe that presently, products and coproducts or powersets and synonymies, respectively, are not further distinguished. The former couple has a sequence of forms, the latter one a single form as coordinator. However, the specific difference will emerge when we introduce the substance-points on the respective form-spaces. 6 We may take the standard set of ASCII characters, but from an informal point of view, just think of usual symbols, such as letters a, b,..., figures 0, 1, 2,..., and other characters, such as brackets, etc. 7 We do not consider infinite decimal digits, such as π = 3.1415926 . . ., in this naive setup. 8 In the general approach to denotator theory, names are also denotators, and the present distinction is obsolete, see appendix G.5.3.
52
CHAPTER 6. DENOTATORS
form name Piano-Note
coordinator forms
type
P
Onset
Pitch
Loudness
Duration
FLOAT
INTEGER
STRING
FLOAT
—
Ÿ
CHR
—
optional
Figure 6.3: The form Piano-Note in graphical representation, including the recursive explication of its coordinator forms Onset, Pitch, Loudness, and Duration. In our naive setting, a denotator is a triple, consisting of the denotator name DenotatorName, the form Form, and the coordinates Coordinates; it is denoted by Denotator-N ame
F orm(Coordinates).
(6.4)
Again, Denotator-Name is a character string, Form is a form, and Coordinates is an object which is determined according to the form’s type. If the type is simple then, by definition, Coordinates is an element of the coordinator of the form. For example, if Type = INTEGER then Coordinates is an integer number, i.e. an element of Z. For example, a denotator for pitch middle C on a keyboard could look like ′C′ Pitch(60), see Figure 6.4 for concrete examples of simple denotators. Again, if there is no risk of confusion, we shall identify a denotator by its denotator name, writing C or C Pitch(60) instead of ′C′ Pitch(60), for example. For denotators we cannot however stick to the naming policy of forms since in general, denotator names are of secondary importance. This is so because denotators usually occur in large numbers (such as notes in a score), and we are forced to distinguish them by their coordinates rather than by names. Very often it will even be reasonable to reduce the denotator name to the empty string ′′9 . Let Type be compound10 . For product, and if11 Coordinator = F1 , . . . Fn , we define Coordinates as being any sequence f1 , . . . fn of denotators with forms F1 , . . . Fn , respectively, i.e. f1 9 Be careful in distinguishing the empty string ′′ from the string ′ ′ consisting of one empty space, sometimes also denoted by t! 10 In this naive setup, coordinates are also denotators. This is however different from the formal setup, but there, we need conceptual circularity on the level of forms and denotator namings. We have preferred this naive approach to ease understanding in a first approach. 11 In this book, we make a logical, though not very common usage of the ellipsis symbol “. . . ”: it means that one has started with a sequence of symbol combinations which follows an evident law, such as 1, 2, . . . n, or a1 + a2 + . . . an . The evidence is built upon the starting unit, such as “1,” or “a1 +” in our examples, and then
6.1. UNIVERSAL CONCEPT FORMATS
53
BOOLE
X X F
"L"~>Loudness(mf)
"HiHat-Open"~>HiHat-State(1)
STRING
FLOAT
INTEGER
b b 4 X X Xj X X X & b4 j
b 4 & b b 4 Xj X
"E"~>Onset(0.625)
bb 4 & b4 X X X X X
bb 4 & b4 X
"H"~>Pitch(70)
Figure 6.4: Examples of simple denotators.
= f1 -Name F1 (f1 -Coordinates), ..., fn = fn -Name Fn (fn -Coordinates). For example, if we are given a denotator named ′myNote′12 , and with form Piano-Note, the coordinates are four denotators E, H, L, D with forms Onset, Pitch, Loudness, Duration and Coordinates e, h, l, d in the coordinators R, Z, CHR, R respectively: myNote
Piano-Note(E, H, L, D), E Onset(e), H Pitch(h), L D Duration(d).
(6.5) Loudness(l),
Figure 6.5 shows the graphic representation of the Piano-Note form (see figure 6.3) with the above denotator myNote inserted. If Type is coproduct, and if Coordinator = F1 , . . . Fn , we define Coordinates as being any one denotator fi of form Fi , for an index i = 1, ..., n, i.e. fi = fi -N ame
Fi (fi -Coordinates).
(6.6)
the following unit, such as “2,” or “a2 +”, and then inducing the following units to be denoted, such as “3,” or “a3 +”, “4,” or “a4 +”, etc., until the sequence is terminated by the last symbol, such as “n” or “an ” in our examples. The ellipsis means that the building law is repeated, and as such, it is a meta-sign referring to the inductive offset. Therefore the more common notation 1, 2, . . . , n, or a1 + a2 + . . . + an is not correct. In the limit, for n = 3, it would imply a notation such as 1, 2, , 3 or a1 + a2 + +a3 , which is nonsense. Moreover, in complicated indexing situation, the common notation would be overloaded. 12 If possible, we name denotators with small initials, but exceptions may occur.
54
CHAPTER 6. DENOTATORS Piano-Note
myNote
~>
P
coordinates
types
~>
Onset
E
~>
Pitch
H
~>
~>
Loudness
L
D
Duration
coordinates FLOAT
INTEGER
STRING
FLOAT
coordinators
~>
—
e
~>
Ÿ
h
l
~>
CHR d
~>
—
Figure 6.5: A product denotator named ′myNote′ with its coordinates in a recursively complete graphic representation. In the above example with F1 , F2 , F3 , F4 = Onset, Pitch, Loudness, Duration, let a P ianoSelector (Onset, P itch, Loudness, Duration) be the corresponding coproduct form. Then, with this form, a denotator named ′mySelection′, given by ′mySelection′ PianoSelector(f ), is defined by the selection of either f Onset(φ) or f Pitch(φ) or f Loudness(φ) or f Duration(φ), φ being an element of R, Z, CHR, R, respectively. If Type is powerset, and if Coordinator = F , we define Coordinates as being a finite set S of denotators, all having one and the same form F . For example, if we consider the form Chord→{}(Pitch), then a denotator myChord Chord(S) is defined by a set S of pitches, i.e. a chord in the classical terminology. For a finite set S = {f1 , ..., fk }, we usually omit the curled set brackets in this notation and simply write {}(f1 , ..., fk ) instead of {}({f1 , ..., fk }). For example, a second degree chord13 IIC in C major consisting of pitch denotators D Pitch(62), F Pitch(65), and A Pitch(69) would be denoted by IIC {}(D,F,A). If Type is synonymy, and if Coordinator = F , we define Coordinates as being a denotator f with form F . Hence, if the denotator’s form is New→Syn(F ), then a denotator with denotator name ′myNewName′ is myNewName New(f ). Essentially, this means renaming f 13 We
shall, of course, give rigorous definitions of all musicological terms later in this book.
6.1. UNIVERSAL CONCEPT FORMATS
55
by ′myNewName′ and doing this within the form which is essentially F , but with a new name ′New′. Besides a renaming of the space and the point, nothing really happens—unfortunately, this is a very common situation in the humanities. Notice, however, that synonymy is not symmetric, i.e. the new denotator is recursively later than the old one. This is essential for understanding the construction history of concepts. If a denotator of form AForm and coordinates myCoordinates is given the empty name ′′, we shall denote it by AF orm(myCoordinates) (6.7) instead of ′′ AF orm(myCoordinates). But we shall not introduce analogous notation for empty-named forms since form names are essential as an identification instance.
6.1.2
Interpretations and Comments
Summary. Interpretations of the above concepts and comments on non-trivial implications are given. –Σ– Pointer Character. To begin with, let us stress the pointer character of the denotator concept. The underlying form concept is self-referential: In general, a form is defined by another (bunch of) form(s), only the reference type is explicit. Recursion is supposed to stop at a simple form, and there, we are left with a selection of substance from a common repertoire, viz mathematics and computer programming data types (numbers, digits, strings). Similarly with denotators. We contend that this scheme of recursive pointers is fundamental in human concept construction. Understanding such a concept means navigating on its recursive ramification tree. This is the way we usually execute our understanding of a concept, e.g. “house”: We say that a house consists of a roof, a set of windows and doors, walls, etc. We point at components of the concept. If necessary, we penetrate such components, and prosecute their ramification trees, etc. But we do not point automatically at any possible knot of the entire concept tree! If the concept “house” appears in a discourse, we only delve into a minimal level of explicitness. In our setup, this means that we just take the name, or, if empty, the form of a denotator if this specification suffices. Only in case some request is made for more details do we unveil deeper concept levels and follow the path down to coordinators and coordinates. Such a pointer scheme realizes a very economic handling of concepts. You only “unpack” what is really needed, the rest remains either referred by name or even completely hidden. This approach is similar to the paradigm of object-oriented programming, more precisely, to encapsulation of object data. They have an identity14 , but the contents are not unveiled until really needed. Circularity. Denotators have the outstanding property that they are open to circular definitions, and this is a fundamental extension of concept building rules with respect to the 14 Usually
a pointer to some memory address.
56
CHAPTER 6. DENOTATORS apparently founded recursivity of denotator structures. Let us give a common example to make the point explicit. We want to define the concept of a book with chapters, subchapters, subsubchapters etc. Evidently, a book can have an arbitrary number of chapters, and the depth of sectioning and subsectioning a book is not limited. But we can say that a book always consists of title, text, and a set of chapters. In turn, each chapter again consists of the same data type, except that we have to enumerate the chapters. We thereof deduce a natural ordering of appearance of the particular chapters. Let us therefore complete the above list by a number: 1. number 2. title 3. text 4. a set of chapters The problem with this definition is that we have to declare the “set of chapters”. In the context of denotators, this problem is solved in the following way: The basic form is termed ′book′, and it has this structure: Y ′book′ → (N o., title, text, chapters), (6.8) ′N o.′ → IN T EGER, ′title′ → ST RIN G, ′text′ → ST RIN G, ′chapters′ → {}(book). We have made self-reference to the form book within the powerset factor chapters of the four-fold product. However, this circularity of the form book is not necessarily inherited by substance-points within the form-space. Since a book-formed denotator myBook
book(..., myChapters)
has any set of chapters, we may give it the empty set ∅ of coordinates: myChapters
chapters().
In this case, no further specification is needed, and the denotator is completely determined. This may happen anywhere in the recursive regression of such a denotator and therefore, any finite book tree can be captured in the sense we expect from practice. Nonetheless, circularity of forms and denotators may occur in a much less inoffensive way which takes the very basis of concept construction into task. We come back to this issue in section 6.7. Meeting Encyclopedism. Let us recapitulate the encyclopedic characteristics and their realization on the denotator level. Recursive typology meets unity since we are given a unified construction principle of concepts; this is the yoga of pointers. It is, of course, not
6.1. UNIVERSAL CONCEPT FORMATS
57
possible to demonstrate in a formal sense that our ramification modalities are complete. But in view of the rigorous formal setup to be exposed in section 6.2, we can say that the most general mathematical construction principles of products, coproducts15 and powersets, together with the epistemological factotum of synonymy, gather everything which is known to be relevant for formation of mathematical and database structures16 . And discoursivity is guaranteed by two measures: First, complete freedom of form and denotator construction guarantees recombination. Even operations of semantic completion are feasible in the following sense. Suppose, for example, that an ethnological pitch-related form has not yet been made fully explicit and is merely ‘sketched’ in a simple, STRING-typed form U nknownP itchT ype → STRING within its recursion tree. The denotators of form UnknownPitchType have only a string of characters, no further contents are available. It may happen in a more in-depth research development that the pitch structure can be specified beyond strings. Then the form UnknownPitchType may be replaced by Y KnownP itchT ype → (U nknownP itchT ype, Pitch), (6.9) or even by form KnownP itchT ype → Syn(Pitch),
(6.10)
where Pitch is the integer-valued form discussed in formula (6.2). The first form relates every denotator pitch string to its numerical value, whereas the second form is a restatement of pitch data via a synonym of the already given Pitch form. Whatever procedure is selected, the old form UnknownPitchType can be replaced by the new KnownPitchType form in all forms which make use of it—just as in genetic engineering. And UnknownPitchTypeformed denotators can keep their names, we only need to replace them by synonymous denotators which point at numerical pitch values. For example, it may happen that a pitch is just named ′F]′, but no special value in terms of frequency is specified. This is a common situation in music theory. Then, in a more explicit situation—of instrumentation, say—this ‘symbolic’ data should be realized in a specific tuning. This is precisely the need to replace or enrich the string ′F]′ by more involved information, e.g. KnownPitchType(′F]′, Pitch(66)) as related to keyboard middle C Pitch(60). The second measure to meet discoursivity is a universally defined ordering principle on denotators. This provides us with an ‘omnipresent’ orientation for the discourse on denotators. Observe that this ordering system does not restrict to lexical alphabetic ordering but extends to more natural geometric constructions. We shall introduce (still naively, see section 6.8 for the rigorous version) this topic in the following section.
15 Products
will then generalize to so-called limits, coproducts generalize to colimits. is astonishing that database theorists have not yet learned to make systematic use of mathematical category theory, although the latter is being used in theoretical computer science. 16 It
58
CHAPTER 6. DENOTATORS
6.1.3
Ordering Denotators and ‘Concept Leafing’
Summary. We give a naive approach to the universally defined linear orderings among denotators and illustrate the results for musically meaningful denotators. –Σ– The question of what d’Alembert called the “encyclopedic ordering” is fundamental to any encyclopedic enterprise. We contend that the ‘EncycloSpace of denotators’ should be provided with a linear ordering < (see section C.2 for its definition). Here are three points why such a requirement is indeed fundamental: • One should be able to have a universal orientation, something that could be termed ‘leafing’ on the denotator EncycloSpace. • Orderings on denotators should be a germ for their representation as ‘points in coordinator spaces’. • A denotator search algorithm has to be defined in a universal way, and preceding visualization, since search engines must a priori be able to invoke this algorithm. It was an interesting conflict within Diderot’s Encyclop´edie that the alphabetic ordering was so low level with respect to the order of ideas. This is above all due to the informal presentation of ideas—no database concepts were available in 1750. We introduce the linear ordering on denotators and their forms by recursion, i.e. by inheritance from already defined orderings on the coordinates and their coordinators. To simplify the discussion, we shall stick to non-circular forms. To begin with, if we look at simple coordinators, we are provided with well-known linear orderings: The coordinator CHR of STRING has the lexicographic ordering realized in every dictionary (see example 64 in appendix C.2 for this concept). The coordinator BIT of BOOLE is ordered by 0 < 1, integers and decimal numbers are given their standard orderings. To define a linear ordering among denotators D with D-N ame
D-F orm(D-Coordinates),
(6.11)
we proceed as follows: Suppose that we have defined linear order relation G
transition to dominant tonality F, modulation
(44-46) ? (47-123) G-major
(47-62) second subject (47-123) part with second subject F-major
(63-99) continuation
(63-74)
EXPOSITION
effective tonality circumstances
(75-99)
(100-123) closing group (124-137) introduction: modulation to Eflat-major
(138-188)
(138-212) kernel: construction of the model, its repetition, process of liquidation, here a kind of canon
Eflat-major
(189-197) ? antiworld
world second catastrophe
(198-226) D-major, b-minor? (227-238) Bflat-major
(249-262) ?
world
(263-266) Gflat (267-271) G (272-276) G -> Bflat (277-361)
(213-226) resting on dominant (F) and preparation of recapitulation repetition of exposition, but all in main tonality Bflat-major wit: no modulation back to main tonality since part with second subject is in Bflat-major
(227-268) part with first subject
(269-278) transition (279-361) part with second subject
RECAPITULATION
(239-248) Gflat
DEVELOPMENT
first catastrophe
(124-137), G -> Eflat
Bflat-major (362-405)
CODA
Bflat-major
(362-405) cut-off, first subject and intensification
Figure 28.8: The sonata scheme of the first movement of the “Hammerklavier” sonata op.106, compared to the normed sonata scheme. Our analysis is not thought of as a contradiction but as a confirmation and elaboration of existing analyses. In particular, we take over the fundamental world-antiworld thesis of Erwin Ratz [434] and the “catastrophe theory” of J¨ urgen Uhde [534] which relates to the specific modulatory role of the diminished seventh chords in the sonata. With regard to the formal ex-
28.2. MODULATION IN BEETHOVEN’S SONATA OP.106, 1ST MOVEMENT
605
plicitness of our model, the take-over will however include precision, completion, and unification of aforesaid world-antiworld thesis and catastrophe theory.
28.2.2
The Fundamental Theses of Erwin Ratz and Jrgen Uhde
Summary. This section presents the thesis of Ratz and its weltanschauung, as well as the thesis of Uhde. A restatement of these theses in group-theoretic terms is given. As its consequence, a particular modulation architecture is predicted. This will be verified in section 28.2.3. –Σ– In [434], Ratz expressed the idea that the “empire of tonalities” in op.106 is polarized into a “world” around the pole B[ -major, the sonata’s main tonality, and an “antiworld” around the counterpole b minor. In [534], Uhde supplemented this thesis in so far as two so-called “catastrophes” occur in the Allegro movement when the “world” is left in order to enter the “antiworld”. Catastrophes are dramatic modulatory processes which differ significantly from normal modulations (see figure 28.9). We discuss the following thesis: Thesis 4 Let M0 = {c] , e, g, b[ } ⊂ P iM od12 the local composition of the diminished seventh chord. Then the modulation structure of op.106 is determined by the modulators in the sense of theorem 30 which are contained in Aut(M0 ). What is the relation of this thesis with Ratz’ world/antiworld? To begin with, the group Aut(M0 ) solves the following purely group-theoretic problem: We look for a maximal subgroup M of the group of inversions and translations T I12 on P iM od12 which acts as modulator group (3) on Dia(3) , but under the restriction that no group element transforms B[ to D(3) . Here, the interpretation D(3) stands for the minor tonality b of ionian mode. Under the action of such a subgroup, the set Dia(3) is divided in at least two orbits (one for B[ , and one for b). A priori, it is not clear whether there are several such subgroups, and how many orbits such maximal subgroups will produce. But it turns out that the subgroup is uniquely determined, viz., M = Sym(M0 ) ∩ T I12 , these are the inversions and translations which leave M0 invariant. We have the surjection Sym(M0 ) → Aut(M0 ), and it is easily seen that its restriction M → Aut(M0 ) is an isomorphism. So we may identify these groups. Moreover, there are exactly two orbits under this automorphism group, namely the world W = Aut(M0 ).B[ = {B.D[ , E, G, A, C, E[ , G[ }, and the antiworld W ∗ = Aut(M0 ).D = {D, F, A[ , B}. We are now in state to make more precise Ratz’ hypothesis: The tonalities of W are those of Ratz’ “world”, whereas the complementary set W ∗ defines Ratz’ “antiworld”. This casts our possibilities in two ways, we are given a precise dichotomy W/W ∗ and the admitted modulators, i.e., the elements of the automorphism group Aut(M0 ).
606
CHAPTER 28. APPLICATIONS
A
B
Figure 28.9: Ludwig van Beethoven: Two modulations in the first movement of op.106 [46] (with kind permission of the C.F. Peters Publishers, Frankfurt, Frankfurt/Main). The first (A), from G to E[ , is a common one, whereas the second (B), from E[ to b-minor, is a catastrophe modulation in the sense of Uhde.
The modulatory situation here is that of a restriction of modulators to the group Aut(M0 ). What is the meaning of this condition for the modulation model? As long as the modulator which is described by the modulation theorem is contained in Aut(M0 ), there is no problem to apply the theorem and to look for pivots. But a modulation to the third circle of fourths, like C E[ or A C only admits translations in our restricted framework. In this case we must refrain from rigidity condition (3) (the triviality of the intersection T I12 ∩ Sym(T ∩ Q)) in property 1 of section 27.1.4. In fact, this condition which guaranteed the uniqueness of the modulator is superfluous now: there is only this translation symmetry, no other candidate! But then, there is a corresponding theorem where the modulators are the translations e±3 . Therefore, one should modulate in the sense of restricted modulators and pivots within W and within W ∗ , whereas modulation between two tonalities living in different worlds should yield a catastrophe in the sense of Uhde, whaich means in particular that in such a situation,
28.2. MODULATION IN BEETHOVEN’S SONATA OP.106, 1ST MOVEMENT
607
diminished seventh chords—which are the ‘creators’ of the catastrophe—should become visible at the surface of the score.
28.2.3
Overview of the Modulation Structure
Summary. We present the modulatory architecture and the modulations which split into “ordinary” cases and “catastrophes” in the sense of Uhde. –Σ– We now have to test these postulates in the total plan of modulations of the Allegro movement. The modulation plan looks like this: W
W
W
W∗
W∗
W
W
W
B[
G
E[
D/b
B
B[
G[
G
e−3
Ug
!
Ud/d]
!
Ub [
Ua[ /a
W e3
B[ .
In order to view these modulations on the circle of fourths and relative to the world/antiworld dichotomy, see figure 28.10. For each modulation, the modulators from Aut(M0 ) are indicated. Here are the single modulations in detail:
G(3)
C(3)
F(3)
b(3) ~ D(3) B (3)
A(3) E (3)
E(3) A (3) B(3) G (3)
D (3)
Figure 28.10: The graph shows the modulation plan in the Allegro movement of Beethoven’s op.106 in the tonality system arranged on the circle of fourths. The start switches from B[ to G. The inverse modulation occurs at the end, and both follow the same procedure, see sections 28.2.4 and 28.2.11. Except these initial and terminal movements, the modulation plan (3) (3) is perfectly symmetric around the symmetry axis between B[ /E[ and A(3) /E (3) .
608
28.2.4
CHAPTER 28. APPLICATIONS
Modulation B[
G via e−3 in W
The first modulation B[ G in the transition (bars 39-46) to the second subject could in principle be performed by use of a “pedal modulation” [478]. We do however not encounter this modulation, but ‘merely’ a sequence of V IIG -degrees whose top notes are shifted by minor thirds from each other, i.e., exactly the situation of the pivot V II and the third translation, as predicted by the modulation under restricted modulators.
28.2.5
Modulation G
E[ via Ug in W
This modulation is bipartite (first part: bars 124-127, second part: bars 128-129). Before we encounter the pivots V II − V − V II of E[ according to modulation table appendix N.1 in part two, we hear tone g as an octave interval: pedal and stationary voice. Here the inversion at g is made evident (figure 28.11). On the one hand, this section is a cadence of cmel whose inner symmetry is Ug . One recognizes a bipartite contrapuntal motive structure of which the second part (bars 126-127) is the inversion of the first part (bars 124-125) in pitch c] . But modulo octave (in the pitch class space), this inversion is the same as Ug . Since c] is not contained in the present scales, it is delegated to the octave in g. This means that the modulator Ug is motivically evidenced in this first modulatory section preceding the pivots and the cadence.
Figure 28.11: If we omit tone f in bars 124-127 (it serves for the identification of cmel ) and transpose all pitches into one octave between the two gs (to the right), then we recognize the motivic inversion symmetry between bars 124-125 and bars 126-127.
28.2.6
Modulation E[
D/b from W to W ∗
This modulation is a catastrophe in the sense of Uhde since it leads to the antiworld W ∗ . As we may recognize already from the score (B) in figure 28.9, bars 189-197 are of a dramatic shape. Any elaborate motivic, rhythmic or harmonic effort is postponed in favor of a pertinent presentation of diminished seventh chords. An approach to modulation V IE[ , ID (bars 189192) fails, the resolution of all alteration signs indicates the exit from tonal space. We hear the “generator” of the catastrophe, the diminished seventh chord as such.
28.2. MODULATION IN BEETHOVEN’S SONATA OP.106, 1ST MOVEMENT
28.2.7
609
B via Ud/d] = Ug] /a within W ∗
Modulation D/b
The process is resolved chordically here and impregnated by two simultaneous local meters (figure 28.12): in the left hand a 6/4 meter via triplets, in the right hand 8/8 meter. In both expressivo indications (bars 205, 209) appears (bar 207, and repeated in bar 211) in the triplets in the left hand the motif f] − b − f] and b − f] − b = Ud/d] (f] − b − f] ). And the right hand plays d and d] = Ud/d] (d) in bars 205-209 as well as e and c] = Ud/d] (e). In bar 209 appears at the beginning degree IB and after that in the right hand a chromatic sequence which comprises the seven pitches B ∗ = {c] , d, d] , e, f, f] , g}. Now, B = Ud/d] (D) = e5 .11(D). But by a concatenation with the fifth circle symmetry we produce e5 .5 = Ud/d] .e0 .7 out of Ud/d] . This result lies in Sym(M0 ) (!), and we exactly get B ∗ = e5 .5(B), so that we really are situated in B—up to an inner symmetry of M0 . 1/8
5/8
9/8
13/8
1/2
2/2
3/2
4/2
1/6
4/6
7/6
10/6
1/2
2/2
3/2
4/2
y = 2 , z1 = 8
y = 2, z2 = 6
Figure 28.12: Example of two simultaneous local meters, corresponding to the left and right hand in bars 209-210 of the first movement of Beethoven’s op.106.
28.2.8
Modulation B
B[ from W ∗ to W
The next modulation leads us back from the antiworld W ∗ to the world W (bars 214-226). Corresponding to the short stay in W ∗ the return is an easy business. There follows a sequence of arpeggiated intervals which is strongly based upon M0 and ends on IB[ : six four four four
twice times times times times
e, c] g, e g] , e b[ , e b[ , f
within M0 within M0 within M0 IB[
610
28.2.9
CHAPTER 28. APPLICATIONS
Modulation B[
G[ via Ub[ within W
For this modulation within the world W , we can apply the normal modulator. The modulation is a fast one (bars 238-239), although the involved tonalities B[ and G[ are separated by four fourths. At the end of bar 238, the neutral degree IB[ is followed by its inversion V IG[ = Ub[ (IB[ ), so the modulator is again put into evidence. Immediately after that, at the beginning of bar 239, follows {b[ , f, a[ } which lies in III ∪ V of G[ and corresponds to the pivots.
28.2.10
Modulation G[
G via Ua[ /a within W
This second Uhde catastrophe (bars 249-262) is highly dramatic. At the beginning (bars 249250), we remain within the large orbit W : I of G[ . As with the first catastrophe, we then encounter a pronounced series of diminished seventh chords. In bars 259-262, it terminates in the intersection B ∩G[ : we do not know whether the change or orbits has been successful or not. The decision is only taken in bar 263 with pitch f which is not in B, but in G[ . So we actually did not really leave the world, at least not unambiguously. This ‘delusion’ is particularly refined since we hear the “fanfare” with IIIG which could also be viewed as I of b-minor. But this third degree corresponds to a pivot of our model, the inversion of the bass in the moment of the forte onset at the end of bar 266: b = Ua[ /a (f] ) shows the same symmetry as for the above modulation D/b B within the antiworld after the first catastrophe. So we do not move from B to D as it is suggested by the change of signature in bar 267 and from the pitch material before the appearance of f in bar 264. This superficial impression is an “allusion” to the antiworld situation, unveiled however as being an illusion and is resolved in a consistent way. We have moved from Gf lat to G.
28.2.11
Modulation G
B[ via e3 within W
This last modulation proceeds completely regularly according to the scheme that we already know from the inverse modulation in the transition to the second subject as discussed above.
28.3
Rhythmical Modulation in “Synthesis”
Summary. The modulation model does not restrict to time from its logical structure. This fact is exploited in a composition of rhythmical modulation for percussion ensemble and piano. –Σ– The harmonic modulation model is based upon the pitch class construction modulo octave, i.e., the space P iM od12 based on the module Z12 derived from the semi-tone pitch space based on the integers, modulo the submodule generated by the octave quantity 12. We have viewed the tonalities as being scales, together with their triadic interpretations by the seven wellknown degrees. But there is nothing substantial to the choice of these spaces. The mathematical framework is completely insensitive to the forms and denotators which implement the underlying parameter and class spaces.
28.3. RHYTHMICAL MODULATION IN “SYNTHESIS”
611
So why not apply the modulation theory to a concept framework which is mathematically the same as for harmony, but which is semantically different, more precisely: the new approach presents a space and its denotators in the context of onset time and its rhythms. We shall develop this switch from pitch to time in the next section and apply this theory to a rhythmical modulation in movement No. 1 of Guerino Mazzola’s jazz concert “Synthesis” [339] for piano, large percussion ensemble, and e-bass. Before delving into technicalities, we should however observe a fundamental difference between two time qualities involved in musical composition. We have known time as a form related to onset and duration. These dimensions are part of what defines events in scores. They are comparable to other event parameters such as pitch, loudness etc. A local or global composition is a configuration of events which represent the composition’s substance. This material data is without any logical or strategic specification. It does not include the composer’s poietic construction plans or the analyst’s structural evaluation. But modulation is a structure that is not only defined on the level of the effective neutral, pivotal, and cadence degrees, it is rather built on the tripartite strategy Neutralization—Turning Point—Cadence as defined by Sch¨ onberg. This is a logical construction on the syntactical level: a sequence of three functional units in the syntagmatic string of musical development. Like abstract logical schemes, such as modus barbara (“a implies b” and “b implies c” implies “a imples c”), this is not a priori a syntagmatic string in the material musical time. It regards logical time rather than material time. In harmonic modulation, we have three logical stages in the pitch domain, and their unfolding on the material time line is only a representation of an abstract process, not the process as such. Substantially, in harmonic modulation we have an excellent example for Augustinus’ definition of music as an art of instantiation of good rational strategies. As already mentioned in the previous discussion of Augustinus’ definition, modulation could also regard rhythmical strategies, for example. It is very important to distinguish logical time from material time in this case since here, the syntagmatic string which embodies the logical time is superposed to the material time of rhythmical structures.
28.3.1
Rhythmic Modes
Summary. This section describes the transfer of harmonic modality to rhythmic categories, and from this derives the modulations in the rhythm domain. –Σ– The transfer of harmonic dimensions to rhythmic ones must deal with the semantic specifications of these dimensions. In fact, pitch is a space category which carries a strong connotation of sound quality, i.e., of instrumental realization. No abstract pitch quality has ever been used in harmony as soon as realistic music parameters are to be described. We could of course say that the abstraction from instrumental realities is only a question of habituation, and that abstract time could also be thought of in an abstract comprehension which already worked for harmony. However, it is easy to imagine a more or less good orientation in the pitch space whereas it is difficult to imagine an orientation in an abstract time space, since the distinction between later and earlier time onsets is only relative and risks failure if no supplementary time-dependent
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structures can be found. Moreover, the superposition of logical and material time layers asks for a clear marking of material time events. This is why we would like to add supplementary, logically redundant markers of time such that one can use such redundancies to shape time in a rhythmical modulation theory. The idea is as follows: Suppose the onset time is parametrized by integers, OnsetZ −→ Simple(Z). Id
Suppose further (in analogy with the octave period) that we have the translation e12 in time on OnsetZ in the sense of a time period. This could be realized by a 12/8 meter, for example, where the eighth duration stands for the unit time in OnsetZ . As a marker parameter, we select the form P ercussion −→ Simple(Z) whose zero-addressed denotators are percussion instruments Id
as represented in a list with integer indexes. This could for example be a list of percussive sounds in an MIDI-environment where the numbers stand for the program changes (see below for more concrete setups). Our events are situated in the combined space ∼
P erOns −→ Simple(Z ⊕ Z) → P O −→ Limit(OnsetZ , P ercussion) Id
Id
and will be called percussion events in this context. But we want more, i.e., macro percussion events, in order to construct profiled markers. So we consider these macro percussion event spaces as they were introduced in section 13.4.3.1: KnotP erOns −→ Limit(P erOns, M akroP erOns ) Id
with M akroP erOns
∼
−→
Power(KnotP erOns )
f :F →2F K ΩF K
and F = F un(M akroP erOns ), F K = F un(KnotP erOns ). A macro rhythmic germ R∗ is a finite local composition in the ambient space KnotP erOns . And an infinite macro local Para-rhythm is paraphrased by the 12-periodic local composition e[−∞,∞]12 R∗ . In the sequel, we shall only consider germs R∗ which are in bijection to their time ∗ ∗ projections ROns , and such that ROns are contained in the period interval [0, 12[. This means that we may equivalently consider the rhythm classes modulo the time period 12. These objects are the residual classes of the macro rhythmic germs R∗ which we now identify with the germs because of our additional assumptions. So we are working in the macro class space: KnotP erOns12 −→ Limit(P erOns12 , M akroP erOns12 ) Id
with M akroP erOns12
∼
−→
f :F →2F K ΩF K
Power(KnotP erOns12 )
and F = F un(M akroP erOns12 ), F K = F un(KnotP erOns12 ), deduced from P erOns12 −→ Simple(Z12 ⊕ Z) Id
in an analogue construction to scales in pitch class spaces. Intuitively, these macro rhythmic germs in M akroP erOns12 are just time scales whose points are loaded with a rhythmic satellite object each. As to modulation theory with such objects, we can apply the modulation model for 12tempered tuning, but we have to take care of the satellites! There may be many rhythmic scales
28.3. RHYTHMICAL MODULATION IN “SYNTHESIS”
613
with the same time projection! This means that the translation and time-inversion symmetries will have to carry over the (unaltered) satellites. This is as if the harmonic modulation would be carried out with “colored” pitch classes, whereas the inversion of pitch classes would preserve colors.
28.3.2
Composition for Percussion Ensemble
Summary. This final section discusses the concrete realization of the rhythmical modulation model, essentially by use of a large ensemble of percussive instruments. –Σ– The composition in question here is a rhythmical modulation in movement No. 1 of Mazzola’s jazz concert “Synthesis” [339]. The modulation takes place after the exposition in the sonata scheme of this movement (3:18-5:48 on piece # 1 of [339]). This developmental start is written in 12/8 measure, in fact the measure which we need for the rhythmical modulation in a 12-periodic scheme as exposed above. The modulation is built upon the rhythmic scale (the time projection of the macro germ) corresponding to the complement c 62 of No. 62 the chord classification list in appendix L.1. The macro germ G∗ above c 62 has the following shape. For each onset x ∈ c 62, we have a corresponding denotator ((x, px ), Satx ) of form KnotP erOns12 . The px -coordinate is an integer value for a percussion sound, whereas the satellite Satx is a zero-addressed denotator of form M akroP erOns12 . We choose for each satellite a zero-addressed three-element motif of form P erOns12 , Satx = {(0, 0), (do1x , dp1x ), (do2x , dp2x )}, with first element the origin (0, 0) ∈ Z12 ⊕ Z (see also figure 28.13). The choice of these threeelement satellites is due to the general construction principle of the “Synthesis” concert, i.e., the use of the 26 isomorphism classes of three-element motives for all melodic and rhythmic structures. This principle was already encountered in section 11.6.3 where we discussed the third movement of the concert. The modulation goes from the rhythmic macro germ G∗ to its symmetric image H ∗ = ∗ R(G ) under the retrograde motion R which fixes the seventh tone (first tone of the seventh degree). The lower half of figure 28.13 shows the retrograde germ with the transported satellites. This is the germ for the “rhythmic target scale”. According to the modulation theory (welltempered case), there is a determined set of pivots related to a selected cadence in the target scale, and we can get off with the explicit construction. In the “Synthesis” concert, we first set up six bars in order to define the start rhythm scale. In each bar 1-5, a new tone of the start rhythm scale is added, and two tones are added in bar 6, so we have the complete scale. Observe that this addition of tones means that each added tone can be repeated in the successive bars and thereby enriches the previous rhythms. From bar 7 to bar 12, the tones of G∗ are successively removed so that we have a neutralization process here. The modulator is made evident by a rhythmic motif which is built around the above retrograde symmetry R, including its repetition during the whole second modulation segment presenting the pivots, from bar 13 to bar 23.
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CHAPTER 28. APPLICATIONS
Figure 28.13: Above: The rhythmic macro germ G∗ of the rhythmical modulation in the first movement of the “Synthesis” concert [339]. Below: The retrograde of G∗ with the satellites being carried over to the transformed onset-percussion events. Already in bar 21, the new tonic of H ∗ , which is (by definition) the image of the old tonic—with its characteristic rhythmic motif satellite—, is played and remains alive until the end of the modulatory process. This means that the new first onset is no more the old one, but occurs on onset 9/8 (modulo 12/8) of the old bar onset. This change is realized by a new bar onset at onset 9/8 of bar 24. From this point on, we have 5 bars to confirm the target rhythm and to terminate the modulation. Both cadences, that of the start and that of the target rhythm, are accompanied by a kind of regular falling drop sequence of two eights duration tones in order to stress the beginning of the respective bar units. Recall from section 13.4.3.1 that the macro rhythms can be flattened down to the sets in the effective form P erOns12 of rhythmic events. But the construction of the effective material had to be performed on a higher conceptual level of macro events. Probably this kind of conceptual upgrading of modulatory processes could also be used to understand existing and construct new harmonic modulations, or even melodic modulations if a modulation model of motivic structures (in OnP iM odn,m , say) is available.
Part VII
Counterpoint
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Chapter 29
Melodic Variation by Arrows punctus est cuius pars nulla est Euclid [215, book I] Summary. The ideas of tangent objects described in section 7.5 are applied to define contrapuntal intervals as tangential “arrows” from the cantus firmus to the discantus tone. This formalism fits with the idea that the discantus is a kind of melodic variation around the cantus firmus line. The ring structure of the set of such arrows is discussed and motivated from the musical perspective. –Σ– The mathematical theory of counterpoint is an excellent subject to illustrate the idea of mathematical conceptualization of musical and musicological objects. It shows that mathematical subtleties are no formal overhead and can help to grasp music(ologic)al differences at the highest level. After the introduction of arrows as a formal restatement of the contrapuntal interval concept, we shall see in section 30.2.1 that this setup provides us with an astonishing relation between harmonic and contrapuntal objects, a relation which was never observed by musicological tradition. This insight could have deep consequences for the understanding of the hitherto unresolved transition process from polyphony to homophony, from counterpoint to harmony as a basis of musical composition.
29.1
Arrows and Alterations
Summary. Arrows are a conceptual refinement of ordered intervals in the pitch domain. We compare these and related concepts and work out their specific differences. –Σ– Most of the common modules of simple space forms describe aspects of elementary music objects which can be related to an abstraction from physical events. However, musical reflection 617
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CHAPTER 29. MELODIC VARIATION BY ARROWS
is a mental and symbolic approach, and this is not only the case on the higher level of compound object interaction as it was developed in the theory of global compositions. We already observe such a symbolism on the most elementary understanding of what we mean when talking about a tone. We are referring to the fact that in systematic musicology, some German specialists (see e.g., [205]) distinguish between “Ton” (tone) and “Tonort” (a tone’s place), when discussion alterations. For example, in the 12-tempered system, the Tonort of f] is the same as the Tonort of g[ , whereas these two tones are thought to be different since their origin, one f , one g, expresses a musical thinking that cannot be traced anymore on the level of the common Tonort. This understanding is based on the concept of musical objects as being grown, dynamic entities and not merely static states. In this approach, a tone is not a point but a variation of a point, something that has a direction from past to present, in short: an ‘arrow’. This unveils a fundamental stream of musical thinking: that of varied structures. In this part, we want to discuss a central instance of this type of thinking, namely counterpoint. The objective is to interpret contrapuntal intervals as “arrows” and to deduce the basic rules of counterpoint. We cannot present a general theory of varied structures yet. Instead, we shall give a prototypical sketch of how such a theory could look. In the following, we suppose that we are working in a form space Sq of simple type, ∼ module1 M and identifier automorphism q : M → M (giving rise to the functorial identifier automorphism @q). For example: SId = EulerM odule with M = Q3 , or S1 = P iM od12 with M = Z12 , or S(7) = F iP iM od12 = P iM od12,(7) , the fifth identifier pitch class space, etc. In contrapuntal reasoning, one is not primarily interested in single pitches, but in “arrows” starting from a basic pitch x: the cantus firmus and ending in a target pitch y, the discantus. In order to catch this information, one could come back to the arrow approach exposed in section 7.2.3, i.e., to the Z-addressed denotators D : Z Sq (ON = x, OF F = y). For the following reason we refrain from this point of view2 : The Z-addressed denotators have too little canonical algebraic structure for our purposes. We shall see this in a moment. The formalization we choose for counterpoint is the dual numbers approach described in section 7.5. The module M is enriched to the dual numbers module M [ε], and we also admit ∼ the presence of the identifier module automorphism q : M → M which induces the canonical automorphism q[ε] of M [ε]; call the corresponding space Sq [ε]. For example we would have EulerM oduleq [ε] −→ Simple(Q3 [ε]), more generally, with the notations of appendix G.5.3: @q
Sq [ε]
−→ Id(Sq )[ε]=@(q[ε])
Simple(coord(Sq )[ε])
for an automorphism q of the coordinator module coord(Sq ). In this setting, if T is an address, a T -addressed contrapuntal interval of space form Sq is a T -addressed denotator D : T Sq [ε](x+ ε.i), x + ε.i ∈ T @M = T @coord(Sq ). If q = IdM , we also omit the q-index, and without any further address specification, we tacitly suppose the zero address. There are four canonical surjections 3 pcf , pint , α+ , α− : Sq [ε] → Sq 1 Always
over a commutative ring R in this context. point of view is however the adequate for harmonic considerations as we have already learned in the self-addressed theory. 3 Pay attention to the fact that these surjections are operating on the form functors and not on the frames. 2 This
29.2. THE CONTRAPUNTAL INTERVAL CONCEPT
619
associated with the first (cantus firmus) and second (interval) projections M [ε] → M and with the synonymous alterators of sweeping and hanging orientation α+ , α− introduced and motivated musicologically in section 7.5. In section 29.5 we shall also consider more general addresses (in particular self-addressed contrapuntal intervals), but for the time being, we stick to the zero-addressed contrapuntal intervals. The enrichment of the dual number modules M [ε] des not lie in the R-linear structure, it is based on the R-algebra structure of R[ε] with ε2 = 0. In other words, the well-known possibility of address killing ˜ q = X B@Sq X@B @S described in section 11.2 could lead to an identification of Z-addressed arrows in Sq with zeroaddressed arrows in Z@Sq , but we need an additional algebraic structure on the latter form space.
29.2
The Contrapuntal Interval Concept
Summary. This section presents a formal definition of contrapuntal intervals or arrows, together with the two possible orientations of sweeping and hanging counterpoint. –Σ– The point of departure of this theory is the fact that the known physical and mathematical concepts of consonance and dissonance are not adequate to the musical paradigm of an antagony of interval categories since they only conceptualize gradual changes in sonance. This unsatisfactory situation was also recognized by Hugo Riemann [456] in his critique of Euler and Helmholtz. Our problem is to turn this antagonistic paradigm into a mathematical concept framework in order to deduce the announced counterpoint model. Our requirements would not be satisfied if we thought of an interval as being a set of two pitch events. This would in particular not do justice to the concept of voices. The cantus firmus could not be distinguished from the discantus, and the crossing of voice could not be conceived. It is adequate to contrapuntal reasoning to distinguish a basic cantus firmus tone from the dependent discantus tone, i.e., to consider arrows as defined above, and to indicate the orientation. In the sense of this musical requirement, we shall now describe an oriented contrapuntal interval formally as a pair (α± , D : 0 Sq [ε](x + ε.i)) of a sweeping or hanging orientation plus a contrapuntal interval D as defined above. When the rest is clear, we shall also reduce the description to the simple pair (α± , x + ε.i) or even x + ε.i, if the orientation is clear. Whereas here, x is the cantus firmus of the contrapuntal interval, the quantity x ± i is the discantus of the interval. Within a given orientation, the two-part style “note against note” is now interpreted as a sequence xs + ε.is , s = 1, 2, 3, . . . of arrows. If a change of orientation happens, the indexes have to be split into regions of constant orientation. The historical development of the counterpoint of many parts is now reflected in
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CHAPTER 29. MELODIC VARIATION BY ARROWS
the historically variable characteristic types of geometric variations of the Gregorian chant of the cantus firmus’ melodic line by arrows. Other tone attributes, such as duration or loudness, are neglected in this elementary motivic exposition. Except some minor remarks concerning just pitch spaces, we shall focus our discussion to the 12-tempered pitch class space P iM od12,q and its contrapuntal space P iM od12,q [ε] of pitch class arrows, q being an automorphism , e.g., one of the four circle automorphisms q = (1), (5), (7), (11), of Z12 . This is quite congruent with the contrapuntal reasoning. For example, the octave extension of a perfect consonance is again perfect, idem for imperfect consonances and for dissonances [468]. So after this reduction we are left with 144 contrapuntal intervals for each orientation, the arrows in P iM od12,q [ε].
29.3
The Algebra of Intervals
Summary. The set of arrows is canonically provided with the structure of an algebra. The formal definition and first mathematical properties are exposed. –Σ– The module Z12 [ε] of P iM od12,q [ε] is not only a Z12 -module but also a Z12 -algebra, i.e., a module with a bilinear, associative and unitary multiplication defined by the nilpotent element ε with ε2 = 0 (see also example 76 in appendix D.1.1). The product of contrapuntal intervals a + ε.b, c + ε.d is defined by (a + ε.b).(c + ε.d) = ac + ε.bc + ε.ad + ε2 .bd = ac + ε.(bc + ad) which yields a new contrapuntal interval. An invertible element of Z12 [ε] is an element a + ε.b with a = 1, 5, 7, 11, and the inverse element is (a + ε.b)−1 = a − ε.b. Corresponding to the symmetries in Z12 , the symmetries in Z12 [ε] are the affine endomorphisms (over the ring Z12 [ε]) ea+ε.b (u + ε.v) which is invertible iff u = 1, 5, 7, 11.
29.3.1
The Third Torus
Summary. The third torus is the geometric representation of Z12 associated with its Sylow decomposition4 . We discuss the mathematical structure and its metrical properties. –Σ– ∼
The module isomorphism T : Z12 → Z3 ⊕ Z4 : z 7→ (z(mod3), −z(mod4)) with its inverse T (z3 , z4 ) = 4.z3 + 3.z4 identify the cyclic module Z12 with a discrete torus. Under this ∼ isomorphism, we have an isomorphism of forms T : P iM od12 → P iT hirds3,4 with the third class form P iT hirds3,4 −→ Simple(Z3 ⊕ Z4 ). Hereby, the first component of a T -transformed −1
Id
pitch class is the number of its major thirds while the second component is the number of minor thirds that add to the given pitch when counted from zero. For example, T (7) = (1, 1), i.e., “fifth = major third plus minor third”, see figure 29.1. 4 See
theorem 41 in appendix C.3.
29.3. THE ALGEBRA OF INTERVALS
621
8 11 0
3
4
7 5 6
2
9 1
10
Figure 29.1: On the torus of thirds, the third relations among pitch classes and intervals are represented in a geometric way. The sequence of semi-tone steps appears as an entwined closed spiral. To describe ‘pure intervals’ in pitch classes, starting from the form IntM od12 −→ Syn(P iM od12 ) Id
for interval quantities in pitch classes, we also use the third torus structure as a synonym for interval quantities in terms of thirds, i.e., IntT hirds3,4 −→ Syn(P iT hirds3,4 ). Id
As in differential geometry, one has the space of contrapuntal intervals IntM od12,q [ε] or the corresponding third tangent torus IntT hirds3,4,q [ε] in terms of a cantus firmus torus where twelve “tangent” tori are attached: The contrapuntal intervals are viewed as tangents to their cantus firmus points: A tangent torus Ix = x + ε.Z12 is attached at each of its points x, see figure 29.2.
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CHAPTER 29. MELODIC VARIATION BY ARROWS
8 11
0
3
4
7 6
5
2
9 1
10
Figure 29.2: The contrapuntal intervals viewed as tangents to their cantus firmus points. −→ Let us review the affine automorphism group GL(Z12 ) in terms of the Sylow representation of Z12 as a discrete torus. The group is evidently generated by these four symmetries: c1 c2 c3 c4
= e0 .11 multiplication by − 1; inversion = e0 .5 multiplication by 5; make fourth circle = e3 .1 addition of constant 3; minor third chain = e4 .1 addition of constant 4; major third chain
−→ This system is not only musically meaningful and therefore turns the group GL(Z12 ) into a musically meaningful group by the concatenation principle 2, it also shows that all automorphisms preserve the third distance in the following sense: Let x, y ∈ Z3 ⊕ Z4 . Define d(x, y) as the minimal number of ascending or descending minor or major third steps on the third torus to move from x to y; for example: d(0, 9) = 1 since 0 − 3 = 9, d(0, 1) = 2 since 0 + 4 − 3 = 1, etc. The distance is just the minimal number of edges on the discrete third torus that connects the two points x and y. This distance, in fact a metric, is left invariant under each of the four generators c1 to c4 , see figure 29.3 for the evidence.—Summarizing: −→ Proposition 49 The group GL(Z12 ) leaves invariant the third distance, i.e., it is a group of isometries of the interval torus IntT hirds3,4 .
29.4
Musical Interpretation of the Interval Ring
Summary. This section deals with musical interpretations of the operations in the interval ring. We take this occasion to deepen the topic of mathematical “overhead” structures and their role in building musical concepts. –Σ–
29.4. MUSICAL INTERPRETATION OF THE INTERVAL RING
a)
b)
4
4 8
7
10
8
7 0
11
3
1
2
10
5 9
0
11
3
6
1
2
5 9
6
4
4
8
7
10 6
2
8
7
0
11
3
c)
623
1
10
5 9
0
11
3
6
2
1 5 9
d)
Figure 29.3: The elementary symmetries on the third torus. The generator c1 is a 180◦ -rotation around the axis through 0 and 6; c2 is a reflection at the torus’ equatorial plane; c3 is a 90◦ rotation around the polar axis; c4 is a tilting movement in by 120◦ . Each of these generators preserves the third distance on the torus.
In this section we only deal with invertible symmetries. We want to understand what could be the musical meaning and interpretation of multiplication in P iM od12 [ε], or, mathematically equivalent, in IntM od12,q [ε]. This is indeed a crucial situation. A priori, music theory never considered such a multiplication of intervals, and a conservative attitude could very well prohibit such an extension as an inadmissible mathematical overhead. But this is a classical situation with mathematical objects in the sciences, be it in physics, chemistry or musicology: Mathematical structures have more properties than their application needs. So there is always a degree of mathematical overhead. The question is rather whether ingredients of that overhead can be enriched by meaning within the applying science. More precisely, we are not preconizing the power of a conceptual oracle with mathematical properties. We simply associate mathematical properties with musical properties (in our case) and try to profit from the semantical added value in favor of a richer mathematization of musical phenomena.
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CHAPTER 29. MELODIC VARIATION BY ARROWS
As already exposed in section 8.3, we shall again use the concatenation principle 2 from −→ section 8.3 here. This means that understanding a symmetry in et .g ∈ GL(Z12 [ε]) means understanding its factors in a determined factorization. As in section 8.3, we shall present a system of ‘elementary’, musically understandable symmetries which generates all symmetries. To begin with, we shall treat the translations et = et .1 and the multiplications g separately. For a translation quantity t = a + ε.b, we have et = ea+ε.b = ea .eε.b . Furthermore, g = u + ε.v = u.(1 + ε.uv) = u.(a + ε.1)uv . This yields the following system: 1. s1 = ea , 2. s2 = eε.b , 3. s3 = u, u = 1, 5, 7, 11, 4. s4 = 1 + ε.1. Here are the musical interpretations of the system’s symmetries: 1. s1 = ea . Symmetry s1 causes a transposition if the interval by a since we have s1 (x + ε.y) = (a + x+) + ε.y. 2. s2 = eε.b . Symmetry s2 causes a transposition of the discant voice by b while the cantus firmus remains unaltered, i.e., s2 (x + ε.y) = a + ε.(y + b). this constant enlargement of the distance between cantus firmus and discantus corresponds to a traditional technique of double counterpoint. An example of an interesting symmetry, a concatenation of types 1 and 2, is given for t = a ± ε.1. For t = a + ε.1, minor intervals are transformed into corresponding major intervals under the transposition by a. Conversely, t = a + ε.1 transforms major intervals into corresponding minor ones under the transposition by a. This means that also not ‘strict’ parallels of thirds, such as 2 + ε.3, 7 + ε.4 = e5+ε.1 (2 + ε.3), 11 + ε.3 = e4−ε.1 (7 + ε.4) are viewed as translation symmetries. 3. s3 = u, u = 1, 5, 7, 11. Symmetries of this type multiply the cantus firmus, and the distance between the voices, and hence also the discantus, by u. Since we cannot reduce this discussion to the involved pair of tones, and hence not to the basic pitch domain Z12 , we must have a closer look at this situation. 3.1 u = 11. The cantus firmus is reflected at the origin c = 0 whereas the voice distance 11.b means the octave complement to b. For example, the sweeping minor third d, f (= 2 + ε.3) is transformed into the sweeping major sixth b[ , g(= 10+ε.9). Observe that, by definition, symmetries do not alter the underlying orientation. 3.2 u = 5. Musically, the distance b of the interval tones can be of equal interest as a multiple of minor seconds (= 1) or of fourths (= 5). The symmetry u = 5 connects intervals of equal values in these two perspectives. In a multiplication by 5, the minor second distance b of an interval a + ε.b is transformed into the fourth distance 5.b.
29.5. SELF-ADDRESSED ARROWS
625
3.3 u = 7. For this value, we have an analogous argument as for the preceding case, with fifths instead of fourths. One could also invoke the concatenation principle and use the factorization 7 = 5.11 to reduce this case to the cases 3.1 and 3.2. 4 s4 = 1 + ε.1. With this symmetry, the cantus firmus remains fixed whereas the voice distance increases by the distance of the cantus firmus from the origin c = 0. Thereby, the cantus firmus acquires a new function, i.e., to appear itself as a ‘discantus’ with respect to the origin. Thus its distance to the origin is added to the given discantus as a fixed reference quantity. If we repeat the application of this symmetry to the resulting interval, the same reference quantity is again added to the discantus, etc. Repeated application of s4 therefore generates a circle on the discantus of the original interval, a circle whose step width is defined by the cantus firmus’ distance to the origin. Musically speaking, the origin could be imagined as being a tonic. The appearance of the origin as a reference pitch is only justified by our choice of the generator s4 in the given system. If we took e−ε.b .s4 instead, then b would play the role of the reference pitch. Example 48 3 + ε.4 7→ 3 + ε.7 = (1 + ε.1).(3 + ε.4) 7→ 3 + ε.10 = (1 + ε.1).(3 + ε.7) 7→ 3 + ε.1 =(1 + ε.1).(3 + ε.10) 7→ 3 + ε.4 = (1 + ε.1).(3 + ε.1)
Example 49 6 + ε.3 7→6 + ε.9 =(1 + ε.1).(6 + ε.3) 7→ 6 + ε.3 =(1 + ε.1).(6 + ε.9)
By the way, this symmetry type is the only one within our generator system which connects the cantus firmus and the discantus components in an irreducible manner. This symmetry type crystallizes an essential difference to the common reasoning in the pitch class space P ichM od12 .
29.5
Self-addressed Arrows
Summary. Since arrows formally behave much like tones, self-addressed arrows can be introduced as a natural generalization of self-addressed tones and (ordinary) arrows via address change according to 8.3.4. This extension is described—together with a canonical projection which will play a major role in the theory of consonances and dissonances in section 30.2.1. –Σ–
626
CHAPTER 29. MELODIC VARIATION BY ARROWS
In this section, we want to concentrate on the algebraic relations between self-addressed tones and contrapuntal arrows. Fix an arbitrary commutative ring R and consider the canonical R-linear injection i : R → R[ε]. We also have the functorial R-linear injection [ε] : R@R R → R[ε]@R[ε] R[ε] : f 7→ f [ε] defined in section 7.5. Furthermore, we have two R-linear address ∼ ∼ change injections c : R → 0R @R R → R@R R, c[ε] : R[ε] → 0R @R R[ε] → R[ε]@R[ε] R[ε]. This entails the following commutative diagram R cy
i
−−−−→
R[ε] c[ε] y
(29.1)
[ε]
R@R R −−−−→ R[ε]@R[ε] R[ε] of R-linear injections. Whereas the left lower corner parametrizes the self-addressed tones in a simple form of module R, and the right upper corner parametrizes the contrapuntal intervals in the module R, the right lower corner parametrizes the self-addressed contrapuntal intervals, as opposed to the zero-addressed tones in the module R, or the prime counterpoint intervals, respectively. The four-dimensional R-module R[ε]@R[ε] R[ε] can also be viewed as a left R[ε]-module by the ordinary composition a.x = e0 .a.x with linear endomorphisms e0 .a, a ∈ R[ε]. Then we have a direct decomposition in one-dimensional R[ε]-modules: R[ε]@R[ε] R[ε] = R[ε].e1 0 ⊕ R[ε].e−ε 1
(29.2)
i.e., two two-dimensional R-modules. Moreover, this decomposition is also a kernel-image decomposition with respect to the idempotent5 right multiplication endomorphism ?.eε 0, of R[ε]modules, i.e., Ker(?.eε 0) = R[ε].e−ε 1, Im(?.eε 0) = R[ε].e1 0. Now, the image is exactly the image of R[ε] under c[ε], whereas ?.eε 0 maps the image of R@R R under [ε] isomorphically (as R-module) onto the image Im(?.eε 0). in other words: Theorem 32 With respect to the embeddings of diagram (29.1), the projection ?.eε 0 associates bijectively the self-addressed tones with the contrapuntal intervals and leaves the original tones in R invariant. This means that self-addressed tones and contrapuntal intervals are put under a canonical algebraic correspondence in the large space of self-addressed arrows
29.6
Change of Orientation
Summary. Orientation within a contrapuntal sequence may change, and enforce a special treatment of contrapuntal steps from sweeping orientation to hanging orientation or vice versa. We show how such a change can be reduced to an orientation preserving situation by means of the regular embedding of the algebra of dual numbers in the linear endomorphism ring. –Σ– 5 See
also appendix C.2.3, example 68.
29.6. CHANGE OF ORIENTATION
627
The left-regular embedding6 (?) : R[ε] M2,2 (R) identifies the two-dimensional dual number algebra with a subspace of the ring of two-by-two matrices over R. In this embedding, the dual number algebra is generated by the dual number multiplication ! 0 0 (ε) = 1 0 with the relation (ε)2 = 0. The sweeping and hanging orientations are interpreted as a projection ! ! 1 1 1 −1 α+ = , α− = 0 0 0 0 which are related by the multiplicative relation α− = −α+ (1 + ε)−2 .
(29.3)
The matrix algebra M2,2 (R) is generated by two indeterminates (ε), α+ with relations (ε)2 = 2 0, α+ = α+ , and α+ .(ε)+(ε).α+ = (1+ε), and it is spanned by the linear basis 1, (ε), α+ , α+ .(ε). Relation (29.3) can be used to reinterpret hanging counterpoint in terms of sweeping counterpoint as follows: Suppose that we have a sequence x1 + ε.i1 , x2 + ε.i2 with the first interval in sweeping, but the second one in hanging orientation. This means that the evaluation via orientation projections produces the two discantus instances α+ (x1 + ε.i1 ), α− (x2 + ε.i2 ). In order to change the interpretation of orientations, use (29.3) and rephrase the second discantus as α− (x2 + ε.i2 ) = −α+ (1 + ε)−2 (x2 + ε.i2 ) = α+ (−(1 + ε)−2 (x2 + ε.i2 )) so that we are dealing with sweeping orientation related to the new interval −(1 + ε)−2 (x2 + ε.i2 ). This technique will be used to deduce contrapuntal steps while orientation changes. Observe that the mediator factor (1 + ε) in formula (29.3) enhances the musical meaning of the fourth generator symmetry s4 in section 29.4: The multiplication by generator s4 helps to reinterpret hanging orientation in terms of sweeping orientation.
6 See
appendix D.1.
Chapter 30
Interval Dichotomies as an Expression of Contrast enim vero sicut vitium mala virtus a nullo umquam morali philosopho dictum fuit, ita nec musicus umquam litteratus discordiantiam malam concordantiam nuncupavit. Johannes Tinctoris [527, p.90] Summary. For contrapuntal composition and theory, consonant and dissonant intervals are a dichotomic concept. We present the mathematical restatement and fundamental properties of the basic concept of an interval dichotomy. For the classification of dichotomies, a strong condition on unique symmetries of polarity between the two halves of dichotomies is added. It reveals a distinguished role of the consonance/dissonance dichotomy of classical counterpoint and of the major dichotomy (associated with the major scale). We discuss evidence of the consonance/dissonance dichotomy from theoretical and empirical points of view and open the discourse to an intercultural perspective guided by the classification of dichotomies, as investigated by Jens Hichert [223]. –Σ–
Remark 16 In the following counterpoint chapters, we shall tacitly work in the fifth pitch and interval spaces, i.e., we take the automorphism q = (7) in the identifiers. We therefore shall—for example—speak of the fifth when addressing the unit 1. Whenever we deviate from this convention, the reader should be warned. We shall also stick to the forms built upon Z12 and tacitly carry over the distance structures of the isomorphic Sylow torus representation Z3 ⊕ Z4 in order to keep notation simpler. 629
630
30.1
CHAPTER 30. INTERVAL DICHOTOMIES AS A CONTRAST
Dichotomies and Polarity
Summary. The technical definition of interval dichotomies in IntM od12,q and in the counterpoint arrow form IntM od12,q [] is given1 . We discuss canonical polarities, in fact automorphisms of such dichotomies, as well as their topological behavior which is measured by diameter and span of a dichotomy. This is used to classify the system of dichotomies. Among the 26 classes, six classes admit unique polarities. These classes of strong dichotomies show a topologically distinguished position of the consonance/dissonance and the major dichotomy, the latter being defined by the six tonic-rooted (proper) intervals of major tonality. Among the strong dichotomies, the consonance-dissonance dichotomy itself (not only its class) is distinguished by a number of characteristics. –Σ– Let A be an addressed Z-module. Then an objective, A-addressed local composition X in ambient space IntM od12,q is called a (A-addressed) marked interval dichotomy iff it is equipollent2 to its complement C(X) in the total local composition A@IntM od12,q . Observe that the latter is not necessarily finite. Hence we have the complement action i 7→ C i (X) of Z2 on the set DiM (A, IntM od12,q ) of A-addressed marked dichotomies. An interval dichotomy is a Z2 -orbit of marked interval dichotomies, the orbit set is Di(A, IntM od12,q ). −→ A second left action is defined by the automorphism group GL(Z12 ) of the ambient space −→ IntM od12,q . If g ∈ GL(Z12 ), X ∈ DiM (A, IntM od12,q ), then we set g.X = {g.x|x ∈ X}, and the set CiM (A, IntM od12,q ) denotes the orbit space of this action, its elements are called the (Aaddressed) marked dichotomy classes. Clearly the two actions commute, i.e., we have a left action −→ −→ of Z2 × GL(Z12 ) on DiM (A, IntM od12,q ), an induced action of GL(Z12 ) on Di(A, IntM od12,q ), and one of Z2 on the space of marked dichotomy classes CiM (A, IntM od12,q ). The total orbit space −→ Ci(A, IntM od12,q ) = Z2 × GL(Z12 )\DiM (A, IntM od12,q ) is the space of (A-addressed) dichotomy classes. The same constructions (mutatis mutandis) can be made on the space of counterpoint arrows IntM od12,q []. Let A be an addressed Z-module. Then an objective, A-addressed local composition X in ambient space IntM od12,q [] is called a (A-addressed) marked counterpoint dichotomy iff it is equipollent to its complement C(X) in the total local composition A@IntM od12,q []. On the set DcM (A, IntM od12,q []) of A-addressed marked counterpoint dichotomies, we have the complement action i 7→ C i (X) of Z2 . A counterpoint dichotomy is a Z2 -orbit of marked counterpoint dichotomies, the orbit set is Dc(A, IntM od12,q []). −→ A second left action is defined by the automorphism group GL(Z12 []) of the ambient space −→ IntM od12,q []. If g ∈ GL(Z12 []), X ∈ DcM (A, IntM od12,q []), then we set g.X = {g.x|x ∈ X}, and the CcM (A, IntM od12,q []) denotes the orbit space of this action, its elements are called the (A-addressed) marked counterpoint dichotomy classes. Clearly the two actions commute, −→ i.e., we have a left action of Z2 × GL(Z12 []) on DcM (A, IntM od12,q []), an induced action 1 This
means the interval form with the identifier defined by the automorphism q of Z12 , i.e., IntM od12,q −→
Syn(P iM od12,q ), etc. 2 ...has same cardinality as...
Id
30.1. DICHOTOMIES AND POLARITY
631
−→ of GL(Z12 []) on Dc(A, IntM od12,q []), and one of Z2 on the space of marked counterpoint dichotomy classes CcM (A, IntM od12,q []). The total orbit space −→ Cc(A, IntM od12,q []) = Z2 × GL(Z12 [])\DcM (A, IntM od12,q []) is the space of (A-addressed) counterpoint dichotomy classes. Definition 92 A marked interval dichotomy X is called autocomplementary if it is isomorphic to its complement C(X), i.e., iff its dichotomy class coincides with its marked dichotomy class. The marked dichotomy X is called rigid if its symmetry group is trivial. It is called strong if it is autocomplementary and rigid. If a marked dichotomy X is autocomplementary, so is its complement. If a marked dichotomy is rigid, so is its complement. Hence, if X is strong, so is its complement. So autocomplementarity, rigidity, and strength are invariants of the dichotomy classes. From the classification of zero-addressed objective local compositions in P iM od12 in appendix L.1, we know that there are 34 classes of zero-addressed marked interval dichotomies (26 classes numbers 63 to 88, complements not counted twice, count twice the 8 numbers without ∗ ). There are 26 interval classes, 8 autocomplementary classes, and 6 strong classes. We often denote a marked dichotomy by (X/C(X)) and its class by [C/C(X)], whereas a dichotomy is denoted by (X|C(X)) and its class by [C|C(X)]. Since the group of symmetries acts on the set of dichotomies, one can say that a symmetry stems not only from an isomorphism of the underlying marked dichotomies but is also associated with an isomorphism of the associated dichotomy. More precisely, if we are given a symmetry f : (X/Y ) → (U/V ) between two marked dichotomies, we get an isomorphism of the interpretations that are associated with the partitions X tY and U tV . Conversely, an isomorphism between such two interpretations is induced by two isomorphisms on two pairs of charts. But each such isomorphism gives automatically rise to an isomorphism of the other chart pair—the only thing which we lose is the order of charts, i.e., we are left with a transformation among the (non-marked) dichotomies. In this sense the unique non-trivial inner symmetry p of a strong dichotomy (X|Y ), i.e., p(X/Y ) = (p(X)/p(Y )) = (Y /X) is also called its polarity. Example 50 Here are the six strong (marked and unmarked) dichotomies, the numbers referring to the classification table of local compositions in appendix L.1. The polarity of dichotomy number n and identifier number q is denoted by pq,n , or pn for q = 7, if the context is clear. If an index is omitted, we tacitly suppose the fifth representation with q = 7. 1. The dichotomy Nr. 64 ∆7,64 = (I7 /J7 ) = ({1, 2, 3, 4, 5, 11}|{0, 6, 7, 8, 9, 10}) with polarity p7,64 = 7.p1,64 .7 = e11 .11 corresponding to the dichotomy ∆1,64 = (I1 /J1 ) = ({2, 4, 5, 7, 9, 11}|{0, 1, 3, 6, 8, 10}), in the semitone representation (q = 1), with polarity p1,82 = e5 .11. The dichotomy ∆7,64 arises when considering all proper (non-vanishing) intervals in a major scale when counted from the tonic.
632
CHAPTER 30. INTERVAL DICHOTOMIES AS A CONTRAST
2. The dichotomy Nr. 68 ∆7,68 = ({0, 2, 7, 8, 9, 11}|{1, 3, 4, 5, 6, 10}), with polarity p7,68 = 7.p1,68 .7 = e6 .5 corresponding to the dichotomy ∆1,68 = ({0, 1, 2, 3, 5, 8}|{4, 6, 7, 9, 10, 11}) in the semitone representation with q = 1, with polarity p1,68 = e6 .5. 3. The dichotomy Nr. 71 ∆7,71 = ({0, 2, 7, 8, 9, 11}|{1, 3, 4, 5, 6, 10}), with polarity p7,71 = 7.p1,71 .7 = e5 .11 corresponding to the dichotomy ∆1,71 = ({0, 1, 2, 3, 6, 7}|{4, 5, 8, 9, 10, 11}) in the semitone representation with q = 1, with polarity p1,71 = e11 .11. 4. The dichotomy Nr. 75 ∆7,75 = ({0, 2, 4, 7, 8, 11}|{1, 3, 5, 6, 9, 10}), with polarity p7,75 = 7.p1,75 .7 = e5 .11 corresponding to the dichotomy ∆1,75 = ({0, 1, 2, 4, 5, 8}|{3, 6, 7, 9, 10, 11}) in the semitone representation with q = 1, with polarity p1,75 = e11 .11. 5. The dichotomy Nr. 78 ∆7,78 = ({0, 2, 4, 6, 7, 10}|{1, 3, 5, 8, 9, 11}), with polarity p7,78 = 7.p1,78 .7 = e3 .11 corresponding to the dichotomy ∆1,78 = ({0, 1, 2, 4, 6, 10}|{3, 5, 7, 8, 9, 11}) in the semitone representation with q = 1, with polarity p1,78 = e9 .11. 6. This is the classical dichotomy Nr. 82 ∆82 = ∆7,82 = (K7 /D7 ) = ({0, 1, 3, 4, 8, 9}|{2, 5, 6, 7, 10, 11}). of contrapuntal consonances (left) and dissonances (right) in the fifth system. Its polarity is ‘the’ autocomplementarity function p7,82 = 7.e2 .5.7 = e2 .5 deduced from the known3 autocomplementarity function p1,82 = e2 .5 of the consonance-dissonance dichotomy ∆1,82 = (K1 /D1 ) = ({0, 3, 4, 7, 8, 9}|{1, 2, 5, 6, 10, 11}) in the semitone representation, as discussed in section 24.1.1. 3 See
[336], for example.
30.1. DICHOTOMIES AND POLARITY
633
These strong dichotomies can also be represented as partitions of the interval torus in the Sylow representation IntT hirds3,4,q −→ Syn(IntT hirds3,4 ) with the q-identifier (usually the (q)
fifth). Recall that we have the third distance d(x, y) on this torus, and that it is the same as the distance with identifier q = 1. Definition 93 Let (X/Y ) be a strong marked dichotomy in IntT hirds3,4,q . Then its diameter is defined by 1 X δ(X/Y ) = d(u, v). 2 u,v∈X
By use of the polarity of a strong dichotomy one sees that δ(X/Y ) = δ(Y /X) so that we may define this number as the diameter δ(Y |X) of the strong dichotomy (X|Y ). Since evidently all dichotomies of a dichotomy class have the same diameters (recall that symmetries of the torus are isometries for the distance), we may define the diameter δ[X|Y ] of a dichotomy class [X|Y ] by the diameter of any of its representatives. The diameter measures the average distance between points of one half of the dichotomy. Definition 94 Let (X/Y ) be a strong marked dichotomy in IntT hirds3,4,q with polarity p. Then its span is defined by X σ(X/Y ) = d(u, p(u)). u∈X
Since any polarity is an involution in our context, one sees that σ(X/Y ) = σ(Y /X) so that we may define this number as the span σ(X|Y ) of the strong dichotomy (X|Y ). Since evidently all dichotomies of a dichotomy class have the same span (recall that symmetries of the torus are isometries for the distance), we may define the span σ[X|Y ] of a dichotomy class [X|Y ] by the span of any of its representatives. Diameter and span of our six strong classes are visualized in figure 30.1. Intuitively, the minimality of δ(K/D) means that the subsets K and D are separated in an optimal way on the torus (figure 30.1). The maximality of σ(I/J) means that I and J are optimally mixed on the torus. If we stay on a point on I, and we want to go to another point of I on a shortest path, then we often have to traverse a point of J, a phenomenon which never happens for (K/D): Between any two consonant intervals there is always a shortest path which does not leave the consonant half. By the given polarities, all these statements about K and I are also valid for D and J. The possible connections within K and within I (and the complements, respectively) are shown in figure 30.2. In contrast to the graph of I, the graph of K has no inner symmetry, i.e., every consonant interval is uniquely determined by its position on the graph. This means that the consonancedissonance dichotomy has a privileged position among all strong dichotomies. But this does not yet exhibit the precise representative (K/D) within the class [K|D]. The selection of the marked dichotomy (K/D) is in fact realized by an algebraic condition: to require that the first half X of an element (X/Y ) of the class [K|D] be a multiplicative monoid. This condition was discovered by Noll [400]. This in fact exhibits the marked half of consonances against dissonance, these two parts are not equivalent from this point of view. Until now it is however not clear what are the structural consequences of the predicates which uniquely exhibit the consonances.
634
CHAPTER 30. INTERVAL DICHOTOMIES AS A CONTRAST
s(X) 16
82
68
10
24
a)
75
71, 78
64
25
28
29
4
d(X)
K
8
7
0
11
3 b)
1 10
5
2
9
6 D
Figure 30.1: (a) Span and diameter of the six strong dichotomies. The polar position of the consonance-dissonance dichotomy (Nr. 82) against the major dichotomy (Nr. 64) is visible. (b) The geometric meaning of the minimal diameter and the maximal span in (K/D) is evidenced by an optimal separation of the two halves of the dichotomy.
30.2
The Consonance and Dissonance Dichotomy
Summary. The consonance-dissonance dichotomy is in a canonical bijection with the Riemann consonances. We also give empirical evidence of this dichotomy from brain research. The section concludes with a cognitive interpretation of the specific role of music for the individual psyche. –Σ– Recall from section 24.1.1 that the consonance-dissonance dichotomy also appears in the context of just intonation and relates to the 12-tempered case by means of the enharmonic projection. We now want to investigate further remarkable properties of the (K/D) dichotomy. To begin with, we define a (marked) dichotomy (X[ε]/Y [ε]) of contrapuntal intervals for every (marked) dichotomy (X/Y ) on the space form S by the rule (X[ε]/Y [ε]) = (coord(S) + ε.X/coord(S) + ε.Y , where coord(S) is the coordinator module of S.
30.2. THE CONSONANCE AND DISSONANCE DICHOTOMY 4
7
11
2
5
1
9
2
class 64
0
3
7
2
6
2
1
class 71
10
1
5
4
635 8
0
class 68
2
5
8
4
1
class 75
0
4
7
0
3
8 6
class 78
9
3 class 82
0
Figure 30.2: The graphs of all strong dichotomies, regarding the third connections. The graphs among the I (major, class 64) and the K (consonance, class 82) intervals show characteristic differences. While in I, all intervals are positioned on a line, the consonance graph shows two minimal paths (0 → 3 → 7; 0 → 4 → 7) between 0, 7, for example. Observe that when starting from the prime, all imperfect consonances (thirds, sixths) can be reached directly, whereas the perfect fifth is the only non-neighbor of the prime.
30.2.1
Fux and Riemann Consonances Are Isomorphic
Summary. We present the Riemann dichotomy and its 1-1 correspondence with the Fux dichotomy based on the diagram discussed in 29.5. –Σ– Recall the commutative diagram (29.1) of Z12 -linear injections for the special ring R = Z12 . Z12 cy
i
−−−−→
Z12 [ε] c[ε] y
(30.1)
[ε]
Z12 @Z12 Z12 −−−−→ Z12 [ε]@Z12 [ε] Z12 [ε] We may vew this configuration as a realization of determined forms as follows: 0@P itchM od12,q cy
i
−−−−→
[ε]
0@IntM od12,q [] c[ε] y
Z12 @P itchM od12,q −−−−→ Z12 [ε]@IntM od12,q []
(30.2)
636
CHAPTER 30. INTERVAL DICHOTOMIES AS A CONTRAST
Observe that the space in the right bottom corresponds to the larger space Z12 [ε]@Z12 [ε]. Also, the lower horizontal arrow is not a natural one in terms of ambient morphisms f /α for local compositions. To begin with, let us look at the Riemann dichotomy which was first defined in [400]. For a justification of this naming that relates to Riemann’s concept of relative consonances and dissonances, see [400]. The Riemann dichotomy is the monoid T rans(D, T ) generated by the “transporter” set T r(D, T ) consisting by definition of all not necessarily invertible symmetries f : D = {1, 2, 5} → T = {0, 1, 4} from the dominant triad D to the tonic triad T (in the fifth system). The monoid T rans(D, T ) has 72 elements. So it naturally gives rise to a dichotomy on Z12 @P itchM od12,q and also defines a point in the intension topology InT op(P itchM od12,q ). Exercise 67 Show that card(T rans(D, T )) = 72. On the other hand, consider the consonant contrapuntal intervals K[ε] = Z12 + ε.K in 0@IntM od12,q []. The intension Int(K[ε]) is canonically identified to a local composition in Z12 [ε]@IntM od12,q [] since the address Z12 [ε] is faithful (take the identity). The intersection W of Int(K[ε]) with the subspace Z12 @P itchM od12,q consists of those endomorphisms ex y of P itchM od12,q that induce endomorphisms of K[ε]. Using the isomorphism ∼
ν =?.eε 0|Z12 @Z12 Z12 : Z12 @Z12 Z12 → Z12 [ε] from section 29.5, W identifies to the monoid R ∈ InT op(P itchM od12,q ) of left stabilizers of ν −1 (K[ε]). In [400] it is shown that T rans(D, T ) = W . More precisely, with the above identification, we have this theorem: Proposition 50 Let ex .y be an endomorphism of P itchM od12,q . Then it is in the left stabilizer4 Q xi −1 x of ν (K[ε]) iff it is a product e .y = i e .yi of endomorphisms5 which transport the dominant triad D = {1, 2, 5} into the tonic triad T = {0, 1, 4}. In particular, every element of ν −1 (K[ε]) can be written as such a product of transporter endomorphisms. The last statement follows from the fact that the consonance interval numbers K are a multiplicative monoid, and therefore, each element ex .k, k ∈ K is a stabilizer of ν −1 (K[ε]). This implies the following: Corollary 19 Let f = 0 + ε.1 the fifth interval at the C tonic 0. For any consonant interval c = x+ε.k, there is a sequence ti = exi .yi , i = 1, . . . m of transporter endomorphisms ti : D → T such that c = tm .tm−1 . . . . t1 (f ). Evidently, there is a deep relation between Riemann theory (as it appears in Noll’s perspective) and the Fux dichotomy of consonance-dissonance. Presently, we do not know more about the harmonic/contrapuntal implications of the above results. The straightforward hope is that the transition from polyphonic counterpoint to homophonic harmonic relates to the above mathematical facts. But neither the systematic nor the historical consequences of these facts 4 I.e., for all endomorphisms et .k with consonant component k, ex .y.et .k = es .k 0 has also a consonant component k0 . 5 Including the “empty” product, i.e., the identity.
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637
are at reach. It is however true that musicology has never understood the theoretical relations between counterpoint and harmony. There must be a fundamental relation because the development of homophony out of the polyphonic tradition cannot be a rupture without any inner coherence. Even if such a rupture were a historic fact, it would be a primordial question of systematic musicology to explain the structural, system-immanent rationales for such a rupture. The present results give first hints for the explanation of this lacuna.
30.2.2
Induced Polarities
Summary. We describe the autocomplementary functions induced on the contrapuntal dichotomies (X[ε]/Y [ε]) which are deduced from strong dichotomies (X/Y ). –Σ– Suppose that we are given a strong dichotomy (X/Y ) which bears the polarity eu .v. Then the contrapuntal dichotomy is no longer rigid, but still autocomplementary. The precise situation is described as follows. Proposition 51 Let ∆ = (X/Y ) be a strong dichotomy with polarity p∆ = eu .v. Choose a cantus firmus point x. Then there is exactly one symmetry px∆ on the counterpoint interval space IntM od12,q [ε] which is a polarity of (X[ε]/Y [ε]) and fixes the “tangent space” Ix = x + ε.Z12 at x. Call this the polarity at x. We have px∆ = ex(1−v)+ε.u .v, and px+y = ex .py∆ .e−x , ∆ and under this polarity, a tangent space Iy is mapped onto the tangent space Ix+v(y−x) . Exercise 68 Give a proof of proposition 51.
30.2.3
Empirical Evidence for the Polarity Function
Summary. A review of neurophysiological verifications of the presence of the Fux polarity in human depth EEG is exposed. –Σ– Although this book is not a report on physiological or psychological correlates to music structures, it is important to give an overview of a pronounced evidence of electrophysiological correlates of the consonance-dissonance dichotomy. This is by no means a justification of even a proof of the adequacy of the mathematical investigations, but it must be considered as a fundamental relativization of traditional consonance-dissonance theories. These ideas never produce dichotomies but yield degrees of consonance or dissonance, a quality which is completely irrelevant to the musical counterpoint dichotomy. The aim of a project at the Neurology Department of the Z¨ urich University Hospital which the epileptologist Heinz-Gregor Wieser, the author, and their collaborators conducted during the years 1984–1988 was to test mathematical principles of classical counterpoint by means of depth EEG responses to musical stimuli. In particular, it was planned to test the
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relative results of the EEG to different musical inputs (consonances vs. dissonances) and not the relation of responses to musical stimuli versus non-musical stimuli. The latter problem has been investigated with much success by Hellmuth Petsche and his collaborators [415]. It is important to stress that from our results, we do not draw any kind of conclusions concerning a possible genetic nature of musical understanding or a possible universal validity of classical European interval categories. Our investigations show that in some defined regions of the brain of some European humans, certain significant reactions take place—nothing more and nothing less. There is no reason to generalize whatsoever, but there are enough reasons to try to repeat these investigations in other research sites with a comparable infrastructure. This is all the more desirable since the qualitative results of the investigations by Wieser and the author (namely the prominent role of limbic structures for the judgment of musical pleasantness) have been confirmed by others; see [58], for example. For a more complete report of our results, we refer to [336, 337, 353, 570, 571, 572]. In this short review, we shall restrict ourselves to the two subtests concerning (a) isolated successive intervals and (b) the polarity between simultaneous consonances and dissonances. Our results confirm our hypothesis on (1) a significant differentiation of EEG responses to consonant vs. dissonant intervals in limbic and auditory brain areas and (2) a pronounced sensitivity of these areas in EEG responses to the fundamental polarity between consonances and dissonances. In particular, the quantitative measurement of these responses by use of the “spectral participation vector” has confirmed our belief that this vector may carry some of the semantic charge of EEG signals. 30.2.3.1
The EEG Test
The test concerned different contexts of consonances and dissonances as well as the test of the polarity e2 .5. We used EEG from the scalp (Hess system), stereotactic depth EEG following [569], and multipolar foramen ovale recordings [573]. The tests were applied to the rare cases of patients suffering from medically intractable complex partial epilepsy seizure of suspected mediobasal temporal lobe origin and underwent presurgical evaluation with a view towards surgical epilepsy therapy. None of the total 13 patients considered the voluntary 30 minute music test through monophonic earphones as being disagreeable. There are several reasons why, despite the particular state of epileptics, the tests remain comparable to tests with normal humans which cannot be conducted for evident reasons. First, our tests were performed during interictal periods. Second, localization of the focus gives a good estimation of its possible influence. Third, epileptiform potentials are easily distinguished from others by the expert. For each patient, we recorded 700 time windows for fast Fourier transform (FFT) spectral analysis with 256 samples per second, each window for different EEG channels, different power windows (δ = 0 − 4Hz, θ = 4 − 8Hz, α = 8 − 14Hz, β = 14 − 40Hz) and repetitions, totally 11’000 raw spectral data per patient. Unfortunately, the project could not be completed for extrascientific reasons and hence, only two patients have been thoroughly evaluated. These patients were C. J.-L., a 35-year-old academic, and V.S., a 31-year-old artisan. Both are male Europeans ad prefer standard classical, light, and folklore music. Figure 30.3 shows the positions of bipolar depth EEG recordings which we are going to discuss. Notice that C. j.-L.’s recordings RCA, RH, LCA are homologous to V.S.’s recordings 4, 10, 14. Recording RCA lies within the
30.2. THE CONSONANCE AND DISSONANCE DICHOTOMY
639
right hippocampus, recording RH is positioned within Heschl’s gyrus (auditory cortex), and recording LCA lies within the left hippocampus.
Figure 30.3: Implantation scheme for the two patients C. J.-L. and V.S. following X-rays showing the topographic position of depth electrodes. The three homologous regions are indicated by dashed ovals. For C. J.-L., RCA is electrode 2/1-3, RH is 6/5-6, and LCA is 8/1-3. For V.s., 4 is electrode 2/1-2, 10 is 4/5-6, and 14 is 6/1-2. (Numbers after the slashes indicate precise positions on the electrodes, where 1 = deepest position and 10 = position near surface. Nevertheless, we have been able to observe a great deal of visual evidence for EEG response to music stimuli in neo- and archicortical regions of all patients; for details, see [336, 570]. 30.2.3.2
Analysis by Spectral Participation Vectors
We used four sounds for these tests: piano, sine wave, cello (without vibrato), and “test”, a clear, organ-like sound, all synthesized from a Yamaha TX7 synthesizer and CX5M voicing program in order to avoid possible emotional artifacts associated with natural sounds from sociocultural premises. The music program was written on a precursor of the commercial composition software prestor [338]. The spectral analysis was executed on a CDC Cyber computer. We made use of the spectral participation vector S(E) = (P (E), P (E)/Pδ (E), P (E)/Pθ (E), P (E)/Pα (E), P (E)/Pβ (E)) of an event E and its associated participation value v(E) = P (E)/Pθ (E) + P (E)/Pα (E) + P (E)/Pβ (E)
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which is a length measure. Here, P (E) is the total spectral power [423], Pδ (E) is the δ-power of event E, Pθ (E) is the θ-power, Pα (E) is the α-power,and Pβ (E) is the β-power, see also [336]. From the results obtained thus far we conclude that this representation is well suited to give an adequate picture of possible semantic charge of EEG signals. It is a measure motivated by,among others, the well-known vigilance-related α-participation Sα = P (E)/Pα (E) observed by Berger. We also use the delta participation Sδ = P (E)/Pδ (E), the θ-participation Sθ = P (E)/Pθ (E), and the β-participation Sβ = P (E)/Pβ (E), the latter has been recognized as being strongly related to higher cognitive brain activity by Giannitrapani (see [182]) and Petsche et al. (who also focus on the γ-band 30 − 50Hz) [416]. 30.2.3.3
Isolated Successive Intervals
We first focus our attention on a subtest concerning musically isolated successive intervals, i.e., the two tones of an interval are played one after the other without interruption. All the intervals were played in three orders: (1) all consonances, ordered according to their size; all dissonances, also ordered according to their size. (2) All consonances, ordered according to complementarity size (if possible); then all dissonances, ordered according to complementarity (if possible). (3) A mixed succession of all intervals according to a particular dodecaphonic all-interval series. Having fixed a frequency band, θ, say, and an interval with first tone event E1 and second event E2 , we consider the quotient Qθ (E1 , E2 ) = Sθ (E1 )/Sθ (E2 ) of the theta participations of the first and second tones. If Qθ (E1 , E2 ) > 1 or Qθ (E1 , E2 ) < 1, respectively, then theta participation lowers or increases, respectively, from the first to the second tone. In order to compare these ratios for consonances and dissonances, we take the quotient Qθ (K/D) = Sθ (K)/Sθ (D) of the mean value Sθ (K) of all values Qθ (E1 , E2 ) for consonances (E1 , E2 ) and the analogous mean value Sθ (D) for dissonances. This construction is repeated for all recording positions and all frequency bands, including θ, α, β; band δ is omitted since it may be affected by noise. This test was performed four times with patient C. J.-L. and six times with V.S. A one-sided Wilcoxon test shows significantly higher quotients for consonances compared to dissonances, i.e., Q? (K/D) is significantly larger than 1 for many recording positions and frequency bands, see figure 30.4, and observe the similarity of distribution of these quotients for our patients with respect to homologous recordings and frequency bands. This means that for consonances, participation lowers more when the second tone appears than for dissonances. 30.2.3.4
Polarity
To test the response to the (K/D)-polarity, we confronted each consonant interval X with all the dissonant intervals Y , and we looked for particular responses in cases where Y was the interval which should correspond to X according to the polarity formula Y = 2X + 5. Here, we looked at simultaneous intervals. For each given consonance X, we played a sequence of six confrontations, i.e., immediate successions (X, Y1 ), (X, Y2 ), . . . (X, Y6 ) of X with each of the six dissonances Y1 , Y2 , . . . Y6 , see also figure 30.5. The duration of each interval was 0.68 seconds.
30.2. THE CONSONANCE AND DISSONANCE DICHOTOMY
641
1.0
b a
a=0.05
a=0.05
4 10 14
a=0.05 a=0.1
q
V.S.
1.0
b a
a=0.05
RCA RH LCA
a=0.05 a=0.1
q
C.J-L.
Figure 30.4: Graphical representation of the quotients Qθ (K/D), Qα (K/D), and Qβ (K/D) for all locations and patients C. J.-L. and V.S. These numbers show that for consonances, participation lowers more when the second tone appears, compared to dissonances. The 1-level is indicated in the graphics, observe the places where this level is exceeded. We looked for the least participation values among the dissonant intervals Y1 , Y2 , . . . Y6 , when confronted with a fixed consonance X. We then compared the effective hits to the a priori chance to hit the correct dissonance. This method was applied to every recording position and to the three above frequency bands. Figure 30.6 shows the numbers compared to 100%, the measure for a priori chance. Due to the small number of samples for this subtest, we did not apply any statistical test here. However, as figure 30.6 shows, the results are remarkable and similar for both patients, and we conclude that this pilot investigation strongly supports the presence of the (K/D)-polarity as a foundation of contrapuntal processes.
30.2.4
Music and the Hippocampal Gate Function
Summary. The neurophysiological results, in particular their localizations in the emotional brain and the auditory cortex, are interpreted from the cognitive perspective. The gate function of the hippocampal formation suggests a key function of music in opening subconscious— preferredly emotional—memory contents. –Σ–
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0.5 sec
Figure 30.5: The score of the confrontation test of each consonance with all dissonances; duration of a note in the test is 0.68 seconds. Music and emotions are intimately related, this is common knowledge. The above results suggest a mechanism that could explain this relation on the neurophysiological and cognitive level. We have seen that the emotional brain in its hippocampal structures has a pronounced response to elementary structures of harmony and counterpoints: the intervals in their consonance-dissonance dichotomy, and this is so independently of any sound color physics. Now, the classical thesis of Papez and MacLean [315] states that the limbic system, a prominent part of the archicortex, is responsible for emotional human behavior, this is why it is also called the emotional brain. So the hippocampal sensitivity to consonances vs. dissonances could relate to the emotional function of music, i.e., of musical intervals in our case. The question is, how musical signs which are by no means emotions by themselves (although Sch¨onberg and other prominent music experts constantly evoke the notes’ emotional and erotic life) can evoke and signify emotions in humans, and why this is done in such a way that the same music may evoke a great variety of such reactions and significations. Evidently these outputs are the result of a determined sample of music plus an individual human ingredient. The point is that the hippocampal formation has been recognized as a key structure for memory [501]. The neuroscientist Jonathan Winson has proposed a more specific theory of the hippocampal memory function [579], in that he argues that the hippocampus performs a gate
30.2. THE CONSONANCE AND DISSONANCE DICHOTOMY
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100%
b a 4 10 14
q
V.S.
100%
b a RCA RH LCA
q
C.J-L.
Figure 30.6: Graphical representation of hitting frequencies for the polarity subtest for patients C. J.-L. and V.S., three homologous locations as well as θ, α, β frequency bands. The frequent and pronounced values above 100% show that for both patients there was a strongly affirmative EEG response to the test of the correct values for the polarity. In addition, the topographic/spectral distribution of values for α and β bands is comparable for these patients. function to the subconscious (he even evokes Freud’s “Unbewusstes”), i.e., to memory contents of emotional character. This means that the hippocampus is a structure that plays the role of a gateway to hidden memory contents. It is well known that humans do not have a free or controlled access to their memory contents, in particular not on the level of long-term and emotional memory, concerning early childhood, for example. This suggests that special mechanisms must be activated in order to open the hippocampal gate to unveil locked memory contents. It is straightforward from our neurophysiological findings and the gate function of the hippocampus that its musical stimulation could yield such a “key” to open the gate to hidden memory contents. If this were the case, two specifica of the relation of music and emotion would be explained at once: (1) The emotional contents are not generated by music, they are merely retrieved and evoked from a memory database, whence the individual emotional response to one and the same music would receive a logical explanation. (2) The musical stimulation of the hippocampus is very probably not independent of the human individual who undergoes this process, in other words: If the music is a key, each individual is likely to have his/her individual key to the “subconscious”. This would explain why there are so many different musical tastes—beyond musical education and culture. This would also
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explain why it is often a specific tune or musical mood that is the personal preference: If this tune played a role in the encoding of a specific emotional memory content, the same key-tune could play a role in the decoding process. Summarizing, we have this thesis: Thesis 5 Consonant and dissonant intervals and associated harmonic or contrapuntal structures evoke a hippocampus state/process which activates a gateway to mainly subconscious memory contents. In other words, Winson’s gate hypothesis of the hippocampal formation must also be stated in the sense of the existence of a musicogenic key to the gate. This thesis does not mean that music produces emotions, it only retrieves and reactivates them from a memory database. So it acts on the brain like a drug and produces psychic effects. In this metaphor, the ‘chemical formula’ of the music drug corresponds to the involved musical structure.
Chapter 31
Modeling Counterpoint by Local Symmetries Der Rangunterschied zwischen den perfekten und den imperfekten Konsonanzen erm¨ oglichte die Formulierung genereller Konsonanzfolgeregeln f¨ ur eine indeterminata positio. Klaus-J¨ urgen Sachs [468, p. 114] Summary. This chapter presents the counterpoint model in form of a counterpoint theorem which guarantees the existence and exhibits an arsenal of admitted contrapuntal steps that come in extremely close to the rules of classical counterpoint. The theorem is based on the concept of a contrapuntal symmetry and follows the paradigm of local symmetries as a rationale for forces in physics. Because of its generic concept framework, the theorem, which in this general form1 was proved by Jens Hichert [223], is also valid for non-European scales. We discuss these extensions. –Σ–
31.1
Deformations of the Strong Dichotomies by Contrapuntal Symmetries on IntM od12,q [ε]
Summary. In the core theory of counterpoint [468], the concept of contrapuntal “tension” between successive, perfect and imperfect consonant intervals plays a crucial role. This idea is made precise in the framework of contrapuntal symmetries which deform the strong dichotomy. The separation property of contrapuntal symmetries is proven. –Σ– 1 The
original theorem is presented in [336, 340] and deals with the classical consonances and dissonances, when applied to European diatonic scales.
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The following counterpoint theorem (section 31.3) is concerned with the elementary and core situation of classical counterpoint: “note-against-note”. It is a theory which describes a system of rules for “allowed” sequences (xs + ε.is , αs )s of oriented intervals, a system which essentially boils down to a set of rules for allowed successor pairings (xs + ε.is , αs ), (xs+1 + ε.is+1 , αs+1 ) within such contrapuntal sequences. We shall also restrict our investigation to sequences of constant sweeping orientation αs = α+ = const., changes in the orientation are handled as described above in 29.6. The first elementary rule of counterpoint “note-against-note” says that we are not allowed to take other intervals than the consonances ξ ∈ K[ε]. This seems evident, but it imposes a strong obstruction against another, more hidden directive: the idea of creating a tension between each interval and its successor. More precisely, the meaning of “contra” is not only that of a vertical opposition between cantus firmus and discantus. As Sachs has remarked in [468], the preposition “contra” equally means a horizontal opposition between successive intervals in the given sequence. This requirement is not very explicit, but it is reflected in the distinction between perfect consonances (prime, fifth, octave) and the others, the imperfect sixths and thirds, and the idea of changing between perfect and imperfect consonances in order to create tension. This conceptual distinction seems to evoke a dissonant ingredient in the consonant character, although it does not really abolish the consonance, it is a kind of coloring effect. So, the idea of contrapuntal tension is in some sense a contradictory requirement against the primordial rule of forbidden dissonances: We should like to behave as if there were dissonances within consonances and to create a tensed movement from consonances to dissonances and vice versa. In order to solve this requirement in our mathematical remake of the contrapuntal rules, we introduce this technique: Given a symmetry g of IntM od12,q [ε], we may apply g to the consonance-dissonance dichotomy (K[ε]/D[ε]) and “deform” it to the dichotomy g(K[ε]/D[ε]). In general, the deformed dichotomy will have its parts in such a position that some real consonances ξ are also g-deformed consonances, i.e., ξ ∈ g(K[ε]), and some are g-deformed dissonances, i.e., ξ ∈ g(D[ε]). This implies that we may restate the directive of creating contrapuntal tension in the sense that for a given pair of successive consonances ξ, η, the first one is a g-deformed dissonance while the other is a g-deformed consonance (or vice versa) for a determined symmetry g, in which case we say that the (unordered) pair ξ, η is g-polarized. We shall see below in section 31.2 that the symmetries which we shall exhibit for this role are indeed local symmetries which in physics are responsible for creation of forces, i.e., deformational tension. Of course it is not evident that there is always a symmetry g that polarizes two consonances ξ, η in the above sense. Let us first discuss this topic. It shows that the strong dichotomies are exactly what is needed to guarantee this polarization property. Proposition 52 Let (X[ε]/Y [ε]) be a strong dichotomy and let ξ, η be two different intervals. Then there is a symmetry g such that the pair ξ, η is g-polarized. If ξ, η lie in different halves of (X[ε]/Y [ε]), then g = Id does the job. So we may suppose that both, ξ and η lie one half, say in X, the other case is settled by a transformation of the pair of intervals to a pair within X via the polarity of (X[ε]/Y [ε]). Let ξ = i + ε.j, η = k + ε.t with j, t ∈ X. The symmetry is g = el+ε.m (n + ε.o), n2 = 1. Then we have these applications: g.ξ = l + ni + ε(m + jn + oi), g.η = l + nk + ε(m + tn + ok).
31.2. CONTRAPUNTAL SYMMETRIES ARE LOCAL
647
We want the coefficients of ε to stay in X and Y , respectively. If we try with o = 0, this means that em .n(j) and em .n(t) stay in different halves. Suppose that both are always in the same half. Try further n = 1. Then we have j + m, t + m ∈ X or j + m, t + m ∈ Y for all m ∈ Z12 . Take any a ∈ X and set m = a − j. Then j + m = a, and t + m = (t − j) + j + m = (t − j) + a. So adding the difference t − j to any a ∈ X is again an element of X. But then the symmetry et−j is a non-trivial automorphism of X, contradicting the rigidity of X, and we are done. QED. We shall give another proof of this fact below. Let us now formulate the properties of symmetries which we want to use for the deformations for a counterpoint rule set: Definition 95 Let ∆[ε] = (X[ε]/Y [ε]) be a strong dichotomy. Let ξ = x + ε.i ∈ X[ε]. A symmetry g is contrapuntal for ξ iff (i) ξ 6∈ g(X[ε]), (ii) px∆ is a polarity of g.∆[ε], (iii) The cardinality of g(X[ε]) ∩ X[ε]) is maximal among those g which have properties (i) and (ii). The reason for these requirements is this. We have seen that every pair of intervals can be polarized by a specific symmetry. But we are interested in a rule set which guarantees more than the mere possibility of separation. We want to have those symmetries which admit a maximal number of polarized couples starting from a fixed interval. The second condition is introduced in order to relate the polarizing symmetry g to the given polarity px∆ of the dichotomy at the cantus firmus x. It can be shown [333] that for x = 0 this property is also equivalent to the commutativity condition g.p0∆ = p0∆ .g, so it is a generalized commutativity condition. Definition 96 If a strong dichotomy ∆[ε] = (X[ε]/Y [ε]) and an interval ξ ∈ X[ε] are given, an interval η is called an admitted successor of ξ if it is contained in an intersection g(X[ε]) ∩ X[ε] for a contrapuntal symmetry g for ξ.
31.2
Contrapuntal Symmetries Are Local
Summary. A closer look at contrapuntal symmetries shows their local character. In complete analogy with modern physics, local symmetries produce the looked-for forces of contrapuntal tension. Thus, the melodic variation of the cantus firmus in counterpoint is perfectly interpreted as a deformation caused by the “forces” from local symmetries. –Σ– Before we deal with the counterpoint theorem, we should explain the local character of contrapuntal symmetries. We shall consider the example of classical consonances and dissonances, i.e., the dichotomy ∆[ε] = (K[ε]/D[ε]). We compare two interpretations of the local composition of all zero-addressed intervals I = 0@IntM od12,q [ε]. The first I C1 is induced by ∆ an has the atlas Ix , Kx = x + ε.K, Dx = x + ε.D, x ∈ Z12 .
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The second I Cg is the analogous construction built on the deformed dichotomy g.∆[ε], i.e., its atlas is Ix , gKx = Ix ∩ g.K[ε], gDx = Ix ∩ g.D[ε], x ∈ Z12 . Although the autocomplementary function px∆ for the cantus firmus x is also a polarity of the g-deformed dichotomy g.∆[ε] according to definition 95, its action on the two interpretations is qualitatively different (see also figure 31.1).
Ÿ12[e] gD (0)
f
gK
Iw
I5w
Figure 31.1: The autocomplementary symmetry on the deformed consonance-dissonance dichotomy is a local symmetry, whereas it is a global one on the original dichotomy. This fact resembles physical forces being induced by local symmetries. Consider the example g = eε.8 (5 + ε.4) for the consonance ξ = ε.9, i.e., at cantus firmus x = 0. We have g.K[ε] = (1 − ε.4)Z12 + ε.e8 .5K. For a cantus-firmus point w, this means gKw = w + ε.(5K + 8 − 4w). But under p0∆ , w is transported to 5w, and we have gK5w = 5w + ε.(5K + 8 + 4w). So we recognize the following: On the first interpretation, p0∆ acts on Kw via translation on the cantus firmus: K5w = e4w (Kw ), followed by the autocomplementarity symmetry on I5w . But on the second interpretation, p0∆ does not operate on gKw by a translation plus autocomplementarity symmetry on I5w , in fact, gK5w is different from e4w (gKw )! This can be understood in the sense that p0∆ acts on the second interpretation via g-deformation in the spirit of physics, i.e., as a local symmetry, instead of a global symmetry as it is the case for the first interpretation. The latter can be viewed as a ‘spiral turn’: ‘rotation’ (=autocomplementarity symmetry on
31.3. THE COUNTERPOINT THEOREM
649
each chart Iw ) plus translation (K5w = e4w (Kw )) ‘along the rotation axis’. The former however does not shift the part gKw of Iw to e4w (gKw ), but deforms it to gK5w = 5w + ε.(5K + 8 + 4w) = eε.8w (e4w (gKw )) by the factor ε.8w. This phenomenon is analogous to physics in the sense that forces in physics are induced by local symmetries [162]. In this understanding, local symmetries on the interval space seem to be responsible for the tension which controls the progression from interval to interval in our model.
31.3
The Counterpoint Theorem
Summary. This section presents the counterpoint theorem and its corollaries. –Σ– In this section, we prove the general counterpoint theorem. “General” means that we deal with all strong dichotomies and construct the lists of admitted interval successors for these cases, including the classical case as a special item. The general theorem was proved in Jens Hichert’s thesis [223] and sheds a new light on the general problem of what counterpoint is about. We shall discuss the special case of the consonance-dissonance dichotomy in section 31.4.1. Remark 17 In order to cope with Hichert’s calculations and tables we shall use the interval space relative to the identity identifier, and not to the fifth identifier. In particular, the consonance quantities are {0, 3, 4, 7, 8, 9} in this section.
31.3.1
Some Preliminary Calculations
Summary. We prove technical lemmata for the exhibition of Hichert’s algorithm to be introduced in section 31.3.3. –Σ– We fix a marked dichotomy ∆ = (X/Y ). Let us first come back to the rigidity property of X[ε]: Lemma 44 The symmetry group of X[ε] is Sym(X[ε]) = eZ12 . Proof. Clearly eZ12 ⊂ Sym(X[ε]). Therefore, ez+ε.t(u+ε.v X[ε] = X[ε] iff eε.t(u+ε.v) X[ε] = X[ε]. This implies t + vz + uk ∈ X for all z ∈ Z12 , k ∈ X. Whence t = 0, u = 1 since X is rigid. Therefore, for z = 1, we imply v + k ∈ X, all k ∈ X, i.e., v = 0. QED.
650
CHAPTER 31. MODELING COUNTERPOINT BY LOCAL SYMMETRIES
Proposition 53 Let H = eε.Z12 .GL(Z12 [ε]). Then X[ε] is H-rigid, i.e., the orbit application Z
H → ObLoc0 12
[ε]
: g 7→ g.X[ε]
is injective, i.e., a cadence of the group H. Exercise 69 Show that each consonant part Xx = x + ε.X of the tangent space Ix is rigid. Lemma 45 With the notation of corollary 53, if g = et (u + ε.v) ∈ H, and if z ∈ Z12 , we set g (z) = g.eε.vz Then we have (i) (g z1 ) )z2 = g z1 +z2 , (ii) ez .g.X[ε] = g (z) .X[ε], and ez .g.Y [ε] = g (z) .Y [ε]. Proof. The first formula is clear. The second follows from ez g = ez+ε.t (u + ε.v) = eε.t (u + ε.v).ez(u−ε.v) = g (z) .ezu . QED. −→ Corollary 20 For g ∈ GL(Z12 [ε]), there is a symmetry h ∈ H such that g.X[ε] = h.X[ε]. In fact, there is a u ∈ H such that g = ez .u, so by lemma 45, g.X[ε] = u(z) .X[ε], and we have the solution h = u(z) . −→ ∼ Lemma 46 Let ξ = x+ε.k, g ∈ GL(Z12 [ε]), and z ∈ Z12 . Then, if ξ 6∈ g.X[ε] and px∆ : g.X[ε] → ∼ z+x z z z z z g.Y [ε], we also have e ξ 6∈ e .g.X[ε], p∆ : e .g.X[ε] → e .g.Y [ε], and e .g.X[ε] ∩ X[ε] = ez .(g.X[ε] ∩ X[ε]), in particular card(g.X[ε] ∩ X[ε]) = card(ez .g.X[ε] ∩ X[ε]). Proof. It is clear that ez ξ 6∈ ez .g.X[ε], while by lemma 51, z z x −z z pz+x .e .g.X[ε] = ez .px∆ .g.X[ε] = ez .g.Y [ε], ∆ .e .g.X[ε] = e .p∆ .e
and finally, ez .g.X[ε] ∩ X[ε] = ez .g.X[ε] ∩ ez .X[ε] = ez .(g.X[ε] ∩ X[ε]). QED. Proposition 54 The contrapuntal symmetries can be calculated if one knows the contrapuntal symmetries g ∈ H at cantus firmus x = 0. More precisely, if ξ = x + ε.k ∈ X[ε] and if g is any symmetry, such that properties (i) through (iii) of g in definition 95 are true, then they are also true with unchanged set g.X[ε] = h.X[ε] for a symmetry h ∈ H. Furthermore, to check this property for h, we may verify the properties (i) through (iii) for the interval ε.k, the symmetry (0) h(−x) ∈ H, and the polarity p∆ . Finally, the intersection h.X[ε] ∩ X[ε] coincides with the translate ex (h(−x) .X[ε] ∩ X[ε]), which means that we may just look for relative cantus firmus steps when building the rules of admitted steps. Proof. The replacement of g by h follows from corollary 20. By lemma 45 we have e−x .h.X[ε] = h(−x) .X[ε], and by lemma 46, with z = −x, we can verify the contrapuntality of h on e−x ξ = ε.k, (−x+x) (0) and on p∆ = p∆ . The last statement follows from the translation formula for intersections in lemma 45. QED.
31.3. THE COUNTERPOINT THEOREM
31.3.2
651
Two Lemmata on Cardinalities of Intersections
Lemma 47 Let (X/Y ) be a strong dichotomy in 0@IntM od12,1 , and 0 ≤ i ≤ 6 and integer. −→ Setting Gi = {g ∈ GL(Z12 )| card(g.X ∩ X) = i}, we have card(Gi ) = card(G6−i ). Proof. Let p be the polarity of (X/Y ), the right multiplication with p induces a permutation −→ ∼ −→ of order 2 ?p : GL(Z12 ) → GL(Z12 ). For g ∈ Gi , consider the dichotomy g(X/Y ) and its intersection with X. This gives X = g.X ∩ X ∪ g.p.X ∩ X, whence card(g.p.X ∩ X) = 6 − card(g.X ∩ X). Therefore, g.p ∈ G6−i , and by the order two permutation ?p and the finiteness of all the involved sets, we have p.Gi = G6−i . QED. The next lemma basically guarantees the existence of admitted contrapuntal successor intervals, as we shall see in the next section. ∼
Lemma 48 Let K be a zero-addressed objective local composition in a finite cyclic group M → Zn and let U ∈ GL(M ). Then X
card(em .U (K) ∩ K) = k 2 .
m∈M
Proof. Let U (K) = {u1 , . . . uk } the image set with its k = card(K) elements. Let r be a generator of M . Then X card(em .U (K) ∩ K) m∈M
=
n−1 X
card(etr .U (K) ∩ K)
t=0
=
n−1 k XX
card({etr us } ∩ K)
t=0 s=1
=
n−1 k XX
χK (etr us )
t=0 s=1 k n−1 X X = ( χK (etr us )) s=1 t=0
with the characteristic function χK for an element being in K or not. Since r is a generator of Pn−1 M , the expression t=0 χK (etr us ) adds up to k, and we have the result. QED.
31.3.3
An Algorithm for Exhibiting the Contrapuntal Symmetries
Summary. This section discusses Hichert’s algorithm for the calculation of all contrapuntal symmetries and admitted contrapuntal steps by use of a specific software. –Σ–
652
CHAPTER 31. MODELING COUNTERPOINT BY LOCAL SYMMETRIES
This section restates the three conditions (i) through (iii) of the contrapuntal symmetries in order to provide an algorithm for the calculation of all admitted contrapuntal interval successors. Given the strong dichotomy ∆ = (X/Y ), and according to proposition 54, we may start from an interval ξ = ε.k, k ∈ X and restrict to symmetries g = eε.t (u + ε.uv) ∈ H (uv instead of v without restriction). Let us first reformulate conditions (i) and (ii). We have g.X[ε]
=
[
g.(x + ε.X)
x∈Z12
=
[
ux + ε.(uvx + t) + ε.uX
x∈Z12
=
[
y + ε.(vy + t) + ε.uX
y∈Z12
=
[
y + ε.(evy+t u.X).
y∈Z12
Setting h(y) = evy+t u, we have [
g.X[ε] =
y + ε.h(y).X.
y∈Z12
Therefore, we have g.X[ε] ∩ X[ε] =
[
y + ε.(h(y).X ∩ X).
y∈Z12
This means that condition (i) is equivalent to k 6∈ h(0).X which means k ∈ h(0).p.X, and this is equivalent to ∃s ∈ X such that k = h(0).p(s) = u.p(s) + t. This is equivalent to the statement that there is an s ∈ X such that [ g.X[ε] = y + ε.(evy+k−u.p(s) u.X)
(31.1)
y∈Z12
so that we have card(g.X[ε] ∩ X[ε]) =
X
card(evy+k−u.p(s) u.X ∩ X).
(31.2)
y∈Z12
Further, condition (ii) means p0∆ .g = g.p0∆ , see the remarks after definition 95. If we have p = er .w, this means that wt + r = ur + t. (31.3) Therefore, conditions (i) and (ii) are equivalent to equations 31.1 and 31.3, whereas the maximal number is calculated upon formula 31.2. In order to calculate the number 31.2, we have to distinguish three cases concerning the values of v:
31.3. THE COUNTERPOINT THEOREM
653
1. v is invertible. Then we have card(g.X[ε] ∩ X[ε]) =
X
card(ey .u.X ∩ X).
(31.4)
y∈Z12
2. v = 0. This gives card(g.X[ε] ∩ X[ε]) = 12card(ek−u.p(s) .u.X ∩ X).
(31.5)
3. v = ±ρ, ρ ∈ {2, 3, 4, 6}. This gives 12/v
card(g.X[ε] ∩ X[ε]) = v
X
card(e(j−1)v+k−u.p(s) .u.X ∩ X).
(31.6)
j=1
If we recall lemma 15, the first case with formula 31.4 implies: Fact 15 There are always at least 36 successors for a fixed given interval ξ ∈ X[ε]. We are now ready to state the algorithm which was implemented on Turbo-Pascal by Hichert [223]. The algorithm starts from the fixed value k ∈ X and first calculates all the possible coefficients of g. This means that we have to go through the loop which transgresses all g = eε.t (u + ε.uv) via u ∈ {1, 5, 7, 11}, s ∈ X, v ∈ Z12 , t = k − up(s), and then for each such value set calculate card(g.X[ε] ∩ X[ε]) according to the three cases 31.4, 31.5, 31.6. For each case, we update the set Ck of intermediate candidates for contrapuntal symmetries, i.e., we add a new g to the existing set if its intersection number is maximal among the already given candidates in Ck , and we remove all previous g where intersection cardinalities are smaller than the actual maximum. Exercise 70 Write a C or Java program which implements the above algorithm. If you do not speak C or Java, write a Mathematica or Maple program. If you do not speak these languages either, interrupt reading this book, learn one of these languages, and proceed. The complete lists of all contrapuntal symmetries, together with their intersection configurations, can be found in appendix O.1. The admitted successors will be listed below. This information yields the following Theorem 33 Let ∆ = (X/Y ) be a strong dichotomy, and let ξ ∈ X[ε]. The number of admitted successors of ξ is always at least equal to 36, in particular, there exists always a contrapuntal symmetry g for ξ. An admitted successor of ξ also always exists if one prescribes the cantus firmus of the successor interval.—For each of the six strong dichotomy classes, a list of forbidden successor intervals is exposed below, after this theorem. We shall say that an interval ξ ∈ X is a “cul-de-sac” under determined conditions if there is no admitted successor under these conditions. The following table is meant as follows: Each table section is related to a fixed representative (X/Y ) of the indicated class. The first column indicates the interval quantity k of
654
CHAPTER 31. MODELING COUNTERPOINT BY LOCAL SYMMETRIES
the “consonant” start interval ξ = x + ε.k ∈ X[ε]. The obstructions to successor interval η = y + ε.l ∈ X[ε] are visible in columns 2–13 where the difference d = y − x = 0, 1, 2, . . . 11 leads the column in the first row. For each couple k, d, we see the forbidden interval quantities l of the target interval. For example, in class 64, the steps x + ε.4 7→ x + 6 + ε.l is forbidden for l = 2, 4, 7, 9 and admitted for all other l.
k
0
1
2
3
4
5
6
7
8
9
10
11
-
Forbidden Successors for Dichotomy Class Nr. 64 2
2
-
2
-
2
-
2
-
2
-
2
4
2,4,7,9
-
2,4,7,9
-
2,4,7,9
-
2,4,7,9
-
2,4,7,9
-
2,4,7,9
-
5
5,11
5,11
5,11
5,11
5,11
5,11
5,11
5,11
5,11
5,11
5,11
5,11
7
2,4,7,9
-
2,4,7,9
-
2,4,7,9
-
2,4,7,9
-
2,4,7,9
-
2,4,7,9
-
9
9
-
9
-
9
-
9
-
9
-
9
-
11
11
11
11
11
11
11
11
11
11
11
11
11
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
-
1
-
1
-
1
-
1
-
1
-
2
2,8
2,8
2,8
2,8
2,8
2,8
2,8
2,8
2,8
2,8
2,8
2,8
3
1,3,5
-
1,3,5
-
1,3,5
-
1,3,5
-
1,3,5
-
1,3,5
-
5
5
-
5
-
5
-
5
-
5
-
5
-
8
8
8
8
8
8
8
8
8
8
8
8
8
0
0,3
3
0,3
3
0,3
3
0,3
3
0,3
3
0,3
3
1
1,2,7
-
1,2,7
-
1,2,7
-
1,2,7
-
1,2,7
-
1,2,7
-
2
2,3
-
2,3
-
2,3
-
2,3
-
2,3
-
2,3
-
3
2,3
-
2,3
-
2,3
-
2,3
-
2,3
-
2,3
-
6
3,6
3
3,6
3
3,6
3
3,6
3
3,6
3
3,6
3
7
1,2,7
-
1,2,7
-
1,2,7
-
1,2,7
-
1,2,7
-
1,2,7
-
Forbidden Successors for Dichotomy Class Nr. 68
Forbidden Successors for Dichotomy Class Nr. 71
Forbidden Successors for Dichotomy Class Nr. 75 0
0
-
-
-
-
-
0
-
-
-
-
-
1
0,1,4,5
-
0,1,4,5
-
0,1,4,5
-
0,1,4,5
-
0,1,4,5
-
0,1,4,5
-
2
2,5
-
-
2,5
-
-
2,5
-
-
2,5
-
-
4
0,1,4,5
-
0,1,4,5
-
0,1,4,5
-
0,1,4,5
-
0,1,4,5
-
0,1,4,5
-
5
2,5
-
-
2,5
-
-
2,5
-
-
2,5
-
-
8
2,8
2
2
2,8
2
2
2,8
2
2
2,8
2
2
Forbidden Successors for Dichotomy Class Nr. 78 0
0,2,6
-
0,2,6
-
0,2,6
-
0,2,6
-
0,2,6
-
0,2,6
-
1
1,2
-
1,2
-
1,2
-
1,2
-
1,2
-
1,2
-
2
1,2
-
1,2
-
1,2
-
1,2
-
1,2
-
1,2
-
4
1,4
-
-
1,4
-
-
1,4
-
-
1,4
-
-
6
0,2,6
-
0,2,6
-
0,2,6
-
0,2,6
-
0,2,6
-
0,2,6
-
10
1,10
-
1
1,4
1
-
1,10
-
1
1,4
1
-
Forbidden Successors for Dichotomy Class Nr. 82 0
0
-
-
-
-
-
0
-
-
-
-
-
3
3,9
-
-
3,9
-
-
3,9
-
-
3,9
-
-
4
0,4,8
-
0,4,8
-
0,4,8
-
0,4,8
-
0,4,8
-
0,4,8
-
7
7
7
7
7
7
7
7
7
7
7
7
7
8
8
-
-
-
-
-
8
-
-
-
-
-
9
3,9
-
-
3,9
-
-
3,9
-
-
3,9
-
-
31.4. THE CLASSICAL CASE: CONSONANCES AND DISSONANCES
31.3.4
655
Transfer of the Counterpoint Rules to General Representatives of Strong Dichotomies
Summary. The above counterpoint theorem was made explicit for one selected representative of a strong dichotomy. Following [223], we give rules to transfer these results and tables to arbitrary representatives of strong dichotomies. –Σ– Let ∆ = (X/Y ) be a strong dichotomy with polarity p∆ . Take an interval ξ = ε.k ∈ X[ε], and let R ⊂ H be the set of contrapuntal symmetries of ∆ and ξ. Take any symmetry g = −→ et u ∈ GL(Z12 ), set gε = eε.t u, and consider the transformed dichotomy (L/M ) = g.∆. We have gε .X[ε] = L[ε], gε .Y [ε] = M [ε]. Proposition 55 [223, Satz 3.2] With the above notation, the conjugate set Rg = gε .R.gε−1 is exactly the set of contrapuntal symmetries of g.∆ and ξg = gε ξ. The number of admitted successors of ξ and of ξg coincide. Attention: In general, the fact that η is a successor of ξ does not imply that gε η is a successor of gε ξ! This result can be used to transform the tables of forbidden successors as given in the table after the counterpoint theorem 33 for other representatives of the dichotomy classes. To this end, suppose that in such a table (for dichotomy representative δ = (X/Y )), the couple ξ = ε.k, η = b + ε.j is forbidden. This means that in our table, on the location of row with interval quantity k and column with interval quantity b, the coefficient j appears as a forbidden quantity. According to proposition 55, in the transformed table, the row g.k and the column g.b must show a forbidden quantity g.j. So we have a recipe for transforming a table for ∆ under a symmetry g = et u: 1. Permute the 12 columns of the given table by a multiplication of the column head numbers 0, 1, 2, 3, . . . 11 by u (mod 12) and rearrange the new numbers by increasing values. 2. Replace the leading column interval numbers k by g.k and rearrange the corresponding 6 rows by increasing values of the leading numbers. 3. Replace each forbidden item j by the item g.j.
31.4
The Classical Case: Consonances and Dissonances
Summary. This section deals with the classical case of the counterpoint theorem for the consonance-dissonance dichotomy of Palestrina–Fux Theory. –Σ– We give a specialized counterpoint theorem for the consonance-dissonance dichotomy, the sweeping orientation, and relating to ecclesiastical modes as defined in section 13.4.2.
656
CHAPTER 31. MODELING COUNTERPOINT BY LOCAL SYMMETRIES
Theorem 34 Let ∆ = (K/D) be the consonance-dissonance dichotomy, and let ξ ∈ K[ε]. The number of admitted successors of ξ is always at least equal to 36, in particular, there exists always a contrapuntal symmetry g for ξ. An admitted successor of ξ also always exists if one prescribes the cantus firmus of the successor interval.—If one restricts the admitted pitch classes to an ecclesiastical mode (see section 13.4.2), then an admitted successor of ξ always exists, even if the cantus firmus is prescribed. So, under these conditions, there is no cul-de-sac. Parallels of fifth (x + ε.7 7→ y + ε.7) are generally forbidden. For all other parallels, no general obstruction exists. The admitted relative progressions are listed in the table after theorem 33, Class Nr. 82, whereas the progressions for the C-major scale are listed in appendix O.2. We shall discuss below the relation between strong dichotomies and scales, in particular the appearance of culs-de-sac.
31.4.1
Discussion of the Counterpoint Theorem in the Light of Reduced Strict Style
Summary. The concluding section gives an overview of the (strong) commonalities with and (weak) diversities from the classical rules of the Palestrina—Fux system and its reduction to pitch classes. –Σ– 287 cases
54 inadmissable, 21 of them forbidden (black)
alltogether 37 forbidden
Figure 31.2: Out of the 287 possible progressions with a mode (modulo translations of the cantus firmus), the reduced strict style exhibits 54 inadmissible cases. According to the mathematical model of counterpoint, 37 progressions are not admitted. Out of these, 21 cases are inadmissible in Fux’ sense. intuitively, the commonalities of the two approaches can be described by use of a probabilistic argument: If somebody tries to hit at least 21 of the 54 inadmissible cases of the reduced strict style without knowing anything about counterpoint by 37 trials, the chance is less than 2.10−8 . In this discussion, we refer to the codification of the strict style by Fux [174] (see also [528]) since its explicit and rigorous rule system is particularly useful for a qualitative and quantitative comparison of the mathematical model with the classical counterpoint rules. In order to establish a basis of comparison for the mathematical model, we first have to transfer
31.4. THE CLASSICAL CASE: CONSONANCES AND DISSONANCES
657
the Fux rules to the interval space IntM od12,1 . This yields a model of Fux’ rules ‘modulo octave’, a rule system which we call the reduced strict style. The upshot of a detailed investigation [342]2 states this: Fact 16 In the reduced strict style, only the rule of forbidden fifth parallels and the tritone rules have an unrestricted validity. Within an ecclesiastical mode, there are 287 a priori possible progressions [342]. According to the consonance-dissonance counterpoint theorem 34, 37 of them are forbidden. Among them, 21 coincide with the 54 Fux-inadmissible progressions. The remaining 16 forbidden progressions of the mathematical model deal with progressions which are bad or allowed by Fux. Out of the ten allowed ones, four concern tritone movements of the cantus firmus. Three of the remaining obstructions (in the mathematical model) concern progressions from the major third into the prime, from which one may lead to an “ottava battuta”. The remaining three obstructions deal with progressions which leave unaltered the pitch material, see also figure 31.2. We should stress that the mathematical model does not formalize a switch between perfect and imperfect consonances. The concrete shape of a polarization which induces a determined progression is redefined for each individual progression. Further, the model can be applied to any scale. This unveils an interesting fact concerning the dominant role of the major scale: In the analysis for the three seven-element scale classes which consist exclusively of minor and d major, 47.1: d melodic minor, and c major second steps (classes 38.1: 62: whole-tone scale, extended by one pitch class) the major scale is by far optimal for the degree of freedom in the choice of admitted successors. There is no cul-de-sac. Only for two progressions with prescribed start interval and cantus firmus progression is the successor uniquely determined. The melodic minor scale has less successor freedom, but there are no culs-de-sac. In 16 cases there is only one possible solution. In the extended whole-tone scale, there are 18 culs-de-sac. The freedom of choice is minimal. We shall discuss this item in more generality in section 31.4.2.
31.4.2
The Major Dichotomy—A Cultural Antipode?
Summary. Among the six strong dichotomy classes, we look for others than the Fux dichotomy and discuss their relative positions. Among the possible alternatives we especially focus on the major dichotomy, a topological antipode to the Fux dichotomy. The possibility to associate the major dichotomy with classical Indian scales is discussed. –Σ– In his thesis [223], Hichert has observed a number of interesting topological properties of the six strong dichotomies. Here is a representative record of these observations: The major and consonance-dissonance dichotomies are not only polar with respect to diameter (definition 93 in section 30.1) and span (definition 94 in section 30.1), see figure 30.1. They are also polar with regard to their number of contrapuntal symmetries and interdictions, see figure 31.3. In other words: 2 The paper [342] was accepted for publication in the journal Musiktheorie, but never published for marketing reasons (!), an anecdote about German musicology which shares a particular flavor.
658
CHAPTER 31. MODELING COUNTERPOINT BY LOCAL SYMMETRIES contrapuntal symmetries
11
82
9
78
6
75
71
68
4
64
50
78
80
82
96
interdictions
Figure 31.3: From the proof of the counterpoint theorem, one deduces the numbers of contrapuntal symmetries and interdictions. These again position the major and consonance-dissonance dichotomies in a polar relation. Fact 17 The consonance-dissonance dichotomy has a maximum of contrapuntal symmetries and a minimum of interdictions, as opposed to the polar major dichotomy. We have already observed in section 31.4.1 that, among the ‘diatonic’ scales (only minor and major second steps), the major scale has a maximum of freedom of choice for the consonance-dissonance dichotomy. Conversely, the major scale has only culs-de-sac for the major dichotomy. So we are also interested in the scales with seven tones with respect to the major dichotomy! It is a further fact that among these scales with no cul-de-sac for the major dichotomy no European scale appears. There is one such scale, namely K ∗ = {0, 3, 4, 7, 8, 9, 11} of class c 60, which is very interesting. To begin with, it is the consonance set K plus an added ‘leading’ note 11. So if the major scale is good for consonances and dissonances, the major dichotomy is good for a scale which is intimately related to consonances! This scale really shows a character which relates to Indian raga music. The basic melakarta framework for ragas is known [103, Bd.8, p.265ff] to be built by the 72 melas. The small number of seven of these mela scales have been used until the present days. One of them, “mayamalavagaula”, N r.15 = {c, d[ , e, f, g, a[ , h}, which is class c 61 in our chord classification, can be represented by {0, 3, 4, 7, 8, 9, 1}. But this is very similar to the above scale K ∗ , we only have to switch the 11 to the 1, as opposed to 11 in 0. So we are led to the question of how far there is a polarity not only in mathematical relations between consonances-dissonance and the major scale as opposed to the major dichotomy and a ‘consonance-dissonance’ scale K ∗ which is akin to mela Nr.15 in raga music, but also a global polarity in musical cultures between European and Indian tradition. See also figure 31.4 for this polarization. This concluding discussion is an excellent example of the anthropic principle on the level of scale and interval interplay. Seemingly, the historical selection tends to optimize certain abstract
31.4. THE CLASSICAL CASE: CONSONANCES AND DISSONANCES
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strong dichotomies K/D
28
I/J
12
6 cul-de-sacs
7-scales K*...(7 items) EXOTIC
whole-tone mel. minor major +1 tone DIATONIC
Figure 31.4: Polarity between seven-tone scales and strong dichotomies shows a polarization between European and exotic (in particular: akin to Indian) scales. The size of darkened disks shows the number of progressions while the stars show culs-de-sac. a priori properties of topological and transformational character. The possibility to compare such different musical cultures as European and Indian traditions opens a wide field of comparative musicology (relative ethnology) which is based upon systematic results instead of historiographic and ethnographic contingencies.
Part VIII
Structure Theory of Performance
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Chapter 32
Local and Global Performance Transformations C’est l’ex´ecution du po`eme qui est le po`eme. Paul Val´ery Summary. Performance includes a non-trivial transformation from the mental reality of a score to the physical reality of acoustic and, in the limit, gestural realization. We discuss a model of local and global transformation structures and present a preliminary discourse on the need to “shape mental reality” in performance. Local performance transformation structures, together with their syntactic combination to global structures, are formally developed. They involve extrapolation from discrete to continuous or differentiable data; the latter are induced by use of spline techniques. We give a justification of such a procedure from the musical and mathematical points of view, in particular with stress on expressive coherence. –Σ– This part, structure theory of performance, is a turning point in the entire theory of the topos of music. In fact, the preceding parts dealt with general structure theory and then, on a more musicological and music-theoretic focus, with mental perspectives of rhythm, motives, harmony, and counterpoint. In contrast, performance is concerned with the transformation of mental structures into physical ones. This is what traditionally happens in a concert where human artists are performing on physical instruments, including all the richness of human expression on the gestural, emotional, or structural level of physiological, social, and physical parameters. Evidently, a comprehensive theory of performance is out of reach as long as major constituents have not even been attacked on a scientific level. For example, there is no deeper understanding of the emotional function of music, one knows some extremely elementary facts, for example those concerning the emotional impact of contrapuntal intervals on the emotional brain, as described in chapters 29, 30, and 31. Also on the level of instrumental factors in the quality of a performance, very little is known, for example, concerning the role of instrumental parameters in the communication of musical contents. Worse than that: There is not even a 663
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commonly accepted structure theory of performance, i.e., a theory which deals with the precise and general description of what is a performance in its most elementary shape. For example, the discussion of tempo has not yet been carried to a point of general acceptance of what tempo can be, including a fundamental disagreement on its hierarchical ramifications. One of these discussions could germinate around the distinction between mental (symbolic, logical, call it as you wish) time and physical time. Naively, mental time, as it is encountered on the score notation, looks like something discrete, to be encoded on integers, or some isomorphic submodule δ.Z of the real numbers, whereas physical time is parametrized by the full line of real numbers. This is also the point of view of Desain and Honing in [125]: They reduce mental time to discrete time intervals of a metrical structure, leaving the smooth part to the continuous time scales of tempo changes and expressive timing. This procedure is mathematically incorrect because • the metrical time is infinitely divisible in itself: No positive lower limit for mental durations has ever been envisaged, metrical time is a topologically dense, not a discrete set in the field of real numbers. Hence, any reasonable (more precisely: uniformly continuous) time function from mental time E to physical time e can uniquely be extended to a time function on the reals (see [261]). There is no conceptual reason to restrict metrical time to a discrete subdomain of the reals. • Tempo does not deal with something more continuous than metrical time. It is another concept1 : the inverse differential quotient of a function E 7→ e(E) between two copies of the real number axis with irreducibly different ontological specifications, namely the musical mental status of the score and the physical status of performed music. Other misunderstandings floating around in musicological environments [189] maintain that tempo is a locally constant function, i.e., a step function, much like the medieval theories on velocity in the spirit of Oresme (see also our discussion in chapter 4). He would decompose accelerated movements into a succession of uniform movements [485, contribution of Isabelle Stengers on Galileo Galilei]. In what follows, we have not tried to downsize the complexity of performance, it is indeed the most complex subject in musicology. It involves all kinds of considerations concerning the three basic realities of physics, mentality, and psychology. But beyond this ontological diversity, it is of a sophisticated structural nature, involving differential geometry, ODEs, and PDEs, and evidently all the music theory which is presupposed in any reasonable performance theory (though: not in everybody’s performance, but blurred gesticulations of musicians “playing their ass off”2 are not the subject of any performance theory). We shall however not discuss proper psychological aspects, but restrict ourselves to the mental and physical perspectives. For psychological concerns, refer to [272]. It may appear that we thereby omit an essential and basic point of view, and that such a restriction would 1 We
shall introduce and discuss such concepts in section 33.1.1. of free jazz saxophonist Werner L¨ udi to Cecil Taylor’s question:“What’s your concept?”
2 Answer
32.1. PERFORMANCE AS A REALITY SWITCH
665
hamper the entire discourse. This is true insofar as we omit an essential perspective. But it is false that this hampers the discourse. The performance discourse has to deal with structural descriptions of performance: What happens if a score is played in acoustical and gestural reality? This is beyond psychology. Also do the rationales of a performance not uniquely rely on the psychological reality (of emotions as stressed in the naive romantic approach), but, in the spirit of Theodor Wiesengrund Adorno or Walter Benjamin, on analytical facts or, in another approach, on gestural paradigms, as investigated by Johan Sundberg, for example. Performance research is only in its initial phase, in particular with respect to the high level of performing artists. But this is no reason to abbreviate the scientific path with the risk of misunderstanding the beauty and complexity of performance, and to fall into some crevasse of oversimplification. Hopefully, this should help the reader to follow chapters on performance theory with due patience.
32.1
Performance as a Reality Switch
Summary. There is a sharp dichotomy of realities, communication, and semantical levels between a score and its performances. We expose these facts and their consequences for a theory of performance versus mental music theory. –Σ– Performance is more than simply playing an instrument “all improviso”; even in free jazz (if it merits that name), musicians always refer to an inner score. The concept of a score is used here in its generic meaning, i.e., a generic score is any written, imagined or conceived scheme for the execution of a musical composition (see [361] for details). In European or Japanese classical music for example, a score is realized as a denotator structure of more or less complex form (see [378], for an explicit score form englobing classical Viennese composition). In improvised music, the scheme is less of a structural nature than of a processual one. The jazz musician, for example, follows action and reaction patterns and rules according to blues schemes and individual dictionaries of motivic, rhythmic, and harmonic elements. We propose this informal but reliable definition:
Definition 97 Performance is defined as the physical realization of an interpretation of a generic score. We leave the meaning of “interpretation” open in its common understanding, but we do include, and this is a core issue of the subsequent discourse, interpretation in the technical sense of interpretations of local compositions. We also stress the generic attribute “physical” which does include the acoustic realization, but does not exclude further performance parameters, such as the gestural dynamics of a performer. But we do, as already mentioned, exclude psychic parameters and therefore mean strictly physical performance. Notice that this subsumes intermediate technological strata as a special subsystem of physical realization.
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CHAPTER 32. LOCAL AND GLOBAL PERFORMANCE TRANSFORMATIONS
Why Do We Need Infinite Performance of the Same Piece?
Summary. This section deals with the a priori necessity of infinite performance. The argumentation relates to infinite analysis—due to the Yoneda philosophy—and to its communication on the rhetoric level of expressive performance. –Σ– The basic problem here is a very common one: Why do we need infinite performance? We are talking about real performances in real concerts. Do we, and why do we need new performances, again and again? Couldn’t it arrive that all possible or—at least—all relevant performances are definitively played at a given moment, and that all successive performances are doomed to the existence of superfluous variants of the arsenal of core performances. Seemingly, a musical composition, as it is fixed in a score and an associated denotator, is a finite object. So it would be a logical consequence of this finiteness that infinity is not inscripted in a musical composition and its performance. We therefore have two questions: Is there a substantial infinity in the interpretative variety of a given score? And, if this is the case, is there an associated infinity of performances, and why should such varieties have a sense for the listener or for the artist? The first question is easily answered: Yes, there is an infinity of interpretations of a given finite score denotator. This was already shown in the discussion of iterated interpretations in section 13.4.2. This is an affirmative answer on the mental level, and one may easily add an infinity of evaluations of any such finite or infinite interpretation, for example on the level of rhythmic, motivic, or harmonic analyses as described previously in the respective chapters. One may also enrich the possibilities of viewing a given interpretation by a variation of the address, a technique which evokes Yoneda’s lemma, of course, but which is also very concretely developed in the context of harmonic topologies. We shall see later in chapter 44.7 on performance operators that the analytical propaedeutics to performance also includes an important class of analyses: the analytical weights, i.e., numerical functions associated with more abstract analyses (such as harmonic topologies). This type of numerical evaluation which was already introduced in the power series λw0 of section 13.4.2 is not just a “boiled-down” version of ‘serious analysis’, it is just another way of looking at complex configurations. The point is that the categories of ‘understanding’ a (finite) musical composition are of an infinite character in many respects, and that there is no reason to claim complete understanding by any finitistic argument. So the expression of mental content relies upon an infinite arsenal: Fact 18 Performance conveys an infinite message. But infinity in performance also comes in from a second point of view: Performance is inevitably an experiment in understanding or advancing comprehension. The realization of a piece of music in physical spaces creates a view, a flight through a virtual landscape which may reveal new insights to the involved actors, insights which are not anticipated but emerge from a particular visual angle of a determined ‘performance flight’. Such an experience is a mental
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experiment as it has been described in chapter 4, and as such it is a creative investigation, not only a reproductive activity for the sake of social coherence and psycho-hygiene. In this sense, performance need not be agreeable or pleasant—recognizing new aspects within an infinite repertory of analytical structures can be painful, but healthy. Glenn Gould’s work is a brilliant piece of history in this investigative field of musical performance. Even his most awkward performances of Beethoven’s piano music (such as the strange, cranky performance of the “Hammerklavier” sonata op. 106 or the funny and blasphemic review of the “Appassionata” sonata op. 57) are masterpieces in revealing new aspects of compositions, be it solely to learn how not to play them...
32.3
Local Structure
Summary. Just as with the analytical work, the performance task is also composed of local subtasks which constitute the morphemes of performance transformations. This section describes the emergence of such units and their structure. –Σ– In a first approximation, performance p could be seen as a set map which associates with every element X of a local composition (in a score-related form) a physical event, encoded by an element x = p(X) in a second local composition whose parameters pertain to a form of physical signification. The coherence of such a performance map is however very rarely a global one. For example, the tempi may change instantaneously, by the indication istesso tempo, forcing the artist to restart from the initial tempo after a deformation via a sequence of agogical indications. Or else, left and right hands may follow their separate tempi—except for some meeting points where the onsets should coincide (in Chopin rubato, for example). In orchestral music, the tuning is a function of the instrument, although everything is written in the same orchestra score. Or it may happen that the local performance of an ornament (a complicated trill, say) is an autonomous shaping map, operating independently of the map on the events which neighbor the ornament. We therefore observe a global structure, a patchwork of local chart maps, much like morphisms on global compositions.
32.3.1
The Coherence of Local Performance Transformations
Summary. Arguments for building local units—based on local compositions—of performance transformations are given. The idea of making performance on local units coherent is explicated: The mapping on the unit’s domain is defined by a coherent rule, typically described by continuously differentiable transformations on connected neighborhoods of the local unit. –Σ– In the theory of local and global compositions, the definition of morphisms was roughly this: You are given a set map between two sets of music objects and you want to express that this map is not just any set-theoretic map, but shares a type of coherence. This is done by the extensibility of such a set map to the ambient space by an affine space map. This is one
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way of expressing coherence between the images of different points of the local composition domain: they are mapped under a formula of affine type. This kind of coherence is also required in classical algebraic geometry when defining morphisms between algebraic sets. For performance maps, the narrow affine formalism and also the formalism of polynomial maps are both too algebraic in nature. Let us give some arguments to make this clear. If we want to model a fermata tempo shape, we can evidently not restrain to linear expressions since the fermata tempo must decrease in the beginning and increase towards the end of the fermata duration. Moreover, the cognitive rules of logarithmic perception of differences3 would enforce changes in tempo in the sense that the change of physical onset as a function of mental onset, i.e., de/dE at onset E proportional to the actual value of physical onset e, i.e., de/dE ∼ e which means that dE/de ∼ 1/e and therefore E ∼ ln(e), i.e., e ∼ exp(E) so we cannot use polynomials for such a shaping of physical onset. The general shape of physical parameters against mental ones is also strongly documented in glissando and crescendo effects, i.e., changes of pitch h or l as functions of mental time E. These two effects are also a strong argument for continuous mental time: Glissando and crescendo must be defined at each moment of a continuous time parameter, otherwise they cannot be realized, not more than the continuous (even differentiable) movement of the conductor’s baton. Of course, such general function requirements could be worked around by gluing of polynomial functions to polynomial spline functions. But this is exactly what we do not want here: We want a unique class of functional expressions for the local coherence. It is this principle: What we can achieve by gluing together local pieces should be left to the global theory instead of hiding the gluing technique on the level of function classes!
32.3.2
Differential Morphisms of Local Compositions
Summary. This section describes the categories of local compositions with differential morphisms on real vector spaces. This formalism captures unreflected prerequisites of musicological approaches on performance instances, such as tempo or tuning. –Σ– Before constructing the categories of differential morphisms, we should specify their objects. As the existing performance theory is only developed for the zero address and for objective compositions, we shall only look at such objects. In general, score denotators live in complex form spaces, including components which are far from being performable, such as bar-lines or pauses. We will not consider these full-fledged structures, but only local compositions which consist of objects that are candidates for events to be performed in physical spaces. Such local compositions live in space forms which are limits (products are sufficient here), whose factor spaces are submodules of real vector spaces. For example, pitch can be encoded by integers (such as is the case for MIDI key numbers), or by tuples of rationals (as is the case with the Euler spaces built from (just) octaves, fifths, thirds, sevenths etc.), whereas onset is encode by a module Z.x or Z[1/2] or Q. Dynamics is usually encoded by integers (such as suggested by the MIDI key velocity numbers 0, 1, 2, . . . 127. With these usual values, one may consider local compositions (K, M ) with M a submodule of Rn . The same observations are valid 3 See
also appendix A.2.2.
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669
when representing local compositions which are related to physical parameter space forms. We shall therefore, without loss of generality, assume that we are given local compositions (K, S) in simple space forms S −→ Simple(Rn ) of modules4 Rn , n ∈ N. Later, we shall specify the Id
form names when more concrete parameters will be discussed. Let us first explain the general construction of tangent compositions and their morphisms:
Definition 98 Given an objective, zero-addressed commutative local composition (K, S) in a simple ambient space S −→ Simple(M ) with R-module M , then the tangent composition T K Id
of K is the composition (K × R.K, S 2 ), where S t −→ Limit(S, . . . S) is the t-fold product. Id
The local composition K is called the basis of T K. For a natural number t, the t-fold tangent composition T t K is the tangent composition T (T t−1 K) = K × R.K × (R.K × R.K)t−1 ) ⊂ S 2t , with special value T 0 K = K. The tangent space Tk K of K at point k ∈ K is the subset {k} × R.K, identified with R.K for its module structure. Given two tangent compositions T K of K and T L of L (both compositions over the same ring R). A tangent morphism T f : T K → T L is a set map T f : T K → T L between local compositions T K, T L which factors through the canonical projections pK : T K → K, pL : T L → L and a (necessarily uniquely determined) map f : K → L, and such that all the fiber maps T fk : Tk K → Tf (k) L are linear. For 1 < t, a t-fold tangent morphism T t f : T t K → T t L is a tangent morphism T (T t−1 K) → T (T t−1 L) whose basis map is a (t − 1)-fold tangent morphism. The obvious category of t-fold tangent compositions and t-fold tangent morphisms is denoted by TantR . Lemma 49 Let ComLoc0R be the category of commutative objective local compositions over the commutative ring R. Let f : (K, S) → (L, U ) be a morphism in 1 ComLoc0R . Then the map f 7→ T f = f × R.f : T K → T L defines an injective natural transformation ComLoc0R → Tan1R onto the subcategory of those tangent morphisms T f : T K → T L between tangent compositions such that the fiber maps T fk are all the same linear map T , and we have f (k1 ) − f (k2 ) = T (k1 − k2 ) for all couples k1 , k2 ∈ K, i.e., the map f is defined via T and the value on one single point. The proof of this lemma is left to the reader. For the real number field R = R, we are not only interested in the tangent categories, but in those morphisms which extend to differentiable maps on the underlying vector space, or, more generally in maps which extend to any maps of a specific category Cat: Definition 99 A t-fold tangent morphism T t f : T t K → T t L of t-fold tangent compositions with positive t in TantR is said to be t-fold differentiable iff there is a t-fold differentiable map F between the underlying vector spaces such that T t F |T t K = T r f . For 0 ≤ t, Such a map is said to be C t iff it may be extended to a C t map F between the underlying vector spaces. 4 The case n = ∞, the countable direct sum of copies of R for possible Fourier coefficients or similar parameters in sound color spaces will not be considered except for some explicitly described special considerations
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Clearly, for positive t, the t-fold differentiable and the C t morphisms, respectively, define subcategories LocDifft and LocC t of TantR , respectively. More generally, if Cat is a category of morphisms on real vector spaces, we denote by TantCat the category of t-fold tangent morphisms which extend to morphisms from Cat. In practice, this mathematical catechism is not very practical, we shall rather use the wording of “an extension of a given t-fold tangent morphism to a morphism of the respective category Cat”. Example 51 In traditional performance research, tempo is an important feature. It is usually described via M¨ alzel’s metronomic tempo indication (M.M.) of the type “x quarters per minute”. This means that we compare mental time E (quarters) to physical time e (minutes) via the quotient ∆E/∆e as a function of E. Forgetting about the other parameters for the sake of simplicity, we have a (not necessarily finite) local composition of (real-valued) mental onsets M yOnsets : 0 OS({E1 , . . . Ei , . . .}) with form OS −→ Power(Onset) (see list of mental forms in formula Id
6.69 in section 6.6). Suppose that we have a performance set map p associating each Ei with a physical onset p(Ei ) = ei of a denotator M yP hysOnsets : 0 P hysOS({e1 , . . . ei , . . .}) in the form P hysOS −→ Power(P hysOnset) Id
over the physical onset form P hysOnset −→ Simple(R) (see 6.6). Suppose that the tempo Id
indications are given on M yOnsets by the values Ti at the onsets Ei . This data defines a tangent morphism T p : M yOnsets → M yP hysOnsets whose linear fibers over Ei are the linear maps E 7→ Ti .E For the other onsets, tempo is not declared. This is the usual situation in music scores. However, it is not clear what should be the tempo in between the indicated onsets. The point is that one would like to turn the tangent morphism into a differential morphism, at least into a piecewise differential morphism, ie., one that extends to a piecewise differentiable function having the derivatives Ti as required at the arguments Ei . Some musicologists even do not agree with this extensibility requirement, but maintain that tempo is not defined except at the given arguments. We have already countered that position, however, if tempo has to be extended in the sense that we ask for a differentiable morphism giving rise to the present tangent morphism, there are many ways to do so. For example, tempo could be set to a piecewise constant function, giving rise to a step tempo curve which integrates to a piecewise linear map P that extends p. As to the mental contents of a score, this question could be viewed as secondary, but if we agree with Paul Val´ery in saying that c’est l’ex´ecution du po`eme qui est le po`eme (see the catchword of this chapter), then the total content of a composition must include the extension of p to P . It should also be stressed that the European understanding of musical time is a type of negative account, time is only interesting if it is over, i.e., the time between two successive notes is non-existent, or, at least, of no existential relevance. Tempo does not exist between two notes since there is no time feeling, performed music has no time except when a note onset intervenes. This is a severe lack of understanding of what musically happens. Such a music understanding is poor and proves a negative, immature relation to time. Positively stated, European time is more a kind of trigger, a Turing machine unit slot transporting the logical processes (typically in harmony).
32.3. LOCAL STRUCTURE
671
It is the merit of performance research, in particular the Swedish school of Johan Sundberg and his collaborators, to have pointed at this delicate time question. We shall come back to the extension type problem for tangent morphisms in chapter 36 on expressive performance. Exercise 71 Let T f : T K → T L be a tangent morphism of Tan1R over a basis K = {x, y} of cardinality 2 in R. Then T f is extensible by exactly one 1-fold differential morphism whose coordinate functions Pi (t) are polynomials in t of degree three. Moreover, these polynomials are rational functions in x, y, fi (x), fi (y), t which are polynomials for fixed x, y. If T fx = T fy = 0, we have min(fi (x), fi (y)) ≤ Pi (t) ≤ max(fi (x), fi (y)) for x ≤ t ≤ y. Exercise 72 Let p : K → L be a set map of local compositions with card(K) = m + 1, then there is a polynomial morphism extension P of p whose coordinate functions have degree ≤ m. (Hint: Find a straight line through the origin and not parallel to any of the finitely many lines connecting pairs of points of K. Take the projection q of K into the orthogonal space to this line. This is a bijection onto the image q(K). Within this orthogonal space, repeat the same procedure until you have projected K into a one-dimensional subspace, call K ∗ the image of K in this subspace. Then, looking at the map p∗ : K ∗ → L induced by p on this one-dimensional projection of K, the claim is a classical result of polynomial interpolation.) Exercise 73 If the points of K in exercise 72 are in general position, then p is a morphism of local compositions and therefore automatically t-fold differentiable for every t. 32.3.2.1
A Recursive Interpolation Algorithm
The next result is related to what we shall call analytical weights, a key structure in the theory of performance operators. But it pertains to the theory of differential morphisms, so we present it right here. Since the essential statement is a recursive algorithm which is of essential use in programming performance theory, we do not present the result as a lemma but as a construction. The situation is this: We are given a finite local composition in a simple ambient space S −→ Simple(Rn ), consisting of points K = {x1 , . . . xs }, and contained in the open n-cube Id
C n =]u1 , v1 [× . . .]un , vn [ of Rn . For m = 1, . . . n, we denote by C m =]u1 , v1 [× . . .]um , vm [ the projection onto the first m coordinates, and by Cm =]um , vm [ the projection onto the m-th coordinate. On K, we are given a tangent morphism T f : T K → T L, where the codomain tangent composition T L sits over a local composition L in an ambient space W −→ Simple(R) of Id
“weights”, and where the linear components T fk vanish for every k ∈ K. We want to find a C 1 extension of T f which adds no extra extrema to the extrema required per definition on the K points, and which evaluates to the constant value 1 on the complement of the frame cube C n . The meaning of this requirement is that we want to extend the “discrete weight” f to a function F on the entire space S such that the extension is C 1 , normalizes to the constant value 1 when the arguments tend to infinity, and has values very close to the given values in a small neighborhood of K without producing new extremal values not defined on K. For the following construction, we concentrate on the module Rn and forget about the underlying space forms. The construction is a recursive one, and we start with n = 1. In this case, we suppose that the s points of K are ordered by size, i.e., we have these n + 2 points
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on the real axis: u1 < x1 < . . . xs < v1 . Extending f to the frame arguments u1 , v1 , these arguments are mapped to the values 1 = f (u1 ), f (x1 ), . . . f (xs ), 1 = f (v1 ). From exercise 71 we have a (unique) cubic polynomial extension of the tangent map on every pair of successive points {u1 , x1 }, {x1 , x2 }, . . . {xs−1 , xs }, {xs , v1 } with zero fibers T fu1 = T fx1 = . . . T fxs = T fv1 = 0. We now consider the spline function defined by gluing all the polynomial extensions at the common points of the pairs, i.e., at the elements of K. Outside the frame, we extend this spline function by the constant function of value 1. This function is C 1 and adds no extra extremal values to the values on K inside the frame. Suppose now that we have succeeded by induction to construct the following extension: We are given the decomposition xi = (zi , wi ), zi ∈ Rn−1 , wi ∈ R. Let pn (K) = {wi1 < wi2 < . . . wir } be the n-th projection of K with the different values in increasing order. For every hyperplane Hx = p−1 n (x), x ∈ {un , wi1 , wi2 , . . . wir , vn }, we are given either the recursively defined C 1 -functions Px : Hx → R, or the constant function 1 for the frame points x = un , vn . For any point v = (z, w) ∈ Rn−1 × R, the value F (v) is defined as follows. If w 6∈ [un , vn [, we set F (v) = 1. Else, there is exactly one interval [a, b] from the successor pairs (un , wi1 ), (wi1 , wi2 ), . . . (wir−1 , wir ), (wir , vn ) such that a ≤ w < b. We then evaluate to the value F (v) = Pa,b,Pa (z),Pb (z) (w) of the rational function P defined in exercise 71. Since a, b are fixed here, the function is a polynomial in the arguments Pa (z), Pb (z), and w. So the function is a C 1 -function in v = (z, w). On each hyperplane Hx , the derivatives of neighboring functions coincide, and the entire function is C 1 . Finally, since the values are constant 1 outside the frames on the hyperplanes, the constant 1 value is guaranteed outside the n-cube C n and we are done. This construction has the disadvantage of depending on the order of the coordinates used in the recursion. For every permutation π of the n coordinates, we have such a function, call it Fπ . Each such function extends one and the same original tangent function and is 1 P outside 1 the cube C n . So we can symmetrize the construction by the weighted sum Fsym = n! π Fπ . However, for programming tasks, such a symmetrization is very time-consuming.
32.4
Global Structure
Summary. The global structure of a performance transformation is a patchwork of local performance transformations. The combination and gluing data of local units expresses the syntax of performance. In turn, this performance syntax defines an interpretation of what has been recognized on the analytical level. This “interpretation of the interpretation” obeys its own rules and constitutes a relatively autonomous rhetorical shaping of the given “text” and its analytical comprehension. –Σ– Evidently, the local performance structures described above are far from sufficient to grasp realistic performance situations. Basically, there are four reasons for this insufficiency:
32.4. GLOBAL STRUCTURE
673
• Instrumental variety The events of a score are not always in one and the same parameter space, especially if we deal with compositions for different instruments. We typically work in a colimit space ScoreOrchestra −→ Colimit(ScoreInstr1 , . . . ScoreInstrk ) Id
of instrumental spaces ScoreInstri −→ Simple(Rni ) Id
with individual coordinator modules. For example, the piano notes have a four-dimensional space associated with pitch, onset, loudness, and duration, whereas the violin space adds crescendo and glissando to the piano space. Ditto for the physical codomain spaces P hysInstri −→ Simple(Rmi ) Id
where the mental instrumental events are mapped. But here, the codomains may be very different in dimension according to the real physical instrument which is addressed on the mental score. The typical colimit space over the physical instruments is P hysOrchestra −→ Colimit(P hysInstr1 , . . . P hysInstrk ). Id
This means that performance starts from a local composition K in the ambient space ScoreOrchestra and maps into the local composition of physical events L in the ambient space P hysOrchestra. The local compositions K, L are disjoint unions of the subcompositions Ki , Li , i = 1, . . . k corresponding to the instrumental cofactors in both, K, L, and the performance splits into a coproduct of k individual local instrumental performances pi : Ki → Li . Even if the mental score level composition K is not a disjoint union of instrumental subcomposition, i.e., a proper colimit, it can be lifted to the coproduct of its cofactors to obtain a disjoint family of performances, so our hypothesis is not restrictive. On the other side, the proper colimits on the physical level are superfluous since it is always (technically) possible to realize the disjoint union of instrumental voices if necessary. It follows that the above splitting is necessary and sufficient for the orchestral globalization of local performances. • Gluing of local extension strategies The local approach is insufficient when sudden, non-continuous changes of the performance map happen. For example, if starting with tempo M.M. quarter = 120 per minute, and then performing a chain of accelerandi and rallentandi, it may be asked by the composer that we reset the tempo after this tempo variations, by the command “istesso tempo”, meaning to return to the original M.M. quarter = 120 per minute. It would be artificial to construct a continuous tempo curve of transition to the reset value. In this case, it is natural to split the performance map into two contiguous domains with their own tempi. Or else, it may happen that the extension category Cat is rather strict, allowing only polynomial maps, say. Such a restriction may intervene for reasons of cognitive nature, or because there are hypotheses about the dynamics of performance, such as mechanical
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CHAPTER 32. LOCAL AND GLOBAL PERFORMANCE TRANSFORMATIONS models (see the work of McAgnus Todd [532], for example). Then it is better to glue the performance map from parts which are conformal with Cat and coincide on the intersection of charts of the atlas patchwork. This latter strategy is nothing else than the well-known spline approach to the extension of discrete map data, see [499, 500] for a typical reference.
• Special roles of selected parameters of a given local performance map (hierarchy) This type of globalization effects is perhaps the most interesting. We stated that in performance theory, we are only interested in the performance map on the mental event which will effectively be played, and not in abstract objects such as bar-lines or pauses. But abstract objects may emerge from effectively played event in the following sense: Suppose that a local composition K in some space S has to be performed, for example a space parametrizing onset E, duration D, and pitch H and suppose that a space S 0 of selected parameters of S, for example onset and duration. We then have the projection πS 0 : S → S 0 and the associated projection K → πS 0 (K). In a great number of performances, it happens that the performance of K factorizes through πS 0 , which means this: We look at the performance map p : K → L, and if L lives in the corresponding physical space P hS with the corresponding projection πP hS 0 : P hS → P hS 0 and the induced projection L → πP hS 0 (L). Factorization means that we have a performance map pS 0 : πS 0 (K) → πP hS 0 (L) which commutes with the projections, i.e., πP hS 0 ◦ p = pS 0 ◦ πS 0 . Here, the time events in S 0 are not really played, but their performance determines the time performance of the really played events. Hence, it may be reasonable to add the projected mental event set πS 0 (K) to the real set K (plus some projection and commutativity conditions as shown above) in order to describe the overall situation. This idea is the basis of the so-called performance hierarchies which control the special roles of parameters. • Stemmatic deployment of performance Performance is never realized on the spot or by means of a unit process which grasps the analytical data and presents their “rhetorical” shaping at once. Humans have to rehearse again and again, continuously refining their results until they reach the final (or provisionally final) performance. This stemmatic deployment process from the sight reading (primavista) performance to the artistically well-devised presentation could be just a human learning process which a machine can achieve at once if it is able to learn the performative substance in abstracto. But this is erroneous, since the logic of performance, the anatomy of the shaping process is a stemmatic one, a multilayered unfolding of deformations of the mental score symbols. Rather than a learning process rehearsal is a meditation on the refinement of understanding. So performance is also global in the sense of a multilayered time-dynamic process. Each stemmatic layer is a logical step in the understanding of the rhetorical expressivity.
32.4.1
Modeling Performance Syntax
Summary. We review the syntactical mechanism of global performance transformation: orchestration, contiguity, hierarchy, and stemmatic layers. –Σ–
32.4. GLOBAL STRUCTURE
675
As was shown above, performance syntax is a bundle of four streams of completely different nature. Contiguity is perhaps the most obvious stream. It reminds us of the syntactic juxtaposition of units within the stream of ordinary language. The orchestration stream is much more difficult to understand since it conveys a layering of the performative mapping which creates the interplay of voices. The structure of this interplay must be viewed in a multidimensional space of geometric parameters, such as onset, duration, pitch, loudness, of sound color parameters for envelopes, Fourier, FM, or wavelet coefficients and the like, and of gestural coefficients for curves of body movements. The hierarchical stream is of high cognitive relevance since it describes the leading and slave parameters of performance. Such a hierarchy is likely to produce orientation in the perception of a performance. But it is also of a purely structural relevance since hierarchies are the turning point and key to the stemmatic layering logic. In fact, the successive refinement process of performance is often managed via deformations of hierarchies, i.e., adjustments of relations and functional dependencies among the members within a given hierarchy. Whereas the three preceding streams are “in praesentia” as semioticians would say, the fourth: stemmatic layering, is of completely different nature. It is not, and this is decisive, of paradigmatic nature, since it describes a development of logical enchainment, i.e., of a juxtaposition of logical stages. But this type of syntagm is not unfolded in time, it is a kind of encapsulated history whose presence cannot be unveiled without a huge amount of ambiguity and uncertainty (see chapter 47 on inverse performance theory), its presence is virtual. The complexity of global performance is not only due to the four-fold stream of global structures, it is enforced by the completely divergent extension morphisms underlying any two different local charts within a particular stream, and even more dramatically when distributed over different streams.
32.4.2
The Formal Setup
Summary. This section introduces the formal performance transformation setup via the category GlTantR of global tangent compositions with t-fold tangent morphisms and, for R = R, with the associated subcategories GlDifft and GlC t of GlTantR . –Σ– From the four types of globalization phenomena, instrumental variety, gluing of local parts, hierarchy of parameters, and stemmatic deployment, the first three will be covered by the following categorical description, while the last one—being of more processual nature—will be treated in chapter 38. Here is the definition which is coined on the definition 36 of a global objective composition. Definition 100 For a positive integer t, a global t-fold tangent composition over a commutative ring R is defined by the following data: (i) A set G and a finite, non-empty covering I of G, (ii) a family (T t Ku , Su )u∈U of local t-fold tangent compositions T t Ku over R, with bases Ku , (iii) a surjection I? : U → I : u 7→ Iu ,
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CHAPTER 32. LOCAL AND GLOBAL PERFORMANCE TRANSFORMATIONS ∼
(iv) a bijection φu : T t Ku → Iu for each u ∈ U , (v) for each couple u, v ∈ U , the induced subcomposition T t Ku,v = φ−1 u (Iu ∩ Iv ) equals the tangent subcomposition of T t Ku induced on the basis Ku,v = pT t Ku (T t Ku,v ), (vi) for each couple u, v ∈ U , the induced bijection t t φu,v = φ−1 u ◦ φv : T Ku,v → T Ku,v
is an isomorphism of local tangent compositions. The data (ii) - (vi) are called an atlas Φ for the covering I of G. Two atlases Φ, Ψ for the covering I of G are called equivalent iff their disjoint union is also an atlas for the covering I of G. A global tangent composition over R is a covering I of G, together with an equivalence class of atlases, a fact which we abbreviate by the symbol GI or even by G if the atlases or the covering, respectively, are clear from the context. If two global tangent compositions GI , H J over R are given, a t-fold tangent morphism from GI to H J is a couple (f, ι) where 1. f : G → H is a set map, 2. ι : I → J is a set map such that f (i) ⊂ ι(i) for all covering sets i ∈ I, 3. for any atlases (T t Ku , Su )U for G and (Lv , Mv )V for H, if we take the chart isomorphisms ∼ ∼ φu : Ku → Iu and ψv : Lv → Jv for some pair u, v of indexes which correspond under the map ι (i.e., ι(Iu ) = Jv ), then the induced maps fu : T t Ku → T t Lv define a morphism of local tangent compositions. The global t-fold tangent compositions and their morphisms define the category GlTantR . For the real number, R = R, we may consider those global t-fold tangent compositions and morphisms which are t-fold differentiable or C t , respectively, on the local data, and thereby define the subcategories GlDifft and GlC t of GlTantR . Example 52 Let GI be zero-addressed objective global composition which is interpretable by a simple ambient space S of module M over the commutative ring R. WLOG we have G ⊂ M , and the atlas is I = {G1 , . . . Gn }. The transition morphisms φi,j between intersections Gi,j = Gi ∩ Gj are the identities. We then have the inductive system G.. = (Gi , Gi,j , ρi,j , φi,j ) with the inclusions ρi,j : Gi,j ⊂ Gi , and G = colim(G.. ). Using the functor from lemma 49, we have an induced inductive system T G.. = (T Gi , T Gi,j , T ρi,j , T φi,j ), together with a morphism of inductive systems p.. : T G.. → G.. . This gives the colimit map p : colim(T G.. ) → G. The domain is T GI = colim(T G.. ) is covered by the images of the charts T Gi . Using theorem 12, we see that the canonical maps φi : T Gi → T GI are injective. In fact, the transition morphisms are the identities, therefore the relation ∼ of that theorem is an equivalence relation, and the
32.4. GLOBAL STRUCTURE
677
theorem can be applied. Call T GI the tangent interpretation associated with GI . This is an interpretation with ambient space S × S, and the projection p : T GI → GI is a morphism of global compositions. Since the transition morphisms are the identities here, one may also refine the tangent construction to one of differentiable nature. 3
3
3
g5 g6 g0 g1
t
2
g2 K1
g0
g0
g1
g5
-t
1
2
t g3
g2
g4 1
K2
t 2
g3
g4
g6 1
K3
Figure 32.1: The seven-element global composition of which the three charts are shown here is a M¨obius bottle. It may not be extended to a global tangent composition by the obvious colimit of its tangent charts and their intersections since, for example, a tangent vector t is equivalent to its negative −t under the colimit construction.
Exercise 74 For a non-interpretable composition the colimit does not yield a reasonable global tangent composition. The following example illustrates this fact, see also figure 32.1. We consider a global composition GI whose support G = {g0 , g1 , g2 , g3 , g4 , g5 , g6 } consists of seven different points, and whose covering consists of three 5-element charts Gi having the common point g0 : G1 = {g0 , g1 , g2 , g5 , g6 }, G2 = {g0 , g1 , g2 , g3 , g4 }, G3 = {g0 , g3 , g4 , g5 , g6 }. They are in bijection with these local compositions Ki in R3 : φ1 : G1 → K1 : g0 7→ (0, 0, 0), g1 7→ (0, 1, 0), g2 7→ (1, 1, 0), g5 7→ (0, 0, 1), g6 7→ (1, 0, 1) φ2 : G2 → K2 : g0 7→ (0, 0, 0), g1 7→ (0, 1, 0), g2 7→ (1, 1, 0), g3 7→ (0, 0, −1), g4 7→ (1, 0, −1) φ3 : G3 → K3 : g0 7→ (0, 0, 0), g3 7→ (1, 0, −1), g4 7→ (0, 0, −1), g5 7→ (0, −1, 0), g6 7→ (1, −1, 0)
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The transition isomorphisms are these: φ12 = Id φ23 : g0 7→ g0 , (0, 0, −1) 7→ (1, 0, −1), (1, 0, −1) 7→ (0, 0, −1) φ13 : g0 7→ g0 , (0, 0, 1) 7→ (0, −1, 0), (1, 0, 1) 7→ (1, −1, 0)
In the colimit of T Ki , T Kij , the tangent t = (g0 , (1, 0, 0)) ∈ T K1 identifies to T φ12 (t) = t, and then to T φ23 (t) = −t ∈ T K32 , whereas T φ13 (t) = t ∈ T K32 , which means t ∼ −t, i.e., the canonical maps T Ki → colim(T Ki , T Kij ) are not injective—gluing tangent spaces is not feasible. Clearly, this phenomenon is due to the non-interpretability of GI , in fact, this is a kind of M¨obius bottle on which any global affine function must identify the values of the pairs (g1 , g2 ), (g3 , g4 ), (g5 , g6 ). Exercise 75 However, if any global composition is such that its nerve is one-dimensional, the colimit construction yields a global tangent composition with the expected chart injections. Give a proof of this fact. Let us discuss the reason why the three globalization perspectives are included in the above definitions. Clearly, the disjoint union of the interpretation morphisms on different instrumental parts is represented by a disjoint union of morphisms on local (or global, if they exist) tangent compositions on each of the instrumental parameter spaces. Let us give an example of a global gluing operation between local tangent compositions. To keep the discussion simple, let us consider the performance map on the onset axis. Suppose that G = {E0 , E1 , E2 } is a sequence of three increasing onsets E0 < E1 < E2 which are mapped onto the set of the three physical onsets g = {e0 , e1 , e2 }, ei = P (Ei ), under the performance map P : G → g. Suppose that the musical conditions are such that one starts with an initial velocity de/dE|E0 = V0 , de/dE|E2 = V2 , whereas the velocity is de/dE|E1 = V1 when we arrive at onset E1 . But the successive path should be started with velocity de/dE|E1 = V1∗ , and end on velocity de/dE|E2 = V2 . This apparent incompatibility at onset E1 can be resolved by the coverings I = {G1 = {E0 , E1 }, G2 = {E1 , E2 }}, i = {g1 = {e0 , e1 }, g2 = {e1 , e2 }} of G, g, respectively. Consider the tangent compositions T GI , T g i of the two interpretable compositions GI , g i , respectively, see figure 32.2. Consider an extension of the global morphism T P : T GI → T g i associated with the map P and the above velocity conditions as linear maps on the four tangent spaces T G1,E0 , T G1,E1 , T G2,E1 , T G2,E2 to a given category of maps on the real axis, C 1 , say. Then we have a “gluing” of two velocities, one from below, one from above, at onset E1 .
32.4. GLOBAL STRUCTURE
679
TG2 TG1 E0
E1
E2
Figure 32.2: The global tangent composition built from two charts T G1 , T G2 of successive tempo regions.
32.4.3
Performance qua Interpretation of Interpretation
Summary. We finally compare the analytical interpretation in the framework of categories Glob of global compositions with associated global performance transformations. –Σ– Recalling the analytical interpretation of a given score denotator, this one a priori regards a variety of involved object-types, such as bar-lines, notes, macros, rests—whatever is needed. Such an interpretation is guided by the analytical approach, such as the topological interpretation using local meters, or motives, or chords, as exposed in the respective chapters. In contrast, the performance-guided interpretation of the given score is firstly restricted to special event types. Usually, for example, bar-lines are not subjected to performance transformations. Secondly, the interpretation within a performable event-type is not primarily guided by analytical considerations, rather it is related to considerations of coherence in the performative process. For example, the agogical variations in a tempo curve, say, can be shaped by complex weight functions as defined via the maximal meter topology or the nerve topology on the composition’s nerve. Such a tempo curve will not refer to the complex covering by maximal meters, but define a unique tangent morphism on a single tangent chart for performance. So the interpretations of analytical and performative nature can and usually will be very different. Nonetheless, the latter interpretations are related in a complex way to the analytical background interpretations. We shall devote the following chapters to the explication of these relations which belong, to our belief, to the most fascinating challenge in mathematical music theory.
Chapter 33
Performance Fields La musique math´ematiquement discontinue peut donner les sensations les plus continues. Paul Val´ery [538, I] Summary. Performance fields are the core of an in-depth theory of performance structure. They are a distinguished type of vector fields which give an infinitely precise, i.e., infinitesimal, account of the ‘shaping forces’ of a given local performance transformation. Although performance fields are not recognized as such in musicology and traditional performance research, they arise in a completely natural way in the traditional context of tempo, intonation, and dynamics. We give a careful account of this basic fact. A closer look at articulation and further sound parameters (apart from onset, pitch and loudness which are used for tempo, intonation, and dynamics) reveals that performance fields should be viewed within a fairly general approach. We define the formal setup. In order to provide a deeper understanding of the semiotic signification process of performance fields, we review the performance philosophies of Theodor W. Adorno, Benjamin, and Diana Raffman. –Σ–
33.1
Classics: Tempo, Intonation, and Dynamics
Summary. This section recapitulates and analyzes the concepts of tempo, intonation, and dynamics. We take the opportunity to make the point of blurred concepts in musicology, in this case in the sense of a fascinating quest for expressive precision, paired with denial of formal explicitness. –Σ–
33.1.1
Tempo
Summary. Tempo is one of the best ‘known’ features of performance. However, its concept is blurred and far from standardized among musicians and performance scientists. We analyze the 681
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state of the art and its deficiencies. It is shown that tempo is a local concept which is charged with a large amount of semantics. We proceed in filtering out the semantics from the structural data. After this, a precise definition of tempo as a one-dimensional performance field in the onset axis is given. –Σ– Tempo is the Italian word for “velocity”. In musical notation and performance, tempo refers to the pace at which the mental events, typically written on a score or imagined by the performer, are projected into physical reality. This is done either to describe the relative velocity or the velocity change with respect to a given absolute tempo situation; typically indicated by notation such as verbal annotation “accelerando”, “rallentando”, or by corresponding pictorial signs. Or else, the tempo is described with an absolute meaning. The prototype of this second situation is M¨ alzel’s Metronome (M.M.), quantified by a quotient such as “quarter = 120”, i.e., a quarter note is given the physical duration of 0.5 second, resulting in 120 quarters per minute. In contrast to this precise indication, a relative tempo sign within a M¨alzel context does not define precisely how much the tempo should change, and in which way this should happen., e.g., by a linear curve or in a quadratic way, etc. The relative tempo signs are massively ambiguous. We shall have to spend quite some time in order to set up a reliable handling of such blurred signs in algorithmic contexts. Absolute tempo is also indicated by more poetic indications, such as “andantino”, “prestissimo”, “maestoso”. Clearly, such wordings are loaded with a good portion of connotation which exceeds the mere tempo and targets emotional refinement, expressed via other parameters, such as articulation (legati, staccati), dynamics (loudness variations) or intonational deformations (on violins or for the human voice). Whatever the specific expression, tempo relates to the transformation of mental onsets to physical onsets. Note that this does not imply that such a transformation is independent of the other event parameters. We shall learn that the performance of onsets may involve all other parameters. A simple example of such a relation is given by the so-called “Chopin rubato” which lets the left hand perform a “mother” tempo while the right hand performs local deviations of the left-hand tempo in order to generate the effect of temporal tension (“daughter” tempi). In this case, the common onset of left and right hand notes may lead to different images in physical time, as a function of the interpretation of the total score by the left-hand and right-hand charts. Formally, local tempo is related to a tangent morphism T f : T K → T k of local tangent compositions T K, T k derived from compositions K ⊂ Onset, k ⊂ P hysOnset in onset spaces. Tempo is then defined iff the linear fiber maps T fX , X ∈ K are invertible and there, we define T empoX = (T fX )−1 , the inverse slope at X. Moreover, tempo is always supposed to be positive, while negative tempi are without evident musical meaning. In such a situation, tangent morphisms are extended to differentiable morphisms with positive, continuous derivatives at every point. The tempo curves associated with such differentiable morphisms are the continuous, positive functions whose values are the inverse derivatives at all points in an open interval containing the points of K. Usually, the tangent morphisms are not of immediate interest in performance theory, this means that one is given differentiable morphisms, ie., the morphisms F inducing tempo curves as their inverse derivatives T empoX = (T F (X)−1 ). However, it may then also be required that these functions be differentiable extensions of tangent morphisms, but this is not the mandatory situation.
33.1. CLASSICS: TEMPO, INTONATION, AND DYNAMICS
683
Nonetheless, the music(ologic)al point of view which negates differentiable extensions is not correct: tempo is also present in between the points of K, and its structure is an essential information of the performance maps. Formally speaking, we are dealing with a local composition K ⊂ Onset, a local composition k ⊂ P hysOnset, two closed intervals OF rame = [A, B] ⊂ Onset, P Of rame = [a, b] ⊂ P hysOnset such that K ⊂ OF rame, k ⊂ P Of rame, and ∼ a C 1 diffeomorphism F : OF rame → P Of rame with positive derivative T F (X) at each point X ∈ OF rame, and such that F (K) = k (and possibly F extending a given tangent morphism ∼ T f : T K → T k if that is required). The tempo field of F on OF rame is the continuous field T empo with values T empoX = (T F (X)−1 ), X ∈ OF rame. By construction, if the performance map x0 = F (X0 ) is defined on any point X0 ∈ OF rame, the performance F (X) on X ∈ K is defined by the integral1 Z
X
F (X) = x0 + X0
1 T empo
(33.1)
of the inverse tempo function. Musically, the “initial value” x0 = F (X0 ) means that the “conductor” defines a starting time x0 of performance from which the remaining performance onsets can be deduced by means of the given tempo curve. Exercise 76 With the above notation, calculate the onset function F for these tempo types: (1) T empoX = q0 + q1 X, (2) T empoX = q0 + q1 X + q2 X 2 , (3) T empoX = eX , (4) T empoX = 2 + sin(X). Discuss the possibilities of coping with these tempo types with tangent conditions. In general, the tempo must also cope with more than one given performance value, x0 = F (X0 ), x1 = F (X1 ), . . . xt = F (Xt ), and then, we have to ask for the conditions Z
Xj
xj − xi = Xi
1 T empo
(33.2)
for i, j = 0, 1, . . . t to guarantee compatibility of the tempo integral with the given values.
33.1.2
Intonation
Summary. Intonation deals with pitch just as tempo deals with onset time. The definition of intonation as a one-dimensional performance field in the pitch axis is given. –Σ– Intonation, including the specialization of tuning, deals with the relation between mental pitch and physical pitch. It is analogous to tempo insofar as it suggests a map from the pitch space P itch to the pitch space P hysP itch. But the situation is not so easy, if we observe the common pitch spaces in music theory. Recall that symbolic pitch is usually represented in the Euler module space EulerM odule, and that we have a space morphism E2M : EulerM odule → M athP itch defined in equation (6.28). This is an injective Q-linear map on the supporting 1 We follow the Douady notation and do not write the old-fashioned infinitesimal “dY ” in the integrand if this one is clear.
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modules Q3 , and R, respectively. If we want to apply calculus to this situation, we have to extend the scalars to real numbers and then consider the induced map E2M ⊗ R : EulerRM odule → P itch with underlying vector spaces R3 , and R, respectively. This latter morphism is no longer injective (in fact: surjective with two-dimensional kernel pv ⊥ , see formula (6.27)). If we consider the performance map composed from the given performance of pitch, p : P itch → P hysP itch, and the above E2M ⊗ R, we are confronted with a performance map p ◦ E2M ⊗ R from Euler space to physical pitch which is far from being a diffeomorphism. This cannot be repaired since there is no physical space corresponding to the Euler construction, at least not in the simple physical pitch dimension. The solution could consist in a construction of a direct performance map from EulerRM odule to a physical “pitch” space of higher dimension. This one would evidently not restrict to mere physical pitch (logarithm of frequency, see appendix A.2.3), but include other parameters, such as sound color or the like. This is however an open problem in performance research: How should we perform Euler space pitch? We will stick to the state of the art and build pitch performance upon the maps of type F : P itch → P hysP itch. Such maps should be the C 1 -extensions of tangent maps T f : T S → T s over bijective basis maps f : S → s. Here, S ⊂ P itch, s ⊂ P hysP itch are a number of pitches and corresponding physical pitches. This situation is formally equivalent to the tempo situation. We may indeed take over mutatis mutandis the statements and formulas established for tempo as follows. We are given a mental pitch frame P F rame = [U, V ], a physical pitch frame P P f rame = [u, v], inclusions S ⊂ P F rame, s ⊂ P P f rame, and a C 1 -diffeomorphism ∼ G : P F rame → P P f rame, the intonation curve, with positive derivative T G(X) for every ∼ X ∈ P F rame (and possibly extending a given tangent morphism T g : T S → T s if that is required). The intonation field of G on P F rame is the continuous field Intonation with values IntonationX = T G(X)−1 , X ∈ P F rame. By construction, if the performance map x0 = G(X0 ) is defined on any point X0 ∈ P F rame, the performance G(X) on X ∈ S is defined by the integral Z X 1 G(X) = x0 + (33.3) X0 Intonation of the inverse intonation function. Musically, the “initial value” x0 = G(X0 ) means that the piece is played from a starting pitch, i.e., the “chamber pitch” x0 of performance from which the remaining performance pitches can be deduced by means of the given intonation curve. In general, the intonation must also cope with more than one given performance value, x0 = G(X0 ), x1 = G(X1 ), . . . xt = G(Xt ), and then, we have to ask for the conditions Z Xj 1 xj − xi = (33.4) Intonation Xi for i, j = 0, 1, . . . t to guarantee compatibility of the intonation integral with the given values. This is, in particular, the case when we are given a specific intonation between semitones in a fixed tuning mode, such as well-tempered or just tempered tunings. Again, musicological approaches only deal with intonation values on the given tangent composition T S, while the intermediate values of the intonation curve are neglected (or even negated—in the worst case). However, if glissando effects are present, discrete intonation fails to give information about the intermediate values, much like the intermediate tempo is necessary to perform glissandi in their development along the time axis!
33.1. CLASSICS: TEMPO, INTONATION, AND DYNAMICS
33.1.3
685
Dynamics
Summary. Dynamics deals with loudness just as tempo deals with onset time, and intonation deals with pitch. The definition of dynamics as a one-dimensional performance field in the loudness axis is given. –Σ– Dynamics is the physical shaping of loudness symbols such as ff, mf, ppp, mp, sf and words such as sforzato, meno forte, diminuendo, crescendo. In this classical setup of score notation, the scope is much less quantified than with tempo or intonation. There is no such norm as M¨alzel’s metronome or the chamber pitch in dynamics. Also, the shaping of dynamics is dramatically different, i.e., refined, with respect to the written prescriptions. It may happen that a section is written in mezzo forte, but within this section, the performance of mezzo forte is quite variable, within a certain tolerance bandwidth of what mezzo forte can be felt. However, just for these reasons, it is much more accepted by musicologists that dynamics is a continuous phenomenon when compared to tempo and intonation. Sticking again to the state of the art, we build loudness performance upon the maps of type F : Loudness → P hysLoudness. Such maps should be the C 1 -extensions of tangent maps T f : T L → T l over bijective basis maps f : L → l. Here, L ⊂ Loudness, l ⊂ P hysLoudness are a number of loudness values and corresponding physical loudness (logarithms of pressure amplitude, see appendix A.2.2). This situation is also formally equivalent to the tempo situation. We may indeed take over mutatis mutandis the statements and formulas established for tempo as follows. We are given a mental loudness frame LF rame = [U, V ], a physical loudness frame P Lf rame = [u, v], inclusions L ⊂ LF rame, l ⊂ P Lf rame, and a C 1 -diffeomorphism ∼ H : LF rame → P Lf rame, the dynamics curve, with positive derivative T H(X) for every ∼ X ∈ LF rame (and possibly extending to a given tangent morphism T h : T L → T l if that is required). The dynamics field of H on LF rame is the continuous field Dynamics with values DynamicsX = T H(X)−1 , X ∈ LF rame. By construction, if the performance map x0 = H(X0 ) is defined on any point X0 ∈ LF rame, the performance H(X) on X ∈ L is defined by the integral Z
X
H(X) = x0 + X0
1 Dynamics
(33.5)
of the inverse intonation function. Musically, the “initial value” x0 = H(X0 ) means that the piece is played from a starting dynamics, i.e., the “mezzo forte” x0 of performance from which the remaining performance dynamics can be deduced by means of the given dynamics curve. In general, the dynamics must also cope with more than one given performance value, x0 = H(X0 ), x1 = H(X1 ), . . . xt = H(Xt ), and then, we have to ask for the conditions Z
Xj
xj − xi = Xi
1 Dynamics
(33.6)
for i, j = 0, 1, . . . t to guarantee compatibility of the dynamics integral with the given values. This is, in particular, the case when a specific dynamics range between successive symbols, such as ppp, mpp, pp, mp, p, mf, f, mff, fff, is required for acoustical reasons. In the precise technical
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sense this is a rare situation which occurs more in studio situations than in live performances. But also in common (what is that in the age of complex man-machine interfaces?) human performance, it may occur that one musician or instrument dictates the dynamical ranges by individual approaches, and that other musicians in the orchestra have to cope with these constraints. For example, the enhanced dynamics of a strong beating drummer can trigger dramatically the dynamical range of a whole jazz band.
33.2
Genesis of the General Formalism
Summary. Tempo, intonation and dynamic fields are one-dimensional special cases of performance fields. We discuss the a priori two-dimensional performance field of tempo and articulation. From this special case, a general type of performance fields is deduced and motivated by a set of representative examples. We conclude the section with a rigorous definition of the concept of a performance field. –Σ– If we consider a local composition K ⊂ Onset⊕P itch (see section 6.4.1 for this space), the ∼ performance of such a composition is a bijection h : L → l with codomain k ⊂ P hysOnset ⊕ P hysP itch. Suppose that this bijection is induced by bijections on the onset and pitch axes as described above. With the above notation, we consider the projections pOnset : Onset⊕P itch → Onset, pP itch : Onset ⊕ P itch → P itch. Consider the two projections L → K = pOnset (L), and L → S = pP itch (L). Suppose that these projections are performed conforming to the above rules, i.e., there are frames OF rame, P F rame with K ⊂ OF rame, and S ⊂ P F rame. Suppose also that we have diffeomorphisms F, G as above. Let our performance h be defined by the projection morphisms, i.e., for (X, Y ) ∈ L, h(X, Y ) = (F (X), G(Y )). This means that we are given a ∼ product C 1 -diffeomorphism H = F × G : OF rame × P F rame → P Of rame × P P f rame which restricts to h on L. Evidently, the two one-dimensional vector fields T empo, Intonation induce a two-dimensional vector field T empo × Intonation on the product frame OF rame × P F rame, see also figure 33.1. Whereas the factor fields are derived from the tangent morphisms T F, T G, the product vector field is evidently not derived from the tangent morphism T H. In fact, the latter is a two-dimensional linear transformation T HX,Y = T FX × T GY and not a vector. We recognize however that the product field verifies this equation: T empoX × IntonationY = (T HX,Y )−1 .∆
(33.7)
where ∆ = (1, 1) is the diagonal unit vector in the tangent space at the H-image (x, y) of (X, Y ). This means that the tempo-intonation field is derived from the diffeomorphism H which extends the map h on the underlying composition L via the inverse image of the diagonal field ∆ on the image product frame P Of rame × P P f rame.
33.2. GENESIS OF THE GENERAL FORMALISM
687
pPitch OFrame ¥ PFrame
Pitch
PFrame
S
Tempo ¥ Intonation
L
Pitch Onset pOnset
Tempo
K Onset OFrame
Figure 33.1: The two one-dimensional vector fields T empo, Intonation induce a two-dimensional vector field T empo × Intonation on the product frame OF rame × P F rame.
33.2.1
The Question of Articulation
Summary. Special attention is given to the delicate relation between tempo and articulation. It reveals that one-dimensional performance fields are a priori insufficient to describe local performance transformations. Articulation requires a priori two-dimensional fields. –Σ– It may appear as if the two-dimensional tempo-intonation field were an artificial generalization of an essentially one-dimensional situation. This is however not the case as we now show for the two-dimensional performance on the plane of onset and duration. This situation is as follows. Suppose that we have a local composition K ⊂ Onset ⊕ Duration which should be performed with respect to a tempo performance of the onset projection. More precisely, call such a performance F (E, D), (E, D) ∈ K, and recall the sweeping alterator already used in counterpoint (and introduced in section 7.5): α+ : Onset ⊕ Duration → Onset : (E, D) 7→ E + D
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with the omission of the dual number formalism here because we do not need this algebraic enrichment of structure in performance theory. Suppose that the onset projection KO = pOnset (K) of K is performed according a performance map f : KO → kO onto an image set kO . Where should we map the duration components? The canonical recipe is to consider the “offsets” of our events, i.e., for (E, D) ∈ K, we take its alteration Of f = α+ (E, D) = E + D and look for its image f (Of f ). We may then calculate the duration d in the image (e, d) = F (E, D) by the formula d = f (Of f ) − f (E). This is however not well defined if we do not know what is the image of onsets outside KO ! This is another strong argument for the existence of tempo outside a composition’s onsets. So we have to embed the onset performance f in a C 1 -differentiable extension, to be defined on an onset frame F rame as above. Call this extension again f . Observe that F rame must now also contain the local composition of alterates α+ (K), i.e., KO ∪ α+ (K) ⊂ F rame.
(33.8)
With these conventions, we may calculate the tangent morphism T F associated with the map
Figure 33.2: This parallel articulation field ∂T empo is derived from the tempo curve T empoE = 1 + 0.4 sin(E). The horizontal axis is onset, the vertical axis is duration. F (E, D) = (e(E, D), d(E, D)) = (f (E), f (E + D) − f (E)). The tangent map at (E, D) is described by the Jacobian JF(E,D) =
∂E e ∂E d
∂D e ∂D d
! =
1/T empoE 1/T empoE+D − 1/T empoE
0 1/T empoE+D
!
33.2. GENESIS OF THE GENERAL FORMALISM
689
which yields the inverse value (JF(E,D) )−1 =
T empoE T empoE+D − T empoE
0 T empoE+D
!
whence the inverse image vector field of the constant diagonal field ∆ on the physical plane: ! T empoE −1 (JF(E,D) ) .∆ = = ∂T empoE,D (33.9) 2T empoE+D − T empoE which we call the parallel field ∂T empo of articulation since the duration component is calculated in parallel to the onset by use of offset values. Figure 33.2 shows such a parallel field. We see that the direction of such an articulation field is from −π/4 to π/2. From this example, we learn that the performance field of the parallel articulation map on the onset-duration plane is not a product of one-dimensional fields, even under the completely innocuous assumption of durations being induced by offset data. Exercise 77 Calculate the performance field of articulation if the performance map has a dilatation by 0 < λ in the duration component, i.e., F (E, D) = (f (E), λ(f (E + D) − f (E))). The example of a parallel field in articulation can be taken over to the pitch domain: Instead of duration we have to think of glissando. This is the proportion of pitch shift with respect to the pitch coordinate if the pitch at the end of an event must have a different pitch with respect to the onset pitch. This generates the parallel glissando performance field ∂Intonation with exactly the same formalism as for articulation. The same method can be applied to generate a parallel crescendo field ∂Dynamics if Dynamics is the field associated with loudness performance. The details are left to the reader. Whenever we deal with basis and pianola spaces, we use this notation: If B = B1 ⊕ . . . Bk is a product space of basis parameters (such as onset, pitch, loudness), and if P = P1 ⊕. . . Pk is the product space of corresponding pianola parameter spaces, we denote by α+ the alteration map B ⊕ P → B defined by (b, p) 7→ b + p on each basis-pianola component, whereas pB is the first projection. So parallel performance maps and the corresponding parallel performance fields are defined by use of the alteration α+ . If the basis field at the basis point X is ZX , the parallel field at the basis-pianola point Q is ∂ZQ = (ZpB (Q) , 2.Zα+ (Q) − ZpB (Q) ),
(33.10)
a linear operator in Z. We shall use this operator not only for basis-pianola couples, but in general situations of a direct product of isomorphic simple spaces B, P with the associated alteration map.
33.2.2
The Formalism of Performance Fields
Summary. This section puts the previous considerations and special cases into a generic formalism, the concept of a performance field in general musical parameter spaces. –Σ–
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The general situation is this. We suppose that a local composition K ⊂ S is contained in a closed rectangle R = [a1 , b1 ] × . . . [an , bn ] of the underlying real n-space. We also suppose that a bijective performance map ℘ : K → ℘(K) onto a local composition ℘(K) ⊂ P S in a simple physical space P S of same dimension n is given, and that this map can be extended to a C 1 -diffeomorphism ℘ : R → ℘(R), i.e., a C 1 -diffeomorphism which is defined on a neighborhood of R. The performance field of this diffeomorphism is a continuous vector field2 Ts on R which is defined by the inverse Jacobian −1 Ts℘ .∆ X = (J℘X )
(33.11)
applied to the diagonal unit vector ∆ = (1, . . . 1), i.e., Ts℘ = T ℘−1 .∆.℘. R R If x0 is a point in P S, the integral curve3 x0 ∆ of ∆ through x0 evaluates to x0 ∆(t) = ℘ x0 + t.∆. For the existence and uniqueness of maximal integral curves of Ts , we suppose that the performance field Ts℘ is locally Lipschitz4 . Then, evidently, integral curves of Ts℘ are transformed into integral curves of the diagonal field under ℘. Therefore, if X ∈ R, if R X0 = X Ts℘ (t), and if the image xo = ℘(X0 ) is known, then we have ℘(X) = x0 − t.∆.
(33.12)
This means that the performance map ℘(X) can be calculated via the integral curve X Ts℘ of the performance field if there is at least one point on such a curve for which the performance map x0 = ℘(X0 ) is known. This generalizes what we have already learned for tempo curves and initial performances on selected onsets. Therefore, performance can be calculated from a Lipschitz-continuous performance field on all points X whose integral curves hit a set Initial ⊂ R of points whose performance is known in advance. We may now forget about the performance map ℘ and start from the performance field and a ‘good’ initial set, defining the performance map via the integral curves and equation (33.12). This performance map will usually only be calculated on the local composition K ⊂ R, or at least for a selected set of points of that frame, and not for any point in R, but this is exactly what we want, see figure 33.3. R
33.3
What Performance Fields Signify
Summary. The very complexity of performance fields parallels a strong impact on semantic layers of performance. We want to lay bare this crucial relation. In a first approach, we discuss the contributions of Adorno, Benjamin, and Raffman to the very ineffability of performance nuances. We then deduce the adequacy of performance fields to deal with this “ineffability” and to control it on the level of the sophisticated language of calculus. Finally, we deal with the tension between structural and performative parameters in music. We expose and discuss Helga de la Motte’s thesis that, historically, there is an increasing number of performance parameters being transformed into structural score parameters. –Σ– 2 Ts
stands for German “Tempo-Stimmung” and is symbolized by the Hebrew letter “tsadeh”. 3 For integral curves of vector fields, see appendix I.2.3. 4 See appendix I.2.1.
33.3. WHAT PERFORMANCE FIELDS SIGNIFY
691
∆ X
x Ts
℘ x0
X0
Figure 33.3: Performance x = ℘(X) can be defined upon the performance field Ts and on an initial set (left polygon) where the performance is known in advance.
33.3.1
Th.W. Adorno, W. Benjamin, and D. Raffman
Summary. Adorno and Benjamin [110] have associated performative adequacy with an activity of “infinitesimal precision”. We make plausible that their language suggests the language of vector fields—though not explicitly stated by these authors. Diana Raffman’s argument for ineffability of musical nuances in performance [432] is discussed. We relate this admittance of ineffability to the search for a powerful language as an extension of the powerless common language. –Σ– The previous discussion of musicological approaches to performance might have given the impression that in general, musicologists share the tendency to oversimplify the complexity of performance, be it in discrete tempo concepts, be it in correspondingly simplistic understanding of intonation. In fact, the usual understanding of articulation is not better than that. It is defective to the point of not realizing that performance of duration is related to the onset performance plus some deformation of the duration according to articulation rules. We have not encountered any such structural description to date—worse: discussions about these phenomena were dominated by a complete ignorance of this kind of effects. However, on a non-quantitative level, very intelligent observations have been advanced by the most sophisticated theorists of performance: Theodor W. Adorno, and Walter Benjamin in [7]. Here is their basic text which introduces the “micrological procedure”: (Dieses mikrologische Verfahren) darf nicht als ein dem k¨ unstlerisch produktiven Entgegengesetztes verstanden werden. Walter Benjamin hat ‘das Verm¨ ogen der Phantasie’ als ‘die Gabe, im unendlich Kleinen zu interpolieren’ definiert. Das beleuchtet blitzhaft die wahre Interpretation. Der Forderung, Phantasie, als Medium des Lebens der Werke, und Genauigkeit als das ihrer Dauer, zu vereinen,
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CHAPTER 33. PERFORMANCE FIELDS der Grundfrage, welcher der verantwortliche Interpret sich gegen¨ uber sieht, wird gen¨ ugt nur durch den gebannten und bannenden Blick auf den Notentext der Werke. In seinem dicht gewobenen Zusammenhang sind die minimalen Hohlr¨ aume zu entdecken, in denen sinnverleihende Interpretation ihre Zuflucht findet. (...) Das Medium k¨ unstlerischer Phantasie ist nicht ein Weniger an Genauigkeit sondern das noch Genauere.
As the wording is chosen, micrologic is a logic in the smallest dimensions of a composition and its performance. The text suggests that this procedure could be misunderstood as opposed to artistic fantasy. Adorno evokes Benjamin’s observation that fantasy is involved in the infinitely small (“im unendlich Kleinen”), more precisely in the interpolation towards the infinitely small. For Adorno, this is a revealing insight into the ultimate, true performance. Adorno asks for the discovery and inspection of the innermost interspaces (“sind die minimalen Hohlr¨aume zu entdecken”). Infinite interpolation is the tool to do so. And this is not contrary to artistic fantasy, it is, so to speak, the strongest microscopic instrument we have, and should use. Artistic fantasy is not the pseudo-romantic blurredness, but a maximum of precision, of intensity and interplay of minimal movements and forces. This absolutely central insight of Adorno and Benjamin is not only astonishing in the musicological environment (though not as a category of Adorno’s and Benjamin’s discourse), it is also a very problematic approach insofar as the humanities—where their text belongs—do not have any means of making such allusions precise. The text is a kind of schizophrenic claim of non-mathematical experts in the words of mathematical concept frameworks: Interpolation, infinitely small, etc. To the mathematically trained, the allusion to calculus is straightforward. No doubt, the language of the infinitely small is calculus. Is it this kind of language which Adorno and Benjamin were aiming at? What is intriguing is that they are talking of infinite interpolation. Between what? The score is a radically discrete symbolism. The infinite interpolation is not a priori inscripted into the score structure. And, what are those cavities (“Hohlr¨aume”) in terms of music parameters and processes? In between the discrete score events, there must be some infinitely divisible space which encompasses the cavities, Adorno and Benjamin are zooming in and penetrating. A solution of this conceptual approach could in fact be the continuous and differentiable interpolation suggested in the previous considerations of extensions of local compositions and maps. We claim that our theory is the mathematically adequate concept framework to the Adorno-Benjamin approach. More precisely, performance fields which include infinitesimal information about the performance process “between the score units” seem to conceptualize this world of infinitely refined reading of what is happening in the cavities of time, pitch, loudness, articulation, glissandi, and crescendi. Without anticipating the expressive power of performance fields, it appears that performance fields in their very rich structure could englobe a deep semantic richness towards human expression of all the intentions which human performance commits. We have to imagine that given a score with its discrete event set, the layer of a performance field which is superposed to that score adds an infinitely fine interpolation in the sense of Adorno–Benjamin. It is like an optical lens system which deforms the “mechanical, rigid” score data into a rhetorical expression of the interpreter’s understanding.
33.3. WHAT PERFORMANCE FIELDS SIGNIFY
693
Performance fields implement a powerful language of performative rhetorics which transcend the discrete vocabulary of common scores. This opens a substantial discussion about ineffability in music. Recall that musical performance is still a strong argument for the ineffability of musical reality: In [432], Diana Raffman has argued that ineffability is a characteristic feature of musical expression, and that this is related to the quale objects as defined by Clarence Irving Lewis in [302]. Quales are those qualities of immediate human experience which cannot be conceptualized and are of private, individual, irreproducible and antilexical nature, such as colors, sounds, hunger, anger, sadness, or happiness. Raffman argues that musical experience is strongly related to quales and therefore shares strongly ineffable characteristics. However, ineffability means escaping the power of language. And this is the critical point: which language is escaped, transcended, what is the boundary to effability? Clearly, the infinitely small, the infinite interpolation are such ineffabilities to the common language. But they are not to the mathematical language of calculus. Ineffability is a relative concept and not a static verdict. This means that ineffability is a challenge for language extension: Are we able to find a richer language which captures phenomena which were—hitherto—ineffable? Principle 24 We argue that performance fields are precisely such an extension of the music description language which turns ineffable instances of musical expressivity into regions which may be controlled by such a language enrichment. After all, vector fields are a very romantic subject: The experience of wind and weather, of stormy rains, of water streams and lava breakouts is a valid metaphor of the forces and processes of our souls. It is not miraculous that performance fields are the exact counterpart of musical expressivity in its most refined appearance as preconized by Adorno and Benjamin in their visionary text.
33.3.2
Towards Composition of Performance
Summary. Helga de la Motte has hypothesized [122] that, historically, there is an increasing number of performance parameters being transformed into structural score parameters. We discuss this argument and make a picture of its consequences for the composition of performance fields; a language reflecting such a refined performance data must be radically different from the known “digital” language as is common in western score notation. –Σ– It is undeniable that from the early days of neumes to the present, or at least, to the classical European notation, an increasing number of music parameters can be observed in the score notation. For example, the bar-lines were only introduced around 1420, whereas the dynamic signs or the instrumental specification were not present in Bach’s Art of Fugue, and the metronomic indications became standard only after M¨alzel’s invention (though sometimes in a problematic way, such as in Beethoven’s “Hammerklavier” sonata). This fact must be seen in the context of the very concept of a score. The concept of a score is that of a mediator between musical ideas and their physical execution. As such, the score of a music piece cannot be narrowed to the level of a notated sign system. The score begins in the mental layers of the composer and musician and is only supported by, not identified with, the
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schematic and digitized material form. The distribution between exterior and interior5 score depends on the specific music culture. In jazz, the major part of the score is present on its interior level, the rudimentary lead-sheet notation sketches only the most elementary ingredients such as harmonic progression and melodic core structures. This has deep implications on the concept of a composition. If one agrees that a composition is all that is fixed on the “neutral level” of the score, the stratification into inner and exterior score makes the composition concept a bit difficult: Where does the composition begin, where does it end? In fact, the interior part of the score may include mental sketches as well as schemes of performance or construction, such as realized in jazz improvisation or processual schemes in the sense of Cage. If we agree that the composition is everything that can be traced either on the exterior or the interior score, then composition may very well include performative instances if the interior score shapes the mapping from mental to physical reality by a well-defined concept. In this sense, Helga de la Motte’s observation of an increasing shift of performative instances to the score level can be made more precise: It is a shift from inner score instances to those of the exterior score. Now, such a shift is a function of two factors: First, as in jazz, and even more in free jazz, it may not be the objective of a musical approach to aggregate exterior score signs, and then, no such shift is needed. Nonetheless, transcriptions of jazz improvisations may be desired (such as the famous transcriptions of John Coltrane soli by Andrew White), and also to a very refined degree. Second, the composer may want to give a precise description of instances of hitherto interior scores, such as happened with the performance signs of accelerando, crescendo, etc. In both cases, the shift can only be achieved by a refined language since the interior score may comprise ineffable spots, regions that cannot be conceptualized on the verbal level: Non-verbal concepts are a widespread phenomenon among musicians! Performance fields may be an extension of the exterior score language which helps shifting non-verbal concepts to the verbal level. But we do not insist on the verbal character of a vector field, at least not in the common sense of verbalization. Mathematical concepts are beyond common language. And they are also beyond quales, they are an effective extension of language which can help thinking things which in the past were completely out of reach to the human intellectual power. Evidently, the score concept which includes such sophisticated objects as vector fields will look completely different from the traditional discrete system. Perhaps it will also per default and as a mandatory condition be related to computer-aided representation and editing. In this sense, the medium computer may help to profile a message which musicians of all cultures have always dealt with and tried to communicate so desperately.
5 To my knowledge, the term “interior score” was introduced by the Jazz theorist and musician Jacques Siron in his remarkable book “La partition int´ erieure” [487] on jazz theory.
Chapter 34
Initial Sets and Initial Performances Jeder Anfang ist ein Ende. Hermann Hesse (1877–1962) Summary. Performance has to start somewhere. The theory of this initialization deals with initial sets and initial performance. Naively speaking, initial sets are the first notes of a performance. Semiotically, initial theory describes a turning point from lexicality to reality in music which is supported by shifters; we explain this rationale. Since music deals with many parameters, initialization has to be specified in all dimensions. We first comment on the classical initial sets in onset, pitch, and duration. On an initial set, the performance cannot be calculated from previous performance, an initial performance has to be defined. We discuss ways to do so. On a more technical level, we introduce the hit point theory, a mathematical account for the control of performance field flows as their curves approach initial sets. Strategies of guessing best approaches to initial sets are presented. –Σ– We now situate our investigations in the framework developed in section 33.2.2. This means that we are given 1. a frame R = [a1 , b1 ] × . . . [an , bn ] in the n-space Rn , 2. a local composition1 K ⊂ R, 3. a performance field Ts, i.e., a locally Lipschitz vector field defined2 on R, 1 We are sloppy about the underlying forms here and just consider the coordinates and modules since the forms are not of primary interest here. It is however subtended that everything happens within well-defined forms that may be evoked if necessary, e.g., if the mathematical methods need a justification via parameter forms beyond numerical coordinates. 2 By definition, this means that the field is defined in a neighborhood of R. But the portions of integral curves of this field contained within the frame R are only a function of the field values within the frame.
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CHAPTER 34. INITIAL SETS AND INITIAL PERFORMANCES
4. and an initial set I ⊂ R. According to the previous theory, we shall suppose that the points X0 of the initial set I are given an initial performance ℘I (X R 0 ), and that we may define the performance ℘(X) for X ∈ K by means of the integral curve X Ts if this one hits a point X0 ∈ I via formula (33.12). In this chapter we shall first discuss the meaning of the initial set approach and then techniques to find initial points for polyhedral initial sets.
34.1
Taking off with a Shifter
Summary. When music performance takes off, a magic moment takes place. This well-known effect relates to shifter signs in the semiotic system of music. We make a point of this magic, in fact a moment with deep consequences for performance as a whole which is fully appreciated by the auditory on an emotional level. This “juncture of fiction and reality” is singular but does in fact happen in several places of a composition’s performance. The magic is distributed among the entire performance, we introduce this subject as an interface between semiotics and psychology. –Σ– Even before discussing the different parameter-specific initial sets, it is important to understand the deep meaning of the fact of initial performance. We have known performance as a transitional process (formally described by the performance map ℘) from mental to physical reality. On the level of documentation this is a transition from the score to the acoustical realization, to be archived on sound media such as a CD. Following the valid doctrine—as preconized by Val´ery3 and Adorno4 , for example—the performance is an integral part of the work of art, and this means that, in the sense of communication theory of art as described by Jean Molino (see our introduction in section 2.2), performance is part of the semiosis of the work, its meaning is not complete except when it is performed. Put it the other way round: The mental score (interior as well as exterior) conveys a part but not the whole significate, and only via performance can we complete the work’s semiosis. Performance englobes a kind of usage of the mental score sign by a performer, much like the usage of a sign in the pragmatic dimension is part of its semiosis. More specifically, those signs whose significate are not only instantiated but substantially depend on the user, are the well-known shifters. Performance of a mental score is such a shift from lexicality to full-fledged meaning, since the pure score is essentially less than the work of art. In other words, performance is a shifter characteristic of the score semantics. Production of full-fledged meaning is only possible by means of performance, and this adds a semantic aspect to the sign which is a non-trivial function of the performer(s). So the anchorage of a score in the physical performance is not only a transformation but also a completion of the score’s meaning. It is a completion of shifter type, i.e., adding a new, user-dependent value of the score sign, a value which turns the abstract score structure into a concrete, existential entity. Within this dramatic transformation process which the conductor Sergiu Selibidache has so violently defended against the musical reproduction industry [476], there is a particularly 3 “C’est 4 “Die
l’execution du po` eme qui est le po` eme.” [538] Idee der Interpretation geh¨ ort zur Musik selber und ist ihr nicht akzidentiell.” [6]
34.2. ANCHORING ONSET
697
dramatic moment of initiation of the existential kernel. This is what happens in the beginning of a performance. This beginning is when the conductor (or the soloist in the case of a solo performance) appears on stage, steps to the conductor’s desk, takes the baton and freezes every movement in order to get off with the first onset gesture. The moment, when the conductor lowers the baton and unfreezes the time process, is the real magic of performance: Some moments ago, the work was still in its lexical potential state, everything could have happened. But now, we are getting off into reality. The first note puts an end to the potentiality, the shifter level has come into life. We do not know whether it is this switch of existentialities which so incredibly fascinates the auditory (and the orchestra), but it is an objectively dramatic event. It might be compared to the reading of the article on the first person singular pronoun “I” in an encyclopedia, and pronouncing the word “I” as a living person with all the shifted meaning pointed at when you say and mean “I”. The question of what really initiates when the performance gets off is not simple. It was possibly suggested in the previous discussion that it is only a time initialization, but this is as wrong as it would be wrong to claim that music reduces to onset time. In the initial moment of a musical performance, many different settings are instantiated, in fact as many as we have parameters to describe the sound events (and the gestural parameters in an extended performance theory) in our piece of music. In what follows, we want to investigate these initializations in more detail.
34.2
Anchoring Onset
Summary. The most elementary and important initialization is that of onset. We give the overview of its structure and function, including the shifter nature of onset initialization. The problem of multiple onset initialization is discussed. –Σ– This section uniquely deals with the Onset space form and performance fields on this space, i.e., tempo curves. Musically speaking, this approach is a naive one, in fact more involved performance of time does not happen independently of other parameters and therefore cannot be described by tempo curves. For example, if we are involved in a performance where onset is a function of pitch, a situation which may happen in an arpeggio. This is viewed as a temporary onset distortion as a function of pitch. A typical such performance map ℘ is as follows: 2
℘(E, H) = (E − 4e−(H−5) , H). The corresponding performance field Ts℘ is shown in figure 34.1. Observe that onset components of the field may also be negative, according to the retard of onset in the middle arpeggio region of pitch around H = 5 in this generic example. The data for our pure onset performance is as follows: We are given a frame interval R = [a, b] ⊂ Onset, a tempo “field”, i.e., a continuous positive tempo curve Ts(E) = T (E) defined on R, and a finite initial set I = {E0 < E1 < . . . Ef } ⊂ R within the frame. On this set, we are given performance data ℘(Ei ) = ei , i = 0, . . . f with e0 < e1 < . . . ef . The meaning of this data is that the composition K’s onset set KE = pE (K) is also a subset of
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CHAPTER 34. INITIAL SETS AND INITIAL PERFORMANCES
Figure 34.1: The performance field on the plane of onset (horizontal axis) and pitch (vertical axis), corresponding to a prototypical arpeggio. This field cannot be built upon a tempo curve. the frame, KE ⊂ R, and that its performance is calculated by the performance formula (33.1). More precisely, in the common situation, the initial set is the singleton I = {E0 } consisting of the composition’s first referential onset. This could be the first element in KE or else the first bar-line’s onset. In this case, no further discussion is required, the integral exists, and we can calculate all required onset performances. However, realistic situations are more involved. It may happen that one is given several initial points, for example if the left hand plays in constant tempo, whereas the right hand is allowed to vary locally against the left hand in a “Chopin rubato”, i.e., in such a way that the onset performances coincide on each bar-line, but it may differ locally. In this case, the initial points for the left hand would have a fixed performance whereas the right hand tempo curve would have to fit the left hand onset values. This will in fact happen in our discussion of tempo hierarchies, see section 38.2. This means that we have additional conditions, the integrals of inverse tempo must coincide with the given initial onset performances, i.e., for 0 ≤ i < j ≤ f , we must have Z
Ej
ej − ei = Ei
1 . T
(34.1)
34.3. THE CONCERT PITCH
699
Under these conditions, the calculation of any onset performance ℘(E) is clearly independent of the reference initial point Ei ∈ I. In this case we say that the tempo curve is adapted to the initial performance ℘I on the initial set I. Proposition 56 With the above notation and hypotheses, there is always a unique continuous tempo curve T (E) which is linear on each interval [Ei , Ei+1 ], i = 0, . . . f − 1, constant outside [E0 , Ef ], and prescribes an arbitrary positive tempo T (Ei ) for one index 0 ≤ i ≤ f . Proof. We prove the case of a fixed start tempo, the general case works in complete analogy. By induction on f , we may restrict the proof to f = 1. It is sufficient to show that we may find a final positive tempo value x = T (E1 ) such that the linear tempo Tx (E) = (x − T (E0 ))(E − RE E0 ) + T (E0 ) fulfills d = e1 − e0 = q(x) = E01 T1x for any positive value d. But q(x) = (E1 − (E0 )) −E0 E0 ) log(x)−log(T , which is a continuous function of x, converging to ET1(E as x → T (E0 ), x−T (E0 ) 0) and ranging from ∞ to 0 for positive x. Therefore there is exactly one positive x1 such that d = q(x1 ), and T (E1 ) = x1 solves our problem, QED.
Corollary 21 Suppose that for the increasing sequence E0 < E1 < . . . Ef of symbolic onsets, we are given positive tempi Ti = T (Ei ), i = 0, . . . f , and that the tempo curve T (E) on the RE interval [E0 , Ef ] is the polygon through these values. Let ∆ = E0f T1 , and select a positive real scalar σ. Then there is a positive scalar τ such that the polygonal tempo curve Tτ for the vertex RE values Tτ (E0 ) = T0 , Tτ (Ef ) = Tf , Tτ (Ei ) = τ Ti , 0 < i < f , has E0f T1τ = σ.∆. Proof. The integral
R Ef −1
1 Tτ
=
1 τ
R Ef −1
1 T
assumes any positive value if τ varies. Further, RE RE proposition 56 guarantees that the initial and terminal integrals E10 T1τ , Eff −1 T1τ can also be adapted to any positive value with varying τ , so we are done. This means that given a tangent morphism T g on the initial values E0 , Ef , and a polygonal tempo curve T which extends T g, we can “deform this curve” to a new polygonal curve Tτ without altering the tangent data T gE0 , T gEf , but such that the total duration is stretched by any positive value σ. In practical situations—such as the composition software prestor ’s Agologic module (see also chapter 49)—this has the following application: We are given a RE polygonal tempo curve as in the corollary, with duration ∆ = E0f T1 . By graphically interactive editing, the curve may be altered in that the vertical position (the tempo coordinate) of one inner vertex of the polygon is augmented or diminished. This changes the duration from ∆ to ρ.∆, but we do want to conserve duration. This can be achieved by a deformation of the graphically altered polygon as in the lemma, setting σ = 1/ρ, and we recover the original duration. So the new shape of the polygon can be conserved as far as possible.
34.3
E1
E1
The Concert Pitch
Summary. In the pitch dimension, initialization deals with concert pitch, i.e., initial intonation or tuning. We compare this initialization with the onset’s take-off and discuss the specific difference in deicticity: Concert pitch is dominated by a lexical dimension. –Σ–
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In this first approach, we shall view pitch performance as a parallel situation to onset performance as discussed in the previous section. This means that pitch performance is viewed independently of other parameters, in particular, independent of time. As with tempo curves, this is a musically naive, but basic approach. In fact, pitch performance may be a function of onset, according to the tonal system of the given composition, for example in modulatory parts. So we are given a frame R = [a, b] ⊂ P itch, an intonation field, i.e., a continuous intonation curve5 Ts(H) = I(H) on R, and a finite initial set I = {H0 < H1 < . . . Hf } ⊂ R. On this set, we are given performance data ℘(Hi ) = hi , i = 0, . . . f with h0 < h1 < . . . hf . In the standard case, this initial set is not as variable as for tempo. In fact, there are several constraints of tuning which are above all given by the orchestral instrumentation. For the piano, to begin with a common reference for orchestral tuning, we are given all semitone pitches I = {H0 < H1 < . . . Hf } of the chromatic 88 keys6 , together with their rigid performance pitches which are ℘(Hi ) = hi = h0 + i. log(2)/12 in the common 12-tempered tuning. The underlying intonation curve must cope with all 88 values. We have the following result which replicates proposition 56 for intonation: Proposition 57 With the above notation and hypotheses, there is always a unique continuous intonation curve I(H) which is linear on each interval [Hi , Hi+1 ], i = 0, . . . f − 1, constant outside [H0 , Hf ], and prescribes an arbitrary positive intonation slope I(Hi ) for one index 0 ≤ i ≤ f. The less rigid case is the tuning of the (continental European) chamber pitch a0 ∼ 440 Hz, together with the octave periodicity condition on the integral, i.e., Z H+12 1 = log(2) I H for all pitches H and the semitone encoding of pitch. Such a tuning is independent of local variations of intonation due to intonation specificities for just fifths and the like which violinists and singers may prefer. It is also open to glissandi which include all pitch values in a determined real interval. There is, however, a qualitative difference between intonation initialization and onset initialization. The latter is a “magic shifter” which by its very construction instantiates the fictitious music score time in real time. The former correspondingly instantiates fictitious score pitch in physical pitch, but its value is not a question of individual construction, it is lexicalized on the standards of music culture and tradition. It is also lexicalized by the absolute pitch perception of a number of musicians, for example Herbert von Karajan, which makes it virtually impossible to alter this conventional initialization on a shifter basis of individual, spontaneous usage. The usage of intonation and its initialization is therefore much more restricted than the usage of onset initialization. One may distinguish the local initialization within a fixed tonal context, between two successive modulations in just tuning of octaves and selected fifths (tonic, dominant for example), say, and the more individual shaping of the individual intonation of less prominent intervals within one such tonal context. 5 We are sorry for the homonymous symbol I for initial sets and intonation fields, a confusion is however very unlikely. 6 For the added keys on B¨ osendorfer’s Imperial model, extend the key numbers by −1, −2, . . ..
34.4. DYNAMICAL ANCHORS
701
It is clear that such a one-dimensional intonation field component is as artificial as the one-dimensional tempo field. It is not artificial in the sense of a useless academic exercise, but in the sense of first approximation to artistic performance—something like a zero-state of performance—which needs refinement. This is an important observation insofar as it suggests an investigation of the problem of refining performance, of seeking for paradigms of unfolding performance shaping. The theory of performance stemmata in chapter 38 will deal with this approach.
34.4
Dynamical Anchors
Summary. In the loudness dimension, initialization deals with reference dynamics. This is a dramatic and completely shifting phenomenon which stays in contrast to the onset’s take-off and the lexical pitch initialization. –Σ– Instead of repeating the propositions presented in the previous sections, which are, of course still valid mutatis mutandis, we should rather focus on the specific character of dynamical initialization. Above all, the performance of loudness in its physical expression is a complete shifter: There is no lexical normalization since every concert determines its dynamical initialization as a function of the concert hall, the orchestra, its disposition, and the public. Whereas the intonation curve may remain more or less the same over different performances of a determined orchestra, the initial anchoring will vary considerably, though not as fundamentally as with onset, because onset will be an existentially different one in each performance. There is another shifting character of dynamics initialization: the reproduction of a performance or the live broadcast from an electronic media, such as a radio, TV or internet concert broadcast or the simple playing of a CD on the private sound equipment. In these cases, the individual user’s preferences may initialize a very specific dynamic anchorage. And, more dramatically: During the ongoing performance, the initialization may be reset according to hearing dispositions and temporary irritations from disturbing environmental noise, including the abrupt lowering of dynamics while a phone-call or a verbal intervention of another person happens. Similar to onset initialization, dynamics initialization is quite strong and shifting, in contrast to pitch initialization. But it is also more existential, together with onset initialization: It reflects the human condition of when and how and why music is enjoyed. In this shifter process, evidently, the lexical musical content seems not to suffer, it is more the anchorage in human life which is profiled. Whereas absolute pitch seems to play a certain role for the understanding of the musical message—even to the vast majority of non-absolute pitch listeners—absolute onset and dynamics play the contrary role: They give the listener his/her coordinates of existence where they want to meet this particular music.
34.5
Initializing Articulation
Summary. The initial theory becomes less trivial when applied to articulation. Initial articulation reveals the complex recursive structure within initial data, i.e., sets and performance.
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We explicate different approaches to initial articulation, as based upon initial onsets. –Σ– Whereas initialization on one-dimensional performance fields is a question of selecting finite point sets, initialization on higher-dimensional frames is dramatically more complex. Let us have a look at the most elementary two-dimensional situation: the articulation field, see figure 34.2.
Duration Frame R (Eb,Db)
Xv (Ea,Da)
Xh
Onset
Figure 34.2: The integral curves of an articulation field may typically hit boundary points of the given frame on different positions: the horizontal lower boundary line or the vertical left one. This entails dramatically more complex initialization data than for the one-dimensional case of tempo and intonation. In this example, we are given a frame R defined by its low left vertex (Ea , Da ) and the high right vertex (Eb , Db ). (In this figure, we even suppose that Da = 0, but this is not the general case.) Let us first look at the right integral curve (terminating at the point on its arrow-head to be performed) which hits the frame boundary on Xh = (Eh , Da ). Initialization can be decomposed in two partial problems: Initialization of onset can be reduced to the onedimensional case since we have a tempo curve here. This means that we may consider the projection of the articulation field onto the tempo curve and accordingly the projection of
34.6. HIT POINT THEORY
703
the integral curve onto its onset component. Suppose further for simplicity that the onset initialization is defined on the frame boundary value Ea . Then we are done with the onset performance, and we may proceed with duration. Let us look at initial duration in Xh . Here, we have the onset performance and should define performance of duration Da . If this value vanishes, it seems natural to perform it to physical duration zero, too. But if the lower bound Da is positive, it is not clear what should be done. The canonical idea would be to identify Da as a difference Da = (Eh + Da ) − Eh = α+ (Xh ) − pE (Xh ). We could then apply the well-known construction of duration performance as a difference of onset performances, i.e., ℘(Xh ) = (℘E (pE (Xh )), ℘E (α+ (Xh )) − ℘E (pE (Xh )))
(34.2)
which is built upon the onset performance ℘E via the tempo curve and the initial performance on Ea . This is not mandatory though: Initial duration performance need not be without articulation, be it legato, be it staccato. Whatever: We have to be sure that not only the onset component of Xh is on the frame, the alteration component α+ (Xh ) must also be within the frame’s onset bounds! This implies that score points to be performed on this basis must stay to the left of the descending diagonal through the lower right corner of the frame. Other points must be given a different initial performance data. If, on the other hand, the frame is hit in a point Xv = (Ea , Dv ) on the left boundary line, its (initial) performance can be settled by the same formula as above ℘(Xv ) = (℘E (pE (Xv )), ℘E (α+ (Xv )) − ℘E (pE (Xv ))),
(34.3)
but now, the initial duration can be any long duration, not just the lowest admitted duration of our frame. In this setup, the initial performance of the left boundary line is a function of the onset performance of the whole onset interval [Ea , Ea + (Db − Da )], and not only of the initial onset performance at Ea . This is quite dramatic as a contrast to the formula (34.2) which is local on the onset of Xh and the slightly shifted onset α+ (Xh ). Here, we have to know a lot about the future onset performance for the initial values at the beginning of the frame. An even more dramatic effect happens if the frame has an upper boundary line which enforces integral curves to hit the upper boundary when reaching the rectangle’s boundary ∂R. In our example, this could happen if the frame ended at the half height (with the same field). There, the initial performance of duration can fail to be controllable by onset performance alone, and we have to design new initial performance strategies. Difficult initial performance problems can also happen if the field doesn’t have a positive onset component, as in figure 34.1. Here, initial performance must be defined on the right boundary hyperplane of the frame, i.e., for the last, not for the first events. And if the performance configuration is such that initial values happen to be positioned anywhere within the frame, initial performance has to be defined by use of general strategies which work for any initial set configuration. We shall review this topic in chapter 38 after the discussion of the basic problem: How can we effectively know where an integral curve hits the initial set?
34.6
Hit Point Theory
Summary. Hit point theory deals with the control of access to initial sets along the integral curves of performance fields. These curves describe the flow associated with performance fields.
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CHAPTER 34. INITIAL SETS AND INITIAL PERFORMANCES
In general, they can be quite “wild” so that a generic strategy of seeking hit points of such curves with initial sets is required. –Σ– The general setup for initial sets is a (locally Lipschitz) performance field Ts defined on (a neighborhood of) a (closed) frame R in a parameter space which—up to isomorphism of functors—is defined by Rn . In many situations, the initial S set for the performance ℘ is not any set, but will be a polyhedron defined as a union I = i Si of a finite family (Si )i of simplexes Si ⊂ R. Each simplex Si is given by a sequence si . = (si0 , si1 , . . . sin(i) ) of (pairwise different) points in R. The points need not be in general position, i.e., dim(R.si .) < n(i) is admitted7 . If we view this data as being the vertexes of the simplex Si = {Σj λj sij |0 ≤ λj ≤ 1, Σj λj = 1}, this means that we admit degenerate simplexes as constituents of initial sets. Also is it not required that the simplexes build a simplicial complex, their intersections can be arbitrary. We also do not require that these simplexes have n + 1 vertexes: it is allowed to have a number of isolated points and straight lines in three-space, for example. Clearly, this type of initial sets allows virtually every shape—up to approximations by triangulations. The previous examples are: a series of zero simplexes in R for tempo and intonation curves, or the sequence (∂E R, ∂D R) of left and lower sides of R in onset-duration space. More generally, given a coordinate Y (or its index j if Y is indexed with respect to Rn ) we shall denote by ∂Y R (or ∂j R) the simplex defined by the vertexes of R which have the lowest Y -coordinate (or j-th coordinate). Correspondingly, highest Y -coordinates define the simplexes denoted by ∂ Y R (or ∂ j R). Observe that these simplexes are degenerate for n > 2. The hit point problem is this: Problem 1 Given an integral curve, we have to decide whether and for which curve parameter R value the curve will hit the initial set I. More precisely, since an integral curve XRTs is defined starting R from an event X ∈ R, we ask for the smallest parameter |t| such that X Ts(t) ∈ I, i.e., X Ts(t) ∈ Si , one of the initial set’s simplexes.
34.6.1
Distances
Summary. This section is dedicated to the calculus of distances between points on integral curves and polyhedral initial sets. –Σ– Given an polyhedral initial set I which is defined by a family (Si )i of simplexes, and any point X ∈ R, the distance d(X, I) is the minimum of the distances d(X, Si ), therefore distances of points X and simplexes S must be calculated. Suppose that S is given by the sequence (s0 , s1 , . . . sm ) of points in Rn . If S is degenerate, it is the union of its non-degenerate sub-simplexes. This follows from the fact that it is a projection of a non-degenerate simplex, and that any point of this projection stems from a point of a side of the non-degenerate pre-image. So we are left with the calculation of the distance d(X, S) for a non-degenerate S. If dim(S) = n, clearly, either X is in S or the 7 The
vector space R.si . is the module associated with the local composition defined by the vertexes si . of Si .
34.6. HIT POINT THEORY
705
distance is achieved on a point of ∂S. So in the latter case, we are in a recursive situation (the zero-dimensional simplex being trivial). To decide upon the position of X relative to S, we consider the unique representation X − s0 = (X − s0 )⊥ +
m X
λj .(sj − s0 )
(34.4)
j=1
of the difference X − s0 as a linear combination of the basis vectors sj − s0 and the vector (X − s0 )⊥ of the (Euclidean) orthogonal space R.s.⊥ . Considering the non-singular symmetric quadratic form of scalar products Q = ((sj − s0 ) · (si − s0 ))i,j=1,...m and the vectors U = ((X − s0 ) · (si − s0 ))τi=1,...m , Λ = (λj )τj=1,...m , we have Λ = Q−1 .U. With this, the barycentric coordinates are defined by adding the coefficient λ0 = 1−Σi=1,...m λi ; we have m X X = (X − s0 )⊥ + λi .si , i=0
Pm and the component y = i=0 λi .si in the affine space spanned by S is in S iff 0 ≤ λi for all i = 0, . . . m. We now have d(X, S) = k(X − s0 )⊥ k + d(y, S). If the second distance is not zero (y 6∈ S), we can proceed with the recursive calculation of d(y, ∂S), and we are done. Observe that this algorithm also gives us the coordinates of a point of S which has minimal distance to X. In computer programming practice, however, it is not reasonable to check for vanishing of distances d(X, S) because of rounding and number representation errors. We therefore should prefer to calculate whether d(X, S) < for a selected positive neighborhood variable . The corresponding routines are obvious. Denote the resulting point in S, which is found by this algorithm, and which has the shortest distance of points within S to X by M in(X, S), whereas its barycentric coordinates are denoted by M in(X, S)i , i = 0, 1, . . . m, i.e., M in(X, S) =
m X
M in(X, S)i .si .
(34.5)
i=0
As a consequence, one can use this algorithm to calculate the distance of a straight line L ⊂ Rn to a simplex S as follows: Take the projection p(S) of S on an affine hyperplane HL orthogonal to L. Then p(S) is defined by the projected points p(si ), and our algorithm applies to the singleton Y of the projection p(L) = {Y } and to p(S), giving d(L, S) = d(M in(Y, p(S)), p(S)). Moreover the coordinates M in(Y, p(S))i give us a point M in(L, S) =
m X
M in(Y, p(S))i .si ,
(34.6)
i=0
which evidently lives in S and has minimal distance to L. In our applications for integral curves, the line L is parametrized by a point X and a directional vector D, i.e., L(t) = X + t.D. Then the parameter λ on L such that we have d(L(τ ), M in(L, S)) = d(L, S) is denoted by τ (X, D, S).
706
34.6.2
CHAPTER 34. INITIAL SETS AND INITIAL PERFORMANCES
Flow Interpolation
Summary. This section deals with selection algorithms for searching points on an integral curve of the performance field which are successively approximated to initial sets. –Σ– After splitting the approximation to an initial set I to the approximation of one of its n simplexes, the problem is this: We are given a positive , a point X, and a simplex R S in R , and we have to decide whether, and in which parameter value, the integral curve X Ts hits the neighborhood8 U S of S. Theoretically, one could just calculate the integral curve and search for a parameter that does the job—if there is any. But in computer programs, such a procedure is illusory. Numerical integration of ordinary differential equations is a time-consuming, expensive task. One cannot afford to calculate all curve points for any score event of a normal composition, which usually contains 104 − 105 events. We shall see further in section 39.4.4 that fields of any complexity may occur by chains of successive reshapings of given fields via arbitrary performance operators—same for the complexity of initial sets. Also the performance fields and the initial sets are of a completely arbitrary relative position. So the situation is this: We are given a point X and a simplex S. We know in what direction—namely TsX —the integral curve starts (with positive or negative curve parameter values). Nothing more. So we have to guess where the curve could approach S as well as possible. We have to evaluate the guess, and then start with another guess, etc. Eventually, we find a curve point in U S and we are happy, or else we will have to give up the search and decide that the curve did not hit U S. This could be a wrong decision, but (calculation) time runs out and we have to resign. Intuitively, the search is best described by the following scenario: We have a paper dragon being suspended at a fixed position in the air. We also have a nervous fly, flying around in its random zig-zag (however differentiable) manner where you never know which will be the next turn. This scenario is traced on a video recorder, and the video really shows whether and when the fly hits the dragon. But we are not in the state of viewing the entire film. Rather are we given a determined sequence V F (0) = (V ideoF rame(ti0 ), V ideoF rame(ti0 +1 )) of two successive frames. This gives us the fly’s position and its velocity vector at time ti0 (supposing that the video is binocular...). We now have to guess, which video frame sequence could be the next best that shows the fly as near as possible to the dragon. We then forward the video to this next frame sequence V F (1) and judge the new situation, and so on, until we find a hit point time or else we run out of time. The first action is to guess a good curveR time from the starting point X and the field vector TsX . Denote by x(t) the integral curve X Ts(t). As nothing is known about the curve’s future directions, we draw the straight line LX (t) = X + t.TsX and look for the parameter value t1 = τ (X, TsX , S) defined above, which gives the nearest point to S on LX , see figure 34.3. We now have to compare d(x1 , S), and d(X, S). If the former is smaller than the latter, we may proceed, if not, we are in a bad position. Of course, this linear first approximation is not necessarily the best one to get off, since the field may be rather circular than linear in this region. We could therefore try other first approximations, such as a circle in the plane9 spanned by the barycenter of S, X, and TsX , for example the circle through the barycenter of S and X, 8 This 9 Since
is U S = {x| d(x, S) < }. the linear approximation failed, this must be a plane!
34.6. HIT POINT THEORY
707
S
LX
Ts0 = TsX
X = x0 = x(t0)
x1 = x(t1) Ts1
Figure 34.3: The first approximation to a hit point on a simplex S when starting from the point X of the given composition is found by a linear approximation LX . and tangent to TsX . We could follow this alternative, but we refrain from this because it does not demonstrate qualitatively new problems. For the following interpolations, recall exercise 71 in section 32.3.2 on cubic splines. This tells us that, given any two points x(s), x(t) on our integral curve, there is exactly one cubic interpolation function Ps,t : [s, t] → Rn with P (s) = x(s), P (t) = x(t), and tangents T Ps = Tsx(s) , T Pt = Tsx(t) . We shall use these approximation curves to guess optimal points since we cannot calculate all curve points for an interval [s, t]. We now proceed as follows. We are given the curve parameter t1 whose point x1 = x(t1 ) is nearer to S than x0 . We now repeat our linear approximation procedure with the line Lx1 and get a new curve parameter t∗ . We also calculate the cubic interpolation point (this is a cheap calculation effort) for Pt0 ,t1 at parameter (t0 + t1 )/2. If x(t∗ ) is nearer to S than x1 and Pt0 ,t1 ((t0 + t1 )/2), we set t2 = t∗ , else, if Pt0 ,t1 ((t0 + t1 )/2) is nearer than x1 , we calculate x((t0 + t1 )/2), and, if this latter is still nearer than x1 , we set t2 = (t0 + t1 )/2. Else, we are stuck and have to quit. The general situation is this: We have calculated a sequence X = x0 , x1 , . . . xk of successively nearer points with curve parameters ti , i.e., x(ti ) = xi . Essentially we now have to check what happens to the left and to the right10 of the best point xk . Suppose that tr , ts are the right and left neighbor times to tk . We now calculate the interpolation points Ptk ,tr ((tk + tr )/2) and Ptl ,tk ((tl + tk )/2). If one of them, with parameter t∗ , is better than xk , we calculate nearer one’s curve point x(t∗ ). If this is still nearer than xk , we have found tk+1 = t∗ . If not, we are stuck and give up. This procedure will be repeated until a maximal admitted number σ of steps, and as long as the points xk , k ≤ σ are not in U S. The procedure stops if we run out of steps or if we eventually hit the given neighborhood of the simplex. This algorithm has been implemented in 10 It could happen that x is the right or left extremal one. Then, we have to make linear interpolation on k that side, but this situation has been explained for the construction of the third point x2 above.
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CHAPTER 34. INITIAL SETS AND INITIAL PERFORMANCES
the PerformanceRUBETTEr of the RUBATOr software, see section 41.4. It seems that the general situation of performing points X meets serious problems for points whose integral curve does not hit any initial point (or, at least, a point in a small neighborhood of the initial set I). This is however not so tragic since in reality, performance is not a one-step process: Stemma theory (chapter 38) will show us that any performance is unfolded from a previous, less artistic performance, starting on the quasi-mechanical reproduction of the score. Therefore, if at a given stage, a point X cannot be performed, its performance which is already defined on a less artistic level can always be called in order to save the performability of the given score. A final technical remark about the above algorithm: We implicitly supposed that the space where point X and simplex S live is the space of a performance field Ts. However, we used only an integral curve through X, and not the performance field itself. This means that we only needed a curve x(t) through a point X, and no information about the curve’s genealogy. So the algorithm is also valid for any such “isolated” curve. This will be used later in our discussion of hierarchies of local performance scores in chapter 35. In theRnext chapter, we shall use R− R this notation: If X is in the given frame R, we denote + by X Ts ( X Ts) the restriction of X Ts to the maximal interval contained in the interval ] − ∞, 0] ([0, +∞[) such that all values of the curve on this interval are contained in R. Given a performance field Ts, ∈ {+, −}, and an initial set I, we set: Z Z I Ts = {X ∈ R| I ∩ Ts 6= ∅} (34.7) X
Here is a sorite concerning this symbol: Sorite 11 In the following statements, direct products of performance fields Ts1 , Ts2 , initial sets I1 , I2 , and frames R1 , R2 refer to the product (limit) space SS built from the given factors SS1 , SS2 . R (i) The operator I 7→ I Ts conserves inclusions, commutes with unions and is idempotent. R R R (ii) Setting I1 I2 = I1 × (I2 Ts2 ) ∪ (I1 Ts1 ) × I2 , we have Z Z Z Z (I1 I2 ) Ts1 × Ts2 = I1 Ts1 × I2 Ts2 . Proof. Statement (i) is straightforward by the uniqueness of integral curves. We show statement (ii) for = −, the other case is analogous. For the inclusion “⊂”, suppose WLOG that we have R− R− R− Ts1 ×Ts2 . This means that we can reach a point of I1 ×(I2 Ts2 ) (X, Y ) ∈ (I1 ×(I2 Ts2 )) R− from (X, Y ) on the integral curve (X,Y ) Ts1 × Ts2 . Since integral curves of a product of vector R− fields project to integral curves of their factors, and by the idempotency of Ts2 , this implies R− R− (X, Y ) ∈ I T s × I T s . As to the inclusion “⊃”, suppose that (X, Y ) is such that 1 1 2 2 R R R T s (s) ∈ I , V = T s (t) ∈ I , and WLOG s ≤ t ≤ 0. Setting W = T s (t) we conclude 1 1 2 2 1 X Y X R− Ts1 × I2 . By the following lemma, we can reach (X, Y ) from (W, V ) on an (W, V ) ∈ I1 R− R− integral curve of Ts1 × Ts2 from t to 0, and we conclude (X, Y ) ∈ (I1 I2 ) Ts1 × Ts2 , QED.
34.6. HIT POINT THEORY
709
Lemma 50 If we are given two locally Lipschitz vector fields Ts1 , Ts2 on respective domains D1 , D2 and two (maximal) integral curves x1 : J1 (0) → D1 , x2 : J2 (0) → D2 , the diagonal curve x : J1 (0) ∩ J2 (0) → D1 × D2 : t 7→ (x1 (t), x2 (t)) is the (maximal) integral curve of Ts1 × Ts2 on D1 × D2 through the couple x(0) = x1 (0), x2 (0)) of initial points. Proof. Clearly, the diagonal is an integral curve since differentiation goes factorwise. On the other hand, if we had a proper extension of x in the product space, its projections p1 ◦ x, p2 ◦ x would yield two integral curves, one of which would have an extended domain, which contradicts the choice of x1 , x2 , QED.
Chapter 35
Hierarchies and Performance Scores On trouve toujours l’homog`ene ` a un certain degr´e de division. Paul Val´ery [538, I, p.209] Summary. As a synthesis of the structural parts described in chapter 32 through chapter 34 we establish the overall structure of performance. The objects of performance structure are understood as being an additional score type, called performance score, layered over the given “symbolic” score like a system of optical lenses which ‘deform’ the rigid configuration of note symbols. The performance score is a global object built from an atlas of local performance scores. A local performance score is built from a hierarchy of performance cells, the very core of performance structures. We first describe the category of performance cells. Local performance scores are defined by a hierarchical construction principle: They are particular diagrams in the category of performance cells. The conceptual and musical background of such local hierarchies is evidenced through a series of examples, including the piano and violin hierarchies. We end up with the definition and exemplification of the concept of a (global) performance score. –Σ–
35.1
Performance Cells
Summary. Performance cells are the very local data of performance. They comprise the cell’s frame (a domain of definition), the symbolic kernel (a set of prima vista objects), the performance field, the initial set, and the initial performance. –Σ– The innermost local structure of performance is the performance cell. We have known all of its ingredients and shall now set up the formal definition of such a cell. 711
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Definition 101 Given ∈ {+, −}, a performance cell is a five-tuple C = (K, R, Ts, I, ℘I ) consisting of the following objects: 1. a local composition K ⊂ SS whose space is simple of underlying module Rn , K is called the symbolic kernel of the cell; 2. a closed frame R = [a1 , b1 ] × . . . [an , bn ] contained in Rn and containing K; 3. a locally Lipschitz-continuous performance field Ts which is defined1 on R; R 4. an initial set I such that K ⊂ I Ts (see equation (34.7)); n 5. an initial performance ℘I : I → P S, with R codomain a physical space R P S with module R R , and such that for any point X ∈ K ∩I Ts and any two points a = X Ts(α), b = X Ts(β) in I, we have ℘I (b) − ℘I (a) = (α − β).∆, where ∆ = (1, . . . 1) is the diagonal vector in P S’s Rn .
Often, I is given as a union of a finite family (Si )i of possibly degenerate simplexes Si , but this is not the general case. A performance cell is visualized by a tetrahedron as shown in figure 35.1. We therefore also call (R, Ts, I, ℘I ) a performance body for K.
℘I
R
I Ts Figure 35.1: Visualization of a performance cell by a tetrahedron, the inner ball symbolizes the cell’s symbolic kernel, whereas the 4-tuple of the tetrahedron’s vertexes is called the performance body for the symbolic kernel. Given a performance cell C, we automatically have a well-defined performance map ℘C : K → P S which we shall always refer to when talking about the performance ℘ associated with C. Without stressing the contrary, all our cells will be −-cells, the theory for = + is the same. We shall only occasionally consider mixed signatures. 1 Recall
R.
that this means that Ts is defined on an open neighborhood of R, but we only identify the field on
35.2. THE CATEGORY OF PERFORMANCE CELLS
35.2
713
The Category of Performance Cells
Summary. Performance cells constitute the objects of the category PerCell of performance cells, the morphisms being the technical expression of relations between performance data on different parameter spaces. –Σ– In nuce, we have already seen phenomena of related performance cells in the discussion of parallel fields as they arise in the Onset ⊕ Duration space. There, we had the projection pOnset : Onset ⊕ Duration → Onset which was compatible with the parallel field ∂T empo and the tempo curve T empo. We shall now set up the formal statement behind those incipits. What we want is a concept of a morphism C1 → C2 which defines the category PerCell of performance cells such that the associated performances are compatible with this morphism. Definition 102 (Recall that we have fixed the signature = − in the following discussion!) Let C1 = (K1 , R1 , Ts1 , I1 , ℘I1 ) and C2 = (K2 , R2 , Ts2 , I2 , ℘I2 ) be two performance cells living in spaces SS1 and SS2 . Suppose further that we have a projection p : SS1 → SS2 of the parameter space SS1 onto the parameter space SS2 such that the underlying projection p : Rn1 → Rn2 is the projection onto a subset of coordinates. Then p is a morphism of performance cells p : C1 → C2 iff the following conditions are verified: 1. p(K1 ) ⊂ K2 (in other words: p : K1 → K2 is a morphism of local compositions); 2. p(R1 ) ⊂ R2 ; 3. T p ◦ Ts1 = Ts2 ◦ p, i.e., a morphism of vector fields p : Ts1 → Ts2 ; R 4. p(I1 ) ⊂ I2 Ts2 ; 5. p ◦ ℘I1 = ℘2 ◦ p|I1 (here, p denotes the corresponding projection on the physical spaces p : P S1 → P S2 ); Lemma 51 With the above notation, if p : C1 → C2 is a morphism, then we have p ◦ ℘1 = ℘2 ◦ p.
(35.1)
with the homonymous p symbol for the mental and physical projections. R R Proof. Let X ∈ K1 , and suppose that X Ts1 hits I1 at the point Y = X Ts1 (t). (By axiom 5 of the definition 101 of a performance cell, it does not matter, which hit point we are selecting.) Then, ℘1 (X) = ℘I1 (Y ) − t.∆1 , and by linearity of p, p ◦ ℘1 (X) = p ◦ ℘I1 (Y ) − t.∆2 . But p ◦ ℘I1 (Y ) = ℘2 (p(Y )) = ℘I2 (Z) − s.∆2 , if Z is the hit point for p(Y ) in I2 , s is the curve parameter for this hit point. Hence, p ◦ ℘1 (X) = ℘I2 (Z) − (s + t).∆2 . But integral curves of Ts1 are projected into integral curves of Ts2 by axiom 3 of definition 102 and by the uniqueness of integral curves (see the fundamental theorem of ODE 78, appendix I.2.2). Therefore, the curve R parameter s + t is also the parameter of the integral curve p(X) Ts2 where it hits Z, and this means that ℘I2 (Z) − (s + t).∆2 = ℘2 (p(X)), QED.
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This means that performances of performance cells behave in the sense of natural transformations with respect to the category PerCell. Exercise 78 Work out the example of a parallel articulation field projection and its associated performance body data, as discussed in section 34.5, to obtain a morphism of performance cells. The next proposition gives us means to construct product performance bodies and product cells from two given cells. Proposition 58 Suppose that we are given two performance cells C1 = (K1 , R1 , Ts1 , I1 , ℘I1 ), C2 = (K2 , R2 , Ts2 , I2 , ℘I2 ) living in spaces SS1 and SS2 . Let SS be the product of SS1 and SS2 , with projections pi : SS → SSi , i = 1, 2. Then the following data defines a performance cell C on SS: 1. the frame is R = R1 × R2 ; 2. the field is Ts = Ts1 × Ts2 ; 3. the symbolic kernel is any sub-composition K ⊂ K1 × K2 ; R 4. the initial set is I = I1 I2 ; R 5. on I, the initial performance ℘I is defined as follows: if (x, y) ∈ I1 × I2 TsR2 , then we set ℘I (x, y) = (℘I1 (x), ℘2 (y)), and symmetrically for the other case (x, y) ∈ I1 Ts1 × I2 . With his data,we have two morphisms pi : C → Ci , i = 1, 2 of performance cells. R Proof. As to the construction to check the relation K ⊂ I Ts1 × Ts2 . From R R of C, we only Rhave sorite 11, we know that I Ts1 × Ts2 = I1 Ts1 × I2 Ts2 . But the latter evidently contains K1 × K2 , and a fortiori K, whence the claim. The morphisms pi are now straightforward by construction, QED. The performance cell C, together with the two morphisms p1 , p2 will be called the product of the cells C1 , CR2 and denoted R by C1 × C2 |K, or simply C1 × C2 if K = K1 × K2 . Although the initial set I1 Ts1 × I2 Ts2 of the product performance cell is not a union of simplexes in general, the product formula of sorite 11 guarantees that the calculation method for hit points on simplexes discussed in section 34.6.2 may be applied to each component of an event X = (X1 , X2 ) if the factor cells have initial sets which are built from simplexes.
35.3
Hierarchies
Summary. Hierarchies are space diagrams arising in local performances. They trace the functional organization among the arguments of performance fields. –Σ–
35.3. HIERARCHIES
715
In order to obtain more concrete results, we shall first consider a particular system of symbolic and physical spaces. We suppose that we are given a series Bi , i = 1, 2, . . . of pairwise different simple spaces, called basis spaces, as well as an equipollent series Pi , i = 1, 3, . . . of spaces, called pianola spaces, with Pi −→ Syn(Bi ), while Bi −→ Simple(Rni ). Often, we shall Id
Id
consider finite products Bi1 ×Bi2 ×. . . Bik ×Pj1 ×. . . Bjl of such spaces, and always ordered with basis spaces first, and pianola spaces second, and each space type ordered by increasing index (no repetitions!). This means that we in fact parametrize such products with finite subsets of the name set BP = B ∪ P, B = {Bi |i = 1, 2, . . .}, P = {Pi |i = 1, 2, . . .}. We shall then identify these product spaces with the simple space Bi1 ⊕ Bi2 ⊕ . . . Bik ⊕ Pj1 ⊕ . . . Bjl −→ Simple(RN ) associated with the direct sum RN of all involved copies of real Id
vector spaces. If the defining sequence Bi1 , . . . Bik , Pj1 , . . . Pjl is denoted by U , we shall also rename the space Bi1 ⊕ Bi2 ⊕ . . . Bik ⊕ Pj1 ⊕ . . . Bjl by ⊕U . By definition, the space associated with the empty sequence ∅ is the simple space ⊕∅ −→ Simple(R0 ) of the zero module. Id
This generalizes the nomenclature introduced in the description of standard spaces (see ∼ equations (6.69) ff.), such as the piano space Onset ⊕ P itch ⊕ Loudness ⊕ Duration → Onset × P itch × Loudness × Duration referring to (symbolic) onset, pitch, loudness, and duration, and represented by R4 . For any subsequence of symbols V = Bu1 , Bu2 , . . . Bur , Pv1 , . . . Bvs of U = Bi1 , Bi2 , . . . Bik , Pj1 , . . . Bjl , we have a canonical projection pU,V : ⊕U → ⊕V of such standard spaces, also denoted by pV or p, if no ambiguities are possible. For any two such sequences V, W , we denote by V ∪ W (V ∩ W ) the sequence defined by the union (intersection) of the basis and pianola symbols in BP. Similarly for other set-theoretic operations, such as complement in BP, differences V − W , etc. This means that we consider the Boolean algebra Sub(BP). When considering the spaces ⊕U associated with such sequences U , we often speak loosely about the sequences and mean the spaces, for example, we speak of the “union of spaces” ⊕U, ⊕V and mean the space ⊕(U ∪ V ) = (⊕U ) ⊕ (⊕V ), etc., but no confusion should occur... Definition 103 Given a finite space collection BP, a space hierarchy in BP is a non-empty sublattice H ⊂ Sub(BP) (closed under finite unions and finite intersections), with maximal element (the top space) T op(H); H is viewed as a category with U → V iff V ⊂ U . The minimal nonempty elements of H are called fundamental spaces, their set is denoted by F und(H). For any non-empty space U ∈ Sub(BP) which is contained in T op(H), we denote by ClH (U ) the unique smallest space in H which contains U , and call it the hierarchy closure of U . A hierarchy space U ∈ H is called indecomposable if it is not the union of two disjoint non-empty subspaces of H. The motivation for the hierarchy concept is that we want to consider systems of performance cells which are related by morphisms, such as the parallel field morphisms or the product cells and their projections. Moreover, we have to group performance cells which are dominated by a root cell since all score events will be performed in one big space whose projections are however compatible with the root space.
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Lemma 52 Given a space hierarchy H, let F und(H) = {F1 , . . . Fk } be the set of fundamental spaces of H. Then (i) Each Fi is contained in a unique maximal indecomposable space T (Fi ) of H. (ii) Any two different maximal indecomposable spaces in H are disjoint. (iii) The top space T op(H) is the disjoint union of its maximal indecomposable subspaces. Proof. The union of two indecomposable superspaces of each Fi in H is indecomposable, therefore there is a unique maximal indecomposable superspace T (Fi ) for each Fi . Evidently, the maximal indecomposable spaces are mutually disjoint, i.e., for T (Fi ) 6= T (Fj ), T (Fi ) ∩ T (Fj ) = ⊕∅. The existence of a decomposition of T op(H) as a product of maximal indecomposable subspaces follows by induction on the cardinality of T op(H), QED. The set of fundamental spaces contained in one maximal indecomposable space of H are called the blocks of the fundament. Hence the blocks are in bijection with the maximal indecomposable subspaces. Definition 104 If an indecomposable space contains a unique (proper) maximal subspace— including the zero space—it is called irreducible. Hence, every indecomposable space is either irreducible or it is the union of its maximal subspaces. Fundamental spaces are irreducible, the zero space is not. With this we may now proceed to define the performance hierarchies. To this end, we consider the zero performance cell C∅ which is defined by a frame over the zero space ⊕∅ and has everything trivial: unique zero field, trivial frame, one-point initial set, zero initial performance. Here is the relation of space hierarchies to performance theory (see also figure 35.2): Definition 105 Given a space hierarchy H, a cellular hierarchy is a diagram (in fact, a functor on the category H) h : H → P erCell with values in the category of cells P erCell such that for each U ∈ H, h(U ) is a cell in the space ⊕U , with morphisms being denoted by h(pU,V ), pU,V , pV , or p if no ambiguities are likely. The domain H of h is called the type of the cellular hierarchy. The meaning of a cellular hierarchy is that we are given a performance on the kernel of the top cell T op(h) = h(T op(H)), which is compatible with the performances on all other cells h(U ) of h. So the parameters of top kernel events which are also grouped on lower hierarchy spaces can be performed independently of the other parameters. Correspondingly, the performance field components grouped on a lower hierarchy space are independent of the other space parameters, a situation already encountered for the articulation hierarchy Onset⊕Duration → Duration, for example. Moreover, if we have two cells h(U ), h(V ), their performance fields TsU , TsV evidently define performance fields TsU ∪V , TsU ∩V on the arguments of the union and intersection of their spaces, and this is met by the lattice structure of H. The signification of a space being irreducible now also becomes more evident: If a hierarchy space is the union of two proper subspaces, any coordinate of the performance of an event may be calculated via projection of the event into an appropriate subspace. However, for an irreducible
35.3. HIERARCHIES
717 D
H
EHLD
E
L
EHD
D
ELD D
E
E
L
H
ED D
E
E E
Figure 35.2: A cellular hierarchy is shown, together with the initial sets (blue simplexes), the performance fields, the projections, and the symbolic kernels (red points). space, we have to make an extra effort to know how its kernel events are performed since no subspace will give us full information. The parallel field construction from section 33.2.1 is a standard example for this: The duration component of the articulation field is not reducible to the cell of a subspace, whereas the onset component is. So the hierarchy does not tell us how to compute the duration component. The parallel construction is one possibility among an infinity of others to deduce the duration component from the fundamental tempo field by means of a special “formula” which in more generality has been explicated in formula (33.10). If we are given a space hierarchy H with T op(H) = ⊕BP = ⊕B ⊕ P , we shall henceforth assume that for each pianola component Pi of P , the corresponding basis component Bi is contained in the basis B, whereas a basis component Bi may happen to live alone without its pianola counterpart in P . Such hierarchies are called standard, if we have to distinguish them from other, non-standard hierarchies. For a standard space hierarchy, we therefore always have
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CHAPTER 35. HIERARCHIES AND PERFORMANCE SCORES
the alteration projections α± : ⊕BP → ⊕B. It is the usual alteration on the couples ⊕Bi ⊕ Pi , and the identity on the unparalleled basis components.
35.3.1
Operations on Hierarchies
Summary. Hierarchies can be altered and recombined according to a set of standard operations. Such operations intervene in practical calculations of performance transformations in software algorithms. –Σ– Given a cellular hierarchy h, if U is any subspace of the underlying space hierarchy H, we may restrict h to the sublattice H|U of H whose top space is U . We may then restrict the cellular hierarchy h to this sublattice. The restriction is denoted by h|U . If we are given two cellular hierarchies h, k with space hierarchies H, K, and whose spaces pertain to the given space collection BP , and such that their top spaces are disjoint, then we have the product cellular hierarchy h × k whose domain is the space lattice with spaces U ∪ V, U ∈ H, V ∈ K. Clearly, in this product space hierarchy, a pair U ∪ V extends U 0 ∪ V 0 iff each component does so in the respective space hierarchy H, K, respectively. So for each such relation and the associated morphism p : U ∪ V → U 0 ∪ V 0 , we have the corresponding morphism of performance cells p : h(U ) × k(V ) → h(U 0 ) × k(V 0 ). The only non-trivial statement within this R fact concerns property 4 of definition 102 and follows from the idempotency of the operator ? Ts (sorite 11, (ii)). Given a space U in the name space B of basis spaces, we call a parallel space to U and denote by ∂U the space U ∪ P |U consisting of U and the space P |U of all the corresponding pianola space components. If a cellular hierarchy h is such that for each projection ∂U → U of spaces in its space hierarchy H, the performance field over ∂U is the parallel field ∂Ts to the field Ts of U (i.e., of the cell h(U )) in the sense defined in formula (33.10), then we say that h is a parallel hierarchy. Parallel hierarchies are the default hierarchies where performance is initiated.
35.3.2
Classification Issues
Summary. For small sets of parameters, frame structures of cellular hierarchies are completely classified. We describe the classification for hierarchies involving tempo, intonation, and dynamics. –Σ– We do not claim classification of the full-fledged cellular hierarchies, this is a much too difficult task, and it is not of primary interest; it is easier and perhaps more relevant for practical reasons to classify “frame structures” for concrete cellular hierarchies. Such a frame structure is the hierarchical organization of the hierarchy’s performance fields. We shall present a complete classification of hierarchies sitting over the set B = {E, H, L} of the three usual basis spaces E =
35.3. HIERARCHIES
719
Onset, H = P itch, L = Loudness. For each hierarchy space U = E, H, L, EH, EL, HL, EHL, we denote the corresponding field as built from the symbols T = T empo, I = Intonation, D = Dynamics. Hence, T is the field over E, I over H, D over L, whereas T D denotes the field over EL, etc., and T ID the field over EHL. We write T I × D for a product field corresponding to the space EHL which is decomposable into the subspaces EH and L in the given hierarchy h. Figure 35.3 shows the classification Hasse diagram, with a straight line from every hierarchy to its next specializations. Here, specialization means that field components become independent from certain parameters with increasing split space hierarchies. TID TID
TID
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T
I
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T¥I¥D T¥I
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Figure 35.3: The complete classification Hasse diagram of basis hierarchy frame structures for onset (E), pitch (H), and loudness (L) in terms of corresponding fields, including specialization (straight lines) to more split fields according to functional independency of parameters. For a performance field Ts = TsU , U ∈ H of a cellular hierarchy h with space hierarchy H, we may ask for its functional dependence within h. Let Ts = TsU be the field of the maximal subspace U of U in H. This is called the territory of Ts, and describes the portion
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of Ts which is completely determined by lower hierarchy data. By the extraterritorial part of Ts, denoted by Tsex , we mean the projection of Ts to the complementary space U − U , so that we have Ts = (Ts, Tsex ). In general, the extraterritorial part of Ts is a function of its territory and of a system P ara of parameters which are external to the hierarchy; we write Tsex = Tsex (Ts, P ara) to indicate this dependency. So the performance fields of the fundamental spaces, the fundamental fields, play a primordial role in the construction of the hierarchical architecture, which is enriched by external parameters in the case of irreducible spaces (for others, the parameters are non-existent). Example 53 Suppose we are given a parallel hierarchy with just one parallel projection ∂Ts → Ts over the space projection ∂U → U , according to the formula ∂Ts = (Ts◦pU , 2.Ts◦α+ −Ts◦pU ) (33.10). Here, we have the empty parameter set P ara = ∅, and ∂Ts = Ts, ∂Tsex = 2.Ts◦α+ −Ts. If we generalize this configuration by the extraterritorial part, then ∂Tsλ,ex =
1+λ .Ts ◦ α+ − Ts. λ
The parameter set is P ara = {λ}, and we recover the old situation by a specialization λ → 1. We shall see in the discussion of performance operators that this parametrization is a natural one in the context of operators for articulation (legato, staccato). These concepts apply to the standard situations while performing special musical effects which we list here: 1. Pianola deformations2 of parallel pianola fields of tempo: Ts = (T, Zex (T, P ara)) with specialization to ∂T . This is used for articulation as discussed above, for local performance cells of ornaments and ties. 2. Deformation of dynamics over tempo: T D = (T, T Dex (T, P ara)) with specialization to T ×D. This is used for ondeggiando (bow vibrato) effects and prima vista dynamics accentuation following bar-lines and time signatures. 3. Deformations of tempo over dynamics: T I = (I, T Iex (S, P ara)) with specialization to T × I. This is used for arpeggios if we agree to quit simultaneity of arpeggio events and to change their physical onsets. 2 Deformation and specialization are reciprocal processes. The first is the embedding of a particular structure in a topologically dominant set of variants (usually in some Zariski topology of algebraic geometry), the second is the restriction of an irreducible set of variants to a special, closed subset, or even a single point, usually by specialization of parameters, see also Appendix F.
35.3. HIERARCHIES
721
The last two deformation types are shown in the lower part of figure 35.3. More systematically, this classification shows these types of specialization (we refer to figure 35.3 in this list): 1. Type: shown on the lowest line, no extraterritorial part, only the one-dimensional fundamental fields T, I, D. 2. Type: shown on the second lowest line, on top no extraterritorial part, in the middle of the hierarchy one two-dimensional field (e.g., ID = (D, IDex (D, IDex (D, P ara)) in the left extreme hierarchy) with a one-dimensional territory (in the example: D), and two fundamental spaces (in the example: T, D). 3. Type: shown on line three from below. One dimension is extraterritorial and we have two fundamental one-dimensional fields (e.g., D in the left extreme hierarchy, with3 T ID = (T, I, T IDex (T × I, P ara)) and the fundamental fields T, I.) 4. Type: line three from above. Only one fundamental field, above which a one-dimensional extraterritorial field component is built, and the same for the top space, it is a onedimensional extension of its two-dimensional territory (e.g., the left extreme hierarchy with T ID = (T I, T IDex (T I, P ara)), T I = (T, T Iex (T, P ara))). 5. Type: second line from top, right half. The fundament is two-dimensional and the top field has a one-dimensional extraterritorial part (e.g., the right extreme hierarchy with T ID = (T D, T IDex (T D, P ara)). 6. Type: second line from top, left half. The fundament is a one-dimensional field, and the top field has a two-dimensional extraterritorial part (e.g., the left extreme hierarchy with T ID = (T, T IDex ((T, P ara)). 7. Type: top line. Here we are left with the single total field T ID without any subspaces; it is at the same time its own irreducible territory and fundamental field. Nonetheless, complete classification of cellular hierarchies should be a major issue of future research, although a difficult one. To this end, one notices that a morphism h → k between cellular hierarchies should be defined as a natural transformation of these functors which, by definition, starts from one and the same space hierarchy. However, the “horizontal” natural morphisms of performance cells should be generalized beyond simple projections of parameter spaces since it is natural to say that a performance 3 We
just write down the whole sequence of territorial fields in this formula.
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cell is isomorphic to another cell, if the second is generated by any reasonable4 diffeomorphism between their frames. So cellular hierarchies have only projections as morphisms, while the general concept of a morphism between performance cells is a more general one. We cannot digress on this issue since nothing non-trivial is known to the date. Let us nonetheless denote the category of cellular hierarchies by H although the morphisms are not made precise here, whereas the objects are.
35.3.3
Example: The Piano and Violin Hierarchies
Summary. We describe the default hierarchy associated with piano and with violin scores. –Σ– ∂T × D
∂T × I × D
∂T
∂T × I
T×D
T×I×D
T T×I
I×D
D I Fundament
Figure 35.4: The default piano cellular hierarchy. Figure 35.4 shows the default hierarchy for piano music. This is the hierarchy which one has to start with, when shaping performance. We see that pianola parameters are not given, except for duration, since on the piano, glissando or crescendo parameters for single notes are not feasible. The basis fields are also completely independent (no coupling), and we have no parameters for extraterritorial parts, since the parallel field ∂T is determined by the underlying territory T . Figure 35.5 shows the default hierarchy for violin music. It is first characterized by a double parameter extension: on one hand, we now have the crescendo extension, on the other, we have the glissando extension. Thirdly, the fundament is reduced to onset E and pitch H. This system also includes two primavista parameters λ, Λ. The point here is that there are primavista predicates for violin, such as “ondeggiando”, an E-periodic change of loudness, which ask for refined parametrization already in primavista hierarchies. Same for the articulation field ∂Tλ . Details of these performance operators will be explained in chapter 44.7. 4 For
example an affine deformation of the frames.
35.4. LOCAL PERFORMANCE SCORES
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∂TDλ, Λ × ∂I Root
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Figure 35.5: The default violin cellular hierarchy.
35.4
Local Performance Scores
Summary. Cellular hierarchies are the core ingredient of the local performance units, but we are still not in state of controlling all the locally relevant parameters. Moreover, we have to get prepared for the future unfolding processes (stemma theory). Local performance scores are the complete local structures for performance, we give the technicalities in the language of the space form LocPerScore of local performance scores. –Σ– So far, a cellular hierarchy lacks several specifications which are mandatory in order to be able to perform on an instrumental basis, and in order to hook a single hierarchy into a chain of unfolding performance stages. We also do not have any reference to parameters which might contribute to the specific performance hierarchy. In what follows, we shall for several reasons set up such an environment in the language of denotators. First, we ought to englobe the language of performance theory in the general denotator concept framework, homework which we have not done to this point. This is particularly important for any software developments of performance tools since the denotator language is the lingua franca of all our theoretical perspectives when they are implemented on the software level. Second, it turns out that the very definition of a local performance score is circular and therefore cannot be expressed in usual terms of mathematical theory. It can be easily expressed on the level of object-oriented programming, and this is one reason why denotators are so useful: Their formalism fits perfectly in the object-oriented paradigm.
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The following definition of the performance score space will be given top-down in the sense that it has certain spaces in its ramification tree which are not yet made fully explicit. See figure 35.6 for its visualization by a double tetrahedron. This will be completed in subsequent chapters, but this is the right place to introduce this tree structure since the role of its components will be clearer if we have presented them in their functionality rather than in a full-size space definition.
Mother
Instrument
Hierarchy Operator Weights
Daughters
Figure 35.6: The visualization of the six instances of a local performance score (LPS) on a double tetrahedron. We shall henceforth abbreviate LPS = Local Performance Score. Here is the space for LPS. We thereby still suppose that we are working in a double sequence B, P of basis and pianola spaces, as introduced in section 35.3, including this notation. LocP erf ScoreBP −→ Limit(D) Id
D =M other, Daughters, CellHierarchyBP , Instrument, Operator, W eightListBP .
(35.2)
Here is the meaning of these factors: Mother. The mother form is the reference to another LPS, from which the given one may inherit a number of properties. It may also happen that there is no mother, i.e., our LPS is already a “primary mother”, in which case a denotator of a score form ScoreF orm, such as the common piano score form defined in [378], will be set instead of the referenced
35.4. LOCAL PERFORMANCE SCORES
725
mother, i.e., M other −→ Limit(ScoreF orm, LocP erf Score) Id
(35.3)
ScoreF orm −→ . . . (. . .) (any adequate score form). Id
Daughters. The present LPS may be related to a finite set of other LPS which are derived from this LPS. The members of such a set are termed daughters of this LPS, the corresponding form is that of finite local compositions over LocP erf Score: Daughters
−→
F in(F )ΩF
Power(LocP erf Score)
(35.4)
with F = F un(LocP erf Score). CellHierarchyBP . This space parametrizes cellular hierarchies. A cellular hierarchy may be parametrized as a set of performance cell denotators h(U ), U ∈ H, H the space hierarchy of h, such that the space of any such denotator also identifies the space name U . Since space names are supposed to be unique for spaces, we may just retain the space names U to nominate these spaces. In this nomenclature, we shall also denote by P U the physical space associated with U within our total space system BP . So cellular hierarchies are modeled by CellHierarchyBP
−→
F in(F )ΩF
Power(CellBP )
with F = F un(CellBP ) and CellBP −→ Colimit(D) Id
(35.5)
(35.6)
with D = (CellU )U ∈BP . So we are left with the space CellU of performance cells of space U . By definition, we have this structure: CellU −→ Limit(InitSetU , F rameU , KernelU , F ieldU , InitP erfU ). Id
(35.7)
The initial set space InitSetU parametrizes either sets of simplexes or just any local composition in U , i.e., InitSetU −→ Colimit(IU , SimplexesU ) Id
IU
−→
Power(U )
(35.8) (35.9)
2F un(U ) ΩF un(U )
SimplexesU
−→
F in(F )ΩF
Power(SimplexU )
with F = F un(SimplexU ) SimplexU −→ Power(U ) F in(G)ΩG
with G = F un(U ).
(35.10)
(35.11)
726
CHAPTER 35. HIERARCHIES AND PERFORMANCE SCORES The frame is just a pair of points in U , i.e., F rameU −→ Limit(U, U ), Id
(35.12)
designating the lower and upper extremal points Rmin , Rmax of the cell frame. The field is something rather mathematical which we leave in its encapsulated form as an element of the vector space5 Der(R) of vector fields over the function on the frame R of the cell, i.e., we have this space form: F ieldU −→ Colimit(D)
(35.13)
with D = (F ieldU,R )frame R⊂U F ieldU,R −→ Simple(Der(R)).
(35.14)
Id
Id
The kernel is a local composition in U , i.e., we have this space for kernels: KernelU
−→
Power(U ).
(35.15)
F in(F un(U ))→ΩF un(U )
The initial performance is given by its graph, i.e., as a (usually infinite) local composition in the product space of U and P U : InitP erfU
−→ Power(U ⊕ P U )
2F →ΩF
with F = F un(U ⊕ P U ) U ⊕ P U −→ Limit(U, P U ). Id
(35.16)
(35.17)
Instrument. The space for instruments is not specified in this general setup. However, if a concrete instrumental specification is needed, we shall make this component more precise. For instance, if the output could be intended to be piped to an MIDI device, a MusicN family member, a physical synthesis device, or a real physical instrument like a piano, a violin, etc. WeightListBP . The weight space W eightListBP has as denotators finite lists of weights on symbolic spaces U ∈ BP . A weight is not really something new, it is only a special kind of textual predicate which we discussed in chapter 18. The relation is this: Given a predicate E and a denotator x, we associate a truth value x/E in TA I . This truth value can be any abstract truth-oriented data, but the generality of the truth modules also includes fuzzy and similar evaluation. In particular, if we select I = R, the truth value may be any subset of R (for the zero-address), and, more specifically, any open interval ] − ∞, a[. If we identify the latter with the upper bound a, this just means that the predicate is a realvalued weighting of denotators. This is the interpretation of the weight concept here. But we do include predicates in their truth-oriented meaning. For example, one may define a set of notes K in the space U , i.e., a local composition which is the symbolic kernel of 5 The
space of derivations on the functions R is identified with the space of vector fields, see Appendix I.2.4.
35.4. LOCAL PERFORMANCE SCORES
727
some cellular hierarchy. This can be achieved by a characteristic function χK : U → TR which takes the value χK (k) = ∅ iff k 6∈ K, and χK (k) = R else. But this also opens the path to weights which also have relevant values between the notes where they originated. Then we have W eightListBP −→ List(W eightBP )
(35.18)
W eightBP −→ Colimit(W eight(U ), U ∈ BP )
(35.19)
Id
Id
with the space macro List for finite lists over a given space. The space W eight(U ) is this: W eight(U ) −→ Colimit(W eightn (U ), n = 1, 2, 3, . . .) Id
(35.20)
with the indexed spaces W eightn (U ) −→ Power(W Pn (U, TR )),
(35.21)
W Pn (U, TR ) −→ Limit(Un , TR )
(35.22)
Un+1 −→ Limit(U n , Un ), with U1 = U
(35.23)
U n −→ Power(Un ),
(35.24)
Id
Id
Id
Id
where values of weights are given by truth denotators on TR , and the weight arguments live in the ambient space U or in one of its powers Un . So the weight predicate is evaluated on single objects in the parameter space U , or in one of its mixed powers: local compositions in U , local compositions of local compositions, mixed with points in U , etc. Usually, weights reflect structures stemming from rhythmical, melodic, harmonic and similar music analysis, see section 44.7 for details. Operator. Operators are new in this setup. The point is that the cellular hierarchy which as such completely describes the performance—together with the instrumental data. But we have not dealt with the problem of generating such cellular hierarchies from system data. The operator instance has precisely this functionality: To define the cellular hierarchy. We shall deal with this very complex component in section 44.7. Mathematically, a performance operator Ω is a map Ω : H × W → H,
(35.25)
where W is the space of weight lists. The first argument, a cellular hierarchy, will be taken from the mother’s data and inherited to the actual LPS by the operator Ω. The second argument, a selected list of weights, is usually conceived as a contribution of given musical analyses to the shaping of the present cellular hierarchy. This subject will also be dealt with in section 44.7. But it is also conceived as a source of symbolic kernels when we need to access them via their characteristic function, see section 38.3.2 for this approach.
728
35.5
CHAPTER 35. HIERARCHIES AND PERFORMANCE SCORES
Global Performance Scores
Summary. Global performance scores are atlases of local performance scores, defined for reasons of performance syntax, such as, for example, instrumentation. We describe this global approach. –Σ– We have already stressed in section 32.4 that performance is a four-fold global phenomenon: There is instrumental variety, gluing of local charts, hierarchies of parameter sets, and stemmatic inheritance. The concept of a cellular hierarchy meets the hierarchical aspect, whereas the broader LPS concept meets also the inheritance aspect by the instances of mother and daughters, and the operator—together with its weights. So we are left with the local character regarding a) instrumentation, b) local charts and strategies. With respect to these two local aspects, a global performance score should be a finite local composition of LPS, i.e., a finite set of LPS which cover different instrumental specifications as well as local charts of a global composition which is to be performed. So we may state formally the space of global performance scores: GlobP erf ScoreBP
−→
F in(F )→ΩF
Power(LocP erf ScoreBP )
(35.26)
with F = F un(LocP erf ScoreBP ). Henceforth, we abbreviate “global performance score” by “GPS”. The moral of this construction is that it serves to perform all the kernel events within each top space in the respective cellular hierarchies of the LPS.
35.5.1
Instrumental Fibers
Summary. If we have the same instrument appearing for several LPS, limit constructions are necessary. –Σ– GPS denotators are just sets of LPS without any specific instrumental relations. If we however want to group several LPS around one and the same instrument, we have to look for limit tools. In fact, if we have k LPS with a common instrument, this is controlled by the projection pBP onto the instrumental form. We Instrument : LocP erf ScoreBP → Instrument Q k BP,k then have to take the k-fold fiber product LP SInstrument = pBP . The Instrument LocP erf ScoreBP coproduct a BP,k BP LP SInstrument = LP SInstrument k=1,2,...
of all these k-fold fiber products gives us the possibility to build global performance scores with instrumental grouping specifications on the space type BP . If we accumulate all space types which are of interest, in a sequence BP. = BP1 , . . . BPn , say, we get the more general coproduct a BPi ,k BP. LP SInstrument = LP SInstrument k=1,2,..., l=1,...n
35.5. GLOBAL PERFORMANCE SCORES
729
and finally the global performance score space
GlobP erf ScoreBP.
−→
F in(F )→ΩF
BP. Power(LP SInstrument )
BP. with F = F un(LP SInstrument ).
(35.27)
Part IX
Expressive Semantics
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Chapter 36
Taxonomy of Expressive Performance This last album is not titled as a memorial album or as an album in tribute because it was titled by Coltrane himself the Friday before his death on Monday, July 17, 1967. He and Bob Thiele were considering words that might apply to the sense of this album, and finally Coltrane said, “Expression. That’s what it is.” Nat Hentoff [219] Summary. Performance structure describes a semiotic fact: expression of meaning by shaping of score data. These expressive semantics are classified according to the three layers of reality captured by the topographic cube: Psychic, physical, and mental (see section 2.4). The first one means that performance expresses emotions, the second one deals with expression of gestural contents, and the third one—the musicologically most interesting one—aims at giving one’s understanding of the musical text a rhetoric expression. In a realistic performance, all three expressive semantics will participate, however, a theory of expressive semantics must first of all deal with the “pure” types which are, each in its own way, difficult subjects of ongoing research. We will not deal with the psychological, cognitive or neurophysiological esthesic aspect of performance since this is part of music psychology and would exceed our subject. –Σ– To the common music lover, it is by no means clear whether and in which way performance should express contents, and what kind of contents could be addressed. The most widespread belief is that music expresses emotions, or even that “music is” emotions. The latter approach is sometimes contended by music psychologists (see 36.1). This is due to the common usage of music as a carrier for external contents. Music often just ornaments events, ceremonies, feasts, and as such it is not intended to ask music for whatever content. Perhaps the most interesting such phenomenon is film music. It is a common saying among film music experts [118] that the best film music is the one of which the spectator does not even take notice while watching the movie. But this is a very superficial judgment, since it is in contradiction to the fact that muting the music channel in a movie virtually destroys 733
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the movie. Without music, most of its message vanishes. This is not only the case where in a criminal story of one of those cheap TV productions music has to announce that something dangerous is imminent. Omitting music evaporates the very atmosphere, the parfume of the best film! So music is quasi-absent but essential to the movie. The point of this apparent contradiction is that the common music expert saying does not really observe what we experience while watching a movie: why are we really sitting in a movie show? First of all, we replace everyday reality by an artificial, virtual reality which is projected into the dark environment of the classical camera obscura. Such a reality switch needs a strong booster to work, and this booster is music. Music is well known as the exemplary environment for a counter-world, a force that lets us forget about the common things and transports us to lost, hidden, and subconscious layers of existence, see also our comments on the Depth-EEG experiments in counterpoint theory in section 30.2.4. In reality, the core of a movie is its music, not its visual process and textual story, the latter are only the pretext of what is being communicated. Pretext in its literal sense: ante textum, not irrelevant, but not the kernel either. One could even require from good film that as a work of art it should transfigure the visual process and textual story into music. In short, this pleading for music states that music expression in performance is expression of something, not pure, self-sufficient artistry. If this is acceptable, performance as a rhetoric category should take care of how to express the contents. Therefore, on the one hand, the question must be posed about which contents can be conveyed in performance, and, on the other, about the equilibrium between contents and the medium of performance. The following discussion deals with the basic strategies of creating performance of something. It does not claim exhaustivity, but sketches prototypical approaches to contents of expression in order to position our research approach in the field of the young science of performance. For a more extended discussion of performance research history and also psychological streams, see Reinhard Kopiez’ excellent survey in [272].
36.1
Feelings: Emotional Semantics
Summary. Several authors, such as Susan Langer, Manfred Clynes, Alf Gabrielsson, J¨org Langner, and Reinhard Kopiez, have also focused their research ona relation between emotions and performance. Whereas Gabrielsson [177] contends with Langer [288] that there is an isomorphism between musical structure and emotion, Langner and Kopiez [270] develop a theory of oscillating systems that is supposed to find a physiological counterpart on the neurological level. –Σ– In [288, p.27], Susan Langner states that “music is a tonal analogue to emotive life”, a statement which is interpreted by Alf Gabrielsson in [177, p.35] as the basic idea of “an isomorphism between the structure of music and the structure of feelings”. This is also the doctrine which Gabrielsson adapts: “In summary, we may consider emotion, motion, and music as being isomorphic.” This latter statement also includes motion as one of the isomorphic structures. This is related to Manfred Clynes’s stress of emotion and its expression as an integrated system where motion, i.e., the gestural dimension, plays a crucial role [91]: Emotion, he calls it a “sentic
36.1. FEELINGS: EMOTIONAL SEMANTICS
735
state”, may be expressed by “gestures, tone of voice, facial expression, a dance step, musical phrase, etc.” While this conjecture may please psychologists, it is completely useless to scientific investigation. In fact, such an isomorphism is a piece of poetic literature as long as the components, emotions, motion (gestures), and music, are not described in a way to make this claim verifiable. Presently, there is no hope for a realistic and exhaustive description of emotions. Same for gestures (see next section 36.2), and as to music, the mathematical categories of local and global musical objects are so incredibly complicated that the claim sounds far-out, see also figure 36.1 for an attribution of emotions to articulatory ambitus. For example, the number of isomorphism classes of 72-element motives in pitch and onset (modulo octave and onset period) is ≈ 2.23.1036 as we have seen in section 11.4.1.3. How could the claimed isomorphism fit in this virtually infinite arsenal? Such a terrible simplification however does not contradict the
Figure 36.1: Attribution of emotions to the articulation ambitus [177, p.42] for the song “Oh, my darling Clementine”.
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CHAPTER 36. TAXONOMY OF EXPRESSIVE PERFORMANCE
generic insight that expression in performance may be motivated and grounded in emotional categories. But we should be careful on this point: Our present discussion is not about emotional effects of music in the listener’s psyche, we are talking about emotional rationales for expressive performance, i.e., the question of how performance could be induced by emotions. This aspect is typically addressed when the score annotations require performance actions in the mood of “amoroso”, “languissant”, “beklemmt”. Here, the performing artist has to play in and through such a mood. To be honest, to play in such a mood is not a concrete way of telling a performer what to do. The mechanism very probably works by a feedback: The performer plays and hears his/her performance so that the output may be adapted to the received impression in the artist’s own ears compared to what the artist conceives as being an impression of the given type. Evidently, such a rationale is extremely difficult to handle on a scientific level. And it is also difficult to understand a relation to the given score which transgresses the explicit annotations mentioned above. Since general feelings as a motor for performance are no good point of departure, one should (truly in the spirit of Langner, Clynes, and Gabrielsson) at least try to relate the score’s contents (not the annotations, the structural facts) to emotions, which, in turn, could then be used to shape performance via the above feedback mechanism. But this is a very complex task. How should a determined emotion (and which?) be incited by a given structural fact, a cadence, a melody, a harmonic configuration, a rhythmic process? So the emotional rationale for performance splits into the • association problem between emotions and performative shaping; • and the association between non-emotive score contents and emotions. This story has not yet been written. A less totalitarian and more quantified approach than Langner, Clynes, and Gabrielsson to emotional rationales for performance has been proposed by J¨org Langner and Reinhard Kopiez [270, 271, 272] with what they call TOS (=Theory of Oscillating Systems). The TOS postulates a system of 120 oscillators with a determined frequency each, reaching from 8 Hz to 0.008 Hz, and distributed in logarithmic steps. In a kind of Fourier analysis (the authors have not to date published the precise formulas), the dynamical curve of a recorded piece of music is decomposed and shows the contributions of these oscillators to the given curve. The TOS is not only meant as a formal description but the authors argue that the musical progression really triggers a series of oscillators in the cognitive stratum of the human brain. It is contended that the TOS spectrum is stored in human memory and that comparison of performances is enabled by comparison of such spectral data. The authors conclude [271, p.33] that “production and perception of expressiveness in music are essentially one entity.” No reference is however made to the score as such, i.e., TOS just measures the performance output and represents the spectral development in time by graphical means (called the oscillogram, see figure 36.2). So TOS does not measure the performance map as such, only the image of the map. It is evidently subtended in the TOS that the spectral decomposition bears a semantics of time processes which—independently of the hidden score—transports the neural activities of the musical brain. So expressivity is correlated to a neuronal oscillator system (expressed by firing rates of neuron populations). This is a type of rationale which refers to a score-independent instance which encodes expressivity. This is a dramatic tournament since
36.2. MOTION: GESTURAL SEMANTICS
737
Figure 36.2: Oscillogram (above) and loudness curve (below) belonging to quarter notes played in a 4/4-time signature by a drum computer with an additional accelerando at the end. This accelerando leads to a parallel upward movement of the dark bands in the oscillogram, which means that the activation changes to higher oscillator frequencies.
semantics, i.e., expressivity of something, is uncoupled from the text. This rationale is a kind of “pure expressivity”, not a result of reflection or analysis, nor a result of gestural structures. It must therefore be an emotional rationale, although the authors are not precise on the cognitive category and the topographic origin of such neuronal oscillators (it could be that the limbic system is meant), and an experimental verification of the existence of such oscillators in human brain outstanding.
36.2
Motion: Gestural Semantics
Summary. Besides—but connected to—emotions, motion, as mediated by gestural paradigms, is a widespread rationale of expressive semantics. Neil McAgnus Todd [532], then Johan Sund-
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CHAPTER 36. TAXONOMY OF EXPRESSIVE PERFORMANCE
berg, Violet Verillo [518], and Ulf Kronman [280], and also David Epstein, Jacob Feldman, and Whitman Richards [146] have proposed physically motivated descriptions of retards at group endings. Richard Parncutt, Jan Sloboda, and Eric Clarke [412] have added the concrete aspect of anatomical constraints in piano performance. We describe and analyze these approaches. We shortly account for the work by Shuji Hashimoto and Hideyuki Sawada [210] concerning the Japanese kansei processing research in the musical domain of gestures. –Σ– The emotional argument of performance is often paralleled by the argument that gestures, “motion”, are also present as a parallel phenomenon to emotion, we have already seen this in Gabrielsson’s isomorphism statement in section 36.1. It is also stressed by Clynes’ sentic concept which measures the human input by a gestural device of joystick-character, and by Kopiez–Langner [271, p.32]. Citing Francois Delalande [115], they state that “body movements and gesture are in close relation to musical timing. Our neuro-psychological oscillation model implies that body movements can be triggered by musical events.” And of course vice versa: Body movements trigger expressive performance. Also, in the framework of Japanese “kansei information processing”1 , Shuji Hashimoto and Hideyuki Sawada stress that “gesticulation is often employed in musical performance to express the performer’s emotion.” Using the dataglove, they have implemented applications which transform gestures into “MIDI control units to improve the performance in real-time” [210]. This regards expression of crescendi, vibrati, or pianissimo, for example. Richard Parncutt has investigated the intuitively evident fact that expressive performance is strongly conditioned and induced by physical constraints from fingering [411, 412]:“good fingering is a crucial ingredient in the preparation of performances that are both technically reliable and appropriately expressive.” In [410], he has also maintained the thesis that “the most important sounds conditioning the perception of rhythm may be the sounds associated with the heartbeat and walking movements of a mother, as heard by her unborn child.” So good performance should been adapted to this motion trigger. In accordance with these general observations, several performance scientists have proposed models of performance which derive performance fields from mechanical principles of accelerated motion. In particular, this has been set forth regarding tempo curves. Analyzing the first experimental studies of final retard phenomena [518] by Sundberg and Verillo, Kronmann and Sundberg [280] propose a model of final retard which is completely derived from the mechanical analogy of constant deceleration of tempo, as if it were induced by the action of a constant force upon a physical mass. However, tempo T is seen here as a kind of velocity that is a function of physical time, not symbolic time, as it is standard. So what is constant is the force as a function of physical p time. We then have T (e) = dE/de = c.e, c = const.. The resulting formula is T (E) = T0 1 − E/E0 , where the tempo T (E) at onset E is related to the starting tempo T0 , the total onset interval E0 until total stop (which is not included in the really onging music, but has to be set as an ideal endpoint of motion). If such a formula is derived, by the same reasoning, one could also derive formulas with different, not necessarily constant force function; this has also been observed by these authors. However, there is 1 This is the term coined to stress a special application of information technology to implementation of emotional contents against the classical Artificial Intelligence and other, more logically and mathematically oriented approaches.
36.2. MOTION: GESTURAL SEMANTICS
739
no indication of which force should act and why so. So in principle, any tempo curve can be constructed by an adequate acceleration and therefore force function. More generally, it is not clear, why the supposed mass should be constant. As with special relativity, the mass could vary as a function of tempo. It seems that the mechanical motion model is only the construction of an intermediate layer to the real question: What are the basic forces which shape tempo? If the straightforward mechanics (constant mass, constant deceleration) are maintained, however, this explanation completely standardizes the final retard phenomenon and uncouples it from the underlying score and composition. This type of approach has also been proposed by Jacob Feldman, David Epstein, and Whitman Richards [146]. Their paper models tempo T (E) as a velocity function of symbolic time E, and its derivative is meant to be determined by a quadratic force function F (E) ∼ E 2 . The Newtonian equation F (E) = m.T (E) yields a cubic polynomial function T (E) = a.E 3 + . . . for the tempo. Of course, this is a completely different mechanical situation, here the force really acts on the symbolic level instead of the physical action described by Kronman and Sundberg. While that one means T (E) ∼ E 1/2 , this one yields T (E) ∼ E 3 . Unfortunately, the latter approach is not congruent with the examples shown in [146]: They refer to the reciprocal value 1/T (E) instead of T (E)! So we should have T (E) ∼ e1/3 , but that requires another mechanical situation. In terms of Kronman and Sundberg, this requires a force which, as a function of physical time e, is proportional to e1/2 . A less simplistic approach which also includes structural analysis (see also below in section 36.3) and not only mechanical generalities is presented in Neil McAgnus Todd’s paper [532]. He rightly observes that the final retard is only a very special agogical situation, and therefore models his tempo curves according to a superposition of accelerando/ritardando units which are defined by a triangular sink potential V . Accordingly, tempo is defined as a velocity v, and the total energy of the system E = 21 mv 2 + V , supposed to be constant (why?) gives the p velocity formula v = 2(E − V )/2. Todd further supposes that there is an intensity variable I for loudness, with a relation I = K.v 2 that is common to many physical systems. This yields P the relation I = 2K(E − V )/m and sums up to an aggregated formula I = l 2K(E − Vl )/ml if the grouping of the piece is taken into account. Thus, the idea is that there is a physical energy and intensity parameter system that controls the “surface” of the tempo (= velocity) via classical energy formulas. So the background structure is an energetic one, i.e., the tempo curve and loudness are an expression of mechanical dynamics. The author comments on his method as follows [532, p.3549]: The model of musical dynamics presented in this paper was based on two basic principles. First, that musical expression has its origins in simple motor actions and that the performance and perception of tempo/musical dynamics is based on an internal sense of motion. Second, that this internal movement is organized in a hierarchical manner corresponding to how the grouping of phrase structure is organized in the performer’s memory. The author also suggests a physiological correlate of this model (loc. cit.): ...it may be the case that expressive sounds can induce a percept of self-motion in the listener and that the internal sense of motion referred to above may have its origin in the central vestibular system. Thus, according to this theory, the reason why expression based on the equation of elementary mechanics sounds natural is that the vestibular system evolved to deal with precisely these kinds of motions.
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CHAPTER 36. TAXONOMY OF EXPRESSIVE PERFORMANCE
Todd refers to insights of neurophysiologists according to which the vestibular system is also sensitive to vibrational phenomena. So the musical expressivity is understood as an effect of transformed neurophysiological motion. The drawback of this approach is that finer musical structures are not involved in the structuring of the energy which shapes tempo/intensity. And even if that could be done, there is an essential kernel of this shaping method which should be based upon paradigms of motion. These paradigms do however not appear clearly in the above approach. More precisely: The complex motion dynamics of the vestibular system cannot easily be mapped onto the structures of performative expressivity. What is the operator which transforms whatever structures of motion into expression parameters? If music were isomorphic to motion, no such isomorphism could be recognized from Todd’s clever approach. Beyond general motion paradigms there is the more visible level of gestural structures which can be implemented in operators for musical expressivity. Unfortunately, the existent classification of gestures is anything but detailed. Classification of gestures is only settled on a prototypical basis. But this is precisely not what music needs so urgently. For example, the extremely refined gestures of Glenn Gould’s performance2 , the movements of his hand and arms, his head and thorax, this is beyond any scope of present classification.
Figure 36.3: Glenn Gould while performing. It is the same with the incredibly refined movements of Herbert von Karajan’s hands, and especially their single fingers: It is known that these gestural directives to the members of the 2 The reader should be careful on this point: In [115], Francois Delalande has recognized the expressive role of Gould’s gestures (analyzing a film record of Gould’s performance of Bach’s Kunst der Fuge, see figure 36.3) with respect to musical structure. This is an extension of the performance parameters from sound to gestural parameters, but this is not what we are discussing here. We are discussing the role of gestures as a cause for musical performance, not a media thereof. Evidently, a pianist’s (and a fortiori a violinist’s) performance is strongly and essentially driven by gestural shapes which are not only mediators of structural facts.
36.3. UNDERSTANDING: RATIONAL SEMANTICS
741
orchestra were observed and followed with extreme attention. To measure such details is not what we can control at present. To be clear: We are not contending that gestural and motion information is irrelevant to performance. On the contrary, this is an essential contribution, but it is too difficult for scientific research as long as classification of gestures is so far from being settled. And that alone does not solve the problem, since operators for shaping performance parameters must be defined from the information provided by gestural input data. All this seems to be a bit easier than the far-out emotional rationales, but still is subject of advanced research.
36.3
Understanding: Rational Semantics
Summary. As opposed to “low level” emotional and gestural expressivity, rational semantics deal with expression of rational interpretations of the score structure. This means that the text is analyzed from different points of view, such as harmony, rhythmics, motivic content. These analyses are used as an input to shape the performance structure. This aspect is dealt with in research by the group of Anders Askenfelt, Anders Friberg, Lars Fryd´en, and Johan Sundberg [23, 163, 164, 166], then by Neil McAgnus Todd [530], Gerhard Widmer [567, 568], then by Jan Beran, Guerino Mazzola, Joachim Stange-Elbe, and Oliver Zahorka [346, 347, 348, 349, 350, 357, 360]. –Σ– Already Hugo Riemann [452] had stressed that the scope of rehearsal should be to support the communication of the motives’ comprehension. And it is a fact of music psychology (see for example [87] or the excellent overview [272]) that performance as an expression of the score’s structure is better understood than performance which disregards structure. Also, in Theodor W. Adorno’s theory of performance [6, 7], the analytical point of view, i.e., the purpose of performance to transmit analytical insights, is prominent. In fact, the most explicit starting point of any performance is the given score. This is a text that abounds with structure that must be shaped in a physical performance space. The reference to this structure as a rationale for performance is a straightforward logic which is completely standard in literature: Interpretation of a text is one of the most recognized and widely practised methods in text performance, especially in the actors’ interpretation of dramas. Besides emotion and motion it is therefore logical to refer to the rational text analysis in order to shape performance. Probably the first explicit and quantitatively stated contemporary approaches in this vein is the “analysis-by-synthesis” method of Sundberg and his collaborators which was first presented in [517]. Analysis-by-synthesis means that a bunch of performance rules for the shaping of different parameters is defined in a software environment, and then applied to the production of a synthetic performance (on an MIDI-instrument, say). The result is then analyzed by an expert (in Sundberg’s group this was the professional violinist Lars Fryd´en) who proposed alterations of the given rules and/or new ones to the programmer. In this experimental cycle, the rules are always of a general character which is based on structural data, not on direct—emotionally or gesturally driven—interventions on the performance data of the individual composition in question. This approach has been implemented on the basis of eleven rules
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in the computer application Rulle, later, renamed to Director Musices [166], with an extended repertory of rules. The rules of this approach are grouped into three categories: the • differentiation category, including these rules: Duration contrast, High sharp, Double duration, High loud, Accents, Melodic charge, Melodic intonation; • the grouping rules, including these rules: Punctuation, Phrase arch, Leap articulation, Phrase final note, Leap tone duration, Harmonic charge, Faster uphill, Chromatic charge, Amplitude smoothing, Final retard, In´egales, Repetition articulation; • the ensemble rules: Melodic synchronization, Bar synchronization, Mixed intonation, Harmonic intonation. We shall discuss these rules in section 37.1.1 of chapter 37 on performance grammars. Let us just give two examples here: The rule “Faster uphill” requires in a melodic context, that if a note is followed by a higher pitch note, its physical duration is shortened by 2.k ms, where k is a system variable in order to give its rules a variability in strength, default is k = 1. A second example is “Melodic charge”, which depends on a non-negative numerical weight attribution for every pitch as a function of its position in the selected tonality. The weights are roughly proportional to the distance of the pitch from the central pitch of the tonic (its weight is zero). Proportionally to the weight, amplitude (in Decibel), duration (percents), vibrato amplitude (percents), and unevenness smoothing are effected, also multiplied with the omnipresent strength factor k. These rules are all very elementary in their mathematical as well as in their music-theoretic approach, but they are completely concrete, and this makes this early attempt so precious. However, the present presentation and formal statement of these rules lacks a clear-cut distinction between symbolic and physical reality. For example, the above mentioned rule “Faster uphill” does not ask for the absolute duration of these notes, the change of duration is independent of the physical data, and produces either too large or too small duration changes in extremal tempi. And this is the delicate point here: A distinction between symbolic and physical reality must rely on absolute tempi. Tempi are however not mentioned in the entire machinery, although in his PhD thesis, Anders Friberg makes a comment on tempi. These rules act not in the sense that they define a map from the symbolic reality into the physical one, but they act on an already given physical image of the symbolic reality. This seems to be a kind of “prima vista” performance, but no information on this is given. There is no performance of a score which lives outside some tempo specification3 , there is only performance of performance. Moreover, the lack of agogical shaping operators is also manifest from the absence of onset in the tone parameters. The shaping of onset as such (independently of articulatory shaping) is not defined. The example of a motion-triggered performance model by Todd which was discussed in section 36.2 above is a realization of Todd’s generic approach to rational semantics in performance [530] which we shall now describe. The background structure of that motion paradigm in fact relies on structural data (of grouping), as we have already mentioned above. Todd’s generic performance model is designed upon a bidirectional transformation pairing from a score representation Ψ to a performance P and backwards by means of: 3 Or
something of this type if simple tempo curves are not possible in complex hierarchies.
36.3. UNDERSTANDING: RATIONAL SEMANTICS
743
1. a performance procedure Π acting on Ψ and an encoding function γ: P = Π(Ψ, γ), 2. a listening procedure Λ acting on P and a decoding function δ: Ψ = Λ(P, δ). In this generality, “the theory. . . is sufficiently general to cover any variable of expression. At the same time, it is agnostic as to what is being communicated, be it structure, emotion, or extramusical reference” [530, p.407]. The generic character of Todd’s approach hides an asymmetry of the transformation pairing which is due to its semiotic background; see also [361] for a modern survey on music semiotics. In fact, performance is a poietic process issued by the performer from the composer’s score. In other words, a performance is caused by its creators and must be understood by the listener, not vice versa. Hence, the performance transformation has to be specified as a semiotic mechanism. This is the difficult part of the business. Without entering into details here (see chapter 46 for a detailed discussion) it can be said that the critical subject of performance theory—a problem which Todd thematizes in the spirit of cognitive science—is a reconstruction problem: Given a performance P , how many representations Ψ and encoding functions γ can you find such that P = Π(Ψ, γ)? In mathematical terms, we are looking for the fiber Π−1 (P ) over P . This is the so-called inverse image of P , and therefore, this branch of performance theory is called inverse performance theory. The listening procedure in [530] is just a formal setup for a section Λ to Π, i.e., the selection of an element in the fiber over P as a function of the decoding data δ. Clearly, the fiber cannot be described in effective mathematical terms if one does not assume a well-defined transformation model. And even for very special models, the so-called locally linear performance grammars (see [352] and section 46.2), fibers turn out to be highdimensional algebraic varieties. Further, the encoding function must be meaningful enough to reflect the score’s structure and its relations to the above categories of expressive semantics. Otherwise, performance cannot claim to interpret the selected score. In other words, the big problem of performance theory is to propose models of adequate generality that cope with expressive semantics. In Todd’s singular example to his theory, he restricts to hierarchical grouping data for the shaping of duration. Commenting on the inverse problem of listening procedure, he states that “the durations used in the calculations are from only one metrical level. Much information about tempo is given at metrical levels below the tactus and in the durations of actual notes. The representation needs to be extended downwards to include note timing, which would mean that a rubato handler would have to work in cooperation with a metrical parser, one feeding the other. Clearly, a lot of work is needed in this area.” Concluding, he notes that “the known algorithms make no reference to any tonal function. Therefore, a rubato handler could be a vital component of any theory of grouping in the perception of atonal music. A complete theory must of course include dynamics, articulation and timbre.” Methodologically, this approach is tightly bound to cognitive science in that any algorithm is first of all tested upon its immediate fitting into human perception mechanisms, within real-time constraints, say. We believe this is a too narrow approach for two reasons: First of all, the investigation of general structural facts must be carried out before any relevance to human perception is taken into account. There is the general problem of getting an overview of possible models and their classification. Second, the cognitive knowledge is all but settled, more
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precisely: We do not know, by what processes cognition of performative expression is handled in the human brain. It could happen that a rather abstract invariant of the geometric structure of a mathematically complex fiber Π−1 (P ) can easily be detected by the cognitive machinery, but that this invariant would not have been detected if we were only permitting fibers which allow an immediate access by the cognitive capacities. For example, the mathematical structure of a M¨obius strip shaped fiber may be too complex to be grasped by the cognitive machinery, whereas its lack of orientation may be an easy task to be tackled by a small test routine built on a neuronal basis. Gerhard Widmer’s work (e.g., [567, 568]) is based on the machine learning method. In this context, structural features of given scores, such as chords, or small motives, are correlated to the performance data of a given human performance. This data is then used to shape a second score. Here, the performance rule is found by an investigation of a given performance. However, there is no background theory to this, it is purely imitative with respect to human competence. We cannot see any deeper value of such an approach since nothing is really learned beyond parrot-style imitation, although such an imitation may sound quite attractive. A completely orthogonal approach, which is more akin to Todd’s intelligent setup, was undertaken by the Zurich school4 of the author and his collaborators in [50, 51, 52, 346, 347, 348, 349, 350, 357, 360]. In our setup, the structure theory of performance in its full-fledged concept framework as developed in chapter 35 is used as an output level for performance, whereas the LPS operators are designed to implement different output data from musical analyses, such as metrics/rhythmics, motives/melodies, harmony, counterpoint, or grouping. This approach was first sketched in [341], then presented at the SMAC in [345], at the ICMC in [347], in a paper [346] concerning tempo hierarchies, and above all in the SNSF reports “Geometry and Logic of Musical Performance I,II,III (1993–1995) [348]. This approach is completely general in the sense of Todd’s scheme [530] described above. However, we have stressed a specific approach to the communication of analytical facts to performance operators which eases many methodological questions concerning the interplay and concerted action of an entire collection of analytical results. This is the method of weights, i.e., it is required that the analysis of the music-theoretic procedure A be delivered in the form of weight functions wA : K → R, where K is a local composition in some parameter space associated with the score, and more directly with the underlying cellular hierarchy of the LPS which implements a performance operator. For example, a metrical weight function wM : KE → R typically associates a non-negative weight wM (E) for each onset E in a given local composition KE of (symbolic) onsets. At first sight, this seems to be utterly restrictive, but the explanation is this: Ultimately, performance has to be defined via numerical indications of how to specify the instrumental parameters. So even if the analysis is of a more symbolic character, it has to be filtered or transformed into numerical values sooner or later. Of course, this could be done in the innards of a specific operator, and there is no obstruction to defining such operators. However, if we want to combine different analyses to be used as nurture for a determined operator, the problem of uniformity of such an input combination arises. But if the input is a priori a weight, the analytical arguments of an operator can be designed in a much more weight-independent format. For example, one may then feed a linear combination of a number of given weights into an operator, without being concerned about where these weights stem from. 4 The
term was coined by Thomas Noll.
36.4. CROSS-SEMANTICAL RELATIONS
745
Apart from these analytical input strategies, the approach of the Zurich school also stresses the generative nature of performance. Its differentia specifica to other approaches is that the unfolding of performance has been thoroughly formalized in the concept of a performance stemma, the genealogical tree of performance rehearsal and development history (see chapter 38 for details). So Todd’s hierarchical examples of refined grouping are not only extended to a cascade of LPS, their hierarchical nodes are also turned into autonomous agents of performance shaping processes which trace and group the entire performance unfolding as a compound historical and logical process in the large. This refined input policy, extending the elementary grouping rationales as exposed by Todd, yields a more sophisticated option to test correlations between measured performances and analytical insights. In this vein, different investigations have been executed with a good success by Beran and Mazzola [50, 51, 52] on the level of statistical methods. The point of this approach is that performance is so complex that it requires a faithful representation of the analytical structure of a score. And that this analytical structure must be an essential input qua significant function of the given individual score. We argue that principles of performance shaping which are completely unspecific towards the concrete score structure cannot provide us with relevant performance directions.
36.4
Cross-semantical Relations
Summary. Since music has a communicative dimension between poiesis and esthesis, each of the three above semantical directions on the poietic level may influence the esthesic level in one or several of the other semantical directions. For example, a rational expressivity may produce a gestural understanding or vice versa. We give an account of these “cross-semantical” phenomena. –Σ– As was already observed above, it may (and will in many practical situations) very well happen that one type of semantic rationale may act only indirectly upon the shaping of performance. For example, an emotional rationale may first “shape” a gestural object which in turn will shape musical performance. There are no restrictions to that. It is only important that there be one semantical modality at least which is able to convey its contents to the effective shaping of musical parameters. The rest may be arbitrarily complicated: For instance, a rational instance (score analysis, say) may produce an emotional object as its consequence, and the emotional object may produce in turn a gestural object which may evoke a second emotional object, completely different from the first! And then it may happen that this second emotional instance acts directly upon the shaping of performance. It is not clear at present, how much and on which basis the interrelation of different types of performance rationale could be implemented as computer applications, since the cross-modal assignment procedures (e.g., emotion from ratio, ratio from motion) may be hard to realize. Undoubtedly, this question is very interesting, be it for Japanese kansei research (as discussed in Hashimoto’s approach in section 36.2), be it for the general problem of harmonizing divergent semantic directions in music, above all: harmonizing the emotional direction with the rational semantics of the text.
Chapter 37
Performance Grammars Why care for grammar as long as we are good? Artemus Ward (Charles Farrar Browne) (1834–1867) Summary. The idea of basing performance on rules in analogy with linguistic grammar goes back to Mathis Lussy [311]. In modern performance research, this terminology was recovered by Johan Sundberg and his school [520]. We discuss the principles for a grammar of performance and give an overview of representative approaches to this theory. –Σ– To our knowledge, the term performance grammar was coined by Johann Sundberg on the occasion of a performance theory conference in Aarhus [520]. The reason for such a conceptualization is that the specialists became aware that performance can be shaped in various ways, but not from an amorphous design rationale, on the contrary: It became evident that there are entire organisms for shaping performance from the given score, its different semantic approaches, and the way these approaches are transformed into concrete performance instructions. The idea that performance should be executed along certain regular patterns that remind us of a language structure comes from the fact that performed music is viewed as a rhetoric vehicle of contents, and that these contents are, by the very nature of musical semantics, hidden, difficult and ambiguous. In other words, the way they are expressed is an essential condition for their communication. For a number of expressive methodologies, their architecture in fact resembles a language although, at present, only very elementary grammatical patterns are known. In [519], Sundberg proposed the creation of a dictionary of expressive rules where the patterns of the performance language can be looked up. We distinguish rule based approaches from rule learning procedures (and we shall not deal with chaotic ad hoc performance being taught in the vast majority of music conservatories!). Representative research is reviewed and classified according to the semantical perspective, as discussed in chapter 36. But the very need for a performance “language” has also a deeper explanation. If we listen to performance, it is not just the concrete piece being performed and the concrete way of performing without further context which are perceived and judged. In fact, one cannot understand the expression of a coherent and extended text without having an access to the 747
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method of shaping contents. Understanding performance really means that we gain access to a system which is applied in this concrete expressive work, and this system is a language with its grammatical rules which shape the message. In other words, understanding the performance of a musical composition amounts to understanding its system, the language and, in particular, the grammar which guides performance.
37.1
Rule-based Grammars
Summary. The generic scheme of rule-based grammars and three representative directions in rule-based grammars are presented: the KTH school, Niel Todd’s approach, and the Zurich School. –Σ– The known rule-based performance languages share a generic grammatical structure which can be described as follows. Basically, there are three components which shape the performance transformation ℘ : K → ℘(K) of a given local composition K, i.e., the kernel of the top cell in a hierarchy, see figure 37.1. The first component, which we call the rationale Rat, yields the
Rat
K Op
sb
Ratio Arg mr Emotion
p
ph Motion p(K)
Figure 37.1: The three components of rule-based performance grammar: the rationale Rat, the argument Arg, and the operator Op. They contribute to the construction of the performance transformation ℘ of the kernel K. The action of an operator on the performance map can be either symbolic (sb), morphic (mr), or physical (ph). raw material which is intended to act on ℘. As we have seen in chapter 36, Rat is a complex organism which may include emotional, motional/gestural, and rational (analytical) agents which may interact and result in a final statement to be delivered to the subsequent shaping actors. The output of these operations is the second component, call it Arg, the argument of performance. For example, in the analytical approach, this may be a weight function. In the motional situation, it could be an object which parametrizes a physical movement, and in the
37.1. RULE-BASED GRAMMARS
749
emotional setup, this may be a verbal description of a feeling, for example. The third component is an operator Op which “understands” the Arg and which, when fed with this argument, yields a determined function Op(Arg) which defines the performance transformation ℘ = ℘(Op(Arg)). If we view this situation in the rich context of LPS theory, the argument may of course include the mother LPS and thereby determine ℘ not only from the LPS’ proper rationale, but also from the LPS’ inherited data from previous shaping activities as they are traced on its mother LPS. The operator’s action may typically be targeted to one of the three ingredients of ℘: its domain K, its codomain ℘(K), or the map (the functional expression) as such. The first type is called a symbolic operator because it alters the domain. This is a very strong action since the original notes, i.e., the score’s genuine structure, are changed. Symbolic operators create a new composition such that performance is defined on new input data. Whether this type is really a case of performance seems somewhat critical. But suppose we are given the portion of a score which is written in fortissimo, and suppose that this dynamical attribute is a part of the kernel’s specification. Then, if an additional dynamical annotation, such as diminuendo, is inserted in the score text, this may be seen as a performative prescription: Change the dynamics of the specified group of notes by a successive lowering of fortissimo dynamics in some determined range. In this case, a symbolic operator would do the job since it is an action which is required before any artistic shaping begins. We shall call this a primavista operator in the operator theory of chapter 44.7. The second operator type acts directly upon the given physical output ℘(K), this is why it is termed physical operator. It may deform the physical data without altering the kernel K or the function ℘. We have to explain this seemingly contradictory argument, because it effectively changes ℘. We have in mind that a physical operator is a successive map ph : ℘(K) → ph(℘(K)), i.e, the original map as such remains what it is, but it is composed with the physical operator’s action and yields ph ◦ ℘. The third operator type alters the functional description of ℘ into Op(℘) and therefore is termed morphic. This is typically the case if ℘ is defined via a performance field as it is implemented in the cellular hierarchy of an LPS. Summarizing, we may restate this principle for a rule-based performance grammar: Principle 25 The described scheme englobes the generic framework wherein the performance language is structured. The contents which are shaped by this grammatical structure are cast in the argument Arg, whereas the grammatical structure is centered around the operator instance Op. The operator is the rhetoric element, it tells how the shaping works, whereas the argument Arg is the codified message to be conveyed after an encapsulated process of semantical elaboration in Rat.
37.1.1
The KTH School
Summary. The KTH school’s system is dominated by local rules which are based on low-level structural analysis of the text. Semantics of this analysis are rational and—to a lesser degree— gestural. They are found by the characteristic empirical “analysis-by-synthesis” method. –Σ–
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The grammatical structure of the KTH school has been described very clearly in [165], see figure 37.2 for a reproduction of that grammatical scheme. According to this scheme, the surface
input
surface level
underlying level
music performance grammar duration amplitude pitch etc.
score
phrase analysis harmonic analysis
melodic gesture analysis
transformation rules
melodic charge harmonic charge chromatic charge
k-values
Figure 37.2: The KTH scheme following [165] shows a clear congruence with the general scheme which was described above in 37.1. level shows the transition from the symbolic score to the physical performance (horizontal arrow). This transition (in fact: the performance map ℘) is shaped by an analytical rationale (left lower group in the underlying level) which comprises phrase, harmony, and melody. The output of this rationale is given in the charges for melody, harmony, and chromatics. These charges are then fed into the operator unit, the transformation rules (right lower group). The performance transformation (horizontal arrow) is visibly factorized via the transformation rules, i.e., these rules define the entire performance map. It is however not clear from this scheme, which kind of action: symbolic, morphic, or physical is taken in the concrete cases. As was already mentioned in the first discussion of the KTH system, the original transformation of symbolic data into physical data is not made explicit in these rules. It is supposed that by
37.1. RULE-BASED GRAMMARS
751
some underlying procedure, there is already a physical image of the score symbols before the explicit rules are activated. So the explicit rules seem to be physical operators. They act on a “primavista performance” which is implicitly assumed. The rationales are purely analytical in their majority, but melodic rules also refer to gestural rationales. For example, “the leap tone duration modifies duration of tones in singular leaps” ([163, Rule GMI 1B]) in the sense of an expression of a gestural constraint of hand and arm movements when playing a keyboard. In fact, the duration of a high tone after a low one √ (or vice versa, same rule) is shortened by ∆DR = 4.2 ∆N .k msec, with the absolute pitch difference ∆N of the leap, ad k the system constant. The structural penetration of these rules is however quite poor. For example, the harmonic analysis does not take into account longer syntagmatic units of harmony. Only values of isolated chords within a predefined tonality are evaluated. Cadences or modulations are not considered. Moreover, rhythmic structures are completely neglected as a rationale for dynamic accents, for example. Ditto for contrapuntal structures. Further, these rules are not inductive in the sense that they are not built to shape already given performances. They just act on the physical output and do not take into consideration the special character of the already given data. And an interaction between different arguments in combined rules is not developed. A hierarchical perspective is also not envisaged in the KTH approach. But this grammar is nonetheless a very clear scientific method that can be verified/falsified upon the audible quality of its output.
37.1.2
Neil P. McAgnus Todd
Summary. Todd’s approach is backed by a systematic formalism of performance as a function of structure and specific grammatical arguments. It relates simple structural data, such as grouping boundaries, to expression by means of physically oriented transformation rules. –Σ– As we have seen in section 36.3, Todd’s approach to performance is a symmetric one using a performance procedure Π which acts on the input Ψ and an encoding function γ, whereas its inverse is a listening procedure Λ which acts on a performance output P and a decoding function δ. The latter is in fact meant to be a section which determines the parameters of the encoding function that lead to the given encoding values. This distinguishes Todd’s approach from the KTH approach since the requirement of a listening procedure which accompanies the performance procedure is a strong restriction to the entire theory. However, this restriction is not really needed, i.e., one may also investigate the performance procedure without knowing whether there is an inverse solution. This is also the way Todd has viewed his scheme in [532]. As an example of an analytical rationale, Todd has described a rubato encoding formula ρB (t, φ. ) = φ1 + at time t, and with parameters 1. φ1 = tempo,
φ2 t (φ4 − 1) { − − φ6 }2 (1 − φ6 )2 φ3 φ5
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2. φ2 = (rubato) amplitude, 3. φ3 = length of phrase, 4. φ4 = upper limit of boundary strength, 5. φ1 = offset of parabola minimum. in [529]. This formula is determined by a hierarchy of time span reduction segmentations in the spirit of Jackendoff’s and Lerdahl’s Generative Theory of Tonal Music [243], and is applied in a recursive way, between the surface at hidden levels. This formula defines an operator which is qualitatively different from KTH operators: It includes the basic tempo variable φ1 which could be interpreted as a previously defined performance information, i.e., this operator may be seen as a refinement of an already given “mother tempo φ1 ”. Although this variable is not meant as a time-dependent quantity, the formula works independently of such a restriction and therefore is a real refinement operator of the tempo field. This means that in our above scheme, the Todd operator acts in a morphic way on the already given tempo field. Inserted in our scheme, the argument of Todd’s rubato operator is φ., and this one includes the mother tempo φ1 , which is in fact a reference to the mother LPS. This approach to rubato is nevertheless poorly rooted in the composition’s musical structure since it does not include the possibility to have harmonical, melodic, rhythmical or contrapuntal arguments in the rubato formula. The existence of a tempo curve is also not mandatory, as we have already seen in the discussion of arpeggio effects in section 34.2.
37.1.3
The Zurich School
Summary. We give an overview of the approach of the Zurich school. The details will be discussed in the subsequent chapters. –Σ– Essentially this approach is centered around the key concept of weights. These are numerical functions that encode analyses and serve as an input to the core of grammatical instances: stemmata and operators. Accordingly, a performance is generated by the stemma, a genealogical tree of nodes representing local performance scores of successively refined performance quality. The generation of such node “daughters” from antecedent “mothers” involves performance operators in the role of “fathers”. The latter are charged with weights and realize grammatical rules of different flavors. The nature of these rules is not further specified, and may include any of the systems proposed by other approaches as long as they are based upon weights (in particular the KTH and Todd proposals). The qualitative difference to the KTH and Todd systems is that a clear primavista performance is defined as a starting point of successively refined performances, and therefore, the initial transition from symbolic score data to physical data is anchored. Further, the weight system is conceived in such a way that a combinations of weights may define new weights to be fed into an operator, thereby allowing a simultaneous combination of different analyses to act on performance. Further, the LPS approach is so rich that any performance situation can
37.2. REMARKS ON LEARNING GRAMMARS
753
be dealt with: tempo hierarchies, abolition of tempo in arpeggio and rubato effects, combined deformation of parameters, and also the treatment of gestural output (beyond physical sound parameters). Finally, the stemmatic genealogy with sexual propagation from mother LPS and father operator guarantees an in-depth simulation of the process of rehearsal, where the spiritual unfolding of understanding a score may be modeled. All this has been implemented in the RUBATOr workstation, see chapter 40.
37.2
Remarks on Learning Grammars
Summary. This section gives a very short remark about grammatical patterns generated by machine-based learning from empirical performance data. –Σ– For this section, we refer to [520]. We do in fact not believe that machine-delegated statistical methods such as neural networks or proper machine learning algorithms for rule learning are of proper scientific value, since when machines learn, we do not. Of course, this is an ideological point of view, but we cannot follow methods which delegate decisions to structured ignorance: understanding cannot be delegated to engineered devices. For example, Gerhard Widmer’s approach starts with a relatively detailed structural analysis of the score, including motives, groupings etc. It then correlates these structures to empirical performance data, such as dynamics or articulation, in order to apply machine learning algorithms for extrapolation to other scores.
Chapter 38
Stemma Theory O matre pulchra filia pulchrior. Horace (65–8 B.C.) Summary. The stemma theory is introduced from its musicological and practical motivation. Semiotically speaking, performance is a result of a diachronic process. This is traced on the structure of a genealogical tree, the stemma of a performance. The stemma formalizes the diachronic process of rehearsal and practising. We describe the structure of stemmata as “family trees of performance”, together with the corresponding genetic and environmental principles. –Σ– When we compare the performance grammars developed by the KTH school and Todd to what happens when a musician learns to perform a new composition or when a conductor rehearses a composition with his/her orchestra, there is a tremendous difference of procedures. In the KTH system, the analysis-by-synthesis makes this particularly evident: The rules are not given a priori, but have to be derived via human criticisms and successive revision. This diachronic process is however not part of the grammar, it is a meta-theoretic construct. There is no trace of this successive improvement of rules in the system. History is annihilated by the uncontrollable criticism of a human expert (the violinist Lars Fryd´en in the KTH methodology). No trace of how the improved rules are produced from the old ones and their result is retained. The KTH methodology is loaded by a meta-theoretical historical dimension in the analysisby-synthesis loop. This dimension is not present in Todd’s approach, although his hierarchical technique suggests a construction of the surface level (single beats in his prototypical example) from hidden levels. These hierarchies are more of a generative nature from global to more local structures, and not historically guided, though. If one studies the way a musician rehearses a performance, it seems that a decisive component of this process is the successive improvement of performance which is built upon an added value with respect to the respective previous stages of perfection. The formal theory of such a diachronic process is the following stemma theory. This theory is already prefixed in the definition of the LPS spaces, since the variables “mother” and “daughters” create the connection to inheritance structures. This theory cannot grasp the 755
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totality of diachronic strategies, though. While a stemma comprises genealogical LPS trees, the selection strategies for the construction of determined LPS are not formalized in the stemma concept, only the result is. Also, the arsenal of performance operators is not under control in this setup, it is supposed that a number of operators is available, but their systematic construction is not dealt with in this framework. The classification of performance operators is far from settled, see chapter 44.7 for this subject. We are also aware that the present stemma structure is far from general with respect to feedback options, although the circularity of the LPS definition includes quite a lot of feedback techniques. But there is a serious limit to generalization: If one is going to implement denotators for stemmata by way of LPS denotators, their finite character must be assured in order to create performance outputs in a finite calculation time.
38.1
Motivation from Practising and Rehearsing
Summary. This section analyzes the development of an artistic performance through practising and rehearsal. We exhibit structural ramification and hierarchy, together with shaping mechanisms. –Σ– When a performance is realized, this is never the result of a one-step process. Performance results from a development of successively improved intermediate performances. To understand performance, we need to understand its genealogy. A number of experiments have been undertaken in order to analyze the process of performance preparation via rehearsal and preparation. For example, Kacper Mikaleszewski has investigated this process via video recordings of the preparation of the XII Pr´elude from the Second Book of Pr´eludes by Claude Debussy [375] by a professional pianist. Mikaleszewski introduces his investigations by the remark that it is not mandatory that pianists may clearly separate stages of their preparatory work of new compositions. In an investigation of A.A. Wicinski [566], ten famous pianists1 were interviewed on their strategies. Seven of them comprise the first group of pianists to distinguish separate stages in their work. The second group of three pianists were not able to separate any such stages. This does not demonstrate that the pianists of the second group do not really follow unconscious strategies. The incapability of verbal description of such strategies is in fact a problem known among musicians: Very often, they are not able to verbalize their activities. The first observation about this experiment is that “the divisions of the musical material introduced by the subject (the pianist) agree with the basic formal units of the composition, here related tightly to its texture. This characteristic agrees with earlier notions about the role of the structure of music in performance, and the tendency to practise longer compositions divided into shorter units (. . . ). At the same time, more complex textures led to the selection of shorter fragments for separate practice, and to making more divisions of the musical material. (. . . ) we may say that what he (the pianist) was been doing is to prepare effective sub-routines of a more complex programme which in turn would make him able to perform the musical composition at a satisfactory level of proficiency.” [566] 1 Among
them were: Sviatoslav Richter, Emil Gilels, and Harry Neuhaus.
38.1. MOTIVATION FROM PRACTISING AND REHEARSING
757
Although the general picture is somewhat ambiguous, it can be deduced from these findings that the subdivision of the given score structure, together with a development of local strategies of performance, are crucial. This can be viewed as a strategy for the acquisition of a global “performance plan”. The analysis is that “the majority of comments concerning the text of the composition, fingering, hesitations, error corrections, and memorization seem to be in agreement with the general objective of the first stage of work mentioned by Wicinski: to work out a general idea of the composition and to become able to perform it with sketchy interpretation.” In other words, the strategy is not to start working out details and local performance, but to go topdown from the overall picture (as sketchy as it might be) to more and more detailed aspects in the fragments of the subdivision of the given score. So one exhibits a hierarchy of performance development which starts with the global sketch and successively ramifies to sub-routines of local aspects.
38.1.1
Does Reproducibility of Performances Help Understanding?
Summary. Psychologically, the structure and function of performance generation is far from inscripted within a conscious memory. We discuss the value of an explication and memorization of such a process: Why is reproducibility of performance processes of scientific interest? Memorization relates to the question of identity of a performance (process). This leads to the question whether human precision is different from “machine” or “mathematical” precision. –Σ– As we have already mentioned above, a number of excellent musicians cannot (or do not want to) control their performance generation, they just rehearse by some instinctive activity and do not care about strategies and conscious plans. So why try to make such processes explicit, since a good number of artists just do not care. The question really is whether there can be a culture of performance without reflection of the conditions of good performance, good in the sense of Adorno: expression of contents that are discovered via analyses of the underlying text. Now, if this goal is accepted, we need to know about the unfolding of such a performance along the nerve of the inner logic of an artist’s elaboration—supposing that such an inner logic subsists. We insist that the absence of such a logic would result in a random walk to performance, an agnosticism driven by blind admiration of an artist’s instinct, genius, call it as you like. But then, understanding performance would reduce to plain admiration of a miraculous phenomenon which does not meet our concept of a performance culture. The scientific treatment of performance culture seems to face still another objection, i.e., the problem of objectivization of performance structures: The uniqueness and magic of artistic performance is sometimes viewed as being in contradiction to objective description, of conceptualization in the framework of scientific experimentation where reproducibility of objectively given conditions is mandatory. This skepticism culminates in the claim that objective, “mathematical” precision misses the precision of a human artist, that the latter precision is of another nature. This is to say that you may draw a faithful trace of a performance and miss the essence, because a human may reproduce the performance in another way, however maintaining the core of “human precision” as opposed to machine precision. This is however not an argument against “mathematical” precision since its claim is an invariance argument: Although mathematical
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changes in a performance may occur, the “human precision” is maintained. This means that we would have to search for invariants within the scientifically precise variety of performances. So this is not in contradiction to “mathematical” precision, it simply states that the latter may be too precise. But as long as we do not know what the “human precision” is about, we better stick to the “mathematical” precision which englobes a faithful trace of performance. Another argument against scientific performance analysis is that its very concept of performance is illdefined. But this means that something has been forgotten in this conceptualization, i.e., one ought to include other components of a performance to grasp its identity. This could be: the audience, the space (concert hall, studio, etc.) of performance, the individual conditions under which a performance is perceived (listening in a bad mood, listening repeatedly to a recording, etc.), and the general historical and cultural background of such an experience. It is evident that the effect of a performance depends on these factors. And that therefore, a complete image of the phenomenon must include these factors. But this does not change the problem of giving a precise description and pursuing an in-depth analysis of the performance in the sense developed so far. This is a fundamental objection we make against the methodology of the humanities: that they refuse to investigate parts of a phenomenon because they are related to other parts. We do not contend that understanding the whole can be reduced to the understanding of its parts, we contend that understanding the whole cannot refrain from understanding its parts, and that this latter task is the first step of any scientific procedure.
38.2
Tempo Curves Are Inadequate
Summary. We discuss the conceptual and technical inadequacy of “flat” tempo structure for performance construction and derive the ramified hierarchical tempo trees as realized on the prestor software. –Σ– It is a commonplace in performance research that a single tempo curve cannot control non-trivial tempo configurations. We therefore implemented a module for hierarchical tempo configurations, called AgoLogic, within the composition software prestor which was developed for the Atarir computer from 1988 to 1994 [335, 338, 340]. Tempo hierarchies2 are a preliminary version of stemmata in the onset domain. In order to make clear the scope of tempo hierarchies, we first give three examples of tempo hierarchies which are of practical significance in the historical context. After that, we shall give a formal definition of a tempo hierarchy in terms of corresponding denotator spaces. Example 54 The first example is an exercise in tempo curves taken from Carl Czerny’s “Pianoforte Schule” [98], see also figure 38.1 . Czerny’s exercise proposes to play this short composition with different tempo curves: first without any tempo change, second with an accelerando in the middle, third with a strong rallentando at the end. We have simulated these proposals 2 This terminology is antiquated now, since we reserve hierarchies for cell hierarchies, and what we call a tempo hierarchy here is in fact a tempo stemma in the present terminology. We nonetheless conserve this antiquated terminology in this special discussion of tempo in the prestor software.
38.2. TEMPO CURVES ARE INADEQUATE
759
120 110 100 M.M.
90 80 70 60 bar1
bar2
bar3
bar4
Figure 38.1: Czerny’s exercise for tempo curve testing. on prestor , and it turned out that the result is quite deceiving: No really relevant tempo experience results. In order to construct a less poor tempo structure, we have split the tempo levels in order to achieve the so-called Chopin rubato3 . The three tempo setups as well as the Chopin rubato version can be heard in the first four samples on the audio-file Czerny on the book’s CD-ROM, see page xxx. This is a very classical technique of tempo shaping. In prestor , this works following a hierarchical construction. In our example, we want the left hand to play a constant master tempo T . The right hand is the slave in tempo; we ask that the right hand tempo may vary anyhow under the condition that both hands coincide on each bar-line. To do so in prestor ’s AgoLogic module, we may split the onset domain I = [a0 , an [ of the right hand into a sequence of one-bar portions I1 = [a0 , a1 [, I2 = [a1 , a2 [, . . . In = [an−1 , an [. Then we have a hierarchy I → I1 , I → I2 , . . . I → In . On each bar portion Ii of the right hand, the user may reshape tempo via graphically interactive editing. The user can define any polygonal tempo R 1 curve R 1 Ti within the limits ai−1 , ai of this interval, provided the integral is conserved, i.e., = , see figure 38.2. The routine taking care of the boundary condition of invariance Ii T Ii Ti of the above integrals was presented in corollary 21 of section 34.2. Of course, the mother tempo of the left hand need not be constant, any polygonal tempo curve can be produced for the left hand, and the daughters intervals IRi can be given corresponding polygonal variations with the R above boundary condition Ii T1 = Ii T1i . 3 Sometimes also called “bound rubato” since one hand is playing a trigger tempo, whereas the other is bound to cope with the master every bar-line onset, say. In contrast to this concept, “free rubato” means that both hands play rubato, but exactly the same, and this frees them from following a master tempo.
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melody line
chord
tempo before refinement of time granularity
rubato for melody line
tempo for left hand (chords)
Figure 38.2: Czerny’s example with a depth 1 hierarchy of tempo curves where each right hand bar is a daughter tempo polygon of the right hand tempo which is constant here.
Example 55 The second example is taken from the Chopin Impromptu op.29, refer to figure 38.3, bars No.78-80. This situation shows a series of trills and arpeggi. The tempo of bars SS is confronted with the tempi of the trills and the arpeggi. This is not the situation of a bound4 Chopin rubato, since the trill ornament is only one portion of a number of notes to be played by the right hand, it is more of a hierarchy in a scenic arrangement where the ornamental notes add a supplementary level of structure. The trill notes are not even explicitly denoted, the artist has to fill up the trill sign in the spirit of Chopin style tradition. The second auxiliary structure in this example is the arpeggio. This is also an incomplete notation insofar as you have an anchor note and several “satellite” notes, i.e., the chord’s notes attached to the anchor note. These must be played in a temporal succession. The temporal succession and the anchor note are not always clear: It could be that the arpeggio succession is read top-down in pitch, or vice versa. Also could the onset of the last or that of the first note be the anchorage onset. These things being selected, the speed and shape of temporal succession 4 A bound Chopin rubato is one where one hand plays a trigger tempo whereas the other hand’s tempo may vary locally, as a tempo slave, but coping with the right hand trigger tempo on a number of master events. A free Chopin rubato is one where the rubato is played synchronically for both hands.
38.2. TEMPO CURVES ARE INADEQUATE
tr
761
tr
tr
Figure 38.3: Chopin’s Impromptu op.29, bars No. 78-80 shows a series of trills and arpeggi. in the arpeggio are not well defined. As with the trill notes’ temporal distribution, the arpeggio development is a “satellite” phenomenon in the note hierarchy. We have chosen this hierarchy in prestor ’s Agologic module (see again figure 38.3): The mother tempo is present on the top level represented by the half notes in the graphic below the score. The top level has two daughter tempi: the trill daughter (to the right below the half notes) for the trill tempo, as well as the arpeggio daughter (to the right below the half notes). The latter controls the tempo of the descending interval notes. It has a daughter tempo for the expression of the arpeggio tempi. This hierarchy does not determine the concrete tempo curves on the mother level, on the trill, and on the arpeggios; to this end, the graphically interactive input by the user is needed. This means in particular that the same tempo hierarchy can express very different ways of performing a piece: from the beginner to a virtuoso. On the book’s CD-ROM (see page xxx), four samples of performances of this tempo hierarchy are traced. They are heard on the last four samples (after the Czerny samples) of the audio-file Czerny. Example 56 The third example illustrates the tempo hierarchies as they are needed in the performance of a large orchestra with special instrumental groups. Our example is a large orchestra which is controlled by a conductor. Within this orchestra, we suppose given a string group (violins, say) which has to obey the conductor’s indications, but within two time windows (the curva1, and the curva2 windows) may follow an individual tempo curve (as indicated by the concert master, say). Within each such slice, on a small time window (curvetta3, curvetta4), there is a soloist part (played by a first violin, say) which may realize a cadence-like small expression, but has to cope with the string group (curva1 for curvetta4, curva2 for curvetta3, respectively). The tempo window of prestor shows the individual curves with their local deviations which, according to the implemented algorithm, yield the same total physical durations
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presto®
curvetta 4
mamma
curva 1
curvetta 3
curva 2
Figure 38.4: The tempo hierarchy of an orchestra is shown. The mother tempo curve (mamma) is ramified into two daughters (curva1, curva2) which in turn have one daughter each (curvetta4, curvetta3, respectively). This interdependence of local and global tempi is used to differentiate roles in orchestral time control. as their respective mothers.
38.3
The Stemma Concept
Summary. The formalism of performance stemmata is introduced. A stemma is a rooted directed graph which carries on its nodes local performance scores. The generative principles as well as the entire structure are modeled after a matrilineal scheme: There are only mothers and daughters on the stemma, starting with the primary mother. But the proliferation of these families is sexual: Fathers, formally represented by performance operators, do contribute to their daughters together with their mother. The detailed “sexual behavior” is described and turns out to be quite similar to the veritable biological/sociological behavior in life. –Σ–
38.3. THE STEMMA CONCEPT
763
This section terminates the chapter with a formal construction of stemmata. This is a remarkable subject for three reasons: First, the stemma concept is probably one of the first to grasp historical processes on the formal level in the mathematical sense, where rather than “formal” we should say: “precise”, as contrasted to the notoriously poor conceptualization in the humanities. This has tremendous consequences for the experimental modeling of historical processes: One can now perform historical developments which never have taken place, more concretely: rebuild real and simulate fictitious performance history of the rehearsals of a pianist, compare these directions and draw conclusions on the quality of the real history against the way of fiction: Are we, and why are we living in the best of possible worlds. Second, this formalization is not only formal in the mathematical sense, but also on the level of implementation in computer software. In fact, we have seen in the course of the intricate definition of an LPS in section 35.4 that this concept can only be defined by means of a circular form LocP erf ScoreBP , since a standard mathematical definition would not allow circularity, circularity being only an accepted technique in implicit equations, not in conceptualization— although, as we have seen in chapter 9, set theory (probably unconsciously) conceptualizes sets in a circular form. But this nature perfectly fits the nomenclature of object-oriented programming languages: An instance variable can very well be an object of the same class as the object whose instance variable it declares. This observation once more evidences a turning point between mathematics and object-oriented programming, where the denotator and form concepts have been derived. Third, the very nature of this formalization of historicity appears to fit with biological inheritance principle of sexual propagation. This is not only a happy coincidence, rather is it a mandatory direction if one wants to model learning: By sheer life experience, inheritance and evolution are the best proven models of successful learning. Although it is possible from the preset concepts, we do not diverge on global performance score constructions for the stemma theory and leave this segment to future research. All the LPS denotators will be situated at the zero address as long as we do not stress the contrary.
38.3.1
The General Setup of Matrilineal Sexual Propagation
Summary. This section describes the overall mechanism of stemma construction. In particular, we discuss the reason for the matrilineal approach. –Σ– Given a sequence of basis and pianola spaces B, P , recall the definition of the LPS space (35.2) with the six factors Mother µ, Daughters ∆, CellHierarchyBP h, Instrument ι, Operator Ω, Weight w. Definition 106 If Λ : 0 LocP erf ScoreBP (µ, ∆, h, ι, Ω, w) is a local performance score, let Λ ↓ (resp. Λ ↑) the directed graph of all LPS that can be reached by finite descent from Λ to the daughters, to their daughters, etc. (resp. the set of all LPS that can be reached by finite ascent from Λ to the mother, to its mother, etc.), together with the mother-daughter arrows. Denote
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by Λ l the union of the graphs Λ ↓ and Λ ↑. Call Λ∞ the directed graph of all LPS that can be reached from Λ by finite ascent and descent, together with the mother-daughter arrows. Definition 107 A local performance score Λ : 0 LocP erf ScoreBP (µ, ∆, h, ι, Ω, w) is called a stemma5 if its graph Λ l is finite and defines an undirected tree (no undirected cycles), and if it is not the daughter of its mother, if that mother exists6 . The leaves of Λ l (which are also the leaves of Λ ↓ in this case) are called the leaves of Λ, whereas Λ is called the primary mother of the stemma. If Λ has no mother, it is called a prime stemma. The set of leaves of Λ is denoted by Λ. Intuitively speaking, a stemma is a primary mother, together with its daughters, its granddaughters, etc., until we reach the leaves which are the output LPS that will eventually yield the data to be performed. It is important to have the mother of a stemma being disconnected from its daughter, i.e., not pointing at the latter in its daughter list. The primary mother of a stemma is meant to define a new tree of unfolding LPS which cannot be accessed by any other stemma. Although it can access another stemma, it is invisible to this latter. We have seen a stemmatic structure in the previous discussion of “tempo hierarchies” which are, in the present terminology, a kind of tempo field stemma, although usually there is more in a stemma than just onset-related fields and onset kernels. Exercise 79 Restate the “tempo hierarchies” in terms of stemmata with B = {Onset}, P = ∅. The matrilineal terminology is not really the whole truth here. In fact, each LPS contains its operator, the patrilineal component which is responsible for the generation of this LPS. But the result, more precisely: the hierarchy h, returns the output information for performance. And it is this result which will also be used to produce further daughters, and not the operator. This justifies the matrilineal terminology, whereas the operator is only a hidden generative instance. Let us look at a historical example of a stemma: the stemma for the composition Kuriose Geschichte, the second Kinderszene in Robert Schumann’s synonymous collection op.15 [482]. This stemma was constructed on the performance platform RUBATOr in 1996 at the Staatliche Hochschule f¨ ur Musik, Karlsruhe by the author, Oliver Zahorka, and Joachim Stange-Elbe. It took us three days to realize the whole setup and performance on a B¨osendorfer MIDI grand. The performance of the piece is documented on a CD, see [360], and in a broadcast of the Austrian TV [161]. Although the stemma is quite primitive, the shaping results were satisfactory and taught us a lot about the empirical aspects of computer-assisted performance research. The stemma is visualized in figure 38.5. Although we see that each single refinement layer is controlled by one and the same operator (horizontal arrow), the layer did not form a grouping in the technical sense to be discussed in section 38.3.3: Each daughter had to be performed as an isolated instance, since no grouping methods were implemented at that stage. The construction of this stemma first follows the splitting of right (RH) and left hands (LH), then, after the shaping of primavista dynamics and agogics, global agogics is constructed on these two LH and RH symbolic kernels. The splitting for operators Ω5 , Ω6 , Ω7 regards a small 5 “Stemma” 6 This
is synonymous to “genealogical tree”. is a slightly irritating subtlety of our conceptualization.
38.3. THE STEMMA CONCEPT
765
Mother LH
RH
L1
W1
W2 L2
L3
L4
shaping global agogics
R4 W5 LB5
RA5
LB6
RA6
RA7
fine shaping of dynamics
RB6 W7
LB7
shaping "Rubato" parts
RB5 W6
LA7
primavista agogics
R3 W4
LA6
primavista dynamics
R2 W3
LA5
separation of LH from RH
R1
fine shaping of articualtion
RB7
Figure 38.5: The stemma of the first performance of Schumann’s second Kinderszene: Kuriose Geschichte that was constructed and performed 1996 on the B¨osendorfer grand piano at the Staatliche Hochschule f¨ ur Musik in Karlsruhe. number of bars which have to undergo a more differentiated rubato. The final shaping regards fine “tuning” of dynamics and articulation in all leaves.
38.3.2
The Primary Mother—Taking Off
Summary. The primary mother represents the performance score which is deduced from the score data as they are inscripted on the predicate level. We make the deduction process explicit, together with the set of prima vista parameters. –Σ– The primary mother Λ of the stemma is the starting point of a stemmatic evolution process, it is used to derive all the LPS in Λ ↓, and eventually leave set Λ. There are two situations in such a primary mother: Either it has a mother LPS Λ0 or else it is a prime mother with the score form mother denotator S : 0 ScoreF orm(σ).
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If it has a mother LPS, this means the following: The stemma which it defines is not an autonomous structure, but it is derived from another stemma Λ0 . This other mother stemma can however not access Λ, this one is invisible to Λ0 . The idea of this asymmetry is that we want to separate stemmatic processes in their information flow. A stemma is a closed unit of unfolding performance stages. It can be used to induce a new stemma, but it is not related by a daughter pointer to the new one. This prevents mixtures of stemmatic unfolding: Once a stemma is defined, it can help another stemma to build refined performances, but it does not refer to this one, while vice versa, a new stemma may refer to the old one via the mother pointer of its primary mother. This technique creates a series of successively improved stemma Λ0 , Λ, Λ0 , Λ00 , . . . If Λ has just the score denotator S, there is no reference to other previous stemmata, and we are in the properly termed primavista situation: There is a score, which is formally expressed in terms of the denotator S, and this score is the only reference to create the first performance. This is a complex operator which we shall now discuss. The construction of all data of this prime mother is the scope of a special operator which we call the PrimavistaOperator7 . The PrimavistaOperator is a macro-operator since it has to deal with all possible score signs incorporated in S, but this is more a question of software implementation than a mathematical problem. The problem is this: recall that in the KTH theory, there was no such a thing as a primavista setup. This prime mother construction must be dealt with somewhere in the performance process. In the unfolding of a performance, it is the first action to be taken: to establish a first version of a performance which is uniquely based upon score data. It could be argued that the score data can be supposed to be introduced in advance via the symbolic kernel. This could even be admitted, but then, all the other data, such as rallentandi, fermatas, slurs, etc., where should they be piped in order to contribute to a primavista performance? For example, a fermata sign must be taken into account while it defines a sensible tempo sink, and this must be done before any refined shaping activity of performance is set forth. Of course, it is not sufficient to just know that this fermata sign is situated at a determined place, we also have to provide the data for the exact shape of this specific tempo sink. So, first of all, the input must list all the possible relevant signs, i.e., the primavista predicates in the sense of section 18.3.3. The input data are all based on the first information about the events of the score S in the different parameter spaces of BP that will participate in different special predicates. These may be ordinary notes, bar-lines, pauses, and the like. In order to produce a set of events that cope with the hierarchy induced by condition 1 in definition 102, i.e., that in a cellular hierarchy, kernels project into kernels. This means that the symbolic kernel event set EvtS (U ) associated with space U ∈ BP must project into EvtS (V ) if V ⊂ U . We shall here speak of “events x in space U ” in the sense of zero-addressed denotators x : 0 U (ξ). This induces the Boolean predicate EvtS (U ), U ∈ BP , via Definition 108 For U ∈ BP and an event x in U , we set x/EvtS (U ) = > iff x : 0 U (ξ) is such that there is an event y : 0 W (η) in the score denotator S, living in a (not neces7 In accordance with code naming conventions in object-oriented programming, the nomenclature is this: every operator is named by a special name Specialname, directly followed by the postfix “Operator”, yielding “SpecialnameOperator”.
38.3. THE STEMMA CONCEPT
767
sarily strict) superspace W of U in BP , such that pU (y) = x (i.e., pU (η) = ξ), else we set x/EvtS (U ) = ⊥. This presupposes that the events of the score denotator S have been identified a priori. These are denotators in spaces of BP that can be found upon inspection of the S. This is a knowledge which ultimately exceeds the formalized knowledge base we are dealing with in our mathematical framework, it needs an instance which can create a score denotator S from the given score. This could be an optical character recognition (OCR) software for scores, or any machine that collects events from MIDI files, for example, or just a human expert in score reading. In any case, we may suppose that the score is transformed into a score denotator S in a score form, and that the events are deduced from S. The latter is a standard task in logical and geometric motivation predicates. To define the prime mother LPS, we have to define all its constituents: Instrument, weights, operator, and hierarchy. As to mother and daughters, the first is S, and the latter will be added when the stemma is made explicit in a later stage of the historical process—presently, it is empty. The most important data is the hierarchy. We have to construct it by use of the PrimavistaOperator and the given weights. The hierarchy defines also the performance map on its top space, and we suppose that the instrumental specifications are sufficient to transform the physical parameter vectors of the image of the performance map ℘ into sound objects. So we may concentrate on the hierarchy construction here. In the hierarchy h to be constructed, we have to define the diagram of cells. This first of all means defining the projections of kernels. We instantiate all the event sets as the first bunch of predicates EvtS (U ) in the weight list of our prime mother. For any space U ∈ BP , we select the kernel KU = supp(EvtS (U )), by definition of the predicates EvtS (U ), the kernels map into each other. Also, the frames RU may be pre-defined as the smallest cubes containing all the predicate supports KU . Since we are starting with an instrumentally well-defined local situation (only one instrument), we may also suppose that there is a top space T opS in BP where the kernel KT opS is not empty. To define the space hierarchy, start taking all spaces with non-empty kernel, these are just all subspaces of T opS . This hierarchy is much too large, in general. We first have to restrict to the standard hierarchy requirement described in section 35.3, requiring that for any pianola space within a hierarchy space U , the corresponding basis space must also be in U . For example, in piano music, duration cannot be a reasonable hierarchy space. One further has to restrict this hierarchy to a subhierarchy which is reasonable for the given instrument. There is no general algorithm for such a procedure, one has to observe two things, however: • the hierarchy must be standard with respect to basis and pianola spaces; • if a KU contains events that are not proper projections of other events, this space must be retained in h; • there are default space hierarchies for specific instruments, such as the piano and violin hierarchies described in section 35.3.3. With respect to these constraints, one will choose as space hierarchy the smallest standard subhierarchy h of the total hierarchy of T opS such that it contains the default space hierarchy and the spaces containing events that are not proper projections. For piano music this means
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taking notes in EHLD, pauses, slurs, accelerandi and similar events in ED, bar-lines, time and key signatures in E, but observe that these event types are all contained in one of the spaces of the default piano hierarchy. The selection of such boundary conditions for hierarchies is a typical system parameter in the PrimavistaOperator, a parameter which, together with the weights, creates concrete hierarchies. A further set of system parameters: the selection of initial sets, performance fields, and initial performance, is a bit more delicate. The performance field to start with depends on the physical values attributed to the symbolic ones before any refined performance takes place. Typically, this can be solved by an affine isomorphism if we convene that on the physical level, we have the common space of pitch, corresponding to the logarithm of frequency, loudness, corresponding to the logarithm of intensity and time corresponding to onset units, etc., see appendix A for such common spaces. Then the performance field is a constant one (the derivative of an affine isomorphism being constant, i.e., its linear part). Since in common situations, the linear part of ℘ is also diagonal (just some positive gauging constants), the prime mother performance field is constant with positive coordinates. This setup evidently guarantees that the projections commute with the constant performance fields TsU on the spaces U (condition 3 in definition 102 of performance cell morphisms), which we have to define in their respective coordinates in each space U of the defined space hierarchy. This is what we now assume. Consequently, the entire frame volume can be reached on these constant prime mother performance fields from the left “bottom walls” of the frames R = [a1 , b1 ] × . . . [an , bn ]. There are n bottom walls for R, i.e., the (2n−1 − 1)-simplexes R Wn,i = {(x1 , . . . xi−1 , ai , xi+1 , . . . xn ), with xk = ak or xk = bk , k 6= i}
(38.1)
for i = 1, . . . n; they are degenerate for n > 2. The initial set I on an n-dimensional space U R R with frame R is the family (Wn,i )i of bottom-wall simplexes. Since we have I TsU = R (with ε = +), condition 4 of the definition of a morphism of performance cells is also fulfilled. We are left with the initial performance condition 5 in the definition of a morphism of performance cells. This initial performance can be defined by the same data which we have used to define the performance fields. Therefore, condition 5 is automatically fulfilled. So we are left with the construction of the weight system and the PrimavistaOperator that acts on the defined hierarchy in order to integrate the information from the primavista score predicates. For events with x/EvtS (U ) = >, or for local compositions of such events, or for local compositions of such local compositions, etc., as required by the weight definition of an LPS, we now look for special predicates according to the musical notation conventions8 . We have already listed the common prima vista predicates in section 18.3.3. Here, we give a selection to show how these predicates can be restated in terms of weights. If needed, such weights are added to the weight list of the LPS. Slurs. There are two types of slurs: normal legato slurs and articulation slurs. Both are boolean predicates which are evaluated on sets of events in a space U . Call the legato predicate LegatoSlurU such that x/LegatoSlurU = > iff x is a local composition in U which is embraced by a legato slur. As we are specifically interested in x being also a local 8 By that we mean the conventions of European tradition. For other traditions, special predicates and parameter spaces have to be introduced, but the procedure is the same, though not always an easy one, as the Japanese Noh nomenclature may illustrate (see the Noh example in 18.3.3.2).
38.3. THE STEMMA CONCEPT
769
composition stemming from S, we set LegatoSlurU,S = LegatoSlurU &EvtS (U ). We shall henceforth abbreviate the logical combination ?&EvtS (U ), by an added index ?S ; for example, the articulation slur is denoted by ArtiSlurS,U . Articulation. The following predicates are self-explanatory by their names and all relate to evaluation on single events of U in S: StaccatoU,S , StaccatissimoU,S , M arcatoU,S , T enutoU,S , AccentU,S Fermata. A fermata lives in the U = E space of onset; no duration of the fermata is explicitly defined. The predicate weight is denoted by F ermataE,S , and an onset x is a fermata iff x/F ermataS = >. Value Change. A value change p/q = r/s is a change in the time signature on a special barline, from p/q = 2/4 to r/s = 3/8, say. So we first need a time signature predicate for time signature p/q. This is an onset-located predicate, call it T imeSig(p/q)S , and then a predicate for two-element onset sets x = {a, b} of time signature predicates, i.e., x/V alCh(p/q = r/s)S = > iff card(x) = 2 & ∃a ∈ x, a/T imeSig(p/q)S = > & ∃b ∈ x, b/T imeSig(r/s)S = >, a predicate that is motivated by a mathematical predicate and the already given time signature predicate. Observe that the structure of the mixed powerset spaces U enables us to use ordered pairs of any points of U as they are defined in classical set theory, see appendix C, definition 114. So it is also possible to redefine the above predicate via ordered pairs a, b of objects with a/T imeSig(p/q)S = >, b/T imeSig(r/s)S = >. The details are left as an exercise. So the weights in our weight list of the prime mother LPS is the list of a) the symbolic kernels, and b) the appended list of all primavista predicates, as discussed above, with their restatement as weights with truth values in R. The P rimavistaOperator now has to add operations on the given hierarchy h which stem from a paratextual meaning of the weight predicates, i.e., an interpretation beyond the abstract textual trace which the score denotator has induced. For example, a fermata predicate has to induce a tempo sink. This and all other actions of the P rimavistaOperator will be discussed in chapter 44.7 about operator theory.
38.3.3
Mono- and Polygamy—Local and Global Actions
Summary. The typology of actions that operators may take in order to unfold the stemma includes monogamic coupling with one mother and production of one or several daughters, or else polygamic coupling with simultaneously several mothers. We give the formal description and its justification in terms of practising and rehearsal. –Σ–
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A performance operator Ω of an LPS Λ : 0 LocP erf ScoreBP (µ, ∆, h, ι, Ω, w) has the function to define the cellular hierarchy and the initial performance ι of Λ. This data is calculated by use of the weight input w and—except for the primary mother—by the already calculated mother LPS µ. This is also represented by the equation Ω ∝ µ = Λ. In general, a mother may have several daughters with the same father operator, i.e., Ω ∝ µ = Λi , which seems to be an inconsistent notation since the index is absent in the left part of the equation. In fact, we have to be more precise here. The operator for daughter Λi must have a system parameter i which specifies that daughter from one and the same mother µ. So the correct notation would be Ω(i) ∝ µ = Λi , where the system parameter is specified. This is also what in software design is realized: For each parameter i, the corresponding daughter is instantiated as a function of that operator with the i-value in its instance variables. This ordinary christian family life is however not the most economic and reasonable in many performance situations. For example, it may happen that one has already developed the stemma to a strongly ramified tree, and that one wants to apply the same refinement procedure to all or a large number of leaves Λk , k = 1, . . . K. One could of course apply the same operator Ω to each Λk independently of the other leaves. But then, changing some parameters of this operator would require us to go through each leaf and alter the data step by step. In order to avoid this “copy and paste” process, it is better to let the leaves know that it is one and the same operator and not just a set of clones Ω∗ of Ω which have to produce daughters Ω∗ ∝ Λk . The problem here lies in the definition of a denotator’s identity. In fact, if we assign to all leaves Λk the same denotator Ω as their operator, then any change in this operator is simultaneously carried out on all these leaves. But this is not so trivial: If we had a situation as it is known from object-oriented programming, i.e., if the operator were an instance variable of a LPS class, then its change would be an automatic change in every instance Λk . In the context of denotators, however, we do not have object-oriented structures, and the change has to be declared explicitly according to some kind of concept surgery, as it was discussed in section 6.9. The surgical intervention to be carried out here would be this: Search for all LPS Λk in the given stemma, or else in some more specific hierarchical position, such that their operator is named Ω. Then define new LPS Λ∗k by replacement of the coordinate (named) Ω by a new coordinate (named) Ω∗ . This only works if names identify denotators, otherwise, we have to search for other keys to retrieve the wanted denotators. One could denote this intervention by the symbol Λ∗k = Λk /Ω t Ω∗ . A global notation of a grouping of LPS within a stemma is to write G/Ω or simply G for a set G of LPS which is operated by one and the same performance operator Ω, we call such sets stemmatic groupings, they are evident denotators in the powerset space of the present LPS space. Accordingly, we write G/Ω t Ω∗ for the replacement of Ω by Ω∗ in the grouping G. Clearly, any two groupings within a given stemma are disjoint. Observe also that the mothers of a grouping do not automatically define a grouping for this reason! Moreover, we ask for the following transitivity (collective responsibility) axiom:
38.3. THE STEMMA CONCEPT
771
Axiom 4 If G is a stemmatic grouping, and if a daughter δ of a mother µ ∈ G is a member of a group H, then every daughter of a mother in G is also a member of the group H. For example, if G is grouped by a tempo operator which imposes a new tempo curve on each member of G, and if we create a group H of daughters with a refinement of their tempo curves, then, if a daughter δ ∈ H has its mother µ ∈ G, it is reasonable to have the same refinement of the tempo curve for all daughters of all members of G, because the shared tempo curve from G should inherit its refinement. Axiom 5 It is always possible to resolve a grouping in the sense that any further changes to the stemma only affect the grouping’s former members instead of the entire grouping. However, the descendants of a grouping’s members are not affected in their grouping memberships. This is a typically historical process in the stemmatic construction: The final stemma is a sequence of intermediate stemmata, i.e., we really have the time parameter entering the world of denotators, or, rather (and more precisely) the world of predicates.
38.3.4
Family Life—Cross-Correlations
Summary. Apart from genetic interaction as described by mono- and polygamic propagation, the shape of a single daughter can also be determined by cross-correlations with its sisters or other relatives. We again give the formal description and its justification in terms of practising and rehearsal. –Σ– Suppose that a mother µ in a stemma Λ has a number δ1 , . . . δm of daughters. What is the common situation for such a family? We have seen in the example of the composition Kuriose Geschichte (figure 38.5) that such a set of daughters can occur if we have to split the whole composition or a part of it into mutually disjoint subcompositions, such as left hand and right hand, which must be treated in individual ways. Or if some bars require a different performance shaping than the other bars, as is the case for the splitting of left and right hand, respectively, in that stemma. This procedure might also be applied in a more systematic way with respect to natural grouping structures of a composition: For example, in Schumann’s Kinderszene 7: Tr¨ aumerei, we have four periods A1 , A2 (= the repetition of A1 ), B A3 (= recapitulation of A1 ), eight bars each. This could be the basis of an operator which splits the entire composition into these four daughter. In each period, we have the grouping into the eight bars, and that can provide us with granddaughters A1,1 , A1,2 , . . . A1,8 ,A1,1 , A2,2 , . . . A2,8 , B2 , . . . B8 , A3,2 , . . . A3,8 . This situation is however not what is really relevant in a realistic performance. The point is that we need operators that deal with relations between different sisters and not only between daughters and mothers. Musically speaking, this means that an artist must take into account what will be played in future bars and what has been played in previous bars of the present period, when shaping a particular bar performance. One must also take into account what is the performance of future and of previous periods qua periods. So we have to face operators which take into account more distant relatives in this large family of stemmatic nodes.
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CHAPTER 38. STEMMA THEORY
In fact, from the artistic point of view, a performance must testify a sophisticated coherence in the large and also within more restricted neighborhoods of musical events. This is a common place in musicological performance theory and in the feuilletonistic music criticism. But such a general insight is very difficult to make precise for several reasons. First, it is not clear which performance structures should be involved when being set into a coherent whole. Must we think of coherence among local tempo curve segments, or, more generally: coherence among local parts of performance fields? Or should we separate coherence questions along the strata of cellular hierarchies? And if we are searching for such a coherence, how should the analytical background, more precisely: the weights which aliment operators, be screened for coherence? In other words: What is a coherent analysis? Evidently, without such a concept, coherent performance fields can be defined, but they risk failing in their rational task: to reflect the analytical background. Can they be coherent just with respect to themselves without expressing coherence of analyses? Second, the mathematical variety (in the non-technical sense of the word) of coherence structures is virtually infinite: One could imagine linear, polynomial, differentiable, analytical, and—h´elas—statistical, any kind of non-linear relations that would be involved to define coherence. The influence of other family members on a determined LPS could also come from its sisters, from cousins, from more distant relatives, anything is imaginable. We shall make explicit models of stemmatic coherence in chapters 44 and 46. These models are essentially linear models on every set of sisters, but in their combination among the whole stemmatic inheritance, they accumulate non-linear phenomena which lead to non-trivial algebro-geometric phenomena, and phenomena pertaining to second-order differential operators. These perspectives should make clear that we scarcely understand the genuine concepts of performative coherence in their musical phenomenology and accordingly cannot construct mathematical models on the basis of such a blurred phenomenology. Adorno and Benjamin have given us the catchwords for a deeper investigation of performance, catchwords which we could very well transform into adequate mathematical concepts. But these theorists did not elaborate their conceptual germs to a degree of differentiation that could help describing and understanding the concrete artistic shaping of performance. We can hardly understand and even less forgive the tremendous lack of musicological conceptualization and knowledge about performance in view of the overly fluffy and too often ridiculously blas´e music criticism in the feuilletons of our newspapers.
Chapter 39
Operator Theory If I chance to talk a little wild, forgive me; I had it from my father. William Shakespeare, Henry VIII. Summary. Operators are the substance of shaping performance. They refer to mothers, but they are the only instances capable of altering, refining, or ruining what has been achieved. They are also the pipes where exterior information, be it from score predicates, from analytical data, or from general system parameters, can be channeled and transformed into performance structures. We describe and motivate the concept of a weight (function). This is the turning point between “exterior” and “interior” strata to performance. We discuss different exigences for performance operators to cope with ‘primavista’ and ‘analytical’ data. A series of common primavista and analytical weights is discussed. We then expose a taxonomy of operators, followed by some special examples of the existing realizations, regarding tempo and articulation, as well as theoretically founded generalizations which are based upon Lie derivatives. The chapter concludes with a discussion of more “social” types of operators which correspond to “family life” correlations introduced in section 38.3.4. The final subject is a prospective to ‘continuous stemmata’, i.e., generalized stemmata based upon infinitely small coupled space portions. –Σ– Whereas the previous theory culminated in the matrilinear stemma theory, we now have to face the masculine contribution to sexual propagation of performance shaping. This part is driven by the performance operators Ω which are a coordinate of each LPS δ and generate this LPS from its mother µ: δ = Ω ∝ µ. The information which is used to feed an operator is a list of weights in the sense of the space form W eightListBP of section 35.4. Recall that a weight is a real-valued predicate on an iterated powerset of local compositions derived from the kernel in the hierarchy spaces. We have to justify this approach and will do that in section 39.1. But from our discussion of expressive performance in chapter 36 it is clear that operators have to use some rationale to shape their LPS’ performance maps. In our following examples, we shall restrict such rationales to rational operators, i.e., such operators which use exclusively score-related primavista and analytical information. 773
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Despite this restriction, we are far from understanding the general nature of rational operators. Clearly, the primavista operators are not the problem: they are straightforward (though not always easily formalized) translations of musical score competence into mathematically valid operations. The problem is rather on the side of general formalisms which could englobe the processing modes of analytical weights for producing reasonable (expressive) performance deformations of a given mother LPS data. A coarse qualitative classification of operators is nevertheless possible according to where the operator intervenes in the entire transformation process from the mental (symbolic) kernel, through the performance field (and the associated performance transformation), to the physical output data. It is also possible to show that a number of special formulas for rational operators, such as the tempo operator, or some types of articulation operators, are special cases of the so-called Lie-type operators (section 39.7). “Lie-type” means that the Lie derivative of a weight function along a performance field is responsible for the field deformation of the daughter with respect to its mother. These Lie operators then specialize to basis operators and pianola operators according to their action being on basis or pianola parameters, respectively. The chapter is concludes with a subject that extends the discrete character of a stemma to continuous parameters for the description of families of daughters. It is in the same vein as the introduction of continuous performance maps to replace discrete maps on discrete sets of notes: Although the material is discrete, the mental construction behind the shaping of this material is more than that: Human cognitive activities can operate on a continuous (or differentiable, etc.) paradigm when generating discrete effects. In fact, we could state this as a very principle of musical activity: Principle 26 Music notation and also the very expression of musical material is essentially a trace of infinitesimal forces on a discrete reduction1 . This is, what Val´ery was alluding to when he stated2 : “La musique math´ematique-ment discontinue peut donner les sensations les plus continues.” In our understanding, this is also valid in the sense of a poietic principle, the words “donner les sensations” being replaced by “provenir des forces”.
39.1
Why Weights?
Summary. Within the transformation process from abstract and conceptual analysis and representation of score data, weights are an inevitable instance charged with the production of numerical performance data. –Σ– Usually, in musicology, analysis does not end up with numerical data. A harmonic analysis, for example, yields a sequence of harmonic functions which take their values in abstract symbols, such as DF , Am7. Or a motivic analysis ends up on a verbal description of certain important motives. However, when an artist has to perform a score, the abstract description level is of 1 This 2 See
reduction is in fact—among others—a reduction of originally gestural neumatic signs! the catchword of chapter 33.
39.1. WHY WEIGHTS?
775
no direct use: Only the numerically quantized information can immediately help in defining shaping processes of instrumental parameters. The final condensation of any analytical facts in performance has a numerical appearance. Our presentation of weights is in some sense an ideal compromise between abstract predicates, such as harmonic symbols, and the naive numerical evaluation of analytical properties. We understand that weights are predicates, i.e., truth-valued functions on local compositions and their powerset constructions, but the truth module is at the same time a numerical one, or at least a truth value can be associated with a numerical value if required.
39.1.1
Discrete and Continuous Weights
Summary. A priori, weights are functions on discrete sets of points within determined parameter spaces. In order to insert weights into general tasks of performance, in particular those referring to infinitesimal or continuous character, one has to consider extrapolation methods. –Σ– We recall that we have defined a cubic interpolation formula of class C1 for a real-valued (discrete) weight w with zero derivative on local compositions K in an n-dimensional simple form space S over the reals, according to defintition 99 of section 32.3.2. We also supposed that K is contained in the n-cube C n and that the interpolation is constantly equal to 1 outside C n . In our present situation, we want to make a slight generalization to general weights with truth values in TR . We suppose that the truth values w(x), x ∈ K are all non-empty sets, i.e., do not have the value ⊥, and that K is still in the cube C n . Usually, this is the case since we are given weights which are open intervals ] − ∞, a[ with non-negative upper bound a. Suppose that wx ∈ w(x) is a selection of values from all truth sets w(x), x ∈ K. Supposing that an interpolation function F has been chosen (e.g., a cubic interpolation with respect to a specific permutation of coordinates, or the weighted sum of all such functions, or still another adequate candidate), we have an interpolation function Fw. for each such selection w., which is constant with value 1 outside C n and with zero derivatives in all points of K. Then we obtain a predicate on every element s ∈ S by the definition Y F (s) = {Fw. (s)|w. ∈ w(x)}. (39.1) x∈K
In case the truth values are intervals w(x) =] − ∞, a(x)[, the interpolation yields the value intervals associated with the interpolation values stemming from the upper bounds a(x). The fact that such a generalized interpolation is not defined for higher powerset arguments is not really a problem. Some operators in fact take these weights and boil their information down to more common weights as discussed above. We may then apply interpolation formula (39.1) to these boiled-down weights. Behind such an interpolation procedure there is the non-trivial question about the justification of the usage of continuous weights in performance operators. There are two reasons we can find for this: The first is a very practical one: calculation precision. Suppose that we are given a local composition K and that we want to calculate the weight w(x) at a point x ∈ K. In the context of computer programs, it is often not clear whether we can really catch x in its
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numerical identification, for example for numerical calculation rounding effects. Therefore, it would be good to have the weight being a non-discrete function which varies little around a given argument. The second argument is that the continuous character of performance fields suggests continuous methods to shape such fields. For example, the commonly accepted continuous character of tempo would ask for a continuous weight function in order to determine a tempo change for all values where the tempo curve is defined. The inherent idea is that anaytical facts that are calculated on discrete events of a given score should have their presence in a neighborhood of such events, in the sense of a field action of such predicates.
39.1.2
Weight Recombination
Summary. We describe and motivate a set of procedures for building new weights from given ones. –Σ– A further advantage of the numerical truth value set W −→ Simple(R) for weight prediId
cates is that one may recombine such weights on standard operations from linear algebra. This type of building new predicates from given ones is a special item in the methodology of predicate calculus as exposed in section 18.3.4, i.e., it is a special logical motivation since it operates on the codomain of predicates. Given a scalar λ ∈ R and a weight w on a local composition K, or its differentiable extension Fw by a cubic interpolation formula as described above, we can define λ.w(x) = {λ.ν| ν ∈ w(x)}, λ.Fw (x) = {λ.ν| ν ∈ Fw (x)},
(39.2) (39.3)
and it is clear that λ.Fw = Fλ.w , and (κ.λ).Fw = κ.(λ.Fw ), (κ.λ).w = κ.(λ.w), respectively. If λ is positive, and if w is a weight whose values are intervals ] − ∞, a[, then λ.w is still of this type, viz, its intervals are the shifted intervals ] − ∞, λ.a[. If we are given two weights v, w, then we can define (v + w)(x) = {µ + ν| µ ∈ v(x), ν ∈ w(x)}, (Fv + Fw )(x) = {µ + ν| µ ∈ Fv (x), ν ∈ Fw (x)}; (v.w)(x) = {µ.ν| µ ∈ v(x), ν ∈ w(x)}, (Fv .Fw )(x) = {µ.ν| µ ∈ Fv (x), ν ∈ Fw (x)},
(39.4) (39.5) (39.6) (39.7)
and we also have Fv + Fw = Fv+w , Fv .Fw = Fv.w . Sum and product are associative, but (κ + λ).Fw 6= κ.Fw + λ.Fw , in general. If both, v and w (and therefore also the differentiable extensions) have intervals ] − ∞, a[, then so is their sum. The philosophy of these combinations is that the action of weights may be better if their influence is mixed and weighted by scalars such as with cooking, the delicate dosage can be controlled. A more general type of recombination is a non-linear deformation of a weight according to a deformation function δ : R → R. With the previous notation, we define a scalar multiplication
39.2. PRIMAVISTA WEIGHTS
777
by δ.w(x) = {δ(ν)| ν ∈ w(x)} δ.Fw (x) = {δ(ν)| ν ∈ Fw (x)},
(39.8) (39.9)
and it is clear that for two such functions δ, we have (δ ◦ ).Fw = .(δ.Fw ). In practical cases, we often have the weight w being defined by intervals w(x) =]−∞, a(x)[ for x ∈ K, and therefore also Fw (x) =] − ∞, α(x)[ for any x ∈ S. Since K is supposed to be finite, and since by construction α(x) = 1 outside the defining cube C n , the image set α(x), x ∈ S is a finite interval [αmin , αmax ]. We then consider a continuous deformation function δ(αmin , αmax , τ ) such that δ(αmin , αmax , τ )(t) = t for t 6∈ [αmin , αmax ]. For t ∈ [αmin , αmax ], the deformation parameter τ describes a one-parameter family of continuous, monotonically increasing deformations of the interval [αmin , αmax ] with δ(αmin , αmax , 0) being the identity. Typically, the deformations for τ and −τ are related to each other by a reflection of their graph at the main diagonal in R2 . For example, we may take affine images of hyperbolas y = −1/x on intervals [−u, −1/u], where u = eτ , yielding the typical formula δ(0, 1, τ )(t) = e2τ −t(et 2τ −1) for non-negative τ , and the symmetric (diagonal reflection) formula for negative τ . The philosophy of non-linear deformations is that often, the action of an analytical weight on performance is qualitatively correct, but its quantitative influence should be distorted in order to yield a good perception. This effect can then be achieved via non-linear deformation functions.
39.2
Primavista Weights
Summary. Many of the traditional score data are not codified in numerical values. Numerical quantification is, however, a conditio sine qua non for any performing artist. We give an overview of common transformation procedures for non-numerical score parameters, including their extrapolation to continuous weights. –Σ–
39.2.1
Dynamics
Summary. We discuss the quantification and syntax of absolute and relative dynamics. –Σ– Here, we need some preliminary remarks concerning the parametric interpretation of dynamical score signs since these are very coarse and of different types. We distinguish three types: • Absolute dynamical signs such as ppp, mf, fff, sempre pp, etc. They give information at a determined onset. • Relative punctual signs such as frz, sf, etc. They indicate a momentous change of dynamics as a function of the momentous dynamical level.
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• Relative local signs such as the crescendo signs, giving verbal or pictorial indications on their domain of validity or having the shape of short wedges. To begin with, absolute dynamic signs are ambiguous verbal descriptions of intended loudness. The first step towards a performance-adequate representation of such symbols is to assign them numerical values in a symbolic loudness scale. The good thing to do here is to assign them values which are in an affine relation to physical (Cents) units3 . The precise value relations are however not codified and must be left to the free decision of the formalizing instance (in instrumental practise: the performer, the conductor, in computer-systems: the user). So we start with the usually sufficient setup of an increasing sequence γ(ppppp) < γ(mpppp) < γ(pppp) < . . . < γ(mp) < γ(p) < γ(mf) < γ(f) < . . . < γ(ffff) < γ(mffff) < γ(fffff)
(39.10)
of 19 real numbers in the Loudness space. The naive setting would be an equidistant series, such as suggested by MIDI and other technological codes. If we could limit loudness quantification to such absolute signs, this would be the end of the story: We would just define a parser for verbal dynamics signs into the numerical format of the space Loudness. However, the meaning of the absolute signs is more than the above gauging convention. In fact, when we are given a sequence of absolute dynamic signs, we are also facing relative dynamic signs which have extensions, i.e., onsets and “offsets” within the range of the given absolute signs. For example, we may see a crescendo wedge, followed by a second such wedge, and then by a decrescendo wedge. What is the loudness curve for this situation? The basic data is a sequence of absolute loudness symbols which are specified by their onsets and the symbols, i.e., a sequence of denotators in the limit space AbsDynamicEvents −→ Limit(AbsDyn, Onset, Duration) Id
(39.11)
AbsDyn −→ Simple(Z < U N ICODE >). Id
The duration of such an event is the difference from its onset and the onset of the next event, except for the last event, whose duration defines the end of the given composition. So we have the sequence AbsDynSequ = Evt1 : 0 AbsDynamicEvents(Dyn1 , E1 , D1 ), Evt2 : 0 AbsDynamicEvents(Dyn2 , E2 , D2 ), ... Evtm : 0 AbsDynamicEvents(Dynm , Em , Dm ),
(39.12)
with D1 = E2 − E1 , . . . Dm−1 = Em − Em−1 . This data induces the weight wAbsDynSequ on onset events E with these values wAbsDynSequ (E) =] − ∞, a(E)[: a(E) = −∞ if E < E or E + D ≤ E, 1 m m (39.13) a(E) = γ(Dyn ) if E ∈ [E , E + D [, i = 1, 2, . . . m − 1. i
3 See
appendix A.2.3 for this gauging question.
i
i
i
39.2. PRIMAVISTA WEIGHTS
779
Over this absolute dynamics data, we now tilt the relative dynamic events: On each interval defined by event Evti , we are given a sequence of events of this form: RelDynamicEvents −→ Limit(RelDyn, Onset, Duration) Id
(39.14)
RelDyn −→ Simple(Z < U N ICODE >). Id
The symbols in RelDyn are such as crescendo,molto crescendo, decrescendo, molto decrescendo, for example. Their onset and duration are visible from the position of the wedges or the dashed lines in the score notation. So we are given sequences RelEvti,. = RelEvti,j : 0
RelDynamicEvents(RelDyni,j , rEi,j , rDi,j ),
(39.15)
j = 1, . . . mi , 0 ≤ mi , with mi = 0 for empty sequences. This time, the conditions on the respective onsets and durations are rEi,j + rDi,j ≤ rEi,j+1 , 1 ≤ j < mi , Ei ≤ rEi,1 , rEi,mi + rDi,mi ≤ Ei + Di .
(39.16)
This is the framework for redefining the finer dynamics according to the relative signs sequences. The problem here is that we do not know how much a crescendo may increase absolute dynamics in order to remain within interval Ei , Ei + Di of the given absolute value γ(Dyni ) of the absolute event Evti . To this end, one needs other information which is implicit in the interpretation of a score: this is the tolerance of dynamical variation around γ(Dyni ) such that we still accept the label Dyni . So we have to provide each symbol Dyn of absolute dynamics with a tolerance number 0 < τDyn , and this means that all values in the half-open interval [γ(Dyn) − τDyn , γ(Dyn) + τDyn [ will be accepted as variations of the label Dyni . This does not exclude that these intervals may overlap, so the labels can be ambiguous and contradictory: a high pp value can be higher than a low mp value. Upon this tolerance system, we now have to define the meaning of relative dynamics in the sequences RelEvti,. . To this end, we have to fix a quotient 0 < κRelDyn of dynamical increase or decrease of each relative dynamical symbol. For crescendi, we suppose 1 < κ, and for decrescendi, we want κ < 1. For each relative dynamical event RelEvti,j , we calculate the dynamical values (i.e., the upper bounds b of the predicate’s intervals ] − ∞, b[) at rEi,j and rEi,j + rDi,j , and then, we interpolate linearly between these cornerstones, whereas the value remains constant inbetween two relative dynamics events. The first relative event RelEvti,1 has the starting value at rEi,1 equal to the absolute value γ(Dyni ) of Evti . We now suppose inductively, that the starting value vj of the relative event RelEvti,j has been defined within the open interval ]γ(Dyni ) − τDyni , γ(Dyni ) + τDyni [. vj −(γ(Dyni )−τDyni ) This defines the quotient (γ(Dyn which we want to increase by the factor κRelDyni,j i )+τDyni )−vj defined by the relative dynamical event at index i, j, i.e., κRelDyni,j
vj − (γ(Dyni ) − τDyni ) vj+1 − (γ(Dyni ) − τDyni ) = (γ(Dyni ) + τDyni ) − vj (γ(Dyni ) + τDyni ) − vj+1
(39.17)
defines the new dynamical value vj+1 at the end of the (i, j)th relative sign. The new value is still in the open interval ]γ(Dyni )−τDyni , γ(Dyni )+τDyni [, and we may go on inductively until
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CHAPTER 39. OPERATOR THEORY
all relative dynamical signs are parsed. Observe that a succession of an increase factor κ by its inverse neutralizes the dynamical value change. More generally, the succession of two factors κ1 and then κ2 results in a value change by factor κ1 .κ2 . So we can manage any succession of relative dynamical signs of this “crescendo/decrescendo” type without falling out of the prescribed tolerance interval. Denote the weight on all of the onset axis which is defined in this way by wEvt.,RelEvt.. . So we are left with punctual relative dynamical signs pct at a note x, such as pct = accent or pct = marcato. We may define the predicate {1} if x is not marked by pct, pct(x) = (39.18) {λ(pct)} else, where λ(pct) is the increase factor for pct. Then, the punctual dynamical sign influences the already given absolute and relative dynamical weight dyn by the product pct.dyn of weights. Of course this implementation need not live for ever, but it is one reasonable solution of a non-trivial problem of making blurred score signs precise. Observe that this kind of weight is not a performance map, it is just a weight on the symbolic (mental) events that must be used for operators to be defined later. Observe also that the above weights are not continuous functions of onset if several absolute dynamical signs are present.
39.2.2
Agogics
Summary. We discuss the quantification and syntax of absolute and relative agogical indications. A special attention is payed to general curve types for retards, fermatas, and general pauses. –Σ– These are the common agogical indications: • Absolute tempo, such as M¨ alzel metronome, or anterior verbal indications of type andante, adagio, etc. Formally, they correspond to absolute dynamic signs. • Relative punctual tempo signs such as fermatas and general pauses. Remarkably, there is no relative punctual acceleration sign corresponding to the fermata. • Relative local tempo signs are the following: 1. Coarse indications concerning agogics, e.g., ritardando, rallentando, accelerando, stringendo etc. 2. notation of correspondence between two adjacent tempi, such as “2/4 = 3/8”, 3. rest signs such as a tempo. The essential difference to dynamical signs is that symbolic onset and duration are precisely codified, so agogical signs relate to proper performance transformations, and not to making blurred signs precise.
39.2. PRIMAVISTA WEIGHTS
781
We have already discussed some of the agogical predicates (fermata, value change) in the presentation of weights for the prime mother LPS in stemma theory, section 38.3.2. alzel metronome sign x quarter/M in.. This is a predicate which Let us first look at the M¨ takes its > values exactly on onsets E and tempi x where a M¨alzel indication is x quarter/M in., and ⊥ else. The reader may easily make the underlying forms precise. Verbal absolute tempo indications need a parsing instance to render them in numerical (M¨alzel) terms.
E onset range of fermata D tempo shape
100%
a% b% D.d D.u
Figure 39.1: The interpolation curve of a fermata weight. See the text for the explanation of the symbols. A fermata has only its onset E made precise, the rest is blurred. We need several additional parameters to generate a viable weight. We first need a duration D, then a shape parametrization to describe the fermata’s tempo sink. We may for example take this form: F ermata −→ Limit(Onset, Duration, Bottom, Down, U p, Af ter) Id
(39.19)
with Bottom, Down, U p, Af ter −→ Simple(R). Id
A fermata F erm : 0 F ermata(E, D, b, d, u, a) is parametrized by (1) the percentage number B of maximal tempo lowering with respect to the given tempo, (2) the percentage d of the duration D from the beginning to reach the lowest tempo, (3) the percentage u from the beginning to restart getting back to the following tempo, (4) the percentage a of the original tempo which is resumed after getting back. This data is used to define a cubic spline interpolation as follows: We have four onsets: E, E + b.D, E + u.D, E + D and corresponding relative tempo values
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1, b, b, a. This defines a discrete weight w(F erm) and a corresponding interpolation function Fw(F erm) , see figure 39.1. Of course, many different curves are possible, but with this shape, the main quality of a fermata can be imitated. Weights for accelerando- or ritardando-typed predicates follow the building scheme of a fermata, except that the duration of these signs is defined. We leave this as an exercise to the reader. Exercise 80 Give an explicit description of accelerandi and ritardandi in terms of denotators and associated weights. The general pause (G.P.) predicate is also undetermined like the fermata. It does however not imply a smooth recapitulation of the original tempo: After the general pause, the previous tempo is reset. We will have to deal with this in corresponding operators. The same phenomenon of a tempo reset is the case for the a tempo predicate.
39.2.3
Tuning and Intonation
Summary. We discuss the weights for pitch values. –Σ– Tuning and intonation is a delicate subject for the PrimavistaOperator since it it not clear from the beginning how much the settings depend on the instrument and how they depend on individual instruments. For piano, the situation is easy since we have a fixed tuning and very often, it is even well-tempered. For violins, this is much more complicated: Should the primavista (!) tuning be a just tuning for each tonality which is encountered in the score, or should it be just one “default” just tuning? It is wise to let a special operator, not necessarily the PrimavistaOperator, do the work of delicate tuning and to just operate the minimum on the first process level. The tuning information is twofold: We have the chamber pitch which is the initial set and initial performance, this is ok. And we have the tuning data for all pitch events of a sufficiently large chromatic scale (88 keys for common pianos). This is completely analogous to a step tempo function given by a number of absolute tempo settings. One may use the form T une −→ Limit(P itch, StepT une) Id
(39.20)
with StepT une −→ Simple(R) Id
being the tuning quantity that measures the “pitch velocity” between to neighboring pitches. Examples for default tuning for some classical cases are found in appendix K. The common well-tempered case has the denotators x:0
T une(x, 1/100 Semitone/Ct)
for each pitch x of the well-tempered chromatic scale.
39.2. PRIMAVISTA WEIGHTS
39.2.4
783
Articulation
Summary. We present the parameters for articulation types, from molto staccato to molto legato. –Σ– On one hand, articulation is a predicate type which regards single notes, for example molto staccato, staccato, legato, or molto legato. All these predicate types can be encoded by weights on events x of a space U within the BP hierarchy which contain duration and take their values ] − ∞, a(x)[ according to whether articulation stretches durations or compresses them. For staccato and other compressing signs, we take 0 < a(x) < 1, whereas for legato and other stretching signs, we take 1 < a(x). On the other hand, we have articulation in the sense of grouping of a set G of usually consecutive notes which are grouped by an articulation slur via form ArtiSlurU,S as introduced in section 38.3.2. Such a predicate may be made more precise by a weight which takes into account the group G as well as the single notes x with their position within G. On a space U , this is covered by the form U2 with the notation from formula (35.18). The weight wGrpArti (G, x) = ]−∞, a(G, x)[ is parametrized by the number a(G, x) which tells how much the note’s x duration is altered relatively to its nominal duration.
39.2.5
Ornaments
Summary. Historically, ornaments form a complex set of constructions. We introduce a unified language for ornaments which is based on macro-events (see section 6.7). –Σ– Recall from section 6.7 that we had defined macro-events by the form M akroBasic which is based on an event form Basic. Here, we take Basic = U , one of the event spaces in the hierarchy of our stemma. An ornament (one of the many forms of a trill, for example [66]) of events in U can be described by a special denotator of the form M akroU . An ornament is first of all anchored at an event a of U , this is the note which is ornamented by a specific sign, with a reference onset a, see figure 39.2. So we start our ornament denotator by a singleton: Ornament : 0 M akroU ({D}) with D:0
KnotU (a, SmallAnchors).
Here, the next ramification denotator SmallAnchors encodes the three anchor events of the three structural units O1 , O2 , O3 , each possibly with repetitions within: SmallAnchors : 0
M akroU (O1 , O2 , O3 ).
Each of these knots O1 , O2 , O3 has a reference event a1 , a2 , a3 : Oi : 0
KnotU (ai , Pi ), i = 1, 2, 3,
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CHAPTER 39. OPERATOR THEORY
with macros P1 , P2 , P3 . Each of these macros is a set of knots: Pi : 0
M akroU (qi,j , j = 1, . . . mi )
with multiplicities mi . Each such knot has the shape qi,j : 0
KnotU (di,j , oi ),
where the macros oi are independent of the second index j: They signify a repetition of the same macro according to the shift quantities di,j . The terminal macros oi have this meaning: The first, o1 , is the starting “micro-motif” of the ornament, it is played before the reference onset of a and ends with the beginning of the second micro-motif o2 which usually starts from a. This inner motif is repeated several times, usually something like eight times, and very fast. The ornament terminates on the third micro-motif o3 which is a tail to the ornament and may be played much slower than the middle sequence. If we apply successive flattening operations F lattenn , n =
o1
onset
m1 = 1
o2
o2
o2
o2
m2 = 4
o3
m3 = 1
reference onset tempo
total duration
Figure 39.2: The structure of an ornament, showing a start shape (sequence) o1 , a sequence of middle shapes o2 , and a final shape (sequence) o3 . 0, 1, 2, 3 to the Ornament denotator, we obtain a hierarchy of successively appearing U -events which gives rise to different refinement levels of tempo curves which an OrnamentOperator in the PrimavistaOperator has to manage. This perspective suggests that the hierarchy spaces U in BP should really be extended to the spaces KnotU in order to couple stemmatic refinement with hierarchies of sounds, but this theory is not developed so far. It transpires however that macro-events fit in the general weight scheme which is defined by successive powerset constructions from the spaces U in BP . So we can assign to each of the involved macro-events or knots a weight in TR in order to weight the performance role of the ornament’s components.
39.3. ANALYTICAL WEIGHTS
39.3
785
Analytical Weights
Summary. We give a generic view on analytical weights. Their concrete shape depends on the analysis which is available. Several types are realized on analytical RUBETTEr modules, see chapter 41. –Σ– Analytical weights are crucial to rational performance and intervene from analyses of the score S relating to metrical, rhythmical, motivic, thematic, harmonic, contrapuntal, grouping and other structural perspectives. We shall sketch the construction for metrical, motivic, and harmonic weights. In all these descriptions, we shall abbreviate the nomenclature of predicates and simply write down the numerical values a of such weights, the predicative statement being implicit, either in form of intervals ]−∞, a[ or in form of singleton predicates {a}, the information is the same, and the usage of the operators does not depend on which predicative encoding is chosen. Also we restrict to the discrete weights, the continuous associated weights being automatic from our previous discussions (subsection 39.1.1) S 1. Metrical weights wmetro : We have a sober weight on the onset space U = Onset for which we have prepared a formula in example 43. In that formula, the weight nW (σ) is calculated on a simplex σ of maximal local meters. For an onset E, we then take the simplex σS (E) consisting of all maximal meters within the onset kernel SOnset of S which contain E S and set wmetro (E) = nW (σS (E)) (with the evident singular value nW (∅) = ⊥ for onsets outside SOnset ). S : Following the preliminary discussion in section 22.9, we take an 2. Motivic weights wmotif µ interpretation SOnset⊕P itch of the onset-pitch space of BP by motives. More precisely, µ may be chosen as the set of all motives within a range of cardinality and extent of onsets between the first and last event. We define sober motif weight which is induced by the weight on motives from the atlas µ. So we are left with the definition of the weight on motives M ∈ µ. To this end, we consider a more precise motif theory framework as defined in chapter 22. We choose a shape type Γt and a distance d, on which an equivariant, isometric group action of a paradigmatic group P is defined, and such that we have the inheritance property fulfilled (for example, the elastic and diastematic types with Euclidean distance and counterpoint groups). We then have the epsilon topology T = Tt,P,d and choose a neighborhood radius 0 < . For a motif M ∈ µ, we consider the -disk neighborhood Dµ (M ) of M in the relative topology Tµ . We define the -presence of M by the weighted sum X p (M ) = 2card(M )−card(N ) (39.21) N ∈Dµ (M )
which visibly counts the elements of the -neighborhood of M with a weight according to the cardinality difference to M ’s cardinality. This is roughly speaking4 the presence implemented in the MeloRUBETTEr . 4 In our implementation of this function, we have also taken into account the multiplicity of submotives X of the N in the neighborhood with card(X) = card(M ), which are in the distance less than to M .
786
CHAPTER 39. OPERATOR THEORY We then also define a -content of M function which is the following weighted sum: c (M ) =
X
2card(N )−card(M ) ,
(39.22)
N,M ∈Dµ (N )
and we therefrom get the weight of M , i.e., the product S wmotif (M ) = p (M ).c (M )
(39.23)
which—again roughly speaking the presence function which is also implemented in the MeloRUBETTEr —measures the presence of M combined with the content of M . Intuitively, the weight of M is a measure of where M appears in other motives—up to distance—and how much motives are contained in M —up to distance. The coefficients are a weight for the cardinality distance form card(M ). S : This weight is also a predicate whose values are non-false 3. Harmonic weights wharmo only on the onset-pitch space SOnset⊕P itch of BP . To define it, we have to start from an η interpretation SOnset⊕P itch by chord events. The construction of a chord atlas is however not uniquely determined since it is not clear what is a chord. We may just take those local compositions of all events having one and the same onset. But we may also consider all the onsets where some chord notes end and some others still last, and then take the still lasting notes as constituents for new chords at those ending onsets.
Given such an interpretation η, we take its abstraction in form of a temporally defined sequence a. = a0 , a1 , . . . ak of length k of chords, i.e., local compositions in the P itch space. For any such sequence a., we have defined weight functions in the context of the Riemann algebra, as discussed in section 27.2. There, formula (27.13), with its interpretation as a truth denotator-valued weight, as explained in the example of section 27.2.2, gives us a criterion for finding a best path popt = (v0 , f0 , a0 ) → (v1 , f1 , a1 ) → . . . (vk , fk , ak ) in the Riemann quiver. We now use this best path to calculate the tension tpopt (ai ) of each chord ai with respect to this path. The immediate tension function would be tpopt (ai ) = Ω(popt |ai ), where we denote by popt |ai the optimal path from the beginning to (inclusively) chord ai . However, the global tension is a problematic quantity as discussed in section 27.2.1. We therefore renormalize the function tpopt (ai ) by the unique shearing of its graph in R2 such that the first point (0, tpopt (a0 )) goes to a predefined initial tension (0, t0 ) whereas the last point (0, tpopt (ak )) goes to (0, tk ). This has the advantage that the global tension can be cast to initial and final values which the local methods cannot ,tk predict and control. Call ttpoopt this new tension function. In order to calculate the weight of single notes x within a chord ai , we look at the Riemann matrix predicates T Ff,t (ai ) = [0, φ(ai )[ and T Ff,t (ai \x) = [0, φ(ai \x)[ where ai \x is the chord ai after omission of x. We know that φ(ai ) is not zero. We may then map the interval [0, φ(ai )] onto the interval [IP, 1], 0 < IP < 1 by an affine map Q for a normalization purpose. Then, we get the relative importance rel(x, ai ) of x within ai by the formula rel(x, ai ) =
1 . Q(φ(ai \x))
(39.24)
39.4. TAXONOMY OF OPERATORS
787
This gives us the final expression for the weight of the note x: We take the tension of the underlying chord ai and multiply it by the relative importance of x within this chord, i.e., S ,tk wharmo (x) = rel(x, ai ).ttpoopt (ai ).
(39.25)
This is the harmonic weight which is implemented in the HarmoRUBETTEr . An important technique to produce new weights from analytical weights is the boiling down method: Often, a weight is given on a space where the operator at hand does not work. For example, the TempoOperator (to be discussed later in this chapter) needs a weight on the Onset space. If a weight lives in a space with more dimensions, we should be able to boil it down to a weight on Onset. Here is the procedure: We have two spaces U, V in the system BP , and a projection pV : U → V . A weight w is given on the kernel SU , and we would like to get a boiled-down weight BDV (w). We make these two definitions: X BDV (w)(x) = w(y), (39.26) y∈p−1 V (x)
BDVmean (w)(x) =
1 BDV (w)(x), card(p−1 V (x))
(39.27)
where x ∈ SV , and with the value −∞ for empty fibers. In general, we have this situation: we would like to have a weight w on a specific space V of the hierarchy, but the given weight lives either in a larger space U which projects onto V , or it lives in a smaller space U onto which V projects. The former case is solved by the boiled-down construction BDV (w), BDVmean (w), whereas the latter is straightforward by the formula wU (x) = w(pU (x)), x ∈ V . In the future, whenever we use a weight which is possibly defined in the “wrong” space, these constructions are referred to, and we simply write w instead of the above correct symbols if no ambiguity is likely.
39.4
Taxonomy of Operators
Summary. Though a general description of operator principles is risky (since there is no general theory of how a stemma can be altered), we want to give a preliminary classification of the ways an operator may choose to intervene in the existing configuration. This taxonomy is guided by the generic description of how performance works: as a transformation from mental to physical reality, and by means of its description via performance fields given in section 33. This means that an operator can intervene on the level of mental or physical reality, on the frame of the local performance cell, see chapter 35.1, and on the performance field together with its initial set. –Σ– The basic data of a performance is encoded in the performance map ℘ : K → ℘(K) on the given performance cell. An operator has to change any of the involved structures: K, ℘, or ℘(K). This is understood in the sense that K is a set of arguments upon which the prescription ℘ acts, and a set ℘(K) which is the output of that map.
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CHAPTER 39. OPERATOR THEORY
The first operator type acts on K. It does this in two ways: First, named splitting operator, it just divides the kernel K into sub-kernels, nothing else. Second, it alters the kernel’s events and submits these new events to the given prescription ℘. This one is termed symbolic operator. It is as if the artist would play in the same “mood” with a changed score. The third type takes the input K, the map ℘, and the output ℘(K) as given and now changes the output without regard to the previous process. This is the physical operator type. The fourth type takes the input kernel K for granted and just changes the map’s “formula”. For example it changes the tempo curve or the intonation, etc. This is the so-called field operator type. This is the most complex operator type and has not yet been understood in its different outfits. Let us now have a closer look at these types.
39.4.1
Splitting Operators
Summary. A splitting operator is rather simple. It restricts to partitioning the performance score’s kernel into sub-kernels without any further change on the performance score. Splitting may also be operated on a group of selected instruments in a global performance score. –Σ– The idea of a splitting operator is this: You are given a score with different groups of notes that you want to perform in an individual way. For example: right hand and left hand, or onset-driven grouping: split the score at a given parameter, such as onset or loudness or duration etc. This may happen at a grouping line such as the beginning and the end of an eight-bar period. Very often, this also happens if a special group needs a special performance procedure because of its inherent structure, such as a trill or another ornament. Whatever you will do to this group is irrelevant, you decide later. So the splitting operator is a propaedeutical operator intending to prepare more in-depth operator actions. A prototypical realization of the splitting operator may be implemented by use of an essentially mathematical predicate SplitU,ν , where U is one parameter (such as pitch, onset, etc.) in the given hierarchy h, i.e., of the top space BP , and where ν is a real number parametrizing a coordinate value in the U parameter. Then if X is an event in one space of the hierarchy, the predicate X/SplitU,ν = > iff either X does not share this parameter U , or if it shares it and the U -value XU of X verifies ν ≤ XU . So X/SplitU,ν = ⊥ iff XU exists and XU < ν. We then get the logical combination predicate SplitU,ν,µ = SplitU,ν &¬SplitU,µ which selects those events with either no XU coordinate or else with ν ≤ XU < µ. In the RUBATOr software, the splitting operator has been implemented such that it may perform any logical conjunction of splitting operators of the types SplitU,ν and their negations ¬SplitU,ν . For example, this enables us to select all events within a half-open onset interval [E1 = ν, E2 = µ[ and a half-open loudness interval [L1 = ξ, L2 = χ[, and having durations D with D1 = δ ≤ D. So the splitting operator Split (we omit further specifications in this notation) is defined by a Boolean predicate which extracts a given set K 0 of events from a given kernel K. This means that Split produces two daughters, one with the K 0 kernel, the other (by logical negation of the former) with the remainder kernel K − K 0 . In the nomenclature of section 38.3.3, we have a coupling Ω(i) ∝ µ = Λi , i = 1, 2
39.4. TAXONOMY OF OPERATORS
789
which is also written more intuitively as Split ∝ µ RemainderSplit ∝ µ.
(39.28) (39.29)
This operator may also be applied to a GPS (µi )i just by applying the operator simultaneously to all the member LPS µi of this GPS. We should add that all the other data of the mother(s) in µ (in (µi )i ) are left as they are, and everything still works fine. The new LPS just have some points of the original kernel being removed. In the formal setup of the weight system of an LPS, the predicate Split would be viewed as being a weight of both daughters and the operator would then act on the mother via the Boolean selection and its negation which is defined by this weight.
39.4.2
Symbolic Operators
Summary. A symbolic operator affects the score data before they are performed whatsoever. This means that the operator really changes the composition which is to be interpreted. Such an operator type is seemingly contradictory to the very objective of performance. Nonetheless, primavista weights suggest an intervention of operators before any real performance, i.e., performance is already initiated on the very level of the score’s interpretation. –Σ– Symbolic performance operators are a delicate species since they intervene at a very early stage of the performance process: on the kernel level of the LPS hierarchy h. One could see this fact as a natural completion of the performance philosophy in that composition is the first stage of performance, so why not alter the kernels, i.e., make a new composition out of an old one. We adopted this integrative point of view while developing RUBATOr . From the software engineering perspective, this is indeed tempting since it would in the limit yield an integrated software for performance and composition. In this generic setup the program routine must, however, recalculate virtually every ingredient of the given LPS. The change of the basic score elements in a recomposition entails that the frames, the initial sets, the initial performances, the fields, and the weight list must be updated, or, rather: rewritten from scratch. Potentially, nothing will be the same again. So the operator then would just be a reset of the complete LPS data for a new composition. And this is not what performance was meant to do. After all, having rehearsed on a given piece and then being told that you get a new score, but you may go on with the old tempo curves, is not the kind of thing you will enjoy, since it will not make sense to perform on this schizophrenic data. For example, if a tempo curve has been developed, and then all onsets and durations are reset to half of the former values, the tempo curve becomes useless unless you also redefine the curve by a so-called time stretching operation (multiply the time arguments by 0.5). In order to avoid such risks, one should really recalculate all weights whenever a symbolic operator has changed the kernel data. But even then this would not necessarily be the right solution since some primavista weights of analytical nature really need to be calculated on an unrefined score, i.e., on symbolic data which do not yet share some sophisticated symbolic
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CHAPTER 39. OPERATOR THEORY
explications, for example in dynamics. This problem could be attacked by an implementation of weight calculation routines which average out the over-refined symbolic data, but this is not very elegant. It remains a fact that weights should also be given the parameter of the stemma LPS where the weight is instantiated. By this method, one would then be sure that a weight is related to a determined kernel in the stemmatic inheritance tree and not exclusively to the primary kernel data. With these caveats in mind, we may nevertheless define some useful (though also risky!) symbolic operator. We name it SymbolicBrueF orceOperator to remind you of this dangerous enterprise. It takes as arguments a “directional factor” d ∈ W , W a space of the hierarchy H, with components dR for parameters R of W , and a weight w. Then, if X ∈ U is an event of KernelU , we set SymbolicBruteF orceOperatord,w (X) = wd (X) (39.30) where wd (X) is the new event in U , whose R-coordinate wd (X)R is the product w(X).dR .XR , if R is a coordinate of W , and XR else. So we only change X-coordinates for the directional factor, and there, we scale the coordinates of X by the weight and by a fixed directional coordinate. The risk here is that for d or w(X) values far from 1, the image wd (X) is likely to fall out of the given frame. But one may then easily redefine the frame, the field, etc., since usually this symbolic operator is applied in a stemma stage where we are just given the default data. As we have seen in section 38.3.2, this data can be adapted to special kernels without difficulty: The frame is extended to include new events, the performance field is the constant field, and the initial set and performance are not a function of special kernel data. Under general stemmatic conditions, this operator is risky, however, but it is also an elegant solution of some nasty parameter problems which arise from the fuzzy score notation. For example, if we are given a weight for primavista dynamics wEvt.,RelEvt.. (see section 39.2.1), then the directional factor d = (1) ∈ Loudness gives the complete dynamics values if the given loudness was set to a preliminary value 1, say. Same for other primavista weights relating to symbolic parameter values. For dynamics, it seems to be clear that the symbolic operator is the right one to be applied. For other primavista weights this not so easy to decide. For example, the articulation weights for slurs, legati, or staccati could be seen as symbolic prescriptions to alter duration within the score event framework. We contend that this is not analogous to the dynamics situation because absolute dynamical signs (such as ff, mf) are not quantified and even less are relative dynamical signs (such as crescendo). We therefore have a clear mission to transform such verbal signs into quantitative data. In contrast, duration is a perfectly quantified parameter, so it needn’t be generated on that level. We therefore decided to deal with these weights in the context of field operators which alter the mother’s articulation field. This minor dilemma demonstrates (once more) that one may have the same performance output with very different infrastructure in the transformation process from the score to the physical events. This subject has not been dealt with from a more theoretical point of view: It pertains to the inverse performance theory to be dealt with in part XII, i.e., the theory of the variety of performance scores leading to a given performance output. But there is no result on the variety of local or global performance scores with variable score data inducing a given performance output; everything to date supposes that we are given a fixed score. Especially in ethnomusicological contexts, where the very concept of a score is uncertain, this is not
39.4. TAXONOMY OF OPERATORS
791
acceptable. For completeness of this taxonomy, we should add that we also call symbolic operator an operator which changes the initial set of a performance cell. This may be necessary if we want to enrich certain boundary conditions on the performance of selected groupings of notes.
39.4.3
Physical Operators
Summary. Physical operators affect given results of performance which are inherited from the mother’s data. They act in a very simplistic way since they do not relate to the deeper process of performance; they only take for granted what the mother had already produced as a performance output. –Σ– At the other end of the operator scale, we have the physical operators. While symbolic operators should be applied in the very beginning of a stemmatic unfolding, physical operators should be applied after all other operators have been deployed and no risk to have them applied once more is likely. An exception to this general rule is the case where after a physical operator another operator is applied, but not in any of the coordinates to which the physical operator has made changes. But the one who applies such exceptional stemmatic operations must know very well what is being done—as with pointers in C-style programming. Formally, what we call the P hysicalBruteF orceOperator is quite analogous to the operator SymbolicBruteF orceOperator discussed in equation (39.30). We are again given a directional factor d ∈ W , W a space in the hierarchy H, with components dR for parameters R of W , and a weight w. Then, if X ∈ U is an event of KernelU , we set P hysicalBruteF orceOperatord,w (X) = wd℘ (X)
(39.31)
where wd℘ (X) is an event in the physical space u corresponding to U , whose r coordinate (r corresponding to R according to our general nomenclature) is the product w(X).dR .℘(X)r , if R is a coordinate of W, and ℘(X)r else. So this operator really changes the performance at the point X, but the symbolic operator only affects the symbolic point and does not intervene on the ℘-level or the output. There is another very pleasant difference: Whereas the symbolic operator is risky as to the frame, field, and initial conditions, the physical operator is completely indifferent against these conditions. It only needs the mother’s performance and the rest will always work! This operator is also very fast on the programming level, it produces very little changes and therefore is quite comfortable to handle. This preconizes another application of P hysicalBruteF orceOperator in the sense of a test operator. In some situations, we would just like to know how an alteration of the output looks. Instead of applying a complicated operator (acting on the performance field and consuming much calculation power, say), we first apply this physical operator and obtain a first impression of how the given weight might influence the mother’s performance. If we like the effect, we may go on, first resetting the stemma to the mother’s level, and then apply a more sophisticated operator with the weight in question. For completeness of taxonomy, we should add that operators which alter the initial performance map are also called physical operators. They tell initial events where to be mapped in the
792
CHAPTER 39. OPERATOR THEORY
physical space. This is however not completely satisfactory since it usually depends upon the way the mother told its initial events where to be mapped. And this could have been done on a more field-theoretic level. The actual change could also involve field-theoretic considerations.
39.4.4
Field Operators
Summary. Field operators are the very essence of shaping performance. They include a variety of more or less general operators according to the direction where the field has to be altered. The theoretical complexity is paralleled by a computational complexity (numerical integration of ordinary differential equations) which is very important for software implementation of such operators. –Σ– Whereas symbolic and physical operators are an easy chapter in performance classification, field operators constitute an entire world of various approaches to the definition and deformation of performance fields. There are several reasons for this. The first is the mathematical complexity of fields, the definition and deformation of a field requires differential-geometric tools of local and global nature. The second is that performance fields have to be inserted in the hierarchies of their LPS. Therefore, the change of a field is always coupled to the fellow fields in the hierarchy. The third reason is that the stemmatic inheritance enforces operators which are capable of acting upon given mother fields of quite general nature: the stemmatic framework has to face unexpected field inheritance. Therefore, the definition of field operators is not completed if a particular type of deformation is implemented, one must also allow for mother fields which do not fit in the naive setup of such a deformation type. The fourth reason is that there is no such a thing like a performance field theory in traditional performance theory, and not even in modern approaches, such as the KTH school or the Todd approach. Performance fields are a new paradigm to performance theory, although, as we have shown, they are a completely natural and mandatory setup to control performance on an artistic level and from the more esthetic point of view as put forward by Adorno and Benjamin. We shall start our investigation of performance field operators by the classical tempo operator. We then generalize its setup to cope with more general mother fields. Then we generalize this approach to a more systematic framework, the so-called scalar operator. Another approach will be discussed under the title of basis-pianola operators, which arose from a systematic generalization of a physical operator to quite general situations. It turns out that this type is an interesting general approach involving Lie derivations of weights along the given mother fields. We then specialize this setup to operators for basis parameter field and pianola parameter field deformations. Although the field operators look very attractive, they have a big practical drawback: If one has to compute integrals of ordinary differential equations in order to solve the integral curve problem for performance fields, one has to control the type of vector field. In fact, not every wild field can be integrated by one and the same numerical ODE routine. For example, we have implemented Runge–Kutta–Fehlberg routines in RUBATOr ’s PerformanceRUBETTEr . We often experienced that the application of a field operator, such as the tempo operator or the scalar operator (see sections 39.5 and 39.6, respectively, for these operators), causes a breakdown
39.5. TEMPO OPERATOR
793
of these numerical routines. So the routines should be adapted to the field, but this is beyond this discussion and should be left to numerical ODE specialists.
39.5
Tempo Operator
Summary. The tempo operator is the classical field operator which alters the mother’s tempo as a function of given weights. We expose the different variants of such an intervention. –Σ– This operator acts on the tempo field TsE of a performance cell, so we suppose that E is a member of the space hierarchy H of h. We are given a weight w and set T empoOperatorw (Ts)E (XE ) = w(XE ).TsE (XE )
(39.32)
for a E-event XE . This operator is only defined if the weight is a strictly positive continuous function on the given onset frame, which we shall from now on tacitly assume. If the tempo operator is clear, we shall write the shorthand T empoOperatorw (Ts) = Tsw . This operator is as easy as problematic if we do not take into consideration other parameters which are canonically tied to onset. Duration is such a parameter. Let us look at the combined articulation field that is influenced by the weight in the tempo operator. Let us first start with the parallel articulation field (see section 33.2.1), i.e., TsD = 2.TsE ◦ α+ − TsE . So if we deform the tempo field TsE to Tsw,E , the D-component of the parallel field is deformed to a Dcomponent of the parallel field with the weight contribution, i.e., Tsw,D = 2.Tsw,E ◦α+ −Tsw,E = w ◦ α+ .2.TsE ◦ α+ − w.TsE . However, this formula is not satisfactory since it only works if we are given a parallel field situation. Else, we would destroy the given mother D-component and retain only the mother’s tempo information. In order to take the mother’s duration field component into consideration, we need a more invariant formula. To this end, rewrite 2.TsE ◦ α+ = TsD + TsE
(39.33)
Tsw,D = w ◦ α+ .(TsD + TsE ) − w.TsE = w ◦ α+ .TsD + (w ◦ α+ − w).TsE
(39.34)
and then get the formula
or for the total articulation field Tsw,ED = (Tsw,E , Tsw,D ): Tsw,ED =
w w ◦ α+ − w
0 w ◦ α+
! TsED .
The new articulation field therefore results from the old one by a linear action ! w 0 Qw = w ◦ α+ − w w ◦ α+
(39.35)
(39.36)
794
CHAPTER 39. OPERATOR THEORY
on the tangent bundle of the ED-frame5 : Tsw,ED = Qw .TsED . The shape of the operator Qw is precisely that which one obtains for the parallel field at tempo w. This means that instead of a real tempo, we take the ‘weight tempo’ w and the associated ‘performance’ ℘w . Its inverse ‘is’ Qw , and the new articulation field Tsw,ED is the inverse image of the old articulation field TsED under the ‘weight performance’ ℘w . Usually, the change from the given tempo curve TsE to Tsw,E = w.TsE changes also the RE RE 1 . If the integration limits E0 , E1 are duration ∆ = E01 Ts1E to the duration ∆w = E01 Tsw,E initial onsets, this is bad news, and we have to adapt the tempo deformation to the condition that these integrals should coincide. Moreover, one usually requires also that the tempi at the initial points should not change. To this end, one introduces a continuous support function supp which vanishes outside [E0 , E1 ], is identically 1 in a slightly smaller interval [E0 + β, E1 − β] for a relatively small positive real number β, and is the cubic spline with zero slopes on the boundary intervals [E0 , E0 + β], [E1 − β, E1 ]. Then s(TsE , w, t) = TsE (t)(1 + supp(t)(w(t) − 1)) coincides with w within [E0 + β, E1 − β] and is 1 outside [E0 , E1 ]. We then try the new duration Z
E1
J(TsE , w) = E0
1 s(TsE , w, t)
and compare the durations: r(TsE , w) =
J(TsE , w) . J(TsE , 1)
We then try the obvious correction w1 = r(TsE , w).w and get a weight which gives us nearly the right duration since within the interval [E0 + β, E1 − β], the error is corrected. We look for the new error r(TsE , w1 ), set the new weight w2 = r(TsE , w1 ).w1 , and repeat this procedure until the error becomes small enough. This procedure has been implemented in RUBATOr ’s PerformanceRUBETTEr , but we do not know whether it converges6 .
39.6
Scalar Operator
Summary. The scalar operator is a first generalization of the tempo operator to articulation. The conceptual background is that tone parameters can be split into basis and pianola parameters. Onset, pitch, loudness are basis parameters. The corresponding pianola parameters are duration, glissando, crescendo. Pianola components of performance fields are coupled to the corresponding basis components of the performance fields. This coupling can be distorted by making operator weights act on specific parameter sets. The scalar operator does this job. –Σ– 5 Recall
that performance fields are defined on open neighborhoods of frames. this could be settled by use of Banach’s fixpoint theorem, we do however not know whether the map is a contraction. 6 Perhaps
39.7. THE THEORY OF BASIS-PIANOLA OPERATORS
795
Suppose that we are given a space B of basis parameters, and that P is the corresponding space of pianola parameters. We suppose given a weight w on the B space. The scalar operator ScalarOperatorw deforms a performance field TsBP = (TsB , TsP ) to a new field ScalarOperatorw (TsBP ) = (Ts∗B , Ts∗P ) as follows. We have two parameters: parallel/not parallel, and a two-bit parameter B = yes/no, P = yes/no. The notparallel case is a deformation of the components of the given field according to the second parameter: 1. B = P = no: We leave the field unchanged. 2. B = P = yes: We set ScalarOperatorw (TsBP ) = w.TsBP . 3. B = yes, P = no: We set ScalarOperatorw (TsBP ) = (w.TsB , TsP ). 4. B = no, P = yes: We set ScalarOperatorw (TsBP ) = (TsB , w.TsP ). The parallel case runs as follows. It means that we refer to the parallel field structure in one way or another. The four cases run as follows: 1. B = P = no: No weight influence, but the parallel structure is installed, i.e., ScalarOperatorw (TsBP ) = ∂TsB
(39.37)
2. B = P = yes: This is the generalized tempo operator, i.e., ScalarOperatorw (TsBP ) = Qw .TsBP
(39.38)
with the matrix being understood as a block matrix of scalar endomorphisms on the basis and pianola spaces, respectively. The original tempo operator is the special case B = E, P = D. 3. B = yes, P = no: We set ScalarOperatorw (TsBP ) = ∂(w.TsB ). 4. B = no, P = yes: We set ScalarOperatorw (TsBP ) = w.∂TsB . We should remark that the ScalarOperator and a fortiori the T empoOperator do not change any standard hierarchy. But the initial data could be affected. First of all, the field changes in the articulation plane of onset and duration could throw some kernel event out of the reach of the given initial set, excluding them from being performed. Second, the initial performance could be turned into an incompatible state, a problem we have already dealt with at the end of the T empoOperator discussion. For the general situation of the ScalarOperator, one must develop specific routines to control initial data, not a trivial task.
39.7
The Theory of Basis-Pianola Operators
Summary. The tempo and scalar operators can be generalized to operators which act on the basis and pianola parameter grouping and their coupling on the level of performance fields. The formalism is deduced from the standard situation and stated in its general shape. The point of this approach is the introduction of Lie derivative as a general device for the production
796
CHAPTER 39. OPERATOR THEORY
of deformations of performance fields by use of scalar potential funcions. The Lie formalism is applied when stressing on basis rather than on pianola parameters. Although it is a general formalism for operators, we do not know whether its application can generate all important deformation operators needed for a musically satisfactory performance construction. –Σ– The standard situation which motivates the basis-pianola operators is this: We are given a performance map ℘(X, Y ) = (x(X), y(X, Y )) on variables X, Y of two disjoint spaces U, V , with a hierarchy projection U ⊕ V → U . For example, this could be an articulation (U = E, V = D) or a direct product (U = EL, V = H), etc. We now want to deform this performance on the second factor, and we are using a positive deformation weight function λ on U . The deformed performance map is defined by ℘λ (X, Y ) = (x(X), λ(X).y(X, Y )). So this is a kind of physical operator we are familiar with. Let us calculate the Jacobian of this transformation (supposing that everything is smooth). The Jacobian for λ = 1, the start situation is ! ! ∂X x 0 A 0 J℘1 = J℘ = = (39.39) ∂X y ∂Y y B C where ∂X x is the submatrix with the partial derivatives of all x coordinates with respect to the X arguments, etc. The inverse Jacobian is ! A−1 0 −1 J℘ = . (39.40) −C −1 BA−1 C −1 And the general inverse Jacobian is J℘−1 λ
=
A−1 −C −1 BA−1 − λ−1 C −1 · dλ ⊗ y · A−1
0 −1 −1 λ C
! (39.41)
∼
with the dual gradient dλ = gradλ∗ and the identification7 Lin(U, V ) → U ∗ ⊗ V . The performance field is calculated by use of the diagonal unit vector ∆ = (∆X, ∆Y ) and yields ! A−1 ∆X −1 Tsλ = J℘λ ∆ = . −C −1 BA−1 ∆X − λ−1 C −1 · dλ ⊗ y · A−1 ∆X + λ−1 C −1 ∆Y If we set Ts(X, Y ) = (Z(X), W (X, Y )), and Tsλ (X, Y ) = (Z(X), Wλ (X, Y )), we get Wλ (X, Y ) = W (X, Y ) − λ−1 C −1 · dλ ⊗ y · Z(X) + ( = W (X, Y ) − LZ (Λ)C −1 y + (
1 − 1)C −1 ∆Y λ
1 − 1)C −1 ∆Y eΛ
(39.42)
with the Lie derivative LZ (Λ) of the function Λ = ln(λ), knowing that λ is a strictly positive function. Everything is taken at the argument X, Y , or (X, Y ), respectively. 7 See
appendix E.3.2.
39.7. THE THEORY OF BASIS-PIANOLA OPERATORS
797
This gives us formula TsΛ = Ts − LZ (Λ)iV C −1 y − (1 − e−Λ )iV C −1 ∆Y
(39.43)
where iV is the canonical injection V → U ⊕ V . So we obtain a deformation of the original field Ts by two additive terms LZ (Λ)iV C −1 y and (1 − e−Λ )iV C −1 ∆Y , which we want to interpret in the following sections. Both terms involve the weight function. The first term LZ (Λ)iV C −1 y involves the basis field Z, while the second term only involves contributions from the Y component of the field. Therefore, we call the first term the basis deformation, whereas the second is called the pianola deformation.
39.7.1
Basis Specialization
Summary. The general basis-pianola theory is specialized to basis parameters and explicated through concrete formulas. –Σ– If we have a weight Λ ∈ U (0) in a very small neighborhood of 0 (or else λ in a small neighborhood of 1), we may neglect the pianola term in formula (39.43). If further the performance map in the second variable is affine, i.e., y = A.Y + B, and does not depend on the first variable X as is the case with pianola coordinates Y , then C −1 = const. = A−1 , i.e., C −1 y = Y + const. So we have a deformation of type LZ (Λ)iV C −1 y = LZ (Λ)iV (Y + const.). For example, consider the two-dimensional situation U = E, V = L of two basis spaces. The performance map for a weight Λ ∈ U (0) on E is ℘Λ (X, Y ) = (x(X), eΛ .y(Y )), where we start from a primavista performance map ℘ = ℘0 . We may therefore suppose that y = A.Y . Then, one has the primavista product field Ts = T × D of tempo T and dynamics D. Then the deformed field is dΛ (T × D)Λ = T × D − LT (Λ)iL Y = T × D − T. iL Y. dX In other words, we deform the given tempo by the gradient of Λ and project this component to the loudness field component via a field iL Y . So musically speaking, the loudness field deformation is essentially controlled by the change of Λ and the mother tempo.—Let us now propose a generalization of the basis deformation: We are given • two not necessarily disjoint spaces U, V of the space hierarchy, • a weight Λ on the first (the basis) space U , • an affine directional endomorphism Dir(Y ) ∈ V @V of the V space. With these data, we define a field deformation of the performance field Ts on a superspace of U ∪ V in the hierarchy by the formula TsΛ,Dir = Ts − LTsU Λ.iV Dir
(39.44)
where the argument of Dir is the V -component, and that of the Lie-derivative is the U component of the total argument, whereas iV is the embedding of V in the space of Ts. The following is immediate from the linearity of the Lie derivative in both arguments.
798
CHAPTER 39. OPERATOR THEORY
Lemma 53 With the above notation: (i) The basis operator is linear in the performance fields, i.e., (µTs1 + νTs2 )Λ,Dir = µTs1Λ,Dir + νTs2Λ,Dir . (ii) If U and V are disjoint, and if Λ1 , Λ2 are two weights, then (TsΛ1 ,Dir )Λ2 ,Dir = TsΛ1 +Λ2 ,Dir . Example 57 An example of a non-disjoint union of spaces U, V is the elementary deformation 1 of tempo R 1−γby a positive C weight γ. On onset arguments X in a frame of positive onsets, we set Λ= X . This implies γT = T − LT (Λ).1E , and we have another example of a basis operator. Example 58 For this example, we suppose given a two-dimensional C2 performance map ℘EH : EH → eh on the plane of onset and pitch with a hierarchy EH → E over the onset performance ℘E : E → e with tempo curve T . So the performance field TsEH is C1 . Setting I℘ = (∂H h)−1 , T = (∂E e)−1 , the field reads TsEH = (T, I℘ (1 − ∂E h.T )).
(39.45)
We want to view such a field as a deformation of the primavista product field T × I which is composed of the same tempo factor T and an intonation factor I, which is also C1 and strictly positive, as usual for primavista fields. We are looking for a weight Λ on the onset-pitch frame R and a pitch shift e−Y0 ∈ H@H such that the deformation LT ×I (Λ)iH e−Y0 of the primavista field yields TsEH . This defines a linear partial differential equation (PDE) in the onset variable X and the pitch variable Y : T ∂E Λ + I∂H Λ = (Y − Y0 )−1 (I℘ (∂E h.T − 1) + I) = Q.
(39.46)
This equation has C1 coefficients T, I, Q, if we let Y0 be smaller than the lower boundary value of the pitch frame interval. Using the method of characteristic curves (see appendix I.6) in pseudolinear PDEs, we can see that there is a solution of equation (39.46). In fact, the characteristic curve projection ODE reduces to the pair dt X = T (X), dt Y = I(Y ), whereas the third curve component Z(t) is defined by the ODE dt Z = Q(X, Y ). This means that the characteristic curve projection onto the XY -space is an integral curve of the primavista field T × I. Clearly, a transversal curve Γ to the characteristic curves exists since the Jacobian criterium T Γ ∦ T × I can be met for the non-vanishing field T × I. We therefore have this result: Proposition 59 Let ℘EH : EH → eh be a two-dimensional C2 performance map on the plane of onset and pitch with a hierarchy EH → E over the onset performance ℘E : E → e with tempo curve T . For the primavista product field T × I which is composed of the same tempo factor T and an intonation factor I, which is also C1 and strictly positive, there is a C1 weight Λ on the EH-frame of this performance such that the performance field TsEH is a deformation of the primavista field T × I by a basis operator: TsEH = T × I − LT ×I (Λ)iH e−Y0 .
(39.47)
39.7. THE THEORY OF BASIS-PIANOLA OPERATORS
799
The basis operator is designed for distributing weight information from the “basis” subspace U over any other space V , independently whether this one is also in the hierarchy or not. This creates a considerable freedom of shaping performance. The way this shaping is related to the given weight is the Lie derivative with respect to the performance field TsU on U . This one also measures the angle between the weight’s gradient and TsU . If their mutual position is perpendicular, the derivative vanishes and the operator has no effect. 39.7.1.1
Deforming Hierarchies
Summary. We discuss the change in an existing hierarchy after the application of a basis operator. –Σ– The most dramatic effect of the basis operator is the deformation of the given hierarchy. For example, in the above example of proposition 59, the projection EH → H is destroyed since the weight Λ is not only a function of H or of E, but of both, in general. So only the projection EH → E survives this deformation. Let us describe more systematically which hierarchy spaces disappear a priori by a basis operator deformation. Suppose that we have a hierarchy space U , a weight Λ on U any other space V within the top hierarchy space, a directional endomorphism Dir = eB .A, and any hierarchy space W which we want to test for survival after the subtraction of the basis deformation LTsU (Λ)iV Dir. Consider the projection p : V → V ∩ W onto the intersection space, including the empty intersection which then defines the projection onto the zero space. Let [p, A] = p.A − A.p denote the commutator endomorphism of p and A on V. Proposition 60 With the above notation, the mother hierarchy space W remains a priori alive (i.e., member of the daughter hierarchy deduced from the mother hierarchy) after the basis deformation LTsU (Λ)iV Dir iff either p = 0, i.e., V ∩ W = ∅, or p[p, A] = 0 and U ⊂ W . In particular, for [p, A] = 0 and U ⊂ W , W remains alive. Proof. Suppose that W remains alive. If p 6= 0, we have at least one coordinate that is common to V and W . On that coordinate, the functional dependence of U is inherited via LTsU (Λ). Therefore we must have U ⊂ W . Since no functional dependence from W − V arguments can be the case on the coordinates in V ∩ W , and since the constant part of Dir is irrelevant here, we must have p.A.(1 − p) = 0. On the other hand, if p = 0 everything is clear, and under the conditions p[p, A] = 0 and U ⊂ W , the deformation arguments all stem from W , and we have saved the life of W , QED. Example 59 Let the top coordinate space be all six usual basis and pianola parameters EHLDGC. Let a weight Λ act on U = EH, take V = ED, whereas the directional endomorphism is ! 0 1 eB . −1 0 with a rotation A as its linear part. Suppose that the hierarchy is the parallel hierarchy, i.e., the hierarchy generated by the basis hierarchy T ID, T I, T D, T and the parallel fields
800
CHAPTER 39. OPERATOR THEORY
∂T ID, ∂T I, ∂T D, ∂T with the corresponding projections. Since the rotation A has no proper invariant subspaces, we must have either ED ∩ W = ∅ or else (because of p[p, A] = 0) ED ⊂ W . But every field in our hierarchy contains E, hence U ∪ V = EHD ⊂ W . This is only the case for the sub-hierarchy EHD, EHDG, EHLD, EHLDG, EHLDC, EHLDGC. 39.7.1.2
Lie Derivatives
Summary. Basis-pianola operator theory leads to Lie derivatives as a device for operator definition. We discuss the Lie formalism in its realization as a component of a performance grammar. –Σ– The appearance of the Lie derivative in this context is quite surprising. Its usage is well known from classical mechanics, for example, where the Lie derivative of a function with respect to a Hamiltonian vector field is related to the Hamiltonian function H via the Poisson bracket [2]. Presently, we do not know of any analogous structures of Hamiltonian or Lagrangian type in performance field theory. But it is good to have this perspective in mind for a future ‘dynamics of performance’. We should however observe that the Lie derivative LTs associated with a vector ∼ field Ts induces an isomorphism L? : XR → Der(F(R)) between the real vector space XR of smooth vector fields over the frame R and the vector space Der(F(R)) of derivations on the real algebra F(R) of smooth functions (see appendix I.2.4). So LTs can be identified with Ts, and the basis deformation means taking into account the ‘weight’ LTs (Λ) deduced from the weight Λ. In this sense, weights become natural mathematical objects associated with performance fields: They are just the natural objects, these fields act upon qua derivations, they are not only justified by the quantification argument given when we introduced weights in section 39.1. In other words: Thesis 6 A performance field is not only a construction principle for the performance map, but equivalently an ‘interpretation of weights’—this is effectively the mathematical transfiguration of the rational approach to performance. This thesis suggests that one should study the natural properties of the Lie algebra Der(F(R)) with respect to performance theory, in particular the question of what is the musical interpretation of the Lie bracket [Ts1 , Ts2 ] of two performance fields. We have to pass this subject to future research. Exercise 81 Consider the performance ℘ : ED → ed : (X, Y ) 7→ (x(X), f (d)), d = x(X + Y ) − x(X), which is a functional change in duration, built upon the parallel performance, with a C1 invertible deformation function f of physical duration d. This is an example of a physical operator. Show that its field is ! ! T 1 0 Ts = (f 0 +1) = (1−f 0 ) T ◦α+ ∂T, (39.48) 1 f 0 T ◦ α+ − T f0 T i.e., a linear automorphism of the tangent bundle of which we have already seen an example for the tempo operator. Observe however, that the automorphism is also a function of the tempo
39.8. LOCALLY LINEAR GRAMMARS
801
curve T ! Show by use of the characteristics method for quasilinear PDEs that this deformation of the parallel field ∂T can also be obtained by a basis operator with U = ED, V = D.
39.7.2
Pianola Specialization
Summary. The general basis-pianola theory is restricted to pianola parameters and explicated through concrete formulas. –Σ– Compared to the basis deformation, the second contribution (1 − e−Λ )iV C −1 ∆Y in the general deformation formula (39.43) plays a different role. Whereas the basis contribution is sensitive to the gradient of Λ, i.e., its local changes, the second contribution is sensitive to the absolute values of the weight, so this contribution is relevant if the weight changes little, but has values different from zero. We give a more precise interpretation of this contribution in the case of a basis-pianola-space situation, i.e., U is a space of basis parameters, and V is the corresponding space PU of pianola parameters. We then have the alterator α+ : U ⊕ PU → U , ∼ and we have a canonical isomorphism τ : U → PU . If the original field is defined on U ⊕ PU and −1 is a parallel field ∂TsU , then we have C ∆Y = τ ◦ TsU ◦ α+ . Therefore we obtain the pianola operator for this special space configuration: TsΛ,U = Ts − (1 − e−Λ )iPU ◦ τ ◦ TsU ◦ α+ .
(39.49)
This formula only involves a weight on any subspace of the top space and a hierarchy space U consisting of basis parameters, together with the corresponding pianola space PU which must also be a subspace (but not necessarily hierarchic!) of the top space.
39.8
Locally Linear Grammars
Summary. According to section 38.3.4, interaction between “inherited” performance score structures of sisters—or farther relatives—can be envisaged. We describe this formalism which is a basic approach in inverse performance theory (see section 46.2). –Σ– Until now, we have only considered operators which are directly related to the mother data, and not to farther relatives, such as sisters, or daughters of sister, etc. In the following discussion, we shall present an essentially linear model for such a more global interconnection of a stemma’s LPS. We start with a stemma, i.e., a local performance score Λ whose graph Λ l is an undirected tree. We want to forget about the mother of Λ and concentrate on the tree Λ ↓= T0 . For the following construction, we need the stemma quiver8 T = TΛ associated with Λ: Definition 109 The stemma quiver is a finite directed graph T = (V, A) with vertex set V and arrow set A, including multiple arrows and loops. It is constructed as follows. We start 8 See
definition 123 in appendix C.2.2 for the quiver concept.
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CHAPTER 39. OPERATOR THEORY
with a directed tree T0 = (V, A0 ) with root r, i.e., each vertex can be reached by a unique path starting from the root. If x → y is an arrow of T0 , we say that x is the mother of y and y is a daughter of x (in combinatorics they are known as ‘father’ and ‘son’, respectively, here we try to be politically correct). If we have a path x → y → z, then z is a granddaughter of x, while x is the grandmother of z, and so on. For a vertex x of T0 , the set of vertexes which are daughters of x is denoted by Dx (T0 ). The vertexes x which are not mothers, i.e., Dx (T0 ) = ∅, are called final (the ‘leaves’ in graph theory). Similarly for each vertex x ∈ V (T ) we define Mx (T0 ) ⊆ V (T0 ) as the set of vertexes lying on the unique path from x to the root (x included). In order to define the stemma quiver T the directed tree T0 is enriched by the following set of arrows (no vertexes added): First, each vertex x is given a loop x , and for any couple of sisters x1 , x2 , i.e., of daughters of a common mother y, we add an arrow x1 → x2 . The resulting quiver T = (V, A) is called a stemma quiver of the tree T0 , which is uniquely determined by the stemma quiver and is called the stemma tree. We therefore may define Dx (T ) = Dx (T0 ) and Mx (T ) = Mx (T0 ). Definition 110 With these graph-theoretical data, a locally R-linear grammar is a family of R-linear representations9 of a stemma quiver T which is defined by the following data: 1. For each vertex x ∈ V (T ) let Ax , Bx be two real vector spaces, Bx of finite dimension sx . 2. Each mother–daughter arrow x → y is represented by a surjective linear map rx,y : Ax → Ay . 3. For all x ∈ V (T ) let ϕx : Bx → End(Ax ) be an affine map, i.e., a linear map ϕ0x , followed by a displacement by ϕtx ∈ End(Ax ). So each loop x is represented by a family of endomorphisms (ϕx (b) : Ax → Ax )b∈Bx parametrized by the parameter space Bx . 4. For all pairs of sisters x, y ∈ Dm (T ), the sister arrow x → y is represented by an isomorphism ix,y : Ax → Ay with iy,x = i−1 x,y and ix,x = IdAx . For concrete performance field configurations associated with Λ this axiomatic setup is realized as follows. For the LPS of the stemma tree, we suppose a number n of real parameters for the top space S of the LPS hierarchies. We concentrate on this top space S of all the LPS, and do not discuss the other cell data of these space hierarchies. For each vertex x of T0 , we have the frame Rx = [lx1 , ux1 ] × . . . [lxn , uxn ] of the top space of its hierarchy. For each mother m and daughter x, we suppose that Rx ⊂ Rm , and that for each couple of sisters x1 , x2 , Rx1 ∩ Rx2 = ∅. This corresponds to a restriction of a larger portion of a musical score to a disjoint grouping of smaller portions. Here is the realization of our above system (properties 1–4) of quiver representations: Consider the vector space Fx of C∞ functions on Rx . We then set Ax = Der(Fx ), the space of derivations, i.e., the C∞ vector fields on Rx . The surjective maps rx,y are defined as the restrictions of vector fields on the mother’s rectangle Rx to the daughter’s rectangle Ry . To define the representations for a sister arrow x1 → x2 , consider the unique affine morphism ax2 ,x1 : Rx2 → Rx1 on the sisters’ rectangles such that the respective vertexes are mapped onto 9 These are representations of the quiver algebra over R, see appendix E.2.1 for the concept of a linear representation.
39.8. LOCALLY LINEAR GRAMMARS
803
each other. Then the sister arrow representations are isomorphisms ix1 ,x2 : Ax1 → Ax2 defined by the transport of a vector field Z on Rx1 to Z · ax2 ,x1 on Rx2 . To define the operation of a parameter family, first take a vector field Z ∈ Der(Fx ), and a weight function Λ ∈ Fx . For Dir ∈ S@S, consider the corresponding vector field ΓDir (t) = (t, Dir(t)) on Rx . Then we have a new vector field Z − LZ Λ.ΓDir , corresponding to a basis deformation. This is an R-linear operator on Z, and the ‘deformation’ part LZ Λ.ΓDir is Rbilinear in the weight function and the affine endomorphism. We now take a finite dimensional subspace Wx of Fx which in performance theory represents the weight functions issued from analyses of metrical, motivic, and harmonic structures of the given score. We now set Bx = Wx ⊗R S@S, and we obtain an R-linear map ϕ0x : Bx → End(Ax ) defined by ϕ0x (Λ ⊗ Dir)(Z) = −LZ Λ.ΓDir . Setting ϕtx = IdAx , we have defined the required affine map ϕx = ϕtx + ϕ0x : Bx → End(Ax ) with ϕx (Λ ⊗ Dir)(Z) = Z − LZ Λ.ΓDir . Let m ∈ V (T ) be a mother. Then for each daughter x ∈ Dm (T ), we define a triaffine (affine in each argument) map fx : Am × C#Dm (T ) ×
Y
By → Ax ,
(39.50)
y∈Dm (T )
by (am , (cm y,x )y∈Dm (T ) , (by )y∈Dm (T ) ) 7→ P m y∈Dm (T ) cy,x iy,x (ϕy (by )(rm,y (am ))) = P P m 0 m t y∈Dm (T ) cy,x iy,x (ϕy (by )(rm,y (am ))) + y∈Dm (T ) cy,x iy,x (ϕy (rm,y (am ))). Referring to the above example, this formula describes the following: In order to determine the performance field on the frame Fx , we use the field of its mother m, we first restrict that field to any of its daughters y and get the fields rm,y (am ). These sister fields to daughter x are then deformed under the endomorphisms ϕy (by ) = ϕty + ϕ0y (by ) induced by the system parameters by . These deformed fields are then transported to x and weighted by the factors cm y,x . These sister fields influence the final value of the field at daughter x. Musically, this means that the performance at x is influenced by surrounding sister fields, which are typically the fields of past and future times (past or future periods, bars, etc.). We shall pursue this model in the course of the inverse performance theory of chapter 46. A final word on a perspective of generalized stemmata which seem to be suggested by the locally linear grammars as discussed above: One could envisage continuous stemmata. They are based on a generalization of the stemma’s ramification structure to one-parameter families and narrowing of the daughters’ extents of ‘infinitesimal’ quantities. This construction would take care of the fact that psychologically, interaction between neighboring performance moments is continuously ‘updated’. A theory of continuous stemmata is still pending.
Part X
RUBATO
805
r
Chapter 40
Architecture The most rigorous test of the efficiency of theories in modern cognitive science is the production of a working computer program whose external behaviour mimics that to be explained. John Sloboda [491] Summary. RUBATOr is a metamachine designed for representation, analysis, and performance of music. It was developed on the NEXTSTEP environment during two SNSF grants from 1992 to 1996 by the author and Oliver Zahorka [348, 347, 350, 357, 588, 590]. From 1998 to 2001, the software was ported to Mac OS X by J¨org Garbers in a grant of the Volkswagen Foundation. RUBATOr ’s architecture is that of a frame application which admits loading of an arbitrary number of modules at run-time. Such a module is called RUBETTEr . There are very different types of Rubettes. On the one hand, they may be designed for primavista, compositional, analytical, performance stemma or logical and geometric predication tasks. On the other, they are designed for subsidiary tasks, such as filtering from and to databases, information representation and navigation tasks, or else for more specific subtasks for larger “macro” Rubettes. A RUBETTEr of the subtask type is coined OPERATOR and implements, for example, what we have called performance operators in section 44.7. The RUBATOr concept also includes distributed operability among different peers. This software is conceived as a musicological research platform and not a hard-coded device, we describe this approach. Concluding this chapter, we discuss the relation between frame and modules. –Σ– r
In the original concept of RUBATO [345], we had defined RUBATOr as being a software for analysis and performance, divided into two submodules: one for “structuring” a score, and the other for “shaping” this score. This meant that structuring would yield analytical structures, whereas the other would yield a shaped performance transformation, alimented by analytical data from the structuring process. In the course of the software development, we learned that no data model for music objects known to the developers at that time would be sufficient for all requirements of a comprising 807
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CHAPTER 40. ARCHITECTURE
music analysis and performance. This led to the concept of denotators and forms, as realized under the title of the “PrediBase” database management system (DBMS) of the first RUBATOr implementation in 1994, as described in [589]. At that time, it became clear that under such a universal data model, RUBATOr would split into an application framework comprising the “PrediBase” DBMS and a series of dynamically loadable software modules as implemented in the Objective C language of those NEXTSTEP OS driven NeXT computers. According to the diminutive convention for modules, such a module was coined “RUBETTEr ”. In 1996, at the end of a grant of the Swiss National Science Foundation where RUBATOr was realized, three analytical Rubettes and one for performance have been developed1 , which will be described in chapter 41. The PerformanceRUBETTEr is connected to five OperatorRubettes (at those times still named “OPERATORS”). The PrimavistaRUBETTEr takes care of paratextual score predicates. In 2001, this software has been ported (and improved in many data management aspects) to Mac OS X by J¨ org Garbers and is now available as an open source project on the internet [357], or on the CD appended to this book, see page xxx. The RUBETTEr screenshots in chapter 41 are all taken from this version of RUBATOr . Figure 40.1 shows the info panel of the Mac OS X version. However, the version included in the book’s CD-ROM is the latest version before the book went into production, whereas the screenshots are somewhat older. We hope that the reader will excuse us for this slight asynchronicity.
Figure 40.1: The info panel of the Mac OS X version of RUBATOr . In the following sections of this chapter we shall however not describe the Mac OS X implementation, we will rather expose the more advanced and flexible architectural principles of the ongoing Java-based implementation of the distributed RUBATOr version.
40.1
The Overall Modularity
Summary. RUBATOr is a modular engine for metamachine rationales and because research is 1 For NEXTSTEP, RUBATOr as well as these Rubettes are available on the internet, see [357]. The source code is GPL and is contained in the book’s CD-ROM, see page xxx.
40.2. FRAME AND MODULES
809
itself increasingly modular. Built upon the denotator language, the RUBATOr concept is fully modular, all parts that can be split into modules have been split in this way. –Σ– The modularity of RUBATOr has two aspects: First, it shows a composition of the software from an arbitrary, a priori undetermined, number of functional units—the RUBETTEr modules. Second, the available Rubettes are a dynamic factor: According to the research progress, new Rubettes of any flavor may be added to the existing arsenal. Modularity is an old principle, in fact, the traditional disciplinarity of science preconizes modules of scientific activities which have or pretend to have a relative autonomy in knowledge production. What is new with respect to the traditional disciplinarity is that this modularity is a dynamical one, at any time new modules of knowledge processing may be added or old ones removed. Discipline becomes a task-driven decision instead of being a rigid preset splitting. Such a modularization can only work on the common ground of unrestricted cross-communication among any subgroup of knowledge modules. Without a common language ground, which in our case is the denotator and form data model, dynamic disciplinarity would inevitably collapse since a new module would require language modifications, adaptations, and extensions. To be clear, we view dynamic disciplinarity as the idealized version of inter- and transdisciplinarity. The unity of knowledge cannot be achieved without a temporary and task-driven compartmentation of research fields, grouping and regrouping is inevitable; there is no direct path to the unity of knowledge. This credo can however not be realized without a common language basis. Otherwise, language engineering frictions would paralyze any major effort of dynamical grouping of disciplines. Evidently, the present RUBATOr environment is limited to musical and musicological scopes. Here dynamic disciplinarity is not that utopic. After all, the denotators and forms are universal language approaches issued from music(ologic)al requirements. But it has been shown in [464] that denotators are not only a priori applicable to non-musical concept modeling, but also in concrete cases such as geographic information systems. This suggests that dynamic disciplinarity could be realized on RUBATOr for modules of completely general scopes as long as the denotator data model is joined. This is one of the most intriguing vectors of future developments concerning the RUBATOr environment. The principle of dynamic disciplinarity has its social form: a so-called collaboratory. According to Bill Wulf, this is “a ‘center without walls’ in which the nations researchers can perform their research without regard to geographical location, interacting with colleagues, accessing instrumentation, sharing data and computational resources, and accessing information in digital libraries”[278]. To collaborate in this way requires adequate software platforms, and RUBATOr is precisely this type of software in the field of musicology.
40.2
Frame and Modules
Summary. Modularity has been realized on the basis of a frame application which offers interfaces to an arbitrary number of modules. This is one of the technical core features in the realization of a metamachine. We describe its splitting interfaces and their functional positions. –Σ–
810
CHAPTER 40. ARCHITECTURE DBMS
FS
FS
DBMS PEER 3
PEER 2
DTX
Rubettes
DTX
LoGeo Pr Prima ima Vista Vista
Rubettes
RMI
LoGeo
Rubato
Class Libraries
Pr Prima ima Vista Vista
Rubato
Class Libraries
RMI
RMI
FS
PEER 1
DBMS
DTX
Rubettes LoGeo Pr Prima ima Vista Vista
Rubato
Class Libraries
Figure 40.2: The RUBATOr layers of Rubettes with denotator communication, and the RUBATOr Framework layer with class libraries that are related to different Rubettes. Each peer has one such configuration. Different peers are interconnected via remote method invocation (RMI).
The RUBATOr platform consists of a number of installations of the software on different peers which may communicate via Java’s remote method invocation protocol (RMI, see SUN’s Java documentation on the internet). For each peer, RUBATOr contains two layers: the RUBETTEr layer and the RUBATOr framework layer. The first contains a number of Rubettes which are autonomous Java applications that communicate with each other and with Rubettes of another peer exchanging denotators via RMI. These are instances of the denotator class. The class library on the RUBATOr framework layer contains corresponding basic Java classes for denotators, forms, diagrams of presheaves, and modules. It also contains other classes which provide Rubettes with the necessary routines. The concept of these libraries is that they should contain all classes and methods that are of general interest, while classes and methods with specific interest for a RUBETTEr ’s functionality should be installed in that RUBETTEr . There are a number of mandatory Rubettes: The InfoRUBETTEr (with an “i” in figure 40.2) is the initialization RUBETTEr . It is automatically started when RUBATOr starts and informs the user about available peers and Rubettes on the distributed environment.
40.2. FRAME AND MODULES
811
The visualization of all Rubettes’ content and manipulation structures is managed by the PVBrowserRUBETTEr , whose functionality is to visualize any denotator in 3D space via Java3D classes, in figure 40.2, this RUBETTEr is represented by a lens symbol. The concept of this RUBETTEr has been described in chapter 20. The advantage of this centralization is that no other RUBETTEr designer has to take care of graphical and other multimedia representations, every denotator is piped to the PVBrowserRUBETTEr in case a multimedia representation is required. And every such representation is uniform according to this Rubette’s visualization routines, which makes orientation much easier than individual design for every RUBETTEr . Nonetheless, as explained in chapter 20, the flexibility of the Satellite form for multimedia objects allows an unlimited multiplicity of shape and behavior. Two further Rubettes are devoted to the storage of denotators. The first, to the upper left of the RUBETTEr layer, we have the DenotexRUBETTEr . It takes care of the storage and editing of denotators which are given in the Denotex ASCII format. The second storage RUBETTEr is a filter to a SQL DBMS such that SQL databases can be transformed into denotators for RUBATOr . A last central RUBETTEr is the LoGeoRUBETTEr , as shown by a toothed wheel symbol to the right front of the RUBETTEr layer. It manages the logical and geometric operations on denotators (see section 18.3.4) and can be used by any RUBETTEr for its specific needs. The methods of this RUBETTEr are encoded in the class library of the RUBATOr layer.
Chapter 41
The RUBETTE Family r
V¨ ogel, Vieh und alles, was auf Erden kriecht, die laß heraus mit dir, daß sie sich tummeln auf der Erde und fruchtbar seien und sich mehren auf Erden. The Holy Bible, Genesis 8 Summary. We give an overview of the analytical MetroRUBETTEr , MeloRUBETTEr , HarmoRUBETTEr , the PerformanceRUBETTEr , and the PrimavistaRUBETTEr , which have been realized on the NEXTSTEP and then on the Mac OS X environment. –Σ– Originally, the Rubettes as such were not the central concern of the RUBATOr project, this was rather to establish their collaboration and the realization of the whole transmission process from analytical data to the performance shaping operators of the PerformanceRUBETTEr . Each of these Rubettes was more an experimental prototype without the claim of a high-end tool in the specific domain. The interest in such experiments lies in the fact that when one starts the design of a RUBETTEr , it turns out that musicology and music theory do not offer any reasonable support, be it in conceptual, be it in operational aspects. The path from the given score to a specific analysis reveals an incredible complexity of what in musicology and music theory looks like an easy enterprise. For example, in the design of the HarmoRUBETTEr , the mere definition of what is a chord cannot be traced from traditional literature. Should we only look for local compositions of pitches that stem from notes with a common onset, or should one also consider durational aspects? The standard answer—or rather: excuse—states that it depends on the particular context, and the context of the context, but this is no way out if one has to implement clear concepts and methods rather than rhetorics. So the design of a RUBETTEr is always a very good test of the validity of a model and of its adequacy with traditional fuzzy understanding. But it is also a test for the tradition: After all, who decides what is a good model for harmony? Here, the alternative between general speculative nonsense theories and concrete, but possibly non-sufficiently general implementation and operationalization becomes dramatic. At least, one can hope that this confrontation will force everybody to rethink ill-defined approaches. 813
CHAPTER 41. THE RUBETTEr FAMILY
814
41.1
MetroRUBETTE
r
Die Z¨ ahlzeiten (Schlagzeiten, rhythmische Grundzeiten) gewinnen unter allen Umst¨ anden erst reale Existenz durch ihre Inhalte. Hugo Riemann [453] Summary. The MetroRUBETTEr is an elementary analysis module which shows that seemingly simple approaches yield complex but informative results. We also make evident that operationalization of abstract concepts reveals unexpected insights into generically not foreseeable structures. –Σ–
Figure 41.1: The 31-part score denotator deduced from Richard Wagner’s composition “G¨ otterd¨ammerung”. The MetroRUBETTEr is built on the sober weight calculation which we developed in chapter 21 in the frame of the maximal meter nerve topologies. The formula in example 43 of section 21.2 is realized, except that P there is no upper length limitation (i.e., it is put to ∞). The mixed weight formula W (x) = Xi ∈SpI (x) Wi (x) is also realized in this RUBETTEr . The input is a score denotator (although still in the early shape of the PrediBase data model, which is the very special denotator form of a list of lists, which start from the simple form of strings). Figure 41.1 shows the 31-part score deduced from Richard Wagner’s composition “G¨otterd¨ammerung”. This data is imported to the RUBETTEr and then evaluated according to the said formulas. For example, the weight of the union of all the onsets of the 31 parts is shown in figure
41.1. METRORUBETTEr
Figure 41.2: he weight graphics of the above composition with profile = 2, minimal length = 2. Horizontal axis: onset, vertical axis: (relative) weight. The unit of the onset grid in the graphical representation of the weight is 1/8.
815
Figure 41.3: The mixed weight graphics of four parts clarinet (part 11), bassoon (part 16), horn (part 14), and violin (part 27) with scaling factors 3, 2.5, 1, and 0.1 for the respective parts. The unit of the onset grid is again 1/8, as in figure 41.2.
41.2. Its parameters are: profile = 2, minimal length = 2, and the unit of the onset grid in the graphical representation of the weight is 1/8. The mixed weight is also an option of this RUBETTEr . Figure 41.3 shows the mixture of the parts: clarinet (part 11), bassoon (part 16), horn (part 14), and violin (part 27). The parameters are same as above, the weights are scaled by the factors 3, 2.5, 1, and 0.1 for the respective parts. For a more in-depth application of this RUBETTEr in musicology, we refer to [155]. This paper is an excellent example of an unexpected musicological application of a RUBETTEr which was not intended to be interesting on its own. The metrical analysis turns out to be quite sophisticated, although the concept of a local meter is a very elementary one. The surprising effect of this setup is that the combination of the simplistic concept reveals unprecented insights into the time structure of classical works. This is a good hint to the musicologists, teaching them that interesting insight can result from a complex aggregation of simple ingredients. In this case, the simple elements are provided by the maximal local meters, whereas their complex aggregation is conceived by the nerve of the covering they define. A second very interesting application of this RUBETTEr has been presented in [349]. It was recognized that the longest possible minimal lengths of local meters for the left hand part of Schumann’s “Tr¨ aumerei” yields a 3+5 quarters periodicity over two bars, whereas the same analysis of the right hand yields the expected periodicity of 4 quarters with stress on the barlines. The sonification of this fact can be heard on the audio example in the book’s CD-ROM, see page xxx. In this RUBATOr version, however, the output is a weight which is by no means a denotator. This output is a final data and must be used in its special format. This will be the case for operators of the PerformanceRUBETTEr . For the Java-based distributed RUBATOr (figure 40.2, such a restrictive usage would be forbidden.
CHAPTER 41. THE RUBETTEr FAMILY
816
41.2
MeloRUBETTE
r
In general, the author does not believe in the possibility or even desirability of enforcing strict musical definitions. Rudolph Reti on the concept of a motif [444] Summary. The MeloRUBETTEr is an excellent example of the tension between abstract concepts and operational implementation. We expose the routines for motivic analysis, the interface concept, and we discuss the performance problem, including proposals for performance improvement and their theoretical limits. –Σ–
Figure 41.4: The weight graphics for the celli part 30 in the above score denotator deduced from Richard Wagner’s “G¨ otterd¨ ammerung”. The MeloRUBETTEr refers to the theory of motivic topologies in chapter 22 and, in particular, to section 22.9 about motivic weights. The score is loaded as for the MetroRUBETTEr , its projection to the onset-pitch space is then analyzed and yields a numeric weight for each onsetpitch event. One such weight is visualized in figure 41.4. It corresponds to part 30 (celli) of the otterd¨ ammerung”. The weight values are encoded in gray-levels of the above composition “G¨ discs which represent the events in onset and pitch. The calculation relies on these parameters which relate to melodic topology: • Symmetry Group. This is the paradigmatic group of the shape type. In each group, we include the translation group in pitch and onset. The choice is then between the translation group (encoded by “trivial”), the one generated by the translations plus the retrograde, or the one generated by the translations plus inversion, and the full counterpoint group, i.e., generated by the inversion and retrograde over the translations. • Gestalt Paradigm. This is one of three possible shape types: diastematic, elastic, and rigid. The first candidate is in fact the type which we called “diastematic index shape
41.2. MELORUBETTEr
817
Figure 41.5: The main window of the MeloRUBETTEr . type” in chapter 22. Observe that we have no topology for the diastematic type, but may nevertheless define neighborhoods! • Neighborhoods. This is the neighborhood radius which was used in the expression Dµ (M ) in section 22.9. • Span. This is the maximal admitted onset difference between motive events. • Cardinality. This is the maximal admitted cardinality of motives. Together with the Span, this condition, defines the selection µ of motives which is addressed in the approach from section 22.9. The presence and content functions are defined as follows: 1. Presence. For a motif M , the presence value prµ, (M ) is the sum of all these numbers: For each N ∈ Dµ (M ), we count the number m of times where M has a submotif M 0 of N that at a distance less than from M . We also look for the difference of cardinalities d = card(N ) − card(M ). This gives a contribution p(N ) = m.2−d , and we add all these numbers. 2. Content. Similarly, for each motif N ∈ µ such that M ∈ Dµ (N ), we take p(N ) = m.2−d with d = card(M ) − card(N ) and m the number of times, where N has a submotif N 0 of M that at a distance less than from N . We add all these numbers p(N ) and obtain the content ctµ, (M ).
CHAPTER 41. THE RUBETTEr FAMILY
818
3. Weight. Given a motif M ∈ µ, this is the product nW (M ) = prµ, (M ).ctµ, (M ), i.e., taking the function ω(x, y) = x.y from section 22.9. We have already given musicological comments on this construction in chapter 22. It is however remarkable to see the overwhelming amount of calculations which arise in this routine. For example, we have calculated the number C of comparisons of motives (for distance measurements) Schumann’s “Tr¨ aumerei” (“Kinderszene” number 7) which comprises 463 notes. If we take Span = 1/2 bar and Cardinality = 4, we obtain 250 745 motives and C = 10 0230 4900 904 ∼ 1.023.109 . This is beyond any explicit human calculation power. It demonstrates that the task of finding a dominant motif is a very hard one, and that this one, if it is recognized by a human listener, can at most be present in a very hidden layer of consciousness. This becomes even more dramatic for larger pieces, such as an entire sonata, say! Here, the combinatorial extent of motivic units exceeds any calculation power of humans and machines, as is easily verified. This means that a huge composition bears a motivic complexity that will escape to (human or machine-made) classification forever. Nonetheless, the usage of statistical methods, of simplified approaches to motivic topology, or of topological invariants that are more easily perceived could help find a rough orientation in the virtually infinite motivic variety of music. This implementation also makes evident the tension between fuzzy concepts in musicology and implementation of a precise model. Although the concept of a motif is rather elementary, it entails a very sophisticated motivic analysis which could eventually converge to the intensions hidden1 behind those fuzzy motive theories.
1 We
are not sure whether they are really hidden and not only faked...
41.3. HARMORUBETTEr
41.3
HarmoRUBETTE
819 r
Eine Theorie aber, die gerade dort versagt, wo auch das Ph¨ anomen, das sie erkl¨ aren soll, ins Vage und Unbestimmte ger¨ at, darf als ad¨ aquat gelten. Carl Dahlhaus on Hugo Riemann’s harmony [100] Summary. The HarmoRUBETTEr makes clear that a vague theoretical approach does not reflect a vague phenomenon but an extremely complex one. The implementation of this RUBETTEr reveals several deep deficiencies of traditional “messy” analysis in harmony. We account for this on the level of preferences that have to be defined in order to get off ground with the analysis. The chapter concludes with a discussion of combinatorial problems due to the global complexity of harmony, and to the local character of tonal paradigms. –Σ– The HarmoRUBETTEr is probably the most interesting RUBETTEr , since it is situated on a turning point of several critical issues in harmony. To begin with, the context problem in harmony is a multilayered and ramified one which is (we said it repeatedly) not clarified by music theory. This is manifest in the preliminary question of what is a chord in a given score. Should one only look at groups of notes as a common onset, should one also consider onset groups which are not manifest, but can be deduced from plausible rules, or is the selection of the relevant set of chords within a score also a function of the harmonic statements which could result thereof? We have implemented two variants. The first one takes as the sequence (ai )i all maximal zero-addressed local compositions ai = {ai,j |j = 1, . . . ti } of pitches of note events with identical onsets. The second one is less naive. Within the given score, we take all local compositions ai = {ai,j |j = 1, . . . ti } of pitches with this property: There is at least one onset which is the offset time of an event, and the chord ai is the non-empty set of pitches of all note events which either start or still last at this offset time. This option is chosen by the button “use Duration” on the RUBETTEr ’s main window, see figure 41.6. This second variant encodes all changes of chord configurations, not only the onset-commonalities. Using either of these methods, the generated chord sequence is the basis for the following analysis which at the end will yield a harmonic weight for each note event, and for which we refer to the harmonic tension theory presented in section 27.2.2. According to that approach, each chord ai of the chord sequence (ai )i must be given a Riemann matrix (T Ff,t (ai ) = φf,t (ai )b )f,t , from which we deduce the weights ω(f, t, ai ) = ln(φf,t (ai )) and also call this data the Riemann matrix of ai . According to that discussion, we may also downsize weights below a threshold φmin to −∞. This is what the user sets when defining the “Global Threshold” in the main window. The local threshold is just the same for relative weights within a given Riemann matrix. The percentages in the main window mean that we downsize values below a defined percentage relative to the global or local (only within a fixed Riemann matrix) value range. Following the rules for the value −∞, we may neglect any path through a chord which has this value. So this singular value means that a chord is “inharmonic” insofar as it cannot contribute to a positive harmonic evaluation. This is a mathematical rephrasing of the classical, but fuzzy concept of inharmonic chords. Here it just means that the harmonic weight of a chord is too small to be considered as a contributon to the global harmonic path, and that the minimal size of allowed weights is set without further
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CHAPTER 41. THE RUBETTEr FAMILY
Figure 41.6: The main window of the HarmoRUBETTEr .
theoretical justification. It is a regulatory limit for the sensitivity of the path maximization with respect to the involved weights. We should stress that this matrix (ω(f, t, ai ))f,t defines Riemann weights without any contextual considerations. This is information that comes along from the isolated calculation on the chord ai . There are different calculation methods for this matrix, one by the author, and using chains of thirds, as discussed in section 25.3.3, and one by Noll, using self-addressed chords—the user may choose his preferred method by a pull-down button on the RUBETTEr ’s main window. The Riemann matrix (ω(f, t, ai ))f,t is visualized on the “ChordInspector” window for each chord (see figure 41.7). In this window, the nonlogarithmic values φ(ai ) are shown. The window also shows the chord’s pitch classes as well as all its minimal third chains. Once the Riemann matrices (ω(f, t, ai ))f,t are calculated, the optimal path in the Riemann quiver is calculated. This one follows the method discussed in 27.2.2. To this end, we need preferences for the matrices TVALtype , TVALmode , TTON . The matrices TVALmode , TTON are defined in the windows shown in figure 41.8. To the left, we have TTON , however as a distance value according to the amount of fourth between pairs of tonalities, to the right, we have TVALmode . The matrix TVALtype is shown in the left lower corner of the window in figure 41.9. In the middle lower part of that window, we have check buttons for every Riemann locus (i.e., the position in the Riemann matrix), meaning that if the check is disabled, no path is possible through such a locus. The large upper matrix encodes the third weights needed for chord weight calculations according to the formulas (25.8) and (25.9). With these settings, the best path is calculated. This is however a tedious task, to say the least. In fact, if we are given 200 chords (a very small example), we may choose from a number of
41.3. HARMORUBETTEr
821
Figure 41.7: The ChordInspector of the HarmoRUBETTEr shows each chord of the chosen chord sequence with its pitch classes, the third chains, and the Riemann matrix according to the chosen calculation method. The grey level of the values is proportional to their relative size.
12200 ∼ 6.8588169039290515.10215 paths. This number exceeds any calculation power of present computers. This exuberant number is due to different factors. First of all, no larger paths are taken into account, i.e., we have not implemented cadences as preferred paths, nor have we implemented modulatory constraints. More precisely, we do not give preference to maximal subpaths within a fixed tonality. We only take into account tonality changes a posteriori, i.e., via the weights of paths of length 1, when they show a tonality change. So we have to calculate the entire path and then hope that the negative points for tonality changes rule such paths out. It is also not clear whether human harmonic logic really can take into account such global path comparisons. In other words, it is more likely that humans only consider local optimization of paths. This is what we have in fact implemented in the following sense. In each index i of a chord ai , we consider only a local part of the entire chord sequence. Such a part is defined by two non-negative entire variables CD = Causal Depth, F D = Final Depth. This means that we look at the subsequence of chords from index i − CD to i + F D (inclusive) and therein select an optimal path pi,CD,F D . Within this path, chord ai is positioned at a determined Riemann locus (f (i), t(i)). The path which we finally select is the path p through all triples (f (i), t(i), ai ). The causal part is a tribute to the influence of preceding chords down to index i − CD on the harmonic position of chord ai . The final part influences the harmonic position of ai relating to the future chords up to index i + F D. The result is visualized in the Riemann Graph which is shown in figure 41.10. This is however not the end of the job. We do not have the weights of the single notes. To
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Figure 41.8: This preference window (for the author’s third chain method) shows the tonality distance matrix TVALmode (left) and the mode matrix TTON (right).
Figure 41.9: The upper matrix encodes the weights of thirds (relative to a fixed reference tonic, the lower left shows the matrix TVALmode . The lower middle matrix encodes the a priori allowed Riemann matrix locus position. this end, we first need the globally calculated weight of each chord ai . In a provisional form, the weight of chord ai is defined by the weight ω(pi ) of the chosen path p from the first chord to ai . This value is finally modified by a global slope σ preference such that the final and the first weight can be set to build a defined slope. Thereby, we meet the requirement of a control over the global tension which cannot be deduced from the locally (only 0 and 1 lengths of sub-paths) defined weights. Let us denote by ωi the weight ωσ (pi ) corrected by the slope preference. Using the weight of a chord ai , we finally may define the weight2 ω(x) of a determined event x in ai . To this end, the weight ω(f (i), t(i), ai ) is compared to the weight ω(f (i), t(i), ai − {x}) of the chord ai which contains x. With a positive preference quantity 0 < d ≤ 1, we consider the factor 1 λ(x) = (41.1) ω(f (i),t(i),a i −{x})−ω(f (i),t(i),ai ) d + (1 − d)e 2 In
this notation, we omit all the preferences.
41.3. HARMORUBETTEr
823
Figure 41.10: The Riemann graph is the sequence of chords, together with their functional values as they result from the optimal path. which measures the weight differences. It evaluates to 1 for the difference zero and yields 1/d for the weight ω(f (i), t(i), ai − {x}) = −∞. This means that the influence of x in the building of the chord’s weight is accounted for. If the weight decreases after omission of x, its influence is important and the factor increases λ(x). So we finally get the weight ω(x) = λ(x).ωi . The graphical representation of this weight is the same as for the MeloRUBETTEr , and we may omit this window of the HarmoRUBETTEr . Whereas the Riemann graph is conformal to the usual function-theoretic analysis (although it need not provide the common data in general), the weights of chords and events are far beyond the usual harmonic analysis and therefore cannot be compared without caution to established knowledge in harmony. It is however a common approach to harmony in its performance aspects to weight chords or notes in a more or less metaphoric way. Our present approach in the HarmoRUBETTEr is a concretization of these metaphors and also a point to be discussed with traditional performance theorists.
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41.4
PerformanceRUBETTE
r
All words, And no performance! Philip Massinger (1583–1640) Summary. The PerformanceRUBETTEr is a ‘macro’ RUBETTEr : It manages the stem-ma generation, the weight input and recombination, the operator instantiation, and the production of output of performance data on the level of music technology. –Σ–
Figure 41.11: The main window of the PerformanceRUBETTEr shows the stemmatic inheritance, descending from left to right. In the figure, the “Mother LPS” has a daughter named “PhysicalOperator”, and this one (all mothers are indicated on top of the daughters’ column) has two daughter “SplitOperator 1”, “SplitOperator 2”, generated by the SplitOperator, etc. Originally, the PerformanceRUBETTEr was the very focus of RUBATOr . Its purpose was the implementation of a type of performance logic with arguments from an analytical output. Although the analytical Rubettes have earned a growing importance, one of the cornerstones of analysis is its success in the construction of a valid performance. In fact, playing a good performance is a way of demonstrating one’s understanding of music. Therefore the performance theory implementation is important beyond its autonomous interest. The PerformanceRUBETTEr implements the stemma theory of chapter 38. The starting point is a selection of a score denotator. This will play the primary mother’s role, i.e., we are constructing a primary mother LPS in the sense of definition 35.3. The score form is provided in the same format that we have known as input for the other Rubettes. For this RUBETTEr , the kernel is always given as a (zero-addressed) local composition in the space form EHLDGC.
41.4. PERFORMANCERUBETTEr
825
This local composition is the kernel in the top space of a cellular hierarchy pertaining to the primary mother’s LPS. The primary mother’s LPS is instantiated according to “hard coded” default parameters in the Objective C source code3 . However, for each specific performance operator, the LPS data are adopted and define specific daughters. After setting the kernel (see figure 41.11) of the primary mother, the main window shows the stemmatic ramification with individual names for each LPS and arranged in a browser, stemmatic inheritance running from the left to the right. In the figure, the “Mother LPS” has a daughter named “PhysicalOperator”, and this one (all mothers are indicated on top of the daughters’ column) has two daughters, “SplitOperator 1”, “SplitOperator 2”, generated by the SplitOperator, etc.
Figure 41.12: The Kernel View window shows the top kernel of the hierarchy of a selected LPS (here the LPS named “PhysicalOperator” in the above stemma browser) in common pianola (piano roll) rectangles, loudness being codified by grey levels. The vertical bars are set to four bar intervals in the given score.
3 Objective C is a programming language for the NEXTSTEP-, OPENSTEP-, and Mac OS X-based RUBATOr projects.
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Figure 41.13: In the PerformanceRUBETTEr , weights are used in their splined interpolation shape. Here, we see a metrical weight issued from the MetroRUBETTEr ’s analysis.
Figure 41.14: The Weight Watcher window shows the loaded weights (top), the upper and lower limit of their range, the non-linear deformation, and the Boolean flag of inverting/non-inverting the weight (button in the lower left corner). For each ramification, one or two daughters are generated according to a chosen operator. For example, in figure 41.11, the SplitOperator generates two daughters, whereas the TempoOperator always produces one single daughter. The operators need only be loaded at run-time as they are needed. So this RUBETTEr is non-terminal in the sense that it allows further ramifications via an arbitrary number of dynamically loadable performance operators. For every highlighted LPS in the stemma browser of the main window, we can visualize the top kernel of its hierarchy on the Kernel View window, as shown in figure 41.12, by means of the usual pianola graphics. Here, the grey level indicates the loudness. In order to apply an operator to a given LPS, one next needs a list of weights, this is conformal with the operator theory exposed in chapter 44.7.
41.4. PERFORMANCERUBETTEr
827
The management of available weights as well as their concrete application are the business of the Weight Watcher system. Figure 41.13 shows a metrical weight in its splined interpolation shape. The weights to be used for a given operator can be loaded into the Weight Watcher, see top of figure 41.14. The loaded weights are then added or multiplied (according to the Boolean flag button “Combine as Product” to the right, below the weight list), and the resulting weight combination is applied to the given operator. For each weight, one can set the upper and lower limit of range (High Norm, Low Norm), the non-linear deformation quantity (Deformation), the inversion/non-inversion flag (Inverted Weight button to the lower left corner), the influence in a combination of several weights (Influence), the slope of decrease to weight value 1 as the arguments tend to infinity (Tolerance). The moral of this Weight Watcher system is a gastronomic one: Weights may be mixed and dosed at will in order to experience their influence on a given operator. This is not merely a lack of theory, it is above all an experimental environment for effective performance research. In fact, since virtually nothing is known about the influence of weights on performance, we have provided the user with a great number of possibilities in order to realize an optimal testbed for future theory.
Figure 41.15: The PhysicalOperator Inspector allows us to select a number of physical output parameters where the weight changes the values.
Figure 41.16: The SymbolicOperator Inspector allows us to select a number of symbolic input parameters where the weight changes the values.
Now, given a weight watcher combination of weights, an operator is fed by this combined weight and acts on the mother LPS to yield a new daughter LPS (or two in the case of the SplitOperator, where however no weight is needed). The detailed operation of a specific operator has already been described in chapter 44.7, we need not repeat these details here.
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Figure 41.17: The effect of an operator, here a PysicalOperator, is shown in the Kernel View of the performed kernel. This figure shows the performed symbolic kernel as shown above in figure 41.12.
Figure 41.18: The TempoOperator inspector allows us to select different integration methods. The “Real” method uses Runge–Kutta–Fehlberg routines, whereas the “Approximate” method uses simple numerical integration.
Figure 41.19: The ScalarOperator inspector allows us to select different options as defined in the scalar operator theory, but Runge–Kutta– Fehlberg ODE integration is mandatory in this situation.
41.4. PERFORMANCERUBETTEr
829
Figure 41.20: The performance field of a selected LPS can be visualized. The user may select two parameters whereon the six-dimensional field is projected.
Figure 41.16 shows the inspector of the SymbolicOperator. The weight acts on selected parameters which are defined by Boolean buttons. The same procedure is performed for the PhysicalOperator, whose inspector is shown in figure 41.15. The action of an operator on the symbolic score is shown in the performed kernel in the same pianola representation as for the symbolic kernel. The action of a physical operator is shown in figure 41.17. A funny application of this operator to Schumann’s “Tr¨ aumerei” can be heard on the book’s CD-ROM under the title Alptraeumerei, see page xxx. This piece is the dead-pan version of the score with the melodic weight being applied to pitch via the physical operator, and everything being played with Schumann’s original tempo indication. The inspector windows of the tempo-sensitive operators, the TempoOperator and the ScalarOperator, are shown in figures 41.18 and 41.19, respectively. The TempoOperator implements basically two methods: “Approximate”, and “Real”. The former is a direct integration method, whereas the latter uses Runge–Kutta–Fehlberg numerical ODE routines, including different parameters for numerical precision. The ScalarOperator uses exclusively Runge–Kutta–Fehlberg numerical ODE routines since it is an operator that acts on two or more parameters, where the naive approximation method cannot work. For the visualization of performance fields, the window shown in figure 41.20 is available. Finally, the parameters for the SplitOperator are determined on the window shown in figure 41.21. Here, one may define those lower and upper parameter limits of the total sixdimensional frame, where the subframe of the split daughter is cast.
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Figure 41.21: On this window, the user may define those lower and upper parameter limits of the total six-dimensional frame, where the subframe of the split daughter is cast for the SplitOperator.
41.5. PRIMAVISTARUBETTEr
41.5
831
PrimavistaRUBETTE
r
It is hard if I cannot start some game on these lone heaths. William Hazlitt (1778–1830) Summary. Several musical predicates from score notation are paratextually loaded. The PrimavistaRUBETTEr takes care of the paratextual signification for the most important predicates regarding dynamics, agogics, and articulation. –Σ– r
The PrimavistaRUBETTE serves a different task insofar as it is neither analytic nor performance oriented. It deals with paratextual information as it is provided by verbal indications for dynamics, tempo, and articulation. It basically does this: it transforms verbal information into weights which may then be used to shape the symbolic data and the tempo before performance in the proper sense is shaped.
Figure 41.22: A number for preference windows for dynamics, articulation, and tempo, allow us to define numerical values of paratextual predicates. The input of this RUBETTEr is a local composition whose elements are events with verbal specification such as absolute dynamics (figure 41.22 right preference window), relative dynamics
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(figure 41.22 left middle preference window), articulation (figure 41.22 left upper preference window), and relative tempo (figure 41.22 left lower preference window). The functionality of this RUBETTEr is to transform these data into weights, this is performed on the window for primavista operations as shown in figure 41.23, and according to the numerical data that are defined the above preference windows. These methods have been discussed in detail in section 39.2.
Figure 41.23: The main window of the PrimavistaRUBETTEr manages the transformation of verbal (paratextual) predicates into weights.
Chapter 42
Performance Experiments Learning by Doing. Summary. This chapter traces the analyses and syntheses processes which led to the historically first full-fledged RUBATOr -driven performance in July 1996 on the MIDI-Boesendorfer at the Staatliche Hochschule f¨ ur Musik in Karlsruhe, as well as to the qualitatively high performance of contrapunctus III in Bach’s Kunst der Fuge. We report the technical prerequisites, the analytical background generated by RUBATOr , and the step-by-step realization of the stemma and the overall parametrization. –Σ–
42.1
A Preliminary Experiment: Robert Schumann’s “Kuriose Geschichte” Nicht eine Geschichte wird hier erz¨ ahlt, sondern der Eindruck, den eine solche bei einem Zuh¨ orer weckt, wird charakterisiert. Thomas Koenig [267]
The first realistic experiment with RUBATOr took place in July 1996 on the MIDIBoesendorfer grand piano at the Staatliche Hochschule f¨ ur Musik in Karlsruhe. The experiment lasted three days and was led by the author, together with his assistant Oliver Zahorka and the musicologist Joachim Stange-Elbe. The experiment was executed on a NeXTStation. The results were digitally recorded on DAT and everything was protocolled. The technical support for the MIDI-Boesendorfer was offered by Sabine Sch¨afer. In what follows, we want to give a very sketchy account of that experiment, however recalling the essentials of initial experiments in performance theory and the consequences thereof. The more elaborate discussion of such experiments is left to the following section 42.2. 833
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CHAPTER 42. PERFORMANCE EXPERIMENTS
The preparatory work for this experiment was above all the analytical part, i.e., the metrical, melodic, and harmonic analyses on the respective Rubettes. We also experimented on the stemmatic and operator strategy, including the WeightWatcher mixtures as discussed in section 41.4. The selection of the MIDI-Boesendorfer was also a consequence of preliminary experiments on the MIDI-Yamaha grand insofar as this instrument turned out to respond in a much too coarse way to the input. In contrast, the Boesendorfer has a refined calibrating interface allowing a realistic MIDI input (with roughly a thousand dynamical values instead of 128 from MIDI), in particular for soft dynamics, where the Yamaha grand is completely inappropriate. The first day was used to adapt the different technical conditions, such as the calibration of the Boesendorfer, the weight ranges, and the dynamical limits. The second day was devoted to the evaluation of our preparatory material and strategy. The third day was devoted to the production of all stemmatic levels, as well as their recording up to the final output level. The most significant experiences were twofold: On the one hand, the judgment of the three experts concerning which weight mixture to use at which stemmatic knot, and the way to use it, with all the variables from range to inversion/non-inversion and deformation, was very precious! We learned that the effects of such strategies were almost constantly judged in three different ways—except for trivial failure or success situations. This means that performative coherence is a very personal affair, even when using very explicit techniques, analyses, and shaping tools. On the other hand, we had to face a very disagreeable side effect of such an extensive performance shaping: Let me call it the zoom of supersensitivity. The effect is this: After having listened to performances of a selected part of the piece, the ear begins to recognize a steadily refined differentiation in the different parameters, such as agogics, dynamics, and articulation. For example, if in the beginning phase, one only recognized a huge change of a weight influence to agogics, with progressing trials, one would believe that the slightest further change in weight influence could change the performance of tempo in an unsupportable way. Maybe one was eventually incapable of really identifying the details of a performance, i.e., it is either simply too complex to be grasped by humans because the attention to different parameters is always selective, or it is not possible to memorize the samples and to compare them, or—and this is the worst case—our perception really changes so much form case to case that we cannot rely on the individual samples. The latter would mean that performance is greatly co-determined by the performance of the perceptive system. The result was nonetheless acceptable and strongly differs from the so-called dead-pan version, see [360, CD attachment], or this book’s CD-ROM (cf. page xxx), where the performance and the dead-pan version can be inspected.
42.2
Full Experiment: J.S. Bach’s “Kunst der Fuge”
Die Kunst der Fuge erschien wohl etliche Monate nach Bachs Tode und kostete vier Taler. Sie fand keinen Absatz. (...) Entt¨ auscht verkaufte der Sohn (Emanuel) die Platten, auf denen das letzte Werk seines Vaters ge¨ atzt war, um den Metallwert. Das war das Schicksal der Kunst der Fuge. Albert Schweizer [484]
42.3. ANALYSIS
835
This experiment is fully accounted in [504]. Here, we give a concise presentation, the version of RUBATOr used in this experiment is the one compiled for OPENSTEP/Intel. The “contrapunctus III” in Bach’s Kunst der Fuge has these characteristics: It is a four-voice composition, comprises 72 bars, has time signature 4/4, and tonality d-minor. The main theme of Kunst der Fuge is only used in its inversion and appears the first time in a rhythmically dotted and syncopated variant; the fugue starts with the theme in its comes shape and contains three complete developments (bars 1-19, 23-47, and 51-67).
42.3
Analysis
Summary. We give an account of the metrical and melodic analyses, whereas the harmonic analysis has not been done. –Σ–
42.3.1
Metric Analysis
For the metric analysis of the “contrapunctus III”, the calculations were made for each single voice, including the sum of the voice weights, and for the union of all voices. Please, refer to the discussion of the MetroRUBETTEr in section 41.1 for the following discussion. The settings of the weight parameters are these: Metrical Profile is 2; Quantization is 1/16; Distributor Value is 1. Since the metrical profile of all voices should be viewed under the same valuation, the Distributor Value was set to a common neutral value; the value 2 for the Metrical Profile resulted from several trials of analyses and yields a well-ordered distribution of the weight profile. The value for Minimal Length of Local Meters was successively decremented starting from the length of the largest local meter, descending until value 2 where the smallest cells are caught in their signification for the metrical overall image. When considering single local meters, it is above all the onset time and the step size of the single time grids1 , which matters. Based upon the summation formula, this can however not be fully understood since from a certain superposition of periodical onset sequences, the observation of a single local meter becomes very difficult if not impossible (see also the consideration of the union of all onset times). 42.3.1.1
Single Voices
Like all other contrapuncti, “contrapunctus III” shows no regularities in the compositional structure, in the repetition of whole parts or single bars. At least for the four bars of the theme, the onsets of themes in the developments of fugues do however create structural incisions, which only become relevant for the motivic analysis—the metric analysis remains unaffected. As to the analysis of single voices, it is interesting to observe that the longest found local meters nearly exclusively relate to relatively short parts, including only a few bars. These parts usually consist of uninterruptedly pulsating sequences of quavers or semiquavers, which in their regular sequence of notes differ significantly from the otherwise pronounced principle of tying stressed 1 In
this discussion, “grid” is synonymous with “local meter”.
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CHAPTER 42. PERFORMANCE EXPERIMENTS
bar onsets to unstressed precursors. This principle, having its origin in the shaping of the main theme with its sequences of quavers that lead to the interluding sections, can be recognized as a rhythmical pattern in the interludes and the countersubjects of the theme. By the tying of this sequence of quavers with the preceding half note, the theme breaks the hitherto confirmed bar accent scheme, a peculiarity which will be further confirmed in the sequel of this contrapunctus. It even has effects on the theme onsets which—unusual in a 4/4-bar—are partly shifted by a quarter note (see also the theme variant which is extended by dots and quaver transitions in the bars 23, 29, and 35). When considering the single voices, one realizes that most of the longest grids (i.e., those with highest weights, according to our choice of system parameters) catch exclusively onsets on unstressed bar times in the middle voices. If the initial value lies on a stressed bar time, a dotted duration value as a step length for local meters also weights more onsets on unstressed times. Only the metrical weight of the bass voice acts in a contrary way, in fact an increasing confirmation of the bar meter is observed, see figure 42.1.
Contrapunctus III (Soprano): metrical weight (Min. Length of Local Meter: 11)
Contrapunctus III (Alto): metrical weight (Min. Length of Local Meter: 19)
Contrapunctus III (Tenor): metrical weight (Min. Length of Local Meter: 17)
Contrapunctus III (Bass): metrical weight (Min. Length of Local Meter: 28)
Figure 42.1: Contrapunctus III: metrical weights for the four voices.
42.3.1.2
Weight Sums of All Voices
When considering the weights of single voices, and in particular the sum of their weights, observe that the longest weights are the most important contributions. This means that the most important weights stem from the bass, followed by the alto, the tenor, and then the soprano voice. The latter has a weakened contribution since in the middle of the contrapunctus, there is a pause of thirteen bars, which breaks down the coherence of the soprano to the metrically weakest voice, see figure 42.2 for the lengths of local meters for the four voices. Further, we observe that the bass voice prevails against the other voices in the confirmation of the bar
42.3. ANALYSIS
837
S A
23 22 21
T B
20 28 26 23 22
20
12 11 ...
2
19 18 17
15 14 13 12 11 ...
2
18 17
13 12 11 ...
2
17 16 15 14 13 12 11 ...
2
Figure 42.2: The lengths of maximal local meters for the four voices. meter since it creates high metrical weights by the longest metrical grids. Nearly all peaks of the metrical weight sum—except some cases in the middle part—coincide with stressed bar times. Generally speaking, the ambitus of the metrical weight profile increases towards the middle of the contrapunctus—an observation which is valid for all pieces of the “Kunst der Fuge” which we have analyzed, see also figure 42.3. The reasons for weak initial metrical profile lies in the formal construction of a fugue, where the voices do not appear simultaneously, but one after another, such that the full voicing only appears after a number of bars. Another reason can be seen in the structure of the theme of the fugue, which does not develop its full motional impulse before the last five notes. These are used to shape—until the next thematic onset—countersubjects and interludes. The equally very frequent decrease of the metrical profile towards the
Figure 42.3: The ambitus of the metrical weight profile increases towards the middle of the contrapunctus; minimal local meter length is 2. end of a piece is explained by different formal facts. In the case of the “contrapunctus III”, this is due to long pauses of the single voices—in particular bass, soprano, and tenor—as well as the pedal point in the bass voice (bars 68-71). 42.3.1.3
Union of All Voices
Relating to the union of all onsets, the “contrapunctus III” shows at first sight dotted step lengths in metrical grids. However, by the two longest grids (Minimal length of Local Meters being 502 and 271), starting at the final notes of the exposition of the theme in the tenor and at
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CHAPTER 42. PERFORMANCE EXPERIMENTS
the end of the exposition of the theme in the alto, an uninterrupted pulse of quavers and quarter notes is established, and the only remaining grid (Minimal length of Local Meters is 128) with step length a half note also drops at the non-stressed bar times of 4th and 8th quaver. All other grids have step lengths of different dotted durations, however dominated by the doubly dotted half note, the said dotted quarter note, and the half note with tied quaver, as well as once the dotted half note. What is remarkable in this contrapunctus is the three ‘double grids’ of
Length of metrical grid
Start of metrical grid (bar, note)
Step length
502
9
4th Quarter
1/8
72
3rd Quarter
271
4
4th Quarter
1/4
72
3rd Quarter
172
8
1st Quarter
3/8
72
3rd Quarter
171
8
2nd Quarter
3/8
72
4th Quarter
128
8
4th Quarter
1/2
72
4th Quarter
108
4
8th Quarter
5/8
72
4th Quarter
104
7
3rd Quarter
5/8
72
3rd Quarter
102
8
2nd Quarter
3/8
72
1st Quarter
8
3rd Quarter
3/8
72
2nd Quarter
91
4
1st Quarter
3/4
72
2nd Quarter
78
3
4th Quarter
7/8
72
1st Quarter
75
6
3rd Quarter
7/8
72
2nd Quarter
74
7
3rd Quarter
7/8
72
2nd Quarter
7
4th Quarter
7/8
72
3rd Quarter
8
1st Quarter
7/8
71
8th Quarter
8
3rd Quarter
7/8
72
4th Quarter
1
1st Quarter
4/4
72
1st Quarter
73
71
End of metrical grid (bar, note)
Figure 42.4: Maximal local meters for the union of all four voices. Minimal length of Local Meters equal to 102, 74, and 73, which within the same bar are shifted to each other by a quarter note, and two quarter notes in the last case. Therefore, in the contrapunctus III, the tendency of ‘metrical instability’, which was already encountered in the single voices, persists, since by summing up the weight contributions by the grids with dotted step width, a metrical profile is established which breaks the bar meter as well as the pulse of uninterrupted quavers and quarters.
42.3. ANALYSIS
42.3.2
839
Motif Analysis
For the calculation of motivic weights each single voice of the “contrapunctus III” was analyzed separately. We refrained from a motivic analysis of the union of all voices since by the contrapuntal structure of the single and autonomous voices within the polyphonic setting, a motivic setup across the voices seems rather unlikely and therefore was omitted. The settings for the motivic analysis were chosen as follows: Symmetry Group: counterpoint; Gestalt Paradigm: elastic; Neighborhood: 0.2. By the choice of the counterpoint symmetry group, the theme forms recta and inversa, as well as their (possibly appearing) retrogrades, were considered as being of equal weight. The neighborhood value has been chosen as based upon analytical experiments during the development period of RUBATOr . The elastic gestalt paradigm was preferred against the rigid and diastematic ones in order to obtain a more elastic point of view. As to the values for Motif Limits, compromises with the calculation power had to be made. By the choice of Span equal to 0.625 and Cardinality from 2 to 7, motives within a span of a half note plus a quaver were captured; this corresponds exactly to the duration of the theme where the transition of the virtual theme to the interludes must be recognized.
5 3
8
11
6
7
1
Soprano
12
Tenor
10 2
Alto
4
9
Bass Figure 42.5: Motivic weights for “contrapunctus III”.
With the results of the metrical analysis, some regularities in the microstructures can be read at first sight; herein we find in particular the onsets of the theme within a particular development. The representations of the single motivic weights in each voice with a mean value of 5 for the Cardinality may be elucidated in detail, see figure 42.5. Within these graphics, the theme onsets are numbered according to their temporal order; equal weights can be recognized on the onsets 1,2,3,4,8,9,12, and 5,6,7, as well as on 10 and 11. While further considering these weights, the overly long pauses in the soprano, tenor,
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CHAPTER 42. PERFORMANCE EXPERIMENTS
and bass voices attract attention. Further, in the length proportion of the single weight representations, the succession of onsets of the single voices (tenor-alto-soprano-bass) is reflected. Moreover, a significantly lower motivic profile at the beginning and after the longer pauses of the respective weights can be observed—due to preceding pauses, this is the case of exposed thematic onsets. For the weight values, a neat exposition of the inverted gestalt of the original theme is observed, bearing nearly identical weights at the beginning of every motivic weight, here even the differences of comes and dux forms are visible, since the weights of the tenor (first appearance) and soprano (third appearance) differ slightly by the different initial interval of the theme (descending fourth in the comes, and descending fifth in the dux form) from the weights of the alto (second appearance) and bass (fourth appearance), see figure 42.6.
Tenor
Soprano
Alto
Bass
Figure 42.6: Motivic weights for “contrapunctus III”, bars 1-4. Other clearly visible onsets of the theme in inverted shape are recognized after the long pauses in the soprano (eighth appearance), bass (ninth appearance), and tenor (twelfth appearance). Characteristically, the inverted shape always appears after pauses. At first sight these observations may seem to be tautological. However, if these weights are viewed with respect to their sense and purpose, their force to shape performance, then the transition from a quantitative to a qualitative information content becomes evident: Thus the different onsets of themes can be shaped by these weights in one and the same way; if these weights are used—in inverted form—for the dynamic shaping, then the thematic onsets can be stressed with plasticity. When trying to evaluate motivic data more in detail, a problem appears that is inherently founded in RUBATOr ’s intended purpose: since the platform was originally conceived for performance research, it is sufficient to present the analytical material for its performance shaping and to ascertain its reasonable construction. But these analyses also yield information about the musical structure, which in the case of metrical analysis—by a certain extra effort apart from RUBATOr —can be traced without difficulties. However, for the MeloRUBETTEr , any practical support beyond the brute calculation of weights—which is also sufficient for performance—is absent; the “introspection” of the details of weight calculation is hidden to the user. Thus for the calculation a motif’s weight it cannot be known which other motives are responsible for its presence and content. Although all motives are arranged in the motif browser in their temporal order, but another structured introspection of motives, for example with regard to their weights, is not possible for this version of RUBATOr . This makes the introspection and evaluation of
42.4. STEMMA CONSTRUCTIONS
841
the ‘heaviest’ motives a difficult task. All analytical information is present, but it is hidden; a particular difficulty may be seen in the exuberant effort in complexity which a detailed motivic analysis, which extracts knowledge from the score in an immanent way, is loaded.
42.3.3
Omission of Harmonic Analysis
A harmonic analysis was omitted in this situation since—to our mind2 —the approach of Hugo Riemann which is implemented in the HarmoRUBETTEr is not really suited for Bach’s harmonies. By use of the Riemann theory which was developed from the Viennnes classics, the specific harmonic structures of a contrapuntal maze, where harmony does not result from progression of fundamental chords but from the linearly composed voices, can only be captured in an incomplete way.
42.4
Stemma Constructions
Summary. This section discusses the performance construction via the stemmatic paradigm, and using the analytical results investigated in section 42.3. –Σ– Before the stemmatic construction and thereby in particular the problem of performing “contrapunctus III” are discussed, some general remarks regarding the various performance strategies are necessary. In the course of the single performance parcours, two different approaches resulted which would turn the given analytical weights into expressivity: the targetdriven and the experimental strategy. The target-driven strategy has its roots in the knowledge about existing performances, it is stamped by a preliminary experience of how the piece should sound and has been performed. With this procedure, the weights are used in a way which targets a predefined performance. One—just to name a pithy example—was oriented towards Glenn Gould’s Bach interpretation; the corresponding weights were selected according to these targets to obtain particular effects. In this procedure, however, the intrinsic structural meaning of analytical weights was ignored! Stamped by the knowledge and the expectation of the existing performances, this strategy did not allow one to judge and categorize those performance constructions which did not suffice for the music-esthetic exigencies. The other approach, the experimental strategy, moves the analytical weight to the center in order to investigate how this weight could ‘sound’, and which analytical insight it could convey in the listening. With this procedure, which views the main performing agent entirely within the weight, one has to free oneself completely from horizons of expectation for any particular performance target. The working process on such performances, the acquaintance of experience with the most different weights, and the playing with their effects taught us in the course of many experiments that this strategy would give rise to much more interesting performance aspects. Here we also have the freedom to admit extremal positions which disclose more about the inherent musical structure and as ‘daring ingredients’ may evoke lively musical expression. 2 Decision
by Joachim Stange-Elbe.
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CHAPTER 42. PERFORMANCE EXPERIMENTS
Moreover, the experimental approach to single performance aspects, which starts from curiosity about the sonic realization of analytical weights, conveys a deeper insight into to score’s musical structure. This path has its take-off in a “sonic analysis”, or else in “the sonic analytical structure” and aims at a “musically reasonable performance”. It is centered around the researcher’s curiosity for a sounding and interpretational realization of analytical weights and for “the never heard”, and it is paralleled by a liberation from expectational presets. Moreover, this strategy tries to apply as few weights as possible in order to couple the clearest possible analytical statements with the resulting performance.
Soprano
Soprano inverted
Alto
Alto inverted
Tenor
Tenor inverted
Bass
Bass inverted
Figure 42.7: Metrical weights for “contrapunctus III”.
42.4.1
Performance Setup
Summary. We discuss the detailed performance construction. –Σ– The performance of “contrapunctus III” took place in three parcours, out of which we only report the last two in more detail. The first one was entirely devoted to a target-driven strategy whereas the subsequent ones switched to an experimental strategy which yielded much more successful and conclusive results. Nonetheless, all these approaches contributed results that influenced the final result in a significant way.
42.4. STEMMA CONSTRUCTIONS
843
Generally speaking, the procedure in all these parcours first focused on isolated single aspects of performance (articulation, dynamics, agogics) and then were put together for the final parcours. For the complete description of all these steps, see [504]. 42.4.1.1
Results From First Performance Parcours
As a result of the first parcours which was executed under the paradigm of a target-driven strategy, the usage of motivic weights for a determined shaping of dynamics has been recognized. Using the inverted form of weights in the WeightWatcher for each voice, the thematic onsets for each development of the fugues could be modeled in an excellent way. Simultaneously however, the global usage of these weights lead to a completely disequilibrated passage so that this type of application was eliminated in the subsequent performance construction.
Cpt-03
Prima Vista Score
Mother Level 1 PV-03-Agogik.stemma
PV-03-Agogik
Prima Vista Agogics Division into Soprano/Alto and Tenor/Bass
03-s-a
03-t-b
Level 2
Division into Soprano, Alto, Tenor, Bass
03-s
03-a
03-t
03-b
Uniform loudness for all voices
03-s-vel
03-a-vel
03-t-vel
03-b-vel
Shaping of articulation Shaping of dynamics
03-s-Art-1 03-s-Vel-1
03-a-Art-1 03-a-Vel-1
03-t-Art-1
03-b-Art-1
03-t-Vel-1
03-b-Vel-1
Level 3
Level 4 03-Stimmen.stemma
Level 5 03-Art-1....stemma Level 5 03-Vel-1....stemma
Figure 42.8: Stemma construction for the second parcours.
42.4.1.2
Construction of Second Performance Parcours
In the second parcours, whose rather simple stemma (see figure 42.8) we are going to discuss hereafter, we changed to the experimental strategy, with the aim to investigate to which degree motivic weights can contribute to the elaboration of a determined performance aspect. Those motivic weights of a voice were used which cover a minimal number of motives (Cardinality: 2) within a dotted half note (Span: 0.625); all these weights were applied in inverted form. As with the subsequent third performance parcours, the core shaping work by means of weights (grey shaded levels in figure 42.8) must be preceded by further steps. To begin with, this concerns the shaping of the score’s primavista components, such as dynamic, articulatory,
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CHAPTER 42. PERFORMANCE EXPERIMENTS
and agogical prescriptions. In the case of contrapunctus III, which contains no performance prescription by the author, this task consisted only in the primavista shaping of a short ritardando before the junction to the final tonical chord (Level 1). The next steps within the stemma (levels 2 to 4) relate to the horizontal separation into the four voices (by the SplitOperator), which were identified by specific (artificial!) different loudness values (Levels 2 and 3), and which in the fourth step are reset to a uniform loudness. Upon this basis, an attempt to shape articulation and dynamics was undertaken. In the course of six subsequent performance experiments it turned out that the global application of a single weight was capable of producing three slightly different variants of articulation shaping, however one had to pay attention to change the parameter for the weights’ influence (in the WeightWatcher) in a minimal and systematic way. Figure 42.9 shows the final choice. Built upon these insights we tried to use the same motivic weights, the same systematic High Norm
Low Norm
Influence
Deformation
Invert
Soprano
1.6
0.2
1
-0.75
YES
Alto
1.5
0.3
1
-0.75
YES
Tenor
1.5
0.4
1
-0.5
YES
Bass
1.4
0.6
1
-1
YES
Figure 42.9: Choice of WeightWatcher parameters for motivic weights. and a similar handling of the change of intensities in order to produce comparable results for the shaping of dynamics. Without going into detail, it remains to be stated as a result that these attempts all failed. Typically, the dynamical relations between the voices turned out to be disequilibrated, and although the dynamics was perfectly modeled within single voices, the dynamical profile of thematic parts was sensibly worse. To obtain conclusive dynamics, one had to find another shaping procedure. 42.4.1.3
Construction of Third Performance Parcours
Because of these different dynamical profiles, the principle of former performance experiments— the exclusive usage of a weight and its global extension—had to be given up. In a first step it was recommended to split the single voices at appropriate locations, and in a second step, a regress to the metrical weights already used in the first parcours and their renewed application under other viewpoints (a mixed usage together with motivic weights) seemed reasonable. The shaping of articulation from the second parcours would be conserved. In a preliminary step, a division of the single voices had to be executed. To this end, one had to find structurally legitimate points from the musical context, such as articulation by harmonic incisions or thematic groupings for developments and interludes. The first division of all four voices took place in bar 39, legitimated by a harmonic close to the major parallel of the minor dominant (C-major); at the same time this is viewed as a possible ending of the second (however incomplete) development and a beginning of a four-bar interlude.
42.4. STEMMA CONSTRUCTIONS
845
Prima Vista Score
Cpt-03
Prima Vista Agogics
Mother Level 1 PV-03-Agogik.stemma
PV-03-Agogik
Division into Soprano/Alto and Tenor/Bass
03-s-a
03-t-b
Level 2
Division into Soprano, Alto, Tenor, Bass
03-s
03-a
03-t
03-b
Uniform loudness for all voices
03-s-vel
03-a-vel
03-t-vel
03-b-vel
Shaping of agogics I
03-s-Agogik-1
03-a-Agogik-1
03-t-Agogik-1
03-b-Agogik-1
Level 5
Shaping of agocics 2
03-s-Agogik-2
03-a-Agogik-2
03-t-Agogik-2
03-b-Agogik-2
Level 6
First division of single voices
03-s-I 03-s-II
03-a-I 03-a-II
03-t-I 03-t-II
03-b-I 03-b-II
Level 7
s-I s-II s-III s-IV
a-I a-II a-III a-IV
t-I t-II t-III t-IV
b-I b-II b-III b-IV
Level 8
Articulation shaping s-I s-II s-III s-IV preparation
a-I a-II a-III a-IV
t-I t-II t-III t-IV
b-I b-II b-III b-IV
Level 9
Shaping of dynamics I
s-I s-II s-III s-IV
a-I a-II a-III a-IV
t-I t-II t-III t-IV
b-I b-II b-III b-IV
Level 10
Shaping of dynamics II
s-I s-II s-III s-IV
a-I a-II a-III a-IV
t-I t-II t-III t-IV
b-I b-II b-III b-IV
Level 11
Shaping of dynamics III
s-I s-II s-III s-IV
a-I a-II a-III a-IV
t-I t-II t-III t-IV
b-I b-II b-III b-IV
Level 12
Second division of single voices
Level 3 Level 4 03-Stimmen.stemma
Figure 42.10: The stemma of the third parcours. In order to equilibrate the dynamical unbalances relating to the interludes from bars 19 and 46, a further division of the two halves of the fugue were necessary. A division of the first half was recommended in bar 19, having a close of the first development (exposition of fugue) and its half close on the dominant (A major). Because of the too strong dynamic sink of the three-voice interlude from bar 46/47, the division of the second half had to take place not later than at this point. This division was legitimized by the half close on the minor dominant (A-minor) beginning in bar 46 on the one hand, and the simultaneous ending of the second (then complete) development according to the three-part construction of the fugue. For the subsequent performance shaping, consider figure 42.10. Besides the already known preparatory steps—horizontal division into single voices (Level 3) and equalizing of loudness (Level 4)—two performance steps for the later shaping of global agogics were inserted (Levels 5 and 6). This trick is applied, because agogics needs long calculation time on the global level of single voices and should be calculated after the stemmatically subsequent shaping articulation
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CHAPTER 42. PERFORMANCE EXPERIMENTS
and dynamics. The vertical division of the single voices is applied in the previously described steps (Level 7 and 8). For the subsequent shaping of articulation and dynamics, each voice had to receive its separate and individual performance shaping for the four sections. This enabled us to apply different parameter values for the intensity effects, one per used weight. For the shaping of articulation, the three already elaborated performance steps were inherited. As is seen in the stemma (figure 42.10), the shaping of dynamics was realized in three consecutive steps. Here, besides the known motivic weights, two additional metrical weights were applied. For the first step (Level 10), we applied the metrical weight from the union of all voices with Minimal Length of Local Meters equal to 2, in inverted form, and without deformation, see figure 42.11.
Figure 42.11: Metrical weights in “contrapunctus III”, union of all voices, Minimal Length of Local Meters equal to 2, in original form (top), and in inverted form (bottom). Upon this stemma, the second step (Level 11) applied the metrical weights with value 5 for Minimal Length of Local Meters for each individual voice in inverted form and also without deformation (the weight graphics were comparable to those from the second performance parcours described above). For the concluding shaping of dynamics, the already known motivic weights were applied
42.4. STEMMA CONSTRUCTIONS
847
in order to give the thematic onsets a plastic relief. From the interplay of the various intensity values, we got the constellation documented in figure 42.12. The result of this performance
Level 10 Weight High Norm Low Norm Influence Deformation Invert Level 11 Weight High Norm Low Norm Influence Deformation Invert Level 12 Weight High Norm Low Norm Influence Deformation Invert
Soprano
Alto
Tenor
Bass
Part 1 Part 2 Part 3 Part 4
Part 1 Part 2 Part 3 Part 4
Part 1 Part 2 Part 3 Part 4
Part 1 Part 2 Part 3 Part 4
Metro: St02-p2
Metro: St02-p2
Metro: St02-p2
Metro: St02-p2
1.1 0.6 1 0 Y
1.1 0.6 1 0 Y
1.1 0.6 1 0 Y
1.1 0.8 1 0 Y
Metro: S05-p2 1.0 0.7 1 0 Y
1.0 0.7 1 0 Y
1.0 0.7 1 0 Y
1.0 0.8 1 0 Y
Melo: s-625-2 1.0 0.5 1 0.5 Y
1.2 0.4 1 0.5 Y
1.4 0.4 1 0.5 Y
1.2 0.5 1 0.5 Y
1.1 0.6 1 0 Y
1.1 0.6 1 0 Y
1.1 0.6 1 0 Y
1.1 0.8 1 0 Y
Metro: A05-p2 1.0 0.7 1 0 Y
1.0 0.7 1 0 Y
1.0 0.7 1 0 Y
1.0 0.8 1 0 Y
1.1 0.6 1 0 Y
1.1 0.6 1 0 Y
1.1 0.6 1 0 Y
1.1 0.8 1 0 Y
Metro: T05-p2 1.0 0.7 1 0 Y
1.0 0.7 1 0 Y
1.0 0.7 1 0 Y
1.0 0.8 1 0 Y
1.1 0.6 1 0 Y
1.1 0.6 1 0 Y
1.1 0.6 1 0 Y
1.1 0.8 1 0 Y
Metro: B05-p2 1.0 0.7 1 0 Y
1.0 0.7 1 0 Y
1.0 0.7 1 0 Y
1.0 0.8 1 0 Y
Melo: a-625-2
Melo: t-625-2
Melo: b-625-2
1.0 1.2 1.05 1.2 0.7 0.7 0.7 0.7 1 1 1 1 0.25 0.25 0.25 0.25 Y Y Y Y
1.0 1.4 1.2 1.2 0.75 0.7 0.7 0.75 1 1 1 1 0.25 0.25 0.25 0.25 Y Y Y Y
1.3 1.7 1.3 1.3 0.7 0.8 0.8 0.7 1 1 1 1 -0.25 -0.25 -0.25 -0.25 Y Y Y Y
Figure 42.12: Intensity values for the concluding shaping of dynamics. communicates a relatively balanced dynamics, spread over the whole contrapunctus, the thematic onsets gain a profile, which can also be confirmed in the slight crescendo that leads to the beginning of the third development after the three-voiced interlude (from bar 46/47). Bringing together the dynamic and the already elaborated articulatory aspects, the result can be stated as a complementary shaping of both performance aspects, which on top of that reveals a musical sense in the elaboration of thematic onsets and the three-voiced passages of the interludes. For the shaping of agogics, the said levels 5 and 6 of our stemma were reserved. We did two different subsequent performance parcours with two different metrical weights: the sum of all voice weights (Minimal Length of Local Meters: 2) and the weight of the voice union (Minimal Length of Local Meters: 91 (!)), see figure 42.13. The intensity values are shown in figure 42.14. As a result we may state that according to the weight structure and a minimal intensity effect a slight increase of the basic tempo towards the middle of the piece happens, but it is
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CHAPTER 42. PERFORMANCE EXPERIMENTS
Figure 42.13: “Contrapunctus III”; top: metrical weights, sum of all voice weights (Minimal Length of Local Meters: 2); bottom: weight of voice union (Minimal Length of Local Meters: 91).
balanced by a slightly slower tempo for the initial and final bars. For the resulting performance as traced on the audio track on the CD-ROM attached to this book, see page xxx. The corresponding stemmata are also found on this location of the CD-ROM. 42.4.1.4
Local Discussion
Starting from the experimental strategy,the performance of “contrapunctus III” was first centered around the shaping with a single weight in order to sound the potential of a single weight. The shaping of articulation showed a reasonable local profile for every voice, i.e., a single analytical structure was capable of giving one performance aspect a reasonable expression. The global application of weights and the usage of a single weight showed its limits, as we have learned from the dynamical shaping of “contrapunctus III”. For example, the global application of weights failed in the different grades between the contributions of the four voices.
42.4. STEMMA CONSTRUCTIONS
849
High Norm
Low Norm
Influence
Deformation
Invert
Total Contrapunctus
1.025
0.975
1
0
NO
Total Contrapunctus
1.05
0.95
1
0
NO
Figure 42.14: Intensity values for the metrical weights. Especially with the motivic weights of the tenor and bass voices, different weight profiles become visible which cannot be eliminated even by suitable deformations. These differing profiles of weights do result from the compositional structure. As this one splits into a number of parts— developments and interludes, groupings by harmonic closes and semi-closes—the division of the voices according to such compositional criteria is legitimized. Within these parts, the selected weights can be applied with different intensities and thusly equalize the disparate shapings. Dynamics received a special significance in the shaping process: the elaboration of thematic onsets by an inverted motivic weight—this was justified in the structure of the theme. More precisely, the long durations of the initial notes resulted in a weak weight profile, which could be used to stress these incipits by the weight’s inversion. This principle which remains valid for almost all contrapunti, can however only be proved for the “Kunst der Fuge”.
42.4.2
Instrumental Setup
Summary. The conditions and influence of the instrumental setup are discussed. –Σ– A discussion of adequate instrumentation must be in the forefront of virtual performance work, since the behavior of the respective sound generator is an essential basis for shaping of the single musical parameters. Since the performance results from RUBATOr are encoded in a MIDI file, we have the possibility to access a MIDI-driven acoustical piano or else to use a corresponding digital device. Here, we should observe some relevant differences which—besides the basic difference in sound—the repetition, the dynamic response, the resonance behavior, and the spatial environment which we cannot discuss here. These differences are not only present between the acoustical and the digital instruments, they also act within each category. For pianos and grand pianos, the MIDI-driven access is offered by Boesendorfer and Yamaha models. At the time of our experiments, the comparison between these two brands could a priori not be made in a serious way. Even if the shaping of dynamics admits a limited bandwidth of variation within the 128 MIDI velocity values, the piano and pianissimo ranges (velocities between roughly 30 and 1) for the Boesendorfer were much finer to tune than for the Yamaha piano and grand piano, where these dynamical values make the keys move silently. This defect as well as the extremal differences in the repetition mechanics show the futility of a comparison between such Boesendorfer and Yamaha instruments and imply that the B¨osendorfer is the only reference for a reasonable performance. In this sense, the Boesendorfer was already chosen for the first experiment with Schumann’s “Kuriose Geschichte” as briefly reported in section 42.1.
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CHAPTER 42. PERFORMANCE EXPERIMENTS
In principle, an acoustical piano should be chosen as a reference for computer-assisted performance, but because of restricted availability of such instruments3 , a digital piano had to be selected in our case. Our experiences with digital pianos and their sounds showed significantly different behavior for identical dynamics and articulation when applying identical weights. And it must be remarked that at the present state of performance work, it is difficult to obtain exact information about the dynamics and articulation shaping in the sense of a possibly reliable performance grammar since the judgment of the dynamical and articulatory aspects as a function of the available instruments, see [505] for a detailed presentation of sound experiments. Any conclusion regarding the sources and rationales for the performance shaping of a score are vitally influenced by these instrumental conditions. In other words, within a strictly scientific framework, the MIDI-encoded performance data enforce an instrumental selection. If research can be realized by means of one and the same MIDI piano, comparative statements can be made exactly for this instrument. In a somewhat broader sense, this is also true for digital maps of the acoustical piano, i.e., one has to restrict the research to one and the same digital piano, such as Kurzweil’s Micro Piano as it was used in [504]. Any conclusion from the digital to the analog piano or vice versa is impossible, even among different digital or analog pianos no comparison is possible, see also [505]. One solution out of this dilemma could be the application of the “physical modeling” or “virtual acoustics” technology (see appendix A.1.2.4), where a direct access of the physical technology of an instrument by the virtualization (i.e., the software modeling) of the physical system of an instrument is enabled. This methodology offers flexible instruments which can be deformed seamlessly and can respond without delay. Although only first experiments have been initiated, this perspective opens an encompassing approach of computer-assisted performance research. The sound and resonance environment with its consequences for the shaping of performance aspects would not only yield new insights in the functioning of musical instruments, but also insights in the practice of instrumental playing. In this context, an interdisciplinary research team consisting of instrumentalists/interpreters, musicologists, computer scientists, mathematicians, and physicists would be required.
42.4.3
Global Discussion
Summary. We summarize the insight drawn from this second experiment. –Σ– In the course of the performance experiments, two different approaches and performance strategies crystallized. We tried to give the score’s text an immanent shaping by means of two approaches: • what is the sound of the analytical structure? • can the sounding analytical structure yield a musically reasonable performance? 3 In Germany, the B¨ osendorfer MIDI grand exists only in two locations: at the conservatories in L¨ ubeck and Karlsruhe. In both cases, access to this instrument is virtually impossible since it is located in rooms that are used for ordinary school activities.
42.4. STEMMA CONSTRUCTIONS
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and two contrary performance strategies: • the target-driven strategy, • the experimental strategy. In contrast to objective analytical approaches, for performance, subjective ingredients cannot be completely eliminated. They are present in their feedback with the performance result, while weights and intensity parameters in the WeightWatcher are determined, but they play a fairly reduced role. From the first performance experiments, which have not been discussed in detail here, until the complete performance as described above, we have known milestones which demonstrated several problematic issues: It was not easy to eliminate the impression of an existing performance—in our case by Glenn Gould, say—and to stick strictly to what is written in the score; the performed version of the piece automatically resonates as a comparison while doing the performance work. This was the situation where we started these experiments with the ambitious task of approaching an artistical and esthetical performance as far as possible. Therefore, the target-driven strategy was to a certain degree determined by the comparison with traditional human performances. Under these conditions, weights were applied and results were judged. This turned the tradition into an obstruction, it positioned the expected performance in the foreground and the shaping weight in the background. Only the consequent questioning of the analytical structure and the systematic liberation from traditional performance expectations led to a performance strategy which positioned the analytical weights in the center of the investigation. This experimental strategy was coined by an as unbiased as possible sounding realization of analytical structures, centered around the question of how a weight, when applied to a particular performance aspect, would sound. Within this procedure it was possible to insert ‘unheard’ results, to admit purposed over-subscriptions in the sense of the ‘still more clear’, whereas the question whether an interpreter would play in this way turned out to be completely irrelevant. From this point of departure, how a determined analytical structure would sound, the experimental approach to shaping a musically reasonable performance was sought. This qualitative determination of what is a “musically reasonable” performance is inevitably a subjective one which as such decided upon the subsequent steps towards the final performance. Similarly to the interpreter who puts up for discussion his provisionally final version while performing in concert—where in the last analysis it is more his personality than the musical performance which is judged—in the case of computer-assisted performance, the subject who works with the performance workstation RUBATOr presents his results as a provisionally final contribution to the ongoing discussion. When judging all these performances, one has to take into account that only metrical and motivic weights were applied and the effects of harmonic passages were not included in the shaping of performance (except of the motivations for the not machine-made subdivisions from global to more local applications of weights in the third parcours). Furthermore a certain economy in the choice of weights and their application was applied. In this sense, we first had to check out which weights would entail what type of shaping consequences, and how the change of intensity parameters would influence the musical expressivity. It was only after this preliminary work that a systematic application of the weights and a partially purposed work with their intensity parameters became possible.
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The portability of the presently described performance technique must be deduced from the compositional structure (a fugue in general and the thematic structure of the “Kunst der Fuge” in particular) as well as from the instrumental context. In nuce it can be said that such systematic statements are still premature. Many more analyses and performances would be necessary, but these can only be realized as soon as RUBATOr has become a common tool of musicology. Then the question of whether general recipes which are valid beyond the limits of single compositions can be stated, or whether performance is rather bound to each individual composition, could be restated. Whatever is true for the transformation of the analytical structure in a scientific work targeting an artistically valid esthetic performance, one should not forget about the elimination of (and nonetheless omnipresent) emotional and gestural aspects. The realization of a sonification of analytical structures during the interaction with the computer always bears a degree of emotionality, a phenomenon that should be taken into account as a kind of “uncertainty relation”. The judgment of the performance results took place in the same line as the judgment of a human performance, and the work with RUBATOr was also proposed as a provisionally final contribution to the work’s discussion. While describing the performance results, the stress of a scientific analytical performance was central. The feedback to the analysis has a particular significance in that possibly, the conclusive character of a performance could yield an analytical criterium. This implies an absolutely serious attitude towards analysis, and no disclosure from emergent new aspects and innovative analytical ways of hearing. Therefore we refrain from a discussion of subjects such as “prejudices against results which are produced by a machine”, or “performance and the soul of music versus soulless performance machines”. Instead, we favor representations of procedures and performance strategies, the exemplary demonstration of connections between analyzed structures, performed results, and the attempt at a generalization of these insights in the form of a performance grammar in its dependency on the instrumental conditions.
Part XI
Statistics of Analysis and Performance
853
Chapter 43
Analysis of Analysis O sancta simplicitas! Jan Hus (1370–1415) Summary. Not unexpectedly, weight analysis turns out to be complex information that cannot always be handled intuitively. This suggests techniques that help analyzing weight analysis. The problem may be tackled by use of statistical methods. We expose the subject and Jan Beran’s approach based upon hierarchical decompositions of weights, together with an application to comparison of analyses of Bach, Schumann, and Webern. –Σ– Although the MetroRUBETTEr seems to implement a very simple analysis, the metrical weights turn out to encode quite complex information. For performance applications such as RUBATOr , this may be acceptable, but for an analytical understanding per se, the weights are too complex to be used directly, except for direct visual inspection of evident surface properties. The prejudice that musical analysis should be a simple, intuitive affair, is therefore banned to the fairy tales of auto-incompetent humanities. In this chapter, we give an account of a statistical approach to understanding weights. More specifically, Jan Beran’s method of hierarchical smoothing (see also [50]) is presented.
43.1
Hierarchical Decomposition
Summary. This section describes and motivates the hierarchical decomposition of weights by use of decreasing sequences of time bandwidths, the so-called hierarchical smoothing. –Σ–
43.1.1
General Motivation
Can additional structural insight into weights be gained by suitable analysis of the analytic weight curves? The idea of the following method is to find a “natural” decomposition of the 855
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weight functions in order to find hidden regularities. In time series terminology, the general problem can be stated as follows: Let {xs (ti ), ti ∈ R, s = 1, ..., k, i = 1, ..., n} be a collection of k time series, measured at the time points ti . The aim is to find a decomposition xs (ti ) =
M X
xj,s (ti )
j=1
such that the components {xj,s , s = 1, ..., k} reveal a maximal amount of ‘regular structure’. One of the difficulties is to define what is meant by ‘regular structures’ and to define corresponding meaningful measures of the amount of ‘regular structure’. Here, a pragmatic approach is taken, in that the amount of ‘regular structure’ is judged visually. Clearly, more formal definitions could be used. Before introducing the idea of hierarchical decomposition, a few general remarks should be made: Remark 18 Traditionally, one of the main structures of interest for time series is periodicity. In particular, spectral decomposition based on sines and cosines may be used for this purpose (see e.g., [423], [69]). In our context, this is not applicable, because many compositions are likely to have much more interesting structures than just periodicities. In fact, some scores may not contain any nontrivial periodicities at all. More generally, the problem is that using the same basis of functions, irrespective of the structure of the score, results in focusing on a very limited number of predetermined features that may in fact not be present. Remark 19 As a consequence, a nonparametric approach based on kernel smoothing will be proposed here. In a traditional setting, the bandwidth b is chosen by minimizing a criterion such as the mean squared error as n tends to infinity. In particular, b tends to zero with increasing sample size. This concept is not directly applicable in our context. The main reasons can be summarized as follows: 1. Based on the definitions given above, the metric, melodic and harmonic aspects of a score are characterized respectively by one weight function only. In contrast, a composer is likely to have a hierarchical view. For instance, a piece has on one hand a global harmonic shape that makes the piece coherent as a whole, and on the other hand more local structures. Some composers in fact consciously write a score using a hierarchical approach, first defining a global shape and then refining more and more local structures. Similarly, while rehearsing, a performer is likely to focus first on global features of the score and then successively refine more and more local features. This fact was also used in RUBATO to design the process from a primavista performance to the refined artistic result [357]. Here, a genealogical tree, the stemma of the performance process, is responsible for successive refinement and localization of the performance. In order to obtain a better picture of the structure of a score it is therefore necessary to “extract” the hierarchy that is hidden in the weight functions. For smoothing, this means that there is not just one optimal bandwidth that is of interest. Instead, there is a hierarchy of relevant bandwidths b1 > b2 > ... > bM . Moreover, the structure of the score, rather than an omnibus statistical criterion (such as the mean squared error), is likely to yield the key information about which sets of bandwidths could be interesting.
43.1. HIERARCHICAL DECOMPOSITION
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2. The weight functions obtained from the analysis above are generally rather complex. In particular, the weights often jump abruptly up and down between very small and very large values (see figure 43.4). This can certainly not be carried over linearly to musical performance. For instance the tempo of a “musically acceptable” performance is unlikely to change up and down drastically and repeatedly within a few seconds. It is therefore reasonable to assume that a performance is not a linear function of the weights but rather a weighted sum of non-linearly deformed smoothed versions of these functions. Again, there may be a hierarchy of several bandwidths that need to be considered. These general considerations motivate the idea of hierarchical smoothing and hierarchical decomposition described below.
43.1.2
Hierarchical Smoothing
Let {xs (ti ), ti ∈ R, i = 1, ..., n, s = 1, ..., k} be a k-dimensional time series observed at time points t1 , ..., tn and Kb a smoothing kernel with bandwidth b and support [−b, b]. Applying the smoothing operator n X Kb xs (t) = Kb (t, ti )xs (ti ) i=1
(t ∈ R) for a hierarchy of bandwidths b1 > ...bM , we obtain a hierarchy of k−dimensional curves {xj,s (t) = Kbj xs (t), s = 1, ..., k}, j = 1, ..., M. Here, the Naradaya–Watson kernel i K( t−t b ) Kb (t, ti ) = Pn t−tj j=1 K( b )
with a triangular function K(s) = 1{|s| ≤ 1} · (1 − |s|) was used. For b = 0, we have Kb xs (t) = xs (t). Figures 43.1 and 43.2 display hierarchies of smoothed curves for Schumann’s “Tr¨aumerei” and for Bach’s “canon cancricans” (see also figure 8.7), resulting from the metric, melodic and harmonic weights. The figures illustrate that different bandwidths make different features more visible. In particular, for the metric weights, smoothing highlights places where high values occur more frequently. Also, some remarkable similarities between the metric, melodic and harmonic weights become apparent after smoothing. Remark 20 The statistical technique of using smoothing kernels deserves a comment from the point of view of stemma and operator theory (section 39.8), a comment which strongly relates to the inverse performance theory to be exposed later in part XII. To begin with, taking into account neighboring values of the analyses by kernel smoothing has a musical meaning: The interpreter is rightly supposed to be conscious of what happened and will happen within the time bandwidth b. In inverse performance theory, the idea of kernel smoothing is introduced in the locally linear performance grammars, see section 39.8. There, the mutual influence of different local parts Ci and Cj of the composition C for the performance shaping process is formalized by use of interaction matrices (ci,j ). The coefficient ci,j quantifies the influence of
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Figure 43.1: Smoothed version of metric, melodic, and harmonic weights for Schumann’s “Tr¨aumerei”. Ci on Cj . The general theory of locally linear performance grammars deals with the description of the de facto algebraic variety of interaction matrices inducing a fixed performance. From this point of view the triangular kernel smoothing means a selection of the a priori shape of interaction matrices, showing a peak around the diagonal.
43.1.3
Hierarchical Decomposition
The approach of hierarchical smoothing suggests a decomposition of the weight function into components of varying smoothness. Thus, let {xs (ti ), ti ∈ R, s = 1, ..., k, i = 1, ..., n} be a collection of k time series. As discussed above, the aim is to find a decomposition xs (ti ) = PM j=1 xj,s (ti ) such that the components {xj,s , s = 1, ..., k} reveal a maximal amount of “regular structure”. Structure can be, for instance: symmetry, repeated shapes/periodicities, relationship between different components etc. Note that with respect to cross-correlations, a number of methods are known in the literature for testing dependence between stationary time series
43.1. HIERARCHICAL DECOMPOSITION
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Figure 43.2: Smoothed version of metric, melodic, and harmonic weights for Bach’s “canon cancricans”. (see e.g., [181], [211], [230], also see [423] and references therein). A direct adaptation of these methods is not possible for the following reasons: 1) The series considered here are not stationary in a nontrivial way and can, in particular, not be reduced to white noise by applying a linear filter. 2) The time points are not equidistant. 3) The aim is not only to obtain high crosscorrelations but also to highlight regular features of the individual series. 4) Not only crosscorrelations between “residual” but between all components are interesting. 5) The musical context suggests that the decomposition should be hierarchical in the sense that, with increasing index j, xj,s should contain increasingly local features. We thus define the following decomposition: 1. Define a hierarchy of bandwidths b1 > b2 > ...bM = 0, based on structural information from the score. Pj−1 2. Define the smoothed function x1,s = Kb1 xs and for 1 < j ≤ M , xj,s = Kbj (xs − l=1 xl,s ). It should be noted that this decomposition is only one of many possible decompositions
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of xs . The problem of choosing a meaningful decomposition of a time series is not new. In particular, in the context of regression analysis (see chapter 44), it is a special case of the general problem of defining meaningful explanatory variables in regression models. Here, subjectspecific considerations provide important guidelines. From a pragmatic point of view, a chosen decomposition can be considered reasonable if the subsequent regression analysis leads to meaningful interpretable results. In our context, the above decomposition appears meaningful, since it decomposes xs in a simple additive way into components of decreasing smoothness. This translates, in a straightforward way, the generally accepted fact that a musical composition as well as a performance may be considered as a superposition of a hierarchy of local and global “shaping features”, obtained by different degrees of “zooming in or out”. For a given sequence of bandwidths b1 > b2 > . . . bM = 0, the first component x1,s represents the most global view of the score (or more specifically of the metric, harmonic or melodic structure, respectively), x2,s represents the next step of refinement by considering, in a more detailed fashion with a smaller bandwidth b2 < b1 , the remaining information (obtained by subtracting the “global information” x1,2 , and so on.
43.2
Comparing Analyses of Bach, Schumann, and Webern
Summary. The statistical method developed in the preceding section is applied to a comparative study of RUBATO-analyses of works by Johann Sebastian Bach, Robert Schumann, and Anton Webern. –Σ– Each of figures 43.3.a through 43.3.d displays the melodic (dotted, middle), metric (full, lower) and harmonic (dashed, upper) weights for Schumann’s “Tr¨aumerei” op. 15/7 (Kinderszene No.7), Webern’s “Variationen f¨ ur Klavier” op. 27/II, the “canon cancricans” from Bach’s “Musikalisches Opfer” BWV 1079, and Schumann’s “Kuriose Geschichte” op. 15/2 (Kinderszene No.2). For onset times with more than one value of the melodic and harmonic weight respectively, the average of the values was taken. It is also interesting to look at scatterplots of the three types of weights against each other. The example for Bach’s “canon cancricans” is displayed in figure 43.4. For each of the compositions, some simple regular features of the weights are visible: • Tr¨ aumerei: From the score it is clear that this composition may be divided into four parts Pj , j = 1, 2, 3, 4, corresponding to the onset intervals I1 = [0, 8] and Ij = ((j − 1) · 8, j · 8], j = 2, 3, 4, respectively. Also is it obvious that these four parts are similar to each other, and that P3 differs most from the other parts. In fact, P2 is, by definition, an exact replicate of P1 (except for the slightly different up-beat). In figures 43.5.a through 43.5.c, the weights for the four parts are plotted on top of each other, i.e., onset time is taken modulo 8. The weights are indeed almost identical to each other. Interestingly, the fact that P3 differs most from the other parts shows only for the melodic weights. Also, the scatter plots do not indicate any strong relationship between the three weight functions. The sample correlations are all in the range [−0.01, 0.09].
43.2. COMPARING ANALYSES OF BACH, SCHUMANN, AND WEBERN
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Figure 43.3: Metric, melodic, and harmonic weights for Schumann’s “Tr¨aumerei”, Webern’s “Variationen f¨ ur Klavier”, Bach’s “canon cancricans”, and Schumann’s “Kuriose Geschichte”.
• Variation op. 27/II: With respect to the melodic and harmonic weights and from the score, it is clear that the composition can again be divided into four parts Pj , j = 1, 2, 3, 4, corresponding to a division of the onset time into four intervals of equal length. Clearly, the first two parts are almost identical with respect to the melodic and the harmonic weights. The same is true for the last two parts. For the metric weights, however, P2 is not a simple replicate of P1 . The same is true for the last two parts. Also, for P1 and P2 , the maximal values of the metric weights are much higher than for P3 and P4 . Again, no apparent relationship seems to exist between the three weights (not shown). However, the largest correlation (in absolute value) is much higher than in the previous example, namely −0.31 between metric and harmonic weights. • Canon cancricans: As expected for a retrograde canon, there is an almost exact time symmetry with respect to the middle of the onset axis. The symmetry is not exact, because the retrograde is not just a reflection of onsets but rather a transvection in the
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Bach metric weight
Bach melodic weight
Bach harmonic weight
Figure 43.4: Scatterplots of analytical weights for Bach’s “canon cancricans”.
onset duration space parallel to the onset axis (see section 8.1.1, example 9). Also striking is the clustered nature of the weights and the apparently very regular high frequency oscillation of the metric curve. A high metric weight is almost always succeeded by a low weight and vice versa. Because of the clustered nature of the weights, scatter plots are not very useful in this case. The correlations between the weights are again very small, ranging between 0.03 and 0.04. • Kuriose Geschichte: Here, the score is again divided into four parts corresponding to the onset intervals [0,6], (6,12], (12,21], (21,30], with P1 equal to P2 and P3 equal to P4 . Again, it is difficult to tell in how far the three different curves may be related to each other. Note however that the metric weights are much lower for onset times above 21. Thus, for the metric weights, the correspondence between P3 and P4 is much weaker. The reason is the breakdown of local meters at bar 21. Similarly to Webern, the strongest correlation between the weights is quite remarkable, namely −0.33 between melodic and harmonic weights.
43.2. COMPARING ANALYSES OF BACH, SCHUMANN, AND WEBERN
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Figure 43.5: Analytical weights for Schumann’s “Tr¨aumerei” against onset time modulo 8.
It is also interesting to compare the four compositions with each other. The weights of Bach’s “canon cancricans” exhibit an extreme high frequency oscillation that is not observed for the other scores. Ignoring that onset times are not exactly equidistant, this can be seen for instance very clearly by comparing the sample autocorrelations of the metric weights (figure 43.6). Another property of interest is the marginal distribution of the weight functions. Eliminating global ‘trends’ by taking first differences x(tj ) − x(tj−1 ) the histograms are given in figure 43.7 for the metric weights. For the compositions by Schumann and Bach, the first difference of the metric weights can essentially be classified into three clusters (low, medium, high). For Webern’s score, the distribution is completely different and in fact rather close to a normal distribution. In contrast, the distributions of the differenced melodic weights are qualitatively similar for all four scores. For the harmonic weights, all distributions appear to be essentially symmetric. However, while for Schumann’s “Tr¨aumerei” and the score by Webern there appear to be three clusters, the histograms for Bach and the “Kuriose Geschichte” are essentially unimodal.
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Figure 43.6: Autocorrelograms of metric weights for Schumann’s “Tr¨aumerei”, Webern’s “Variationen f¨ ur Klavier” op.27/2, Bach’s “canon cancricans”, and Schumann’s “Kuriose Geschichte”.
In summary, a first look at the weight functions reveals certain elementary features of the score. In the following it will be demonstrated that a more thorough analysis leads to further new insights about the structure of the scores. In particular, note that the three weight functions were defined in a completely different way. It may therefore be expected that there is no strong relationship between the curves. The scatter plots of the weights seem to support this conjecture. But the following analysis will show that certain components of the weight functions are indeed closely related. Specifically, application to the four examples was carried out using M = 4. This choice was based on musicological considerations (time signature and bar grouping) as explained in the following. In this sense, the analysis here is exploratory, since no statistical selection criterion was used for choosing M . The following notation will be used here: x1 = xmetric =metric weight, x2 = xmelod =melodic weight, x3 = xhmean =harmonic (mean) weight, xj,metric = xj,1 , xj,melod = xj,2 , xj,hmean = xj,3 . The choice of the bandwidths was based on the time signature and bar grouping information. Example Schumann/Tr¨aumerei is written in 4/4 signature, the
43.2. COMPARING ANALYSES OF BACH, SCHUMANN, AND WEBERN
865
Figure 43.7: Histogram of first difference of metric weights for Schumann’s “Tr¨aumerei”, Webern’s “Variationen f¨ ur Klavier”, Bach’s “canon cancricans” and Schumann’s “Kuriose Geschichte”.
grouping is 8 + 8 + 8 + 8. The chosen bandwidths are therefore 4 (4 bars), 2 (2 bars) and 1 (1 bar). Example Webern is written in 2/4 signature, its formal grouping is 1 + 11 + 11 + 11 + 11; however, Webern insists on a grouping in 2-bar portions [562], suggesting the bandwidths of 5.5 (11 bars), 1 (2 bars) and 0.5 (1 bar). Example Bach is written in 4/4 signature, the grouping is 9 + 9 + 9 + 9. The chosen bandwidths are 9 (9 bars), 3 (3 bars) and 1 (1 bar). For example Schumann/Kuriose Geschichte, the time signature is 3/4, the grouping is 8 + 8 + 12 + 12. The chosen bandwidths are 3 (4 bars), 1.5 (2 bars) and 0.75 (1 bar). Figures 43.9 (“Tr¨ aumerei”) and 43.8 (“canon cancricans”) show remarkable regularities that have not been observed for the original weights (same for Webern and Schumann/Kuriose Geschichte, which we omit here). In particular, for all four compositions, much stronger similarities between the metric, melodic, and harmonic components can be observed than for the original weights, especially for j = 2, 3. Moreover, for the first two scores, the same kind of relationship can be observed for j = 2, 3, namely: positive correlation between xj,melod and
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Figure 43.8: Hierarchical components of metric (solid lines), melodic (dotted lines), and harmonic (dashed lines) weights for Schumann’s “Tr¨aumerei”, as defined in section 43.1.2: (a) b = 4; (b) b = 2; (c) b = 1; (d) remaining (residual) series.
xj,hmean , negative correlation between xj,melod and xj,metric , and negative correlation between xj,hmean and xj,metric . Particularly surprising is the fact that Webern’s score shows the same type of association as Schumann’s “Tr¨ aumerei”. This leads to new insights into different approaches to composition. The weight functions are in fact very complex data and deserve a refined “analysis of analysis”. Hierarchical smoothing is a possible approach to this problem. Webern’s piece is written in a completely dodecaphonic way, and thus breaks with harmonic and homophonic tradition. This deserves a special methodological comment. The fact that we have nevertheless applied harmonic analysis could be viewed as being in contradiction to Webern’s rupture with harmony. Now, we do not claim that this analysis corresponds to Webern’s poietic position when composing his “Variationen”. Nonetheless, an objective analysis according to the Riemann approach is reasonable for two reasons: (1) Riemann intended to attribute tonality to any possible chord. The fact that he did not succeed in his goal is no reason for refraining from completion of his sketch. This is what the HarmoRUBETTEr is about: It
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Figure 43.9: Hierarchical components of metric (solid lines), melodic (dotted lines),and harmonic (dashed lines) weights for Bach’s “canon cancricans”, as defined in section 43.1.2: (a) b = 9; (b) b = 3; (c) b = 1; (d) remaining (residual) series. is a proposal to discuss possible completions of Riemann’s theory. (2) Therefore it is also very interesting to discuss its application to apparently atonal compositions. Such an experiment is likely to yield a testbed for the universality of Riemann’s approach. These considerations suggest that the following fact is not completely surprising, although it has not been established explicitly elsewhere in the literature: The correspondence between metric, melodic, and harmonic structure in Webern’s “Variationen” is very similar to Schumann’s “Tr¨aumerei”. It should be emphasized that this conclusion and in particular its quantitative demonstration is new in the musicological literature. Schumann’s “Kuriose Geschichte” also shows a strong correspondence between the three curves for j = 1, 2 and 3. But this time, the relations are different: For onset times below 12, we have the following:
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1. For j = 1, cor(x1,metric , x1,melod ) = 0.83, cor(x1,metric , x1,hmean ) = −0.71, cor(x1,hmean , x1,melod ) = −0.63. 2. For j = 2, we have the following (rounded) correlation values: cor(x2,metric , x2,melod ) = 0.00, cor(x2,metric , x2,hmean ) = −0.31, cor(x2,hmean , x2,melod ) = −0.82. 3. For j = 3, we have cor(x3,metric , x3,melod ) = −0.67, cor(x3,metric , x3,hmean ) = −0.20, cor(x3,hmean , x3,melod ) = −0.61. Observe in particular that, in contrast to the other scores, melodic and harmonic components are negatively correlated. After onset time 12, the correlations are: 1. For j = 1: cor(x1,metric , x1,melod ) = 0.10, cor(x1,metric , x1,hmean ) = −0.38, cor(x1,hmean , x1,melod ) = −0.29. 2. For j = 2: cor(x2,metric , x2,melod ) = −0.47, cor(x2,metric , x2,hmean ) = −0.14, cor(x2,hmean , x2,melod ) = −0.11. 3. For j = 3: cor(x3,metric , x3,melod ) = −0.75, cor(x3,metric , x3,hmean ) = 0.58, cor(x3,hmean , x3,melod ) = −0.69. Finally, for Bach’s composition, the only noticeable correlations occur between metric and harmonic weights, namely: 1. For j = 1 : cor(x1,metric , x1,hmean ) = 0.94, 2. For j = 2 : cor(x2,metric , x2,hmean ) = 0.63, 3. For j = 3 : cor(x3,metric , x3,hmean ) = 0.61. With respect to the shapes of xj,. , for j = 2 and 3, the two scores by Schumann and the one by Webern are clearly more similar to each other as compared to Bach’s shapes. From the point of view of music history, this is quite plausible, since Webern’s organic composition principle is more related to Schumann’s rankly growing romanticism than to Bach’s self-disciplined architectural setup (see also the following remarks).
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869
Finally, note that the scatterplots in figure 43.4 show that Bach’s harmonic weights are highly clustered and the smoothed curves in figures 43.8.a through 43.8.d are more ‘edgy’ than for the other compositions. In this sense, Bach’s composition exhibits a high degree of organization. This confirms the general belief that the principle of architectural rather than processual construction plays a dominating role in Bach’s music. Overall, we may conclude that hierarchical decomposition reveals interesting properties, in particular strong similarities between the metric, melodic and harmonic weights, that were not visible in the original series. The results are musically plausible in that the analysis of Bach’s score turns out to be the most regular one and the analyses of Webern and Schumann appear to be closer to each other than to Bach’s. The results are surprising in that (the analysis of) Webern turns out to be closer to (the analysis of) Schumann than expected. Also, the strong relationship between the three analytic curves could not be expected a priori, since the three weights were calculated using completely different aspects of the score and the scatter plots of the original curves did not show almost any association. Based on the results, one may conjecture that appropriate matching of metric, melodic and harmonic structure plays an important role in music, independently of musical style. The tools introduced here provide the possibility of investigating which types of relationships may exist in which musical and historical contexts. An important task for future research will be to investigate such aspects for a larger variety of compositions.
Chapter 44
Differential Operators and Regression Rejection by common sense, for whatever reason, proves nothing. Other fields of science are built on propositions that seem absurd but in fact are true. Donald O Hebb [213] Summary. We give statistical evidence from 28 performances of Schumann’s “Tr¨aumerei”, as measured by Bruno Repp [438] that the rhythmic, motivic, and harmonic analyses provided by RUBATOr are shaping structures for the agogical streams. The statistical model is based on regression analysis and realizes shaping of agogics by a second degree linear differential operator as a function of analytical weights which are averaged over a natural grouping hierarchy (as described in chapter 43) of the score. –Σ– At present, the best investigated aspect of performance theory—including appropriate software—is timing microstructure, i.e., agogics on the level of tempo curves and their hierarchies, see [272, 346] for further reading. This chapter deals with this topic: agogics as an expression of harmonic, melodic and rhythmic structures. So observe that we do not consider emotional or gestural rationales for agogics. This does not mean that these factors are negated. We merely restrict our investigation to the question whether and how strongly agogics could be explained by exclusive causal reference to structural analysis. Even in this neat reduction is the question neither trivial nor even well defined, since musicology does not offer precise tools for rhythmical or melodic analysis, and even harmonic analysis is far from effective. Therefore, the question is only a scientific one if one specifies the analyses and their output data format. For the general setup for such an explicit and operationalized analysis framework, namely the RUBATOr analysis and performance workstation, we refer to part X. The main concern is an empirical study regarding the basic question whether agogics (the tempo curve of the “Tr¨ aumerei”, also called “timing microstructure”) may be expressed in mathematical terms by use of structural data obtained from a specific set of musicological 871
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analyses offered by RUBATOr ’s modules. The experimental data are taken from Repp’s timing measurements of 28 famous performances of Schumann’s “Tr¨aumerei” [438]. Summarizing our results, we can state that Result 2 There is strong statistical evidence for the timing microstructure in 28 famous performances of Schumann’s “Tr¨ aumerei” as being an expression solely of harmonic, melodic and rhythmic structures furnished by the RUBATOr analysis. In other words: These structural rationales are sufficient for explaining agogics, and emotional or gestural rationales can be disregarded for the present experimental material. In the framework of RUBATOr , the analytical output of any type of analysis is always a (smooth) weight, i.e., differentiable function of one or several note event parameters, such as onset E, pitch H, loudness L, duration D, etc. With this specification, the above result can be restated in more mathematical but less intuitive terms: Result 3 The tempo curve (timing microstructure) can be generated by an agogical operator Ω which is essentially a linear differential operator of second order as a function of harmonic, melodic and rhythmic smooth weights. Statistically speaking, this means that in our empirical context, the fiber (in the sense of Todd’s approach) of the chosen performance transformation is not empty. Observe that the operator Ω which plays the role of the transformation Π in Todd’s theory [530] (see also section 36.3) does not rely on general encoding tempo curve functions. Intuitively, this means that our approach generates agogics from smooth weights as score-specific functions, and not general, score-independent, curve types, as proposed in [518, 532]. This does not contradict usage of general encoding tempo functions, it simply suggests that agogics is a superposition of more “primavista”-like tempo functions and of a strong and differentiated timing microstructure stemming from analytical data.—Thirdly, the statistical results suggest Result 4 Essential commonalities and diversities among tempo curves may be characterized by a relatively small number of analytical weight curves. There is in general no unique way of attributing features of the tempo to exactly one cause (harmonic, metric or melodic analysis). Results depend on which of the three analyses is given priority. However, there appear to be a certain number of canonical curves that are essentially independent of the priority. Overall, a large variety of musically meaningful results is obtained. This is in particular due to the fact that a score-specific basis of curves is used on which the tempo curves are projected. We thus may conclude Result 5 The analytical curves obtained from (1) score-specific harmonic, melodic and rhythmic smooth weights and (2) a score-specific hierarchical decomposition of these weights, yield a natural score-specific linear basis in the space of tempo curves, for performances of the considered score. We should stress that all our results are intimately related to the concrete analyses which RUBATOr produces—together with the underlying theories. There is no unique analysis, and
873 therefore, specification and numerical representation of musical analysis is not secondary and will in any case (!) influence the results. We should also remind the reader trained in natural sciences that musical analysis is not a neutral tool but pertains to the unavoidable “artifacts” of analysis in the humanities. There are no objective laws in human creations which subsist beyond interpretative interaction. This is a caveat to those who believe that performance can be more than a relation between what we understand (rationally, emotionally, gesturally) and how we express this understanding.
44.0.1
Analytical Data
Summary. This section describes the analytical weights used in this analysis. –Σ– We shall omit the notification of the used parameter lists P aram which are related to the specific RUBETTEr , and of the predicate PT r¨aumerei of Schumann’s score and abbreviate xmetric
T r¨ aumerei = xP M etroRubette,P arammetric
xmelodic
T r¨ aumerei = xP M eloRubette,P arammelodic
xhmax xhmean
T r¨ aumerei = xP HarmoRubette,P aramharmonic/max T r¨ aumerei = xP HarmoRubette,P aramharmonic/mean
to denote the four following weights used in our context. We are going to give their description when evaluated on onsets E of note events occurring in PT r¨aumerei : PT r¨ aumerei • xM etroRubette,P arammetric
This is a metrical weight which measures the rhythmic relevance or “weight” of every onset of a note event in the composition in the lines of Riemann [453], Jackendoff–Lerdahl [243], and Mazzola [340]. A detailed description of the MetroRUBETTEr was given in section 41.1. PT r¨ aumerei • xM eloRubette,P arammelodic
This is a “boiled-down” melodic weight1 which measures the sum of the melodic weights w(Evt) of all note events Evt at a given onset E. The calculation is extremely complex and time-consuming and goes back to theories of Reti [444], see chapter 22. See section 41.2 for a detailed description of the MeloRUBETTEr . PT r¨ aumerei • xHarmoRubette,P aramharmonic/max
This harmonic weight measures the harmonic relevance of a chord ch which occurs at an onset E in PT r¨aumerei . It is calculated by the same method as the fourth weight. The only difference is that this weight captures the harmonic relevance of the most important note in ch whereas the fourth weight represents the average harmonic relevance among all notes of ch. See section 41.3 for a detailed description of the HarmoRUBETTEr . 1 See
formula (39.26) in section 39.3.
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CHAPTER 44. DIFFERENTIAL OPERATORS AND REGRESSION
T r¨ aumerei • xP HarmoRubette,P aramharmonic/mean
As already mentioned, this weight is a variant of the third one, the only difference being an averaging instead of maximizing procedure. We refer to the previous discussion for the basics.
44.1
The Beran Operator
Summary. In this section, we define the conceptual setup for the following statistical analysis. –Σ– The general idea is that agogics is to be shaped by use of smoothed versions of the boileddown weights, their first and second derivatives, and corresponding kernel-smoothed versions with respect to hierarchical (triangular) kernel functions.
44.1.1
The Concept
As we know, the general RUBATOr concept of shaping performance is built on smooth (actually C 2 in this context) weights x where dE x, d2E x denote the first and second derivatives with respect to symbolic time E. The kernel smoothing process relates to kernel functions ˆb(s) = 1/b·χ{|s| ≤ b} · (1 − |s|/b) with triangular, zero-symmetric support of extent ±b, and characteristic function χ{P } for a predicate P . The linear smoothing operator b f on a function f is defined by the convolution Z b f (E) = ˆb(t − E) · f (t). (44.1) It averages f around E with weighted center E and bandwidth b. If this function is a weight, this means that the weight’s analysis within the entire bandwidth neighborhood of a given onset is included instead of spiking the analysis to the singular onset. In the following process, this kernel smoothing process has been applied to a hierarchy of bandwidths, starting with b = 4 (= eight bars), then b = 2, then b = 1. The averaging process is taken to define successive remainder functions as follows: f1 = 4 f, f2 = 2 (f − f1 ), f3 = 1 (f − f1 − f2 ), f4 = f − f1 − f2 − f3
(44.2)
This means that the decomposition x = x1 + x2 + x3 + x4
(44.3)
for a smooth weight x defines a “spectrum” of that weight with respect to successively refined neighborhoods of its ambit. Remark 21 Musically speaking, as already observed before, this kernel smoothing process is completely natural. In fact, the kernel function alters the original time function f (E) by a weighted integration of f -values in the kernel neighborhood of a given time E. This means that we now include the information about f from the neighboring times to make an analytical
44.1. THE BERAN OPERATOR
875
judgment. This latter is a well known and common consideration in musical performance: The interpreter looks up a full neighborhood of a time point to derive what has to be played in that point. Moreover, the repeated application of the kernel smoothing process with increasingly narrowed neighborhoods is understood as a succession of a refinement in local analysis: First, the interpreter makes a coarse analysis over eight bars (b = 4), then he/she looks for the remainder f − f1 and goes on with refined actions, if necessary. This procedure is applied to the metric, melodic and harmonic weights and to their first and second derivatives. This gives the following list of a total of 48 spectral analytical functions: xmetric,1 dE xmetric,1 d2E xmetric,1
xmetric,2 dE xmetric,2 d2E xmetric,2
xmetric,3 dE xmetric,3 d2E xmetric,3
xmetric,4 dE xmetric,4 d2E xmetric,4
xmelodic,1 dE xmelodic,1 d2E xmelodic,1
xmelodic,2 dE xmelodic,2 d2E xmelodic,2
xmelodic,3 dE xmelodic,3 d2E xmelodic,3
xmelodic,4 dE xmelodic,4 d2E xmelodic,4
xhmax,1 dE xhmax,1 d2E xhmax,1
xhmax,2 dE xhmax,2 d2E xhmax,2
xhmax,3 dE xhmax,3 d2E xhmax,3
xhmax,4 dE xhmax,4 d2E xhmax,4
xhmean,1 dE xhmean,1 d2E xhmean,1
xhmean,2 dE xhmean,2 d2E xhmean,2
xhmean,3 dE xhmean,3 d2E xhmean,3
xhmean,4 dE xhmean,4 d2E xhmean,4
For which musical reasons are these derivatives added to the analytical input data? The first derivatives measure the local change rate of analytical weights. Musically speaking, this is an expression of transitions from important to less important analytical weights (or vice versa), i.e., a transition from analytically meaningful points to less meaningful ones (or vice versa). This is crucial information to the interpreter: It means that he/she should change expressive shaping to communicate the ongoing structural drama. In the same vein, information about second derivatives is musically relevant since it lets the interpreter know that the ongoing structural drama is being inflected. Evidently, one could add higher derivatives but we argue that an interpreter is already highly skilled if he/she can take care of all these functions, also because different analytical aspects from metrics to harmonics must be observed simultaneously. Besides these analytical input functions, we add three types of ‘sight-reading’ functions. They regard the following three instances: ritardandi, suspensions2 , and fermatas. It is clear that any text-sensitive performance should be aware of such information. 2 Suspensions are notes which are tied by a slur while the harmony changes; we may attach to such events the time interval where the suspension does not start until the harmony change is terminated.
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1. Ritardandi The score shows four onset intervals R1 , R2 , R4 , R4 for ritardandi, starting at onset times Eo (Rj ) (j = 1, 2, 3, 4) respectively. We define the four linear functions xritj (E) = χ(Rj ) · (E − Eo (Rj )), j = 1, 2, 3, 4.
(44.4)
2. Suspensions The score shows four onset intervals S1 , S2 , S4 , S4 for suspensions, starting at onset times Eo (Sj ) (j = 1, 2, 3, 4) respectively. We define the four linear functions xsusj (E) = χ(Sj ) · (E − Eo (Sj )), j = 1, 2, 3, 4.
(44.5)
3. Fermatas The score shows two onset intervals F1 , F2 for fermatas. We define the two support functions xf ermj (E) = χ(Fj ), j = 1, 2. (44.6) Summarizing, we have a total of 58=48+4+4+2 onset functions of analytical and primavista types. Call X the analytical vector of these 58 functions listed in a fixed order. The present approach is to define the tempo function at onset E as being a linear function of these 58 variables. For a ‘shaping’ vector ω ∈ R58 , the shaping operator ΩX ω for the tempo curve is defined by the canonical scalar product of X with the shaping vector ω, ΩX ω = (X, ω).
(44.7)
This means that for every onset E, we have ΩX ω (E) = (X(E), ω). Recapitulating the meaning of the analytical vector X, we are dealing with a second order differential operator which we call “Beran operator” since it was introduced by Jan Beran in [52]. On this basis, the central question of the following is whether tempo curves T of the “Tr¨aumerei” as they appear in the context measured by Repp in [438] may be approximated via ΩX ω by appropriate choice of the shaping vector ω. The main result of this approach states that there is strong statistical evidence for the equation ln (T ) = ΩX ω +C
(44.8)
for the given analytical vector X, a suitable shaping vector ω, and a constant C. This means that the 58 coefficients of the shaping vector ω are random variables and that we prove a significant statistical correlation—in the mathematical form described by the Beran operator—between a certain subset of the analytical vector X and tempo as it is measured for the 28 performances by Repp. Thesis 7 One may therefore try to use the above formula (44.8) to define tempo as a function of analytical score data in the sense of a general performance grammar as described in chapter 37.
44.1. THE BERAN OPERATOR
44.1.2
The Formalism
44.1.2.1
Tempo Information
877
In the following, a more detailed description of the tempo data used for the analysis is given. • Onset times: The onset times are on a grid of 1/8th beats. Thus, for instance, grace notes are excluded. From this set of onset times, we consider only onset times where at least one note is actually played. This results in a set T of n = 212 not equidistant onset times ti (i = 1, ..., n) which are multiples of 1/8. • Log-transformation: Instead of the original tempo y we consider its natural logarithm ln y. Intuitively this can be justified by the expectation that a performer may control the tempo in a relative rather than an absolute way. Also, the statistical results were more satisfactory on the logarithmic scale. In the following we refer to the logarithmic tempo as ‘the tempo curve’. • Standardization of individual curves: The data consist of tempo measurements (or tempo curves) for m = 28 performances. In the current analysis, the interest lies in investigating the shape of the tempo curves rather than the absolute tempo values. Therefore, each of 28 tempo curves is standardized. More specifically, let y ∗ (ti , j) be the (natural) logarithm of the tempo of the j th performance at onset time ti (i = 1, ..., n; j = 1, ..., m). Then the standardized tempo data are defined by y(ti , j) = where y¯∗ (j) = n−1
Pn
i=1
y ∗ (ti , j) − y¯∗ (j) s∗ (j)
y ∗ (ti , j) and
s∗ (j) = [(n − 1)−1
n X
1
(y ∗ (ti , j) − y¯∗ (j))2 ] 2 .
i=1
44.1.2.2
The Explanatory Variables
The following notation is used: Let A be a p × q1 matrix and B a p × q2 matrix, then C = (A, B) denotes the p × (q1 + q2 ) matrix obtained by ‘attaching’ B on the right-hand side of A. The following steps describe the definition of the matrix of explanatory variables in more detail. According to the concept in section 44.1.1, the score data (metric, harmonic and melodic weights, additional score information) are given in the form of a design matrix that is used subsequently in a regression analysis. The following definitions are used: 1. Derivatives. “Derivatives” are defined as finite differences divided by the difference of the onset times. Thus, for instance, dE xmetric,j (ti ) = and d2E xmetric,j (ti ) =
xmetric,j (ti ) − xmetric,j (ti−1 ) (ti − ti−1 )
dE xmetric,j (ti ) − dE xmetric,j (ti−1 ) . (ti − ti−1 )
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CHAPTER 44. DIFFERENTIAL OPERATORS AND REGRESSION
2. Hierarchical smoothing. Each of the weights and their first and second (discrete) derivatives are decomposed into four components of different smoothness as defined by equations (44.2) and (44.3). 3. Additional variables. Additional variables modeling ritardandi, suspensions and fermatas were defined by (44.4), (44.5), and (44.6). The aim is to model these musical events in a “minimal” way. For instance, the resulting linear model for a ritardando is only a crude approximation to a “true” ritardando. The reason for using only the simplest parametrization is that the main purpose here is to examine to what extent the metric, melodic, and harmonic weights alone, together with only absolutely necessary additional information from the score, contain enough information to “explain” the tempo of a performance. 4. Initial design matrix. Using the definitions above, we define for j = 1, 2, 3, 4 the n × 4 matrices Xj (harmo) = (xhmean,j , xhmax,j , xmetric,j , xmelod,j ) Xj (metric) = (xmetric,j , xhmean,j , xhmax,j , xmelod,j ) Xj (melod) = (xmelod,j , xhmean,j , xhmax,j , xmetric,j ) Clearly, there are more possibilities of permuting columns. Here, we consider only the representative permutations above. The first column of Xj (harmo) is xhmean,j so that, due to the orthonormalization to be described in the following section, the main emphases is put on the harmonic mean weights. Similarly, the metric and melodic emphasis are chosen with Xj (metric) and Xj (melod). Furthermore, we define dE Xj (harmo) = (dE xhmean,j , dE xhmax,j , dE xmetric,j , dE xmelod,j ) dE Xj (metric) = (dE xmetric,j , dE xhmean,j , dE xhmax,j , dE xmelod,j ) dE Xj (melod) = (dE xmelod,j , dE xhmean,j , dE xhmax,j , dE xmetric,j ) d2E Xj (harmo) = (d2E xhmean,j , d2E xhmax,j , d2E xmetric,j , d2E xmelod,j ) d2E Xj (metric) = (d2E xmetric,j , d2E xhmean,j , d2E xhmax,j , d2E xmelod,j ) d2E Xj (melod) = (d2E xmelod,j , d2E xhmean,j , d2E xhmax,j , d2E xmetric,j ) and the n × 10 matrix Xadd = (Xrit , Xsus , Xf erm ) where Xrit = (xrit1 , xrit2 , xrit3 , xrit4 ), Xsus = (xsus1 , xsus2 , xsus3 , xsus4 ) and Xf erm = (xf erm1 , xf erm2 ).
44.1. THE BERAN OPERATOR
879
Finally define the n × p matrices (with p = 58) X(harmo) = (X1 (harmo), X2 (harmo), X3 (harmo), X4 (harmo), dE X1 (harmo), dE X2 (harmo), dE X3 (harmo), dE X4 (harmo), d2E X1 (harmo), d2E X2 (harmo), d2E X3 (harmo), d2E X4 (harmo), Xadd ), X(metric) = (X1 (metric), X2 (metric), X3 (metric), X4 (metric), dE X1 (metric), dE X2 (metric), dE X3 (metric), dE X4 (metric), d2E X1 (metric), d2E X2 (metric), d2E X3 (metric), d2E X4 (metric), Xadd ), X(melod) = (X1 (melod), X2 (melod), X3 (melod), X4 (melod), dE X1 (melod), dE X2 (melod), dE X3 (melod), dE X4 (melod), 2 dE X1 (melod), d2E X2 (melod), d2E X3 (melod), d2E X4 (melod), Xadd ). 5. Orthonormalization. Each of the design matrices X(metric), X(harmo), and X(melod) turned out to be singular, since the last column can be expressed as a linear combination of the previous ones. Hence, we omit the last column. For simplicity of notation, the new n × 57 matrices will also be denoted by X(metric), X(harmo), X(melod). Figure 44.1 shows that the corresponding columns of the three matrices are closely related, at least for j = 1, 2, 3. Intuitively this means that it is not possible to distinguish exactly whether certain characteristics of the tempo curve stem from the metric, the harmonic or the melodic analysis. Thus, results may depend on the sequence of orthogonalization. This sequence reflects whether, in our view, the harmonic, the metric or the melodic has priority. Moreover, instead of focusing on names such as “metric weight”, “first derivative of the metric weight”, etc., we will also try to extract typical weight curves (canonical curves) that appear to be important for the tempo, independently of which of the three analytic approaches (metric, harmonic, melodic) have priority. More specifically, the three design matrices are defined in the following way: The columns of X(harmo), X(metric), and X(melod) respectively are orthogonalized and standardized successively. We thus obtain three n × 57 matrices which will be denoted by Z(harmo), Z(metric), and Z(harmo). Each of these matrices has orthonormal columns. The reason for computing three different matrices is that orthonormalization depends on the initial sequence of the columns. An artificial preference of the variables that are accidentally in the first (or first few) column(s) is avoided by carrying out three separate regression analyses with the respective matrices Z(harmo), Z(metric), and Z(melod), and by comparing the common features of the three results.
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CHAPTER 44. DIFFERENTIAL OPERATORS AND REGRESSION
x_{metric,2}, x_{hmean,2} and x_{melod,2} 10
x_{metric,1}, x_{hmean,1} and x_{melod,1}
x_{metric,2} x_{hmean,2} x_{melod,2}
-2
0
0
2
x
x
4
5
6
8
10
x_{metric,1} x_{hmean,1} x_{melod,1}
0
5
10
15
20
25
30
0
5
10
15
20
25
30
onset time
onset time
x_{metric,3}, x_{hmean,3} and x_{melod,3}
x_{metric,4}, x_{hmean,4} and x_{melod,4} 10
x_{metric,4} x_{hmean,4} x_{melod,4}
x -2
0
0
2
2
4
x
4
6
6
8
8
x_{metric,3} x_{hmean,3} x_{melod,3}
0
5
10
15
20
25
30
0
onset time
5
10
15
20
25
30
onset time
Figure 44.1: The four hierarchical levels of metrical, harmonic, and melodic analyses.
44.2
The Method of Regression Analysis
Summary. Inspired by visual comparison of logarithmic tempo cures, where strong similarities between the different performances become visible, we applied the following regression model. –Σ–
44.2.1
The Full Model
Let Z be one of the three matrices Z(harmo), Z(metric), or Z(melod), respectively. The full (i.e., biggest possible) model for the j th individual tempo curve is y(ti , j) = Zi β(j) + (ti , j), (ti ∈ T ),
44.3. THE RESULTS OF REGRESSION ANALYSIS
881
where Zi is the ith row vector of Z, β(j) = (β1 (j), . . . β57 (j))t and (ti , j) (ti ∈ T ) are (for each fixed j) identically distributed zero mean random variables. This means that we assume each performance to be essentially characterized by a 57-dimensional parameter vector β(j). Under the present orthonormalized conditions, this vector corresponds to the shaping vector ω introduced in section 44.1.1. Note that we do not assume the residuals i to be independent, since corrected p-values are used that take into account serial dependence. Also, due to standardization of y and of the columns of Z, there is no intercept in the model. The vector β(j) is the parameter vector corresponding to the performance number j. Therefore, β(j) is assumed to be a random vector, sampled from the space of all “possible” interpretations, with expected value E[β(j)] = β. We then may write β(j) = β + η(j) where η(j) is a random vector with E[η(j)] = 0 and y(ti , j) = Zi β + Zi η(j) + (ti , j). Intuitively this means that, up to a small unexplained deviation (ti , j), the (logarithmic) tempo of the j th performance at onset time ti can be expressed as a “mean performance” Zi β plus an individual deviation from the mean that is equal to Zi η(j). For the mean tempo curve y¯(ti ) = m−1
m X
y(ti , j),
j=1
we then have y¯(ti ) = Zi β + ˜i (ti ) where ˜(ti ) (ti ∈ T ) are identically distributed zero mean random variables.
44.2.2
Step Forward Selection
In the following, the main focus is on the individual curves. Some comments on the mean curve are also given. In order to decide which components of β or β(j) respectively are not zero (i.e., which explanatory variables contribute “significantly” to the tempo curve), stepwise forward selection [84] is carried out with F-to-enter level of significance 0.01. For the individual curves, a separate stepwise regression is carried out for each individual. The statistics software S-Plus [467] was used for the calculations.
44.3
The Results of Regression Analysis
Summary. In the following discussion the main questions are: 1) Is there a relevant association between the analytical weights computed from the score and the observed tempo curves? 2) How complex is the relationship? 3) Are there commonalities and diversities; how can they be characterized? –Σ–
882
44.3.1
CHAPTER 44. DIFFERENTIAL OPERATORS AND REGRESSION
Relations between Tempo and Analysis
Summary. The statistical analysis of the relation of tempo data and analytical weights via the Beran operator is carried out. –Σ–
0 -1 -2 -3 -4 -5
standardized log(tempo)
1
Mean log-tempo curve and LS-fit with Z(harmo), F-alpha= 0.01
5
10
15
20
25
30
25
30
25
30
onset time
0 -1 -2 -3 -4 -5
standardized log(tempo)
1
Mean log-tempo curve and LS-fit with Z(metric), F-alpha= 0.01
5
10
15
20
onset time
0 -1 -2 -3 -4 -5
standardized log(tempo)
1
Mean log-tempo curve and LS-fit with Z(melod), F-alpha= 0.01
5
10
15
20
onset time
Figure 44.2: Mean tempo curve and LS-fit with orthonormalized Z matrices with harmonic, metric, and melodic emphasis. To begin with, it is interesting to learn how much can be ‘explained’ at most by the analytical weights. Recall that in regression, R2 denotes the proportion of the variability of y that is explained by the estimated regression function. Ideally, R2 would be equal to 1.00 which would mean that the (log-)tempo curve can be expressed exactly as a linear function of the analytic information encoded by the design matrix Z. Such an exact correspondence between the analytic curves and each tempo curve can hardly be expected. The maximal achievable values
44.3. THE RESULTS OF REGRESSION ANALYSIS
883
of R2 , which are obtained by using the full matrix Z (i.e., without eliminating nonsignificant variables), are quite high however for the mean curve as well as the individual curves. For the mean curve, R2 is equal to 0.84. For the individual curves, we have 0.65 ≤ R2 ≤ 0.85. It is well known that R2 can be increased by simply including a sufficiently large number of explanatory variables, even if these variables have nothing to do with y. It is therefore necessary to investigate which explanatory variables contribute significantly to the response y. To do this, we first applied stepwise forward selection with F-to-enter α = 0.01. In all cases, all coefficients in the resulting model turned out to be significantly different from zero at the 5% level of significance, even after taking into account the possibility of serial correlations in the residuals. The values of R2 for the mean curve and the individual curves respectively are still remarkably high: For the mean curve, R2 is equal to 0.79 for Z(harmo), 0.79 for Z(metric) and 0.77 for Z(melod). Figure 44.2 shows that the fit (dotted line) to the mean tempo curve (full line) is very good, even if only significant coefficients are used. For the individual curves, we have 0.46 ≤ R2 ≤ 0.79 for Z(harmo), 0.48 ≤ R2 ≤ 0.78 for Z(metric) and 0.36 ≤ R2 ≤ 0.77 for Z(melod). The low value of 0.36 is obtained for Kubalek. Excluding this performance, we have 0.51 as the lower bound Z(melod). Thus, the quality of the fit is also good in general, but varies individually. This is illustrated further by figure 44.3 to 44.3 for Z(melod). The results on commonalities and diversities given below in section ?? yield further evidence for the existence of a meaningful association between y and Z.
44.3.2
Complex Relationships
Summary. We discuss the complexity of the relationships between the individual results.
–Σ–
Even when using only significant coefficients, the estimated models are very complex. As an example, consider the performance by Brendel. With Z(melod), the R2 is in this case equal to 0.76. No. 4.4 in figure 44.3-44.6 confirms the good fit. The following table A summarizes the result:
Table A: Coefficients of explanatory variables chosen by stepwise forward selection with F-toenter=0.01 and Z(melod), for the logarithmic tempo curve of Brendel. (The P -values given here do not take into account serial correlations.)
884
CHAPTER 44. DIFFERENTIAL OPERATORS AND REGRESSION z−variable
est. coefficient
std. error
t-statistic
P-value
zmelod,1 zhmean,1 zhmean,2 zhmax,2 zmelod,3 zhmean,3 zhmax,3 zmelod,4 dE zmelod,1 dE zmelod,2 dE zhmean,2 dE zmelod,3 dE zhmean,3 d2E zmetric,1 zrit3 zrit4 zf erm1
-0.3136 -0.2737 0.2781 0.2659 -0.1258 0.1303 -0.2800 -0.1562 -0.2663 0.1567 0.1143 0.2938 0.1308 -0.1361 -0.1952 0.1032 -0.1425
0.0353 0.0353 0.0353 0.0353 0.0353 0.0353 0.0353 0.0353 0.0353 0.0353 0.0353 0.0353 0.0353 0.0353 0.0353 0.0353 0.0353
-8.8760 -7.7443 7.8713 7.5248 -3.5597 3.6882 -7.9244 -4.4193 -7.5371 4.4337 3.2356 8.3143 3.7024 -3.8508 -5.5249 2.9208 -4.0331
0.0000 0.0000 0.0000 0.0000 0.0005 0.0003 0.0000 0.0000 0.0000 0.0000 0.0014 0.0000 0.0003 0.0002 0.0000 0.0039 0.0001
The number of significant coefficients is very large. The model contains all four weight functions, first and second derivatives of various degrees of smoothness and also two ritardandovariables and one suspension. Also note that all degrees of smoothness are used. Formally, even after adjustment for serial correlations, all p-values are below 0.05. (As a cautionary remark, it should be noted however that p-values obtained after model selection can be used as guidelines only.)
44.3.3
Commonalities and Diversities
Summary. In spite of the high complexity of the selected models for 28 individual tempo curves, there are interesting commonalities and diversities. They are characterized in this section. –Σ– Before going into the details of commonalities and diversities among the 28 given performances, we should make a remark on the performance selection and the performers as made available by Repp [438]. Above all, we should emphasize that Repp succeeds in a choice of first quality pianists, among others the celebrated “romantic virtuoso” Vladimir Horowitz, the “analytical mannerist” Alfred Brendel, or the “perfect but utterly cool” omnipresent Vladimir Ashkenazy, to name just three of them3 . So from the point of view of performance culture, 3 For a complete list of all performers, see [438]. We shall only name selected performers who are relevant to this analysis.
44.3. THE RESULTS OF REGRESSION ANALYSIS
885
4.2 ARRAU
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4.3 ASHKENAZY
4.4 BRENDEL
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4.5 BUNIN
4.6 CAPOVA
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4.7 CORTOT1
4.8 CORTOT2
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Figure 44.3: Fit for Z(melod). the election is unprecedented and representative. But in this scientific context, we must refrain from further judgments. This discussion is not about journalistic criticism. However, we should encourage critics to review their understanding of performance by focusing on the question whether and to what degree the analytical structure of a score may be responsible for agogical expressivity. Our present answer to this question—incomplete as it must remain—may seem to position some of the artists in unexpected relative position to each other. But this is not surprising since we do not claim that the overall judgment from common criticism really does represent the strictly analytical perspective of our approach. As mentioned in chapter 36, one should absolutely add emotional and gestural components to reach a complete description, an objective which was out of reach of this discussion. In spite of the high complexity of the selected models for individual tempo curves, there turn out to be interesting commonalities and diversities that can be characterized by either of the matrices Z(harmo), Z(metric) and Z(melod) respectively. Recall that using Z(harmo)
886
CHAPTER 44. DIFFERENTIAL OPERATORS AND REGRESSION 4.10 CURZON
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4.11 DAVIES
4.12 DEMUS
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4.13 ESCHENBACH
4.14 GIANOLI
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4.15 HOROWITZ1
4.16 HOROWITZ2
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Figure 44.4: (Cont.)
corresponds to an understanding of the score that put a first priority on the harmonic structure. Using Z(metric) corresponds to putting first priority on the metric structure. Using Z(melod) corresponds to putting first priority on the melodic structure. Therefore, depending on which of the three matrices is used, somewhat different results should be expected. The fundamental problem is the ambiguity of a performance. In general, based on one performance, it cannot be decided with certainty whether certain features of the tempo are ‘due to’ the harmonic, the metric or the melodic content. Nevertheless, the results below show a strong similarity between the three regressive analyses. Thus, there appears to exist at least a core of tempo features that are unambiguously attributable to specific weight functions. A number of different aspects of commonality and diversity can be considered. Here, three possible aspects are described.
44.3. THE RESULTS OF REGRESSION ANALYSIS 4.18 KATSARIS
-6
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4.17 HOROWITZ3
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4.19 KLIEN
4.20 KRUST
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4.21 KUBALEK
4.22 MOISEIWITSCH
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4.23 NEY
4.24 NOVAES
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Figure 44.5: (Cont.) 44.3.3.1
Signs of Coefficients
We ask the following question: For which k = 1, . . . p do we have either βˆk (j) ≥ 0 for all 1 ≤ j ≤ m, or βˆk (j) ≤ 0 for all 1 ≤ j ≤ m? In other words, which coefficients have the same sign for all performances? The result is quite amazing: For Z(metric), Z(harmo), and Z(melod), all except 3, 2, and 1 coefficients (out of 57) respectively have the same sign for all performances. Thus, the sign of the coefficients is a very strong commonality. The analytic curves ‘act’ in the same direction. In particular, the following general tendency can be observed: • The tempo decreases as the original (not orthogonalized) harmonic weight increases. • The tempo increases as the original (not orthogonalized) metric weight increases. • The tempo decreases as the original (not orthogonalized) melodic weight increases.
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CHAPTER 44. DIFFERENTIAL OPERATORS AND REGRESSION
4.26 SCHNABEL
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4.27 SHELLEY
4.28 ZAK
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Figure 44.6: (Cont.)
It should be noted however that these conclusions are valid under the assumption that all other variables are kept fixed. As we saw above, for the score considered here, the original weights are strongly correlated. This makes the actual relationship between weights and tempo much more complicated. 44.3.3.2
Frequency of Variable Inclusion Pm ˆ For k = 1, ..., p, let nk = j=1 1{βk (j)6=0} be the number of performances for which the explanatory variable number k was included in the model. Figures 44.7-44.9 show, for Z(harmo), Z(metric), and Z(melod) respectively, the curves of variables that were chosen at least 24 times (out of 28). The curves are multiplied by the sign of the coefficient. At least two types of curves are common to practically all performances, independently of the matrix that is used: 1) very smooth ‘global’ curves, such as zmelod,1 , that shape the overall tendency of the tempo; 2) almost periodic curves, with a period of about four measures, corresponding to the approximate periodicity of the harmonic curve zhmean,2 . Comment on Z(harmo). Note that zhmean,1 is identical with xhmean,1 , see figure 44.1a. Moreover, zhmean,2 is almost the same as xhmean,2 in figure 44.1b. Also, dE zhmean,3 exhibits features that are very similar to xhmean,3 , see figure 44.1c. Thus, analytical weights obtained by local averaging without orthogonalization have a direct impact on the performance. In fact, by the above, the orthonormal curves selected as most relevant by the regression turn out to be closely related to the original curves. Comment on Z(metric). Here, resemblance of curves is similar to Z(harmo). Namely:
44.3. THE RESULTS OF REGRESSION ANALYSIS
Z(harmo): z_{metric,1} chosen 28 times
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Z(harmo): z_{hmean,1} chosen 28 times
889
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Z(harmo): z_{melod,1} chosen 28 times
Z(harmo): d_E z_{hmean,3} chosen 27 times
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Z(harmo): z_{hmean,2} chosen 25 times
Z(harmo): d_E z_{metric,1} chosen 24 times
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Figure 44.7: Frequently selected curves for Z(harmo). zhmean,1 is almost the same as xhmean,1 , zmetric,1 and xmetric,1 are identical, and zhmean,2 is very similar to xhmean,2 . Also zmelod,1 is almost the same as zmelod,1 in Z(harmo), compare figures 44.7 and 44.8. Again, we conclude that the original averaged weight curves influence the performance directly. Comment on Z(melod). Similar comments as for Z(harmo) and Z(metric) apply for Z(melod). Here, zmelod,1 is the mirror image of xmelod,1 . Further, zhmean,1 and zhmean,2 are very similar to xhmean,1 and xhmean,2 , respectively. Comment on non-linear deformations. The melodic weights seem to play the most prominent role. Independently of the emphasis, zmelod,1 is chosen for all 28 performances. For the melodic emphasis, zmelod,1 is obviously identical with xmelod,1 . For the harmonic and metrical emphasis, the corresponding zmelod,1 -curves turn out to be non-linear deformations of xmelod,1 . The method of non-linear deformations of analytical weights as arguments of refined shaping of performance is also implemented in the PerformanceRUBETTEr of RUBATOr .
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CHAPTER 44. DIFFERENTIAL OPERATORS AND REGRESSION Z(metric): z_{melod,1} chosen 28 times
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Z(metric): d_E z_{metric,3} chosen 28 times
Z(metric): z_{metric,1} chosen 26 times
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Figure 44.8: Frequently selected curves for Z(metric). In summary, we conclude that for the score of “Tr¨aumerei”, there is a small number of “canonical” analytical weight curves that are relevant for most performances and essentially do not depend on the analytical emphasis. 44.3.3.3
Largest Coefficients
Since the design matrix Z is orthonormal, the importance of the k th explanatory variable may be assessed by the absolute value of the corresponding k th estimated coefficient (ranked in comparison to the other coefficients). (Also note that, due to orthonormality, all estimated slope components are uncorrelated and their standard deviations are the same.) For fixed j, let rk (j) be the rank of |βˆk (j)| among all coefficients βˆs (j) (s = 1, ..., p). Furthermore, for 1 ≤ l ≤ p, Pm let fk (l) = j=1 1{rk (j) > p − l}. Thus, fk (l) is the number of performances for which |βˆk (j)| is at least the lth largest. Consider first l = 1. Thus, fk (1) is the number of performances for which the k th variable
44.3. THE RESULTS OF REGRESSION ANALYSIS
Z(melod): d_E z_{melod,3} chosen 28 times
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Z(melod): z_{hmean,2} chosen 27 times
Z(melod): z_{hmean,1} chosen 25 times
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Figure 44.9: Frequently selected curves for Z(melod). is most important. It turns out that fk (1) is not zero for a very small set of variables. The results for the three matrices are: Z(harmo) : figure 44.10 displays the four curves for which fk (1)6=0, i.e., variables that are most important for at least one performance. Table B shows the clusters of columns H1 to H4 of performances for which the corresponding variables zhmean,2 (∼ H1), zhmean,1 , zmetric,1 , and zmelod,4 are the most important ones. Thus, using Z(harmo) and therefore an approach that gives priority to the harmonic structure, the first cluster of performances has the 4-measures periodicity of the harmonic structure as the dominating feature. In particular, all Cortot performances are included. For the second cluster that includes in particular Horowitz1 and Horowitz2, the more global shaping curve zhmean,1 is most prominent. For Bunin and Gianoli, a global curve with a peak around the 15th measure is most important. Finally, for the first performance by Horowitz1, the very detailed local structure of the melodic curve zmelod,4 seems to dominate.
892
CHAPTER 44. DIFFERENTIAL OPERATORS AND REGRESSION For Z(harmo), z_{hmean,1} has f_k(1) = 10
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For Z(harmo), z_{metric,1} has f_k(1) = 2
For Z(harmo), z_{melod,4} has f_k(1) = 1
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Figure 44.10: Important variables for Z(harmo).
Table B: Overview of clusters as derived by the above criterion with l = 1.
25
30
44.3. THE RESULTS OF REGRESSION ANALYSIS Artist ARG ARR ASH BRE BUN CAP CO1 CO2 CO3 CUR DAV DEM ESC GIA HO1 HO2 HO3 KAT KLI KRU KUB MOI NEY NOV ORT SCH SHE ZAK
H1
• • • • • • • • • •
• • •
• •
H2
•
H3
H4
M1
M2
• • • • • • • • • • • • •
•
• •
• • •
M5
M6
• •
•
• • • • • • • • • •
•
•
ML2
• •
•
•
•
ML1
• • •
•
• • • • •
M4
• •
• •
•
M3
893
•
•
• • • • • • • •
ML3
•
• • •
•
Z(metric) : The clusters of columns M1 to M6 in Table B show the results for Z(metric) (see figure 44.11). These six columns correspond to the variables zhmean,1 , zmetric,2 , zmetric,1 , zhmean,2 , zhmax,3 , and dE zmetric,3 . Hence, by use of an approach that gives priority to the metric structure, the cluster with Horowitz2 and 3 is exactly the same as for Z(harmo), with the exception of Ashkenazy who
894
CHAPTER 44. DIFFERENTIAL OPERATORS AND REGRESSION For Z(metric), z_{metric,2} has f_k(1) = 7
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For Z(metric), z_{metric,1} has f_k(1) = 4
For Z(metric), z_{hmean,2} has f_k(1) = 4
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For Z(metric), z_{hmax,3} has f_k(1) = 1
For Z(metric), d_E z_{metric,3} has f_k(1) = 1
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Figure 44.11: Important variables for Z(metric). was not included before. Also, it corresponds essentially to the same curve. Similarly, the cluster with the three Cortot performances corresponds to a very similar curve. It is however smaller since it is a proper subset of the previous ‘Cortot’ cluster above. Bunin and Gianoli are again in the same separate cluster, this time together with Capova and Kubalek. The peak of the curve is now around measure 20. The cluster with Argerich contains several of those performances that were previously in the Cortot cluster. The dominating curve is still almost periodic with a period of about four measures. Finally note that Horowitz1 builds again a separate cluster with a locally very refined metric curve corresponding to the derivative of zmetric,3 . Z(melod) : For Z(melod) (figure 44.12), we obtain the clusters shown in columns ML1 to ML3 in Table B. The corresponding variables are: zmelod,1 , zhmean,2 , and zmelod,2 . The melodic approach yields very simple clusters. For almost all performances, including all Horowitz performances, the global shape of zmelod,1 is the most important feature. For Cortot1 through 3, and Krust, the 4-measures periodicity of zhmean,2 is most important. For
44.3. THE RESULTS OF REGRESSION ANALYSIS
For Z(melod), z_{hmean,2} has f_k(1) = 4
1 0
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For Z(melod), z_{melod,1} has f_k(1) = 23
895
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For Z(melod), z_{melod,2} has f_k(1) = 1
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Figure 44.12: Important variables for Z(melod).
Ashkenazy, the 4-measures (almost) periodic curve zmelod,2 is dominating. Note in particular that the minima and maxima of this curve do not occur at the same place as for zhmean,2 . In comparison, zmelod,2 appears to be shifted to the right. Also, zmelod,2 has an extreme local minimum around the beginning of measure 30. 44.3.3.4
Argerich “Versus” Horowitz
The remarkable first performance Horowitz1 from 1947 evidences a preference of very detailed local information, be it from the melodic or metrical analysis—in typical contrast to the highly coherent Argerich performance. This observation is confirmed by an investigation of the correlation coefficients in the algebro-geometric analysis of the performance genealogy in the sense of RUBATOr ’s stemma theory, see chapter 38. When translated into common language these quantitative results are in perfect coincidence with the judgments of experts on Argerich’s and Horowitz’ specific differences in performance [42]. Let us therefore make these findings more meaningful to the common understanding. How would an interpreter such as Horowitz experience his performance? He would look up a few notes ahead and remember just a few of the past note events when hitting a couple of keys in a given moment. He would then realize a couple of neighboring analytical facts, such as a harmonic step or a melodic contour, in his imagination, and then shape the present note event in its tempo, dynamics, and articulation within this minor context, and thusly express his analytical consciousness. Metaphorically speaking this resembles a near-sighted man who can only see and recognize nearby objects. It is as if he had no significant memory of what was happening several bars ago, or of what will happen in the
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CHAPTER 44. DIFFERENTIAL OPERATORS AND REGRESSION
larger time span ahead. In contrast, a performer of the Argerich type would be aware of lots of long-range facts in the overall analytical stream of the piece which is being played. She would then remember and plan everything and therefore hit the present notes in full consciousness of what was and what will be. This is what the semantics of the data tells. Table C: Overview of clusters as derived by the above criterion with l = 3. Artist ARG ARR ASH BRE BUN CAP CO1 CO2 CO3 CUR DAV DEM ESC GIA HO1 HO2 HO3 KAT KLI KRU KUB MOI NEY NOV ORT SCH SHE ZAK
M1
H2
• • • • • • • • • • • • • • • • • • • • • • • • • • • • •
H1
•
• • • • • • • •
•
DM3
• • •
M2
•
•
•
• •
• • • • • • • • • • • • •
HM2
•
• • •
HM3
DDM1
SS2
•
• • •
•
•
DM1
•
•
•
M4
•
• •
• •
DM2
• •
•
•
•
•
Similar results can be obtained for l > 1. Here, fk (l) is the number of performances for
44.3. THE RESULTS OF REGRESSION ANALYSIS
897
which the k th variable is among the k most important ones. Consider, for instance, l = 3. The following partially overlapping clusters corresponding to variables with fk (3)6=0 are obtained (see Table C). The columns correspond to the following variables: M1 ∼ zmelod,1 , H2 ∼ zhmean,2 , H1 ∼ zhmean,1 , DM3 ∼ dE zmelod,3 , M2 ∼ zmelod,2 , DM2 ∼ dE zmelod,2 , HM2 ∼ zhmax,2 , HM3 ∼ zhmax,3 , M4 ∼ zmelod,4 , DM1 ∼ dE zmelod,1 , DDM1 ∼ d2E zmelod,1 , and SS2 ∼ zsus2 . The first cluster consists of all performances except Ashkenazy. Thus, using the ‘melodic approach’, apart from Ashkenazy, the global melodic curve zmelod,1 is one of the three most important factors for all tempo curves. Again, the 4-measure periodicity determines clusters with Cortot performances. It is remarkable that, in spite of the large number of overlapping clusters, there is no cluster—except the first one—that contains at the same time Cortot and Horowitz. Moreover, there is one cluster consisting solely of Horowitz1 through 3 corresponding to the complex local melodic structure of zmelod,4 . Evidently, many more detailed comments about figures 44.10-44.12 could be added. We conclude the analysis by noting that the relative size of the coefficients suggests a natural way of obtaining simplified tempo curves that contain the most important features. For given j and 1 ≤ q ≤ p, let the p × 1 vector γq (j) = [γq,1 (j), γq,2 (j), , ..., γq,p (j)]t be defined by γq,k (j) = βˆk (j)1{rk (j) > p − q}. Then yq (ti , j) = Zγq (j) is a simplified tempo curve that corresponds to using the variables (analytic curves) that are among the q most important ones for tempo curve j, importance being measured by rk (j). Thus, the resulting tempo curve is a simplified curve obtained superposition of the q most important features only. Note that, for q = p, this yields the complete curve fitted by stepwise regression. Nos. 44.13.1-44.16.28 in figures 44.13-44.16 display yq (q = 1, ..., p) for Z(melod) for all performances.
44.3.4
Overview of Statistical Results
The main statistical conclusions from the analysis above can be summarized as follows: • There is a clear association between metric, melodic and harmonic weights and the tempo. • The exact relationship between the analytic weights Z and an individual tempo curve is very complex. However, a large part of the complexity can be covered by our model. • Commonalities and diversities among tempo curves may be characterized by a relatively small number of curves. There is in principle no unique way of attributing features of the tempo to exactly one cause (harmonic, metric or melodic analysis). Which curves need to be used depends partially on which of the three analyses (harmonic, metric, melodic) has ‘priority’. However, there seems to be a small number of canonical curves that are essentially independent of the priorities and which determine a large part of the commonality and diversity among tempo curves. Natural clusters can be defined. • There is a natural way of reducing an individual tempo curve to a series of simplified tempo curves containing an increasing number of features. Overall, the proposed method yields a variety of results that are interpretable from the point of view of music and performance theory. In particular, the hierarchic approach of decomposing each of the weight functions into components of different degrees of smoothness seems to
898
CHAPTER 44. DIFFERENTIAL OPERATORS AND REGRESSION 11.2 ARRAU
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11.1 ARGERICH
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11.4 BRENDEL
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11.5 BUNIN
11.6 CAPOVA
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11.7 CORTOT1
11.8 CORTOT2
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Figure 44.13: (44.13-44.16) Superposition of the q most important features. be appropriate. The different choices of the bandwidth h correspond to a hierarchic approach to musical performance, starting with the most global features of the score and refining the performance successively in greater detail. The results here are closely related to Repp’s work [438]. Repp applied principal component analysis to the 28 tempo curves. One of his main results is that Cortot and Horowitz appear to represent two extreme types of performances. Thus, in a heuristic way, Repp suggested classifying the performances according to their factor loadings into a Cortot and a Horowitz cluster respectively. Our regression analysis confirms the basic findings. Due to the use of weights obtained from a musical analysis of the score, we obtain further information about the nature of the commonalities, diversities and clusters. For instance, as discussed above, for the Horowitz cluster, the most important feature appears to be the overall descending line of z1,melod and very local variations of the tempo that correspond to the local variations of the analytic weights. On the other hand, for the Cortot cluster, the up and down movement of z3,hmean with a period
44.3. THE RESULTS OF REGRESSION ANALYSIS 11.10 CURZON
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11.11 DAVIES
11.12 DEMUS
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11.13 ESCHENBACH
11.14 GIANOLI
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Figure 44.14: (Cont.) of about four measures is the most important characteristic. Result 6 More generally, this approach reveals a set of canonical curves whose combination yields the most important features of a tempo curve. It should be emphasized that these curves are score-specific. Thus, for each score (in our case “Tr¨ aumerei”), a new set of essential curves is obtained. The original weights as well as the decomposition into parts of different smoothness are based on the specific score that is performed. This is a crucial feature that is in sharp contrast to traditional mathematical ‘omnibus-decompositions’ such as provided by Fourier or wavelet analysis. The score-specific choice of the Z−matrix enables us to relate statistical results directly to the musical/analytic content of the score. The main point here is this: We argue that to understand the character of tempo, it is above all important to refer it to a “basis” of score specific analytical curves and not to curves—such as sinoidal curves in Fourier representation—which have a generic type that tells nothing about
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CHAPTER 44. DIFFERENTIAL OPERATORS AND REGRESSION 11.18 KATSARIS
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11.19 KLIEN
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11.21 KUBALEK
11.22 MOISEIWITSCH
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11.23 NEY
11.24 NOVAES
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Figure 44.15: (Cont.) the particular context of the genealogy for the given tempo curve.
11.26 SCHNABEL
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11.27 SHELLEY
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Figure 44.16: (Cont.)
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CHAPTER 44. DIFFERENTIAL OPERATORS AND REGRESSION
Part XII
Inverse Performance Theory
903
Chapter 45
Principles of Music Critique Never trust the artist. Trust the tale. The proper function of a critic is to save the tale from the artist who created it. David Herbert Lawrence (1885–1930) Summary. Inverse performance theory deals with the critic’s problem of how to extract in his critique what is hidden behind a performance output. To initiate this theory, we therefore should question the perspectives which a critic has to envisage. To begin with, we inquire the intriguing task of feuilletonistic critique: why is it a never ending story? Is it substantially necessary— beyond music business? We then position the critic within the sociological context: How do norms intervene in critique? This is exemplified by Glenn Gould’s performative redefinition of classics. The chapter terminates with an ethnomusicological view on historicistic performance as it is typically undertaken by Nicolas Harnoncourt. –Σ–
45.1
Boiling down Infinity—Is Feuilletonism Inevitable?
Summary. It is not accidental that music critique has stuck to feuilletonism. Its scope is an infinite one in several dimensions: It has to cope with the infinite interpretative work with respect to a given text and with the infinity of expressive nuances of each given interpretation. We analyze the necessity of boiling down this infinite challenge in view of the poor tools of traditional musicology. –Σ– We have learned in previous chapters, in particular in section 13.4.1 and section 32.2, fact 18, that performance conveys an infinite message. And that this infinity is a double one1 : that of interpretative perspectives as they are realized in music analysis, and that of performative 1 Leaving
aside the gestural and emotional rationales for the time being.
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shaping variants as they may be expressed on the infinitesimal vocabulary of performance fields. It is not clear whether critics are aware of such a variety of background that may produce concrete performances. In particular with respect to (analytical) interpretation, they preferredly stick to the traditional canon of how the structure of a composition should be viewed and interpreted or analyzed, respectively. Of course, it is not clear whether music critics should be cognizant of possibly new interpretations, but once they have gone into their business, a creative dealing with analytical problems should be mandatory. One may understand that this is not necessary ante rem, but after the event, a re-reading of the text should be considered, be it only for comparative handling of the present performance: Could it be that the artist discovered a new interpretation of the given text? In practice, the selection of an analytical interpretation (in the best case, autoincompetent critics excluded...) is just a matter of limitations of time, energy, and interest, besides ignorance of the infinite variety of interpretations. As to the infinity of performance nuances, this is beyond the vocabulary of music critics and it is also beyond the present measurement technology for such data: In a common concert, no performance field reconstruction is feasible. So critics are nolens volens limited to describing performance by use of common language expressions (“elegant diminuendo blended by a mysterious pedaling cloud...”) which beyond its imprecision cannot relate expression to interpretation. So is feuilletonism inevitable? Or rather: Is such a bad feuilletonism inevitable? Is it necessary to play the game of a unique “best” interpretation whose expression has to move along unreflected paths of prejudices? The alternative would be to embed one’s judgment in the potential infinity of analytical interpretation and expressive performance. And to keep this embedding omnipresent in the critical discourse. We argue that the most precious role of a music critic would be that of putting the infinity of perspectives on a musical work into evidence in every concert or CD review. These would be the crucial points: • Infinity of analytical interpretations, • infinity of expressive performances, • infinity of correlations between interpretative rationales and expressive performance shaping. And it is not the question about bad or good quality in these specifications, in the limit, the only quality is to teach us something about the work in question, and about the relativity of each perspective. Suggesting a boiled-down finitistic or even unicorned view of art is a destructive way of reduction and hinders every deepening or progress in the arts.
45.2
“Political Correctness” in Performance—Reviewing Gould
Summary. Given the infinity of critical understandings of artistic performance, norms are easily infiltrated against unlimited variation of expressivity. We make the point concrete with the example of Glenn Gould’s eccentric (ab-normal...) performance of classics from Bach to Beethoven. –Σ–
45.2. “POLITICAL CORRECTNESS” IN PERFORMANCE—REVIEWING GOULD
907
A testing ground for a valid music critique is Glenn Gould’s performance of classical works from Bach to Webern. His technically unprecedented performances have evoked strong reactions which unveil a number of limitations in common critique styles. Whereas Gould’s Bach performances may be non-conformist, but still acceptable and adequate for Bach’s compositions, his performances of Beethoven’s sonatas is beyond the supportable deviation from common taste. The famous critic Joachim Kaiser has described in [257] the most famous “mis-performance” of a Beethoven sonata on the example of Gould’s presentation of op. 57 “Appassionata”: Bei Goulds Wiedergabe des allegro assai d¨ urfte es sich um die verr¨ uckteste, eigensinnigste Darstellung handeln, die jemals ein Pianist einem Bethoven-Satz hat angedeihen lassen; und das will etwas heißen. Gould h¨ alt es f¨ ur richtig, demonstrativ langweilig und gelangweit den Kopfsatz so zu bieten, als ob ein Beethoven-Ver¨ achter seinen Plattenspieler nur mit halber Geschwindigkeit ablaufen ließe. Tranig langsam, langweilig und gelangweilt, die Triller w¨ ahrend des pp im Schneckentempo, w¨ ahrend der Fortissimo-Stellen etwas rascher, qu¨ alt sich die Musik vorbei. Man meint, der Pianist imitiere ein Kind, das mit erfrorenen Fingern die Appassionata vom Blatt spiele. Nur selten vergißt er dabei, daß er ja vergessen machen wollte, der genialische Glenn Gould zu sein. This critique is strongly based upon the commonly accepted reading of the Beethoven text as a passionate message which calls for temperament and stormy dynamics in performance, and not for analytical cool vivisection of such a vital piece of literature. In Kaiser’s characterization, Gould’s production is like a “child with frozen fingers in a sight-reading performance”. Here, the different and aberrant performance is incorrect, even forbidden. It is a norm which the politically incorrect Gould has broken and thus made the sonata ridiculous; Kaiser even comments that the sonata “remains silent” when confronted with such a misreading. The basic hypothesis behind such an outrageous indignation is that Kaiser knows what and when and how the sonata (which is personalized here) would have communicated, and that crazy Gould just destroyed that known and accepted messaging. Kaiser in fact evokes an installed performance grammar which requires a passionate forte seventh degree cascade towards the piano on the dominant in bars 14-16 of allegro assai. Instead, Gould descends like a noble, bored lady and snobbishly sits down on the boring dominant fermata. No passion whatsoever. The same, even more dramatic viz. ridiculous deformation can be observed in Gould’s performance of sonata op.106, “Hammerklavier”. This case is even worse since one just thinks that Gould did not understand a single word of the text, he simply was too stupid for the performance task. What happened? And why was Gould’s Bach so much more successful? Evidently, Gould’s microscopic performance method works for Bach, and not for Beethoven. Why does this microscopic view fascinate and illuminate Bach’s work whereas it virtually kills Beethoven’s sonatas? The point is that in Beethoven’s work, there is an inbuilt performance grammar which is not engraved in the score but stems from the performance tradition as such, an oral tradition so to speak, an element of rhetoric communication which transcends written code. Instead, Gould reads the same code from the Bach and from the Beethoven scores, and effectively demonstrates,
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that there is a huge defect in Beethoven’s written code; it is quite trivial, at least locally, the written script is simply boring. Gould has effectively given a quasi-mathematical demonstration that the same performance strategy cannot be applied to Bach and to Beethoven, that the same analytical insight and the same rhetoric shaping yield completely different results for these composers. To me, this is a sensational lesson to teach a characteristic difference between Bach and Beethoven. This is very clear in the descending seventh passage on bar 14th bar, which runs on semiquavers after a triggering triplet of quavers at the end of bar 13. Gould effectively takes the double temporal rate of the semiquavers with respect to the quaver triplet, without any tempo increase, without any dynamic profiling, just letting us see the anatomy of this triadic descent structure. The common reading [534] of this passage is that of an explosion:
Die Explosion (a tempo, Auftakt zu T.14) erfolgt im niedersausenden Dominantampft durch einen CArpeggio und f -Sextakkord (T.15), wird aber sogleich abged¨ Dur-Sextakkord, p, T.16.
With Gould, there is no explosion, just the written text, cleanly played, but antagonistic to any such musical drama to which an explosion would testify. The common reading in fact classifies this sonata as a musical drama, and asks interpreters to integrate this semantic into their performance. Gould plays the “Appassionata” minus the commonly implied drama. The question here is whether this dramatic character is implicit in the score structure or whether it is an external determinant which has been added by historical standards—which Gould filters away to lay bare what he believes is a poor structural essence [191]. So the question arises whether the commonly accepted dramatic performance is an expression of Beethoven’s work or of an added character. Let us therefore analyze the specific performative shape of the passage in question. To begin, its agogics is profiled against the temporal neighborhood, i.e., not only is the indication “a tempo” valid from the last three quavers of bar 13, but in bar 14, the resumed tempo is again increased. The dramatic performance contains an increase of tempo, and within that level, also an increase of tempo towards the middle of the descent. Further, the dynamics is not only the forte at the end of bar 13, but the target tones of each descending intervallic movement of the descent is played louder, maybe to a ff or sf. As a whole, this descent (with its added ascending tail in bar 15) is not only a musical structure, but more an explosive gesture whose very beginning goes to the top pitch, falls down and bounces back to the dominant fermata. This is not a written rationale, but it is a semantic unit which can easily be deduced if gestural semantics is to be included in the performance shaping. So Gould’s experiment would demonstrate that Beethoven requires gestural rationales beyond analytical ones. Meaning that Beethoven’s compositions have a performative added value of gestural nature which is not (yet?) virulent in Bach’s architectural music. Observe however that this gestural character is not on the level of the interpreter’s gestures, it is a rationale in the performance grammar, a semiotic layer which is added to the score system. Summarizing, Gould’s politically incorrect performance withdraws from the common dramatizing approach and gives us an insight to Beethoven which would not have occurred otherwise.
45.3. TRANSVERSAL ETHNOMUSICOLOGY
45.3
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Transversal Ethnomusicology
Summary. Ethnomusicology is essentially confronted with the synchronic normalization problem: Given two (simultaneously existing) cultural areas, understanding one of them from the other’s perspective means normalizing fundamental categories, e.g., the “score” concept, such that comparison of different ethnics becomes feasible. When dealing with performance of works which belong to distant epochs, the ethnomusicological problem of understanding a performance is restated as diachronic normalization: what is the common ground for understanding the performance of a work which was written in the spirit of another epoch? We discuss the efforts of Nicolas Harnoncourt in this direction of “transversal” ethnomusicology. –Σ– In its common understanding, ethnomusicology deals with a transformation problem between synchronic music cultures. Such a transformation may cause major problems to the contents and forms of music, for example because of incompatible notation or even incompatible modalities of communication, such as oral traditions, rebuilding instruments for each new musical event, embedding of music in more global forms of art, etc. But synchronic ethnomusicology has the undeniable advantage that a feedback process may help deal with such problems, and eventually solve them in the ideal case2 . This advantage cannot be claimed for diachronic ethnomusicology. What is this type of ethnomusicology? It deals with transformation of music cultures which are at a temporal distance instead of a geographic or social distance of contemporaneous cultures. For example, if we play an opera of the late Renaissance composer Claudio Monteverdi in the 20th century, this is a diachronic type of ethnomusicology. In fact, the cultures of Monteverdi’s time and of the 20th century are very different, and the communication between them is restricted to the historical proliferation channel. The historical distance influences strongly the communication of forms and contents. It is by no means automatic that everything is transmitted without loss of information. Above all, the socio-cultural background of a musical composition is easily blurred by historical filters. But also, the instrumental practice and technology impose dramatic deformations of what was reality in a historically distant context. Nicolas Harnoncourt has with great success restarted a dialog with historically distant traditions in the sense of understanding those conditions and not imposing ours to a diachronically distant culture. His approach is based upon the basic position that instrumental constraints are very important for shaping one’s performance and expressivity. Restriction gives one a clear frame, a limited field of activity where a composition must unfold its semantics, and not an unlimited Wagner-tailored orchestra, where the quasiinfinity of instrumental power and colors competes with and actually substitutes the efforts for better expressivity. All this intelligent effort does not solve the communication problem in the historical dimension: in contrast to synchronic ethnomusicology, diachronic communication is unidirectional: 2 However, the real case may be far from ideal. For example, if we want to initiate inverse ethnology, i.e., the review of our own music culture from the point of view of an ‘exotic’ culture, major obstacles will occur, from financial ones to the intolerance against another culture which tries to relativize the usual perspective of occidental supremacy.
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No real feedback from ancient times is possible, we only have the sources and must try to understand without really having answers to our experiments. Harnoncourt’s experiments may be an attempt to transform our performance practice back to Monteverdi’s, but it is not a demonstration of anything. So performance is not only a hic et nunc affair, but also a process which is coupled to spatio-temporal distances of cultures. Within this transformational framework, there are different serious asymmetries of communication. In the synchronic direction, feedback can be dealt with—however in the limits of sociocultural asymmetries for inverse ethnology. In the diachronic direction, the communication is intrinsically unidirectional, and performance remains a challenge for the adequacy of cultural transformation.
Chapter 46
Critical Fibers To see a World in a Grain of Sand, And a Heaven in a Wild Flower, Hold Infinity in the palm of your hand, And Eternity in an hour. William Blake (1757–1827) Summary. Stemma theory offers a model for a critical understanding of performance as a complex process between interpretation and the rhetorics of expressive performance grammar. We make the model and its limitations explicit. –Σ–
46.1
The Stemma Model of Critique
Summary. Modeling performance through stemma theory allows us to define inverse images of performances within a well-defined variety of stemmatic situations leading to the given performance. A critical fiber is such a variety. A critique is a choice of a point within this fiber. We discuss criteria to make such a choice, i.e., to select one critique among all possible critiques within the stemma model. –Σ– In section 32.4 we discussed the four global aspects of performance, in particular the fourth point: stemmatic deployment of performance. In that discussion, we stressed the fact that stemmata are not just a learning process but much more a logical unfolding of deformation strategies. If we take this point of view for granted1 , music critique should deal with the stemmata which could possibly lead to a given performance, since they would unveil the anatomy and genealogy of a performance, and this must be the central issue of a critique which merits that name. 1 Other models of performance genealogy are at hand, but we do not know of any such model that is technically as explicit as the stemma model.
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Probably the very construction of a stemma with its genealogical factors from cell hierarchies to performance operators and ramification architecture would help a critic to shape his/her style and criteria for judging performances. But beyond that, the general framework of stemma theory is far too generic for any concrete goal. What is the concrete goal? Principle 27 Try to construct a stemma that produces a given performance from the existing score! In this form, the problem has trivial solutions if we admit that any performance field is admitted without further articulation and operator constraints. The (non-trivial!) construction of a performance field from experimental performance data (e.g., on a MIDI file record) has been implemented by Stefan M¨ uller, see section 46.3, and can be used to construct a depth-one stemma whose operator just produces this performance field on a monolithic LPS. Such a solution is however not what a critical understanding of performance would preconize since this is nothing more than the brute ‘sampling’ of a performance transformation, and would not lay bare any of the semantically valid rationales. The crucial point is that one should impose constraints on the admitted tools for stemma construction. This would lead to the more reasonable Principle 28 Given a defined set of constraints regarding the construction tools for stemmata, try to construct a stemma that produces a given performance from the existing score! In this form, the principle becomes a challenge for critics since it thematizes the strategies a critic could imagine to be used by a specific artist. Only after such a strategic preset can the reconstruction of a possible generating stemma be seriously tackled. The lesson to be learned from such an ‘exercise’ is that in criticism, one should learn to reflect very cautiously the conditions under which critical judgments are made. The critical business now splits into two subtasks: comparative criticism of one and the same performance, and comparative criticism of a number of different performances of the given score. The first means that we are given a fixed performance and compare different reconstructions of backing stemmata: Which one is acceptable, which one is simpler but still adequate, in what respect could two criticisms be considered as being isomorphic, etc. The second one means that the phenomenological difference of performances is lifted to a genealogical difference of backing stemmata, and this one is the core activity of a music critic. However, it presupposes that usually, the constraints on the stemma reconstruction potential are the same, a condition for comparability which is not automatic.
46.2
Fibers for Locally Linear Grammars
Summary. Locally linear grammars are a special approach to stemma varieties, see section 39.8. We give a first description of critical fibers in this context. They turn out to be varieties in the sense of algebraic geometry and are called grammatical varieties. –Σ–
46.2. FIBERS FOR LOCALLY LINEAR GRAMMARS
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The general inverse problem of reconstructing a stemma for a given performance of a defined score is not solvable for two reasons: we first do not have a general formalism and second, even a reasonable general formalism would imply wild2 mathematical classification problems. We therefore want to consider a more tractable situation: the locally linear performance grammars which were introduced in section 39.8. We refer to that section and keep its notation, recalling in particular from definition 110 that a locally R-linear grammar is a family of R-linear representations of the stemma quiver T which are parametrized by R-vector spaces Bx of finite dimension sx , x ∈ V (T ), the vertex set of T , the parametrization being given by affine maps ϕx : Bx → End(Ax ). Within this very precise setup, where the stemma and the locally linear grammar are fixed, we may ask for the structure of the fibers lying above the performances which are defined on the stemma’s leaves by the given locally linear grammar. Technically speaking, we proceed as follows: For each final vertex z ∈ V (T ) let mz0 = r, mz1 , mz2 , . . . mznz = z, be the ordered sequence of elements of Mz (T ). For the root r and for each final vertex z ∈ V (T ) let us fix arbitrary er ∈ Ar , ez ∈ Az , respectively. The main task of this inverse performance theory is to study the set of solutions of the system of equalities ez
= m z −1 fmznz (fmznz −1 (. . . fmz1 (er , (cry1 ,mz1 ), (by1 )), . . .), (cynzn ), (bynz )) z ,mznz
(46.1)
where for k = 1, . . . nz the vertexes yk lie in the set Dmzk−1 (T ). For the explicit calculation, we select a basis vx1 , . . . vxsx of the vector space Bx . This means that for every vertex x ∈ V (T ) we may consider the linear operators Oxi := ϕ0x (vxi ) ∈ End(Ax ), i = 1, . . . sx as a part of our data. Whence, if we identify Bx with Csx through the basis vx1 , . . . vxsx , we can define the homomorphism (39.50) as a triaffine map Y
fx : Am × C#Dm (T ) ×
Csyy → Ax
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sy X
cm y,x byj iy,x (Oyj (rm,y (am ))) +
y∈Dm (T ) j=1
X
t cm y,x iy,x (ϕy (rm,y (am ))).
y∈Dm (T )
The equations (46.1) become: ez =
(46.2) X
m z −1 cynzn b z ,mznz ynz jnz
· · · crmy1 ,z1 by1 j1 mz,y1 ,...ynz ,j1 ,...jnz
+
... + X m z −1 cynzn · · · cry1 ,mz1 mz,y1 ,...ynz z ,mznz 2 In algebra, a classification problem is said to be wild, if its solution would imply the classification of any module category.
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where the sum is over all y1 ∈ Dr (T ), y2 ∈ Dmz1 (T ), . . . ynz ∈ Dmznz −1 (T ) and all jk = 1, . . . syk , for k = 1, . . . nz . The leading vector summand of this linear combination equals mz,y1 ,···ynz ,j1 ,...jnz = iynz ,mznz (Oynz jnz (rmnz −1 ,ynz (. . . iy1 ,mz1 (Oy1 j1 (rr,y1 (er ))) . . .))).
(46.3)
The general vector summand refers to a choice of endomorphisms X... = Oynz jnz or X... = ϕtynz etc. and is equal to iynz ,mznz (X... (rmnz −1 ,ynz (. . . iy1 ,mz1 (X... (rr,y1 (er ))) . . .))).
(46.4)
In order to describe the solution variety we can interpret (46.2) as a system of linear equations with parameters, m
X
ez =
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cy,znz −1 lyz .
(46.5)
(T ) −1 m
Observe that ez , lyz are vectors in Az , so the solutions cy,znz −1 of equations (46.5) result from simultaneous solutions in the vectors’ coordinates if we are given a basis of each Az . The coefficients lyz equal linear combinations of the vector summands (46.3) and (46.4) with coefficients which are monomials in the remaining variables c..,. and the b.. . m If we now assume that all variables czynz −1 are independent, we see that for any two different z ∈ Dmnz −1 (T ), the solutions of the equations (46.5) do not interact with each other. This means that the solutions of the system (46.5) for z ∈ Dmnz −1 (T ) is either empty (if the value ez is not in the image of the linear map (46.5)) or equals Y z∈Dmn
z −1
Lz ⊆ (T )
Y z∈Dmn
z −1
C#Dmnz −1 (T ) (T )
where Lz are linear subspaces of C#Dmnz −1 (T ) . Their codimension can be read from the matrix zk of the coefficients. Since there is no other condition on the other variables cm y,x we obtain altogether the following result: Theorem 35 The solution space of (46.2) is a linear fibration over some appropriate affine space. Corollary 22 The dimension of the non-empty fibers of this fibration is generically minimal, and it increases along some finite union of proper algebraic subschemes (the loci where the minors of the coefficients’ matrices vanish). This means that the non-empty fibers F ib(e. ), i.e., the solution spaces defined by (46.1) over a given output set e. = (ez )z are all generically isomorphic, i.e., isomorphic when restricted to appropriate open subschemes. However, the configuration of the specialization subschemes is not evident and depends on the particular vector summands. Also, the condition for non-empty fibers is not evident in general.
46.2. FIBERS FOR LOCALLY LINEAR GRAMMARS
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How can we interpret this result in musicological terms? A quantitative measurement of a performance can be done by recording the values of the parameters that characterize a note, i.e., loudness, duration, and so on. In the last paragraph we have denoted this data by the vectors ez . Equation (46.2) explains that in order to produce given values of the parameters from the zk weight functions, one has to find suitable values for all cm y,x . It would be nice if the choice of these coefficients could be small, because this would mean that in order to produce a given performance one does not have a lot of freedom in the choice of system parameters. However from this model this is not the case. In fact, corollary 22 tells us that for a fixed performance one has either no one or else infinitely many possibilities to choose system parameters which produce the given performance. Moreover, we learn that all performances with non-empty fibers F ib(e. ) have open dense subsets U (e. ) ⊂ F ib(e. ) which are all isomorphic with each other, so these generic subsets U (e. ) are qualitatively equivalent, see also figure 46.1. In other words, the non-empty fibers only differ on their special loci apart from generic open subschemes U (e. ). Lie operator parameters: weights, directions
Output fields e. Affine transport parameters
Fib(e.)
Figure 46.1: All performances with non-empty fibers F ib(e. ) have open dense subsets U (e. ) ⊂ F ib(e. ) (elliptic regions) which are all isomorphic with each other In musical terms, a fiber F ib(e. ) could be called a critical fiber because its points are the possible background parameters—in the present stemmatic model—which lead to the given performance output data e. , such as local tempi, articulations, dynamics, detunings, etc. So the fiber really includes the possible ways of understanding why a performer is playing his actual performance. In fact, finding out which parameters the interpreter could have used is (or should be) the core activity of a music critic. Having generically isomorphic fibers means that in any
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two critical fibers of two given performances, there are “dominant” open sets of “criticisms” (i.e., points in the fiber!) which are isomorphic with each other. This does not mean that the criticisms are the same for the two given performances, but that their structural contexts can be identified. Again, this does not mean that the relevant criticisms may be identified—on the contrary: maybe, these isomorphic contexts just describe the criticisms which are what everybody could say if no supplementary information about the specific performance culture of the interpreter is known. So we should not discard this model as insignificant, instead we need to look at it in the right way. In fact, it is like pretending to explain the geometry of smooth plane curves, just using lines. The theory of lines can be useful for local questions, but not for global ones. The same thing happens here. The great flexibility of this model enables us to adapt it to the questions that one tries to answer, but they can not be too general. Indeed, one can select subschemes that are appropriate for the study of a particular problem. For example, if one is interested in comparing several performers under the point of view of the local/global way of playing, one could try to restrict the research to the level of the daughters of the root, and to use some more of the structures of the vector space Ax (if any). Another question that could be asked is that of the final/causal way of playing. This requires one to impose mutual dependence conditions zk upon the variables cm y,x for varying z ∈ Dmzk (see 46.4).
46.3
Algorithmic Extraction of Performance Fields
Summary. The algorithmic extraction of performance fields is a first step for systematic calculations in inverse performance theory. The extracted fields can further be used for visualization. We describe an approach how performance fields can be calculated from given scores and performances and present the tool that implements the theory. –Σ– In this section, we address the question of inverse performance theory which deals with the reconstruction of a performance field for a given performance on a determined score, including an implementation in the RUBATOr framework, named EspressoRUBETTEr . This question generalizes the well-known problem of constructing a tempo curve from a measured performance to form space S = EHLD... of parameters such as onset E, pitch H, loudness L, etc. Such an attempt is first of all an interpolation task where a continuous performance field must be reconstructed from a discrete data set of performed notes. As such it is subjected to (a) the problem of matching symbolic and performance events, (b) ambiguities in the local definition of field vectors, (c) algorithmic constraints for real-time objectives, and (d) visualization options for performance fields.
46.3.1
The Infinitesimal View on Expression
The concept of expression is ambiguous as far as the content of the expression and its reality layer are not a priori clear, see chapter 36 for the details. If we aim at analyzing expression, this does not regard the psychological perception of a performance by humans. The psychological aspect is a legitimate one, but it touches a category which relates the performed music to human
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categorizations in terms of emotional response. Such a perspective is, for example, dealt with in [231] or in [289]. In contrast, our point of view is expression as a rhetorically shaped transfer of structural score contents by means of the deformation mapping of symbolic data into a physical parameter space. The psychological implications are not the subject of this perspective, it is a purely mathematical description of this mapping, not of the emotional correlates. The theory of performance fields is derived from the general hypothesis that performance is a smooth (continuously differentiable) isomorphism ℘ = R → RP on a frame neighborhood R of the given local composition C ⊂ S. This is of course a strong hypothesis, but it is, at least locally on the given composition, a reasonable one. In our inverse problem, we are not given ℘, but only its restriction ℘|C to the given local composition. Accordingly, we shall not really construct the performance field Ts associated with the unknown map ℘, but a discrete performance field, defined on the points of C, which is determined by the restriction ℘|C . We shall now construct such discrete fields, their interpolation on a neighborhood of C, as well as their visualization by means of color fields. It will also be possible to calculate difference fields in order to compare two performances of the same local composition.
46.3.2
Real-time Processing of Expressive Performance
An implementation of the performance field theory should be able to operate in real-time, especially for interactive applications, where immediate feedback, either visible or audible, is desired. As we shall see, the complete calculation of the performance fields can be split up in dedicated, communicating modules for specific tasks. Particularly important for performance is the extraction of tasks that can be processed in advance. Figure 46.2 gives an overview of input score (i.e. MIDI)
input performance
input filtering input filtering
matching real-time context
basis calculation
field calculation field interpolation visualization
preprocessing (off-line)
real-time processing
Figure 46.2: Overview of the modules and the flow of control (as shown by the vertical arrows) in our implementation of the EspressoRUBETTEr . the modules and the flow of control (as shown by the vertical arrows) in our implementation. Modules are notified by events when new data for processing is ready. The modules themselves are stateless, they share their information with other modules in the real-time context, a data
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structure which contains all relevant information for the whole process, thus minimizing the risk of inconsistency. Of course, asynchronous accesses to the context have to be synchronized using locks or a similar synchronization technique. For increased flexibility and efficiency all modules accept lists of events, therefore making off-line and real-time processing structurally identical. For example, the input filtering modules for the score and for the performance are the same. The former accepts the score as a whole and the latter processes individual events as they are received in real-time. Further, modules can be prioritized and be put to sleep if there is not enough processing power to support all present modules temporarily. The following items give short descriptions for the modules shown in figure 46.2. Details will be presented in the subsequent sections. It is important to see that the described architecture allows the definition of additional modules as needed. This mostly depends on application requirements. Also some application might not need certain already defined modules, i.e., field interpolation in a computer accompaniment system. Input filtering. This module translates incoming note events to the representation defined in the real-time context. It also processes structural information, such as different voices, tempo changes, etc. The input filtering module must be implemented for any external representation (e.g., MIDI or RUBATOr s denotator format). Basis calculation. The calculation of the bases depends only on the input score, not on the performance and can thus be performed off-line. For each event an appropriate basis has to be calculated. Typically this is a time-consuming process. Matching. The incoming performance events have to be matched to the corresponding score events. As we shall see in the designated section, this is a non-trivial task and has developed to a research field on its own. Field calculation. The individual field vectors for each note are be calculated based on the precalculated basis and the given match. Field interpolation. The field vectors calculated by the former step are typically not aligned on a grid. However, for visualization a 2D or 3D grid-like field, with field vectors defined anywhere in this grid is desired. The interpolation step allows the definition of such a grid and performs the translation from the note field to the interpolated field. Visualization. Finally the calculated field is ready for visualization. Here, many user defined parameters such as scaling, color-specification, ranges, etc., have to be taken into account.
46.3.3
Score–Performance Matching
A lot of research effort has been put into score–performance matching techniques. Scorefollowing, the real-time matching and tracking of soloists performing a given score was first published by Dannenberg [108] and Vercoe [544]. Puckette [426] presented the methods used on the IRCAM Signal Processing Workstation (ISPW). Heijink et al. [216] have given an evaluation of different approaches to score-performance matching.
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Literature has typically differentiated between two types of algorithms: For real-time algorithms, mostly used in real-time accompaniment software, good performance had higher priority than matching quality. Off-line algorithms, where calculation time is less important, were mostly used for in-depth analysis applications requiring a high level of matching quality. We however experienced that with todays processing power high quality matching can be performed in realtime, particularly when the algorithms are well suited for extensive preprocessing of the given score. Mathematically, the matching problem is complex and depends upon the difference which one allows between score and performance. For example, if chords remain chords and all notes are played exactly once, the problem is trivial. But normal performance includes more or less strong arpeggiation of chords, omissions of notes or playing additional notes by error or by ambiguous definition of the notes, such as is common for trills and other ornaments. We have implemented an algorithm which will not be described in detail here since it is not our principal subject. The algorithm is a kind of matching along a ‘wave front of notes which are defined by the temporal unfolding of the performance and thereby fits in the real-time constraints. We nevertheless should sketch the principle ideas behind our algorithm. Usually, matching is thought bottom-up in that the performance map of the whole piece is constructed from the performance map X 7→ ℘(X) on the single elements X. We rather tried a top-down strategy, i.e., to rebuild the element images from maps on sets of specific coverings I, J, respectively of the local composition C and its performance D. Typically, one considers the covering of C by hyperplane sections in each parameter (for example onset slices). On D, a covering J is defined which is a more fuzzy version of I, for example neighborhoods of hyperplane sections (for example ε-neighborhoods in the onset dimension). If ℘ exists, different constraints can be imposed on the induced map on the coverings: First, ℘ induces a map n0 (℘) : I → J such that ℘(U ) ⊆ n0 (℘)(U ) for all covering elements U in I. This yields a map n(℘) : n(I) → n(J) of the simplicial nerves, and thus conditions on the map on the covering sets. Second, the sets of these coverings are linearly ordered3 by U ≺ V iff either U ⊂ V or both U − V and V − U are non-empty and min(U − V ) < min(V − U ). In this ordering, we require that U V ⇒ n0 (℘)(U ) ≤ n0 (℘)(V ). Third, if one defines a distance d(U, V ) between the covering sets (for example the elastic shape distance from motif theory, see section 22.2.1.3), one requires that d(U, n0 ℘(U )) < ε for a given positive distance limit ε. With these constraints one may define the map, and then recover ℘ if every point X in C may be seen as the intersection of all covering sets of I which contain the point. This is evidently the case for the hyperplane sections described above. Following these observations, an implementation typically makes use of structural properties of a musical score and a corresponding performance. Further, dynamic programming techniques help in coping with the real-time problem: Multiple possible solutions are created, maintained, and discarded as the matching process is running.
46.3.4
Performance Field Calculation
Let us consider the score space S, the performance space P , the performance transformation ℘ : S → P and the constant vector field ∆(x) = ∆ = (1, ..., 1) for all x ∈ P . Recall from section 3 This
is the usual linear ordering of powerset denotators as defined in section 6.8.
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33.2.2, equation (33.11) that the performance field is the inverse image of ∆ and evaluates to Ts(X) = J(℘)−1 (X)∆ where J(℘) is the Jacobian matrix J(℘)(X) =
xi =e,h,l,d,... ∂xi ∂Xj
Xj =E,H,L,D,...
=
∂e ∂E ∂h ∂E
.. .
∂e ∂H ∂h ∂H
.. .
... ... (X) .. .
at X. In order to calculate the field vectors in an element X of the given local composition C in S, we have to determine J(℘). Now, assume that we are given a matrix UX of not necessarily orthogonal basis vectors based in X. J(℘) can be rewritten as: −1 −1 J(℘) = J(℘)UX UX = VX UX
where the basis matrix VX is the image of the basis matrix UX . Then: Ts(X) = UX VX−1 ∆ = det(VX )−1 UX Adj(VX )∆. The last term is used to identify three cases: 1. VX is regular, thus Ts(X) is defined, 2. det(VX ) = 0, Adj(VX ) 6= 0: only the direction of the vector is given, not its length, and 3. det(VX ) = 0, Adj(VX ) = 0: no information at all is given. While we are now able to calculate the field vectors, the question of how to find the appropriate bases is still open. 46.3.4.1
Obtaining the Bases
The only information available for basis calculations are difference vectors of the given score notes. Basically any difference vectors could be considered as basis vectors, but due to the following restrictions, the candidates have to be selected carefully: First, only notes in a small neighborhood of X should be considered. This principle of locality ensures that the basis consists of notes that are in the local musical context. The second restriction is of a mathematical nature: we have seen that the transformed basis has to be regular in order to be able to calculate the field vector. Because the performance is allowed to have arbitrary deviations from the score, there is no general solution to this problem. What can be done is to decrease the possibility that the transformation of the basis UX yields to a non-regular basis matrix: This can be accomplished by making UX as orthogonal as possible. Thus, the selection of the basis vectors is based on the following two criteria: 1. Locality: |det(UX )| is minimal, and N orm N orm 2. Orthogonality: |det(UX )| is maximal (where UX is the matrix of normalized basis vectors).
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Note that the two criteria are to some extent in competition, so they have to be weighted and combined. Consequently, a basis-calculation algorithm has to select bases by searching for 1 min wloc |det(UX )| + worth N orm ) det(UX with wloc and worth being positive pre- or user-defined weight values. Unfortunately, there is still one case that has to be dealt with: the case where it is not possible to find a regular basis matrix UX in a small neighborhood of X. This may occur if all notes have the same loudness, if the basis has to be calculated for an isolated chord, where all onsets are equal, or for repeated notes with the same pitch. The only option left here is to construct orthogonal basis vectors that ensure that the basis remains regular. Finally, the pseudo-code for a basis-calculation algorithm can be given: for(each Note X in Score S) { List neighbors = S.getNeighborList(X, maxDist); List basisVectors = emptyList; for(each Note N in neighbors) basisVectors.add(N - X); List bases = getCandidates(basisVectors); X.basisCost = infinity; for(each Basis B in bases) { float basisCost = wLoc * abs(B.det()) + wOrth / abs(B.norm().det()); if(basisCost < X.basisCost) { X.basisCost = basisCost; X.B = B; } } The function getCandidates(), whose pseudo-code was omitted here, generates a list of bases containing the permutations of the basis candidates, and also adding constructed basis vectors if necessary. The above algorithm can be optimized by generating the permutations on the fly, the best expected ones first. In that case, the candidate list can be sorted by increasing distance, and the distance is used to cancel the loop, as soon as it is known that a lower basisCost can not be reached anymore.
46.3.5
Visualization
One of the most straightforward applications of a calculated performance field is its visualization. Vector field visualization has been successfully used in many science and engineering domains, i.e., in gas and fluid dynamics. Thus, many different techniques and their corresponding implementations are available. Common to all those methods is that they should be accurate, fast, and display the field in an intuitive way. See [76] for an advanced method that is suited for 2D as well as for 3D visualization. This section shows how the calculated field vectors need to be processed in order to make them available to such standard visualization methods. So far, we have dealt with a score C, consisting of a set of notes, the corresponding performance D, and the associated set of calculated field vectors F . The points in those sets reside in
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an n-dimensional space, n being the number of symbolic sound parameters. For visualization, n will normally be too large, so as a first step it has to be decided which parameters are to be used for visualization. For instance, we may choose onset E as the horizontal axis and pitch H as the vertical axis in a 2D setup. The remaining sound parameters are omitted. Further, the desired field vector components have to be selected, for example E in horizontal direction and D in vertical direction for a tempo-articulation field.
46.3.5.1
Field Interpolation
Typically, when dealing with vector fields, the field vectors are arranged in a grid of a given resolution. In contrast, our setup implies that the score points reside at arbitrary locations, making it impossible to use standard vector field visualization methods. Thus, a conversion from the calculated field vectors to vectors located on a grid is necessary. This can be accomplished through interpolation. At first sight, a triangulation of the given set could be considered, making it easy to calculate the interpolated grid vectors in the resulting triangles. However, since the different symbolic sound parameters have different meaning, triangulation is not well suited here: interpolation should occur in a musically meaningful way. Therefore, it makes sense to perform interpolation in a defined recursive order. For instance, when interpolating an ED field, first the D axis of the grid is considered and then the E axis. More precisely, one draws hyperplanes H1 , H2 , ..., Hk perpendicular to the n-th axis in the symbolic parameter space S such that every point of the given composition C sits in one such hyperplane. By recursion, we suppose that the interpolation is available for the first n − 1 coordinates. To get the interpolation value on an arbitrary point X, one draws the straight line through X and parallel to the n-th axis. This line cuts two neighboring hyperplanes in points P , Q. The values in P and Q are then interpolated by a cubic spline with zero slope in P and Q. For a detailed description see section 32.3.2.1. What happens at the boundaries of the given set? Since no field vectors are available, boundary vectors have to be defined. When having a look at the theory of the former sections, it becomes clear that outside the boundaries a frame of diagonal vectors has to be placed. The distance between the boundaries and the frame is constant and has to be predefined.
46.3.6
The EspressoRUBETTEr : An Interactive Tool for Expression Extraction
The methods for algorithmic extraction of musical expressions have been implemented in a tool called EspressoRUBETTEr . The tool can run as a stand-alone Java application. The Swing and Java2D classes take care of the user interface, and the user can manipulate calculation and visualization parameters through a simple dialog panel. As an alternative and more flexible approach, the software also implements the RUBETTEr interface and can thus be integrated into the Distributed RUBATOr framework, see chapter 40.
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Figure 46.3: A chromatic scale and its performance: Above the field vectors, middle: the color encoded and interpolated field vector, below the performed scale.
46.3.6.1
Example 1: Tempo Field of a Chromatic Scale
Let us now give examples of calculated performance fields. Figure 46.3 shows a chromatic scale and its performance. In this case representation is close to the one of a piano roll: the horizontal axis represents onset, pitch is mapped to the vertical axis. The width of events corresponds to their duration. Note that the EspressoRUBETTEr allows arbitrary redefinition of those mappings. The top section shows the score containing the chromatic scale, twelve note events in increasing pitch order, all with the same duration. A hypothetical performance of the twelve events is shown in the bottom section. The first three events are played at the same speed as the original MIDI score. Then the performance is getting slower, and towards the end it is getting faster again. The last two notes are played faster than the MIDI score. This situation is depicted by the calculated and visualized vectors in the top section: the first, the second and the eleventh vector are diagonal vectors, stating that the notes are played at the given tempo. The angle of the other vectors depends on the local tempo played at a
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given note. The middle section shows the corresponding interpolated field, at a resolution of 400 times 200 cells. Here, the slope of each vector has been mapped to a color, and its length is related to the brightness of a cell. 46.3.6.2
Example 2: Excerpt from Czerny’s Piano School
Figure 46.4: A performance field for an exercise from Carl Czerny’s piano school. The second performance is a real-world example, namely a performance field for an exercise from Carl Czerny [98], as recorded on an MIDI file. Figure 46.4 shows the first two bars of the exercise, axes and note representations are as in the previous example. The upper half shows the performance field on the score space S = EH, the lower half shows the physical space S = eh of the performed piece (physical parameters being written in small letters). The exercise shows the Chopin rubato, i.e., the right hand plays the melody slightly shifted in time against the firm left hand chords in such a way that synchronization is recovered at the end of bars. The field shows the E- and D-components of the four-dimensional EHLD-performance field, as encoded by colors. One recognizes that the left hand notes are quite near in color to the green color
46.4. LOCAL SECTIONS
925
which encodes the diagonal unit field for mechanical, unshaped performance. We see that there is a right hand rubato effect in the middle of each bar, and significantly more in the second bar. The cyclic coloring effects are due to a multiple covering of the color circle in order to make the fine slope differences of the performance field more visible. Remark 22 The results - calculated and interpolated performance fields - contain explicit expressive information and are available for visualization or for other performance analysis tools. The algorithms are not restricted to specific sound parameters, and the method can thus be used for extensive expressive analysis. Currently, basis calculation imposes the biggest limitation. In some cases, the calculated basis of a note does not correspond to its musical context, resulting in field vectors that are - while being mathematically correct - hard to understand. Here, ongoing research will definitely deliver better results. A promising field of further research is also the insight that performance fields are not restricted to musical data. In medical applications and in computer-aided anthropology, the growth information of human bones and organs can be extracted in a similar manner - in which case we may talk about Nature’s Performance.
46.4
Local Sections
Summary. We study canonical sections, i.e., selectors of points in grammatical varieties. The subject deals with problems of how to choose a determined critique out of an entire variety of possible critiques. –Σ– The generic isomorphism of critical fibers as shown in the above corollary 22 suggests that we should look for musically motivated restrictions on the admitted system parameters in order to obtain more specific information. In this section, we propose a model which is based upon the causality and finality of locally linear grammars and gives a manageable approach to the structure of causality and finality in the transition parameters cm x,y from a daughter x of a mother m to its sister y (notation inherited from section 46.2). The idea is to parametrize the transition parameters by a small set of shaping parameters which give the system of all cm = (cm x,y )Dm (T ) a causal-final coherence. Suppose that the daughters in Dm (T ) are linearly ordered with respect to time. For example, this may happen if these daughters are defined by onset intervals of a sequence of adjacent bars or periods. Then we may just enumerate these daughters x1 < x2 < . . . xd(m) in ascending order and temporarily abbreviate cm xi ,xj = ci,j . In this notation, if i < j, this means that ci,j measures the causal influence of the prior LPS xi on the later LPS xj , while i > j yields a measure ci,j for the final influence of the later LPS xi on the prior LPS xj . The diagonal values ci,i measure the “autocorrelation” of LPS xi . With this in mind, we want to give the value matrix cm a simplified shape as follows: Suppose that the index set 1, 2, 3, . . . d(m) is evenly distributed on the interval [1, −1], i.e., λ(i) = 1 − (i − 1)2/(d(m) − 1). We now define the function def ormLiGr on the square [1, −1] × [1, −1]
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on the real plane as follows: spl(x) = e−(0.4x)
2
(1 + 0.1.causal.x2 )spl(x) if 0 ≤ x, LiGramm(causal, f inal, x) = (1 + 0.1.f inal.x2 )spl(x) else, LiGr(causal, f inal, x, y) = LiGramm(causal, f inal, (x + y)/2) def orm(start, end, y) = 0.5((start − end)y + (start + end)) def ormLiGr(causalStart, causalEnd, f inalStart, f inalEnd, x, y) = LiGr(def orm(causalStart, causalEnd, y), def orm(f inalStart, f inalEnd, y), x, y).
(46.6)
Figure 46.5: Nine shapes of the function def ormLiGr for the system parameter 4-tuples causalStart, causalEnd, f inalStart, f inalEnd (from left top to right bottom): (3,3,3,3), (3,3,1.5,1.5), (3,3,0,0),(1.5,1.5,3,3),(1.5,1.5,1.5,1.5),(1.5,1.5,0,0),(0,0,3,3),(0,0,1.5,1.5), (0,0,0,0). The causal extremum (1, d(m)) is to the left, the final extremum (d(m), 1) is to the right of each surface. The horizontal diagonal is the autocorrelation area which has the constant value 1. For each fixed pair x, y, the function def ormLiGr is an affine function of the four system parameters causalStart, causalEnd, f inalStart, f inalEnd. The system parameters contain information on the strength of the causal and final correlations at start and end of the given time interval. Figure 46.5 gives an image of the function for nine different system parameter
46.4. LOCAL SECTIONS
927
combination. The function is evaluated on each pair x = λ(i), y = λ(j) and yields cm i,j as an affine function of the four system parameters. We see that low parameter causal and final parameter values, respectively, give low correlations in causal and final direction, respectively. For example, the top right “flying carpet” is strong in final, but weak in causal direction. This means that we are looking at an algebraic variety which is defined by the substitution of the generic transition coefficients cm i,j by functions m m m m cm i,j (causalStart , causalEnd , f inalStart , f inalEnd )
of four system parameters which are a function of the given mother m. If m(T ) is the total number of mothers of the stemma quiver T , this yields a total number of 4.m(T ) causal-final variables which add to the analytical variables from the modules Bx . To be clear, the shape of a “flying carpet” is a transcendental function involving exponential and quadratic components, but the causal-final system variables which define the actual shape of a carpet are only involved in an affine way.
46.4.1
Comparing Argerich and Horowitz
Summary. This section applies the comparative theory to a particular case of agogics performance in Robert Schumann’s “Tr¨ aumerei” by Martha Argerich and Vladimir Horowitz. It turns out that Argerich’s agogical coherence is global compared to a rather local coherence with Horowitz. –Σ– In this section, we shall apply the preceding approach to the inverse problem regarding tempo curves and their shaping by use of motivic weights and associated operators. The piece is Robert Schumann’s Kinderszene 7 “Tr¨ aumerei”, the performances are those by Martha Argerich (ARG) and Vladimir Horowitz (HO1 in Repp’s list), both measured by Bruno Repp [438] among a total number of 28 performances that were already discussed in chapter 44. For our example, we take the stemma on the onset space which is defined by the primary mother RM frame that extends from the first onset through the end of the last event. This mother has four daughters RA , RA0 , RB , RA00 . Frame RA is the time frame of part A from the very beginning to the end of bar 8. The second frame RA0 is the time frame of the eight bars of the first period, the frame RB is that of the third eight bar period, the last frame RA00 is that of the reprise eight bar period to the end. Each of these daughters is the mother of eight bars, except the first, which has a first daughter including the upbeat quarter in the first full bar. Let us denote the daughters of RA by RA,1 . . . RA,8 , then those of RA0 by RA0 ,1 . . . RA0 ,8 , those of RB by RB,1 . . . RB,8 , and those of RA00 by RA00 ,1 . . . RA00 ,8 . For the analytical parameters, we take a constant vector space Bx = B = hW1 , W2 , . . . Wt i generated by a finite family (Wi )i of global weights on the onset frame RM . For a given onset frame R of our stemma, we have the linear map ϕx : B → Ax into the vector space of C1 tempo curves on the stemmaRframe x. It maps the weight W to the scaling transformation by the quantity ϕx (W ) = |x|−1 x W . This operation leaves constant fields constant, polynomial fields polynomial, etc., it conserves any reasonable special type of tempo curves. Observe that we really restrict global weights to local frames and do not recalculate local weights when averaging.
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Figure 46.6: The restriction of the generic fiber spaces to ‘flying carpet’ coherence domains yields characteristic differences between Argerich’s and Horowitz’ performances. This being the case, we start by a global constant default tempo dT , then take its restrictions to the period daughters, then apply all the scaling transformations to the restricted tempi, then add up all the shifted contributions that are multiplied by the crosscorrelation coefficients from a “flying carpet” with causal-final variables causalStartM , causalEndM , f inalStartM , f inalEndM on the periods. Then we have the 32 restrictions to the single bars, the application of the scalings according to the averaged weight restrictions to the bars, and then ending by the cross correlations according to “flying carpets” for the four periods and eight bars each, with the total of 16 causal-final variables causalStartA , causalEndA , f inalStartA , f inalEndA , 0
0
0
0
causalStartA , causalEndA , f inalStartA , f inalEndA , causalStartB , causalEndB , f inalStartB , f inalEndB , 00
00
00
00
causalStartA , causalEndA , f inalStartA , f inalEndA .
46.4. LOCAL SECTIONS
929
We have applied this system to the one and only motivic boiled-down weight T r¨ aumerei xP M eloRubette,P arammelodic
which was also used in the statistical analysis of chapter 44. This yields a total of 21 variables and a total of 32 cubic polynomials (one for each bar) in these variables, whose solutions yield the variety lying above the average measured tempi on the 32 bars. Therefore, it cannot be expected that we really have zeros of all these equations. With Mathematicar routines, we have therefore calculated local minima of these polynomial equations and found the following results [348, III] in terms of causal-final variable values:
Result 7 Period level: • In the inter-period coherence, Argerich is more final than Horowitz, whereas the causal level is more pronounced by Horowitz. Result 8 Bar level: • Horowitz plays the first period with pronounced causal and final coherence, whereas the causal coherence decreases to a very low level towards the end of the piece. • The repetition A0 of the first period A shows a ‘relaxation of coherence’ which may be justified by the repetitive situation. • The development section B slightly increases the final character. • The recapitulation seems to be quite ‘tired’: the causal character is very low, the final character is decreased. • For Argerich, the first period has a less coherent ambitus than with Horowitz. • In contrast to Horowitz, the final coherence of Argerich increases as the piece goes on. • The development and the recapitulation are pronouncedly final. The development and the recapitulation shows a consciousness of the end of the piece which is absent with Horowitz. • In other words, Argerich’s recapitulation is ‘prospective’ and not ‘retrospective’. These calculations are however only locally relevant, and a global solution space with more subtle estimations should also be calculated. Nonetheless, an identical algorithm is applied to two samples and therefore, comparing these inverse performances is admitted.
Part XIII
Operationalization of Poiesis
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Chapter 47
Unfolding Geometry and Logic in Time Keeping time, time, time, In a sort of Runic rhyme, To the tintinnabulation that so musically wells From the bells, bells, bells, bells. Edgar Allan Poe: The Bells (1849) Summary. Musical poiesis in composition (and performance) is intimately related to a projection of abstract objects into time. We discuss the logical and geometric aspects of this mapping process. The subject is crucial to the entire art of music since music is involved in the creation of autonomous time beyond the physical “tyranny” of real time. This enforces a review of Michael Leyton’s theory of time as a philosophical category which is derived from spatial symmetry transformations [303]. We also discuss the role of unfolding insight within the syntagmatic discourse of music. –Σ– After the development and analysis of the various aspects of musical structure and process in this book, it should no longer astonish that the creation of music involves an incredibly complex semiotic, communicative, and reality-critical construction. Far beyond a simple representation and performance of elementary sound events, music is a narration of strong logical and geometric categories, or, at least: without such an intense existentiality, music would never reach the status of a valid antiworld which takes us to an autonomous time and space. This is valid all the more than straightforward common signification processes fail, and meaning has to be built without external references. So the question is legitimate: What is the story that compositional narration is likely to convey to the listener? This question is critical in two regards: first, the narrative discourse without common, external content is not likely to be a remake of the narrative discourse in ordinary language. Second: The absence of an ordinary content (the story) makes it a questionable point whether narration in music is a reasonable category at all, or whether it is rather 933
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a metaphor which one instantiates ` a d´efaut de mieux, and which one should avoid rather than abuse. Of course, the musical performance is embedded in the physical time-line, just like the ordinary story which is being told in a piece of literature. But in literature, this time surface transports a hierarchy of time strata which owe their existence to told and telling times of a time-sensitive reality, be it a description of events and movements in a fictitious world, be it the story teller’s discursive telling time. In music, this is not possible. Except physical time, there are no timely contents that are referred to and represented, including a beginning, middle, and ending part, as Aristotle has described narration in his poetics. This means that if narration does exist in music, it must not only invent an autonomous content layer, but in addition, it has to create the time hierarchy that organizes the narrative stream. In this short chapter, we want to give some remarks on how such a creative process and its contents could be conceived.
47.1
Performance of Logic and Geometry
Summary. In contrast to linguistic discourse, musical syntax does not, a priori, tell a story which has its own time dimension. We contend that the intrinsic story in music is about performing logic. This includes—in particular—that musical time means above all syntagmatic time, and not the material time shared by the parametrized music events. We relate the subject to Algirdas Julien Greimas’ theory of narrativity [195]. –Σ– Although there is no common storyboard in music, it does nevertheless realize a narrative discourse. We shall see in chapter 52 that the discursive dialog among four humanist persons (as represented by four instruments from the family of violins) is a characteristic feature of the string quartet. This is all the more remarkable that string quartet music is quite the contrary of a program music, it has always been the art form of “absolute music” where an abstract “musical idea” is processed. So what are these musical personalities talking about? In common musicology, one would argue that they are organizing musical ideas. We suggest that more precisely, they are constructing a poetical work along the principles of Jakobson’s poetical function: projecting the paradigmatic axis into the syntagmatic axis, see section 11.6.1 for this function. Such a paradigm could be a pitch or chord class, a tonality, a motif or a melody, a contrapuntal line such as cantus firmus or discant, a rhythm, or an instrumental color, for example. The projection of such paradigms into the syntagmatic axis means (1) selecting representative instances of such paradigms and (2) arranging these instances along the syntagmatic contiguity. So the paradigms in absentia are syntagmatically represented in praesentia. This may be a contiguity in time, such as is the case for a succession of tonalities, or else a contiguity in pitch, such as we have discussed for contrapuntal voices, or a contiguity in a sound color space for a multi-instrumental projection. The point here is that these syntagmatic relations are not linear, i.e., the contiguity can extend to different dimensions. And it can also take place on relatively abstract levels in the sense that, for example, a tonality need not be the concrete score level, but a preliminary organizational level which is a generic scheme for the score realization. We shall also see in
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chapter 51 on the OpenMusic software that the abstract syntagms are completely natural in intelligent implementations of compositional strategies. The syntagmatic arrangement of paradigmatic representatives creates the first instance of Greimas’ fundamental categories of narration, i.e., succession. But we again have to stress that succession is not necessarily one in physical time, it is one in an abstract parameter space, or even in a space form of generic concepts. The concatenation of successive units is given a logical justification by the insertion of transformational process units, such as contrapuntal interval relations, contrapuntal or harmonic rules of progression, modulatory parts between adjacent tonalities, transformations of motives to their variations, for example. This enrichment is associated with Greimas’ category of transformation which, together with the states of succession defines his “programme narratif”1 . In the third stage of the narrative organization, the category of “mediation” is recovered in order to embed the narrative program in a global reasoning, i.e, the declaration of a purpose behind the narrative program. This can be, for example, the overall strategy for a modulatory plan as we have known it in the analysis of the modulatory landscape in Beethoven’s op. 106, see section 28.2. Summarizing, the narrative organization is a performance of the logical strategies which dispatch paradigmatic units in the syntagm of their logical concatenation, the latter being explicated on successive transformations, whereas the whole expresses what one really could call “the musical idea”. But the syntagm of this organization is by no means a linear one, and even less one in the physical time of a typical telling instance. And accordingly, the story being told is much more than a temporal succession of events, it is a logical construction of local and global compositions, of morphisms between such objects, of universal constructions in the corresponding categories, short: of logically and geometrically motivated predicate instances. This is why constraint programming has become an interesting field in computer aided composition, but see chapter 51 for details. At this point we have to a certain degree reached what Eduard Hanslick [206] calls “t¨onend bewegte Formen”. So what is missing? Essentially, there are two major gaps at this stage: • A narrative time concept and its organization is still missing. • The unfolding of the syntagmatic logical display into physical performance time is outstanding. We are now going to discuss these issues.
47.2
Constructing Time from Geometry
Summary. We sketch predicates of logical and geometric nature which are designed to set up syntagmatic time in music. In this context, Michael Leyton’s theory of time is briefly reviewed. –Σ– As the present organism is a purely logical and geometric one, the only hope to generate narrative time is by means of mechanisms which turn logic and geometry into time. On the 1 “On appelle programme narratif (PN) la suite d’´ etats et de transformations qui s’enchaˆınent sur la base d’une relation S-O [Sujet-Object] et de sa transformation.”
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logical level, for example, the implication form A =⇒ B induces a time category where the condition is antecedent to the implied statement. On the geometric level, there are several symmetry-related “time generators”. In fact, a symmetry S can be viewed as a transformation, where objects x are moved around, i.e., x 7→ S(x), whence a time-stamped relation from x to S(x). A radical approach to time is proposed by Michael Leyton [303, 304]: In its cognitive and physical reality, the time concept is viewed as a derived one. Leyton’s theory develops structures to reduce time ontology to a dynamical syntax of spatial symmetries. These symmetry groups are wreath products2 F o C of a control group C and a fiber group F . Leyton’s approach allows us to understand music as a natural reconstruction method of time via syntax of symmetries. This syntax aligns symmetries and their broken variants as a ordering relation: first symmetry, then its broken version. Time is generated via symmetry breaking. This implies that musical time is generated by such symmetry breaking relations. Together with the logical time construction, we obtain a variety of ordering relations which are induced by symmetry breaking processes. More generally speaking, time is generated by logical and geometrical succession relations. For example, if we recall the model for tonal modulation (chapter 27) the modulation quantum is a global composition whose inner symmetries relate the antecedent to the successor tonality, but it is not possible to distinguish these tonalities in their logical role from the structure of this symmetric quantum. Leyton would say [304, symmetry and asymmetry principles p.41] that symmetry is without memory, since it is symmetry breaking that constructs memory. The time ordering of the involved tonalities is only generated by marking the trace of the modulation quantum in the target tonality: which defines in fact the modulation steps. This is also the Greimas transformation part relating the two units in the succession chain. Such a breaking of a symmetry in modulatory degrees creates time and bans the symmetry from the syntagmatic surface. A good example of memory-less construction by unbroken symmetries is Sch¨onberg’s dodecaphonic method. It starts on an original dodecaphonic series and proposes a display of some of its (generically) 48 transformed versions under the known group D12,12, (see the discussion following definition 22 in section 8.1.1). There is no specification of the method regarding the instantiation of the involved symmetries, they are external to the composed syntagm, and there is no symmetry breaking that would create a time-line among the 48 variants in that orbit. So the narrative structure of this method is not specified. Maybe this drawback, which lives in a pronounced contrast to the excellent narrative sonata form, is one of the reasons for the failure of pure dodecaphonism, i.e., dodecaphonism that is not enriched by syntagmatic concepts for the narrative construction. So let us assume that syntagmatic time is generated as an ordering relation between logical and/or geometric units. Such a time is not necessarily linear, there may be several competing time strings. For example, recall from counterpoint theory that “punctus contra punctum” relates to the horizontal interval succession as well as to vertical cantus firmus vs. discantus succession (as the melodic implication). So we have two time strings which are simultaneously present in the “contra” process. In a modulatory process, the time-line of tonalities can be superimposed by a vertical time-line of melodic deduction from the harmonic basement, and a time-line induced by an instrumental hierarchy. Such abstract time-lines are the narrative 2 See
Appendix 73.
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bricks of the musical counter-world. Suppose that we now dispose of such a narrative organization. Then, its unfolding is the following project: In performance, we have to unwind the logical and geometric time-lines in a space of performance time. We have discussed at length the time hierarchies and stemmata of performance in parts VIII and IX. Evidently, these tools must be used to unfold the abstract narrative organization in physical reality. Presently, such a theory is lacking, but the prerequisites are at hand. The main problem with regard to human cognitive capacities is to find an equilibrium between the complexity of the narrative instances of logical and geometric time-lines and the limitation of comprehension of such structures and processes in the physical performance space-time. This means that we have to name principles that define the basic structures of a global composition with regard to its narrative communication. A global composition has a basic syntagm that is defined by its covering and, more specifically, its nerve. This is already present in Sch¨onberg’s harmonic strip between successive chords, which is in fact an elementary but prototypical syntagmatic junction (the transformational part between two adjacent succession units in Greimas’ theory). So the composition’s nerve is an excellent syntagmatic device in that it includes narrative paths (on the nerve’s one-dimensional skeleton) of neighboring local charts in praesentia. For example, if we have the interpretation of a diatonic scale by the seven triadic chords, we obtain the harmonic (M¨ obius) strip as a global composition. The geometric relation of two neighboring degrees on the strip, IC , IIIC , say, can be given a narrative direction in the succession IC6 , IIIC where the common notes e, g of the inversion IC6 and the third degree may be held such that we only have the time event of a downward movement c 7→ b to indicate the time-line from IC6 to IIIC . We contend that any unfolding of the syntagmatic logical and geometric time-lines should locally on the physical time axis produce a small linear storyboard, i.e, minimize the multi-dimensionality of the abstract syntagm, or at least try to stress its most important components. To put it the other way round: Understanding the abstract narrative structure of a composition’s nerve amounts to realizing a sequence of paths through the nerve’s one-dimensional skeleton such that eventually, all important parcours of the nerve’s ‘city map’ have been exposed. The problem here is analogous to the problem of how the eye’s path should look when it observes a painting in order to obtain a good understanding of what is seen, or else to the problem of how we should walk through a house in order to optimally understand it as an architectural organism. It is the problem of curvilinear reduction of high-dimensional objects, a problem which by the human language stream of words has been solved in a very special context and with the success of a very limited textual representation of the world.
47.3
Discourse and Insight
Summary. This final section locates the musical discourse as a means of gaining insight into complex multidimensional structures by syntagmatic sequentialization (linearization). This contrasts with the linguistic discourse, since the latter has rich semantic rootings which are absent in music, must therefore be balanced by explicit discoursivity. –Σ– The advantage of common language above music is the presence of external semantics, the language stream runs around any kind of complex entities that are not part of the language
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structure. This makes language an easy business since it does not include its objects, but only points to them. In contrast music cannot point outside (except for trivial special cases of onomatopoiesis), it has to build everything from an inner discourse. This is also its advantage and scope: to be able to invent worlds instead of pointing to them. In this sense it is an excellent exercise to learn what is really needed to create a ‘world’ in our imagination, i.e., a meaningful spiritual architecture without external roots. The investigation of the narrative discourse should teach us how we gain insight into things when telling their story, meaning: how to understand music while playing it.
Chapter 48
Local and Global Strategies in Composition Aus den Auswahlkriterien also entsteht die Dialektik der upfung von Lokalstrukturen, wobei Abfolge oder Verkn¨ diese Auswahlkriterien bestimmend sind f¨ ur die Eingliederung der Lokalstrukturen in die große allgemeine Struktur, die Form. Pierre Boulez [60, II] Summary. This chapter sketches the compositional process between paradigmatic selection and syntagmatic combination within the musical sign system. Apart from these semiotic perspectives, the process is characterized by a “dialectic” interrelation of local and global criteria. These features—well known from the general structure theory of global compositions—reappear in the special light of poiesis: The construction of a composition resembles the step-by-step completion of a puzzle of logical units, distributed in syntagmatic time, and selected to optimize association to already placed units. This activity is remunerative and fed back by a successive accumulation of poetical semantics. –Σ– We do not intend to describe the psychological path in the composer’s mind, but the fundamental strategic steps, independently of the psychological or cognitive realization. We also do not impose this strategy on any composer, but want to sketch a possible system for composition which can also be implemented on the software level. We shall see that two implementations: prestor and OpenMusic realize quite a portion of these ideas. Essentially, the poiesis of a musical composition is the construction of a global composition, i.e., a patchwork of local charts, together with gluing transformations. Moreover, the charts are selected from a set of paradigms and combined according to a set of syntagmatic rules. But the concrete making of a composition is not a one-step procedure, it results from a successive completion of an ensemble of charts. Ideally, one is 939
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1. given a simplicial complex and would like to realize it as the nerve of a covering by local charts. 2. One also wants the charts to pertain to determined isomorphism classes or else being specializations of representatives of such classes. 3. One finally wants particular gluing transformations on determined parts of the charts. Point two reflects the analytical approach using the paradigmatic theme of Ruwet and Nattiez, see section 11.7. But it extends it in that the nerve is planned as a syntagmatic design pattern, further specialization and not only transformation of basic paradigms is allowed, and finally, the gluing transformations for the syntagmatic combination are thematized.
48.1
Local Paradigmatic Instances
Summary. Local paradigmatic strategies can be split into transformational and topological procedures. In each case, we describe the basic options. –Σ– Recall from chapter 10 that paradigmatic relations can be either transformational or topological—or combinations thereof. Whereas the first is related to symmetries, the latter is related to the idea of variation. Usually, transformations will quite brutally change the auditory impression of a local composition and must be used with care, whereas topological variation is more adequate for instantaneous recognition of similarity.
48.1.1
Transformations
Summary. Starting from Sch¨ onberg’s dodecaphonic method, we describe varieties of local compositions produced by a determined set of transformation groups, the paradigmatic themes of poiesis, acting on a selected set of local “germs”. –Σ– This method starts from a sequence S = (G1 , . . . Gc ) of local germs, for instance thematic units such as the singleton S = (G1 ) containing a dodecaphonic series, or a two-element set S = (G1 , G2 ) containing a main motif and a side motif of a sonata. Then, for each local germ Gi , we are given a transformation group Pi , the paradigmatic group of this germ, which describes the a priori allowed transformations. For instance, for dodecaphonic compositions, we have P1 = D12,12 . It is however not necessary to realize the entire orbits Pi .Gi of these group actions, a composition will usually select a small number of such transformed germs. In the construction of the germinal melody of the “Synthesis” composition in section 11.6.3, see also figure 11.16, we have 26 isomorphism classes of three-element motives and take one representative of each class, where the transformation group is the same for all classes, i.e., the full affine group on Z12 . The main point in this procedure is the instantiation of the transformations within the composition. In fact, it is very rarely sufficient to place just two objects Gi , p.Gi in contiguous position in order to communicate the particular symmetry p which is responsible for the
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succession. It is part of the failure of dodecaphonism that its exponents never did care about instantiating p. A nice example of such an instantiation in Beethoven’s op. 106 was given in section 28.2.5.
48.1.2
Variations
Summary. Instead of transformation groups, a set of local “germs” can be deformed according to a set of similarity paradigms (e.g., gestalt topologies) –Σ– In this approach, there is no group action and the original germ Gi must be associated with one of its specializations by use of deformation procedures. For example, one might superimpose a force field on the given germ and then move its points around according to the local force action. Such a procedure has been implemented in the OrnaMagic module of the prestor software, see section 49.3. This variational change of perspective is very precious for understanding a germ’s potential, but in general will not give us back the original germ. The quantity of change is a function of the used gestalt topology whose choice depends on the composer’s preferences.
48.2
Global Poetical Syntax
Summary. According to rules of contiguity and semantic “added value”, a variety of transformed and/or deformed local germs is distributed along the syntagmatic space. We systematize the possible procedures according to horizontal and vertical poetical functions. –Σ– The construction of a global assembly of local charts which are provided by the local techniques is very delicate, see also Pierre Boulez’ reflections [60, II]. For instance, the germinal melody in “Synthesis” was built from the generic motif class representative, then gluing it with a representative of the first different motif class that fits with the first representative on a subset of two points. But the possibility to find such a solution until every class was represented exactly once is not obvious, it is a constraint to the compositional material that could have ended in an incomplete solution. The normal procedure is a puzzle reconstruction: One begins on any interesting germ and adds other germs at any syntagmatic position (not necessarily adjacent), and successively densifies the display, thereby observing constraints on the nerve that one wants to construct. For example, in the sonata “L’essence du bleu” [368], the plan as described in [328] was to construct a global composition on a motivic zig-zag {c, c] , d, d] , e, d] , d, c] , c} by use of nine three-element submotives in such a way that the nerve turns out to be a M bius strip. The construction of a nerve is not only complicated as a constraint problem, it is also a problem of narrative character: the intersection configurations must also be mediated on the concrete time axis, and in such a way that the transformations between associated units can be made explicit, e.g., the M¨ obius strip must be boiled down to the time axis such that its
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one-dimensional skeleton (we stick to the ideas from section 47.2) is transgressed on reasonable paths. Evidently, the main point of this global patchwork is the syntagmatic distribution of paradigmatic representatives in order to achieve an added poetical value in the sense of Jakobson’s poetical function. This objective splits into two subtasks: horizontal and vertical poetical functions according to Jakobson and Posner, respectively.
48.2.1
Roman Jakobson’s Horizontal Function
Summary. Distribution of a variety of transformed and/or deformed local compositions can follow Jakobson’s correspondence [245] along the syntagmatic axis. –Σ– Jakobson’s poetical function was introduced in section 11.6.1. Counterpoint, harmonic syntax or dodecaphonic row distribution are examples of this mechanism. The most elementary realization of this function is the action of a translation group on rhythmical units in order to produce a cyclic character of our composition. This semantic enrichment creates a coherence of the compositional corpus which is basic to all other elements. It may, for example, connect motives, harmonies and similar units if they are distributed on the regular positions of the rhythmic basis. But it also creates a time circle that abolishes the straight physical time-line and rebuilds and “good” cyclic eternity (in Hegel’s sense) which our finite life was to negate.
48.2.2
Roland Posner’s Vertical Function
Summary. Jakobson’s poetical function is orthogonal to Posner’s vertical function which relates different signification levels in denotation and connotation. We discuss strategies which make use of this functionality in the production of poetical “added value”. –Σ– Jakobson’s poetical function relates signs of the same semiotic system, for example, two phonological signs such as “bad”, “dad” which are positioned in syntagmatically equivalent places, in a rhyme, say. But this structure is enchained with the semantic sign system (by the double articulation of language, see [361]) where the phonological equivalence in the rhyming places produces a paradigmatic equivalence between the bad and dad, i.e., “dad is bad”, a meaning which is neither inherent in “bad” nor in “dad”. So the level of equivalence is shifted from the phonological to the semantic one. This shifting of equivalence in Jakobson’s function illustrates the vertical function as proposed by Roland Posner [421]. In music, such a construction is not obvious since connotative semantic levels are not automatic, but must be constructed or simulated. This can be achieved by a small global composition which is attached to a local ‘signal’ composition whose equivalence to another local ‘signal’ composition on the horizontal Jakobson function induces an equivalence of the attached small global compositions. Such a small global composition can be a configuration consisting of a chord interpretation and a melodic unit, for example. So here, the connotative levels are constructed by successively enriched small global compositions within the total global composition.
48.3. STRUCTURE AND PROCESS
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Structure and Process
Summary. The poietic process is confronted with the resulting ‘score’ structure in order to recall a more general approach to the score concept. –Σ– We have stressed the global composition as a target of compositional strategies. However, the making of this composition can be more relevant than the result. In other words, the general score concept which we discussed in section 33.3.2 may be specialized to the interior score known from jazz: Instead of a more or less fixed score structure in form of a global composition, the processes that lead to the effectively played material are stressed, while the resulting structure is quite secondary (albeit not irrelevant, as some fundamental critics of free jazz have argued) in that it is one possible variant of those processes, and another variant—as an exemplification of the operating processes—would do as well. The situation is comparable to algorithms for complex shapes, such as L-systems or fractals. They are defined as processes, and the resulting variants are all an expression of the same basic process type, as with biological phenotypical expression of the genotype1 . It is a very interesting research area to classify process score types and to implement corresponding tools in music composition software. An example of such an approach (the composition software OpenMusic) is given in chapter 51. The following chapter 49 describes a software (the composition software prestor ) which is mainly based on paradigmatic strategies. As a matter of fact, such strategies are very difficult to handle because a paradigm is, by definition, in absentia to the text, whereas syntagmatic structures are in praesentia, which means that a software that is built on the LEGO-like juxtaposition of bricks will have an easier acceptance than the paradigmatic one. However, if the bricks are no longer concrete musical material, but abstract units of processual character, the ease of the LEGO approach is no longer valid.
1 Phenotype is the “outward, physical manifestation” of the organism, while genotype is the “internally coded, inheritable information” (written in the genetic code) carried by all living organisms.
Chapter 49
The Paradigmatic Discourse on presto r
Damit k¨ onnen Sie mich eine ganze Nacht lang allein lassen! Herbert von Karajan to Guerino Mazzola on the occasion of the presentation of the prestor prototype in Salzburg 1984 Summary. The prestor composition software was developed as a commercial implementation and operationalization of mathematical music theory. Several large compositions have been successfully realized on this software [48, 49, 338]. One of them will be discussed in chapter 50. We describe the overall architecture and functionality of prestor . This is mainly driven by a local/global paradigmatic perspective. The paradigmata of transformation and deformation are realized on (1) the level of modular affine transformations in the four-dimensional space of onset, pitch, duration, and loudness, and (2) the level of variational deformations. Both paradigmata are discussed and exemplified. We conclude the chapter with a remark on the problem of abstraction in paradigmatic composition, since composers tend to have major difficulties to get familiar with abstract paradigmatic structures. –Σ– Although the software was implemented in ANSI-C on now historic Atarir computers, we first describe its functional scheme since it represents a prototypical and still unique way of thinking paradigmatic composition strategies (to our knowledge, no similar strategy has been implemented on software to the present). The prestor software as well as the entire source code are GPL and can be downloaded—together with a manual, examples, and a concept presentation software prestino.prg—from the book’s CD-ROM, see page xxx.
49.1
The presto Functional Scheme r
Summary. We describe the functional scheme of prestor . The scheme is centered around the 945
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back bone of global score, score, and local score. This threefold architecture reflects an early stage of the global-local paradigm from mathematical music theory. –Σ–
Figure 49.1: The functional scheme of the prestor software. It is centered around the back bone of global score, score, and local score. The graphics of this chapter are all from the original prestor manual [338]. The functional architecture of prestor consists of a series of so-called scores, see figure 49.1. The global score, the score, and the local score deal with geometric representation and editing of notes (mental note events) on global to local space levels. The recording score allows for mousedriven and instrumental real-time input, the transformation score deals with graphical input of affine transformations, and the grid score allows for graphical input of ornamental grids. The input goes via recording via mouse or MIDI instrument, loading from MIDI or prestor data files, and painting with the mouse. Editing features split into affine transformations on charts of notes, building of ornaments, all-parametric variations according to ornamental attraction and repulsion fields, instrumental and parametric coloring of local charts, Boolean combination operations on groups of notes, and performance editing, especially construction of complex tempo hierarchies. For the latter, please refer to our corresponding discussion in section 38.2. Output is split into audio and SMPTE, saving is on MIDI and presto formats. Figure 49.2 shows the main window of prestor . The rectangular score window in the middle shows onset (horizontal) and pitch (vertical, can also be set to loudness or duration),
49.1. THE P REST Or FUNCTIONAL SCHEME
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and Schumann’s Kinderszene “Haschemann”. The black subrectangle marks a block which may be copied, cleared, moved, etc. Below the score, the tempo curve is visible. Above the score, the total composition space of the global score (narrow rectangle) is seen, below the tempo curve, six registers for provisional local parts of the score are placed. On top, we see 16 icons for instrumental colors (MIDI program change data), checked means the instrument is active; while editing, one may work on any checked subset. The different score and register windows
Figure 49.2: The main window of prestor . The rectangular score window in the middle shows onset (horizontal) and pitch (vertical, can also be set to loudness or duration), and Schumann’s Kinderszene “Haschemann”. implement some chart types in the global composition strategy. One may perform any Boolean operations on such parts, e.g., on two registers. Evidently, modern object-oriented windowing techniques could vastly generalize this elementary implementation. For a detailed editing of small composition parts, the local score window is available, see figure 49.3. The local score can be opened by double-clicking on a selected position on the score; after editing it can be merged to the score. A miniature view of the local score’s content is visible on the right lower corner of the main window in figure 49.2. The local global paradigm is realized on the local score via the feature of “coloring”. On the local score, we view a four-dimensional cube, representing the four-dimensional discrete torus Z471 , in any of the six relevant projections Onset+P itch, Onset+Loudness, etc. onto two of the four parameters of pitch, onset, duration, and loudness. All parameters are integers modulo 71 and suitably calibrated: Pitch is a MIDI
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Figure 49.3: The local score is the interface for detailed editing of small portions of the actual composition. The darkened polygon marks a region where any operation, such as copy, erase, transform, change the instrumental icon, or simply play can be performed. New events can be defined by mouse drawing actions.
Figure 49.4: The transformation score is the interface for detailed editing of affine transformations of the torus Z271 . The user’s editing acts on a number of standard transformations (buttons to the right, middle), matrix text fields, and a graphical editing option by direct drawing of affine images of the unit square.
key number in the interval [27, 97], loudness is a MIDI velocity value between velocity 0 and 127, equally distributed among 71 values, onset and duration are in integer units that can be set to be different fractions of a bar, depending on the time signature and the resolution preferences for each metrical unit. The local-global paradigm is realized by the coloring feature: The user may define any closed polygon in a 2D projection, and thereby select all events that lie in that region with regard to the given projection. Such a coloring domain can be used to do many different things, such as moving around on Z471 , copying, erasing, transforming, setting new instrumental icons, playing. By mouse-driven drawing actions, new events can be inserted. The parameter plane can be changed at any time. To the right, we see the orchestration (only piano in figure 49.3, see below for complex orchestras), where any instrumental icon can be unchecked for instrumentally specified editing options. Of course selection of a toroidal representation and manipulation (for example for shifting operations, where events exit to the right and reenter from the left) is questionable, but once you are dealing with affine operations, such finiteness decisions have to be taken. The advantage of this selection will become clear in the next section 49.2.
49.2
Modular Affine Transformations
Summary. This section describes the mathematical framework of modular affine transformations in Z271 , their four-dimensional extension and the combination options for such transformations, their action on selected local charts, and the graphical input features. –Σ–
49.3. ORNAMENTS AND VARIATIONS
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The selection of the torus Z471 was motivated by small number of pixels in those early screens, together with the need for icons in order to represent instrumental ‘colors’1 . Now, the selection of a prime number was not by case. In fact, the choice offers us a finite field where non-zero determinants of matrices automatically yield invertible matrices. So a large number of transformations becomes invertible, and therefore a large number of compositional processes become non-destructive. The software offers a transformation interface where any affine endomorphism in Z471 @Z471 can be defined graphically as well as numerically see figure 49.4. We first use the fact (see section E.3.6, theorem 53) that Z471 @Z471 is generated by its transformations which only move around two of the four dimensions. So the user may transform his/her material on the selected plane, then switch to another plane for further transformations, etc. The user may directly define a product of any number of plane transformations (for different planes) by a feature which memorizes a list of plane transformations. We have a total of 7116 .714 = 100 5960 6100 5760 3910 4210 0320 6620 8670 1400 1330 2020 401 ≈ 1.05966 × 1037 elements in Z471 @Z471 , and the number of 100 4450 2600 4660 8320 4830 5790 4360 1910 9050 9360 6400 000 ≈ 1.04453 × 1037 −→ elements in GL(4, 71) according to the formulas in appendix C.3.5. The transformation score in figure 49.4 shows the basic input choices: matrix coefficients, standard transformations, and graphical editing by definition of the images of the three points: origin, head of horizontal unit vector, head of vertical unit vector. The parallelogram image of the unit square is visualized on the local score. The geometric advantages of the prime number 71 are abundant, among others, we have ∼ these facts: Since the multiplicative group 71× is cyclic of order 70, i.e., isomorphic to Z70 → −→ ∼ Z2 ×Z5 ×Z7 , we have fifth and seventh roots of unity in GL(4, 71). We also have SO2 (71) → Z72 (see appendix C.3.5), and therefore, a generator D of the special orthogonal group. Such a generator is ! 30 33 D= 38 30 which means that we may view D as a rotation by 360◦ : 72 = 5◦ . In the software, we have implemented the rotation by the triple, i.e., D3 ∼ 15◦ -rotation. Of course, such rotations in the modular torus can transform harmless compositions into very wild looking variants, but we have always experienced that the transformed version maintains certain regularities that were present in the original form.
49.3
Ornaments and Variations
Summary. Modular affine transformations are used to build ornaments by translation grids of “cells” of notes. Such ornaments are used directly as periodic note sets, for instance in drum patterns. They may also be used as “background” ornaments whose points act as centers of attraction or repulsion for other notes. The latter method is a multidimensional generalization 1 Because
of low resolution, color was not a good commercial option in 1988
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of the well-known variational technique, in particular of the alteration of pitch sets (scales, chords, motives, etc.). –Σ–
Figure 49.5: An ornament is defined by a motif, together with a translation grid and a range in each direction of the defining grid vectors. The grid cell is the parallelogram spanned by the generating vectors. The software’s module for ornaments is termed OrnaMagic. The idea is this: The user first defines a motif M , either a small one on the local score, or an arbitrary large one on the score. M is just a local composition on the four-dimensional EHLD space of the software. Next, two (usually linearly independent) translation vectors gh , gv define the grid, i.e., the group hegh , egv i. This group operates on the motif M and yields a translated motif Mi,j = ei.gh +j.gv .M for each integer pair (i, j), see figure 49.5. The user defines a two-dimensional ornament ofSa special ornament window, the grid score, see figure 49.6. This gives us a ‘grid of translations’ a≤i≤b,c≤j≤d Mi,j . A second method allows the user to define also larger grid vectors on the score level, but the principle is the same. In the composition “Synthesis” to be described in the subsequent chapter 50, we recognize a number of superimposed ornaments of drum sounds; here in the second movement, see figure 49.7. The compositional principles will be described in section 50.3. Apart from this explicit usage of ornaments, there is a second truly paradigmatic, more precisely: topological usage by two-dimensional alterations. Here is the general procedure. We
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Figure 49.6: The grid score is the interface for defining an ornament via its grid vectors (socalled “horizontal” gh and “vertical” gv ) and the range a ≤ i ≤ b, c ≤ j ≤ d, i.e., the interval of horizontal/vertical translation ei.gh +j.gv to be effected on the motif M .
have two local compositions, L, G, where G plays the role of the driving grid, whereas L is the composition which we want to deform. Usually, G is defined by an ornamental construction as described above, but this is not mandatory, i.e., one may also take a large motif M and just apply the grid group on the zero range (a = b = c = d = 0). Given these data, the alteration works as follows. Each event x of L is shifted towards or from G according to a procedure which starts by the calculation of a ‘nearest point’ G(x) to x in G. This point is found starting on x, and then moving along a spiral outward around x, until the first point of G is hit; this is G(x). Following this first algorithm, several alteration strategies are offered. First, one may choose a degree of deformation, say y%, a positive or negative real number. This means that the vector G(x) − x is stretched to y.(G(x) − x), and that the alteration of x is set to x+y.(G(x)−x). However, this is not the last word in alteration, one may also define its direction. Typically, this is vertical (conserving onset time), or horizontal (conserving pitch), etc. But the vector G(x) − x is neither one nor the other. So we have to introduce the projection pt (G(x) − x) of the difference G(x) − x to the direction t. This yields the final alteration At,y,G (x) = pt (y(G(x) − x)) + x of x according to the system variables t, y, G.
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Figure 49.7: Drum patterns in the composition “Synthesis”. This is all implemented in the OrnaMagic module of prestor . Moreover, given a block with onset limits u < v, the user can define a successively increased alteration by setting the percentage to y(u), y(v) and altering from At,y(u),G to At,y(v),G as onset moves from u to v. Such procedures have been used in the composition “Mystery Child” in [49]. This piece can also be heard from the book’s CD-ROM, see page xxx. The effect resembles a morphing operator which lets the background ornament G act on L with successive variation of its “alteration force field”. As a special case, we have the classical pitch alteration that is driven by a background tonality G, as well as the onset alteration, i.e., better known as groove effect on sequencers, defined by an onset grid that is derived from a rhythmical onset configuration. These special effects are implemented as “easy alteration variants” in prestor . Example 60 A special application of pitch alteration is tonal inversion: If we want, for example, to apply tonal alteration of the C-major scale around e (leaving e fixed, exchanging d and f, c and g, b and a), we may first apply the real inversion Ue and then apply the tonal alteration downwards with respect to C-major.
49.4
Problems of Abstraction
Summary. Whereas syntagmatic contiguity is a relation in praesentia, paradigmatic associa-
49.4. PROBLEMS OF ABSTRACTION
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tivity is a relation in absentia. This difference has heavy consequences on the ease of managing paradigmatic composition tools. As a consequence, composers rarely transcend straightforward paradigmatics and prefer sticking to syntactical composition software, such as MAX, or common sequencer software. –Σ– The composition software prestor was designed as a tool for everybody, even for the non-experts in musical notation. The publicity and the product concept tried to invoke the statement “Beethoven’s creativity at everyone’s reach”. The argument was that drawing music (see figure 49.8) is an interface that could give everybody access to the otherwise abstract and heavily codified music. Also, the simultaneous presentation in geometric coordinates of EHLD space, together with the 2D projections, and the presentation in notes (see figure 49.9) was thought to be a big advantage against classical approaches to composition. Now, at the center of this concept are evidently the TransforMaster (symmetries), the OrnaMagic (ornaments), and the AgoLogic (tempo curve hierarchies) modules. The first two of them are paradigmatic tools par excellence. They are powerful, but require strong abstraction capabilities. The absentia of the paradigm enforces a selection process of representatives, and a memorization of their relation to the paradigm. This reality switch is a major obstacle to a breakthrough of such a software concept. It seems to be a major task of future composition software development to build concepts that enable seamless transitions from abstract to concrete composition layers. In the limit, one should have very abstract objects, such as self-addressed compositions or even more generally addressed objects (why not play a functorial global composition, how would one do that?) at one’s fingertips and handle them completely naturally as if they were normal notes or chords, and in fact: they are completely normal, only we did not learn to sonify them in appropriate shape. In this regard, Tom Johnson’s compositorial approach [253] is one of the most promising in trying to morph mathematical objects into musical events.
Figure 49.8: The prestor concept tried to merge Beethoven’s creativity and ease of a graphically interactive interface for musical composition. The local score here shows a drawing of Beethoven in a geometric space, and using instrumental icons.
Figure 49.9: The geometric representation on the local score can also be viewed in classical score signs. One can immediately understand the geometric shape, whereas the common notation yields a completely cryptic object.
Chapter 50
Case Study I:“Synthesis” by Guerino Mazzola Habe die CD nun gr¨ undlich mir angeh¨ ort und frage mich, woher kommt bloss die Kraft, die so was Sch¨ ones schafft. Jazz Saxophonist Werner L¨ udi on “Synthesis” [309] Summary. “Synthesis” is a composition for piano, percussion and e-bass. Its global and local organization was driven by classification of local compositions and modulation theory on one hand, and by the prestor software tool on the other. We describe the overall organization and the four movements. –Σ– “Synthesis” is documented on CD [339]. It is the result of a composition grant of the city of Zurich and was composed and recorded in 1990 using the prestor software, together with this hardware: one Atarir Mega ST4 computer, the synthesizers Roland R-8M (drums and percussion), Yamaha RX5 (drums, percussion, and special sounds), and Yamaha TX802 (bass), and a Steinway grand for the piano part played by the author. The drum, percussion, and bass parts of the composition were written on prestor in four months and then completed by the piano part in an additional two months. The music critics did not recognize on the CD (where no trace of the computerized music was given) that the whole composition’s percussive and bass parts were synthetic. This is what G´erard Assayag rightly calls a Turing test for the viability of computer-aided composition, in particular since in the field of jazz this technology is thought to be an impossible tool. The original presto-files (extender .sto) as well as the audio-files of “SYNTHESIS” are available on the book’s CD-ROM, see page xxx. 955
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CHAPTER 50. CASE STUDY I:“SYNTHESIS” BY GUERINO MAZZOLA
The Overall Organization
Summary. We discuss the overall organization: its material, composition principles in the four movements, and the instrumentation strategy. –Σ– The overall organization is a four-fold view on a generic material: the 26 classes of threeelement motives in OnP iM od12,12 (see appendix M.3 for the class list). The multiplicity of views is given by classical forms: • the sonata form, • the cycle of variations, • the scherzo, and • a finale.
50.1.1
The Material: 26 Classes of Three-Element Motives
Summary. The entire composition is based on the 26 isomorphism classes of three-element motives in OnP iM od12,12 (see section 11.3.8) and its specialization tree. We make precise the different usage modes which have been realized in“Synthesis”. –Σ– After the inspiring analysis of the Schubert–Stolberg work in 11.6.2, the usage of the 26 classes was recommended. The classes were used in different contexts. We refer to section 11.3.8 for the theory and to appendix M.3 for the 26 classes, in particular figures M.1, M.2 for representatives and the specialization Hasse diagram, see section 12.2.2 for specialization. To begin with, representatives of all classes were patched together to build the germinal melody already described in section 11.6.3, see especially figure 11.16. This melody appears explicitly in the introduction to movement four, see figure 50.1. The 26 motives also appear as percussive “phonemes” in the third movement, as already discussed in section 11.6.3. In the second movement, the germinal melody is altered according to ornamental deformation techniques and yields a sequence of characteristic melodies which are played by the bass and harmonically ornamented by the piano. Already the very first percussive motif in movement one (just after the piano solo intro) is the germinal melody, however played with pitches being encoded by percussive sounds. And in movement four, these motives are used to define ornaments and fractal refinements thereof. More schematically, this strategy is shown in figure 50.2.
50.1.2
Principles of the Four Movements and Instrumentation
Summary. The four movements include a sonata form, a variational sequence, a scherzo and a finale. The instrumentation issue is intimately related to the structural aspects. Each movement bears its specific constraints on the role of instrumentation. –Σ–
50.1. THE OVERALL ORGANIZATION
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Figure 50.1: The beginning percussion rolls in movement four of “Synthesis”, as extracted from prestor ’s main window, together with the repeatedly accelerating tempo curve. The germinal melody comes twice just before the piano intro solo. The principle of a four movement concert such as are encountered in the classical concert form is realized as follows, each movement with a total of 122 instruments: First movement: Earthquake/Full Force, duration: 10:46. It is devoted to the Greek element of earth. It is a sonata form which exposes the ideas, followed by modulatory development, a reprise and ending by a coda. Instrumentation: Yamaha RX5, TX802, Roland R-8M (PCM cards: Jazz, Jazzbrush, Ethnic Percussion). Second movement: Liquid Colours/Sea of Faces, duration: 14:05. It is devoted to the Greek element of water. It is a cycle of variations. These variations are taken from the germinal melody and are concatenated in a paradigmatic way. Instrumentation: Yamaha RX5, TX802, Roland R-8M (PCM cards: Jazzbrush, Ethnic Percussion, Mallets). Third movement: Poem of Wind/Fly! Fly! Fly!, duration: 09:27. It is devoted to the Greek element of wind. It is a scherzo. The motivic alphabet of the 26 classes is produced in poetical arrangement of rhythmic patterns.
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26 motif classes GLUING, SYMMETRIES
CORRESPONDENCE
poetical production via Baudelaire
germinal melody SELECTION, SYMMETRIES
3rd movement
SELECTION, REFLECTIONS, ORNAMENTS rhythms and their modulation 1st movement
bass licks all movements
DECOMPOSITION, ALTERATIONS, ORNAMENTS
variations according to Messiaen grids 2nd movement
SELECTION, DILATATIONS, REFINEMENTS
fractal refinements 1st movement
Figure 50.2: The overall strategy in “Synthesis” is driven by exploiting the 26 classes of threeelement motives. Instrumentation: Yamaha RX5, TX802, Roland R-8M (PCM cards: Jazz, Contemporary Percussion, Mallets). Fourth movement: Burning Spears/Interstellar Space, duration: 10:28. It is devoted to the Greek element of fire. It is a finale. Self-similar repetitions and refinements of the germinal structure successively densify the musical material. Instrumentation: Yamaha RX5, TX802, Roland R-8M (PCM cards: Jazz, Jazzbrush, Contemporary Percussion)
50.2
1st Movement: Sonata Form
Summary. The first movement “Earthquake” is a sonata form which uses the modulation theorem (see section 27.1.4) in time dimension. –Σ– This movement is a sonata form for rhythmical structures. After the piano intro, the rhythmical germinal melody is played on toms (pitches are associated with different toms) from
50.3. 2N D MOVEMENT: VARIATIONS
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the RX5 synthesizer. During the exposition, the germinal subject is enriched by drum and percussion, multiplied and profiled by bass lines. The flight of the piano over the rhythmical carpet indicates an intense cadence of the exposition, ending on a fermata. The development consists of three rhythmical modulations (see sections 28.3.1 and 28.3.2). This delicate process includes four different rhythmical macro-scales, which each consists of seven rhythmical three-tone motives from the 26 classes (see also figure 28.13). After the fermata, the first scale is built and cadenced by regular “falling drops” from the ohkawa instrument. Subsequently, its character is neutralized and transformed into a second scale. You hear the cadence of this new scale as being again marked by the ohkawa’s regular “falling drops”, this time enriched by a rain of light piano pearls. After a further fermata, the second scale is altered into a third one, whose cadence coincides with an intermediate climax of the piano. When the piano finally recedes, the third scale is neutralized until a march-like turning point introduces the fourth scale, whose cadence terminates the development by another fermata. The reprise follows in a slightly altered instrumentation. The finale starts after a short fermata. It is recognized on a heavy rock rhythm which is played in reduced tempo on the kicks.
50.3
2nd Movement: Variations
Summary. The second movement “Sea of Faces” follows a syntax of melodic variations of a fundamental melody which entails a particular “harmolodic” color in the vein of Ornette Coleman’s music/theory. –Σ– This adagio is a labyrinthic wandering of piano, bass, barafons, glockenspiel, sanzas, gender, tube bells etc. through seven melodic variations of the germinal melody. Drums and percussion also obey these deformation forces, each harmonic, rhythmical, and melodic change is a specific expression of these forces. Scales and rhythms are known to be periodic structures in pitch and onset. By the choice of two scales from the first three Messiaen scales, one for pitch, one for onset, we generate an ornament, a “harmonical-rhythmical” scale. After certain rotations of such ornaments, the harmonic and rhythmical Messiaen components in onset and pitch are completely mixed. Here, the deformation forces of our germinal theme become manifest: Each tone event is displaced according to the ornament alteration algorithm described in 49.3. The deformation uses the following grids (see figure 50.3): We take the Messiaen scales M1 , M2 , M3 as described in example 13 of section 8.1.1. For each grid, we take a pair (Mi , Mj ), i, j = 1, 2, 3 and build the cartesian product Mi × Mj of two such scales, one in pitch, the other in onset direction, and each of them with a large range such that it acts on the germinal melody as if it were infinitely extended. The action is taken to be 100% without any directional constraints. This yields nine variations of the germinal melody, as shown in figure 50.4.
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Figure 50.3: The original graphics for the grid construction. To the left two Messiaen grids, to the right their transformation via a rotation-dilatation which is used for rhythmical connections. The Messiaen grids are cartesian products of the Messiaen scales and rhythms.
Figure 50.4: The original graphics for the nine variations of the germinal melody G according to the Cartesian product grids Mi ×Mj (upper part). The variations are played in the order shown below, however ending on variation Mi × Mj , for reasons of the length limits of the movement
50.3. 2N D MOVEMENT: VARIATIONS
961
In the second movement, seven of the nine variations are used in the order shown in figure 50.4, lower part. The bass plays these variations in a very extended slow gesture, and the piano ornaments these melodic lines by a harmonization work.
Each couple of such variations is connected by a rhythmically complex intermediate structure, see figure 50.5. The intermediate parts terminating in the Mi × Mj -variation are constructed as follows: We take the Messiaen grid Mi × Mj and operate a dilatation-rotation by 45◦ on this object, see figure 50.3, right part, for two such transformations. From these transformed local compositions (or from a very similar skew transformation) we build an interpretation by a partition into five subcompositions which are again transformed as visualized in figure 50.6. These charts are then taken as motives of different ornaments which are generated by the grids as shown in figure 50.7. The upper part of figure 50.7 shows the superposition of the five ornaments generated from Messiaen grid M3 × M3 . This rhythmical construction is also seen in the right and lower part of figure 50.5.
Figure 50.5: The original graphics for the intermediate structure between two successive variations, here connecting the M2 ×M1 -variation to the M3 ×M3 -variation. The connecting structure is a rhythmical construction that is deduced from the generating Messiaen grid M3 × M3 .
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Figure 50.6: The original graphics for the interpretation of the rotated Messiaen grid and the symmetries applied to the partition’s components.
Figure 50.7: The original graphics for the ornaments that are generated by charts of a rotated Messiaen grid, following different grid translation vectors.
50.4. 3RD MOVEMENT: SCHERZO
50.4
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3rd Movement: Scherzo
Summary. The scherzo “Poem of Wind” is based on a transcription of Charles Baudelaire’s last poem “La mort des artistes” in “Les fleurs du mal”. –Σ–
Figure 50.8: The beginning of the third movement with its breakneck changes in agogics. This movement is in complete contrast to the quiet flowing of that adagio. It resembles Ernst Jandl’s concrete poesy. A poem of wind: without firm ground or romantic sky. Here, the piano builds an expressive dialog with a pointillistic percussion. The first two strophes of Charles Baudelaire’s “La mort des artistes”is the last poem in his famous “Les fleurs du mal”. The transformation of this poem was already discussed in section 11.6.3. This procedure transforms every word into a sequence of rhythmical motives, we hear it as a coherent sound sequence which is separated from its successor by a short fermata. At the end of each verse, longer interruptions are inserted. From this raw material the final composition is constructed by an extremely refined agogical architecture, changing from breakneck accelerandi to stagnating ritardandi. Here, the third movement makes extensive use of the prestor -AgoLogic for tempo curves. The single motives are also enriched by echo-like variations and repetitions, such that their color engraves itself on ones mind. The answer of the piano to this witty poesy is the albatross in the air, which flies, falls and plays with Cecil Taylor’s florescence as if it were an old tale. Just one not too many salty swift—and goodbye.
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50.5
4th Movement: Fractal Syntax
Summary. The fourth movement “Burning Spears” is constructed in a completely non-syntactic way from fractal principles. –Σ– This finale is a steadily evolving rhythmical organism whose form could be compared to Alexander Scriabin’s ecstatic sonata “Vers la flamme”. The movement gets off by a drum intro, followed by a bass intro which announces fanfarelike that something is to happen. It receives a promising answer from the low piano keys until a samba-like rhythm makes the final run. This starting rhythm is again derived from the germinal melody. You can hear its essential elements from the drum intro. After two roll groups, the samba reappears, but now with augmented time values, whereas the now opened gaps are filled up by new micro rhythms (see figure 50.9 where the original motives are seen in the low and high regions, and figure 50.10, where the stretched motives are now also visible in these regions). This process of intensification is cadenced until a cutting bass line terminates the developmental part. The piano dances like an entranced sorcerer over this rhythmical lava. In the reprise, the samba figure reappears, but with a higher tempo. It ends in a strong pulse which leads to a further time augmentation after a bass cadence. This time, the black dancing extension figure is refined by a cackling entwinment by wood blocks which is joined by a fast dialog with the piano. The following earthy rhythm is introduced by a bass drive as if we should be warned that we are now changing the civilizations. Here too, the construction is an augmentation of rhythm and an intensification. What we are hearing is in fact a fractal repetition of self-similar time structures. The last and extremely fast part expresses the force of a burning spear, which is thrown into the sound sky by a delirating dervish dancing on an exploding volcano... So the musical principle of this finale is not architectural, but a self-renewing process.
Figure 50.9: The original motives are seen in the low and high regions of the local score.
Figure 50.10: The stretched motives are also visible in the low and high regions.
50.5. 4T H MOVEMENT: FRACTAL SYNTAX
965
Figure 50.11: The end of the fourth movement shows a particularly dense and polyrhythmic percussive part.
Chapter 51
Object-Oriented Programming in OpenMusic Le probl`eme que l’ordinateur pose au compositeur n’est pas d’abord d’ordre sp´ecifiquement musical, mais avant tout culturel et philosophique. Il implique une refonte compl`ete des rapports de l’abstrait et du concrete. Le musicien, le musicologue, l’auditeur doivent bien se rendre ` a l’evidence de ces changements sans aller pour autant jusqu’` a la maladie de l’adaptation. Hugues Dufourt [130] Summary. OpenMusic is a visual programming language for music composers. It was designed and implemented by the Musical Representation Team at Ircam-Centre Georges Pompidou. OpenMusic is based and implemented on CLOS (Common Lisp Object System) [510]. It shows several original features, such as reflexivity, meta-programming capacities, handling of the duality between musical and computational time, and provides a framework of predefined musical objects for handling sound, MIDI and musical notation. OpenMusic combines different technics of programming, e.g., functional programming, constraint programming and object-oriented programming. We will focus in this chapter on the last one. Object-oriented programming is crucially connected with the categorical approach. Category theory helps formalize in an original way concepts like inheritance, methods, classes, etc. More details on this relation can be found in [306] or in section 9.4.2 of this book. –Σ– Although one may consider music composition to be an important issue in any computer music research or development, the term Computer Assisted Composition (CAC) has taken a specialized meaning during the past years. As opposed to the generation and processing of audio signal, by means of DSP hardware or software technologies, CAC systems such as OpenMusic focus on the formal structure of music. The software technology is rather based on symbolic computation, where the typical data structures (trees, graphs, sets, collections, associative memory, etc.) and algorithms (often issued from discrete mathematics) are suited to handle the complex structures involved in a compositional process. The great diversity of 967
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esthetic, technical, formal (or anti-formal) models, implies that one cannot conceive an environment of CAC as a fixed application which provides a fixed collection of generative procedures and musical transformations. On the contrary, we conceive an environment as a programming language, helping each composer to constitute his personal universe. Of course, there is no sense in providing a traditional programming language, the control of which requires a great technical expertise. For this reason, our purpose is to build a programming language conceived especially for composers. This leads us to think about the various existing models of programming, intuitive graphical interfaces (which enable control of this programming) and the internal as well as external representations of musical structures that will be built and transformed when using this programming. Therefore the main goal is to implement a language containing the concept of notation of the result (a musical score) as well as the concept of notation of the process leading to this result (visual program).
51.1
Object-Oriented Language
Summary. In this section we describe the main entities of our object language. Such language entities are usually called meta-objects. We describe each meta-object in a graphical and formal way. Basic calculus for object-oriented programming inherits the approach imposed by the precursory language Simula and its successors. The attempts to formalize this family of object models used the concept of parametric polymorphism [80]. More recently, languages based on multiple-dispatching (methods that dispatch on a product of types rather than a single type) such as CLOS could be formalized using concepts of overloading or ad-hoc polymorphism, found in λ&-calculus [81]. OpenMusic, which is based on CLOS, may be formally described in this way. Although we do not give here a real formalization of OpenMusic, we use the λ&-calculus to give a general idea of each meta-object. From a visual point of view, the meta-objects of our calculus are represented as graphical entities called frames. –Σ– Object-oriented programming is based on simple concepts. A program can be seen as a set of entities called objects. An object is made of data (slots) and operations applied on it (methods). Objects communicate in a specific way, usually called message passing. Most of the object-oriented languages implement the notion of class in order to abstract similar objects. New classes can be created from existing classes using the mechanism of inheritance. Inheritance allows the extension or partial modification of a class. If a class A inherits from a class B, A is called the subclass of B and B is called the superclass of A. Object-oriented programming offers in a natural way a dynamic management of resources, which means that we can create new objects at any time. The mechanism of object creation from a class is called instantiation. The new created object is called an instance of the class. Meta-objects are represented either as composed frames or simple frames. Several frames (i.e., different point of views) may be produced for the same object. The simple frames which represent an object are called views. They generally appear as icons. The composed frames representing an object are called editors. For more details about formal and graphical description of OpenMusic see [11].
51.1. OBJECT-ORIENTED LANGUAGE
51.1.1
969
Patches
A patch is a meta-object specific to OpenMusic. It reifies the notion of program. A patch is the place where objects will be interconnected in order to specify musical algorithms. From a formal point of view, a patch can be seen as a λ-function. Patches are composed by boxes (icons) and connections between them. Boxes represent functional calls while connections represent functional composition. Figure 51.1 shows the view and the editor of a patch implementing an algorithm for the expression x2 + 2. It can be formalized as the lambda function λx(x · x + 2).
Figure 51.1: View and editor of a patch.
51.1.2
Objects
In the λ&-calculus, objects are represented as registers. A register can be seen as a set of labeled fields l = v where l is called the label and v is called the value. For instance, an object representing the note C3 can be written as: note = hpitch = C, octave = 3i. Figure 51.2 shows the view and the editor of this note object.
Figure 51.2: View and editor of an object. The next two rules define the field selection and the field writing, respectively: hl1 = v1 , . . . , ln = vn i ◦ li 7→ vi , hl1 = v1 , . . . , li = vi , . . . , ln = vn i[li | v] 7→ hl1 = v1 , . . . , li = v, . . . , ln = vn i.
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CHAPTER 51. OBJECT-ORIENTED PROGRAMMING IN OPENMUSIC
Classes
If objects are seen as registers, classes will then be seen as generators of registers. The editor for a class is an ordered collection of views representing slots. Slots contain information about their name, their type (a class icon), a default value and a flag that indicates if the slot is public or private. View and editor for the class of the note object defined in the previous section are shown in figure 51.3.
Figure 51.3: View and editor for a class.
51.1.4
Methods
Methods are simple functions (λ-abstractions) where arguments are typed by classes. The editor in figure 51.4 shows a method with two inputs self and num of type Note and Integer respectively.
Figure 51.4: View and editor for a method. The body of this method increments the value of the slot octave of self by num and returns the note object. This method is formalized by the expression: λself:note λnum:integer self [octave | self ◦ octave + num].
51.1. OBJECT-ORIENTED LANGUAGE
51.1.5
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Generic Functions
A generic function is a collection of methods (&M1 &...&Mm ) where is the empty method. The type of a generic function containing methods Mi of type Ui → Vi , 1 ≤ i ≤ m, is {U1 → V1 , . . . , Um → Vm }. However, not any set of methods can be seen as a generic function. A set of methods with types Ui → Vi is a generic function iff the two following conditions are satisfied for all i,j: Ui ≤ Uj ⇒ Vi ≤ Vj , U is maximal in LB(Ui , Uj ) ⇒ ∃Uk , Uk = U where LB(U, V ) indicates the set of common lower bounds1 V of types U . Figure 51.5 shows a generic function composed of two methods: the first one is the method described in the previous section; the second one is specialized for inputs of type Number.
Figure 51.5: View and editor for a generic function.
51.1.6
Message Passing
Message passing is achieved by generic function invocation. We distinguish the application of a simple function from the application of a generic function, which will be indicated by the operator •. The application of a generic function G to arguments N of type U consists of two steps: selection of a method Mj among the methods of G and normal application of Mj to N . (&M1 &...&Mm ) • N 7→ Mj N. Note that U may not be contained in the set Ui of input types of the generic function. In this case we select the method Mj satisfying : Uj = mini=1...m {Ui | U ≤ Ui }. The CPL of a class is a linearization of its superclasses. The details of the linearization are not crucial for our purposes, see [510].
51.1.7
Inheritance
The inheritance mechanism is defined by the subtyping and the mechanism of method selection. Subtyping is defined by the following rules: 1 The
ordering ≤ is defined by the Class Precedence List (CPL) of the class U .
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U2 ≤ U1
V1 ≤ V2
∀i ∈ I, ∃j ∈ J, Sj → Tj ≤ Ui → Vi
U1 → V1 ≤ U2 → V2
{Sj → Tj }j∈J ≤ {Ui → Vi }i∈I
The rule on the left is the usual contravariant-covariant rule for arrow types. The other one states that an overloaded type is smaller than or equal to another overloaded type if for every branch in the latter, there is a branch in the former smaller than or equal to this one [81]. Graphically, a class inherits from another one when there exists an arrow from the superclass to the subclass, as is shown in figure 51.6. The class rnote extends the class note by adding a new slot called rhythm of type string with default value quarter.
Figure 51.6: Graphic inheritance.
51.1.8
Boxes and Evaluation
Boxes are placed in patches and allow other meta-object to be involved in the calculus. Boxes are composed of an icon and an ordered set of inlets and outlets. There are different types of boxes depending on the referenced meta-object. The user may create boxes in a patch by dragging meta-objects into it. Figure 51.7 shows different types of boxes and their references.
Figure 51.7: Boxes in a patch. Inlets in functional call and generic function call boxes correspond to the function arguments. For factory boxes, inlets and outlets refer to the slots of the class. Boxes and connections in a patch can be seen as a graph of functional compositions. By clicking on the output of any box, an evaluation from the corresponding point into the graph is induced. A box evaluation can give rise to other box evaluations creating a chain corresponding to the execution of a program.
51.2. MUSICAL OBJECT FRAMEWORK
51.1.9
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Instantiation
Users can create instances of a class with the aid of factory boxes (boxes built from a class). A factory contains a number of inputs corresponding to the public slots of the class. There are as many outputs as inputs. When evaluating an output, a new instance is created. The values returned by outlets are, from left to right, the instance itself, then the current value of its public slots. An instance can be visualized graphically as a box which can eventually be connected to other boxes (see figure 51.8).
Figure 51.8: Graphical instantiation.
51.2
Musical Object Framework
Summary. A framework is a set of reusable classes that one can use as building blocks for a specific software [179]. This section describes the musical OpenMusic framework. –Σ– OpenMusic offers a set of predefined classes and generic functions for musical representation and manipulation. This framework can be extended by using inheritance or by defining new methods in generic functions or by writing new generic functions. There are graphical editors for these definitions. In this section we will focus on the structure of musical objects (classes) rather than on their behavior (generic functions). The object-oriented concept of encapsulation can be defined as the separation between the internal representation and the interface of an object. These two aspects will be described in the next two sections.
51.2.1
Internal Representation
In OpenMusic any musical structure is a container which embeds other musical structures. All containers have a temporal extension (e) and an offset (o). e and o are expressed as a multiple of a rational unit v = 1/u, u ∈ N+ , the set of positive integers, v is considered a fraction of the quarter-note. The value u can be redefined at each level of the embedded structure. From the temporal point of view, we have entities coming from different rational time scales. In order to express them in a homogeneous way, we have implemented a hierarchical unit system, which will be discussed briefly by showing an example. Figure 51.9 shows a music fragment and an integer hierarchy describing it.
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Exercise 82 Define the corresponding form whose denotators are these rhythmical hierarchies.
Figure 51.9: Hierarchical representation of a rhythm. We can calculate the extension of each element ei by the recursive formula d(ei ) = where S is the sum of ei ’s brothers including itself, and F (ei ) is ei ’s father. The 4 deepest triplet in our example has an extension given by: 13 25 42 41 = 15 . For this reason, we set u to 15 (the quarter note is equivalent to 15 units). In this scale the new extension of the triplet is 4. Following the idea of a variable scale adapted to each embedded level in a musical structure the rhythm of figure 51.9 can be represented as shown in figure 51.10. ei S d(F (ei )),
Figure 51.10: Temporal organization of musical instances. Some basic operations for container manipulation are described below: • N ewContainer. Create an empty container (u equal 1 by default). • AddT o(c1, c2, at). Set the container c2 into the container c1 at the position at and calculate new values for u, e and o attributes. • RemoveF rom(c1, c2). Remove the container c2 from the container c1. • QReduce(c, [n]). Computes u0 = ppcm(ui ), where ui are units for all subcontainers of c. The attributes u of the container c is set to u0 . New values for parameters ui , oi and ei are calculated too. This operation allows comparison between all parts of a container. If a value n is given, then u0 = ppcm(ui , n). • QN ormalize(c). Sets u value for c and each of its subcontainers to optimal, i.e., the smallest possible integer number, considering the offset and extent values that have to be expressed as integer multiples of 1/u.
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The QN ormalize operation sets all extensions to integer values. If we want to compare all values we need to choose for the whole measure one unit u equal to ppcm(2, 3, 15) = 30. In this case the new values for u are 30 for the last eighth note and 30 for the triplet, see figure 51.11. The choice of using only integers for our representation allows us an easier translation to music notation and a coexistence of objects described in hierarchical time and in continuous time. Moreover, in this way we avoid reversibility problems coming from the translation between reals and integers (continuous and discrete time).
Figure 51.11: A normalized container. This model of containers and their operations represents in a natural way traditional musical structures like polyphony, measures, chords, etc. We will see in the next section how these containers are passed to the composer in an object context.
51.2.2
Interface
A summary of the predefined musical classes available in OM is given in figure 51.12.
Figure 51.12: Musical framework. There are three main classes of musical objects: • superposition, chords and polyphonies are made of other objects placed in parallel.
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• sequence, objects like voices, measures, etc are composed by other objects one after the other. • simple-score-element, these objects are terminals (empty containers). There is a predefined set of generic functions with methods for all musical classes. They are used to apply transformations to musical instances (i.e., transposition, inversion, etc.) or simply to play or visualize them. Editors of musical instances have been replaced by musical notation editors (figure 51.2 is replaced by figure 51.13). We use the paradigm Model-View-Controller, where the model is the musical instance, the view is its representation on the screen as a score and the controller is a user interface allowing us to change slot values.
Figure 51.13: View and editor of a note instance. As we have seen in the previous section, musical entities can be represented as containers. However it is not suitable for composers to have to build musical objects from containers. In order to create a musical instance we must provide to the composer a symbolic abstraction of the real musical object. For instance, MIDIcents are a good way to code pitch values. On the contrary, problems arise when the rhythmic content is taken into account. We propose in the next section a representation of rhythmic information adapted to the composer. 51.2.2.1
Rhythmic Trees
Rhythmic Trees (RT s) are the base of the rhythmic representation in OpenMusic. They must be understood as an alternative description of symbolic rhythmic structures using traditional music notation, and an external specification for container objects. • Syntax An RT is defined as a pair (D S) where D is an integer ratio (> 0) and S is a list of n elements. Each element in S can be either an integer or a RT . Here is an example corresponding to this syntax: (2 ((1 (1 1 1 1)) (1 (1 1 1 1))). Remark 23 Solving the above exercise 82, we have here a nice example of a doubly circular form. To begin with, given a form F , a list form over F is the circular form List(F ) −→ Colimit(Item(F ), T erminal) Id
with the terminal form T erminal −→ Simple(Z) in order to formalize the end of a finite Id
list and to indicate its length with the integer of the coordinator module Z. The form Item(F ) is a Cartesian product Item(F ) −→ Limit(F, List(F )). Id
51.2. MUSICAL OBJECT FRAMEWORK
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Building on this form, we define the form RT by RT −→ Limit(DurationV alue, List(RT )) Id
and a simple duration form DurationV alue −→ Simple(Q). Id
It is useful to visualize this doubly circular form in its construction graph as drawn in figure 6.5. • Semantics For a given RT = (D, S), D expresses a duration and S defines a group of proportions of D. For instance, by taking as unity the quarter note we have for RT = (1, (1, 1, 1, 1)) the rhythm shown in figure 51.14.
Figure 51.14: Rhythm for RT = (1, (1, 1, 1, 1)). RT s allow us to represent, in a homogenous way, different types of musical objects. Polyphonies, voices, measures groups, etc. are expressed as RT s. When the value D is at the measure level, we express it in whole note units. For example, the RT for the next rhythm will be (3/4, (2, 1)).
Figure 51.15: RT = (3/4, (2, 1)). As it was defined, S represents a sequence of proportions of D. The example in figure
Figure 51.16: RT = (4/4, (1, 2, 1)). 51.16 shows the case for RT = (4/4, (1, 2, 1)). Until now, we have presented RT s where S was a list of integer elements, representing notes or beats. For an RT = (4/4, (1, (2, (1, 1, 1)), (1, (1, 1, 1)))), more complex elements appear as shown in figure 51.17. The RT s contained in S represent what we call (rhythmical) groups. In general, groups are graphically represented as beamed notes. As well as measures, groups can contain either notes or other groups. Here is an example for the RT = (4/4, (1, (1, (1, 1, (1, (1, 1, 1)), 1, 1)), 2)):
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Figure 51.17: RT = (4/4, (1, (2, (1, 1, 1)), (1, (1, 1, 1)))).
Figure 51.18: RT = (4/4, (1, (1, (1, 1, (1, (1, 1, 1)), 1, 1))2)). We extend the RT syntax in order to integrate rests and ties which will be represented respectively by negative numbers and floats2 . For RT = (4/4, ((1, (1, 1)), (2, (1.0, 1, −2, 1)), (1, (1.0, −1, 1)))) we obtain the following rhythm:
Figure 51.19: RT = (4/4, ((1, (1, 1)), (2, (1.0, 1, −2, 1)), (1, (1.0, −1, 1)))). RT s may encode the metric intention in a rhythm. The two rhythms in the next figure are encoded by different RT s RT 1 = (2/4, (2, 2, 2, 2)) and RT 2 = (2/4, ((1, (1, 1)), (1, (1, 1))).
Figure 51.20: RT s RT 1 = (2/4, (2, 2, 2, 2)) and RT 2 = (2/4, ((1, (1, 1)), (1, (1, 1))). The symbolic representation is essentially based on the hierarchical musical content. This allows an overall readability of the hierarchical structure and a symbolic format, which can be controlled by algorithms and other transformations such as inversion, recursion, etc. This is made possible by the fact that the format is the abstraction of the object itself. RT s are automatically encoded into optimal container structures.
51.3
Maquettes: Objects in Time
Summary. The maquette is an OpenMusic meta-object aiming at representing, in a same object, patches (musical process) and containers (musical material). It is a new concept of score 2 Perhaps, a better distinction of domains for ties would be to take imaginary rational numbers i.x/y instead of floats, since rational numbers are not really different from floating point numbers. A more musical solution would be to take the form T imeSig defined in formula (6.92) instead of DurationV alue for normal duration values (see also appendix A.2.1), to reserve DurationV alue for ties, and to replace DurationV alue by the colimit of DurationV alue and T imeSig in the above definition.
51.3. MAQUETTES: OBJECTS IN TIME
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where the static description of musical structures and the definition of dynamic computational processes seamlessly coexist. The user may go back and forth between these two metaphors by considering the maquette as a score (in traditional or graphical notation) or as a set of interconnected processes. As external objects like MIDIFiles or SoundFiles may also be imported, maquettes offer an original environment for music creation. –Σ– The maquette is an original concept in OpenMusic which allows us to solve the problem of combining the design of high level hierarchical musical structures, the arrangement of musical material in time, and the specification of musical algorithms. Just like other meta-objects, maquettes may appear as a view or may be opened in a maquette editor, which is basically a 2dimensional surface with time flowing along the x-axis. This surface contains several blocs that we call temporal boxes. In the maquettes the hierarchical imbrication of musical structures and their temporal order can be represented in an explicit visual way. In the musical sketch presented in figure 51.21, made by the composer Mikhail Malt, temporal boxes have been disposed on the maquette surface.
Figure 51.21: Temporal boxes in a maquette. Horizontal temporal box positions correspond to onset values in ‘absolute’ (physical) time. Durations and intensities are given by their horizontal and vertical extensions. Pictures have been associated to the boxes in order to give an elementary musical semiotics. Thus, triangles correspond to chords whose resonance decrease quickly. Multiple triangles are associated with chords ostinato. Other figures are triangles whose intensity follows the geometrical contour of the picture. In figure 51.22, connections between temporal boxes are shown. They represent a different kind of musical information. We can see that temporal boxes are inferred one from another by functional relations. For instance, the ostinatos are linked to the first chord. This level of information bears paradigmatic content, because analogies between any part of the structure can be derived.
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Figure 51.22: Functional relation of temporal boxes.
Figure 51.23: Patch calculating a temporal box.
If we open the editor of the third ostinato (figure 51.23), we can see that the chord coming from the first box (which is represented by the box called input) is transposed by an augmentedfourth (18 half-tons or 1800 MIDI-cents), then repeated six times and finally sent to a factory that builds an instance of the class chord-sequence. Elements of the musical material take place syntagmatically in the final result. Let us take now the first chord that is provided as input for the ostinato. If we open its editor (figure 51.24), we can observe that the generating algorithm is reduced to a given data. This is the fourth level of information which concerns the basic music material representation.
51.3. MAQUETTES: OBJECTS IN TIME
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Figure 51.24: Patch with musical material for a temporal box. Finally, we can toggle the visual representation of boxes by showing the traditional representation of the musical result, see figure 51.25.
Figure 51.25: Musical notation of a maquette. In the previous example, we tried to show how an OpenMusic object scheme enables us to structure the musical information at different levels: • the static level of the form, allowing us to create visual semiotic markers; • the dynamic and paradigmatic level of the form (i.e., its functional relations between the temporal boxes); • the syntactical level, it is the calculus building the musical discourse inside the temporal boxes; • the material level.
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These four levels of information are obviously interconnected. The most important advantage in the maquette concept is to offer a visualization of this interaction and at the same time, interactive control of it. This produces a source of experimentation: • Recombination at the form level: Temporal boxes are moved and stretched in time without changing the other three levels. • Modification of functional relations: We do not change the position of the blocks but their causal relation. • Syntax modification: Algorithms which build the material can be changed according to the compositional goals. • Change of the material: The color of the piece may change while keeping all the formal organization. When combining these procedures, sophisticated musical experiments can be carried out.
51.4
Meta-object Protocol
Summary. Meta-object protocols (MOP) provide an alternative framework that opens the language implementation to user’s intervention. OpenMusic was implemented using metaprogramming technics. We have extended CLOS meta-objects (methods, classes, generic functions, etc.) by adding visual counterparts. In the same way we extended CLOS as such by using the technique of meta-programming; the user can make extensions of the OpenMusic language thanks to the visual MOP. In this section we describe OpenMusic’s graphical MOP, as well as musical applications. –Σ– Originally the MOP was conceived for solving problems in the design and implementation of CLOS [262]. Usually the internal architecture of a programming language is not interesting for the programmer but only for language designers. For object-oriented languages, a MOP of the language is an interface to the language, presented as a framework. Classes, generic functions, patches, maquettes and other entities discussed in the previous sections are simply instances of special classes. These instances are called meta-objects and the classes meta-object classes. The set of these classes, as a framework, constitute the MOP. In addition, a language may support a MOP only if it has two characteristics: reflection and reification. The reflection is the ability of a program to inspect and modify its state at run time. In order to achieve this purpose it is necessary to have a mechanism to represent the program’s data and state, this mechanism is called reification. A MOP is composed by a static and a dynamic part. The static part is given by the hierarchy of meta-object classes. The dynamic part is made of a set of generic functions that can be applied to meta-objects in order to control their behavior. Examples of such functions are: • Get-Elements. We will consider that a meta-object is composed by a set of meta-objects together with a relation between them. For example, a class is composed by an ordered
51.4. META-OBJECT PROTOCOL
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list of slots, a generic function consists of a set of methods, a patch contains a list of boxes, etc. This function returns the list of elements of a meta-object. • Get-View and Get-Editor. It returns the two possible graphical representations of a metaobject. • Open-Editor. It shows the editor for a meta-object. • Add-Element and Remove-Element. They allow editing of any basic object. • Box-Value. It enables us to trigger the evaluation of a visual expression in a patch. Figure 51.26 shows the graphical representation of the dynamic and static parts of the OpenMusic MOP. One may be surprised by the simplicity of the protocol. In general, a protocol which is too much specified is not very modular. It means that changes must be made in several places, and as consequence, modifications are not very reliable. On the other hand, a poor specification of the protocol makes it very difficult to find the place where modification should be included.
Figure 51.26: Dynamic and static part of the OpenMusic MOP. The main tools for the meta-programmer are: subclassing inside the static class part and redefining functions in the dynamic protocol part. It is clear that the user can change some features of the language, but the most interesting point is that he can extend the language without loss of compatibility with old programs. Programmers can profit from customizing the language semantics. The following example in figure 51.27 shows the redefinition of the generic protocol function Box-Value which is called at each evaluation of a box. Graphical redefinition of these two methods changes the behavior of the language by introducing a visual trace of programs. The first method, which is defined with qualifier before, graphically selects any currently evaluated box before its execution. The second method, with qualifier after, deselects the box after its evaluation. These two simple modifications allow us to extend the language by tracing graphically and in an orthogonal way the functional composition.
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Figure 51.27: Changing a generic function of the MOP.
51.4.1
Reification of Temporal Boxes
Finally, this section shows how we have extended the temporal boxes in maquettes by applying meta-programming concepts. Indeed, thanks to the graphical reification of temporal boxes, users have access to the temporal boxes at the same level as other objects like notes, chords, etc. Figure 51.28 shows a first example of the new possibilities. By opening the editor of the temporal box, we can see a new box self (pointed by one arrow) that represents the temporal box itself. The three public slots—available as outlets—are from left to right: the meta-object itself, its offset and its extension. The process associated to this temporal box builds a major chord whose pitches are transposed taking into account the box’s time offset in the maquette (which means that the start time of the box is directly proportional to the transposition interval).
Figure 51.28: Mixing calculations and temporal relations. As has been shown above, temporal boxes may send and receive data by using connections. By reifying temporal boxes, we allow them to have access to other temporal boxes belonging to the maquette. In figure 51.29 the upper temporal box is sent as a whole to the other one,
51.4. META-OBJECT PROTOCOL
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which sets its own offset to the same value as the previous. One evaluation of the maquette will produce two chords and will align them in time.
Figure 51.29: Mixing calculus and temporal relations again. We stress two principal temporal box relations within maquettes: a temporal relation, given by the horizontal position, and a causal relation, established by functional connection between boxes. Examples in figures 51.28 and 51.29 expose how we have combined these relations by using the MOP. In the first example, the result of the calculation is affected by the position of the box. Informally, we can say that time changes calculation. In the second example, the evaluation of the functional composition of the boxes changes their position in time. In this case, the calculation changes the time organization. We finish this chapter with a musical example of composition using OpenMusic.
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51.5
CHAPTER 51. OBJECT-ORIENTED PROGRAMMING IN OPENMUSIC
A Musical Example
Summary. Encore, By Jean-Luc Herv´e for two ensembles, live electronics, and two MIDIcontrolled pianos (commissioned by IRCAM-EIC). Created April 9, 2000 by Ensemble Intercontemporain, Conductor : Patrick Davin. Fr´ed´eric Voisin was musical assistant and designed in OpenMusic the maquette described here. –Σ– This piece is concluded by a cadence for two mechanical pianos (actually two MIDIcontrolled pianos) followed by a short orchestral finale. The cadence is actually an OpenMusic maquette that is played through MIDI . The piece is based on the concept of instrumental gesture. A gesture is a small musical unit with a typical energy profile, such as a glissando, a strong note preceded by a group of grace notes, a repeated note, etc. The gestures performed by the instruments may be continuous, such as a glissandi. The piano cadence is prepared by a densification of orchestral gestures, soon imitated by the piano. At the end of the cadence, the instruments enter back again and the cadence progressively sinks into the orchestral mass. The idea behind the cadence is to accumulate and superpose an enormous quantity of discrete gestures (played by the piano) into a well-defined architecture so that the total construction sounds as a quasi continuous sculpted sound shape. This is why it has been realized as a maquette: it was in fact impossible to write it manually in detail. Rather, the architecture is specified by the hierarchical inclusion of maquettes within maquettes (three levels). At the deeper level, one finds the elementary gestures. Elementary gestures are temporal boxes that contain all the algorithmics necessary to generate them. Groups of gestures inside a submaquette are linked by two kinds of relations. Firstly, temporal logic relations force them to begin and end simultaneously, even if they are moved or stretched in time by the composer. Secondly, they have inputs that are fed by links coming from a special block. This block is not a true temporal block in the sense that it is here only for computing harmonic material from which the gestures are colored. But this block does not generate any in-time music by itself. We will not detail the algorithmics behind gesture generation, we only want to make precise the relation between the computation of music material and the time information. Whenever the composer stretches a temporal block in order to experiment with duration, the note durations inside the block are not extended proportionally, as the default maquette behavior would enforce it. Rather, the composer’s intention is to keep the same density of notes within a different duration. This is achieved by using the ‘extent’ outlet of the ‘self’ reflexive box inside a temporal block. By connecting this outlet to the right place in the visual algorithm, a gesture possessing the same overall profile is generated by inserting more notes picked out from the harmonic reservoirs. The whole process is illustrated in figures 51.30 to 51.36.
51.5. A MUSICAL EXAMPLE
Figure 51.30: The piano cadence maquette of “Encore”.
Figure 51.31: The second maquette level revealed.
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Figure 51.32: The fourth submaquette (starting left) opened.
Figure 51.33: The third maquette level revealed.
51.5. A MUSICAL EXAMPLE
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Figure 51.34: The fifth block (starting left) opened. The block on the top generates the harmonic material. The seventh block underneath are gestures. The vertical lines are temporal logic constraints.
Figure 51.35: A gesture opened. The algorithmics generate in-time music material. The link coming from the ‘self’ box informs the algorithm about time conditions.
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Figure 51.36: A gesture opened in music notation mode. Manual modifications can be achieved at that time.
Part XIV
String Quartet Theory
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Chapter 52
Historical and Theoretical Prerequisites Bei der n¨ amlichen Gelegenheit fragte ich Haydn, warum er nie ein Violinquintett geschrieben habe, und erhielt die lakonische Antwort, er habe mit vier Stimmen genug gehabt. Ferdinand Ries [458, p.287] Summary. This chapter introduces the best evolved theoretical part of instrumentation: string quartet theory. It starts with a short historic synopsis and then reviews Ludwig Finscher’s work [151] on string quartet theory. We then focus on the technical core subject: the violin family as an instrumentation paradigm for the string quartet. The chapter concludes with a general discussion of semantics of sound colors. –Σ– The scope of this part is the elaboration of a systematic foundation of the distinguished role of the string quartet at the end of the eighteenth century. This part is a synthesis of methods and results from modulation theory (chapter 27), classification (chapter 15), and counterpoint theory (part VII), combined with knowledge about the nature of sound parameters (see also appendix A). It is not astonishing that precisely the classical string quartet—one of the absolute highlights of instrumental music—needs a complex theoretical background for its comprehension. But since we deal with a systematically as well as historically founded phenomenon, it is adequate to prepend some remarks on the theory and history of the string quartet, remarks which we shall orient towards Ludwig Finscher’s pioneering work [151]. This art form, whose instruments include two violins, one viola and one violoncello, can be characterized in the following way [374, p.409]:“In the string quartet, individuality and character of the single players are combined to a harmonic whole, where each one finds himself with an added profile. This reaches from a single complex chord which—in contrast to the piano—is not played and nuanced by the hand of one player, but by four players, to the entire performance. In the quality as a whole lies the specificity of the string quartet. In its harmonically contrasting 993
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togetherness, this chamber music corresponds to the ideal world view and to the high humanistic vision of the classical epoch.”
52.1
History
Summary. The history of the string quartet is exceptionally short since this instrumental species almost instantly appeared around 1760 with the works of Luigi Boccherini and Joseph Haydn. It almost instantly installed a leading stream of sophisticated instrumental expression. We discuss the historical background of this phenomenon. –Σ– The prehistory of the string quartet is more complicated than that of any other instrumental art form of the eighteenth century. It cannot be causally deduced from any single one of the threads of tradition from where it comes. To a certain degree it is the creative act, the invention out of a moment of the delicate historic equilibrium, the kairos in the sense of ancient Greek thinking. The prehistory dates only from about 1720 to 1760 when Luigi Boccherini and Joseph Haydn independently invented the string quartet. In 1761, Boccherini wrote his first quartets in northern Italy, they were published 1767–68 in Paris under the name of “quatuor concertant”. Probably Haydn had written quartet “divertimenti” already in the 1750s in Vienna, they were however only well-known in 1760. The sparse regional, instrumental, and stylistic rootedness in the string quartet’s prehistory, from which this new art form has quite spontaneously emerged, provokes the question whether beyond historical rationales a more systematic understanding could better enlighten the ‘string quartet phenomenon’. The problem is to question this precise date (1760) of the rise of this precise instrumental art form (the string quartet) in the context of the European music from the systematic point of view. In this question Dahlhaus [104, p.105,p.119] is fairly right in stating that (...) erst die systematische Konstruktion den Blick daf¨ ur ¨ offnet, welche Tatsachen einer Geschichte angeh¨ oren, die zu erz¨ ahlen lohnend scheint. (...) Daß etwa das Ausmaß in dem die Besetzung von Instrumentalmusik im 18. Jahrhundert gattungspr¨ agend wurde, mit dem Grad ¨ asthetischer Autonomie, mit der Herausbildung musikalischen ‘Formdenkens’ und mit der Festigung der Institution des ¨ offentlichen Konzerts eng zusammenhing, ist keineswegs nur eine geschichtliche Tatsache, die sich empirisch feststellen l¨ aßt, sondern erscheint auch als Sinnzusammenhang, der sich einer ph¨ anomenologischen — also ‘systematischen’ — Analyse erschließt.
52.2
Theory of the String Quartet Following Ludwig Finscher
Summary. In his habilitation thesis [151] Ludwig Finscher investigated the theory of the string quartet and exhibited three perspectives: the texture of four parts, the topos of conversation among cultivated humanists, and the family of violins. We discuss this threefold theory. –Σ–
52.2. THEORY OF THE STRING QUARTET FOLLOWING LUDWIG FINSCHER
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If we have decided to turn the string quartet into the subject of a formally valid investigation, this is because the theoretical reflections on the string quartet have reached a scientifically valid status. Finscher remarks [151, p.279]: Das Streichquartett ist die einzige Gattung der neueren Instrumentalmusik, die eine solche an einem einzigen k¨ unstlerischen Modell entwickelte, vergleichsweise genau und detailliert ausformulierte und als allgemeinverbindlich akzeptierte Theorie ausgebildet hat. This theory is based on two fundamentals: • the four part texture; • the topos of a conversation of four humanistically educated persons.
52.2.1
Four Part Texture
Summary. The texture of four parts is a basic structural prerequisite for the string quartet theory. We review its implications. –Σ– The four part texture was the ideal type of structured polyphony which was oriented on the counterpoint with its long tradition. This is the formal, or better: formalized element of string quartet theory. We have to take it in the full conceptual ambiguity, i.e., on the one hand the texture “note against note” in its linear temporal progression in the sense of classical counterpoint. On the other, it is a texture of vertical units—charts in the terminology of global compositions—as an expression of harmonic relations. In the radically harmonic thinking, which is realized in the work of August Kollmann [268] from 1796, it is even possible to reverse the tradition in that the following thesis is proposed: Counterpoint should not start from the intervallic two-part texture, but from the four-part texture, since “a complete harmony” is fourpart and not two- or three-part. It seems that the formation of theories was already influenced by Haydn’s success with his famous “Russian” string quartets from 1782. It is interesting to observe that this ideal type of an instrumental art form was fixed to exactly four voices, not one more. Especially with Haydn one could imagine that he could have added a fifth voice to “enrich the texture”. But it is reported that he ‘failed’ on several occasions with this ‘experiment’. Ries reports 1838 [151, p.287]: Bei der n¨ amlichen Gelegenheit fragte ich Haydn, warum er nie ein Violinquintett geschrieben habe, und erhielt die lakonische Antwort, er habe immer mit vier Stimmen genug gehabt. Man hatte mir n¨ amlich gesagt, es seien drei Quintette von Haydn begehrt worden, die er aber nie h¨ atte komponieren k¨ onnen, weil er sich in den Quartettstil so hineingeschrieben habe, daß er die f¨ unfte Stimme nicht finden k¨ onne; er habe angefangen, es sei aber aus einem Versuche am Ende ein Quartett, aus dem anderen eine Sonate geworden. Presently, Haydn’s argumentation that he has “enough” with the four voices, cannot be understood. We come back to this point at the end of our discussion in section 54.3.
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CHAPTER 52. HISTORICAL AND THEORETICAL PREREQUISITES
52.2.2
The Topos of Conversation Among Four Humanists
Summary. String quartet tradition is intimately related to non-verbal humanistic conversation. This gives the species a rhetoric characteristic which has important consequences for the instrumentation problem. –Σ– On the one hand, this is a topos which stems from the analogy of contrapuntal texture to the conversation and argumentation of humans. For example, Mattheson [151, p.285] says on imitation: ...daß eine Stimme die andere gleichsam gespr¨ achsweise unterhalte. This topos must be cautiously distinguished from the well-known topos of a “Klangrede”1 , i.e., from the similarity of musical expression or semantics to the common language. In the case of the string quartet, the more important thing than speaking is the dialog, a fact that becomes more evident in the French expression “quatuor dialogu´e” for “string quartet” (in fact the invention of a publisher). The association of a discourse to the string quartet was initiated by the musician Johann Friedrich Reichhardt in 1777 [151, p.287]: Bei dem Quartett habe ich die Idee eines Gespr¨ achs unter vier Personen gehabt.
Like Haydn, Reichhardt also views the number of four as being the upper limit for a good dialog. He tries to add a fifth person to the quartet. But he fails: achs nothDie f¨ unfte Person ist hier ebensowenig zur Mannigfaltigkeit des Gespr¨ wendig, als zur Vollstimmigkeit der Harmonie; und in jenem verwirrt sie nur und bringt Undeutlickeiten in’s St¨ uck. The same happens to Schumann [151, p.289] in a discussion about a viola that was added in a quintet: Man sollte kaum glauben, wie die einzige hinzugekommene Bratsche die Wirkung der Saiteninstrumente, wie sie sich im Quartett ¨ außert, auf einmal ver¨ andert, wie der Charakter des Quintetts ein ganz anderer ist, als der des Quartetts. Die Mitteltinten haben mehr Kraft und Leben; die einzelnen Instrumente wirken mehr als Massen zusammen; hat man im Quartett vier einzelne Menschen geh¨ ort, so glaubt man jetzt eine Versammlung vor sich zu haben. The quartet discourse as a dialog is very well suited to communicate understanding within music. This is the sense of the dialog. Goethe stresses this aspect in an enlightening comment on a concert by Niccol` o Paganini [151, p.288], [185]: 1 German
for “sound speech”, however, difficult to translate.
52.2. THEORY OF THE STRING QUARTET FOLLOWING LUDWIG FINSCHER
997
Mir fehlte zu Dem, was man Genuß nennt und was bei mir immer zwischen Sinnlichkeit und Verstand schwebt, eine Basis zu dieser Flammen- und Wolkens¨ aule. W¨ are ich in Berlin, so w¨ urde ich die M¨ oserschen Quartettabende selten vers¨ aumen. Dieser Art Exhibitionen waren mir von je her von der Instrumentalmusik das Verst¨ andlichste: man h¨ ort vier vern¨ unftige Leute sich untereinander unterhalten, glaubt ihren Diskursen etwas abzugewinnen und die Eigent¨ umlichkeiten der Instrumente kennen zu lernen. F¨ ur diesmal fehlte mir in Geist und Ohr ein solches Fundament; ich h¨ orte nur etwas Meteorisches und wußte mir weiter davon keine Rechenschaft zu geben. The connection between the dialogical discourse and the communication of understanding which Goethe indicates means a valuation of the dialog, its qualification in function of communication of understanding. This valuation in turn is related to the instance of varying competition by Karl Popper [420]: “The value of a dialog depends above all from the manifold of competing opinions.” But this manifold of opinions, always related to a given subject, is nothing else than a variation of points of view, of the perspectives of the participants. Therefore the dialogical principle of the string quartet turns out to be an instance of the Yoneda philosophy discussed in section 9.3: Understanding or classification, respectively, by a variation of the point of view. It is not astonishing that this phrase is thoroughly indebted to humanism with which the string quartet is deeply associated. Result 9 Summarizing, one root of the string quartet, the four part texture, appears as a form which, in the sense of Hanslick, is a carrier of musical spirit. This carrier in turn is made accessible to our understanding by the second root: the dialogical discourse of four violin personalities, by means of a variation of perspectives. At this point of our discussion the question arises, why the string quartet and the end of the eighteenth century are related to each other, more precisely: What is the connection between the four-ness within the family of violins (and its above all violins, not violinists who speak!) and the paradigm of four part texture at Haydn’s time?—This question leads to a mathematically tractable apparatus in the manifold of violin sounds in defined parameter spaces.
52.2.3
The Family of Violins
Summary. We discuss the exceptional role of the family of violins in the building process of the string quartet species. In Finscher’s work, other instrumental families are compared to the violins, we comment on the results of this study. –Σ– The formation of the string quartet would not have been thinkable without the collaboration of the homogeneous sound of the instruments of the violin family, which in the eighteenth century was discovered as an ideal type of four-part music and also perfected on the artisanal level after the violin’s creation from local variants of string instruments in the fifteenth century. The high quality of violins is guaranteed by a technical standard which through its fineness, sometimes also imponderableness such as the choice of woods and the varnish covering, contradicts any normalization. The individual sound color of every good violin is characteristic
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to the family. But the family of violins is also strongly differentiated from other families of string instruments. Finscher describes [151, p.124/125] the characteristic of violins as compared to the gambas, which were the preferred solo instruments in the seventeenth century, as follows: Die Violinen hatten gegen¨ uber den Gamben jedoch noch eine weitere zukunftstr¨ achtige Eigenart: Sie gliederten den Tonraum, der ehemals in Analogie zu den menschlichen Stimmengattungen gebildet, nun aber in der Tiefe wie in der H¨ ohe l¨ angst kr¨ aftig erweitert worden war, klarer und sinnf¨ alliger, mit deutlicherer Individualisierung ihrer jeweiligen Tonbereiche. (...) F¨ ur das klassische Streichquartett, das die Beweglichkeit, den Lagenwechsel, den Kontrast- und Farbreichtum des symphonischen Streichersatzes mit der gr¨ oßtm¨ oglichen Ann¨ aherung an eine streng auskomponierte Vierstimigkeiit zu verbinden suchte, bot sich das vierstimmige Ensemble aus Gliedern der Violinfamilie als das ideale Instrument an. It will be a task of the following chapter to access the specificities of the sound of the violin family via a mathematical description in the frame of parameter spaces.
Chapter 53
Estimation of Resolution Parameters Si tous les instrumets jouent l’accord staccato, je supprime ce moyen naturel d’analyse, et la perception ne peut plus discerner, a l’interieur du bloc sonore, ` de quelle combinaison il s’agit. L’identification d´epend, das ce cas, de la pr´esence ou de l’absence des ´el´ements essentiels pour la perception. Pierre Boulez [61, p.547] Summary. This chapter is a technical account of the variety of sound parameters which intervene for the family of violins. –Σ– Based on the preceding historical, systematic, and philosophical reflection we argue that the classification theory of global compositions (chapters 15 through 17) is an excellent candidate to be applied to the theory of the string quartet. From classification theory it follows that the resolution of a global composition is a formalization and optimization of the process of understanding this composition. We shall equally apply this approach to string quartets, which means that we have to think of the composition as being a global composition GI , whose resolution ∆GI one wants to represent. In chapter 54, we shall specify the global composition associated with a string quartet composition. Our approach to the resolution is essentially realized via a variation of the perspective of the resolution parameters, i.e., by a differentiated change of their dichotomy into essential and accessory parameters. Within the string quartet, this dynamics of weighting of performance parameters is a subtle tool for communication of understanding and articulation as a vehicle of dialog. It is as if you walk around an object—which is the resolution in our case—and observe it from one, and then from another perspective. 999
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CHAPTER 53. ESTIMATION OF RESOLUTION PARAMETERS
If we now envisage the resolution of a string quartet composition, we first have to investigate the possibility of parametrization by concrete physical performance parameters as they intervene on the arsenal of the violin family.
53.1
Parameter Spaces for Violins
Summary. We describe the denotator types for violins, including generic geometric parameters, sound color parameters, and technical parameters, in particular those related to vibrato. –Σ– As we already did in classification theory, we want to work over Q, which is a reasonable postulate for string quartet theory since we are dealing with physical parameters. Here, we envisage a fundamental problem of mathematics since we have to deal with a huge number of independent numerical parameters. If, for example, a sound color has to be represented by means of amplitudes and phases of the Fourier representation, and by the envelope, we easily add up hundreds of numerical parameters. Even though locally, the structure of music thinking may concern a small number of parameters, the techniques of classification theory, by their requirement of points in general position, enforce a number of geometric degrees of freedom that might possibly be far larger. We shall have to work with charts in modules Qn with large dimension n. This should however not prevent us from visualizing essential aspects of our reflections in three space Q3 . The only point here is to build mathematically representative analogies in three space. What is the shape of the physical parameter space of a violin sound? To begin with, we dispose of parameters which we call geometric1 , i.e., • Onset • Duration • Pitch The amplitude (loudness) is omitted in this presentation since the poietic violin parameters are coupled to the way of generation of the sound color aspect. The other attributes of the violin sound span the color space. To begin with, this aspect includes the following ingredients: • Envelope • Amplitude • Fourier spectrum What is the range of variability of these color aspects of a violin sound? Regarding this question, we refer to [371]. For example, we consider the amplitude spectrum for pitch g and g] of a “Guarneri del Ges` u” as compared to an F horn, see figure 53.1. One recognizes that the spectra 1 See
appendix A.1.2.1.
53.1. PARAMETER SPACES FOR VIOLINS
Guarneri
1001
75 dB
g
60 45 30 15 0 75
dB
g#
60 45 30 15 0
F-horn
75 dB
g
60 45 30 15 0 75
dB
g#
60 45 30 15 0
0
1
2
3
4
5
kHz Figure 53.1: We consider the amplitude spectrum for pitch g and g] of a “Guarneri del Ges` u” (top) as compared to a F horn (bottom). One recognizes that the spectra from g to g] are significantly more different for the “Guarneri” than for the F horn. between g and g] are significantly more different for the “Guarneri” than for the F horn. In the representation of the amplitude spectrum as a vector in the color space, one may say that the change from g to g] for the F horn is essentially a dilatation, whereas the corresponding change for the “Guarneri” violin also includes a change of direction (figure 53.2). In other words: The g and g] spectral vectors of the “Guarneri” span a plane in the color space, whereas the corresponding vectors for the F horn lie on a line. This statement evidently is not meant in a strictly physical sense, but in the sense of valence theory (see appendix B.2): For the F horn, the g- and g] -spectral vectors are indistinguishable in the auditory perception from a pair of linearly dependent vectors2 . This instrumental difference is justified by the fact that for winds, the sound color is 2 This discussion is somewhat speculative since precise measurements should be made and relations to valence theory should be investigated in a more quantitative way. However, the means for such an investigation depend on the insight in the basic problem setup.
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CHAPTER 53. ESTIMATION OF RESOLUTION PARAMETERS
g# g g#
f-horn
g
Guarneri
Figure 53.2: In the representation of the amplitude spectra as vectors (represented in 3D space here), the “Guarneri” vector differs beyond the valence limit when comparing the g sound to the g] sound. essentially built from an air column, but not from the material. In contrast, for the violins, sound color is an essential function of the material, i.e., of the corpus’ resonance properties. Moreover, the spectrum depends on the string on which a fixed pitch is intonated (to this end, the spectral envelopes for different strings are compared [371]). All this turns the sound color of violins substantially into a function of the individual construction: from the material through the artisanal manufacturing to the individual history of the particular instrument. This variability is out of the question with wind instruments. For the same reasons, string pianos are inferior to violins. Figure 53.3 shows the principal configuration of the spectral vector for three violins as opposed to three such vectors for string pianos. To these instrumental parameter properties, central techniques of instrumental practice of sound shaping are added, techniques which for several other instruments do not even exist. The string player has the following possibilities to vary parameters: • Bow pressure • Bow velocity • Contact point of bow and string • Bow angle Further, the string player may shape his/her vibrato according to the following points of view: • Delay time with respect to the tone’s onset
53.2. ESTIMATION
1003
Figure 53.3: The instrumental sound parameters of violins are more variable individually than these parameters are for pianos. Here the amplitude spectra of three violins against three pianos are visualized schematically in 3D space. In contrast to the piano spectra, those of violins lie in general position. • Modulation frequency (frequency of the finger’s movement) • Pitch modulation (Range of finger movement on the string) • Amplitude modulation (Contact point of the finger-tip) Compared to the color attributes envelope, amplitude, and spectrum, these vibrato parameters are new. They enable the violinist to realize sound in still larger spaces. Together with the four bow parameters, the four vibrato parameters define an additional eight-dimensional space. These eight parameters which emerge in contrast to the instrumental parameters are called technical parameters. Figure 53.4 shows the effect of bow pressure and contact point variation while bow velocity and sound color remain constant.
53.2
Estimation
Summary. This section is devoted to a theorem giving an estimation of the maximal possible chart dimension ch(n) within a global composition, which can be produced by an orchestra consisting of n individual strings (in the violin family). We make use of the resolution theory for classification of global compositions (see chapter 15.2). –Σ– In the last step of our parameter analysis, we deal with differentiation within the technical parameters. Let us recall that we are searching for parameter spaces, i.e., coordinate functions which are suited for charts of global compositions. But this signifies that one has to distinguish
CHAPTER 53. ESTIMATION OF RESOLUTION PARAMETERS
ght
bri &l
sul ponticello
oun d
bow pressure
1004
rough unstable
unstable without base tone
quiet smooth
max sul tasto
}
bow pressure
min 1 25 bridge
1 contact point 1 10 5 end of fingerboard
Figure 53.4: The effect of a variation of bow pressure and contact point while bow velocity and sound color remain constant. parameters which can be varied independently of each other in short parts of the composition, in fact on charts, from those parameters which may very well vary from player to player or from situation to situation within a larger composition, but which are relatively constant on local regions. The latter, which may strongly depend on the player’s personality, include: • the vibrato parameters. Even for a professional violinist they can scarcely be separated from the personality and are difficult to control; • The bow angle and contact point are relatively inert parameters, therefore not suited for extremely local purposes. The local variability is therefore distributed on two dimensions: • Bow pressure which above all acts on the amplitude. (This is the reason why we did not add amplitude to the geometric parameters here: it is only an aspect of the action of bow pressure!) • Bow velocity which above all acts on the dynamics of partials. Evidently the bow parameters are coupled with each other in their action; we only indicated the main actions. Summarizing, we have found three types of color parameters: 1. Instrumental parameters (Violin type, choice of strings, performance conditions),
53.2. ESTIMATION
1005
2. global technical parameters (vibrato, bow angle, contact point), which are a strong function of the individual player, 3. local technical parameters (bow pressure, bow velocity), which can be steered quite independently of each other. Therefore, the local technical parameters define a plane in the color space. Since these are under general control and can be steered objectively, we may assume that these planes H1 , H2 , . . . for player 1, player 2, ... are one and the same player-independent plane H up to an individual translation (figure 53.5). Observe that our hypothesis is made in the context of the widespread
H H H
Figure 53.5: In the color space of violins a plane H is appended to the heads of individual vectors vs of instrumental and global (inert) color parameters and yields Hs = vs + H for player s. string quartet at the end of the eighteenth century, and that in fact the signification of the string quartet is a pronounced reality of music sociology with regard to normal music practice in bourgeois saloons. It would be unrealistic to model our theory upon elite ensembles in this context. Based on the above observations about the individuality of colors and the violinists’ personalities, we may set forth the following hypothesis in the spirit of string quartet theory: Assumption 3 The n “instrumental vectors” v1 , v2 , . . . which are spanned by the instrumental and global technical parameters can be chosen to be linearly independent of each other and of the common plane H.
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CHAPTER 53. ESTIMATION OF RESOLUTION PARAMETERS
This means that the submodule is spanned by v1 , . . . vn and H has dimension n + 2. In other words: If H is spanned by two vectors h, k, the zero vector and the heads of the vectors v1 , . . . vn , h, k are in general position3 . ∼ If we add the three-dimensional module G → Q3 of geometric parameters to the color space (direct sum), we obtain a total parameter space, where the n instrumental vectors are positioned, of which to each is attached a 2+3 = 5-dimensional space H +G of local parameters, see figure 53.6.
G
G
H
H G
H
Figure 53.6: Viewed as points for the n instruments of the violin family, the sounds are distributed on n affine subspaces H + G + vs , s = 1, . . . n, in the total attribute space. Here G is the three-dimensional space of sound geometry (pitch, onset, duration), H is the plane of the local technical parameters bow pressure and bow velocity, and vs is the sth instrument vector. One may assume that the Q-module spanned by G, H, vs , s = 1, . . . n has dimension n + 5. In view of classification theory our question is how many points in the total parameter space can maximally be distributed in general position on the n affine subspaces H +G+vs , s = 1, . . . n. So every such point has the form x = vs + h + g, h ∈ H, g ∈ G, see figure 53.6. It can be shown (appendix E.2.1, theorem 47) that there are maximally n + 5 points, where 5 means the dimension of G + H. The musicological meaning of this result for classification theory is this: Theorem 36 With n string players from the violin family, charts of global compositions in local parameters can be defined with maximally n + 5 points in general position.
3 See
appendix E.3.4.
Chapter 54
The Case of Counterpoint and Harmony Au fur et a ` mesure que l’orchestre s’agrandit, que le rˆ ole de l’instrument devient, je ne dirais pas flou, mais ductile, multiple, les formes, elles aussi, s’amplifient. Pierre Boulez [61, p.544] Summary. This final chapter on string quartet theory deals with the analytical conditions on global compositions which are powerful enough to comprehend the structural richness of central European music in the epoch of Boccherini and Haydn. These structures are—essentially— counterpoint and harmony. As a germ for a systematic theory of instrumentation we propose an estimation of maximal necessary chart dimension for Fuxian counterpoint and traditional harmony (including cadence and modulation). –Σ– As the last member of our model for the string quartet, we need the initially announced information to solve the question of how a string quartet should be defined as a global composition. To this end we imagine that a score, written by Haydn, say, is given as a local composition. From this one would like to construct an adequate interpretation (in the technical sense of interpretable global compositions). And it is here where the historical moment comes into the game. The interpretation should be such that the European structural music thinking in the four part texture at the end of the eighteenth century is expressed in its essential features. In our presentation of the roots of the string quartet theory, the structural basis was first presented: the four part texture with its polysemic meaning as a contrapuntal as well as harmonic setup. These two main components are therefore to be investigated for the construction of an atlas.
54.1
Counterpoint
Summary. In this section, we calculate the upper limit of chart dimensions for counterpoint. 1007
1008
CHAPTER 54. THE CASE OF COUNTERPOINT AND HARMONY
We make use of the counterpoint model exposed in part VII. –Σ– In this section we shall refer to the discussion of the core theory of counterpoint, “note against note”, as it was presented in chapters 29 through 31. For intervals which are represented by arrows in counterpoint theory, we need two-element charts (admitting that what is physically played are not arrows, but their heads and tails). For the cantus firmus and the discant steps we also need two-element charts, one for each. For the consideration of a progression from interval to interval, one needs four-element charts. It is advantageous to include also the succession of two interval steps, for example regarding hidden/composed tritones, we need six-element charts. This is however not the statement of the rules of counterpoint! As we know, this would be much more complex. But we have defined the “cartographical” setup, and that is what we need. Therefore: Result 10 The classical contrapuntal texture as it was codified by Johann Joseph Fux in 1725, the early days of the prehistory of the string quartet, requires in its core structure maximally 6-element charts. All more complex configurations can be reduced to this core structure: Thereby charts are glued together, but not enlarged.
54.2
Harmony
Summary. In this section, we calculate the upper limit of chart dimensions for harmony. We make use of the cadence (chapter 26) and modulation (chapter 27) models. –Σ– The degree theory of chords as vertical structures interprets a chord as being a subset of a triadic covering (see chapter 25), i.e., by three-element charts. A cadence (see chapter 26) can be thought within the scheme consisting of three degrees (typically: IV-V-I), where the minimal cadential sets (here: IV,V) are completed by the first degree or else by the tonic note of the given tonality. This produces an interpretation of maximal nine tones per chart, consisting of one, three or nine points. The most complex situation in the harmony at the end of the eighteenth century is crystallized in the modulation process which we have formalized in the modulation model that is based on triadic degrees, see chapter 27. In a modulation which is presented as a sequence of neutral degree, modulation degree, cadence degree, we may recognize these three charts, with three elements each. Moreover, like with a cadence, the entire process is collected in a nineelement chart that contains the three triadic degrees. The cadence as such, which we described above as a nine-element interpretation, intersects our modulation in the cadence degree and concludes it as a process. Therefore: Result 11 Harmony has a maximum of nine points for the relevant local charts.
54.3. EFFECTIVE SELECTION
54.3
1009
Effective Selection
Summary. As a result of the instrumental parameter estimation made in chapter 53 and the structural parameter estimation from this chapter, we obtain a global theorem estimating the minimal number four of string instruments (in the violin family) which is needed to express analytical music structures in the compositions of the classical epoch of Boccherini and Haydn. This theorem makes essential use of the resolution theory for global compositions, as exposed in chapter 15.2. –Σ– Finally, we are in a state of indicating the minimal number of instruments from the violin family in order to provide the resolution of an interpretation of a score under the structure as preconized by harmony and counterpoint at the end of the eighteenth century by a sufficient number of free parameters. In sections 54.1 and 54.2 we have seen that a maximal number of nine points per chart are present in such an interpretation. Since in a resolution, all charts have their points in general position, the number n of instruments must, according to theorem 36 at the end of section 53.2, suffice the inequality n + 5 ≥ 9. (54.1) We therefore need at least four instruments for these scopes. As announced, our model of the string quartet has been deduced from the information about parameters, modulation theory, counterpoint, and the classification technique. From this point of view, it is not astonishing that classical string quartet composers such as Haydn did not see any sense in the accumulation of instruments: For a purely economical point of view, they were superfluous. Four string players were perfectly sufficient in order to provide the textural structure with a profiled representation in its resolution.
Part XV
Appendix: Sound
1011
Appendix A
Common Parameter Spaces This appendix chapter is an overview, not an exhaustive treatise of spaces which parametrize sound objects. These spaces where sounds are positioned always define an aspect, never the totality of music thinking, and every attempt to define a preferred space will narrow the music thinking, not the music. The best that can occur is that we offer an encompassing or at least a representative ensemble of parameter spaces which are interrelated by a precise relation. To this end, it is recommended to distinguish topographic positions, above all in their realities and communicative perspectives. This will also entail the corresponding mathematics. We start by the physical descriptions, turn over to more mathematical abstractions and the describe more symbolic viewpoints which we call interpretative since they are not just a new mathematical rephrasing of a priori equivalent physical description, but express abstraction with some mental background constructions.
A.1
Physical Spaces
As a physical object, a sound1 is a more or less regular variation of normal air pressure2 as a function of time. Starting at a determined onset time e sec, it starts from a source Q at position q = qQ m in the ordinary physical space and expands as a wave. At a location x and time t, −2 is perceived as a the pressure variation (the difference from the normal pressure) pQ x (t) N m longitudinal air wave, i.e., with a pressure front perpendicular to the waves expanding direction, see figure A.1. For a punctual sound source, however, the wave front at x 6= q is a spheric surface; we can write −1 pQ pq (t − |x − q|/v), (A.1) x (t) = |x − q| √ where v is the expansion velocity of the wave3 . It is calculated by the formula v = C. T , where T K is the absolute temperature in Kelvin degrees, and C is a constant with value C = 20.1 1 German:
“Klang”. the zero height above sea and zero degrees Celsius, this is ≈ 1.1013.105 N m−2 . 3 It is known [462] that the square of the pression variation is proportional to the intensity, i.e., energy flow per surface and time unit, and the latter, by energy conservation, decreases proportionally to the square of the distance |x − q|, whence the formula. 2 At
1013
1014
APPENDIX A. COMMON PARAMETER SPACES
x
Q q
Figure A.1: The prototypical punctual sound source and the spherical sound wave. for the normal pressure. For normal conditions, we have v ≈ 343 msec−1 . But with complex sound sources and room-specific reflection and refractions, the pressure variation of a sound may be an overlapping of different spheric wave components. If several P sound isources Q1 , . . . Qs . are given, the resulting pressure variation sums up to pQ (t) = 1≤i≤s pQ x x (t), and it is the R integral dpdQ (t) of a family of infinitesimal point sources dQ. In general, from the knowledge x . of pQ (t), one cannot infer the original source functions. This is like with painting where in x general one perspectivic image does not allow us to infer the original object configuration: The esthesic position is not sufficient to reconstruct poiesis4 . The fundamental problem of musical acoustics is that the neutral data, the objectively measurable pressure values, are far from what is intended by musicians, i.e., the neutral data is not what is interesting and what is the message. So one of the most important tasks of musical acoustics is the interpretation of the neutral data, in other words, of what is behind the data, what could be the hidden parameters of the audible phenomenena. And it is one of the worst tragedies of traditional music-acoustics and psycho-acoustics that the fundamental difference of neutral and poietic levels is ignored and disregarded.
A.1.1
Neutral Data
In order to describe the neutral sound data, let us first concentrate on the information at the source location q = qQ of a point source Q, in an idealized model of a single instrument. We shall come back to room acoustics in the next section A.1.1.1. The source sound (variation) 4 See
chapter 2 for the concepts of music topography, such as “esthesic”, “neutral”, or “poietic”.
A.1. PHYSICAL SPACES
1015
event pq (t) is usually a finite event, starting at time e, and ending after the duration d. The variation between these time limits is also limited by the maximal amplitude Am of the total pressure variation. So the function pq (t) is the affine image of a normalized function p0q (t) which starts at time e = 0, has duration d = 1, and amplitude A = 1. More precisely: pq (t) = Support(A, e, d)(p0q )(t) = A.p0q ((t − e)/d).
(A.2)
The operator Support(A, e, d) reduces the unknown sound event to a normalized event. What happens between the normalized unit supports is however completely arbitrary. It may be a percussive sound or the sound of a Stradivari violin. The normalization by the support operator is completely harmless, but not much more than this data can be traced on the neutral level. The theory of all the rest is far from neutral; we are going to deal with this in section A.1.2. A.1.1.1
Room Acoustics
The room is an important part in the information chain from the information source (instrument, speaker, public address system) to the receiver (listener, director, artist). Room sizes vary from small living rooms to huge cathedrals or concert halls. In this section, we want to tackle only basic features, reverberation time and acoustical power. For a large, irregular room, we can visualize the acoustical conditions by imagining a wave traveling inside the room. This wave travels in a straight line until it strikes a surface. It is reflected off the surface at an angle equal to the angle of incidence and travels in this direction until it strikes another surface. Because sound travels about 343 msec−1 , many reflections will occur within a small time span. Absorption. After a wave has undergone a reflection from a wall that is absorbing, its intensity will be less during its next traverse of the room. In a large, irregular room, the number of waves traveling are so numerous that at each surface all directions of incident flow are equally probable. The sound absorption coefficient α is therefore taken to be averaged for all angles of incidence. All materials have absorption coefficients that are different at different frequencies5 [54]. In the frequency range of interest between 250 Hz and 4 kHz, plain walls and floors, as well as closed windows, have absorption indices below 0.2. Higher absorption can be achieved with acoustic tiles and, at least in the upper frequency range, with thick carpets and draperies. The total absorption A of a room is the sum of the product of surfaces Ai m2 and absorption coefficients αi : A = Σi αi Ai . If there is an open window in the room, all the energy incident on its area will pass outdoors and none will be reflected.6 The absorption of an area of acoustical material in a room can therefore be expressed in terms of the equivalent area of an open window. For this reason, the total absorption A of a room can be characterized by its equivalent “open window surface”. Critical Distance. The sound field which builds up in a room is fundamentally different from the free field situation. Let us first assume a sound source in a free field, i.e., a loudspeaker on a high post emitting sound in all directions. If the sound radiation is unidirectional 5 See
section A.1.2 for the discussion of the frequency concept. statement is strictly true only if the window is several wavelengths wide and high, otherwise diffraction will occur. 6 This
1016
APPENDIX A. COMMON PARAMETER SPACES and no reflections occur, the source will emit a spherical wave and sound pressure will decrease inversely proportional with the distance from the source. For loss-less reflecting walls on all sides around the source, the sound waves will be reflected over and over. In the case of no absorption in the air, the sound pressure will now rise and rise, as no energy loss occurs. Small absorption will cause an equilibrium. However, the sound field in the room will no longer be a directional spherical field because the reflected waves will by far dominate over the direct sound wave. The sound field will be diffuse, reflections will arrive from all directions with equal probability. Only in the close vicinity of the small sound source is the sound field directional, because there the sound pressure of the unreflected direct sound wave will dominate over the diffuse sound field. In summary, the sound field in a room will be directional only close to the sound source Q. There, as we know from the above, at point x, the sound pressure falls with 1/|x − q|. The sound field at a large distance from the source is diffuse. Sound pressure is almost constant and much higher compared to a free sound field. The distance from the source, where the transition between these distinct regions, occurs is called critical distance or diffuse field distance rH . If the average absorption α ¯ ≤ 0.4, the critical distance for the absorption A can be calculated with a precision of about 10% [54]: p √ (A.3) rH = A/16π ≈ 0.14 A. By the inverse distance law, the sound pressure decrease factor ρD in the diffuse field of a room can be estimated by the formula: 7.1 ρD = 1/rH ≈ √ . A
(A.4)
Reverberation Time. Temporal effects during the onset and the decay of sound are essential features in rooms and are not present in a free sound field. In large rooms, such as a cathedral, long decay times are apparent.The human auditory system can distinguish whether a sound source is located in a large room, a small room or even whether it is not in a room at all, but outside. The human ear is obviously able to extract information about the room size from the temporal structure of sound. We will therefore discuss the onset and decay of sound. When the sound leaves its source, the direct sound reaches the receiver first, its delay is determined by the distance sound has to travel divided by the velocity of sound. With further delay, reflected sound waves with only a slightly longer travel distance arrive, and later, also reflections with longer routes and multiple reflections arrive. The sound pressure is increasing until it reaches its steady state value. Only the direct sound gives information about the location of the source, which is exploited by our hearing system for sound localization. The build-up of the reverberation is only perceived in highly reverberant rooms. The reverberation is much more perceptible after the source is muted, and the echo may be still noticeable after seconds. The direct sound ceases after the propagation time from the source to the receiver, however, all the reflected sound waves still arrive. Their intensity will be reduced by factor 1 − α ¯ after each reflection on a wall. Therefore, the sound pressure will decrease exponentially.
A.1. PHYSICAL SPACES
1017
The reverberation time T is defined as the time required for the sound to decay by a sound pressure level7 of 60 dB. Let αL m−1 be the absorption in air (a highly frequencyand humidity-dependent variable), and let V m3 be the volume of the room. Then the reverberation time T can be estimated with Sabine’s formula [595]: T ≈
0.161V sec. A + 0.46αL V
(A.5)
In the frequency range below 4 kHz, sound absorption in air is usually neglected, whereas for frequencies above 4 kHz, the reverberation time T is mainly determined by the absorption in air αL . The reverberation time is the most important variable to describe the acoustics of rooms. In rooms with a long reverberation time, sources with a relatively low level yield a high sound intensity, however, speech intelligibility is decreased due to increased temporal masking. As a compromise, the reverberation time must be “appropriate” to the room size. For speech, reverberation time should be between 0.5 sec and 1 sec (increasing with room size), for music presentations 1 sec to 2 sec are acceptable. Beyond the global description of a room by the reverberation time, the temporal finestructure of the reverberation, the temporal incidence of the reflections, is of interest. From the first reflections, the human hearing system is able to extract information about the size of the room. If the first reflections occur very early (1 msec to 10 msec after the direct sound wave), the sound color—especially for music recordings—is altered. Reflections in the time span from 10 msec to 50 msec increase the perceived loudness. Single echoes, arriving with a delay of more than about 100 msec, are perceived as echoes. Very disturbing are periodic echoes, which are generated between parallel walls, for example. For good room acoustics, the reflections should be homogeneous and the intensity should decrease with time. Single echoes should not be larger than 5 dB compared with their temporal vicinity. In concert halls, the reverberation time is measured as a function of frequency and additional sound absorbers and reflectors are placed to improve the acoustics. Remark 24 This sketchy discussion shows that room acoustics is a complex topic, therefore acoustic experts should be consulted already in the planning of music rooms. This need is documented by plenty of examples, where the acoustics of rooms built for the purpose of audio presentations is so bad that speech intelligibility is severely hampered. Thoroughly planned room acoustics is essential not only for concert and lecture halls, but also for most other rooms such as offices and even hallways and production areas, to keep noise levels down and achieve an environment which is pleasing to the ear. Remark 25 We have also included this discussion since it makes plausible that there is no chance to integrate a poor theory of room acoustics in a valid music-theoretic framework, and that great efforts should be made to lift this status in order to give the compositions with room acoustical specifications a firm background. 7 See
section A.2.2 for the definition of loudness.
1018
A.1.2
APPENDIX A. COMMON PARAMETER SPACES
Sound Analysis and Synthesis
This central subject of acoustics relates to the poietic (synthesis) and esthesic (analysis) aspects of neutral sound data, more specifically, of the sound pressure variation function p(t) = p0q (t) at a source location q and cast in a determined standard support (see section A.1.1). More precisely, we are considering a time function p which is defined for all times, vanishes outside a finite time interval, e.g., [0, 1], and has a finite absolute amplitude supremum sup(|p|), e.g., sup(|p|) = 1. Sound synthesis means that we have exhibited an operator σ which for each sequence of its finite or infinite number of numeric (mostly real-valued) parameters x1 , x2 , . . . yields a function p = σ(x1 , x2 , . . .) of the described type. Analysis then means that we are given p and the operator σ and would like to determine a sequence x1 , x2 , . . . of arguments such that p = σ(x1 , x2 , . . .). The general map σ : (x1 , x2 , . . .) 7→ σ(x1 , x2 , . . .) will be neither surjective nor injective. So synthesis is neither a synthesis of any imaginable p, nor is analysis unambiguous, i.e., the fiber of σ could be a large set of parameters. Moreover, one may have a synthesis operator σ1 and an analysis operator σ2 such that the analysis of a tame synthesis may become pathological. In the following sections, we shall discuss four such operators: Fourier, frequency modulation, wavelets, and physical modeling. We shall however not deal with mixed synthesis/analysis problems which, mathematically speaking, are wild ones—let alone the associated technological problems. A.1.2.1
Fourier
The Fourier approach deals with periodic functions, in our case periodic pressure variations p(t) as functions of time t. This however is not the direct approach to produce a pressure function which is inserted into the support operator, since such a p has an infinite support. In order to turn a periodic function f into one with a standard support Support(1, 0, 1), say, it is usually multiplied by an envelope function H, see figure A.2. This is a continuous, piecewise differentiable8 non-negative function on R which fits in the standard support Support(1, 0, 1), i.e., H(t) = 0 outside [0, 1] and kHk∞ = M ax(H) = 1. In most technological applications, H is even a spline function (often even a linear, i.e., polygonal spline) modeling the attack and decay of a sound event. Then, the pressure function is given by p = w.H. Observe that this is already a source of poietic ambiguities: neither w (not even its frequency), nor H are uniquely determined by p. We then say that the standardized pressure function p is defined by the envelope H and the wave w. Fourier’s theorem deals with periodic wave functions w which are piecewise smooth9 waves. For any piecewise smooth function w : R → R, the additive group R acts by translations: (eP .w)(t) = w(t + P ). The group of periods P eriodsw of w is the isotropy group of w under this action. For any non-zero period P , the inverse fP = 1/P is called a frequency of w, its unit is Hertz, Hz. If Pw = inf ({P ∈ P eriodsw , 0 < P }) = 0, w is evidently constant, otherwise, 8 Differentiable
except for a finite set of points except for a finite set of discontinuities (it need not be defined in these points, but the left and right limits of the functions exist in these points), with a continuous derivative, except for a finite set of points (where the derivative is not continuous or even not defined, but the left and right limits exist in all these singularities). Many examples are plain C1 functions, but the saw-tooth function is not. 9 Continuous,
A.1. PHYSICAL SPACES
1019
envelope H
wave w H.w
A
e
e+d
Figure A.2: The envelope H (top left), the wave w (top right), and its combination H.w (bottom left), as well as the affine deformation by the support operator Support(A, e, d) (bottom right).
P eriodsw = hPw i is the discrete group generated by the smallest positive period Pw . To avoid ambiguities in the periods or frequencies of a wave, one addresses this smallest period Pw or frequency fw = 1/Pw if one speaks about the “fundamental period” or the “fundamental frequency” of w (otherwise, not even the period or frequency of a wave would be uniquely determined). Fourier’s theorem is this (for a proof, see [276]):
Theorem 37 If w is a piecewise smooth wave function and P ∈ P eriodsw is a positive period, with f = 1/P the corresponding frequency, then there are two sequences (An , P hn+1 )n=0,1,2,... of real numbers such that
w(t) = Ao +
X
An sin(2πnf t + P hn ),
(A.6)
n=1,2,3,...
i.e., the infinite series converges and represents the wave for every time t for all points where the function is continuous. For the given period, the coefficients An and P hn are uniquely
1020
APPENDIX A. COMMON PARAMETER SPACES
determined and can be calculated as follows: P/2
Z A0 = f
w(t)dt, −P/2
Z
P/2
an = 2f
w(t) cos(2πnf t)dt, 0 < n, −P/2
Z
P/2
bn = 2f
w(t) sin(2πnf t)dt, 0 < n, −P/2
p An = a2n + b2n , 0 < n, P hn = arcsin(an /An ) if An 6= 0 and P hn = 0 else. The An is called the nth amplitude, whereas P hn is called the nth phase of the wave with respect to the selected period. The sequence (An )n is called the amplitude spectrum, (A2n )n is called the energy spectrum since the energy of a wave is proportional to the square of the amplitude, and the sequence (P hn )n is called the phase spectrum. If the period/frequency is the fundamental period/frequency, one omits these specifications. An equivalent representation (with coefficients an , bn , unique for a given period) is obtained for the explication of the sinoidal components via the goniometric formula sin(a + b) = sin(a) cos(b) + cos(a) sin(b) and yields w(t) = Ao +
X
an cos(2πnf t) + bn sin(2πnf t).
(A.7)
n=1,2,3,...
Remark 26 It is well known [307, Thm. 6.7.2], that the function sequences (sin(2πnf t)n=1,2,3,... , (cos(2πnf t)n=0,1,2,... form an orthogonal basis of the pre-Hilbert space10 C0 [−P/2, P/2] of the continuous functions on [−P/2, P/2] for the 2-norm (see appendix I.1.2), where f = 1/P . This follows in particular from the trigonometric orthogonality relations of the defining scalar R P/2 product (f, g) = −P/2 f (t)g(t)dt of functions f, g ∈ C0 [−P/2, P/2], i.e., (sin(2πnf t), cos(2πmf t)) = (sin(2πnf t), sin(2πnf t)) = (cos(2πnf t), cos(2πmf t)) = 0 (A.8) for n 6= m. There is an infinity of such orthogonal bases for C0 [−P/2, P/2], and mathematically, nothing distinguishes the sinoidal basis chosen by Fourier from the other orthogonal bases. Moreover, sinoidal functions are all but elementary. Mathematically, they are very complex, as is evident from Euler’s identity cos(x) + i. sin(x) = eix . A justification for using sinoidal waves lies in the fact that simple mechanical differential equations, such as the spring equation m.¨ x = −k.x, have sinoidal functions as their solutions. But this is a physical argument which must be coupled with a dynamical system of this equational type in order to give these functions any preference. 10 A
normed real vector space whose norm is defined by a positive definite symmetric bilinear form.
A.1. PHYSICAL SPACES
1021
In order to meet the requirement for a unit amplitude, the coefficients of the Fourier representation can be dilated by a common factor, and we are done with the periodic wave. A common generalization of the Fourier representation (A.7) is defined if the frequency and coefficients are also functions of time: f = f (t), An = An (t), P hn = P hn (t), a situation which is also needed to represent sounds of physical instruments with glissandi, crescendi, and their natural damping effects. This construction is the poietic perspective. The esthesic one deals with the problem of constructing an envelope H, a periodic wave w and its Fourier representation (A.7) for a given sound function p. As was already mentioned above, the wave and the envelope cannot be reconstructed unambiguously in general. Even if the wave is known, the envelope is not reconstructible, although a number of obvious candidates can be calculated, e.g., a polygonal envelope defined by the local maxima and minima of the enveloped wave. As to the wave, one candidate for such can be guessed by the analysis of a time window [t1 , t2 ] of p within the supporting duration, such that the local maxima of p are relatively constant (neither at the initial, nor at the decay phase of the sound). One can then take a multiple of the period as a time window, and calculate the Fourier representation of this time window which is interpreted as a finite interval of a really periodic function, i.e., prolongation of this window to infinity. Although this period will not be the fundamental period of the wave, the Fourier representation will yield the right coefficients modulo a multiple of the fundamental frequency. If the fundamental period is small relatively to the total duration of the sound, there is a chance to calculate the underlying wave. In general, this is a highly ambiguous situation. Once the wave is reconstructed, the Fourier coefficients are uniquely determined by the Fourier theorem, and we are done. But this is only the last phase of a highly ambiguous situation. It is of course always possible to find an underlying wave, it suffices to take the total duration of the support and to set it to the wave’s period, i.e., prolongation to infinity of the sound by adding copies of itself to the left and to the right of the sound support. For the relation of these reconstructions to what is heard, see below appendix B. Remark 27 A final remark on the terminology of sound frequencies in music. Usually, when we deal with “the frequency of a sound”, we do not mean that this frequency is a neutral property of the sound (although it could happen that the sound really has a fundamental frequency), but the fundamental frequency of a wave that is used in the standard representation of the sound as a product of its (periodic) wave and a deformed envelope. This is a poietic definition, and this is what we will use because a neutral definition does not exist for general sounds. In this setup, we have the sound function p = Support(A, e, d)(wA.,P h. (f ).H). Here, we may assume that wA.,P h. is the formal representation of the trigonometric sum by the amplitude and phase spectra, and such that the total amplitude maximum is 1; the frequency is given as an additional argument, and the envelope H is given in its standard support. We call the 4-vector (e, f, d, A) the geometric coordinates of the sound, and the pair (wA.,P h. , H) the color coordinates since they are responsible for the sound color (timbre) in this representation. For other poieses of sound which may also use the frequency coordinate (for example those to be discussed in the following sections), this one would also be referred to, but there is no neutral access to this concept. A word of caution: The way humans “detect” sound frequencies is not neutral, this is an esthesic psycho-physiological system whose function is far from understood, so do never mix up neutral facts with poietic or esthesic facts when dealing with sound attributes!
1022 A.1.2.2
APPENDIX A. COMMON PARAMETER SPACES Frequency Modulation
In this section, we have a similar decomposition as discussed before: the sound p is written as a product p = H.w, where H is the envelope, and w is a “wave” function. However, this time we do not use the additive combination of sinoidal functions of Fourier synthesis to build w. The combination is rather a functional concatenation of such functions, i.e., sinoidal functions have in their arguments other sinoidal functions, and so on. This synthesis operator was introduced by John Chowning [86] and implemented first in the legendary Yamaha’s DX7 synthesizers. The formal definition of frequency modulation (FM) functions in terms of circular denotators is given in section 6.7, example 3, and yields this type of expressions: F M sound(myF M Object)(t) =
n X
Ai sin(2πFi t + P hi + F M sound(myM odulatori )(t))
i
where myM odulatori is the FM-Object factor of the limit type denotator Knoti . In the terminology of FM synthesis, the (respective) interior functions F M sound(myM odulatori )(t) are called the modulators with respect to the (respective) exterior sinoidal functions Ai sin(2πFi t + P hi ) which are named the carriers. We symbolize this relation by an arrow myM odulatori ⇒ myF M Object. So the FM functions start as a sum of carriers where modulators are inserted, and these modulators are again of this nature, etc., until no modulator appears and the recursion terminates. In FM synthesis it is also allowed to have circular denotators in the sense that a modulator can be a carrier in one and the same function! Observe that the existence of a denotator myF M Object describing such a “self-modulating” function does not imply the existence of the corresponding function F M sound(myF M Object). However in digital sound synthesis, one often resolves this problem by taking the time argument of the modulator one digital step before the time argument of the carrier, i.e., F M sound(myF M Object)(tn ) =
n X
Ai sin(2πFi tn + P hi + F M sound(myM odulatori )(tn−1 )).
i
In the DX7 implementation, the FM recursion scheme is presented in the graphical block diagram format of a so-called algorithm (not an adequate wording, though); figure A.3 shows Yamaha’s 32 algorithms. Each sinoidal carrier component is written as a block, whereas the modulators of a given carrier are those blocks which are above the carrier and are connected by a line with this carrier. Each of the building blocks can be specified in the respective parameters. The point of the FM synthesis is that it needs a small number of sinoidal functions— in fact only six sinoidal oscillators are needed for the DX7 algorithms—to simulate complex instrumental sounds, which is a theoretical and technological advantage. But the elegant FM synthesis has also drawbacks concerning the uniqueness question: How many different denotators myF M Object do yield the same function F M sound(myF M Object)? A general answer seems difficult, we have two partial solutions. The first regards a tower of modulators: Lemma 54 Let G : R → R be a bounded C1 function. Then G is determined by its value G(0) and by any function F : R → R of the shape F (t) = a + b sin(ct + G(t)), with b 6= 0.
A.1. PHYSICAL SPACES
1
2
10
3
4
11
5
23
29
7
13
9
15
16
21
25
30
8
14 20
19
24
28
6
12
18
17
1023
26
31
22
27
32
Figure A.3: Yamaha’s 32 algorithms for FM synthesis. Only six sinoidal oscillators are required to generate a variety of more or less natural sounds. Proof. Evidently, by the boundedness of G, a + b = max(F ), a − b = min(F ), so a = (max(F ) + min(F ))/2, b = (max(F )−min(F ))/2, are determined by F . Moreover, the continuous function 0 − c is a function of F , and the value G(0), together with the integral of G0 G0 = √ 2 bF 2 b −(F −a)
which is a function of F , completely determine G, QED. Proposition 61 If M = (Mn ⇒ Mn−1 ⇒ Mn−2 ⇒ . . . M0 ) is an FM denotator defined by a sequence of relative modulators for the functions Mi = Ai sin(2πfi t+?), then the resulting sound function F M sound(M ) determines all the modulators. Proof. This follows from lemma 54 by recursion on the modulators, starting with total function F M sound(M )(t) = A0 sin(2πf0 t + F M sound(M1 )), where M1 is the denotator starting from M1 instead of M0 . In the lemma, take F = F M sound(M ), G = F M sound(M1 ), and observe that G(0) = 0, QED. The second solution regards a flat sequence of unrelated carriers: P Proposition 62 Let F (t) = i=1,...k ai sin(bi t) with 0 < b1 < . . . bk , 0 < ai be the function associated with a flat FM denotator. Then the coefficients, and therefore the denotator, are all uniquely determined by F .
1024
APPENDIX A. COMMON PARAMETER SPACES
P 2n+1 n 2n+1 Proof. We have for all n = 1. Consider the function i=1,...k ai bi P (−1) F x (0) = H(a., b.)(x) = i=1,...k ai bi of the real variable x. Suppose we have two sequences (ai )i=1,...k , (bi )i=1,...k and (a0i )i=1,...k , (b0i )i=1,...k such that they yield the same function F . Then we have 0 H(a., b.)(x) = H(a0 ., b0 .)(x) for all x = 2n + 1, n = 1. Write bi = eβi , b0i = eβi , and suppose 0 bk < bk . Then P P P (βi −βk )x (βi −βk )x βi x + ak H(a., b.)(x) i=1,...k−1 ai e i=1,...k ai e i=1,...k ai e P = . = = P P 0 0 0 −β )x 0 0 0 0 0 β x (β −β )x (β k k i i i H(a ., b .)(x) i=1,...k ai e i=1,...k ai e i=1,...k ai e But for x → ∞, the denominator goes to 0 whereas the numerator goes to ak , contradicting the fact that this quotient is 1 for all x = 2n + 1, n = 1. Therefore bk = b0k , so in the above quotient, the limit for x → ∞ is ak /a0k which is also 1, and we have equal coefficients for the index k. Therefore, we may proceed by induction to k − 1 and we are done, QED. These very special results show that given the tower or the flat FM schemes, the functions determine their coefficients (the background denotators) uniquely, but for general FM schemes, there is no such a result. Moreover, if the FM scheme is not known, we have no idea of how the scheme should be determined from the function. This is a drawback compared to the Fourier operator, where the coefficients are always uniquely determined once the fundamental frequency is fixed. In other words, FM synthesis is much more efficient than Fourier synthesis, but one has to pay for this when turning to the respective analyses. The switch between Fourier and FM operators is essentially managed by Bessel functions. These are defined directly from a core situation from FM synthesis, i.e., Definition 111 Let z be a real number. Then the Fourier expansion of the 2π-periodic function sin(z sin(t)) of t is X sin(z sin(t)) = 2 J2n+1 (z)sin((2n + 1)t), (A.9) n=0,1,2,...
whereas the Fourier expansion of the 2π-periodic function cos(z sin(t)) of t is X cos(z sin(t)) = J0 (z) + 2 J2n (z)cos(2nt).
(A.10)
n=1,2,3,...
The functions Jm , m = 0, 1, 2, . . . are called the mth Bessel functions. The above definition relies on the Fourier representation of the respective functions and on their properties as odd or even functions. An alternative definition of Bessel functions is Rπ Jm (z) = π1 0 cos(mt − z sin(t))dt. From definition 111, one obtains the following fundamental equations for Bessel functions: sin(r + z sin(s)) = cos(r + z sin(s)) =
∞ X −∞ ∞ X −∞
Jn (z) sin(r + ns), Jn (z) cos(r + ns).
A.1. PHYSICAL SPACES
1025
In particular, with s = 2πf t, z = I, t = 2πgt, we have sin(2πf t + I sin(2πgt)) =
∞ X
Jn (I) sin(2π(f + n.2πg)t),
−∞
which is a Fourier type linearization, however, it is not a proper Fourier representation since the so-called “nth side band” frequencies f +n.2πg are not a multiple of a fundamental frequency in general. This is rather a reduction of a FM concatenation to a flat FM configuration. Conversely, every (finite) Fourier decomposition is evidently a flat FM configuration. A.1.2.3
Wavelets
Although the FM operator is much more efficient than the Fourier operator, it is still an operator which produces functions with an infinite support, a property which no real sound shares, and we have in fact added an envelope to cope with this requirement for Fourier and FM operators. From this point of view, wavelets are fundamentally better suited for handling finite sound objects without any envelope casting. Refer to [308] and [279] for the wavelet theory and its applications to sound and music. Let f be a square integrable function (element of L2 (R), i.e.,
(1 - t^2)Exp[-t^2/2]
Sin[2 Pi t]Exp[-(t)^2/2]
Figure A.4: Two wavelets: Murenzi’s Mexican hat [387] (left) and Morlet’s wavelet (imaginary part) [192] deduced from the sinoidal function (right). R R
|f (x)|2 dx < ∞). Then its Fourier transform is defined and is the function Z fˆ(ω) = (2π)−1 f (x)e−ixω dx.
(A.11)
R
R ψ| ˆ2 Definition 112 A square integrable function ψ is called a wavelet if 0 < cψ = R ||ψ| dψ < ∞. For a wavelet ψ, the wavelet-transformed of a square integrable function f is the function (of two variables a, b) Z t−b −1/2 −1/2 Lψ f (a, b) = cψ |a| f (t)ψ( )dt, (A.12) a R with a ∈ R − {0}, b ∈ R.
1026
APPENDIX A. COMMON PARAMETER SPACES
Figure A.4 shows two typical examples of wavelets. The wavelet transform is a function of two variables defined on every couple (a, b) ∈ (R − {0}) × R. The point of this representation is that it is a kind of system of coefficients (Lψ f (a, b))(a,b)∈R−{0}×R which is parametrized by two real numbers, corresponding to a scalar product −1/2
cψ
(f, Support(|a|−1/2 , b, a)(ψ))
(A.13)
of the function f with an affinely deformed version11 Support(|a|−1/2 , b, a)(ψ) of the “mother” wavelet ψ. This deformation Support(|a|−1/2 , b, a) is an isometry on the square integrable functions, see also figure A.5 for some deformed wavelets. By the following formula, the wavelet
y((?-b)/a) a
b
Figure A.5: Various affine deformations of the mother wavelet ψ. transformations Lψ f (a, b) redetermine the original function f : Z da db −1/2 f (t) = cψ Lψ f (a, b)Support(|a|−1/2 , b, a)(ψ)(t) 2 . a 2 R
(A.14)
For selected wavelets, it is possible to generate an orthonormal basis of L2 (R) which is defined as a so-called frame. For a0 > 1, b0 > 0, and for the Meyer wavelet12 ψ we consider −m/2 m the frame (Support(a0 , nb0 am 0 , a0 )(ψ))m,n∈Z of deformed versions of ψ. Then this is an −→ that the operator Support(|a|−1/2 , b, a) defines a linear action Support : GL(R) → GL(L2 (R)) : −→ −1/2 2 7→ Support(|a| , b, a) of the affine group GL(R) on the space L (R) of square integrable functions. 12 See [308, 2.1.25]. 11 Observe
eb a
A.1. PHYSICAL SPACES
1027
orthonormal basis of L2 (R) for a0 = 2, b0 = 1. This is an analogous situation as encountered for Fourier series and their sinoidal bases with the known trigonometric orthogonality relations in formulas (A.8). For a comparison of Fourier and wavelet analysis, see [560]. A.1.2.4
Some Remarks on Physical Modeling
The previous operators were directly acting on the production of a sound function. Their poietic nature was a mathematical one, just to construct a time function p(t) by a mathematical procedure from a certain type of “atomic”, i.e., basis functions. In contrast, physical modeling is one step more poietic in that it does not directly deal with sound, but with a physical system that produces sound. On the one hand, this is a strong restriction since sound does not care for the physical system that evokes that sound. It seems however that this drawback is compensated by the fact that musical expressivity is largely determined by the physical device which the artist manipulates when interpreting or improvising music. It is also an important argument that the simulation of the physical instrument and then a possible canonical extension could yield more interesting sounds than just “abstract nonsense” procedures. For a general survey on physical modeling, see [460, chapter 7], we restrict our discussion of this extensive topic to some systematic remarks. The idea is that one considers a physical model of a sound production device and then implements this model as a software which—if sufficient calculation power is available—calculates the physical output on the level of the sound wave that is emitted by the modeled instrument. At present, there are three methodologies for such a modeling: mass-spring, modal synthesis, and waveguide. The mass-spring paradigm just models a physical instrument (a string, a drum) by a finite space configuration of point masses that are related by springs and damping effects [225]. The modeling is built upon the classical mechanics of Newton’s law and the corresponding dynamic behavior that eventually terminates in the air’s vibration. The modal synthesis paradigm [77] reduces the vibrating physical system to a system of vibrating substructures, usually very small in number compared with the mass-spring components. These substructures are characterized by their frequencies, damping coefficients and parameters for the vibrating mode’s shape. This adds up to a sum of modal vibrations. Whereas in simple configurations these data can be obtained from classical literature in equations for vibrating systems, the complex data must be extracted from experimental results. A prototypical implementation of this paradigm is MOSAIC, developed by Jean-Marie Adrien and Joseph Morrison, see [460, p. 276 ff.]. The waveguide paradigm has been implemented in commercial physical modeling synthesizers by YAMAHA and KORG , see [460, p. 282 ff.]. It has mainly been developed by Julius O Smith III and collaborators [494, 495]. The waveguide model implements the traveling wave along a medium, such as a tube or a string. See [496] for an update of physical modeling strategies. Although physical modeling is a successful approach in the simulation of musical instrumental sounds, it is a step back from the neutral sound objects to their generators. This has not only been a technological requirement for performance theory (where physical modeling is a core approach), it is also a consequence of the failure in the understanding of the topological semantics of sound objects as exposed in section 12.3. In particular, it is an open problem to
1028
APPENDIX A. COMMON PARAMETER SPACES
relate the physical modeling theory and technology to the other operators, such as Fourier, FM, and wavelets. The deeper question here is whether the neutral sound objects are really the most relevant ingredients of musical performance, i.e., how strongly the gestural components influence and characterize the sounding reality. It is not clear how deeply a sound conveys the generating gestures in its autonomous structure. To our knowledge, sound classification has not been directed towards a gestural coordinate in the neutral sound description, except, perhaps, in the straightforward envelope component.
A.2
Mathematical and Symbolic Spaces
The physical description of sounds is not what can be used for music theories. This is based on (1) the way humans perceive sounds, (2) the shape of music thinking, and (3) available instrumental technologies. Therefore physical parameter spaces must be transformed into spaces which essentially encode the same information, but do so in a way which is more adapted to music. Basically, we shall present mathematical structures where physical parameters are represented. Based upon these spaces, we shall explain a derived set of representations which encode different music-topographic aspects. Our discussion regards the geometric coordinate pairs (basis coordinate plus associated pianola coordinate) onset and duration (A.2.1), amplitude and crescendo (A.2.2), and frequency and glissando (A.2.3).
A.2.1
Onset and Duration
In music, one speaks about tempo, metronome, quavers, semiquavers, triplets, 3/4 meter, etc. The relation to the physical time parameters is as follows. To begin with, the physical onset time e sec and duration d sec are opposed to a musical onset time E note and duration D note, usually also in real values, and in units such as “note”, meaning a whole note. However, in many practical contexts, the rational number field Q will do. In this latter context, we have the ratios of integer numbers E = w/n, D = z/n, n > 0, and the denominator is of the form n = 2r 3s 5t 7u with natural exponents r, s, t, u, whereas the numerators w, z are integers, i.e., we are working in the ring localization13 Z[1/2, 1/3, 1/5, 1/7] at the primes 2, 3, 5, 7. Musically, this means that one is allowed to add, subtract, multiply such numbers at will without leaving the domain, and also division of such numbers by 2, 3, 5, 7 leaves the domain invariant, i.e., the construction of duplets, triplets, quintuplets, septuplets is possible without any restriction. The musical time shares a mental reality and should not be confused with physical time. Genetically, musical time is an abstraction from physical time, but they are by no means equivalent. This abstraction is also a creation of an autonomous time quality where mental constructions such as a score can be positioned. The relation between these two time qualities is defined by the tempo, usually encoded as a metronomic indication of x quarters per minute according to M¨alzel, meaning a difference quotient (velocity) ∆E/∆e of musical time per physical time. This presupposes that we are given a one-to-one performance mapping E 7→ e(E) from musical time to physical time. It is common in mathematical music theory to write physical parameters in lower case letters, whereas musical parameters are written in upper case letters. 13 Make the integers 2, 3, 5, 7 invertible, i.e., admitting fractions 1/2, 1/3, 1/5, 1/7. But see appendix E.4.1 for a formal definition.
A.2. MATHEMATICAL AND SYMBOLIC SPACES
1029
Let x, y be positive integers. Then every onset E can be written uniquely as E = δ +τ.x/y, where 0 ≤ δ < x/y is in Q, and τ ∈ Z. If δ = q/(ny), we say that in x/y time14 , E is on the (q + 1)st ny-tuplet in bar τ + 1. The additive group Z.1/y is called the meter of the x/y time. Evidently, this initializes a score at time E = 0, but this is pure convention. Pay attention not to view the symbol x/y as a plain fraction, but as a pair x, y giving rise to a fraction. In other words, in music notation, the symbol x/y is the mathematical fraction x/y plus the meter. Example 61 Let x = 3, y = 4, E = 15.375. Then we have E = 3/8 + 20.3/4, i.e., E is on the fourth quaver of bar 21. If we work in Z[1/2, 1/3, 1/5, 1/7], then with E and 1/y, the remainder δ and the meter are automatically in this domain. If we are given a duration D, then durations of form D/n are also called n-tuplets (with respect to D). The reason why the concept of onset time is not common in music lies in the fact that a note’s onset can be deduced from its position in the score. The time signature, bar-lines, and simultaneous or preceding notes help establish the time context of each note. This helps calculate onsets algorithmically by recursion from the first score onset. This algorithm is derived from the linear syntagm of written language, but it sometimes leads to ambiguities.
A.2.2
Amplitude and Crescendo
The amplitude A of a sound relates to the loudness sensation, i.e., to musical dynamics. However, the chain of transformations to the physiologically relevant measures is quite complex [462]. The first member of this chain is the transformation which associates A with the sound pressure level (SPL) l(A) = 20. log10 (A/Athreshold ) dB. (A.15) Here, Athreshold = 2.105 N m−2 is close to the amplitude of the threshold pressure variation for a 1 kHz sound in a young, normal hearing human subject. The unit dB for l is Dezibel. Of course, these constants are pure convention. From the auditory physiology, the shape l(A) = a ln(A) + b is essential, i.e., l(A) is a conventionally normed linear function of the logarithm of the amplitude. The physio-psychological motivation for this approach lies in the Weber-Fechner law according to which the sensation of a difference of sensory stimuli is proportional to the stimuli [242], yielding the logarithmic representation as an adequate encoding of this sensorial modality. Figure A.6 shows some environmental SPL values (the musical units ppp, pp, etc. will be discussed below). For example, the SPL ≈ 120 dB of an air jet is one million times the threshold SPL. Just as duration is the time interval from the onset time to the “offset” time of a sound event, the crescendo parameter c is the difference between the onset SPL and the offset SPL. In normal piano sounds, this vanishes, but for violins, trombones, etc., this is a relevant quantity. In this context, one supposes that the sound representation p = Support(A, e, d)(wA.,P h. (f ).H) has a time-dependent amplitude A = A(t). At the offset time e + d, the amplitude has changed by the amount A(e + d)/A(e), or in terms of loudness, c = l(A(e + d)) − l(A(e)). As with onset and duration, it is not specified what happens in the sound process between the onset and the offset, it’s just the difference that matters. This difference c is termed the (physical) crescendo. 14 We
stick to the continental terminology since “meter” will be reserved for a different concept.
1030
APPENDIX A. COMMON PARAMETER SPACES
threshold of pain
limit of damage risk hearing threshold at 3 kHz ppp
pp
p
mf
f
ff
fff
conversational speech sleep
-12
0
10
20
30
40
traffic
50
60
70
air jet
80
90
100
110
120
130
L(dB)
Figure A.6: Some environmental sound pressure level values.
In the musical abstraction L for loudness, one also works in the field R of real numbers, but the common score notation is not well defined for all real values. More precisely, the first still physically motivated codification is a conventional calibration such as, for example, l : Z → R with l(L) = 10 dB.L + 60 dB such that the values 0 7→ mf 1 7→ f 2 7→ f f 3 7→ f f f
− 1 7→ p − 2 7→ pp − 3 7→ ppp
are arranged symmetrically around the mezzoforte sign. This looks like an identification of the score symbols with precise physical values. But this is wrong, it is a transformation from mental to physical reality. In fact, the affine transformation l(L) is only a “default” assignment of a mental quantity symbolized by a dynamic symbol (which is codified by an integer). This fact is more evident if one recalls the velocity parameter in the MIDI code. This quantity is an integer in the interval [0, 127], but the physical meaning of a velocity value depends on the assignment by specific technological calibrations, in particular by the output chain where the loudspeakers can take values that have nothing to do with the velocities. Units in the mental level of loudness are not standard, but we could, for example, take vel for MIDI’s velocity parameter. The extension of integer values to real or rational values (depending on the specific usage) is a reasonable procedure, for instance preconized by MIDI velocity codification. It is also useful for finer loudness data management with relative loudness signs such as “crescendo”, “diminuendo” which make sense in a mental crescendo parameter which we shall denote by C. See also our discussion of performance transformations and primavista weights in section 39.2.
A.2. MATHEMATICAL AND SYMBOLIC SPACES
A.2.3
1031
Frequency and Glissando
Sounds which share a frequency parameter f are musically important, at least in the European tradition. However, the corresponding mathematics is a bit more involved than for time and amplitude. This is an expression of an intense discussion of traditional harmonies and interval theories with an evolving instrumental technology. To begin with, f behaves like amplitude: For reasons of auditory physiology (see also Appendix B), f is transformed via the logarithmic formula h(f ) = u ln(f ) + v, yielding the physical pitch h of frequency f . The common unit of pitch is the Cent Ct, it corresponds to the logarithm of a relative frequency increase by the factor 21/1200 , i.e., one percent of a welltempered semitone (see below for tuning types); this entails u = 1200/ ln(2), but the constants u, v are purely conventional. Presently, and in the Western framework, the relevant frequencies have the shape f = 132 Hz.2p .3s .5t , where p, s, r ∈ Q. It is based on the chamber pitch 440 Hz of the one-line a, as fixed in London 1939. Instead, we have chosen the (unlined) c with the frequency 132 Hz = 440 Hz.2−1 .3.5−1 as a starting frequency in order to relate the examples with ease to c. Mathematically, the restriction to the first three primes 2, 3, 5 is not essential. One could as well take any sequence p1 , p2 , . . . of mutually prime natural numbers (larger than 1), the rational powers of which are multiplied by 132 Hz. The natural logarithm of f is ln(f ) = p. ln(2) + s. ln(3) + r. ln(5) + ln(132). But music theory is rather interested in the relative pitch, i.e., ln(f ) − ln(132) = p. ln(2) + s. ln(3) + r. ln(5). Moreover, the passage to another logarithmic basis b could be desirable, i.e., ln(b)−1 . ln(f ) − logb (132) = p. logb (2) + s. logb (3) + r. logb (5), so that we are given the linear function u.X + v = ln(b)−1 .X − logb (132) for X = ln(f ). For music theory, the restriction to rational exponents is essential, since this hypothesis enables an unambiguous representation, see Appendix E.2.1. Every frequency f can be replaced by a point x = (p, s, r) ∈ Q3 . Such a point represents the frequency f (x) = 132 Hz.2p .3s .5r , and from f (x) = f (y), we conclude x = y. With this interpretation, Q3 is called the Euler space and a point x = (p, s, r) ∈ Q3 is called an Euler point. This means that a real number p. logb (2) + s. logb (3) + r. logb (5) is viewed as a vector which is a rational linear combination of linearly independent vectors logb (2), logb (3), logb (5). The choice of the three first primes stems from the tradition of just tuning, where for two frequencies f, g, we have: (O)f /g =2/1 :f is the octave (frequency) for g. (Q)f /g =3/2 :f is the (just) fifth (frequency) for g. (T )f /g =5/4 :f is the (just) major third (frequency) for g.
1032
APPENDIX A. COMMON PARAMETER SPACES
This is why 2 is associated with the octave, 3 with the fifth, and 5 with the major third. We therefore call p the octave coordinate, r the fifth coordinate, and s the third coordinate (of the sound with frequency f or of the corresponding Euler point. The Euler point o = (1, 0, 0) is called the octave point, the point q = (0, 1, 0) is called the fifth point, and t = (0, 0, 1) is called the third point. Observe that fifth and third points are not the fifth and octave. The structural meaning of these points is explained in section 6.4.1. The minor just third frequency 6/5 does not add a new prime number. For a fixed chamber pitch, the 2-3-5 just tuning (short: just tuning) is the set of frequencies which are represented by integer coordinates, i.e., the subgroup Z3 ⊂ Q3 of the Euler space. This is the three-dimensional grid which was introduced by Leonhard Euler [143]. The group of those grid points whose third coordinate vanishes is the Pythagorean tuning. The Euler space is derived from the logarithmic representation of pitch, but the coefficients are beyond physical reality, this is why we view this space as a mental space. The same is valid if we consider the codomain space of pitch h, but interpreted with rational coefficients (written as R[Q] instead of the usual real line R, the one-dimensional real vector space15 ). More precisely, one should name H the pitch on the mental pitch space P itch, and h the pitch on the physical pitch space P hysP itch. Whereas the unit for physical pitch is Cent, the unit for mental pitch could be Semitone or MIDI’s key(number), but no standard exists here. In performance, we have a transformation ℘ : P itch → P hysP itch. Both spaces have the real numbers as underlying sets, but the meaning of the spaces is different. The mental pitch space encodes pitch as it is symbolized on the score or as a key number in MIDI code. Ideally, performance transforms this abstract data into mathematical pitch M ath(H) = (p, s, r) in the Euler space, and this is transformed to the pitch ln(f ) = p. ln(2) + s. ln(3) + r. ln(5) + ln(132) in R[Q] . Only after forgetting about the coefficients, we are in P hysP itch. And this is by no means a formal play: It is a dramatic change of reality if one views the reals as an infinite-dimensional space with linearly independent octave, fifth, and third logarithm vectors, or as a line, where everything shares the same direction! In analogy to onset and loudness, one introduces physical glissando g and its symbolic counterpart G. In the common visualization of the Euler space, all grid cells are shown as cubes. This could suggest that angles and distances are relevant to this space. So far, this has however no musicological reason, and it is nonsense to argue with angles and distances as Arthur von Oettingen [406], Carl Eitz[137], and Martin Vogel [547] have done. Evidently, the mathematical structure of just tuning is independent of the historical choice of the sequence 2,3,5. One could as well take any pairwise prime positive numbers 1 < p1 < p2 < p3 and would get a p1 -p2 -p3 just tuning. If the pitch range in Euler space is described by non-integer coefficients, one speaks of tempered tunings. The most well known are defined by a uniform construction mode. For a natural number w > 1 one considers all pitches whose octave coordinate is a fraction of shape p = x/w, whereas the other coordinates vanish. This is called the w-tempered tuning. This ∼ defines a grid Z.1/w.o → Z with step width 1/w of the octave point. By the same recipe, tempered tunings in fifth and third direction can be defined. The 12-tempered tuning is the famous “well-tempered” tuning. The 1200-tempered tuning is less interesting for conventional composers than for measurement techniques, where the unit step is the Ct step defined above. 15 See
Appendix E.2.1.
A.2. MATHEMATICAL AND SYMBOLIC SPACES
1033
More generally, one may define tunings which consist of tempered and just components. The procedure runs as follows. Take three positive integers w1 , w2 , w3 and consider the grid Z.
1 1 1 .o + Z. .q + Z. .t w1 w2 w3
which specializes to the 2-3-5 just tuning as well as the tempered tunings as defined above. Call this construction the w1 -w2 -w3 just-tempered tuning. Historically relevant is the mediante tuning, which is the 1-1-2 just-tempered tuning, and which includes the tempered whole-tone step in the major third. With respect to the auditory psycho-physiology (see Appendix B), we should consider the distribution of the just tuning grid vectors x = (p, s, r) in Z3 with respect to the mathematical pitch H(x) = p. logb (2) + s. logb (3) + r. logb (5). More generally, take any vector x ∈ R3 and denote by Hprime = (logb (2), logb (3), logb (5)) the prime vector. This means that pitch is the usual scalar product H(x) = (Hprime , x) (A.16) of the prime vector with the generalized Euler point x. Therefore, Proposition 63 With the definition (A.16) two vectors x, x0 ∈ R3 have the same pitch H(x) = H(x0 ) iff their difference is orthogonal to the prime vector Hprime . ⊥ Call E = Hprime the plane orthogonal to the prime vector. Then the proposition means that for any generalized Euler point x, x + E is the set of points with same pitch as x. Now, according to what we know, E lies so skew in R3 that for a point x ∈ Q3 ,
(x + E) ∩ Q3 = {x}. Nonetheless, every real number φ can be approached within any given error by points of the just tuning grid: For any positive bound δ, there is x ∈ Z3 such that |H(x)−φ| < δ, see Appendix 73 for a proof. In particular, we have the following proposition which has dramatic consequences for theories of hearing of just tuned pitch (see section B.2): Proposition 64 If φ = H(x0 ) is a mathematical pitch of a grid point x0 , then for any positive bound δ, there is an infinity of grid points x such that |H(x) − H(x0 )| < δ.
Appendix B
Auditory Physiology and Psychology “Music listening” is a metonymy of understanding music: For all participants, the ear functions as an interface for perceiving music between physical, psychological, and mathematical reality. But a metonymy is not the matter as such. This is what deaf Beethoven teaches us impressively: his innermost ear was an organ of imagination that was uncoupled from the material ear. And the physiology of the hearing process teaches us that the neural coupling of the ear to the respective cortical regions is extremely complex and still hardly understood. Section B.1 is written to give an overview on auditory physiology. Beyond receptive processes hearing means also an active shaping according to templates of esthesis and poiesis. One of the difficult basic problems concerning the activity of hearing deals with the compatibility of these templates and the physical input, and in particular the notorious “straightened out hearing” (German: “Zurechth¨oren”) which keeps alive a lot of wishful thinking in music theory. Basically, the problem is that we do not perceive physical sounds, but classes of indistinguishable sounds, what Werner Meyer-Eppler coined “valences” in [372]. Valence theory, which is sketched in section B.2, has dramatic consequences for the relation between mathematical theories and their semantic potential for music. Historically and materially, and in view of the genealogy of a mathematical theory, the subject pairing of “consonance-dissonance” is an excellent illustration of auditory physiology and psychology. We deal with the formal and some physiological aspects of this subject in part VII. In the following section B.3, we want to expose the stratification of the phenomenon of consonance and dissonance. Hereby, the problem setup as well as the approaches to its solution demonstrate a strong dependency of the addressed reality layer. Methodologically, section B.3 is important since it makes evident that known approaches of the mathematical argumentation in musicology turn out to be too narrow with respect to the existing music, and too dogmatic and scientifically unbased with respect to music thinking. 1035
1036
APPENDIX B. AUDITORY PHYSIOLOGY AND PSYCHOLOGY
B.1
Physiology: From the Auricle to Heschl’s Gyri
Two fundamentally different regions of sound processing in the auditory system can be distinguished. In the peripheral region, mechanical preprocessing takes place, especially in the fluid-filled inner ear. The sensory cells encode the preprocessed mechanical oscillations into electrical nerve action potentials. In the second region of the hearing system, neural processing of the sound information is conducted in ascending nuclei, which finally leads to auditory sensation. The hearing system, especially in mammals, has pushed its bandwidth up in a frequency range, where the limits of neural processing in the time-domain are exceeded by far. Instead of processing high frequencies in the time domain, evolution has developed the so-called frequency-place principle. In the inner ear, the frequency contents of sound signals are separated in the spatial domain. The sensitivity of our hearing system is thereby remarkable, in its most sensitive region, the threshold of the hearing system is limited only by thermal noise.
B.1.1
Outer Ear
Sound energy is collected by the outer ear and transmitted through the outer ear canal to the ear drum (figure B.1).
Figure B.1: Schematic view of the outer, middle and inner ear, modified from [594]. The delicate structures of our hearing system are well protected inside the skull. For the sound transmission, the outer ear canal acts like an open pipe with a length of about 20 to
B.1. PHYSIOLOGY: FROM THE AURICLE TO HESCHL’S GYRI
1037
30 mm. Its quarter-wave resonance is responsible for the high sensitivity of our hearing organ in this frequency range, indicated by the dip of the threshold in quiet around 4 kHz. This high sensitivity is however also the reason for high susceptibility to noise-induced damage in the region around 4 kHz.
B.1.2
Middle Ear
The fluids of the inner ear must be excited by the sound-induced vibrations of the air particles in front of the ear drum. The light but sturdy funnel-shaped ear drum (tympanic membrane) operates over a wide frequency range as a pressure receiver. It is firmly attached to the long arm of the hammer (malleus) (figure B.1). The motions of the eardrum are so transmitted via the anvil (incus) to the stirrup (stapes). The stapes foot plate, together with a ring-shaped membrane called the oval window, forms the entrance to the inner ear. The middle ear optimizes the energy flow from air-borne sound in front of the ear drum to fluid motion in the inner ear by a mechanism called impedance transformation. One part of the impedance transformation is based on the lever ratio of about 1.5:1 produced by the different lengths of the arms of malleus and incus [72]. The lever ratio transforms oscillations of the ear drum with small forces into motions of the fluid with large forces. The law of energy conservation implies also that the tiny displacements of the ear drum are transformed into still smaller oscillations in fluid. An even larger transformation of the pressure is due to the ratio of the large ear drum to that of the small oval window. This ratio is about 17 [47]. Through the lever and area ratios, an almost perfect impedance match is reached in man in the middle frequency range between 1 and 4 kHz. It allows optimization of the energy flow into the inner ear, which otherwise would be reflected. The middle ear operates normally when it is filled with air at atmospheric pressure. The Eustachian tube, which connects the middle ear cavity to the upper throat, normally opens and closes periodically, thereby insuring that the static pressure in the middle ear will remain the same as atmospheric pressure. We experience a pressure difference when the Eustachian tube fails to open during ascent or descent in an elevator. For an elevation of 8 m, the change in atmospheric pressure is 100 N m−2 , corresponding to a sound pressure of 130 dB relative to the 20 µN m−2 reference! This pressure causes a static deflection of the ear drum and increases the stiffness of the middle ear transmission system and sound transmission is attenuated.
B.1.3
Inner Ear (Cochlea)
The shape of the cochlea resembles that of a snail shell with two and one-half turns (in humans) and hence its name (figure B.2). The central conical bony core of the cochlea is called the modiolus. The auditory nerve fibers run in this bone and exit the cochlea at its base. The outer wall of the modiolus forms the inner wall of a 30 mm long canal which spirals the full two and one-half turns around the central core. This canal is separated into three partitions called scales: Scala tympani is separated by the so-called cochlear partition, which is formed by a thin shelf of bone projecting from the modiolus (the ossesus spiral lamina), which is connected by the basilar membrane and the spiral ligament to the outer wall of the cochlea. The sensory organ of hearing, the organ of Corti, is located on top of the basilar membrane. As can be seen in the cross section of the cochlear spiral, the cochlear scalae become smaller and smaller in cross-sectional area as the apex is
1038
scala media
APPENDIX B. AUDITORY PHYSIOLOGY AND PSYCHOLOGY
scala vestibuli
e
an
br
'
ers
m me
sn eis
tectorial membrane
stria vascularis
R
spiral ligament
scala tympani
IHC 1 mm
auditory nerve
osseous spiral lamina
OHC
BM 0.1 mm
Figure B.2: Section through the human cochlea (left) and magnified view of the organ of Corti (right). IHC: inner hair cell, OHC: outer hair cell, BM: basilar membrane.
approached. Directly opposed, the basilar membrane becomes progressively wider towards the apex (figure B.2). This is because the osseous spiral lamina is broadest at the cochlear base where the basilar membrane is only about 0.16 mm wide (in humans); at the apex the basilar membrane has broadened to about 0.52 mm. Scala media and scala vestibuli are separated by a thin membrane, called Reissner’s membrane (figure B.2). At the apical end of the cochlea, scala vestibuli and scala tympani are connected by an opening in the cochlear partition, called helicotrema. Scala tympani and scala vestibuli are filled with perilymph, which resembles in its chemical composition other extracellular fluids. Perilymph is characterized by high sodium (Na+ ) concentration of about 140 − 150 mM and its low potassium (K+ ) content of only around 5 mM . Scala media is filled with endolymph, which is unlike any other extracellular fluid found in the body. From its chemical composition, it resembles intracellular fluids. Its predominant cation is potassium with a concentration of about 157 mM ; sodium is very low (1.3 mM ). In addition to its special chemical composition, the endolymphatic space exhibits a considerable positive electrical potential within scala media of about +80 mV relative to scala tympani and scala vestibuli, called the endocochlear potential. The chemical composition and the electrical potential of the endolymphatic space is sustained by active ion transport provided by the cellular layers of stria vascularis. The organ of Corti lies just between the endolymphatic and perilymphatic spaces (figure B.2). Its surface is sealed by tight junctions to keep the fluids separate. The basilar membrane, on which the organ of Corti rests, is composed mainly of extracellular matrix material with embedded fibers. In contrast to the tight surface of the organ of Corti, the basilar membrane is thought to be permeable to perilymph. In the organ of Corti, two types of sensory cells, one row of inner hair cells and three to four rows of outer hair cells are embedded. In humans, there are approximately 3,500 inner- and 12,000 outer hair cells. The membrane potential of inner hair cells is about −40 mV , of outer hair cells even as low as −70 mV . Both types display “hair bundles” or stereocilia, which project into the endolymphatic space. The stereocilia are arranged in several rows, which are graded in size. Stereocilia from different rows are connected by a fine filament, called tip-link. The current theory of transduction assumes
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that hair bundle deflection pulls on the tip-links, opening transducer channels which are close to the attachment points. This concept also explains that the hair bundle is only sensitive to mechanical stimulation in the direction of the tallest stereocilia. The transducer channel is a nonselective cation channel which is very impermeable to anions. Therefore, the transduction current is mainly carried by potassium (K+ ) and calcium (Ca2+ ) cations, driven by the large electrical potential between the endolymphatic space and the receptor cells. The driving potential sums up to 120 mV for inner hair cells and to 150 mV for outer hair cells. Potassium leaves the sensory cells via potassium channels present in the basolateral cell membrane and diffuses into scala tympani. It is interesting to notice that the intracellular concentration of potassium is as high as that of endolymph. Potassium is driven into the cells by the electrical potential and because of the concentration gradient, it can diffuse into scala tympani without requiring energy from the sensory cells. The inner and outer pillar cells, the phalangeal processes of the Deiters’ cells and the cylindrical bodies of the outer hair cells build a complex, three-dimensional truss (see figure B.3). One peculiarity is that outer hair cells are not in contact with other cells along their lateral
IHC
OHC
10 mm DC
Figure B.3: Scanning electron micrograph of the three-dimensional arrangement of the organ of Corti. IHC: inner hair cells, OHC: outer hair cells, DC: Deiters cells [141]. surface but immersed in extracellular fluid. From figure B.3 also the different morphology of the hair bundles becomes apparent: Whereas the bundles of the inner hair cells are arranged in a straight line, these of the three rows of outer hair cells are W-shaped. The hair bundles of outer hair cells are excited by a shearing motion between the surface of the organ of Corti and the so-called tectorial membrane. The tectorial membrane is a gel-like structure composed of extracellular matrix material and it is in direct contact with the longest row of outer hair cell stereocilia. In contrast, the hair bundles of the inner hair cells seem not to be in direct contact with the tectorial membrane. Their bundles are probably driven by fluid
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forces. The transduction current flowing through the stereocilia is converted into a receptor potential in the cell body of the hair cell. At low frequencies, the receptor potential follows the stimulus cycle-by-cycle. Upon mechanical stimulation of the hair bundle in excitatory direction, tip-links are stretched and the transduction channels open. As positive K+ ions are driven into the cell, the potential inside the cell becomes more positive. If the bundle is stimulated in the other direction, tip links relax and transduction channels close. However, as for inner hair cells at rest only 20% of transduction channels are open, the receptor potential is highly asymmetric when stimulated with a tone. On the basal pole of the inner hair cells about 10 to 30 afferent synapses are located, and upon depolarization of the cell membrane, voltage-sensitive Ca2+ channels located in the basal pole of the cell membrane open, the increased Ca2+ level causes vesicles filled with transmitter to fuse with the lateral cell membrane and release transmitter into the synaptic cleft. This transmitter release triggers an action potential in the afferent nerve, and these electrical spikes transmitted to the brain finally lead to the hearing sensations. These mechanisms work well at low frequencies, where the events of neural processing can easily follow the sound stimulus. Because of its refractory period, a single auditory nerve fiber can not respond to each successive cycle of a high-frequency sound. This problem can be partially overcome by the fact that more than one nerve fiber contacts a single inner hair cell. Each fiber is incapable of responding to every cycle of the stimulus, but collectively, they can do so. The temporal structure of the sound is conserved, as each fiber responds in the depolarized state of the inner hair cell. We call this behavior “phase locking”. Even if the firing rate of a nerve fiber is too slow to follow the stimulus, basic features of its temporal structure—like the phase—are still coded in the neural pulse train. Still, the effect of phase locking is limited by several factors. The lateral membrane has a time constant of about 1 kHz, and above this frequency, the AC-amplitude of the receptor potential decreases. Also the synaptic processes like vesicle release and the generation of the postsynaptic potential are limited in speed and accuracy. This leads to a gradual loss of phase locking starting above 1 kHz, and above 3 kHz, phase locking is completely lost in humans. In the frequency region, where the receptor potential can no longer follow the stimulus on a cycle-by-cycle basis, we see a depolarization of the inner hair cells membrane potential while stimulated. This so-called DC-component of the receptor potential follows the stimulus envelope. This is because of the asymmetry of the hair-bundle transduction. The depolarizing currents dominate upon sinusoidal stimulation and the receptor potential, low-pass filtered by the cell membrane, cause a depolarization of the cell membrane. All the information about the sound’s frequency is lost in this signal, therefore, a different way to code high frequencies had to be developed by evolution. This task was achieved by “sorting” sounds by frequency. Sound signals are mechanically preprocessed in a way that they are separated spatially. This concept is well-known as the frequency-place principle where high frequencies are located in the basal part of the cochlea only and low frequencies in the apical part. Different sets of auditory nerve fibers elicit different auditory sensations by virtue of their central connections. We will examine the mechanical frequency separation process in the sec section B.1.4 and to do so, we will have to focus on the mechanical properties of the inner ear. It is however not clear how specific excitations converge to yield well-defined pitch in general. There is increasing evidence that pitch is extracted in the time domain (periodicity analysis [286]). Moreover, pitch was found to be independent of the frequency-place mapping
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of the components of complex tones [287].
B.1.4
Cochlear Hydrodynamics: The Travelling Wave
The cochlea consists of three fluid-filled scalae, but from a mechanical point of view, the elastic properties of the thin Reissner membrane can be neglected compared to the stiffness of the basilar membrane. We can therefore simplify our mechanical investigations to a fluid model with two chambers separated by the cochlear partition (compare figure B.4). The basilar membrane contributes a large part of the elastic properties of the partition. Its width is increasing from base to apex from about 0.16 µm to 0.52 µm (in humans). Its thickness, on the other hand, decreases along the cochlea. Thus, its stiffness decreases greatly along the length of the cochlea. At the basal end of the cochlea there are two openings to the cochlear ducts, one on each side of the cochlear partition, that are covered by membranes. One is called the oval window and, as we already mentioned, it is in contact with the stapes foot plate. The other, the round window, is just below the oval window. Because we can assume the cochlear fluids as incompressible, the round window has to move out of phase if the oval window is driven by stapes motion. If we consider very slow stapes motions, the fluid is pushed along the entire length of scala vestibuli, through the helicotrema and back along scala tympani. Thus, the helicotrema provides a low-frequency shunt for extremely low frequencies. When the stapes moves into the cochlea, pressure builds up in the fluid, which deflects the cochlear partition. Pushing fluid requires overcoming inertial forces generated by the fluid mass. The elastic properties of the basilar membrane in combination with the mass of the surrounding fluid constitutes secondorder resonators. As the stiffness of the basilar membrane is very high at the basal end of the cochlea and much lower at its apical extreme, the resonant frequency of the cochlear partition monotonically decreases from base to apex. Inversely, the time constant of each resonator increases from base to apex. However, the basilar membrane is not under tension, and it does not respond like a series of independent resonators, like the strings of a harp. Instead, each part of the cochlear partition is coupled to the next by the cochlear fluids, and due to the large inherent friction, they are highly damped. For a periodical motion of the stapes, the cochlear partition is first set into motion at the basal extreme, where the mechanical time-constant is smallest. Because the stiffness is very high, the deflection of the basilar membrane is fairly small. The deflection propagates in the form of a wave in apical direction. As the time-constants of the partition increase and the stiffness decreases, the response will be more and more delayed but its amplitude will grow. At the location of resonance, the wave will reach its maximum and lose its energy, its amplitude will drop very rapidly. The location of “cochlear resonance”, the place where the displacement—and therefore the excitation—of the cochlear partition reaches its maximum, depends on the stimulus frequency. Low frequencies will travel along the basilar membrane and reach a maximum close to the cochlear apex, high frequency sounds will exhibit their maximum response close to the cochlea base and fade out. The exact form of the vibration response of the cochlear partition was investigated by Georg von B´ek´esy in human cadaver ears [47], a feat which earned him the Nobel Prize. He found that the deformation of the basilar membrane is a traveling wave. The wave starts at the cochlear base, where the basilar membrane is stiffest. It propagates toward the apex with a time delay that depends upon its own mechanical properties and the properties of the surrounding
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APPENDIX B. AUDITORY PHYSIOLOGY AND PSYCHOLOGY
fluid. Its vibration amplitude is increasing until it reaches a maximum, close to the location of cochlear resonance, and from then on, the wave diminishes rapidly. Because the stiffness gradient of the basilar membrane is approximately logarithmic, the peaks of the excitation patterns of sinoidal tones are located on the basilar membrane with a logarithmic frequency spacing. Figure B.4 schematically illustrates traveling waves elicited by a stimulus composed of three frequencies, together with the envelope of the peak displacement. The peaks of the
stapes
helicotrema 5 kHz
2 kHz
500 Hz
round window
Figure B.4: Schematic illustration of the traveling waves elicited by three pure tones with frequencies of 500 Hz, 2 kHz and 5 kHz. The displacement of the basilar membrane is shown at the instant T0 (solid line) and a quarter of a cycle later (dashed line). The dotted line indicates the envelope of the wave. three waves are clearly separated along the cochlea, however, there is also considerable overlap between the waves and recordings from the auditory nerve have been found to be much more frequency selective, especially close to threshold, than the mechanical responses observed in cadaver ears. Therefore, a second mechanism is required to boost the frequency selectivity of the vibration responses.
B.1.5
Active Amplification of the Traveling Wave Motion
We have so far neglected the function of the outer hair cells, the second group of receptor cells within the organ of Corti. Despite the at about three to four times higher number of outer hair cells, only 5% of the afferent fibers innervate outer hair cells. The fibers are so-called type II fibers, they are highly branched and each fiber innervates dozens of outer hair cells. Little is known about these fibers because they are small and unmyelinated, making it difficult to record their activity. They are expected to be less sharply tuned, since they innervate a broad region of the cochlea, and, because of their lack of myelination, their conduction velocity is likely to be very slow. Most of the frequency selective and time-critical auditory information must therefore be carried by the afferent fibers originating from the inner hair cells. Whereas inner hair cells are not directly innervated by the efferent system, myelinated fibers from the medial to the medial
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superior olivary nuclei make direct synaptic contact with the outer hair cells. Outer hair cells seem to be under neural control, much more like muscles than sensory cells. A final indication of their “active” role was established when their ability to change their length upon electrical stimulation was detected. Modern research assumes that the outer hair cells indeed provide an active, mechanical amplification of the cochlear traveling wave. As neural processing is by far too slow to keep up with high frequency hearing, current concepts assume that amplification relies on a local mechanical feed-back process [107]: Outer hair cells sense motions of the cochlear partition by converting shearing motion between the surface of the organ of Corti and the tectorial membrane into an electrical receptor potential. Upon depolarization, the outer hair cell reacts with a contractile force, which is fed back into the motion of the basilar membrane. From theoretical calculations derived from measurements in the inner ear, it is required that energy is pumped into the vibration of the cochlear partition in a region starting before the traveling wave reaches its maximum up to the place of cochlear resonance. The function of this amplification process is still unclear in its details, but from the observations of active cochlear mechanics it has wide reaching consequences. Measurements in the basal part of the cochlea indicate, that the amplification boosts cochlear sensitivity up to thousand-fold. The amplification is limited to a narrow frequency range, covering only about half an octave. The active traveling-wave response becomes very sharp in the region of cochlear resonance. The amplification of the vibration response is required to achieve the extraordinary sensitivity of the hearing system. The second hallmark of amplification is non-linearity. The amplification process boosts only weak sounds, it saturates at increasing levels. This non-linearity greatly compresses the dynamic range of the mechanical responses. This is important, because the dynamic range of the inner hair cell receptor potential is limited to a range certainly not exceeding 60 dB. This nonlinearity also has unwanted side-effects: If two sinoidal tones are presented, the non-linearities of the hearing-organ generate so-called distortion products, additional tones which we perceive under certain circumstances. In general, however, by virtue of its construction, artifacts due to the non-linearities of the inner ear are surprisingly small. Figure B.5 shows the excitation pattern expected on the human basilar membrane when stimulated with a 3 kHz sinoidal tone. For high sound levels (i.e., 100 dB), the feed-back amplifier is saturated and the traveling-wave is almost purely passive. It is highly damped and its envelope shows the characteristic shallow increase from base to apex and a sharp decay after the maximum is reached.1 The threshold of the auditory nerve is somewhere between a minimal basilar-membrane velocity of 50 µmsec−1 and a displacement of 1 nm, exact values are still unknown. The broad traveling wave at high levels indicates that a large number of nerve fibers are stimulated, especially in the basal part of the cochlea. For faint sounds, the traveling-wave response becomes sharper and sharper and its envelope is almost symmetrical for levels below 40 dB. The location of maximum amplitude for a sinoidal tone at low levels is called its characteristic place. Only nerve fibers originating from a very narrow region around the characteristic place of the cochlea are stimulated. Note that for increasing levels, the maximum of the traveling 1 The data in figure B.5 (dashed lines) indicates that further towards the apex, the wave does not die out completely in this experiment. It is still under debate, whether this remaining response is also present in the intact human cochlea.
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v (mm/s)
100 dB
10000 80 1000 60 100
40 20
10
0
0
5
10
15
20
x (mm)
Figure B.5: Reconstruction of the excitation pattern in the human cochlea for a 3 kHz sinoidal tone. Original measurements were recorded in a chinchilla cochlea at a distance of 3.5 mm from its most basal extreme [392]. The characteristic frequency of this location was 9.5 kHz. Data has been converted assuming a frequency-place map of 8 mm/octave to illustrate the excitation pattern in a human cochlea. The shaded area indicates excitation below neural threshold, which is expected between 50 µmsec−1 and 1 nm. wave shifts considerably in the basal direction. If we analyze the level-dependence of the responses at various locations of the cochlea, we clearly see the effects of the non-linearities of the amplification. At the characteristic place (about 16 mm in figure B.5), the velocity amplitude increases from a value of 42 µmsec−1 at 0 dB to 5 msec−1 at 100 dB. Without amplification, the amplitude of the traveling-wave response would be expected to drop by a factor of 105 , or from 5 mmsec−1 to 50 nmsec−1 from 100 dB to 0 dB! The amplification therefore is almost a factor of 1000 or 60 dB. In addition to enabling the detection of weak signals, the amplification therefore also compresses the dynamic range, again by almost 60 dB. The inner hair cells, at the location of the characteristic place, have to cope only with a stimulus ratio of a little bit more than 40 dB to cover a 100 dB-level change of the sound stimulus.
B.1.6
Neural Processing
The difficulty to understand neural processing of sound strongly stems from the extremely complex innervation from the auditory nerve to primary auditory cortex (Heschl’s gyri) in the temporal lobe (see figure B.6). We have to stress that the image in this figure is considerably simplified, and that in particular, there are also connections from the cochlea to the ipsilateral auditory cortex, as well as efferent nerves from the auditory cortex down to the cochlea. The auditory system includes at least five “relays stations”, whence it is clear that any particular functional decomposition (like Fourier’s) will not be transferred unchanged to the auditory cortex. For example, the tonotopy of the latter is a multiply distributed one, see figure B.7.
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Figure B.6: (With kind permission of the Hallwag-Verlag) This simplified image shows the six relevant relais stations of the auditory path from the cochlea to the Heschl gyri: (1) Nervus cochlearis, (2) nucleus cochlearis, (3) nuclei superiores olivae, (4) colliculus inferior, (5) corpus geniculatum mediale, (6) gyri Heschl.
By tonotopy, one understands the spatial distribution of excitation patterns according to specific pitch. More recent research also demonstrates that outside the auditory cortex, i.e., in the limbic system (the hippocampal formation, to be precise, which plays an important role for emotional and memory tasks), one finds a refined processing of pitch information [336, 337, 570, 571]. The “template fitting model” of Julius Goldstein [187] shows how far we are from understanding the neural pitch processing. In this model, the mathematical principle of a neural “central pitch detector” is proposed, from which the fundamental frequency of a periodic wave can be extracted if its Fourier components are known. The central pitch detector is however charged with the solution of local minima problems for functions in two variables—a rather heroic task for a small neural population. It seems hopeless to identify within the neural network the physical realization of a dynamic system that solves these differential conditions. We presently have no chance to verify the model physiologically, since the human ethics excludes adequate experiments in humans. In view of these facts, it is not only logically erroneous and experimentally very delicate to infer the higher sound processing from the superficial auditory physiology of the ears. And ethically, such an attempt is problematic since one runs the risk to “justify” discrimination of “degenerate music” against the acceptance of so-called “commonly accepted” music. It is also not clear how much such investigations reveal grown and trained configurations instead of
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Figure B.7: (With kind permission of the Thieme-Verlag) Left: tonotopy in the spiral of the cochlea: high pitches are thick points, low pitches are thin points. Right: Corresponding multiple tonotopy in the auditory cortex of the cat. biological inheritance.
B.2
Discriminating Tones: Werner Meyer-Eppler’s Valence Theory
From our everyday experience and from specific experiments it follows that we do not really hear the single tone events or chords, motives or rhythms as they have been parametrized on the level of physical or mathematical description. In fact, despite the very sensitive physiology of Corti’s organ and its hair cells, we cannot distinguish all physically or mathematically distinct sound objects. For instance, sounds with frequency above 20, 000 Hz or below 0 dB are indistinguishable since you cannot hear them. Or two sounds p, q with a phase shift, i.e., p(t + ∆) = q(t) are indistinguishable. More important is that even under ideal condition we cannot distinguish sounds with arbitrary precision. Every singer or violinist who has to adapt his/her pitch to a prescribed context knows this. Investigations on variations of instrumental intonation show a remarkable bandwidth [57]. The same is valid for listening to time values, loudness degrees and instrumental colors. It is mandatory but not easy to take into account these phenomena. Werner Meyer-Eppler [372, 373] has attempted to solve the problem by use of the concept of a “valence”. According to this approach, Definition 113 In a specific context, given two sound objects s1 , s2 , s1 is metamere to s2 with regard to a given predicate P (short: P -metamere, in symbols: s1 ∼P s2 ) iff the s1 cannot be distinguished from s2 with respect to P by human listeners. The set ∼P s of sound objects which are P -metamere to a given sound s are called the valence of s. Since usually the relation ∼P is symmetric, one calls two sound objects s1 , s2 metamere if s1 ∼P s2 . If a sound object s is defined by a sequence P1 , . . . Pk of predicates, one defines its valence as being the sequence of valences ∼P1 , . . . ∼Pk , and we may then reduce the total valence to those components which do not include all sound objects since these predicates are
B.2. DISCRIMINATING TONES: WERNER MEYER-EPPLER’S VALENCE THEORY1047 S not relevant to the distinction of sound objects. The union supporting valences ∼Pij of supporting V valences is in fact the valence of the conjunction predicate i Pi . Predicates which are relevant in this sense are called valence supporting by Meyer-Eppler. In practice, if we are given a sound whose sound color is described by partials with frequencies up to the limit frequency f0 Hz, and which last d sec, Meyer-Eppler deduces a maximal number of numerical predicates (dimensions) that are relevant to the valence, i.e., of valence supporting numerical predicates. This limit is called the maximal structure content, and its value is K = 2.d.f0 . It is however an open fundamental question of sound color theory, which and how many valence supporting predicates must be chosen in order to yield a differentiated perception of sound colors. Probably, these valences define a multiply connected topological space in the physical parameter space, but see also section 12.3. Although Meyer-Eppler’s conceptualization is plausible, it hides two delicate problems. The first concerns the context where valences take place, as related to the predicate in question. If the context’s specification is neglected, the valence concept loses its meaning. Let us make two representative examples concerning pitch and onset. To begin with, we have to agree on who is accepted as a listener. In the sense of a statistical approach, Meyer-Eppler proposes that two pitches should be called metamere if at least 90% of the test subjects cannot distinguish them [373]. Moreover, one has to agree on the parameter(relations) of the test sounds. The result will depend essentially on the choice of instruments, flute, brass etc., and conditions upon the duration and the onset distances of test sounds. For example, the simultaneous presentation of sounds of several seconds duration will yield smaller valences because of beat effects, as compared to a comparison of non-overlapping sound events. Besides the parametric conditions, the context can also depend on the chosen music. Let us discuss this on the onset parameter, for example. In [373], the duration valence density is indicated by 50 − 60 per second. This means that sounds which are less than 1/50 − 1/60 sec apart are perceived as being simultaneous. However, the musical context of such a claim is relevant. Within a very slow piece with a small number of instruments, the temporal variation of 1/8 sec will scarcely be noticed, whereas in a rhythmically very dense and fast piece, 1/60 sec is known to define a quite coarse grid. The second problem is important for the theoretical significance of the valence concept. It relates to the fact that metamery is not an equivalence relation2 in general. More precisely, any of the wanted properties: reflexivity, symmetry, and transitivity, can be violated. Reflexivity can be violated if the comparison of two sounds is temporally so separated, by several hours, say, that the human memory fails to recognize one and the same sound. Symmetry is violated if the order of appearance of sound objects is relevant, for example, a very loud sound, immediately followed by a very soft one, can mask the latter’s properties. The most dramatic failure is the absence of transitivity: s ∼P t, t ∼P u does not always imply s ∼P u. For example, if three pitches are such that the first is perceived as being equal to the second, and the second being equal to the third pitch, this is not entailed by equality of first and third pitch! Therefore, pitch valences are not equivalence classes, they may overlap. This means that the attempt to define pitch by an esthesic position in music psychology must fail. The perceptional concept of pitch is a non-transitive relation among tones, and therefore is not an attribute of tones. You hear that two tones have the same pitch, but you do not hear the pitch. Therefore corresponding 2 See
appendix C.2.
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attempts such as [394] must fail. This has important consequences for musical practice and for theoretical aspects. In practice it is desirable to select grids of sounds such that their valences would not overlap. This needn’t be a grid which is fixed once for ever, it might be a time-dependent construction, but it must yield a locally disjoint valence set. With the common notation, such a grid is realized as an orientation device as well as an acoustic and performative scheme. The continuum of onsets, durations, pitches, and sound pressure values is quantized in the well-known way such that Boulez’ “notched tone space” can be taken as a grid behind the realiter played or heard. The reality layer of a grid is a mental or psychological one. The blurredness of hearing, as it is expressed in valences, has to be subjected to a cognitive interpretation. Semiologically speaking, the valence is the significant, the expression for a meaning which relates to our understanding of music. By use of the grid which is superposed to the valence perception, it is possible to associate the valences of perceived sound to objects of our imagination. The psychological quality of pitch then results from this mapping as a grid object in our imagination. The semiotic power of this signification depends upon the definition of the actual grid. A music-theoretically fundamental grid is the selection of the pitch arsenal, the tuning, wherein tones may be played. Let us first look at the chromatic w-tempered scales (see section 7.2.1.1). If w is not too large, valences of neighboring tones can be separated. From our experience with microtonal music, w = 36, i.e., tempered sixth-tone intervals, is not too large. So the postulate of an adjustment of pitch by inner grids is acceptable for w-tempered scales with w ≤ 36. The situation of just intonation (see section 7.2.1.2 and appendix A.2.3) is much more delicate. For each pitch H(x0 ) of a point x0 in the Euler grid Z3 of just intonation, and for any positive real number , there are infinitely many points x such that |H(x − x0 )| < , see proposition 64 in appendix A.2.3. In particular, infinitely many pitches fall into the valence of x0 . In the valence semiology, just intonation is infinitely homonymic. This problem can be solved by a restriction of the context, where just intonation music is played. As soon as the local context is a small region of the Euler space, for example, a small neighborhood of a determined finite portion of the chromatic scale, valences can be used to distinguish just grid points. A second difficulty for just intonation relates to the perception of pitch differences. A classical argument of just music theory is Euler’s substitution theory [144], according to which intervals are heard in a way such that the frequency relations of their tones form fractions a/b (in reduced representation) with minimal numerator a and denominator b. This is the basis of the classical consonance-dissonance theory which we address in the following section. In the grid of just intonation this means that the pitch difference of the interval is corrected/adapted to an interval which is realized by two points of the just intonation grid under the constraint that their (Euclidean or 1-norm) distance is minimal. Of course, this correction has to happen within a valence in order to be a reasonable process of human hearing. However, it is easy to see that in general, there are several solutions a/b, a0 /b0 , a”/b”, . . . with minimal distance. So Euler’s substitution theory would have to impose a contextual restriction for single tones as well as for intervals. These considerations should be taken for nothing more than they are: esthesic aspects of hearing. Nothing prevents us from doing music theory on neutral and poietic layers without bothering about valences. But then, one has to be conscious of the fact that highly differentiated
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mathematical structures may be blurred by the semiology of valences from auditory psychology and physiology.
B.3
Symbolic, Physiological, and Psychological Aspects of Consonance and Dissonance
For a long time, mathematical reflections in music were centered around the problematic concept couple of consonance and dissonance. This is based on the ancient Greek Pythagorean tradition where consonance and dissonance of intervals was laid in the involved frequency relation. Perfect consonant intervals corresponded to ratios like 2/1 for the octave, 3/2 for the fifth, and 4/3 for the fourth. This simple arithmetic corresponded to the philosophy of the metaphysical tetractys. See [330, 394] for a historical discussion of these roots. Here, we simply want to recall that a unified mathematical foundation of musical thinking in the paradigm of simple consonant frequency ratios could not survive the differentiated development of theories in the contrapuntal setting [468], the psychological foundation of musical relations as introduced by Ren´e Descartes [126], and the discovery of physical partials by Marin Mersenne [310]. According to these more recent positions, the problem of consonances and dissonances changes as a function of the layer of reality where it is investigated—and on each layer it is not a minor one. In the present shorthand presentation, this result seems to be a provocative one since the conceptual unity seems violated. We want to make clear that we really are dealing with three different meanings of the sonance concept—Euler’s gradus suavitatis on the mental layer, Helmholtz’ beat model on the physical layer, and Plomp–Levelt’s psychometrics on the psychological layer. In itself, each of these approaches is consistent, the problem only arises if one attempts to reduce one reality to another one. Following the knowledge about the neural processing of sounds (see the previous discussion in this chapter), it is hardly astonishing that psychological and physiological layers are not congruent: what the ear (in Helmholtz’ model) does not “like” can very well be “agreeable” for the limbic system or the auditory cortex.
B.3.1
Euler’s Gradus Function
Being a number theorist, Euler was interested in prime numbers. A priori, his gradus function Γ [143] is a purely number-theoretic function, it is defined as follows: According to the prime factorization of integers (see appendix D.2), a positive integer a is the unique product a = pe11 .pe22 . . . . penn of positive powers of primes p1 < p2 < . . . pn (with the singular case of zero factors and a product = 1 for a = 1). Euler’s formula, the gradus suavitatis, is Γ(a) = 1 + P 1≤k≤n ek (pk − 1), and more generally Γ(x/y) = Γ(x.y) for a reduced fraction x/y. In just intonation—where Euler’s approach belongs—only intervals of tones are considered whose frequency ratio is a positive rational number x/y. The gradus function is defined for such intervals. Each interval evaluates to a positive integer. The frequency ratios with Γ-values ≤ 10 are listed in table appendix J, see also figure B.8. In just intonation, intervals can be read as differences ∆ = x − y = (e, f, g) of Euler points x and y in the grid Z3 . For the gradus function, we may restrict to non-negative coordinates
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APPENDIX B. AUDITORY PHYSIOLOGY AND PSYCHOLOGY 10 / G
10
5
0
1 1
16 15
9 8
6 5
5 4
4 3
45 32
3 2
8 5
5 3
16 9
15 8
interval
Figure B.8: The agreeableness of intervals within the octave in just intonation according to Vogel’s chromatic [547], see section 7.2.1.2, is represented. It is reasonable to represent the reciprocal values 10/Γ instead of Γ since the latter is rather a ‘gradus dissuavitatis’: small Γ values are taken for the octave and the fifth, large ones for the second and tritone. The factor 10 is only a scaling constant. The order of Euler’s valuation is this: Prime, fifth, fourth, major third/major sixth, minor third/minor sixth/major second, minor seventh, major seventh, minor second, tritone. e, f, g. We then have Γ(∆) = Γ(2e .3f .5g ) = 1 = 1 + (2 − 1)e + (3 − 1)f + (5 − 1)g
(B.1)
or as a scalar product Γ(∆) = 1 + (Φ, ∆) with Φ = (1, 2, 4).
(B.2)
If we compare this formula with the pitch formula H(∆) = (Hprime , ∆) where Hprime is the prime vector from section A.2.3, then we observe a similar construction. The gradus function is something like a pitch function, but the ‘direction’ is Φ instead of Hprime , see figure B.9. The ranking of intervals by the gradus function was already criticized by Euler’s contemporaries Mattheson, Mitzler, and Rameau [71]. But it is the merit of Euler to have defined a ranking by a linear expression which, together with pitch and octave coordinate
B.3. ASPECTS OF CONSONANCE AND DISSONANCE
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third coord.
F t
gradus coord. pitch coord.
h
q o
x
fifth coord.
octave coord.
Figure B.9: The gradus function is something like a pitch function, but the ‘direction’ is Φ instead of Hprime , Both values are obtained from a scalar product of an Euler point x (of a difference point ∆, respectively) with Hprime (with Φ, respectively). However, the point is not uniquely determined by the ‘gradus coordinate’, in contrast to the ‘pitch coordinate’. defines a coordinate system for the Euler space on the one hand, and considers the consonances and dissonance, on the other. The disadvantage of Euler’s approach is that it is based on the valence-theoretically invalid substitution hypothesis (see section B.2).
B.3.2
von Helmholtz’ Beat Model
Hermann von Helmholtz proceeds from the hypothesis that beats between partials of two tones is responsible for sonance phenomena. The fact that he uses partials is bound to Ohm’s postulate that we have a cochlear Fourier analysis [217]. Hence Helmholtz’ approach only regards the cochlear basis of music perception and not the higher limbic and cortical auditory processing. A beat is the periodic amplitude variation which results from the superposition of two sinoidal waves which have a frequency difference ∆ = f − g, the beat frequency3 which is small with respect to their frequencies f, g. Helmholtz calculates the roughness, i.e., the degree of dissonance of an interval which consists of two tones p, q as the sum of the beat intensities In,m , which are associated with the nth partial of p and the mth partial of q, where In,m is supposed to have a strong maximum for beat frequency ∆n,m = 33 Hz. 3 One
uses the trigonometric equation sin(x) + sin(y) = 2 sin( x+y ) cos( x−y ), see figure B.10. 2 2
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APPENDIX B. AUDITORY PHYSIOLOGY AND PSYCHOLOGY
Figure B.10: A beat between two superposed sinoidal waves is characterized by a periodic amplitude variation whose frequency is twice (!) the difference of the given frequencies. Therefore Helmholtz’ dissonance concept depends on the pitches and the involved sound colors. On the example of the violin, Helmholtz obtained good coincidences with Euler’s gradus function. This model is impressive since it explains the experience according to which consonance is a function of the instrument and the absolute pitch of the interval tones. Its experimental verification is somewhat problematic. It essentially depends on the measurability of the beat intensities in the cochlea. Since non-linear distortions on the sound’s way to the cochlea change spectra, one would have to perform ethically problematic invasive cochlear measurements. Moreover, individual statistical variations of the non-linear distortions would make the experiments even less robust. A fundamental doubt on the model’s validity results from binaural experiments in hearing by Heinrich Husmann [242], where the interval tones are presented on separate left and right headphone inputs. In this case, no beats can intervene in the cochlea. Nonetheless, the experiments also revealed consonance as “happy moments of within the general (interval) disaster”. The hypothesis that in the binaural experiments, Helmholtz beats must occur in the relays station of the corpus geniculatum mediale (see figure B.6) is speculative and demonstrates the limit of physiological models.
B.3.3
Psychometric Investigations by Plomp and Levelt
In the psychological reality of interval perception, the judgment of interval qualities looks quite different. In their investigation of “pleasantness” of intervals, Reiner Plomp and Wilhelm Levelt [418] have presented pairs of sinoidal tones and asked for pleasantness as a function of the given interval. The experiment was intendedly performed with musically untrained individuals in order to avoid judgments as a function of musical knowledge. Figure B.11 shows the resulting valuation curve. It is quite different from Euler’s function as shown in figure B.9. Using this curve, Plomp and Levelt have tried to infer a description of Helmholtz’ beat intensities, a procedure which was already recognized as being problematic in appendix B.3.2.
B.3.4
Counterpoint
There is however another—quite remarkable—point of view of the consonance-dissonance phenomenon, which has been poorly recognized within the psychoacoustic discussion, namely the prominent meaning of the concept pairing in the contrapuntal tradition, which was elaborated
percentage of test subjects judging the interval as being consonant
B.3. ASPECTS OF CONSONANCE AND DISSONANCE minor major third third
100
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fourth
fifth
80 60 40 20
400
450
frequency of lower tone
500
550
600
frequency of higher tone
Figure B.11: (From [462],with permission of Springer-Verlag) The psychometric investigation [418] of Plomp and Levelt yields a valuation which differs significantly from Euler’s Γ function and is based on sinoidal tones shown in figure B.9. in the High and Late and Middle Ages, and which was encoded in an exemplary way by Johann Joseph Fux’ Gradus ad Parnassum [174]. Carl Dahlhaus [99] has rightly pointed out that the textural function of the contrapuntal consonance concept is not yet fully understood. Interestingly, in the framework of the core theory of counterpoint, the interval of the fourth is dissonant, in contradiction to the other theories. It is inconceivable that the mathematical and physiological theories of counterpoint never included this perspective. A mathematical model of counterpoint is discussed in chapters 29 through 31.
B.3.5
Consonance and Dissonance: A Conceptual Field
Even on the mental level, the concept of consonance and dissonance is multiply explained, without necessarily leading to contradictions. In fact, Euler’s approach was a neutral mental (number-theoretic) one, whereas the contrapuntal approach is poietic. If we look at the status quo of the consonance-dissonance discussion and the fight for a valid final semantics, we are confronted with a disaster. The mathematically arbitrary ornaments which composers such as Klarenz Barlow [39] or neo-Pythagoreans such as Martin Vogel [549] add to Euler’s formula does not interest psychoacousticians such as Ernst Terhardt [525], who would extrapolate cochlear findings into the auditory cortex, and the latter approach cannot shed light onto the esthetics of music. What is common to all these positions adds to a concept of consonance and dissonance which is a conceptual field within the topography of music, a field with a quite ubiquitous presence. As a musical thought it results from a fundamental linear dynamic between polar extremals. It should be a main task of mathematical music theory to elaborate reliable and semantically reasonable, but esthetically undogmatic, models for such a way of thinking music.
Part XVI
Appendix: Mathematical Basics
1055
Appendix C
Sets, Relations, Monoids, Groups C.1
Sets
The language of sets describes mathematical facts in a classical way. An alternative foundation to sets is the language of categories, see appendix G. A set M is an object which is defined as a collection of uniquely determined objects which are also sets. These objects are called the elements or points of M . Two sets are equal iff they have the same elements. Whenever we say that M is a set, we mean that it is defined in a consistent way, i.e., without causing any contradiction. Existence of a mathematical object means that the object’s definition causes no contradiction in classical logic (A is identical to A, (exclusive) either A or non A, and exclusion of a third). Then, for any set m it is either an element of M or it is not. One writes m ∈ M for “m is an element of M ”, or also “m is a point of M ”. In order to define a set M by its elements m, m0 . . ., one also writes M = {m, m0 , . . .}. Observe that multiple enumeration does not change the set, for example {x, x, x, y} = {x, y}. A set N whose elements are all elements of M is called a subset of M , in signs: N ⊆ M , also sometimes N ⊂ M iff N ⊆ M and N 6= M . Two sets are equal iff they are mutually subsets of each other. The empty set ∅ is defined as having no elements. It is a subset of any set. A set is called finite if is empty or its elements can be indexed1 by a sequence2 0, 1, 2, 3, . . . n of natural numbers. Otherwise it is called infinite. 1 A mathematically correct definition of finiteness is this: A set if finite iff it is not in bijection with any proper subset. 2 Recall that in this book, we make the logical, though not very common usage of the ellipsis symbol “. . . ”: it means that one has started with a sequence of symbol combinations which follows an evident law, such as 1, 2, . . . n, or a1 +a2 +. . . an . The evidence is built upon the starting unit, such as “1,” or “a1 +” in our examples, and then the following unit, such as “2,” or “a2 +”, and then inducing the following units to be denoted, such as “3,” or “a3 +”, “4,” or “a4 +”, etc., until the sequence is terminated by the last symbol, such as “n” or “an ” in our examples. The ellipsis means that the building law is repeated, and as such, it is a meta-sign referring to the inductive offset. Therefore the more common notation 1, 2, . . . , n, or a1 + a2 + . . . + an is not correct. In the limit, for n = 3, it would imply a notation such as 1, 2, , 3 or a1 + a2 + +a3 , which is nonsense. Moreover, in complicated indexing situation, the common notation would be overloaded.
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C.1.1
APPENDIX C. SETS, RELATIONS, MONOIDS, GROUPS
Examples of Sets
Example 62 Z = {0, ±1, ±2, . . . ± n, . . .}, the set of integers; N = {n ∈ Z, n ≥ 0}, the set of natural numbers; the set Q = {p/q, p, q ∈ Z, q 6= 0} of rational numbers; the set R of decimal or real numbers, e.g., x = −741.76, π = 3.1415926 . . .. We have N ⊂ Z ⊂ Q ⊂ R, where integers p are identified with rational numbers of form p/1, whereas rational numbers are identified with periodic real numbers. Example 63 If M, N are sets, their difference M − N or the complement of N in S M is the set of points of M which are not in N . If V = (Mi )i is a family of sets, we denote by V the union of V , whose elements are precisely S the elements collected from any of S the Mi , for finite families V = M1 , . . . Mn , one also writes V = M ∪ . . . M . In particular, ∅ = ∅. A covering of a set 1 n S T X is a family V such that V = X. The intersection of a family V is the set V consisting exactly of those points which areTpoints in any of the family’s member Mi . For finite families T V = M1 , . . . Mn , one also writes V = M1 ∩ . . . Mn . In particular, ∅ = AllSet, the set whose elements are all existing sets. In less universal contexts, one only takes the intersection with regard to a large superset of the family’s members. Observe that there is no reason why AllSet should not exist, however, not any of its subcollections defined by predicates equally exists. For example, the subset of all sets not containing themselves as elements does not. A partition of a non-empty set X is a covering such that any two of its members are non-empty and disjoint, i.e., intersect in the empty set.
C.2
Relations
Definition 114 If x, y are two sets, the ordered pair (x, y) is defined to be the set {{x}} if x = y, (x, y) = {{x}, {x, y}} else.
(C.1)
Lemma 55 For any four sets a, b, c, d, we have (a, b) = (c, d) iff a = c, b = d. A triple (a, b, c) is a pair of form ((a, b), c). Clearly, (a, b, c) = (a0 , b0 , c0 ) iff a = a0 , b = b , c = c0 . 0
Definition 115 Given two sets A, B their Cartesian product is the set A × B = {(a, b)|a ∈ A, b ∈ B} consisting of all ordered pairs (a, b), with the first coordinate a an element in the first factor A, and the second coordinate b an element in the second factor B. A relation from A to B is a subset R ⊆ A × B. One writes aRb for (a, b) ∈ R, and, if the relation is clear, more simply a ∼ b. The inverse of a relation is the set R−1 = {(b, a)|aRb}. If A = B, one also speaks of a relation on A. Definition 116 A graph f : A → B from A to B is a triple (A, B, f ) with f a relation from A to B; it is called
C.2. RELATIONS
1059
(i) total iff every a ∈ A is the first factor of a pair in f ; (ii) functional iff (a, b), (a, b0 ) ∈ f implies b = b0 ; one writes f (a) instead of the uniquely determined b, or also f : a 7→ b. (iii) a function or map iff it is a total functional graph. The set A is the domain of f , whereas B is the function’s codomain. If f : A → B and g : B → C are two functions, their composition or concatenation function g ◦ f : A → C is defined by g ◦ f (a) = g(f (a)). The function f : A → A with f (a) = a for all a ∈ A is called the identity on A and is denoted by IdA . Exercise 83 Verify that g ◦ f is indeed a function. If h : C → D is a third function, we have (h ◦ g) ◦ f = h ◦ (g ◦ f ) and therefore we write h ◦ g ◦ f . Show that IdB ◦ f = f ◦ IdA = f . Definition 117 A function f is called (i) injective if f (a) = f (a0 ) always implies a = a0 ; (ii) surjective, iff for every b ∈ B, there is a ∈ A such that f (a) = b; (iii) bijective, iff f is injective and surjective. Lemma 56 For a function f : A → B, the following conditions are equivalent: (i) There is a function g : B → A such that g ◦ f = IdA and f ◦ g = IdB . (ii) The function f is bijective. The g in this lemma is uniquely determined by f and is called the inverse function of f , it is denoted by f −1 . Bijections f : A → A are also called permutations of A. If there is a bijection f : A → B between two sets A, B, we say that they have the same cardinality, and write card(A) = card(B). On AllSet, the cardinality relation is an equivalence relation, and one may define the cardinality card(A) of a set A as the equivalence class [A] under this relation. A finite set is one whose cardinality is that of a set of form {1, 2, 3, . . . n}, with a natural number 0 ≤ n, where we take the empty set for n = 0. Definition 118 A binary relation ≤ on a set S is said to be (i) reflexive iff x ≤ x for all x ∈ S; (ii) transitive iff x ≤ y and y ≤ z implies x ≤ z for all x, y, z ∈ S; (iii) symmetric iff x ≤ y implies y ≤ x for all x, y ∈ S; (iv) antisymmetric iff x ≤ y and x 6= y excludes y ≤ x for all x, y ∈ S; (v) total iff x ≤ y or y ≤ x for all x, y ∈ S.
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APPENDIX C. SETS, RELATIONS, MONOIDS, GROUPS
Definition 119 A binary relation ≤ on a set S is called an equivalence relation iff it is reflexive, transitive, and symmetric. In this case, the relation is usually denoted by “∼” instead of “≤”. Lemma 57 Let ∼ be an equivalence relation on S. Then the subsets [s] = {t|s ∼ t} are called equivalence classes of ∼. The set of equivalence classes is denoted by S/ ∼. It defines a partition of S, i.e., it covers S, and for any two elements s, t ∈ S, either [s] = [t] or [s] ∩ [t] = ∅. Definition 120 A binary relation ≤ on a set S is called a partial ordering iff it is reflexive, transitive, and antisymmetric. A partial ordering is called linear iff it is total. A linear ordering is called well-ordered iff every non-empty subset T ⊂ S contains a minimal element. Lemma 58 Let ≤ be a binary relation on a set S. Denoting x < y iff x ≤ y and x 6= y, the following two statements are equivalent: (i) The relation ≤ is a partial ordering. (ii) The relation ≤ is reflexive, the relation < is transitive, and for all x, y ∈ S, x < y excludes y < x. If these equivalent properties hold, we have x ≤ y iff x = y or else x < y. In particular, if we are given < with the properties (ii), and if we define x ≤ y by the preceding condition, then the latter relation is a partial ordering. Proof. (i) ⇒ (ii) Clearly ≤ is reflexive. If x < y and y < z, then x ≤ z. If we had x = z, then we were in the second statement of (ii), and it suffices to prove this one. But x < y and y < x implies x = y by the asymmetry of ≤, a contradiction. (ii) ⇒ (i) Transitivity: If x ≤ y and y ≤ z, and either x = y or y = z, we are done. Otherwise x < y and y < z, whence x < z, therefore x ≤ z. Asymmetry: If x ≤ y and y ≤ x, then x 6= y is excluded by (ii) whence the claim. The last statement is clear, QED. Example 64 Suppose that (I, 0, there is a pair n, m of integers such that 0 < n + m · L < δ.
D.5. SOME SPECIAL ISSUES
1081
Proof. We construct a sequence (n1 , m1 ), (n2 , m2 ), . . . (ns , ms ), . . . of pairs such that 0 < ns + ms · L < 1/2s . We may start by (n1 , m1 ) = (2, −1) since L ≈ 1.58. Suppose that we have found (ns , ms ) such that 0 < ns + ms · L < 1/2s . There is a maximal positive integer k such that k · (ns + ms · L) < 1. Then (k + 1) · (ns + ms · L) > 1 since L is not rational by corollary 24. For the same reason, either (k + 1) · (ns + ms · L) − 1 < 1/2 · 1/2s = 1/2s+1 or 1 − k · (ns + ms · L) < 1/2s+1 . Then either (ns+1 , ms+1 ) = ((k + 1) · ns − 1, (k + 1) · ms ) or (ns+1 , ms+1 ) = (1 − k · ns , −k · ms ) solves the problem, QED.
D.5
Some Special Issues
D.5.1
Integers, Rationals, and Real Numbers
Definition 133 The index function index : R → Z is defined by if 0 < x, 1 index(x) =
−1 0
if 0 > x,
(D.3)
if 0 = x
for x ∈ R. A real number x has unique representation x = bottom(x) + x+ with bottom(x) ∈ Z, 0 ≤ x+ < 1; we set top(x) = f loor(x) + 1. Definition 134 The rounding function round : R → Z is defined by f loor(x) if x ≤ 0.5, + round(x) = top(x) else for x ∈ R.
(D.4)
Appendix E
Modules, Linear, and Affine Transformations E.1
Modules and Linear Transformations
Definition 135 Let R be a ring, then a (left)1 R-module is a triple (R, M, µ : R × M → M ) where M is an additively written abelian group and µ is the scalar multiplication, usually written as µ(r, m) = r.m if µ is clear(R is also called the ring of scalars, and M the group of vectors), with the properties: 1. We have 1r .m = m for all m ∈ M . 2. For all r, s ∈ R and m, n ∈ M , we have (r + s).m = r.m + s.m, r.(m + n) = r.m + r.n, and r.(s.m) = (r · s).m. If (R, M, µ : R × M → M ), (R, N, ν : R × N → N ) are two R-modules, a group homomorphism f : M → N is called R-linear (or a module homomorphism if the rest is clear) iff it is “homogeneous”, i.e., f (r.m) = r.f (m), for all r ∈ R, m ∈ M , with the respective scalar multiplications. The set of R-linear homomorphisms from M to N is denoted by LinR (M, N ). It is an additive group under the pointwise addition (f + g)(m) = f (m) + g(m). The settheoretic composition g ◦ f of two module homomorphisms f : M → N, g : N → L is also a module homomorphism, and we have distributivity, i.e., (g1 + g2 ) ◦ f = g1 ◦ f + g2 ◦ f for gi : N → L, f : M → N and g ◦ (f1 + f2 ) = g ◦ f1 + g ◦ f2 for fi : M → N, g : N → L. By the distributivity of composition of module homomorphisms, the group EndR (M ) = LinR (M, M ) is a ring, the endomorphism ring of M , which contains the multiplicative automorphism group AutR (M ) = EndR (M )× of M . An R-linear homomorphism f : M → N has a group-theoretic kernel Ker(f ) and an image Im(f ) which are also submodules. For a submodule N ⊆ M , the quotient group M/N is also an R-module by the scalar multiplication r.(m + N ) = r.m + N . The group-theoretic 1 Right
modules are defined in complete analogy, the scalar multiplication being written m.r instead of r.m.
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APPENDIX E. MODULES, LINEAR, AND AFFINE TRANSFORMATIONS
results proposition 70 and the isomorphism theorems 39, 40 in appendix C, are valid literally if we replace the respective groups by modules (the normality of subgroups is automatic here).
E.1.1
Examples
Abelian groups G are canonically identified to Z-modules by z.g = g + g + . . . g, z times for z > 0, (−z).g = −(z.g), and 0.g = 0G . If M is an R-module, a submodule N ⊆ M is a subgroup that is stable under scalar multiplication, i.e., R.N = N . If S ⊂ M is a subset, thePsmallest submodule containing S is denoted by hSi and consists of all linear combinations i ri .si , ri ∈ R, si ∈ S. If there is a finite set S such that hSi = M , M is called finitely generated. Finitely generated Z-modules are completely classified, see C.3.4.2. If (Mi )i∈I is a family of submodules of M , we denote by S ΣI Mi the module h IMi i. It consists of all finite sums xi1 + . . . xik , xij ∈ Mij . Every ring R is a left R-module R R and a right R-module RR by the given multiplication. For commutative rings these structures coincide. A left, right ideal in R identifies to a submodule of R R, RR , respectively. If ϕ : S → R is a ring homomorphism and M is an R-module, we have an S-module structure M[ϕ] on M via s.m = ϕ(s).m, the module defined by restriction of scalars. If ϕ is clear, one also writes M[S] . For the Z-algebra structure of every ring R, M[Z] gives the underlying structure of an abelian group M back. In particular, an R-algebra S is an R module via (S S)[R] . A dilinear homomorphism from an S-module M to an R-module N is a pair (ϕ : S → R, f : M → N[ϕ] ) consisting of a scalar restriction ϕ and an S-linear homomorphism f . If (ϕ : S → R, f : M → N[ϕ] ), (ψ : T → S, g : L → M[ψ] ) are two dilinear homomorphisms, their composition is defined by (ϕ ◦ ψ : T → R, f ◦ g : L → N [ϕ ◦ ψ]). The set of dilinear homomorphisms from M to N is denoted by Dil(M, N ). If the scalar restriction is fixed by ϕ, we denote the corresponding set by Dilϕ (M, N ), and the special case DilIdR (M, N ) is just LinR (M, N ) as above. Q For any family (Mi )i∈I of R-modules, we have the product module I Mi . This is the product of the underlying groups, together with coordinatewise scalar multiplication r.(mi ) = (r.m direct sum module L i ) The submodule of those (mi ) with only finitely many mi 6= 0 is the Q M . For every index j, one has the canonical (linear) projections π : j I Li I Mi → Mj and π : M → M , via π ((m) )) → 7 m , as well as the canonical (linear) injections ιj : Mj → j i j j i j I L M with ι (m) having zero coordinates except for coordinate index j where the value is m. i j I Lemma 72 (Universal limit property of direct products of modules) For every familyQ(fi : X → Mi )i of linear homomorphisms there is exactly one linear homomorphism f : X → I Mi such that fj = πj ◦ f for all j ∈ I. (Universal colimit property of direct sums of modules) For every L family (fi : Mi → X) of linear homomorphisms there is exactly one linear homomorphism f : I Mi → X such that fj = ◦f ◦ ιj for all j ∈ I. A sum ΣI M Li of submodules Mi of a module M is called (inner) direct, iff the linear homomorphism I Mi → M which is induced by the inclusions Mi ⊆ M is an isomorphism. For two positive integers m, n denote by m×n = [1, m]×[1, n] theLset of all pairs (i, j), 1 ≤ i ≤ m, 1 ≤ j ≤ n. For a ring R, we have the direct sum Mm,n (R) = m×n R whose elements
E.2. MODULE CLASSIFICATION
1085
are written in the matrix notation
r1,1 (ri,j ) = . . . rm,1
... ri,j ...
r1,n ... rm,n
whose rows or columns are the submatrices with constant first or second index, respectively. If (ri,j ) ∈ Mm,n (R) and P (sj,k ) ∈ Mn,l (R), we have the matrix product (ri,j ) · (sj,k ) = (ti,k ) ∈ Mm,l (R) with ti,k = j ri,j ·sj,k . Whenever defined, the product is associative. It is also 0 0 distributive, i.e., ((ri,j ) + (ri,j )) · (sj,k ) = (ri,j ) · (sj,k ) + (ri,j ) · (sj,k ), and (ri,j ) · ((sj,k ) + (s0j,k )) = 0 (ri,j )·(sj,k )+(ri,j )·(sj,k ). For m = n, one has the identity matrix Em = (δij ) with the Kronecker delta δii = 1, δij = 0 for i 6= j. With this identity and the matrix addition and multiplication, Mm,m (R) is a ring. With the matrix multiplication as scalar multiplication, Mm,n (R) becomes a left Mm,m (R)-module and a right Mn,n (R)-module. If R is commutative, Mm,m (R) is an R-algebra via r 7→ r.Em , i.e., the scalar multiplication of the R-module Mm,m (R) coincides with the multiplication with R-elements from the algebra embedding.
E.2 E.2.1
Module Classification Dimension
L For any set C and ring R, we have the free R-module C = RC of rank card(C) which is the direct sum of C copies of R R (for C = ∅, we take the zero module 0R ). A free module M is one that is isomorphic to a free module RC . It is well known that the rank card(C) is then uniquely determined and called the dimension dim(M ) of the free module M . If R = F is a skew field, an F -module is called a vector space (over F ), and we have the main fact of linear algebra: Theorem 46 Every vector space M over the skew field F is free, and the dimensions are a complete system of invariants of isomorphism classes of vector spaces. The proof of this theorem is based on the concept of linear (in)dependence in a module. A family P (mi )i of elements mi ∈ M is called linearly independent iff any (finite) linear combination 0 = j=1,...k rj .mij implies rj = 0, for all j. Otherwise the family is called linearly dependent. A base of a module is a family of linearly independent elements which generates the module. The main theorem 46 is proved by the exchange theorem which states that any family (mi )i of linearly independent vectors can be inserted in a given basis by exchanging some of its elements with the (mi )i . Example 77 dim(R[Q] ) = card(R) = 2ℵ0 , and the sequence of b-logarithms (logb (p))p= prime is linearly independent by corollary 24 in appendix D.2. This means that for any finite increasing sequence p. = (p1 , p2 , . . . pk ) of primes and the corresponding sequence Hp. = (logb (p1 ), logb (p2 ), . . . logb (pk )),
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APPENDIX E. MODULES, LINEAR, AND AFFINE TRANSFORMATIONS
the scalar product map H : Qk → R[Q] P with H(x) = (Hp. , x) = i logb (pi )xi is a linear injection. The special case of the first three primes and Hprime = (logb (2), logb (3), logb (5)) was discussed in section A.2.3. Here we can prove the result needed for proposition 64: Lemma 73 For any positive real bound δ and every real number φ, there is x ∈ Z3 such that |H(x) − φ| < δ Proof. WLOG, we can work with logarithm basis b = 2. We know from lemma 71 in appendix D.4 that there is x0 = (n, m, 0) ∈ Z3 such that 0 < H(x0 ) < δ. Clearly, there is an integer multiple x = z.x0 which does the job, QED. ∼
∼
Corollary 25 (of theorem 46) If f : N → Rn and g : M → Rm are two free modules of finite ranks n, m, isomorphic to the free modules Rn , Rm via isomorphisms f, g, then the linear homomorphisms are described by matrices: If h : N → M is a linear homomorphism, there is a uniquely determined matrix H ∈ Mm,n (R) such that for x ∈ N , we have h(x) = g −1 (H · f (x)), where f (x) is written as a column matrix in Mn,1 (R) which canonically identifies to Rn . And conversely, each such matrix defines a linear homomorphism. In other words, there is an isomorphism ∼ LinR (N, M ) → Mm,n (R) (E.1) of additive groups. If R is a commutative ring, then an R-algebra S is also an R-module. We have an injective homomorphism of R-algebras Λ : S → EndR (S) Λ(s)(s0 ) = s · s0
(E.2)
which is called the left regular representation of S. If S is a free R-module of dimension n, then the isomorphism (E.1) induces the left regular representation in matrices: λ : S → Mn,n (R).
(E.3)
More generally, a linear representation of an R-algebra A is an algebra homomorphism f : A → EndR (M ) into the R-algebra of endomorphisms of an R-module M . Definition 136 The points of a k-element local composition K in Rn is in general position iff dim(R.K) = k − 1. Theorem 47 Let v1 , v2 , . . . vn be n vectors in a Q-module, and take two submodules G, H of dimensions g = dim(G), h = dim(H). Suppose that the module which is generated by the vectors vi , G, and H, has dimension n + g + h. Then we can have at most n + g + h points in general S position in the union i vi + G + H.
E.2. MODULE CLASSIFICATION
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Proof. WLOG, one may suppose v1 = 0 after a shift. Take bases x1 , . . . xg , y1 , . . . yh of G, H, respectively. Then, the vectors 0, v2 , . . . vn , x1 , . .S . xg , y1 , . . . yh are in general position. Conversely, if the vectors z1 , z2 , . . . zm in the union i vi + G + H are in general position, then dim(Q.{z1 , z2 , . . . zm }) = m−1. But hz1 , z2 , . . . zm i is contained in the module which is spanned by v2 , . . . vn , G, and H, whose dimension is n + g + h − 1, i.e., m − 1 ≤ n + g + h − 1, QED.
E.2.2
Endomorphisms on Dual Numbers
For a commutative ring R, we have the commutative R-algebra R[ε] of dual numbers (see example 76 in appendix D.1.1). As an R-module, it has dimension 2 and is isomorphic to R2 under the map a + ε.b 7→ (a, b). By the above corollary 25, the R-linear endomorphism ring of R[ε] identifies to the four-dimensional matrix ring M2,2 (R). In this situation, the left regular representation of R[ε] is the homomorphism of R-algebras λ : R[ε] → M2,2 (R) with λ(a + ε.b) =
! a 0 , b a
which represents the linear endomorphism of multiplication by a+ε.b. We have shown in section 29.6 that M2,2 (R) is generated by the four R-linear basis elements λ(1R ), λ(ε), α+ , α+ · λ(ε), where α+ is the sweeping orientation ! 1 1 α+ = , 0 0 and that the R-algebra M2,2 (R) identifies to the quotient 2 Rhλ(ε), α+ i/(λ(ε)2 , α+ − α+ , α+ · λ(ε) + λ(ε) · α+ − λ(1 + ε))
of the polynomial R-algebra in the two non-commuting variables λ(ε), α+ .
E.2.3
Semi-Simple Modules
A module M 6= 0R which has no submodules except 0R , M is called simple. A module M is called semi-simple iff it has the following equivalent properties: Lemma 74 Let M be an R-module. The following statements are equivalent: (i) Every submodule of M is a sum of simple submodules. (ii) M is the sum of simple submodules. (iii) M is the direct sum of simple submodules. (iv) Every submodule N of M is a direct summand (i.e., there is a submodule N 0 of M such that M = N ⊕ N 0 .
1088
APPENDIX E. MODULES, LINEAR, AND AFFINE TRANSFORMATIONS The following is immediate:
Lemma 75 A linear homomorphism between simple R-modules is either an isomorphism or zero. Hence the endomorphism ring EndR (M ) of a simple module M is a skew field. For example, the left Mm,m (F )-module Mm,n (F ) and the right Mn,n (F )-module Mm,n (F ) of m × n-matrices over a skew field F is semi-simple. For the left module, the n columns are the simple submodules, whereas for the right module, the m rows are the simple submodules. Moreover, the only left and right submodules of Mm,n (F ) are the zero module and Mm,n (F ). In particular, the only two-sided ideals in the ring Mm,m (F ) are 0 and Mm,m (F ), i.e., this ring is simple. Theorem 48 (Wedderburn) The semi-simple rings are the finite products matrix rings Mmi ,mi (Fi ) over skew fields Fi .
Q
i
Mmi ,mi (Fi ) of
If G is a finite group, and if K is a commutative field, we have defined the monoid algebra KhGi in D.1.1, it is called the group algebra in this case. Here are the semi-simple group algebras over commutative fields: Theorem 49 (Maschke) The group algebra KhGi is semi-simple iff char(K) - card(G).
E.2.4
Jacobson Radical and Socle
Definition 137 The intersection of all maximal submodules of an R-module M 6= 0 is called the Jacobson radical of M , it is denoted by Rad(M ). Sorite 12 let M, N be R-modules. Then: (i) For f ∈ LinR (M, N ), we have f (Rad(M )) ⊆ Rad(N ). (ii) We have Rad(M ⊕ N ) = Rad(M ) ⊕ Rad(N ). (iii) We have Rad(M/Rad(M )) = 0. (iv) We have Rad(R R).M ⊆ Rad(M ). (v) If M is semi-simple, then Rad(M ) = 0. (vi) If a submodule N of M has M/N semi-simple, then Rad(M ) ⊆ N . For a ring R, one may look at its left radical Rad(R R), or at its right radical Rad(RR ). Fortunately, there is no difference in that: Proposition 79 The left and right radicals of a ring R coincide, Rad(R R) = Rad(RR ), and this (two-sided) ideal Rad(R) is the maximal ideal I which annihilates every semi-simple module M , i.e., I.M = 0.
E.2. MODULE CLASSIFICATION
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For a ring R, we set Jr = {r ∈ R|1R 6∈ r · R}, Jl = {r ∈ R|1R 6∈ R · r}. Proposition 80 For a ring R the following conditions are equivalent: (i) The quotient ring R/Rad(R) is a skew field. (ii) We have Jr = Rad(R). (iii) We have Jl = Rad(R). (iv) The set Jr is additively closed. (v) The set Jl is additively closed. Definition 138 A ring with the equivalent properties of proposition 80 is called local. In particular, a commutative ring R is local iff it has a unique maximal ideal. The length l(M ) of a module M is the maximum length l of finite chains 0 & N1 & N2 & . . . Nl = M of submodules (if that maximum is ∞, we set l(M ) = ∞). A module is called indecomposable iff it is not the direct sum of two proper submodules. Lemma 76 If M is an R-module with local endomorphism ring, then M is indecomposable. Conversely, let M be an indecomposable module of finite length l. Then EndR (M ) is a local ring, and the radical S = Rad(EndR (M )) is nilpotent, namely S l = 0. Proposition 81 Let X ⊂ M, Y ⊂ N be two non-zero submodules of indecomposable R-modules M, N of finite lengths. If F : M → N, g : N → M are linear maps which induce mutually inverse ∼ ∼ isomorphisms f |X : X → Y, g|Y = (f |X)−1 : Y → X, then f, g are isomorphisms. Proof. Since f · f restricts to the identity of X, so does any of its positive powers. So non of them can be the zero endomorphisms of M . But from lemma 76 we know that End(M ) is local and that a non-nilpotent endomorphism must be invertible. A symmetric argument yields inversibility of f · g and therefore of both, f and g, QED. Lemma 77 (Fitting’s lemma) Let f : M → M be a linear endomorphism of a module M of finite length. Then there is a positive power f n and a direct decomposition M = N ⊕ Ker(F n ) such that f n is an automorphism on N . Definition 139 The socle Soc(M ) of a module M is the sum of all its simple submodules. Theorem 50 For a module M of finite length, the following three statements are equivalent: (i) M is semi-simple. (ii) Rad(M ) = 0. (iii) Soc(M ) = M .
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APPENDIX E. MODULES, LINEAR, AND AFFINE TRANSFORMATIONS
E.2.5
Theorem of Krull–Remak–Schmidt
Theorem 51 (Krull–Remak–Schmidt) Let M1 , . . . Mk , M10 , . . . Ml0 be modules with local endomorphism rings (in L L particular these modules are all non-zero). Suppose that the direct sums i=1,...k Mi and j=1,...l Mj are isomorphism of R-modules. Then k = l, and there is a per∼ 0 mutation σ of the indices such that we have isomorphisms Mi → Mσ(i) for all i = 1 . . . k. Corollary 26 A module of finite length is a direct sum of indecomposable submodules in a unique way up to permutation and isomorphisms of the summands. This follows from theorem 51 in view of lemma 76, QED.
E.3
Categories of Modules and Affine Transformations
See appendix G for a reference to category theory. For an additive group M and an element m ∈ M , the translation by m is the set map em : M → M : x 7→ em (x) = m + x. The exponential notation is chosen because e? : M → SM is an injective group homomorphism. We denote by eM the group of translations on M , a group which is isomorphic to M . Definition 140 For two rings R, S, an R-module M and an S-module N , a diaffine homomorphism f is a map of form en · f0 , where en is a translation on N and f0 ∈ Dil(M, N ). The set of diaffine homomorphisms f : M → N is denoted by M @N . If we fix the underlying scalar restriction ϕ : R → S and only take f0 ∈ Dilϕ (M, N ), the corresponding set is denoted by M @ϕ N . In particular, if ϕ = IdR , we write M @R N for the set of (R-)affine homomorphisms. Sorite 13 If R, S, T are rings and M, N, L are modules over these rings, respectively, we have the following facts: (i) If f = en · f0 ∈ M @N , then n = f (0), and f0 = e−n · f . So the translation part en and the dilinear part f0 are uniquely determined. (ii) If f = en · f0 ∈ M @N, g = el · g0 ∈ N @L, then the set-theoretic composition g · f ∈ M @L and g · f = el+g(n) · g0 · f0 .
(E.4)
(iii) The diaffine f = en · f0 ∈ M @N is an isomorphism iff its dilinear part f0 is so, and then −1 the inverse is f −1 = e−f0 (n) · f0−1 . (iv) If f, g ∈ M @ϕ N , then so is the pointwise difference f − g, hence M @ϕ N is an additive group. If S is commutative, M @ϕ N is an S-module and also an R-module under ϕ. −→ (v) For M = N, ϕ = IdR , the general affine group GL(M ), i.e., the group of affine automorphisms is isomorphic to the semidirect product M oφ Gl(M ) of M with the general linear group GL(M ) of linear automorphisms of M under the group action φ = IdGL(M ) .
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The category of modules and diaffine homomorphisms is denoted by Mod, whereas the subcategory of (left) R-modules and R-affine homomorphisms is denoted by ModR . If f : S → R is a ring homomorphism, we have the scalar extension functor S⊗R ? : ModR → ModS : M 7→ S ⊗R M
(E.5)
which acts by scalar extension of the linear parts of morphisms and by the canonical map M → S⊗R : x 7→ 1 ⊗ x on the translation part. See [63, II.5.1] for details. In Mod, we have to add an additional object ∅R for each ring R. This is the empty set plus the unique possible scalar multiplication. This is not a module in the usual sense since it is not even a group! But there are important category-theoretic reasons to introduce these objects. Observe that for an S-module N , ∅R @N is in bijection with the set of ring homomorphisms Hom(R, S), whereas N @∅R is empty if N is not empty. By Mod@ , we denote the category of set-valued presheaves on Mod, i.e., the contravariant set-valued functors F : Mod → Sets. In particular, the Yoneda embedding Mod → Mod@ yields the representable presheaf @M for a module M , with @M (X) = X@M for X ∈ 0 Mod. This is one reason why we also write X@F for the evaluation F (X), even if F is not representable. By M @ we denote the covariant functor with M @(X) = M @X. In the context of presheaves, we often call a module X that is an argument of such presheaves an address; the reasons for this wording are made explicit in the musicological chapter 6 on forms and denotators.
E.3.1
Direct Sums
Proposition 82 Let A be an R-module, and n a natural number. Then there is a canonical ∼ isomorphism @A⊕n → (@A)n , i.e., A⊕n represents the n-fold product functor. Proof. Let X be any module. Then every affine homomorphism f = et · f0 : X → A⊕n projects to the n factors fi = pi · f via the respective projections pi : A⊕n → A. Also, the dilinear part f0 projects to the n dilinear factors f0,i : X → A. Let ti be the ith component of t. Then we ∼ have fi = eti · f0,i . This yields the desired bijection X@A⊕n → (X@A)n , and this is functorial in X. QED.
E.3.2
Affine Forms and Tensors
In this section we suppose that all modules have a commutative coefficient ring R, i.e., we work in the category ModR . Tensor products2 are automatically taken over R. By X ? we denote the R-linear dual LinR (X; R) of the R-module X. For the R-module AF of affine forms on an R-module (address) A and an R-module M, we have a canonical linear injection M AF ⊗ M . In fact, there is an R-linear isomorphism ∼ ∼ AF → R ⊕ A? : er · x 7→ r + x, and we deduce an R-linear isomorphism AF ⊗ M → M ⊕ A? ⊗ M , whence the above injection; it maps m ∈ M to e1R .0 ⊗ m. With the above notation, fix an R-module A (an address). We have the subfunctor A@R : ModR → Sets : M 7→ A@R M 2 See
[63] for tensor products.
(E.6)
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APPENDIX E. MODULES, LINEAR, AND AFFINE TRANSFORMATIONS
of A@, and induced on the subcategory ModR of Mod. We further have the functor AF ⊗ : ModR → Sets : M 7→ AF ⊗ M.
(E.7)
This functor acts as follows on affine morphisms F = en · F0 : M → N . Identify n with the canonically associated element of AF ⊗ N . Then, we define AF ⊗ F = en · AF ⊗ F0 ,
(E.8)
in other words, (AF ⊗ F )0 = AF ⊗ F0 . For the composition F
G
et+G0 (n) · G0 F0 : M −→ N −→ T of affine morphisms F = en · F0 and G = et · G0 this implies AF ⊗ GF =et+G0 (n) · AF ⊗ G0 F0 =et · eG0 (n) · AF ⊗ G0 · AF ⊗ F0 =et · AF ⊗ G0 · en · AF ⊗ F0 =AF ⊗ G · AF ⊗ F, whence the claimed functoriality. Lemma 78 With the above notation, there is a natural transformation θ : AF ⊗ → A@R .
(E.9)
If M is an R-module, it is defined by its action on pure tensors x ⊗ m ∈ AF ⊗ M by θ(x ⊗ m) : A → M : a 7→ x(a)m.
(E.10)
The natural transformation θ is an isomorphism if A is a finitely generated projective module. Proof. The formula (E.10) is an extension of a classical formula in the linear case, see [63, II.74]. ∼ ∼ In fact, write AF ⊗ M → M ⊕ A? ⊗ M , and then A@M → M ⊕ LinR (A, M ). Then the classical formula θ0 : A? ⊗ M → LinR (A, M ) : x ⊗ m 7→ θ0 (x ⊗ m) with θ0 (x ⊗ m)(a) = x(a)m extends to the linear map θ0 (m + λ) = m + θ0 (λ) which for special pure tensor arguments x ⊗ m = er · x0 ⊗ m yields θ0 (rm + x0 ⊗ m) = rm + θ0 (x0 ⊗ m), corresponding to erm · θ0 (x0 ⊗ m), and the latter evaluates to (erm · θ0 (x0 ⊗ m))(a) = rm + x0 (a)m = x(a)m
E.3. CATEGORIES OF MODULES AND AFFINE TRANSFORMATIONS
1093
which means θ0 = θ. Let us then prove the naturality of θ. Let F = en · F0 : M → N be an affine morphism. We have to show that the diagram θ
AF ⊗ M −−−−→ A@M AF ⊗F y yA@F
(E.11)
θ
AF ⊗ N −−−−→ A@N is commutative. It suffices to verify it for pure tensors x ⊗ m ∈ AF ⊗ M . Take a ∈ A. Then A@F (θ(x ⊗ m))(a) =F (θ(x ⊗ m)(a)) =F (x(a)m) =n + F0 (x(a)m) =n + x(a)F0 (m) =θ(n) + θ(x ⊗ F0 (m))(a) =θ(n + x ⊗ F0 (m))(a) =θ((AF ⊗ F )(x ⊗ m))(a) and we are done with diagram (E.11). If A is finitely generated projective, the classical linear map θ0 is iso, and hence so is θ. QED. Observe that the special case A = 0 of zero address is included in the lemma and that in this case, θ identifies to the identity transformation on the forgetful functor.
E.3.3
Biaffine Maps
In this section we again suppose that all modules have a commutative coefficient ring R, i.e., we work in the category ModR . In classical module theory, the tensor product is a universal construction relating to bilinear maps. The extension to biaffine maps runs as follows. Definition 141 Let U, V, W be modules in ModR . A map f :U ×V →W is called biaffine if it is affine in each variable, i.e., if fu : V → W : v 7→ fu (v) = f (u, v) and f v : U → W : u 7→ f v (u) = f (u, v) are all affine, i.e., fu ∈ V @R W and f v ∈ U @R W . The set of all biaffine maps f is denoted by A2 (U, V ; W ). Lemma 79 For R-modules U, V, W in Mod, there is a canonical bijection ∼
A2 (U, V ; W ) → U @R (V @R W ).
(E.12)
If these sets are given their canonical structure of R-modules, bijection E.12 is an isomorphism of R-modules.
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APPENDIX E. MODULES, LINEAR, AND AFFINE TRANSFORMATIONS
Proof. Let f : U × V → W be a biaffine map. Then the associated map fu , u ∈ U by definition stays in V @R W . So we have a map f? : U → V @R W . Let us show that this map is affine. Set λ(v) = f (0, v), this affine map is the constant part of our candidate f? , i.e., we claim that fu − λ is linear in u. But fu (v) − λ(v) = f v (u) − λ(v) = f v (u) − f v (0) is linear in u and we are done. Conversely, each g ∈ U @R (V @R W ) defines g˜(u, v) = g(u)(v) with g˜ ∈ A2 (U, V ; W ), and this clearly is an inverse to the map f 7→ f? . That we have a module isomorphism is clear. QED. Proposition 83 Let U, V, W be R-modules in Mod, and define the affine tensor product U V = U ⊗ V ⊕ U ⊕ V . Then we have a canonical bijection ∼
: A2 (U, V ; W ) → (U V )@R W,
(E.13)
i.e., U V is a universal object in the affine category ModR like the tensor product is for R-linear maps. If f ∈ A2 (U, V ; W ), then its image f applies a typical element u ⊗ v + r + s to f (u ⊗ v + r + s) = f (u ⊗ v) + f 0 (r) + f0 (s) − f (0, 0) (E.14) where f is the linear map associated with the bilinear map f (u, v) = f (u, v) − f0 (v) − f 0 (u) + f (0, 0).
(E.15)
The universal map i : U × V → U V is defined by i(u, v) = u ⊗ v + u + v. Proof. The proposition follows directly from lemma 79, the definition of the affine tensor product and the universal property of the linear tensor product. We then have f (i(u, v)) = f (u ⊗ v + u + v) = f (u ⊗ v) + f 0 (u) + f0 (v) − f (0, 0) = f (u, v) + f 0 (u) + f0 (v) − f (0, 0) = f (u, v) − f0 (v) − f 0 (u) + f (0, 0) + f 0 (u) + f0 (v) − f (0, 0) = f (u, v). QED. Definition 142 For modules U, V, W, X in ModR and affine maps f : U → W, g : V → X, the affine tensor product map f g :U V →W X
(E.16)
is defined as the canonical affine map h according to proposition 83 which is associated with the biaffine map h : U × V → W X : (u, v) 7→ i(f (u), g(v)). Sorite 14 For modules U, V, W, X in ModR and with the notation of section E.3.8, we have: ∼
(i) U V → V U . ∼
∼
(ii) U 0R → 0R U → U . ∼
(iii) U (V W ) → (U V ) W , i.e., we can identify these products and write U V W . ∼
(iv) (U q V ) W → U W q V W .
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1095
For a module M in ModR , the functor @R M : ModR → Sets : B 7→ B@R M contains redundant structure in B since there are elements in B which are annihilated by all linear maps into M . We want to reduce B to a module where this annihilator set is the zero submodule. T Definition 143 With the above notation, set An(B, M ) = k∈B@M ) Ker(k0 ), denote B/M = B/An(B, M ) and write /M : B → B/M for the canonical projection. The module B/M is called the M -reduction of B. The following lemma is clear: Lemma 80 The assignment ?/M : B 7→ B/M defines a functor on ModR . The projection /M : B → B/M and the uniquely defined commutative diagrams /M
C −−−−→ C/M f fy y /M
(E.17)
/M
B −−−−→ B/M which are associated with affine homomorphisms f : C → B define a natural transformation on IdModR →?/M . ∼
red Proposition 84 Let @red R M = @R M ·?/M Then we have a natural isomorphism @R M → @R M .
Proof. In fact, the natural transformation /M : B → B/M induces an isomorphism (of R∼ modules) B@red R M → B@R M . QED. red
˜ ˜ Corollary 27 With the above notation, the functors B @M and B @ canonically isomorphic.
˜ M = B/M @M are
Proof. If A ∈ 0 ModR , we have red
˜ A@B @
˜ M = A@B/M @M = A B/M @R M ∼
→ A@R (B/M @R M ) ∼
→ A@R (B@R M ) ∼
→ A B@R M ˜ = A@B @M. Let M, A be modules in ModR . Then we have Lemma 81 There is an isomorphism of R-modules ∼
LinR (M, A@R R) → A@R M ? which is functorial in both, A and M .
(E.18)
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APPENDIX E. MODULES, LINEAR, AND AFFINE TRANSFORMATIONS
Proof. We have these functorial isomorphisms: ∼
LinR (M, A@R R) → LinR (M, A? ⊕ R) ∼
→ LinR (M, A? ) ⊕ M ? ∼
→ (M ⊗ A)? ⊕ M ? ∼
→ (A ⊗ M )? ⊕ M ? ∼
→ LinR (A, M ? ) ⊕ M ? ∼
→ A@R M ? . Proposition 85 Let M be as above, and A = Rn , 0 ≤ n, then we have canonical R-linear maps
u : A@R M ? → (A@R M )? , d : A@R M → (A@R M ? )? ,
(E.19) (E.20)
which are isomorphisms for M finitely generated and projective. Proof. As to the first map, a linear map v : LinR (A, M ? ) → LinR (A, M )? is defined as follows: For g : A → M ? and f : A → M , we have the composition f ? · g : A → A? , which is a bilinear form on A, and we may set e(g)(f ) = tr(f ? ·g), a linear function of g, calculated in the canonical bases of A and A? . This map is canonically extended by the identity on M and we are done for u. For the second map, we have the canonical bidual linear map l : A@R M → A@R M ?? , and we may apply the first map to the bidual of M . The statement concerning finitely generated projective modules is standard. QED
E.3.4
Symmetries of the Affine Plane
We consider symmetries, i.e., affine transformations D = et · H on R2 . From the geometric point of view, the set R2 @R2 of these maps is described as follows. Fix a (zero-addressed) local composition ∆ = {u, v, w} in the real plane R2 , with three points in general position (see appendix E.2.1). Then we know from section 15.2.1 that the map B(∆) : R2 @R2 → (R2 )3 : D 7→ (D(u), D(v), D(w))
(E.21)
is a bijection. We use this bijection to describe some special transformations: Shearings. Let G be a straight line in R2 (not necessarily through the origin), and let u, v, w be in general position such that u, v lie on G, whereas w does not. A shearing S relating to G is a symmetry which leaves both, u, v, fixed and transforms w into w + r.(v − u), r ∈ R. Then G remains fixed identically, and w is shifted in parallel motion with respect to G. The nth power S n of S is the shearing which fixes G and transforms w into w + nr.(v − u).
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Dilatations. For a point u ∈ R2 , and two scalars δ, σ ∈ R, a dilatation D by factors δ, σ and centered in u is defined by the prescription that D(u) = u, and that for two other points v, w such that u, v, w are in general position, we have D(v) = u + δ.(v − u), D(w) = u + σ.(w − u). A dilatation with δ = σ = −1 is called point reflection with center u, it corresponds to a rotation by 180◦ around u. Glide Reflections. Let G be a straight line, and take u, v, w as in the above paragraph E.3.4 about shearing. A glide reflection P is a symmetry, such that P (u) lies on G, and P (v) − P (u) = v − u, P (w) − P (u) = u − w. Therefore, P (v) lies also on G, and P (w) lies on the “opposite side” of w on the line through w, u, i.e., P is a translation by P (u) − u, followed by a ‘skew’ reflection in G in the direction of the line through w and u. Especially for the diagonal G = R.(1, 1) and u = P (u) = 0, w = (−1, 1), we obtain the exchange of coordinate axes: the parameter exchange.
E.3.5
Symmetries on Z2
Theorem 52 Every symmetry f ∈ Z2 @Z2 is the product of some of the following symmetries: 1. a translation T = e(0,1) , 2. a shearing S which leaves the first axis Z.(1, 0) fixed and transforms (0, 1) to (1, 1), 3. the parameter exchange P , 4. the reflection K at the second axis, 5. the dilatations Dm , 0 ≤ m in the direction of the first axis by factor m. −→ The general affine group GL(Z2 ) is generated by T, S, P, K. Proof. The statement concerning the general affine group is immediate from the first part of the theorem. To show the latter, we say that a symmetry X is “good” if X can be written as a product of symmetries of the required shape. Here are the matrices for the generators: ! ! ! ! 1 1 −1 0 0 1 m 0 S= ,K = ,P = , Dm = . 0 1 0 1 1 0 0 1 ! a b We first show that all 2 × 2-matrices, i.e., all linear maps X = are products of c d symmetries of type S, K, Dm , P . Observe that P 2 = K 2 = E2 . 1. If P · X is good, then so is X = P 2 · X = P · (P · X). The same is valid for X · P . Here, P · X is the exchange of rows in X, whereas X · P is the exchange of columns in X. ! ! 1 −1 1 n 2. We have S −1 = K · S · K = . Therefore the powers S n = are good for 0 1 0 1 ! ! integer n. Further, P · S ±1 · P =
1 0 ±1 1
and therefore (P · S · P )n =
1 n
0 1
is good.
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APPENDIX E. MODULES, LINEAR, AND AFFINE TRANSFORMATIONS
3. For m ≥ 0, P · Dm · P =
! 1 0 and P · K · Dm · P = 0 m
! 1 0 are good. 0 −m
4. If a coefficient of X vanishes, one can enforce c = 0 by a row or column exchange. Because of ! ! ! ! a b 1 0 1 b a 0 = · · 0 d 0 d 0 1 0 1 and the preceding results, such an X is also good. 5. If no coefficient of X vanishes, one can apply ±E2 and exchange of columns that satisfy c ≥ d > 0. Applying the Euclidean algorithm (section D.3), we can write c = nd + r, 0 ≤ r < d. This yields ! ! ! ! a b 1 0 a − nb b a0 b · = = . c d −n 1 c − nb d r d If r = 0,!we are in case 4 above. Else, we have d > r > 0. Via column exchange we obtain b a0 , and we may set c0 = d, d0 = r. But d0 < d, so the algorithm leads to case 4 after d r a finite number of steps. This settles the linear maps. For translations, observe the following identities: e(1,0) = P · T · P, e(−1,0) = K · e(1,0) · K, e(0,−1) = P · e(−1,0) · P. Therefore all transpositions are good since we have natural numbers x, y such that e(±x,±y) = (e(±1,0) )x · (e(0,±1) )y . This settles all the cases, QED.
E.3.6
Symmetries on Zn
For integers n ≥ 2 and 1 ≤ i < j ≤ n, we have the diagonal embedding ∆i,j : GL(Z2 ) GL(Zn ) defined by ! a b ∆i,j ( ) = (xu,v ) with c d xi,i = a, xi,j = b, xj,i = c, xj,j = d, xu,u = 1 for u 6= i, j, and xu,v = 0 else. Theorem 53 For an integer n ≥ 2 the group GL(Zn ) is generated by the diagonal embedded groups ∆1,j GL(Z2 ), 1 ≤ j ≤ n. The proof goes by induction on n and uses the Euclidean algorithm, we leave it as an exercise.
E.3. CATEGORIES OF MODULES AND AFFINE TRANSFORMATIONS
E.3.7
1099
Complements on the Module of a Local Composition
Lemma 82 Let A be an address module over the commutative coefficient ring R and (K, A@M ) a commutative local composition. Then: (i) R.K ⊂ hKi. (ii) R.K = hKi iff K is embedded. (iii) If f : K → L is a morphism of embedded commutative local compositions (K, A@R M ), (L, A@R N ) at the same address A. Then any underlying symmetry F : M → N restricts to R.K and R.L, i.e., F (R.K) ⊂ R.L, and this restriction is uniquely determined by f . We denote this affine map by R@f : R.K → R.L. Proof. The first statement is clear since R.K = hx − x0 | x ∈ Ki. If R.K = hKi, then obviously K ⊂ hKi = R.K, and K is embedded. Conversely, if K ⊂ R.K, then also hKi ⊂ R.K and equality follows from (i). As to (iii), observe that we have a linear application R.f : R.K → R.L which is induced by the linear part F0 of F , and which is only a function of f , by lemma 6 of chapter 8. Further, if F = en · F0 , and if k ∈ K, we have n = F (k) − F0 (k) = f (k) − R.f (k) since K ⊂ R.K; in other words, n = nf is only a function of f , and not of the underlying F . Therefore, since both, f (k) and R.f (k), are elements of R.L, nf ∈ R.L. This means that for x ∈ R.K, F (x) = nf + Fo (x) = nf + R.f (x) = R@f (x) ∈ R.L, QED.
E.3.8
Fiber Products and Fiber Sums in Mod
Theorem 54 The category Mod of modules and diaffine transformations has arbitrary fiber products. Proof. We are given a fiber product diagram K→M ←L f
g
(E.22)
of modules over the fiber product diagram A→C←B u
v
(E.23)
of corresponding coefficient rings. If any of these modules K, L or M is empty, or if intersection Im(f ) ∩ Im(g) is empty, then the empty module over the fiber product A ×C B of coefficient rings does the job. So we may suppose that neither of these four spaces is empty. Consider the dilinear parts f0 and g0 of f and g. Then we have the dilinear homomorphism d : K ⊕ L → M : (k, l) 7→ f0 (k) − g0 (l) with regard to the fiber product ring homomorphism A ×C B → C. Take any couple (k, l) ∈ K ⊕ L with f (k) = g(l). Then the set-theoretic
1100
APPENDIX E. MODULES, LINEAR, AND AFFINE TRANSFORMATIONS
fiber product ∆ ⊂ K ⊕ L equals Ker(d) + (k, l). This implies that the diaffine embedding ∼ e(k,l) : Ker(d) → ∆ ⊂ K ⊕ L, followed by the projections to K and L defines a fiber product p2 ·e(k,l)
Ker(d) −−−−−→ p1 ·e(k,l) y
L g y
(E.24)
f
−−−−→ M
K
of modules and diaffine transformations. QED. Theorem 55 The category Mod has fiber sums for all pushout diagrams of modules over a fixed coefficient ring, i.e., where the scalar restrictions are the identity. Proof. We are given a fiber sum diagram K←M →L f
g
(E.25)
of modules over coefficient ring A. If M is empty we have to construct the sum K q L, and we may suppose that both summands are nonvoid, the other cases being trivial. Consider the direct sum S = K ⊕ L ⊕ A and the affine injections i1 : K S : k 7→ (k, 0, 1) and i2 : L S : l 7→ (0, l, 0). Suppose we are given two diaffine transformations f : K → X, g : L → X, with factorizations f = ex · f0 and g = ey · g0 and scalar restriction s : A → B. Define a dilinear map h0 : S → X : (k, l, t) 7→ f0 (k) + g0 (l) + s(t)(x − y). Then we have a diaffine transformation h = ey · h0 which does the job, in fact, h · i1 = f , and h · i2 = g. Since i2 is linear, we have h(0) = h(i2 (0)) = g(0) = y. Hence the affine part of h is uniquely determined. If we had two candidates h and h∗ for universal arrows, they would only differ in their dilinear parts h0 and h∗0 . But then, their difference d = h0 − h∗0 would vanish on all elements of shape (0, l, 0), l ∈ L and on all (k, 0, 1), k ∈ K. The latter implies that d(0, 0, 1) = 0, and by dilinearity of d, d(k, 0, 0) = d((k, 0, 1) − (0, 0, 1)) = 0, whence the uniqueness of h. On the other hand, if M is non-empty, so are K and L. We then have two arrows u = i1 · f, v = i2 · g : M ⇒ K q L from the diagram f
M −−−−→ gy i
K i y1
L −−−2−→ K q L
(E.26)
E.4. COMPLEMENTS OF COMMUTATIVE ALGEBRA
1101
and we are done if we can show that there is a coequalizer3 of the couple u and v. If we have the factorizations u = et ·u0 and v = es ·v0 , take the quotient module E = K qL/A(t−s)+Im(u0 − v0 ). Clearly, the projection p : K q L → E equalizes u and v. If r : K q L → X is any diaffine transformation with scalar restriction s : A → B and equalizing the couple u and v, then r has a unique factorization through E. In fact, we may suppose without loss of generality that r is dilinear. In this case, r has the required factorization since it annihilates A(t − s) + Im(u0 − v0 ). With this construction we define K qM L = E and obtain a commutative diagram f
M −−−−→ gy
K p·u y
(E.27)
p·v
L −−−−→ K qM L which is the required pushout diagram in Mod. Observe that this proof technique—build the sum and then the coequalizer—is a special case of the fact that existence of fiber sums is equivalent to existence of sums and coequalizers provided that we have an initial object, see appendix G.2.1. QED. Proposition 86 A dilinear morphism f : M → N over scalar restriction g : A → B is mono iff f is diinjective, i.e., iff f, g are both injective. Proof. If both, f, g are injective, then clearly the dilinear morphism is mono. If f is not injective, there are two different affine morphisms ki : 0Z → M, i = 1, 2, which are equalized by f . If the scalar restriction g is not injective, there are two different ring homomorphisms r1 , r2 : Z[X] → A on the polynomial ring Z[X] with r1 (X) ∈ ker(g), i = 1, 2,, and the zero morphism = 0Z[X] → M for these two scalar restrictions does the job.
E.4
Complements of Commutative Algebra
In this section, all coefficient rings are commutative.
E.4.1
Localization
See also [64, II] for concepts and facts described in this section. Let S be a multiplicative subset of a ring A, i.e., st ∈ S for all s, t ∈ S, and 1 ∈ S. The localization S −1 A is the set of equivalence classes of A × S modulo the relation (a, s) ∼ (a0 , s0 ) iff there is t ∈ S such that t(as0 − a0 s) = 0. The equivalence class of (a, s) is denoted by the fraction a/s or as . It is a ring by the well-defined addition a/s + a0 /s0 = (as0 + a0 s)/ss0 and multiplication a/s.a0 /s0 = aa0 /ss0 . The canonical map iS : A → S −1 A : a 7→ a/1 is a ring homomorphism with the universal property that for any ring homomorphism f : A → B such that f (S) ⊂ B × , there is a unique ring homomorphism j : S −1 A → B such that f = j ◦ iS . The ring S −1 A is called the localization of A in S. Classical example: A is a domain (no zero divisors), S = A − {0}, whence S −1 A is the classical field f r(A) of fractions over A. 3 See
appendix G.2.1.
1102
APPENDIX E. MODULES, LINEAR, AND AFFINE TRANSFORMATIONS
If M is an A-module, the localization S −1 M is the set S −1 M of equivalence classes of M ×S for the equivalence relation (m, s) ∼ (m0 , s0 ) iff there is t ∈ S such that t(ms0 − m0 s) = 0. The addition m/s + m0 /s0 = (ms0 + m0 s)/ss0 and the scalar multiplication r/s.m/t = rm/st makes S −1 M a S −1 A-module. One has the canonical dilinear homomorphism iM : M → S −1 M : m 7→ m/1 with respect to the homomorphism iS . It has this universal property: For every homomorphism f : M → N of A-modules, such that every dilatation s? : N → N : n 7→ s.n is bijective, there is a unique homomorphism of A-modules j : S −1 M → N such that f = j ◦iM . It is easily seen that the tensor product S −1 A ⊗A M , together with the canonical homomorphism of A-modules M → S −1 A ⊗A M : m 7→ 1 ⊗ m is isomorphic to the localization S −1 M . For the multiplicative set Ss = {1, s, s2 , s3 , . . .}, s ∈ A, one writes As , Ms instead of −1 Ss A, S −1 M . For a prime ideal q ⊆ A, the complement S = A−q is multiplicative by definition. And one then writes Aq , Mq instead of S −1 A, S −1 M . By the universal property of localization, if f : A → B is a ring homomorphism and if S ⊆ A, T ⊆ B are multiplicative sets such that f (S) ⊆ T , then there is a canonical ring homomorphism fS,T : S −1 A → T −1 B which extends f . If g : M → N is a dilinear homomorphism over f , it extends uniquely to a dilinear homomorphism gS,T : T −1 M → T −1 N over fS,T . For a multiplicative set S ∈ A let S 0 = {t ∈ A|there exists a ∈ A, s ∈ S such that s = at} be the saturation of S. Then the canonical homomorphisms S −1 A → s0−1 A and S −1 M → s0−1 M are isomorphisms and we may identify the two corresponding localizations. In particular, if s ∈ A, we identify As , Ms with Ss0−1 A, Ss0−1 M . So if for s, t ∈ A, one has Ss0 ⊆ St0 , one has canonical homomorphisms As → At , Ms → Mt . Proposition 87 If iS : A → S −1 A is the localization homomorphism, the inverse image q 7→ −1 i−1 A S (q) is an order preserving bijection from the set of maximal (resp. prime) ideals in S to the set of maximal (resp. prime) ideals in A which are disjoint from S. In particular, if S = A − p for a prime ideal p, the localization Ap is a local ring with maximal ideal mp = pp and the residue field κp = Ap /mp is isomorphic to the field of fractions f r(A/p).
E.4.2
Projective Modules
Definition 144 An A-module P is projective iff it is a direct summand of a free A-module. Equivalently, it is projective if for each pair of homomorphisms u : P → N , v : M → N , v an epimorphism, there is a homomorphism w : P → M such that u = v ◦ w. Let U 7→ U ?? be the bidual functor on A-modules. Then a direct summand U ⊂ V is mapped into a direct summand U ?? of V ?? . Now, if U is projective, it is a direct summand of a free module R(n) . It is easily seen that the bidual map R(n) → (R(n) )?? is injective. Therefore, if U is projective, the bidual map U → U ?? is also injective. If U is finitely generated and projective, the bidual map is an isomorphism. Let i : N → M be the inclusion of a submodule N of an R-module M . For any x ∈ M and positive exponent r, let ∧r x :
^r
N→
^r+1
M : y 7→
^r
i(y) ∧ x
the linear map defined by the rth exterior power of i and the wedge product with x in the
E.4. COMPLEMENTS OF COMMUTATIVE ALGEBRA
1103
exterior algebra of M . This defines a linear map ∧r : M → LinR (
^r
N,
^r+1
M)
(E.28)
which has the following property: Lemma 83 With the above notation, if N is a direct factor of M which is locally free of rank r, then Ker(∧r ) = N . ^r Proof. Let x ∈ M . In the special case where N is free of rank r, if N = R.u for a basis ^r r vector u of N , the claim x ∈ Ker(∧ ) is equivalent to u ∧ x = 0. But this follows from [63, ch.III, §7, no.9, Prop.13]. In the general case, clearly the condition is sufficient. Conversely, suppose ∧r x = 0. Recall from [64, ch.II, §5 no.3, Th.2] that an R-module is projective of rank r ∈ N iff it is locally free of rank r. Notice that localizing commutes with exterior powers ([64, ch.II, §2 no.8]), so we may localize in f ∈ R such that for the localized element xf ∈ Mf , we have ∧r xf ∈ Nf and Nf is free of rank r, whence xf = 0. As there is a cover of Spec(R) with basic open sets D(fi ) associated with localizations Rfi such that Nfi is free, we deduce that x ∈ N . QED.
E.4.3
Injective Modules
Proposition 88 ([63, ch.II, §2, exercise 11]) For an R-module M , the following properties are equivalent: (i) The functor LinR (?, M ) is exact. (ii) The functor LinR (?, M ) is exact on short exact sequences. (iii) For every R-module E and every linear injection F E, every linear map F → M extends to a linear map E → M . (iv) For every ideal a ⊂ R and every linear map f : a → M , there is m ∈ M such that f (a) = a.m for all a ∈ R. (v) M is a direct factor of every module which contains it. (vi) For every R-module E which is sum of M and of a module I with one generator, M is a direct factor of E. Definition 145 An R-module M is said to be injective iff it has the equivalent properties of proposition 88. The ring R is said to self-injective iff it is injective as a (left) module over itself. Exercise 87 Show the following statement: A direct sum of R-modules is injective iff each factor is.
1104
APPENDIX E. MODULES, LINEAR, AND AFFINE TRANSFORMATIONS
Example 78 Let 1 < N be a natural number. Then the Ring ZN is self-injective. In fact, let ∼ N = pn1 1 · . . . pnr r be its prime factorization (see appendix D.2). Then ZN → Zpn1 1 × . . . Zpnr r , n and injectivity can be checked on each factor, so suppose N = p . We apply criterion (iv) of proposition 88. An ideal a in Zpn is generated by pm , m ≤ n, and we have the isomorphism ∼ of Zpn -modules a → Zp(n−m) . Then a linear map f : Zp(n−m) → Zpn evaluates to f (a) = f (a.1Zp(n−m) ) = a.f (1Zp(n−m) ), and we are done.—In particular, every free ZN -module ZrN is injective. The self-injectiveness of Zpn also follows from this: Proposition 89 ([139, proposition 21.5]) Let R be a zero-dimensional local ring. The following are equivalent: (i) R is self-injective. (ii) The socle of R is simple. In fact, the socle of Zpn is isomorphic to Zp , the simple group of order p and it is well known that Zpn is zero-dimensional. Proposition 90 A finitely generated Zpn -module M is injective iff it is free of finite rank. Proof. Clearly, by example 78, a free module of finite rank is injective. Conversely, by the main ∼ theorem on finitely generated abelian groups E.2 we have M → Zpn1 × . . . Zpnk , ni ≤ n. By exercise 87 above, it suffices to see that Zpm cannot be injective if m < n. In fact, if the injection Zpm ← Zpn : x 7→ pn−m .x had a left inverse h, we would have x = h(pn−m .x) = pn−m .x which only works for x = 0, a contradiction. QED.
E.4.4
Lie Algebras
Definition 146 For a module L over commutative ring R, a Lie algebra structure is an Rbilinear multiplication [ ] : L × L → L, the Lie bracket, such that [xx] = 0 identically, and the Jacobi identity [x[yz]] + [y[zx]] + [z[xy]] = 0 holds for all x, y, z ∈ L. A homomorphism of Lie algebras f : L1 → L2 is a linear homomorphism such that f ([xy]) = [f (x)f (y)] for all x, y ∈ L1 . The corresponding category of Lie algebras over R is denoted by LieR . ∼
Example 79 If L → Rn is free, and if (xi ) is a basis, a Lie algebra structure on L is defined by bilinearity P and skew symmetry (which follows from [xx] = 0) of the Lie bracket if the Lie brackets k akij xk = [xi , xj ], i < j are known. The condition for such a bracket to generate a Lie algebra is akii = 0, all i, k,
(E.29)
akij
(E.30)
akji
+ = 0, all i < j, k, X k m k m akij am kl + ajl aki + ali akj = 0, all i, j, l, m. k
(E.31)
E.4. COMPLEMENTS OF COMMUTATIVE ALGEBRA
1105
The coefficients akij are called structural constants of the Lie algebra in the given basis. Example 80 For any module L, the R-algebra of linear endomorphisms End(L) becomes the general linear algebra gl(L) by the bracket [xy] = x◦y−y◦x. A sub-Lie-algebra of a general linear ∼ algebra is called a linear Lie algebra. If L → Rn is free of rank n, we also write gl(L) = gl(n, R). Its subalgebra of endomorphisms with vanishing trace (check that it is a sub-Lie-algebra!) is called the special linear algebra and denoted by sl(L) or sl(n, R). Example 81 Let L be any R-module with a bilinear product x·y (no other conditions required). A derivation is a linear endomorphism D : L → L such that D(x · y) = x · D(y) + D(x) · y. The set Der(L) is a submodule of End(L), and in fact a Lie subalgebra of the general linear algebra gl(L). In particular, if we take the Lie algebra structure on L, Der(L) is another Lie algebra. Observe that for x ∈ L, the left multiplication ad(x) = [x?] : y 7→ [xy] is a derivation by the Jacobi identity. One has the Lie algebra homomorphism of adjunction ad : L → Der(L),
(E.32)
a representation of L in the general linear algebra of gl(L). A derivation ad(x) is called inner, any other is called outer derivation. Proposition 91 If x is a nilpotent endomorphism in a linear algebra, then its adjoint ad(x) is also a nilpotent endomorphism. See [239, p.12] for the easy proof. Suppose that L is a linear algebra in End(V ) for an R-module V , and that the conjugation Inte , e ∈ GL(V ) leaves L invariant. Then evidently the conjugation is an automorphism of L. Suppose now that R is a Q-algebra. If ad(x) is nilpotent, then the exponential exp(ad(x)) = 1 + ad(x) + ad(x)2 /2! + . . . ad(x)k /k! + . . .
(E.33)
is defined. We have: Lemma 84 If ad(x) is nilpotent, then edp(ad(x)) is an automorphism of the Lie algebra L. Moreover, if x is nilpotent, exp(x) is defined and we have Intexp(x) = exp(ad(x)). See [239, p.9] for the proof.
Appendix F
Algebraic Geometry For this chapter, we refer to [64, 123, 198, 199, 140].
F.1
Locally Ringed Spaces
Given a topological space X, its system of open sets OpenX is viewed as a category with inclusions as morphisms. If f : X → Y is a continuous map, the inverse image map U 7→ f −1 U defines a functor Openf : OpenY → OpenX . This defines a functor Open? : Top → Cat into the category of categories and functors1 . Let C be a category of sets with some additional algebraic structure, such as the categories Mod, Mon, Gr, Ab, Rings, ComRings of modules, monoids, groups, abelian groups, rings, or commutative rings, respectively. A contravariant functor F : OpenX → C is called a C-space (this is a presheaf plus the algebraic morphism conditions). For example, a ringed space is just a Rings-space. In the present context, we always suppose that a ringed space is one with values in ComRings, i.e., a commutatively ringed space. The set of C-spaces on X is denoted by Cspaces . The contravariant functor Open? induces a functor Cspaces : X 7→ Cspaces . It maps X X ? spaces the continuous map f : X → Y to the set map Cf : F 7→ F ◦ Openf . The image F ◦ Openf is denoted by f∗ F and is called the direct image of F . If F : OpenX → C, G : OpenY → C are C-spaces and f : X → Y is continuous, then an f -morphism h : F → G is a natural transformation h : G → f∗ F . These morphisms define an evident category, the category Cspaces of C-spaces. Suppose that the category C has colimits for filtered diagrams2 , such as Rings, ComRings, Mod, Gr, Ab, and take a C-space F . For each point x ∈ X, the filtered system OpenX,x of open neighborhoods of x defines an object Fx = colimU ∈OpenX,x F (U ), the stalk of F at x. So for a (commutatively) ringed space, this is a (commutative) ring. Let h : F → G be an f morphism for f : X → Y . For x ∈ X, we have the restriction Openf,x : OpenY,f (x) → OpenX,x 1 Restricted to a universe, if the limitless collection bothers the reader, or even the category of partially ordered sets with order preserving maps, to stick to reality. 2 Meaning that for any two objects in the diagram quiver, there are two arrows with these domains targeting at a common codomain.
1107
1108
APPENDIX F. ALGEBRAIC GEOMETRY
of Openf to the neighborhood systems OpenY,f (x) and OpenX,x . This induces a C-morphism hx : Gf (x) → Fx . The subcategory LocRgSpaces of ComRings-spaces consists of all ringed spaces F which have a local ring Fx with maximal ideal mx in each point x ∈ X, and of those morphisms h which induce local morphisms hx in all stalks, i.e., h−1 x (mx ) = my . It is called the category of locally ringed spaces. The residue field in a point x of such a space F is the field κ(x) = Fx /mx . For a section s ∈ F (U ) over the open set U , we denote by s(x) the canonical image of s in κ(x). For the category ComMod of modules over commutative rings and dilinear homomorphisms, a ComMod-space F can also be described by the underlying ringed space R and the abelian group space F (same notation), together with a scalar multiplication R(U ) × F (U ) → F (U ) in each open set U , and the evident dilinear transition maps. One therefore also says that F is an R-module. Same wording, mutatis mutandis, for an R-algebra or for an R-ideal. If we are given a C-space F on X which is a sheaf, and if B is a topological base for X, the restriction F |B to this subcategory of OpenX completely determines F . If U ∈ OpenX , U = colim(B ∈ B|B ⊆ U ), and F (U ) = lim(F (B), B ∈ B). Conversely, if we are given a contravariant functor F : B → C, we obtain a C-space F 0 by F 0 (U ) = lim(F (B), B ∈ B) and by the universally given transition morphisms. This presheaf is a sheaf if F is a sheaf on B, i.e., if Q for every covering (Bi ) of B ∈ B by elements of the base, the canonical application F (B) → i F (Bi ) is a bijection x 7→ (x|Bi ) onto the tuples (xi ) such that for every i, j and base element B 0 ⊆ Bi ∩ Bj , we have xi |B 0 = xj |B 0 .
F.2
Spectra of Commutative Rings
Definition 147 The (prime) spectrum is a contravariant functor Spec : ComRings → LocRgSpaces which is defined as follows: Let A, B be commutative rings, and let f : A → B be a ring homomorphism. 1. The topological space consists of the set Spec(A) = {p a prime ideal in A}. The closed sets are the sets of the form V (E) = {p|E ⊆ p} for a subset E ⊆ A. Equivalently, a base of open sets is given by the system Df = {p|f 6∈ p}, f ∈ A, and we have Df ∩ Dg = Df g . 2. For the base D = {Df |f ∈ A}, we have a sheaf on D, which is defined by Df 7→ Af , the localization at the saturated multiplicative set S(f ) defined by f , a well-defined setup since Df = Dg iff S(f ) = S(g) see [198, I.1.3.2]. This presheaf is a sheaf on D, and the associated sheaf on Spec(A) is denoted by A˜ and called the ring sheaf associated with A. ∼ ˜p→ If p ∈ Spec(A) is a prime ideal, we have (A) Ap , i.e., A˜ ∈ LocRgSpaces.
F.2. SPECTRA OF COMMUTATIVE RINGS
1109
3. For the homomorphism f : A → B, the inverse image map on prime ideals Spec(f ) : Spec(B) → Spec(A) : p 7→ f −1 p is defined, and we have Spec(f )−1 (Dg ) = Df (g) , Spec(f )−1 (V (E)) = V (f (E), i.e., Spec(f ) is continuous. Furthermore, we have a canonical map fg : Ag → Bf (g) which is natural and therefore induces a morphism Spec(f ) : ˜ over the continuous (synonymous) map Spec(f ). The stalk homomorphism fp : A˜ → B ˜ ˜Spec(f )(p) , colimit of the natural homomorphisms fg : Ag → Bf (g) , is local. One Ap → B therefore has a contravariant functor Spec as announced, and one often denotes Spec(A), when meaning the locally ringed space A˜ over Spec(A). Theorem 56 The functor Spec is fully faithful, the inverse global section functor Γ of a ˜ → A˜ is given by the ring homomorphism u(Spec(A)) : A = LocRgSpaces-morphism u : B ˜ ˜ A(Spec(A)) → B(Spec(B)) = B. See [198, I.1.6.3] for a proof. Since one often writes F (U ) = Γ(U, F ) and calls the elements ˜ = section above U , the theorem’s notation is justified by the global section notation Γ(A) ˜ Let Aff be the full subcategory of LocRgSpaces consisting of the objects Γ(Spec(A), A). which are isomorphic to prime spectra. These spaces are called affine schemes. We therefore have that the map Spec : ComRings → Aff is an equivalence of categories. ˜ ˜ , whose sections on the base D are defined If M is an A-module, we have a A-module M ˜ by Γ(Dg , M ) = Ag ⊗ M = Mg , the localization of M at the multiplicative set S(g). ˜ is an exact3 and fully faithful functor from the category Proposition 92 The map M 7→ M ˜ of A-modules ModA to the category ModA˜ of A-modules. It also commutes with colimits of modules, with tensor products, Hom-modules, with sums and intersections of submodules. ˜ 7→ Γ(Spec(A), M ˜ ), which is The inverse to this functor is the global section functor M also exact. See [198, I.1.3] for a proof. The modules in ModA˜ , which are hit by this tilding process are the quasi-coherent ones: A module M over a ringed space A over a topological space X is quasi-coherent iff there is a covering Xi of X such that each restriction Mi = M|Xi is the cokernel of a homomorphism fi : AIi i → AJi i , where Ai = A|Xi . ˜ ˜ iff it is quasi-coherent. Theorem 57 An A-module M is isomorphic to a module M See [198, I.1.4.1] for a proof. This means that we have an equivalence of categories of quasi-coherent modules over A˜ and ModA (with linear homomorphisms). ∼ For a ring element f ∈ A, one has Spec(Af ) → Df . When restricting the associated ring ˜ M ˜ to basic open sets Df , this yields ring and module sheaves which are and module sheaves A, ˜ ˜ isomorphic to Af , Mf . Theorem 58 Let M be an A-module. The following conditions are equivalent: 3A
f
g
sequence K → M → L of linear homomorphisms of modules is exact in M iff Im(f ) = Ker(g). Such exact sequences are preserved by the functor.
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APPENDIX F. ALGEBRAIC GEOMETRY
(i) M is projective4 and finitely generated. (ii) There is a finite family (fi ) of elements of A which generate the ideal A, i.e., Spec(A) = S ˜ D i fi such that the localizations Mfi = Γ(Dfi M ) are free of finite rank over Afi . This is why one can also define a finitely generated projective module as being a locally free module of (locally defined) finite ranks. If the locally constant rank is constant n, the module is said to be locally free of rank n.
F.2.1
Sober Spaces
A topological space X is irreducible - iff every non-empty open subset is dense, or, equivalently, if any two non-empty open sets have a non-empty intersection. A subset of a topological space is called irreducible if it is so with its relative topology. A point x of an irreducible space X is said to be generic iff its (always irreducible) closure {x}. We say that a point x dominates a point y, in signs x > y, iff {y} ⊆ {x}. This is a partial order relation on X. An irreducible component of a space X is a maximal irreducible subset. Sorite 15 These are the sorite properties concerning irreducibility: (i) A subset of a topological space is irreducible iff its closure is. (ii) Irreducible components are closed. (iii) Every irreducible subset is contained in an irreducible component, in particular, a topological space is the union of its irreducible components. (iv) The image f (E) of an irreducible subset E ⊆ X under a continuous map f : X → Y is irreducible. Definition 148 A topological space X is sober iff each closed irreducible subset has a unique generic point. Call Sob the full subcategory of the category Top of topological spaces consisting of sober spaces. T If A is a commutative ring, and if E ⊆ Spec(A), then we denote J(E) = p∈E p, and E = V (J(E)). This ideal is prime iff E is irreducible. In this case, E = {J(E)}. In fact, for two points p, q in Spec(A), p > q iff p ⊆ q. In particular, Spec(A) is a sober space. Its irreducible components correspond to the minimal prime ideals. Proposition 93 The canonical injection j : Sob → Top has a left adjoint ?s : Top → Sob. 4 See
definition 144 in appendix E.4.2.
F.3. SCHEMES AND FUNCTORS
1111
Proof idea. This adjoint associates with any X a sober space X s which is defined as follows. Its points are the irreducible closed sets in X. The open sets are the sets V s = {YS ∈ X s |Y ∩V S 6= ∅}, where V varies over all open sets in X. Clearly, (V ∩ W )s = V s ∩ W s and ( Wi )s = Wis for any family (Wi ) of open sets. On continuous maps f : X → Y , the functor acts via f s : X s → Y s : E 7→ f (E). One has a canonical continuous map qX : X → X s : x 7→ {x} and a commutative diagram of continuous maps: f X −−−−→ Y qY (F.1) qX y y fs
X s −−−−→ Y s s The map qX : X → X is a homeomorphism if X is sober. The adjunction is given by the mutually reciprocal maps Top(X, j(Y )) → Sob(X s , Y ) : f 7→ qY−1 ◦ f s and Sob(X s , Y ) → Top(X, j(Y )) : g 7→ g ◦ qX . Lemma 85 The canonical continuous map qX : X → X s is a quasi-homeomorphism,. i.e., the s inverse image map 2X → 2X is a bijection between the open sets of X s and those of X.
F.3
Schemes and Functors
A scheme (X, OX ) is a ringed space OX on X which locally is isomorphic to a spectrum of a commutative ring, i.e., there is an open covering (Xi ) of X and a family Ai of rings ∼ such that (Xi , OX |Xi ) → Spec(Ai ). The category Schemes of schemes is the subcategory of LocRgSpaces whose objects are schemes. By Yoneda, we have a fully faithful functor Y : Schemes → Schemes@ . Proposition 94 The restriction YAff : Schemes → Aff@ is fully faithful. Equivalently, the corresponding functor YComRings : Schemes → ComRings@ into the category ComRings@ of covariant set-valued functors on ComRings is fully faithful. This means that we may consider schemes as special covariant functors on the category of commutative rings. The functors which correspond to schemes are characterized by a sheaf condition: Property 3 We are given a functor G ∈ ComRings@ . For every ring A ∈ ComRings, and every finite family (fi ) of elements of A which generate A as an ideal, the diagram Y Y G(A) → G(Afi ) ⇒ G(Afi fj ) i
i,j
is exact, we say (by abuse of language, but theoretically justifiable) that G is a sheaf.
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APPENDIX F. ALGEBRAIC GEOMETRY
Then the full subcategory of ComRings@ consisting of sheaves G in the sense of property 5 3, together with the property that there S is a family of rings Rt and morphisms at : Rt @ → G such that for every field K, G(K) = t Rt @K, comprises the functors which are isomorphic to images of schemes under the Yoneda map YComRings , see [198, I.2.3.6] and [140, VI.2] for details. This means that schemes are characterized without any reference to the geometry of ringed spaces. See also [314, III.3] for the relation of this setup to the systematic topos-theoretic restatement of the schemes in terms of the Zariski site. The most important universal property of the category of schemes is that it has fiber ∼ products. In the affine case, we have Spec(A) ×Spec(C) Spec(B) → Spec(A ⊗C B).
F.4
Algebraic and Geometric Structures on Schemes
If a scheme is viewed as a set-valued functor on rings, the sets may also be enriched by algebraic structures, such as groups, monoids, etc., to yield a category C. We then view a scheme as a functor S : ComRings → C, and say that S is a C-scheme, for example, an abelian groupscheme if C = Ab. Example 82 For 0 ≤ n, we have the additive group scheme An whose functor is An (R) = Rn , with the canonical addition of this free module, and the canonical transitions An (R) → An (S) for a ring homomorphism f : R → S. Example 83 The n-dimensional linear group scheme is given by the functor R 7→ GL(n, R) ⊂ 2 ∼ Mn,n (R) → An , together with the canonical map GL(n, R) → GL(n, S) for a ring homomorphism f : R → S. The set GL(n, R) is defined as the set of n × n-matrices M with invertible determinant: det(M ) ∈ R× . The functor is represented by the affine scheme GLn = Spec(Z[Xij , 1 ≤ i, j ≤ n]det ), where det = Det(Xij ). The group structure is the multiplication of invertible matrices.
F.4.1
The Zariski Tangent Space
For a field K, we are given a K-scheme X, i.e., a scheme s : X → Spec(K) in the comma category Schemes/Spec(K) (see section G.2.1). Suppose that we have a K-rational point x : Spec(K) → X, i.e., a section of s. This means that the corresponding K-algebra OX,x has ∼ an isomorphism K → κX,x = OX,x /mX,x . The Zariski tangent space in x is the linear K-dual 2 TX,x = (mX,x /mX,x )∗ . Consider the K-scheme DK = Spec(K[ε]) over the dual numbers K[ε], see example 76 in appendix D.1.1. It has the K-rational point ε : Spec(K) → DK corresponding to the projection K[ε] → K : ε 7→ 0. Proposition 95 With the above hypotheses and notation, there is a bijection of the Zariski elements t of the tangent space TX,x and the morphisms τ : DK → X of K-schemes which map the K-rational point to the K-rational point x. 5R
t@
is the covariant functor on rings, i.e., Rt @R = HomComRings (Rt , R).
F.5. GRASSMANNIANS
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See [140, VI.1.3] for a proof. In particular, if the scheme X is given by its functor on rings, this means that the tangents are special elements of the evaluation of the functor in dual numbers X(K[ε]). For example, if X = A1K = A1 ×Z K, we have tangents x + ε.τ, τ ∈ K, over ∼ the rational point x ∈ K → A1K (K).
F.5
Grassmannians
A subfunctor G → F in ComRings@ is open iff for every morphism a : R@ → F (corresponding to an element a ∈ F (R) via Yoneda), the fiber product projection G×a R@ → R@ is isomorphic to the functor of an open subscheme of Spec(R). Clearly, then, if b : X@ → F is a morphism from a representable functor X@ of a scheme X, then the projection G ×b X@ → X@ is isomorphic to an open subscheme of X. An open covering of a functor F in ComRings@ is a family (gi : Gi → F ) of open subfunctors of F such that the fiber product projections Gi ×b X@ → X@ for morphisms b : X@ → F from the representable functor of a scheme X define an open covering of X. For example, the open subfunctors of an affine scheme Spec(R) are the functors FI : ComRings → Sets of form FI (S) = {f : R → S|f (I)S = S}, where I is an ideal in R. The Grassmann scheme Grassr,n is defined for any couple 0 ≤ r ≤ n of natural numbers by the functor Grassr,n (R) = {V ⊆ Rn |Rn /V locally free of rank r}, which for a ring homomorphism R → S maps the exact sequence 0 → V → Rn → Rn /V → 0 to the exact sequence 0 → Im(S ⊗R V ) → S n → S ⊗R (Rn /V ) → 0, where the image of the tensorized space S ⊗R V is the image of V under this map. The locally free quotient remains locally free since the localization on R carries over to a localization over ∼ S: For f ∈ R, and its image f 0 ∈ S, we have (S ⊗R Rn /V )f 0 → Sf 0 ⊗Rf (Rn /V )f . The functor Grassr,n is covered by the following open subfunctors. Let i. = i1 , i2 , . . . ir be an increasing subsequence 1 ≤ i1 < i2 < . . . ir ≤ n. We have the affine L open subfunctors Grassn,r,i. (R) of those submodules V ⊂ Rn such that the factor Ri. = j=1,...r R.eij of Rn projects isomorphically onto the quotient Rn /V . If i0. denotes the complementary increasing 0 sequence, Grassn,r,i. (R) identifies to the set of graphs Γf of linear maps in LinR (Ri. , Ri. ), i.e., ∼ to n × (n − r)-matrices with columns (ei0k , f (ei0k ))t . In fact, the isomorphism Ri. → Rn /V corre0 0 ∼ sponds to an isomorphism V → Ri. , and this makes V a graph of a linear map in LinR (Ri. , Ri. ). The fact that these open subfunctors (represented by affine schemes Ar×(n−r) ) cover the Grassmannian results from the situation over a field, where the covering is evident. Proposition 96 Let n be a positive natural number. Then the subfunctor Bn : R 7→ {x ∈ Rn | x is part of a basis of Rn } of the affine n-space An over Z is an open subscheme.
(F.2)
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APPENDIX F. ALGEBRAIC GEOMETRY 2
Proof. Consider the open subscheme GLn of An . Then Bn is the image of GLn under the projection onto the first column (n1j ) which by [199, IV/2, 2.4.7] is (universally) open. QED. Lemma 86 If X is an S-scheme, E a quasi-coherent OX -module, then any section s : S → Grassr (E) is a closed immersion. Proof. By [198, Proposition (9.7.7)], Grassr (E) separated over S, and by [198, Corollaire (5.2.4)], a section of such a structural morphism is a closed immersion. QED. Lemma 87 If R is a product of local rings of finite length, then for two elements x, y of an R-module M , R× x = R× y iff these elements generate the same space, i.e., R.x = R.y. Without loss of generality, we may suppose that R is local with maximal nilpotent ideal m. Clearly, the condition is sufficient. Suppose now that R× x 6= R× y, and therefore R× x∩R× y = ∅. Then, R.x = R.x implies R× x ⊂ m.y, since R = R× ∪ m. But then we have x ∈ m.y, and symmetrically y ∈ m.x, which gives x ∈ mk .y for all powers k, and m being nilpotent yields x = y = 0, a contradiction. QED.
F.6
Quotients
If G is a finite group, and if (X, OX ) is a scheme, a group action of G on X can be given by a group homomorphism α : G → Aut(X). This can also be seen as a morphism of schemes α0 : GZ ×Spec(Z) X → X with the functorially described axioms of group actions associated with the functors of the schemes GZ , X, the scheme GZ = Spec(ZG ) is a group scheme whose multiplication is associated with the group multiplication µ : G × G → G via the ring ho∼ momorphism µ0 : ZG → ZG×G → ZG ⊗Z ZG . The scheme GZ is finite and locally free over Z. The set-theoretic orbits of the action α are the equivalence classes defined by the relation on the product set X × X, image of the set map G × X → X × X : (g, x) 7→ (α(g).x, x). If we use the schema-theoretic map α0 : GZ ×Spec(Z) X → X, the cokernel functor of the pair pr2 , α0 : GZ ×Spec(Z) X ⇒ X of functor morphisms, if it exists, is called the scheme functor of orbits of X under the action of G. We have this particular case of [123, III,2.6.1]: Theorem 59 With the above notation, if G is a finite group and α0 : GZ ×Spec(Z) X → X the group action associated with an ‘abstract’ action α : G → Aut(X) on the scheme X, such that every set-theoretic orbit is contained in an affine open subscheme of X, then there is a scheme-functor of orbits Y = coker(pr2 , α0 ) and the associated diagram of schemes (qua locally ringed spaces) G ×Spec(Z) X ⇒ X → Y is exact.
Appendix G
Categories, Topoi, and Logic For a comprehensive introduction to category theory, see [313]. For topos theory and sheaves see [314], for topos theory and logic, see [186].
G.1
Categories Instead of Sets
One may rebuild mathematics from categories rather than from sets. In this framework, the most radical approach is the arrow-only definition of a category1 : Definition 149 A category C is a collection of objects f, g, h, . . . which are called morphisms, together with a partial composition f ◦g which yields morphisms of C. An identity is a morphism e such that, whenever defined, we have e ◦ f = f and g ◦ e = g. We have these axioms: 1. Whenever one of the two compositions (f ◦ g) ◦ h, f ◦ (g ◦ h) is defined, both are defined and they are equal; we denote the resulting morphism by f ◦ g ◦ h. 2. If f ◦ g, g ◦ h are both defined, (f ◦ g) ◦ h is defined. 3. For every morphism f there are two identities, a ‘left’ identity eL and a ‘right’ identity eR , such that eL ◦ f, f ◦ eR are defined (and necessarily equal to f ). It is easily seen that two right (left) identities of a morphism f are necessarily equal; they are called the domain of f (codomain of f ) and are denoted by dom(f ) (codom(f )). To make domain and codomain evident, one also writes f : a → b with a = dom(f ), b = codom(f ) instead of f . For two morphisms a, b, the collection of those f with dom(f ) = a, codom(f ) = b is denoted by Hom(a, b), HomC (a, b), C(a, b), . . . according to the specific situation. Evidently, no morphism can be a member of Hom(a, b) and of Hom(a0 , b0 ) if either a 6= a0 or b 6= b0 , i.e., the Hom collections form a partition of C (in the non-set-theoretic common sense). 1 Mac Lane calls this type of set-less categories “metacategories”, and reserves the proper term “category” for metacategories which are built upon sets. We do however preconize the foundational character of metacategories and therefore omit the “meta” prefix. However, we then should provide a germ for existing categories, in order to get off ground as with axiomatic set theory. See [314, VI.10] for a discussion of the foundation of mathematics via topoi.
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APPENDIX G. CATEGORIES, TOPOI, AND LOGIC
Exercise 88 Two identities e, e0 of C can be composed iff they are equal, and then e ◦ e = e (identities are idempotent). In a more conservative understanding of categories, the identities are associated with the “objects” of a category, which are a second type of concepts, but do not enrich the category except in the way it is constructed. The identification of objects and identities is carried out as often as possible in our present text. In either case, the collection of identities (qua objects) is denoted by 0 C or Ob(C), whereas the morphisms are denoted by 1 C or M or(C). To stress the morphic character of an identity e (in contrast to the underlying object in the conservative understanding), one also writes Ide instead of e. In a category, a morphism f is mono, a monomorphism, iff for any two compositions f ◦ g, f ◦ g 0 , the equality f ◦ g = f ◦ g 0 implies g = g 0 . The morphism f is epi, an epimorphism, iff for any two compositions g ◦ f, g 0 ◦ f , the equality g ◦ f = g 0 ◦ f implies g = g 0 . The morphism f is called a section if there is a left inverse g, i.e., g ◦ f = dom(f ); f is called a retraction if it has a right inverse h, i.e., f ◦ h = dom(h). A morphism f that is a section and a retraction is iso, an isomorphism. If dom(f )) = codom(f ), the morphism is called endo, an endomorphism. An endomorphism which is an isomorphism is called auto, an automorphism. The collection of endomorphisms for a domain c is denoted by End(c), whereas the collection of automorphisms for c is denoted by Aut(c). If these collections are sets, they define monoids End(c) and groups Aut(c) with the identity Idc as unit. Exercise 89 The composition of two monomorphisms, epimorphisms, isomorphisms, endomorphisms, and automorphisms, if defined, shares, each of these properties.
G.1.1
Examples
Example 84 The category Sets of all sets. The morphisms are the set maps between existing sets, and the composition is the usual composition of set maps. Remark 28 Usually, the delicate comprehension axiom which can cause contradictory constructions of sets, is avoided by a strong restriction of the available sets. One takes a very large set U , which has the properties of a “universe”, i.e., it is stable in the following sense: • If x ∈ U , then x ⊂ U ; • If x, y ∈ U , then {x, y} ∈ U ; • If x ∈ U , then 2x ∈ U (the set of all subsets, the powerset); • A set of all natural2 numbers N is element of U ; • If f : x → y is a surjective function with x ∈ U, y ⊂ U , then y ∈ U One then restricts the Sets objects to the elements of the universe U and says that these are small sets. We denote such a category of small sets by SetsU . 2 For
example the set of finite ordinals 0 = ∅, 1 = {0}, 2 = {0, 1}, . . . n, n+ = n ∪ {n}, . . .
G.1. CATEGORIES INSTEAD OF SETS
1117
Example 85 Given a quiver Q = (head, tail : A ⇒ V ) (see section C.2.2), the path category P (Q) has the paths as morphisms, the identities are the lazy paths, and the composition is the path composition. Here, the vertexes are separate concepts which can be identified (and in fact are identified in our construction) with the lazy paths. All paths are mono and epi, but only the identities are isomorphisms. The terminology “quiver” stems from algebra, in category theory, a quiver is more known as a “diagram scheme”. Relations among paths give rise to quotient categories as follows: Suppose that we are given any binary relation ∼ between some paths of equal domain and codomain. Consider the smallest equivalence relation ∼0 among paths which contains ∼ and is a ‘two-sided ideal’ in the sense that for f ∼0 g with dom(f ) = dom(g) = d, codom(f ) = codom(g) = c and h, k with dom(h) = c, codom(k) = d, we have c ◦ f ∼0 c ◦ g and f ◦ d ∼0 g ◦ d. Then we obtain a new category, the quotient category P (Q)/ ∼, and its morphisms are the equivalence classes of paths, while the composition is the composition of representatives of these classes. In the language of category theory, the relation ∼ is called a commutativity relation of the given diagram scheme. Example 86 Fix a ring R, the matrix category over R is the collection MR of all m×n-matrices M = (mi,j ) with coefficients in R and for any row and column numbers m, n, together with the usual matrix multiplication M · N as composition. The identities are all the identity matrices En , n = 1, 2, . . . (over R). We evidently have HomMR (En , Em ) = Mm,n (R). In particular, the vectors in Rn are identified with the morphisms in HomMR (E1 , Em ) = M1,n (R). Example 87 Given a category C, the isomorphism classes of C-objects define a skeleton category C/iso: For each isomorphism class, select a representative and then consider the full subcategory3 of C on these representative objects. Clearly C/iso is defined up to isomorphism of categories (see below G.1.2) and no two skeleton objects are isomorphic. Example 88 Common examples of categories are the categories Mon of monoids, Gr groups, Rings of rings with ring homomorphisms, LinModR R-modules with linear homomorphisms, LinMod modules with dilinear homomorphisms, ModR R-modules with affine homomorphisms, Mod of modules with diaffine homomorphisms, or Top of topological spaces with continuous maps. Example 89 For every category C we have the opposite category Copp . Its morphisms are the same, but composition works via f ◦opp g = g ◦ f , i.e., it is defined iff the composition with opposite factors is defined in C. This opposite construction exchanges the domains and codomains of morphisms. Intuitively, an arrow f : x → y in C becomes a arrow f : y → x in Copp .
G.1.2
Functors
Functors are the morphisms between categories: Definition 150 If C, D are categories, a functor F : C → D is a function which assigns to every morphism c in C a morphism F (c) in D such that 3 See
example 90 below.
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APPENDIX G. CATEGORIES, TOPOI, AND LOGIC
(i) F (c) is an identity if c is so, (ii) if c ◦ c0 is defined in C, then F (c) ◦ F (c0 ) is defined and F (c ◦ c0 ) = F (c) ◦ F (c0 ). In particular, functors carry isomorphisms to isomorphisms. Moreover, the composition F ◦ G : C → E of two functors F : C → D, G : D → E is a functor. Two categories ∼ C, D are called isomorphic if there exists a functor isomorphism, i.e., two functors F : C → ∼ −1 −1 −1 D, F : D → C such that F ◦ F = IdC , F ◦ F = IdC . A functor is called full iff the F (HomC (x, y)) = HomD (F (x), F (y)) for all object pairs c, d. It is called faithful iff F : HomC (x, y) → HomD (F (x), F (y)) is injective for all pairs c, d. It is called fully faithful iff it is full and faithful, i.e., the map F : HomC (x, y) → HomD (F (x), F (y)) is a bijection. Functors are also called “covariant” since they are opposed to functors F : Copp → D, which are called “contravariant” but then also denoted by F : C → D. One often considers systems of morphisms in a category C, which are defined by a graphical approach: diagrams. Here is the precise definition. Definition 151 A diagram in a category C is a functor ∆ : P (Q) → C, where Q is a quiver. The diagram ∆ is said to commute with respect to a relation ∼ among Q-paths, iff ∆ factorizes through P (Q)/ ∼. If the relation is maximal (it identifies all paths having common domain and codomain), then the diagram is said to be commutative without specification of ∼. By the very definition of a path category, diagrams are given by systems of morphisms in C which cope with the domain-codomain configuration in the underlying quiver (i.e., diagram scheme). Example 90 If C is a category, a subcategory is a sub-collection C0 of C such that for each morphism f in C0 , its domain and codomain are also in C0 , and such that for any two f, g in C0 such that f ◦ g is defined in C, the composition is also a morphism in C0 . A category can be defined by an arbitrary selection of objects (identities) out of C and the full collections of morphisms having these identities as domains or codomains. Such a subcategory is called a full subcategory of C. A subcategory obviously induces an embedding functor C0 → C by the identity on the morphisms in C0 . For any collection S of morphisms in C, the smallest subcategory of C containing S is denoted by hSi and called the subcategory generated by S. Example 91 If C, D are two categories, the product category C × D consists of all ordered pairs (c, d) of morphisms c in C and d in D. The composition (c, d) ◦ (c0 , d0 ) is possible iff it is possible in each component and then evaluates to (c, d) ◦ (c0 , d0 ) = (c ◦ c0 , d ◦ d0 ). One has the canonical projection functors p1 : C × D → C, p2 : C × D → D with p1 (c, d) = c, ps (c, d) = d. The same procedure allows the definition of any finite product of categories.
G.1.3
Natural Transformations
Natural transformations are the morphisms between functors. Definition 152 If F, G : C → D are two functors, a natural transformation t : F → G is a system of morphisms t(c) : F (c) → G(c) in D, for each object c in C, such that for every
G.1. CATEGORIES INSTEAD OF SETS
1119
morphism f : x → y in C, we have G(f )◦t(x) = t(y)◦F (f ). One can also rephrase this property by requiring the following commutative diagram in D: t(x)
F (x) −−−−→ G(x) G(f ) F (f )y y
(G.1)
t(y)
F (y) −−−−→ G(y) Natural transformations can be composed in an evident way, and the composition is associative. For every functor F we have the natural identity IdF .We therefore have the category F unc(C, D) of functors F : C → D and natural transformations N at(F, G) between two functors F, G : C → D. Properties between such functors are said to be natural if they relate to the ∼ category F unc(C, D), for example, F → G is a natural isomorphism iff it is an isomorphism among the natural transformations from F to G. If two categories C, D satisfy the following properties, they are called equivalent, equivalence is an equivalence relation which is weaker than isomorphism. Lemma 88 For categories C, D the following properties are equivalent: ∼
∼
(i) There are two functors F : C → D, G : D → C such that G ◦ F → IdC and F ◦ G → IdD , where these isomorphisms are natural. (ii) There is a functor F : C → D which is fully faithful and essentially surjective, i.e., every object (identity) in 0 D is isomorphic to an image F (c) of an object of C. Example 92 If C is a category with sets as hom collections Hom(x, y), we have two types of hom functors as follows: For fixed object x, we have the functor Hom(x, ?) : C → Sets : y 7→ Hom(x, y), which sends a morphism f : y → z to Hom(x, f ) : Hom(x, y) → Hom(x, z) : u 7→ f ◦ u. We further have the contravariant functor Hom(?, y) : Copp → Sets : x 7→ Hom(x, y), which sends a morphism f : x → z to Hom(f, y) : Hom(z, y) → Hom(x, y) : u 7→ u ◦ f . The category F unc(Copp , Sets) of contravariant set-valued functors on C is denoted by C@ ; its elements are called (set-valued) presheaves over C. In the theory of denotators, one works with Mod@ and for a module M , we have the notation HomMod (M, ?) = M @, whereas the contravariant hom functor is HomMod (?, M ) = @M . Example 93 For two categories C and D and an object S of D, we have the constant functor [S] : C → D with [S](X) = S and [S](f ) = IdS for all X ∈ Ob(C) and all f ∈ M or(C). In particular, if S is a set, then we write [S] for the constant functor in Mod@ if the contrary is not stressed. Given a quiver G, if we fix an object c in a category C, we have the constant diagram ∆c = [c]. It associates every vertex of G with c and every arrow with Idc . For a diagram ∆ in C, a natural transformation [c] → ∆ is called a cone on ∆, whereas a natural transformation ∆ → [c] is called a cocone on ∆. In a cone, all arrows starting from c must commute with the arrows of the diagram, whereas in a cocone all arrows arriving at c must commute with the arrows of the diagram.
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G.2
APPENDIX G. CATEGORIES, TOPOI, AND LOGIC
The Yoneda Lemma
For a given category C with sets as hom collections Hom(x, y), the Yoneda embedding Y is the functor Y : C → C@ : x 7→ Y (x) = Hom(?, x) (G.2) with the natural transformations Y (f : x → y) : Y (x) → Y (y) being defined by u 7→ f ◦ u for u : z → x ∈ Y (x)(z) = Hom(z, x). For C = Mod, we also write Y =?@, i.e., Y (M ) = @M for a module M . ∼ A functor F in C@ is called representable iff there is an object c in 0 C such that F → Y (c). Yoneda’s lemma states that the full subcategory of representable functors in C@ is equivalent to C, and that such an equivalence is given by the Yoneda embedding. More precisely: Lemma 89 For every functor F in C@ and object c in 0 C, the map : N at(Y (c), F ) → F (c) : h 7→ h(c)(Idc )
(G.3)
is a bijection. The proof is an easy exercise, but see also [198, 313]. In particular, if F = Y (d), we ∼ have a bijection : N at(Y (c), Y (d) → Hom(c, d). More precisely, this means that the Yoneda functor Y is fully faithful, so we obtain an equivalence of categories as announced. For C = Mod, we also write F (A) = A@F , even if F is not representable. We then have the bijection ∼ N at(@A, F ) → A@F . This means that the evaluation of F at “address” A is the same as the calculation of the morphisms from @A to F . This is a justification of the name “address” for the argument A: Evaluating F at A means “observing F under all morphisms” when being “positioned on (the functor @A of) A”. And the Yoneda philosophy means that F is known, when it is known while observed from all addresses.
G.2.1
Universal Constructions: Adjoints, Limits, and Colimits
Definition 153 We suppose that for two categories C, D, the hom collections are sets. Given two functors F : C → D, G : D → C, we say that C is left adjoint to D or (equivalently) that G is right adjoint to F , in signs F a G iff the functors HomD (F (?), ?) : Copp × D → Sets and HomC (?, G(?)) : Copp × D → Sets are isomorphic. One also writes this fact in these symbols: c → G(d) F (c) → d meaning that morphisms in the numerator correspond one-to-one to morphisms in the denominator. In particular, if we are given an adjoint pair of functors F a G, when fixing the variable d in D, the adjointness isomorphism means that the contravariant functor c 7→ HomD (F (c), d) is representable by the object G(d).
G.2. THE YONEDA LEMMA
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Example 94 For C = D = Sets, fix a set A. We have the functors A×? : Sets → Sets : X 7→ A × X and ?A : Sets → Sets : X 7→ X A , which are an adjoint pair A×? a?A via the isomorphism that sends f : A×X → B to f a: X → B A : f a (x)(a) = f (a, x). This adjointness property is crucial in the definition of exponential objects in topoi. The “exponential set” B A represents the functor X 7→ Hom(A × X, B). See section G.3.2 for this subject. A terminal object 1 in a category C is one that admits exactly one morphism, denoted by ! : x → 1 from each object x of C. An initial object 0 is a terminal object in the opposite category. For example, in Sets, every singleton, such as 1 = {0}, is a terminal object, while the empty set 0 is initial. Example 95 A terminal object in C@ is defined by the constant 1C@ = [1] (of the set 1). For a presheaf P ∈ C@ , a global section γ is a natural transformation γ : 1C@ → P . In other words, the global sections Γ correspond to the hom functor Γ(P ) = N at(1C@ , P ). The global section functor Γ : C@ → Sets is right adjoint to the constant functor [ ] : Sets → C@ . Universal objects in categories, such as limits and colimits, are related to terminal or initial objects as follows. Given a “basis” object b of C, the comma category C/b has all morphisms f : x → b as objects, and for two objects f : x → b, g : y → b, we have HomC/b (f, g) = {u|g ◦ u = f }, the set of “commutative triangles above b” with the evident composition. The cocomma category C/opp b is the comma category (Copp /b)opp , in other words, for two objects f : b → x, g : b → y, we have HomC/opp b (f, g) = {u|u ◦ f = g}. Given a quiver Q and a category C, we have in the category F unc(P (Q), C) of diagrams in C, and given such a diagram ∆, the comma category F unc(P (Q), C)/∆. In this category, take the full subcategory cones(∆) of cones [c] → ∆. Then a limit of ∆ is a terminal object lim(∆) in cones(∆). Since terminal objects are evidently unique up to isomorphisms, a limit is also unique up to isomorphism. A colimit colim(∆) of a diagram ∆ is an initial object in the subcategory cocones(∆) of cocones on ∆ in the cocomma category F unc(P (Q), C)/opp ∆. If the diagram is a pair f : a → c, g : b → c, the limit is called the fiber product or pullback of f, g, or (more sloppily) of a and b if f, g are clear; it is denoted by a ×c b. If the diagram is a pair f : c → a, g : c → b, the colimit is called the fiber sum or pushout of f, g, or (more sloppily) of a and b if f, g are clear; it is denoted by a tc b. The limit of two isolated objects a, b (discrete diagram with two points) is called the (cartesian) product of a, b and denoted by a × b. The colimit of two isolated objects a, b is called the (disjoint) sum of a, b and denoted by a t b. Theorem 60 The category of sets Sets has arbitrary limits and colimits. For a category C with sets as hom collections, the category C@ of presheaves over C has arbitrary limits and colimits. Q Proof. If ∆ is a diagram of set morphisms fi,j,k : Xi → Xj , the limit is the subset in i Xi consisting of all families (xi ) such that for any pair (xi , xj ) ∈ Xi × Xj and any fi,j,k , we have
1122
APPENDIX G. CATEGORIES, TOPOI, AND LOGIC
fi,j,k (xi ) = xj . The projections lim(∆) → Xi are the restrictions of the canonical projections ` from the product to Xi . The colimit is the set colim(∆) of equivalence classes i Xi / ∼ defined by the equivalence relation generated by the relation xi`∼ xj iff there is fi,j,k (xi ) = xj . The morphisms Xi → colim(∆) are the injections Xi → i Xi , followed by the quotient map ` X → colim(∆). The universal properties are immediate and left as an exercise. i i For a diagram ∆ of presheaves Fi , we take for each argument c in C the set-theoretic limit or colimit, respectively, of the set diagram ∆(c) of the sets Fi (c) and the corresponding maps to define the limit or colimit of ∆, respectively, QED. The following proposition makes sure that the category of presheaves over C is not too large with respect to its Yoneda embedding of C: Proposition 97 Every presheaf F in C@ is a colimit of representable presheaves. See [314, pp.41/42] for a proof. The idea of this proof uses the so-called category of elements F of a functor F . Its objects are all pairs (C, p) where C is an object of C, and p ∈ C@F . The morphisms (C, p) → (C 0 , p0 ) are the morphisms u : C → C 0 in C such that p0 .u = p. R
C
Definition 154 A category is called hfinitelyi (co)complete iff it has (co)limits for all hfinitei diagrams hdiagrams with finitely many objects and arrowsi. Proposition 97 turns out to make the Yoneda embedding into a universal device for making a category C cocomplete: Proposition 98 For each functor f : C → E to a cocomplete category E, there exists an essentially unique colimit preserving functor L : C@ → E such that f = L ◦ Y . See [314, p.43] for a proof.
G.2.2
Limit and Colimit Characterizations
Proposition 99 For any category C, the following statements are equivalent: (i) C is finitely complete. (ii) C has finite products and equalizers4 . (iii) C has a terminal object and fiber products. For a proof, see [481, I, 7.8.8]. Proposition 100 For any category C, the following statements are equivalent: (i) C is finitely cocomplete. (ii) C has finite sums and coequalizers5 . (iii) C has an initial object and fiber sums. 4 An 5A
equalizer is a limit of a pair f, g : x ⇒ y of arrows. coequalizer is a colimit of a pair f, g : x ⇒ y of arrows.
G.2. THE YONEDA LEMMA
1123
This is just the dual statement of proposition 99. Proposition 101 Let C be a finitely complete category. A morphism f : A → B in C is mono iff the canonical projections p1 and p2 in the pullback p1
X −−−−→ p2 y
A f y
(G.4)
f
A −−−−→ B coincide and are isomorphisms. Proof. Clearly, if f is mono, then X = A and p1 = p2 = 1A define a fiber product. Conversely, if p1 = p2 is an isomorphism, then any couple u, v : Z → A with f · u = f · v creates factorizations u = p1 · t and v = p2 · t through t : Z → X which therefore also coincide. QED. Therefore we have the dual result: Corollary 28 Let C be a finitely cocomplete category. A morphism f : A → B in C is epi iff the canonical morphisms i1 and i2 in the pushout f
A −−−−→ fy
B i y1
(G.5)
i
B −−−2−→ X coincide and are isomorphisms. Proposition 102 For any category C, let f : H → G be a morphism in C@ . Then: (i) The morphism f is mono iff A@f : A@H → A@G is injective for all objects A of C. (ii) The morphism f is epi iff A@f : A@H → A@G is surjective for all objects A of C. (iii) The morphism f is iso iff it is mono and epi iff A@f : A@H → A@G is bijective for all objects A of C. Proof. Observe that C@ is finitely complete and cocomplete and that limits and colimits are calculated pointwise. Let us first look at point (iii). Clearly, f is iso iff its evaluations A@f : A@H → A@G are all bijective. Further, we know from proposition 101 that f is mono iff the fiber product projections p1 and p2 coincide and are iso. But with (iii) this is true iff this is true for all evaluations at objects A of C, i.e., iff this is true set-theoretically, and this means having an injection for every object A of C, and (i) is done; the dual argument shows (ii). Finally, iso always implies mono and epi; conversely, mono and epi means being in- and surjective, i.e., bijective at every object A of C, whence f is iso. QED.
1124 G.2.2.1
APPENDIX G. CATEGORIES, TOPOI, AND LOGIC Special Results for Mod@
Lemma 90 Let H be a functor in Mod@ , M, N be two addresses, and f : N → M a morphism of addresses. Then the map M 7→ M @F in(H) := {F ⊂ M @H, card(F ) < ∞},
(G.6)
together with the maps f @F in(H) : M @F in(H) → N @F in(H) : X 7→ f @H(X),
(G.7)
defines a functor F in(H) in Mod@ . Lemma 91 For any functor H in Mod@ and address M , the maps singH (M ) : M @H → M @F in(H) : x 7→ {x}
(G.8)
defines a monomorphism singH : H F in(H) of functors. Lemma 92 The map F in : Mod@ → Mod@ : H 7→ F in(H)
(G.9)
@
defines an endofunctor on Mod , and the monomorphism sing defines a natural transformation sing : IdMod@ F in.
(G.10)
Lemma 93 Let D = H0 → H1 → H2 ... be a natural sequence diagram in Mod@ . Then we f0
f1
have
∼
colim(F in(D)) → F in(colim(D).
(G.11)
This yields an important proposition for the construction of circular forms and denotators. Proposition 103 Let H be a functor in Mod@ . Then there are functors X and Y in Mod@ such that ∼
X → F in(H × X) and ∼
Y → H × F in(Y ).
(G.12) (G.13)
Proof. For the first isomorphism, let (Xn )0≤n be the following sequence of functors in Mod@ . We recursively define ∅ for n = 0, Xn = (G.14) F in(H × X for n > 0. n−1 ) Then we have a diagram of subfunctors fn : Xn ,→ Xn+1
(G.15)
G.3. TOPOI
1125
for all 0 ≤ n. In fact, clearly X0 ,→ X1 . Now, let 0 < n and take an address M . We have M @Xn = M @F in(H × Xn−1 ) = F in(M @H × M @Xn−1 ). Since by induction Xn−1 ,→ Xn , we have M @Xn−1 ⊂ M @Xn and hence M @Xn ⊂ M @Xn+1 . Now, we know from [481] that the product commutes with the colimit over a sequence diagram. Taking the diagram D = X0 → f0
X1 → X2 ... and setting X = colim(D), lemma 93 yields f1
F in(H × X) =
(G.16) ∼
F in(H × colim(D)) → ∼
F in(colim(H × D)) → ∼
colim(F in(H × D)) → colim(D) = X and we are done for the first isomorphism. For the second, take ∅ for n = 0, Yn = H × F in(Y for n > 0. n−1 )
(G.17)
(G.18)
We again have a diagram E = Y0 → Y1 → Y2 ... of subfunctors g0
g1
gn : Yn ,→ Yn+1
(G.19)
for all 0 ≤ n. Setting Y = colim(E), our second isomorphism results: H × F in(Y ) =
(G.20) ∼
H × F in(colim(E)) → ∼
H × colim(F in(E)) → ∼
colim(H × F in(E)) → colim(E) = Y
(G.21)
and we are done.
G.3
Topoi
Topoi are special categories which imitate the crucial constructions of set theory, such as cartesian products, disjoint unions, power sets, and characteristic maps. In our context, topoi play two roles: (1) the role of basic mathematical realities which are instantiated to get off ground in denotator theory, i.e., to build compound concept spaces and their points; (2) the more technical role of topoi of sheaves associated with presheaves for Grothendieck topologies.
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APPENDIX G. CATEGORIES, TOPOI, AND LOGIC
G.3.1
Subobject Classifiers
Definition 155 Given a category C which is finitely complete, with the terminal object 1, a monomorphism true : 1 Ω in C is called a subobject classifier iff given any monomorphism σ : S X in C, there is a unique morphism χσ : X → Ω such that the diagram σ
S −−−−→ !y
X χσ y
(G.22)
true
1 −−−−→ Ω is a pullback. Subobject classifiers are unique up to isomorphism. If a subobject classifier exists, the ∼ morphism χ must be the same if we replace σ by σ ◦ q for any isomorphism q : S 0 → S since an isomorphic object to a pullback is also a pullback. A subobject of X is an equivalence class of monomorphisms σ : S X under the relation σ ∼ σ 0 iff there is an isomorphism q such that σ 0 = σ ◦ q. Suppose that the collection of subobjects of X is a set SubC (X) for each object X in C. Then this is a presheaf in C@ by this map: take a morphism f : Y → X. Then we define SubC (f ) : SubC (X) → SubC (Y ) : σ 7→ σf where σf : S ×X Y → Y is the canonical projection of the pullback under f, σ. It is straightforward that this is a monomorphism. Then we have: Proposition 104 A category C which is finitely complete and such that the subobject presheaf ∼ SubC is defined, has a subobject classifier iff the SubC is representable, SubC (X) → Hom(X, Ω) for all X. If so, the subobject classifier can be set to the inverse image Ω of IdΩ in SubC (Ω). See [314, p.33] for a proof. Example 96 In the category Sets, the ordinal number inclusion true : 1 2 = {0, 1} : 0 7→ 0 is a subobject classifier. Since the subobjects of a set X in Sets identify to the subsets S ⊆ X, we have the classical result that subsets S of X are characterized by their characteristic maps χS : X → 2, a fact that is also traced in the notation 2X for the set of subsets, the powerset of X. Example 97 The equivalence classes of monomorphisms of presheaves S X in C@ are ∼ defined by their images Im(S) ⊆ X (take everything pointwise). So subC@ (X) → {S ⊆ X}, the set of subfunctors of X (supposing that it exists as a set). By the Yoneda lemma, if a subobject ∼ ∼ classifier in C@ exists, we must have6 SubC@ (@Y ) → Hom(@Y, Ω) → Ω(Y ). So the functor Y 7→ SubC@ (@Y ) is a canonical candidate for Ω, and it in fact does the job, see [314, pp.37/38]. The final presheaf being the constant presheaf 1C@ : X 7→ 1, we get the true morphism (natural transformation) true(0) = @Y . A subfunctor of @Y is called a sieve in Y , so a candidate for the subobject classifier is the functor of sieves (verify that it is a functor!). 6 Writing
the shorter @Y instead of Hom(?, Y ).
G.3. TOPOI
1127
Exercise 90 The categories Ab of abelian groups or ModR of R-modules have no subobject classifiers. In denotator theory, sieves and more general subfunctors replace local compositions (which are essentially subsets of ambient modules) in the functorial setup. This is also necessitated since module categories are no topoi (since they have no subobject classifiers, see definition 156 in appendix G.3.3), so the passage to the presheaves over modules, i.e., the category Mod@ is mandatory in order to recover the subobject classifier structure.
G.3.2
Exponentiation
Recall example 94 in appendix G.2.1 of exponential sets. More generally, a category C is called cartesian closed iff it has finite products7 and each element A is exponentiable, which means that the functor A×? has a right adjoint ?A , i.e., we have an adjoint pair of functors A×? a?A . Example 98 The category of sets Sets is cartesian closed. And any product of cartesian closed categories is cartesian closed. Example 99 A category of presheaves C@ is cartesian closed by the following discussion. Again, we use the Yoneda lemma to find a canonical candidate of the exponentiation X Y of ∼ ∼ two presheaves X, Y . If the exponential X Y exists, we must have U @X Y → N at(@U, X Y ) → N at(@U × Y, X). So one canonical definition must be U @X Y = N at(@U × Y, X)
(G.23)
for any object U in C, which is evidently a presheaf. The proof that this formula does the job is found in [314, p.47]. In every cartesian closed category C one has these standard formulas ∼
∼
∼
∼
1X → 1, X 1 → X, (Y × Z)X → Y X × Z X , X Y ×Z → (X Y )Z , which follow from the universal adjointness property of exponentiation.
G.3.3
Definition of Topoi
There are several equivalent definitions of a topos, which we first summarize in the following proposition: Proposition 105 For a category C, the following group properties are equivalent: 1. (a) C is cartesian closed, (b) C has a subobject classifier 1 Ω. 2. (a) C is cartesian closed, 7 Equivalently:
binary products and a terminal object.
1128
APPENDIX G. CATEGORIES, TOPOI, AND LOGIC (b) C is finitely cocomplete, (c) C has a subobject classifier 1 Ω.
3. (a) C has a terminal object and pullbacks, (b) C has exponentials, (c) C has a subobject classifier 1 Ω. 4. (a) C has a terminal object and pullbacks, (b) C has an initial object and pushouts, (c) C has exponentials, (d) C has a subobject classifier 1 Ω. 5. (a) C is finitely complete, (b) C has power objects8 , Definition 156 A category C which has the equivalent groups of properties in proposition 105 is called a (elementary) topos. Here are some general properties and examples of topoi: Proposition 106 For a topos C a comma category C/b is also a topos. Proposition 107 For a category C the presheaf category C@ is a topos. This is immediate from the previous discussion of the presheaf category. Proposition 108 Let C be a topos. Then we have these properties: (i) Every morphism f has an image, i.e., factors as f = i ◦ e with i mono and e epi. For any two such factorizations f = i ◦ e, f = i0 ◦ e0 , there is an isomorphism t such that e0 = t ◦ e, i0 = i ◦ t. (ii) A morphism is iso iff it is mono and epi. (iii) The pullback of an epi is an epi. (iv) Every arrow X → 0 is iso. (v) Every arrow 0 → X is mono. Definition 157 Logical morphisms between topoi are functors which preserve (up to isomorphism) finite limits, exponentials, and subobject classifiers. For example, the canonical base change functor C/b → C/c of comma topoi for a base change morphism c → b is logical, see [314, p.193]. 8 See
[186, p.106] for this group of properties
G.4. GROTHENDIECK TOPOLOGIES
G.4
1129
Grothendieck Topologies
Grothendieck topologies and associated topoi of sheaves are a classical example for the geometric aspects of topoi. Here is the context. Given a finitely complete category (a small one for those who like universes) C with the subobject classifier functor of sieves X@Ω = SubC@ (@X) (see example 97). Recall that given a morphism f : Y → X the functor maps a sieve S ⊆ X to the pullback sieve f ∗ (S) = S ×X Y . Definition 158 A Grothendieck topology on a category C is a function J which for each X is a subset X@J ⊆ X@Ω of sieves in X with these properties: (i) @X ∈ X@J, (ii) (Stability) If S ∈ X@J, then for f : Y → X, f ∗ S ∈ Y @J, (iii) (Transitivity) If S ∈ X@J and R ∈ X@Ω with f ∗ R ∈ Y @J for all f : Y → X in S, then R ∈ X@J. A site is a pair (C, J) of a Grothendieck topology J on a category C. A sieve in X@J is called a covering sieve, one also says that “it covers X”. The first two requirements mean that J is a subfunctor of Ω through which the true arrow factorizes. Very often, Grothendieck topologies are not given directly, but via a so-called basis: Definition 159 For a finitely complete category C, a basis (for a Grothendieck topology) is a function K which assigns to each object X a collection K(X) of families of morphisms with codomain X such that: ∼
(i) For every isomorphism f : X 0 → X, the singleton {f } is in K(X); (ii) (Stability) If (fi : Xi → X) ∈ K(X), and h : Y → X, then (h∗ fi : Xi ×X Y → Y ) ∈ K(Y ); (iii) (Transitivity) If (fi : Xi → X) ∈ K(X) and, for each index i, (fij : Xij → Xi ) ∈ K(Xi ), then (fi ◦ fij : Xij → X) ∈ K(X). A pair (C, K) is again called a site (see below for a justification!); whereas the families in the sets K(X) are called covering families. Here is the relation to Grothendieck topologies: Given a basis K as above, one defines JK (X) = {S| there is R ∈ K(X) with R ⊆ S}, (G.24) S where R ⊆ S means that R is in the union of the evaluations C Z@S of S. And the converse: Given a family R of morphisms with codomain X, we denote by (R) the sieve generated by R, i.e., the smallest sieve in X containing all arrows of R. Then a Grothendieck topology J can be defined by the following basis K which is this set at X: K(X) = {R ⊆ @X|(R) ∈ X@J}.
(G.25)
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APPENDIX G. CATEGORIES, TOPOI, AND LOGIC
G.4.1
Sheaves
Definition 160 Given a site (C, J), a presheaf P in C is a sheaf for J iff for every covering ∼ sieve S ⊆ @X, the inclusion induces a bijection N at(@X, P ) → N at(S, P ). This condition can be rephrased for a basis K of J in a more effective and classical way. To this end recall that a (co)equalizer of a pair f, g : x ⇒ y of parallel arrows is the (co)limit of this diagram. Proposition 109 A presheaf P on C is a sheaf for the topology J, iff for any covering family (fi : Xi → X) ∈ K(X), the canonical diagram Y Y P (X) → P (Xi ) ⇒ P (Xi ×X Xj ) (G.26) i
i,j
is an equalizer. Here the two arrows to the right stem from the two projections from Xi ×X Xj to Xi and to Xj , whereas the arrow to the left stems from the covering morphisms fi . Definition 161 A Grothendieck topos is a category which is equivalent to the full subcategory Sh(C, J) of sheaves in C@ The justification of this terminology lies in the following theorem: Theorem 61 A Grothendieck topos is an elementary topos. The proof evidently splits in the verification of finite completeness, existence of exponentials and of a subobject classifier. For details, see [314, pp.128-144]. Finite completeness is easy since: Lemma 94 Limits of sheaves are sheaves. Lemma 95 If P is a presheaf and F is a J-sheaf in C@ , then the presheaf exponential F P is a sheaf and therefore an exponential in Sh(C, J). Definition 162 A sieve L ⊆ @X is closed with respect to (C, J) iff f : Y → X with f ∗ (L) ∈ Y @J implies f ∈ Y @L. Proposition 110 The function X 7→ X@ΩSh = {closed sieves in X} ⊂ X@Ω contains @X defines a subpresheaf of the subobject classifier of C@ and is a sheaf. Together with the morphism true : 1C@ → ΩSh it defines a subobject classifier of Sh(C, J). The topos Sh(C, J) is a subtopos of C@ with the natural inclusion i : Sh(C, J) → C@ . This natural transformation has a left adjoint of sheafification which we shall discuss now. If P is a presheaf over C, the sheafification operator P 7→ P + evaluates as follows. For an object Q X ∈ C and a sieve S ∈ X@J, consider the limit M atchP (S) = f :Y →X∈S Y @P . Consider the diagram (M atchP (S))S∈X@J with canonical restriction maps M atchP (S) → M atchP (T ) for
G.5. FORMAL LOGIC
1131
T ⊆ S, and define X@P + = limS∈X@J M atchP (S). For a morphism g : X1 → X2 , we have a map P + (g) : X2 @P + → X1 @P + . It takes a “matching family” (xf )f ∈S to the matching family (xg.h )h∈g? S . This evidently defines a presheaf, and we have a canonical morphism η : P → P + . Then: Theorem 62 With the above notation, we have (i) The presheaf P + is separated9 . (ii) The presheaf P is separated iff η is mono. (iii) The presheaf P is a sheaf iff η is iso. For a proof, see [314, III.5]. In particular, the double application aP = ((P )+ )+ yields a sheaf and a natural presheaf morphism P → aP . We have Theorem 63 The map P 7→ aP defines a left adjoint of the inclusion i; a a i. The composition a ◦ i is isomorphic to the identity on the sheaf category Sh(C, J). For a proof, see [314, III.5, Theorem 1] and [314, III.5, Corollary 6]. Corollary 29 If f : F → G is a morphism of sheaves, f is mono iff it is an injection for each argument. The proof follows from the fact that this is true for presheaves, and that by the adjunction theorem 63, i preserves and reflects10 monomorphisms, QED.
G.5
Formal Logic
Formal logic does not replace absolute logic which is built upon the non-formalizable theorem of identity (A is identical to A), of contradiction (A and non-A exclude each other), and of the excluded third (there is no third choice except A or non-A). It does however model the way a specific domain of knowledge can handle its formal truth mechanisms.
G.5.1
Propositional Calculus
Sentences in propositional calculus are defined from a set Φ = {π0 , π1 , . . .} of symbols, called propositional variables; a set Ξ = {!, &, |, −>} of logical connective symbols11 (! for negation, & for conjunction, | for disjunction, and −> for implication); a set ∆ = {(, )} of two brackets; mutually disjoint from each other. Let EX = F M (Φ ∪ Ξ ∪ ∆) be the free monoid of word expressions above these symbols. Let S(EX), the set of sentences, be the smallest subset of EX with these properties: 9 The
left arrow in equation (G.26) for P + is injective. functor f reflects a property of morphisms if the property for the image morphism f (x) implies the property for x. 11 Our symbols are near to programming symbols, where the formalism is really needed. 10 A
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APPENDIX G. CATEGORIES, TOPOI, AND LOGIC
Property 4 Given the symbol sets Φ, Ξ, ∆, we require: (i) Φ ⊂ S(EX); (ii) if α ∈ S(EX), then (!α) ∈ S(EX); (iii) if α, β ∈ S(EX), then (α)&(β) ∈ S(EX); (iv) if α, β ∈ S(EX), then (α)|(β) ∈ S(EX); (v) if α, β ∈ S(EX), then (α) −> (β) ∈ S(EX). Clearly, in S(EX), the building blocks of a sentence are uniquely determined, so it makes sense to define set-valued functions : S(EX) → A on such sentences by recursion of the building blocks. Suppose that A is a lattice, i.e., a partially ordered set (A, ≤) with a join operation ∨ : A × A → A, a meet operation ∧ : A × A → A, minimum (False) ⊥, a maximum (True) >, further a unary negation operation ¬ : A → A, and a binary implication operation ⇒: A × A → A. Call such an A a logical algebra. Then any set function 0 : Φ → A extends in a unique way to the evaluation = (0 ) : S(EX) → A by these rules: Property 5 For all sentences α, β, we set (i) (!α) = ¬(α); (ii) ((α)&(β)) = (α) ∧ (β); (iii) ((α)|(β)) = (α) ∨ (β); (iv) ((α) −> (β)) = (α) ⇒ (β). Propositional calculus deals with the evaluation map on special logical algebras. A sentence α is called A-valid, in symbols: A α, iff it (α) = > for all evaluations 0 : Φ → A on the propositional variables. It is called classically valid or a tautology iff it is 2-valid for the wellknown Boolean algebra 2 = {0, 1} of classical truth values, where we set > = 1, ⊥ = 0. The symbol for classical validity is α. Here are typical classes of logical algebras: Boolean Algebras. A Boolean algebra is a distributive logical algebra such that x ∨ ¬x = > and x ∧ ¬x = ⊥. Distributivity means that x ∧ (y ∨ z) = x ∧ y ∨ x ∧ z and x ∨ (y ∧ z) = x ∧ y ∨ x ∧ z. Further, implication is defined by x ⇒ x = ¬x ∨ y. In a Boolean algebra (BA), one has these properties: ¬¬x = x, x ∧ y = ⊥ iff y ≤ ¬x, x ≤ y iff ¬y ≤ ¬x, ¬(x ∧ y) = ¬x ∨ ¬y, ¬(x ∨ y) = ¬x ∧ ¬y. Heyting Algebras. A Heyting algebra A is a partially ordered set which, as a category whose morphisms x → y are the pairs x ≤ y, has all finite products and coproducts, and which has exponentials, so it is cartesian closed. In other words, a Heyting algebra is a lattice with minimum ⊥ and maximum > which has exponentials xy . One writes the product as meet (∧) and the coproduct as join (∨). The exponential xy is written as y ⇒ x, and the adjunction property of exponentiation reads z ≤ y ⇒ x iff z ∧ y ≤ x.
(G.27)
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For a Heyting algebra, we define a negation ¬x = x ⇒ ⊥, which is equivalent to y ≤ ¬x iff y ∧ x = ⊥. Proposition 111 For a Heyting algebra, we have these identities: x ≤ ¬¬x, x ≤ y implies ¬y ≤ ¬x, ¬x = ¬¬¬x, ¬¬(x∧y) = ¬¬x∧¬¬y, (x ⇒ x) = >, x∧(x ⇒ y) = x∧y, y ∧ (x ⇒ y) = y, x ⇒ (y ∧ z) = (x ⇒ y) ∧ (x ⇒ z). Proposition 112 A Heyting algebra is distributive, and it is Boolean iff x = ¬¬x for all x, or iff x ∨ ¬x = > for all x. Proposition 113 For a presheaf category C@ , the partially ordered set SubC@ (P ) of an object P is a Heyting algebra. The connectives are defined as follows. If S, T are two subfunctors of P , then: (i) X@(S ∨ T ) = X@S ∪ X@T ; (ii) X@(S ∧ T ) = X@S ∩ X@T ; (iii) (S ⇒ T )(X) = {x ∈ X@P | for every morphism f : Y → X, if x · f ∈ Y @S, then x · f ∈ Y @T }; (iv) X@(¬S) = {x ∈ X@P | for every morphism f : Y → X, x · f 6∈ Y @S}; More generally (see [314, IV.8] for details): ∼
Theorem 64 For every topos C, the partially ordered set Sub(X) → Hom(X, Ω) of subobjects of X is a Heyting algebra. To the left, this structure stems from the canonical Heyting algebra structure on the subobjects of X. To the right, this structure is induced by the following operations on Ω: 1. Negation ¬ : Ω → Ω is the characteristic map of the false arrow f alse : 1 Ω, which is the characteristic map of the zero arrow 0 1. 2. Disjunction ∧ : Ω × Ω → Ω is the character of the diagonal morphism ∆(true, true) : 1 → Ω. 3. Conjunction ∨ : Ω × Ω → Ω is the character of the image of the universal morphism a t b : Ω t Ω → Ω × Ω, where a = ∆(IdΩ , true) : Ω → Ω × Ω and b = ∆(true, IdΩ ) : Ω → Ω × Ω. 4. Implication ∨ : Ω × Ω → Ω is the character of the equalizer of p1 , ∧ : Ω × Ω ⇒ Ω. If a sentence α is valid for all Boolean algebras, we write BA α. If it is valid for all Heyting algebras, we write HA α. If it is valid in the Heyting algebra 1@Ω of a topos C, we write C α. Validity is also described by a recursive construction process of valid sentences. One gives a set AX of sentences, called axioms, and defines theorems as those sentences s which are at the end of proof chains, i.e., finite sequences of sentences (s0 , s1 , . . . sn , s) such that each member of this sequence is either an axiom or can be inferred from earlier members by a set RU LES of rules. The classical setup is this. AX consists of 12 types of sentences:
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APPENDIX G. CATEGORIES, TOPOI, AND LOGIC
Axiom 7 Axioms of classical logic (CL): (i) α −> (α&α) (ii) (α&β) −> (β&α) (iii) (α −> β) −> ((α&γ) −> (β&γ)) (iv) ((α −> β)&(β −> γ)) −> (α −> γ) (v) β −> (α −> β) (vi) (α&(α −> β)) −> β (vii) α −> (α|β) (viii) (α|β) −> (β|α) (ix) ((α −> β)&(β −> γ)) −> ((α|β) −> γ) (x) (!α) −> (α −> β) (xi) ((α −> β)&(α −> (!β))) −> (!α) (xii) α|(!α) The system CL has one single rule of inference: Principle 29 (Modus ponens) From α and α −> β, β may be derived. The property of a sentence α of being a CL-theorem is denoted by |—— α. Following CL
Heyting, the intuitionistic logic (IL) is the (CL) with the axiom (xii) omitted, and the same inference rule. Theorem 65 The following validity statements are equivalent: (i) |—— α. CL
(ii) α. (iii) There exists a Boolean algebra B such that B α. (iv) BA α. See [186] for details. We have this weaker relation: Theorem 66 The topos validity C α implies classical validity |—— α. CL
Definition 163 A topos is Boolean iff the Heyting algebra Sub(X) of each object X is Boolean. Theorem 67 The following statements for a topos C are equivalent:
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(i) C is Boolean. (ii) Sub(Ω) is a Boolean algebra. (iii) true : 1 → Ω has a complement in Sub(Ω). (iv) f alse : 1 → Ω is the complement of true in Sub(Ω). (v) true ∪ f alse = IdΩ in Sub(Ω). (vi) C is classic, i.e., true t f alse → Ω is iso. (vii) The first inclusion ι1 : 1 → 1 + 1 is a subobject classifier. For a proof, see [186, p.156 ff.]. Theorem 68 If the topos C is Boolean, then C α|!α for all sentences α. Theorem 69 For a topos C, the following are equivalent: (i) C α iff |—— α for all α. CL
(ii) C α|!α for all α. (iii) Sub(1) is a Boolean algebra. Theorem 70 We have HA α iff |—— α. IL
Theorem 71 For all topoi C, the validity |—— α implies C α. IL
G.5.2
Predicate Logic
Predicate calculus generates a richer set of sentences whose validity is a function of the interpretation of predicate variables and individual variables and not only of abstract propositional variables. We are given a set Υ = {ι` 0 , ι1 , . . .} of individual variables, a set Ξ = {!, &, |, −>, ∃, ∀} of predicate connectives, a set Π = i=0,1,2,... Πi , Πi = {Ai , B i , . . .} of i-ary predicate variables, and a set ∆ = {(, )} of brackets as above. Within the free monoid P EX = F M (Υ t Ξ t Π t ∆) of predicate expressions, one exhibits the subset F O(P EX) of formulae as follows: Property 6 Given the symbols Υ, Ξ, Π, ∆, we require (i) (Atomic formulae) Ai ιi1 ιi2 . . . ιii ∈ F O(P EX) for any Ai ∈ Πi and ιik ∈ Υ, and 0-ary predicate variables are constants.
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(ii) (Propositional formulae) If α, β ∈ F O(P EX), then
(!α) ∈ F O(P EX), (α)&(β) ∈ F O(P EX), (α)|(β) ∈ F O(P EX), (α) −> (β) ∈ F O(P EX). (iii) (Quantifier formulae) If α ∈ F O(P EX), x ∈ Υ, then (∀x)α ∈ F O(P EX), (∃x)α ∈ F O(P EX). An individual variable x which appears in the formula after an expression of type (∀x) or (∃x) is bound, otherwise it is free. A model M of a predicate logic F O(P EX) is a set M , together with i-ary relations ai , bi . . . ⊆ M i (elements a0 ∈ M for constants). In such a model, the formal predicate expressions are interpreted via the interpretation of atomic formulae Ai ιi1 ιi2 . . . ιii by the truth value of (xi1 , xi2 , . . . xii ) ∈ ai . Whereas the interpretation of a quantifier formula (∀x)α means “true” if the truth value of the interpretation of α is “true” for all valuations in M of all occurrences of the variable x, and the interpretation of a quantifier formula (∃x)α means “true” if the truth value of the interpretation of α is “true” for at least one value in M of the occurrences of the variable x—of course, this is only decided if no free variables are left, in which case one calls the formula a sentence, otherwise, no truth value is defined and the formula is just a truth-valuefunction of the left free variables. We write M α[x] for truth value “true” for the evaluation [x] of the free variables of α. The recursive calculation of truth values of compound formulas relates to the Boolean algebra Sub(M ) as follows: To begin with, if a logical combination of two formulae is considered, one may suppose that both variable sets of these formulas coincide by just taking their union if they do not coincide. If we fix such a variable set of cardinality m, say, the truth evaluation of a formula α with (at most) these m free variables can be described by the inverse image supp(α) ⊆ M m of true. Then evidently, supp(!α) = M m − supp(α), supp((α)&(β)) = supp(α)∩supp(β), supp((α)|(β)) = supp(α)∪supp(β), supp((α) −> (β)) = supp(!α)∪supp(β). For the quantifiers, we have this situation: If a variable x is bound by a quantifier, we have the support supp(α) ⊆ M m of the given formula α and the support supp((∀x)α) ⊆ M m−1 or supp((∃x)α) ⊆ M m−1 , respectively. Suppose that x is the ith variable, then we have the projection pi : M m → M m−1 , which omits the ith coordinate, and the inverse image map m−1 m p∗ : 2M → 2M . If S is a support of a formula α in M m , then the support (∀x)(S) of (∀x)α is the set {(y1 , . . . , ym−1 )|(y1 , . . . yi−1 , x, yi+1 , . . . ym−1 ) ∈ S for all x ∈ M }, while the support of (∃x)(S) of (∃x)α is the set {(y1 , . . . ym−1 )|(y1 , . . . yi−1 , x, yi+1 , . . . ym−1 ) ∈ S for at least one x ∈ M }. m
m−1
is a right adjoint Proposition 114 The functor of partially ordered sets ∀x : 2M → 2M m−1 m m m−1 of p∗ : 2M → 2M , while ∃x : 2M → 2M is its left adjoint, in other words, p∗ (X) ⊂ Y iff X ⊂ ∀(x)(Y ) and Y ⊂ p∗ (X) iff ∃(x)(Y ) ⊂ X.
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The topos-theoretic generalization of this result is the following theorem; for a proof, see [314, p.209,p.206]. Theorem 72 If f : A → B is a morphism in the topos C, then the functor Sub(B) → Sub(A) of Heyting algebras (which are viewed as categories via their partial orders as morphisms) associated with the natural morphism Ωf : ΩB → ΩA has a right adjoint functor ∀x and a left adjoint functor ∃x . In order to rewrite the predicate calculus in general topoi, one uses the characteristic maps associated with supports of predicates as follows: If M is a non-zero object of the topos C, we consider the characters χam : M m → Ω of the “supports” am ⊂ M m of m-ary predicates. Their recombination via logical constructions runs as follows: Using the morphisms of negation, conjunction, disjunction, and implication defined above in section G.5.1 for Ω, one has the evident combination of supports of formulas via their characters. The new thing here is the definition of quantifier supports. Given a character χα : M m → Ω, we have the adjoint morphism adi (χα ) : M m−1 → ΩM with respect to the ith coordinate. Then we have two arrows ∀M , ∃M : ΩM → Ω. The first ∀M is the character of the adjoint of the composite true◦! ◦ prM : 1 × M → M → Ω. The second ∃M is the character of the image of the composed map prΩM ◦ εM : ε ΩM × M → ΩM , where εM : ε ΩM is the subobject whose character is the evaluation map evM : ΩM × M → Ω adjoint to the identity on ΩM . We then have these formulas: ∀(x)α has the character ∀M ◦ adi (χα ), ∃(x)α has the character ∃M ◦ adi (χα ).
G.5.3
A Formal Setup for Consistent Domains of Forms
Since forms do not automatically exist if we allow circularity, it is important to set up a formal mathematical context in order to describe what a logically consistent domain of forms should be. This mathematical formalism turns out to be valid in an interesting general context. We have been working in the topos Mod@ of presheaves over the category Mod, where we have the Yoneda embedding Y : Mod → Mod@ . Without loss of generality, we may identify Mod with the full subcategory of represented presheaves @M, M a module. More generally, we may consider Yoneda pairs R ⊂ E, where R is a full subcategory of a topos E, R playing the role of represented modules (we also say that R is a Yoneda subcategory). This means that we require that the canonical Yoneda functor E → E @ → R@ be fully faithful. By Yoneda’s lemma, we may identify the evaluation M @F of a “presheaf” F ∈ E at a “module” M ∈ R by the morphism set from M to F : M @F = HomE (M, F ). This setup in particular includes the classical case of E = Sets and R the one-element category consisting of a singleton 1 (the terminal object in Sets), say 1 = {∅}, and its unique identity morphism. In this case, we may identify 1@F and the set F . To achieve the intended formalism, we consider the set M ono(E) of monomorphisms in E. We further consider the set T ypes = {Simple, Syn, Limit, Colimit, Power} of form types. And we need the free monoid N ames = F M (U N ICODE) over the U N ICODE alphabet (which is an extension of the ASCII alphabet to non-European letters). We next need
1138
APPENDIX G. CATEGORIES, TOPOI, AND LOGIC
FORMS
Names
F sem Dia(Names/E)
sem(F) = (typeF, idF, IDF) Mono(E) Types
Figure G.1: The formal setup of a semiotic of E-forms. the set Dia(N ames) of all diagram schemes with vertexes in N ames. More precisely, a diagram scheme over N ames is a finite directed multigraph whose vertexes are the elements of a subset of N ames, and whose arrows i : A → B are triples (i, A, B), with i = 1, . . . natural numbers to identify arrows for given vertexes. Next, consider the set Dia(N ames/E) of diagrams on Dia(N ames) with values in E. Such a diagram is a map dia : D → E which to every vertex of the diagram scheme D associates an object of E and to every arrow associates a morphism in E between corresponding vertex objects. So i : A → B is mapped to the morphism dia(i) : dia(A) → dia(B). We also will identify two such diagrams iff their arrows for given names A, B are permutations of each other, i.e., we only consider the orbits of diagrams modulo the permutation group of arrows on given names. Why? Because any construction of limits or colimits is invariant under this group since the limit condition is a logical conjunction which does not depend on the numbering of the arrows. So this identification will always be valid unless explicitly suspended. Observe further that a multiple appearance of a vertex in a diagram scheme is not allowed, so when constructing diagram schemes upon form names, one must add synonymous forms when multiple appearance of one and the same form in a diagram is desired. This is the advantage of form names: the annoying indexing of mathematical names can be absorbed by intrinsic renaming on the level of form names. With these notation, we can define a semiotic of E-forms as follows (see also figure G.1): Definition 164 A semiotic of E-forms is a set map sem : F ORM S → T ypes × M ono(E) × Dia(N ames/E) defined on a subset F ORM S ⊂ N ames with the following properties (i) to (iv). To ease language, we use the following notation and terminology:
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• An element F ∈ F ORM S is called a form name, and the pair (F, sem) a form, • pr1 · sem(F ) = t(F ) (=type of F ), • pr2 · sem(F ) = id(F ) (= identifier of F ), • domain(id(F )) = f un(F ) (= topor12 or “space” of F ), • codomain(id(F )) = f rame(F ) (= frame or “frame space” of F ), • pr3 · sem(F ) = coord(F ) (= coordinator of F ). Then these properties are required: (i) The empty word ∅ is not a member of F ORM S (ii) For any vertex X of the coordinator diagram coord(F ), if X ∈ F ORM S, then we have coord(F )(X) = f un(X). (iii) If the type t(F ) is given, we have the following for the corresponding frames: • For Syn and Power, the coordinator has one vertex G ∈ F ORM S and no arrows, i.e., coord(F ) : G → f un(G), which means that in these cases, the coordinator is determined by a form name G. Further, for Syn, we have f rame(F ) = f un(G), and for Power, we have f rame(F ) = Ωf un(G) , if coord(F ) : G → f un(G), as above. • For Limit and Colimit, the coordinator is any diagram coord(F ) whose names are all in F ORM S. Further, for Limit, we have f rame(F ) = lim(coord(F )), and for Colimit, we have f rame(F ) = colim(coord(F )). • For type Simple, the coordinator has the unique vertex ∅, and a value coord(F ) : ∅ → M for a ‘module’ M ∈ R (i.e., a represented presheaf M = @X in the case of presheaves over Mod), or, in a more sloppy notation: coord(F ) = M . Here, circular forms are evidently included via form names which refer to themselves in their diagrams or in deeper recursion structures. With this definition we may discuss the existence and size of form semiotics, i.e., the extent of the F ORM S set, maximal such sets, gluing such sets together along compatible intersections, etc. However, we shall not pursue this interesting and logically essential branch for simple reasons of space and time. 12 The
functor in the special case E = Mod@ .
1140 G.5.3.1
APPENDIX G. CATEGORIES, TOPOI, AND LOGIC Morphisms Between Semiotics of Forms
Although the theory of form semiotics is in its very beginnings, it is clear that two form semiotics with intersecting domains F ORM1 and F ORM2 nee not be contradictory even if the semiotic maps do not coincide on the intersection F ORM1 ∩ F ORM2 . In fact, it could happen that on this intersection, the maps are just “equivalent” semiotics. More generally, it could happen that two form semiotics have subsemiotics which are in complete correspondence and therefore we may glue them to a global semiotic structure. In other words: It is reasonable and feasible to consider morphisms and then categories of form semiotics and therefore isomorphisms of semiotics, which enables us to construct global semiotics just by gluing together local “charts” as usual. Let us abbreviate Sema(E) = T ypes × M ono(E) × Dia(N ames/E), Sema being an abbreviation for semantic target space. Suppose that we are given two form semiotics sem1 : F ORM S1 → Sema(E1 ), sem2 : F ORM S2 → (E2 ). We correspondingly denote by f un1 , f un2 , t1 , t2 , id1 , id2 , f rame1 , f rame2 , and coord1 , coord2 the respective maps. Consider pairs (u, v) where u : F ORM S1 → F ORM S2 is a set map, and where v : E1 → E2 is a logical functor (see appendix G.3) sending R1 to R2 . We say that the pair (u, v) is morphic (for F ORM S1 , F ORM S2 ) iff 1. We have u(∅) = ∅. 2. The functors commute with u, v, i.e., we have v · f un1 = f un2 · u. 3. The type is invariant and u, i.e., t2 · u = t1 . In particular, mono- and epimorphisms on E1 are preserved (see appendix G.2.2). Suppose that we are given a diagram scheme C = coord1 (F ) : D → E1 (modulo permutations on the numberings of the arrows between fixed names, as announced!) associated with the form name F ∈ F ORM1 . Let |D| be the vertex names of D. We define a diagram scheme E as follows. Its vertexes are the image |E| = u(|D|). For every vertex pair (X, Y ) of |E| we take all arrows i : A → B with X = u(A), Y = u(B). By lexicographic order on the triples (A, B, i), we can order all these arrows and index them with positive natural numbers j = 1, . . . n(X, Y ). This defines a unique new diagram scheme. Secondly, we define a new diagram C 0 : E → E2 as follows: If the arrow i : A → B gives arrow j(i) : X → Y , the new diagram C 0 maps this arrow to the morphism v(C(i)) : f un2 (X) = v(f un1 (A)) → v(f un1 (B)) = f un2 (Y ). Denote this diagram by (u, v)(C). Clearly, since we only retain orbits of diagram schemes, we have functoriality, i.e., if (u1 , v1 ), (u2 , v2 ) are two such morphic pairs for F ORM S1 , F ORM S2 , and F ORM S2 , F ORM S3 , respectively we have (u2 , v2 )((u1 , v1 )(C)) = (u2 · u1 , v2 · v1 )(C).
(G.28)
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Definition 165 A morphic pair (u, v) for the pair F ORM S1 , F ORM S2 is said to be a morphism of form semiotics (u, v) : sem1 → sem2 if the following semiotic data with each given form name F ∈ F ORM S1 are verified: 1. Let F be simple, i.e., F −→ Simple(M ). Then u(F ) −→ Simple(N )13 , and we require 0 Id
Id
that • N = v(M ), • Id’=v(Id), i.e., the monomorphism of Id0 is the v-image of the monomorphism of Id—the domains and codomains are already the right ones, only the morphism (a monomorphism by the conservation of limits) has to fit—so Id0 = v(Id) : f un2 (u(F )) = v(f un1 (F )) N = v(M ). Syn(G0 ), and we require that 2. Let F be synonymous, i.e., F −→ Syn(G). Then u(F ) −→ 0 Id
Id
G0 = u(G) and Id0 = v(Id) = f un2 (u(F )) f un2 (u(G)). 3. Let F be of power type, i.e., F −→ Power(G). Then u(F ) −→ Power(G0 ), and we require 0 Id
Id
that G0 = u(G) and Id0 = v(Id) = f un2 (u(F )) v(Ωf un1 (G) ) ∼ → Ωv(f un1 (G) = Ωf un2 (u(G)) . 4. Let F be of limit (resp. colimit) type, i.e., F −→ Limit(C) (resp. F −→ Colimit(C)) Id
Id
with C = coord(F ). Then we have u(F ) −→ Limit(C 0 ) (resp. u(F ) −→ Colimit(C 0 )). 0 0 Id
Id
We then require that C 0 = (u, v)(C) and that Id0 = v(Id) = f un2 (u(F )) = v(f un1 (F )) ∼ ∼ v(lim(C)) → lim(v · C) → lim((u, v)(C)) (resp. the analogous expression with colimits). In any case, the associated form u(F ) is related to its ingredients through the given functor v and the recursive constructions on the coordinators via u. Clearly, the evident composition of two morphisms from formula (G.28) is again a morphism, and we obtain the category ForSem of form semiotics. G.5.3.2
Local and Global Form Semiotics
It is clear what one should understand by a global form semiotic: This is a set G, together with a ∼ covering I and an atlas fi : Ii → F ORM Si of bijections onto domains of form semiotics semii : ∼ F ORM Si → Sema such that all the induced bijections ui,j : F ORM Si |j → F ORM Sj |i extend to isomorphisms (ui,j , vi,j ) of form semiotics. This means in particular that all intersections F ORM Si |j are form domains of form sub-semiotics in semi , and that the underlying functors on Ei are compatible. We leave the details to the interested reader. 13 Observe that in the case R = Mod, E = Mod@ of presheaves, we usually write the module as a coordinator, but we mean its represented functor @M .
1142 G.5.3.3
APPENDIX G. CATEGORIES, TOPOI, AND LOGIC Connotator From Semiotics
Denotator and form names were very simple word objects in the previous setup. But name spaces may also be required to encompass more articulated structures, in other words: we want names to be denotators as well, thereby turning the denotator concept into a ‘connotator’ concept. Here is the formal setup. We again suppose given a Yoneda pair R, E. We also retain the set M ono(E). We are given two sets D of denotators and F of forms, they are supposed to parametrize denotators and forms according to the following system of maps. We have three maps on D: coordinate : D → E, f orm : D → F, denotatorN ame : D → D. The coordinate C of a denotator is supposed to be any morphism with domain A within R, which is called the denotator’s address. We require that a denotator be uniquely determined by its coordinate C, form F , and denotatorName N . This is why denotators are also written as quadruples N : A@F (C), where the address is denoted for comfort since it is important information. The denotator’s form mimics the space where the denotator lives. To this end, we need two more sets. The set of types is T = {Limit, Colimit, Power, Simple}, it contains the basic constructors of objects in a topos. But we omit synonymy in this generic setup because it can be mimicked by a limit with just one vertex. We also need the set Diagrams(D/E) of finite diagrams whose vertexes are denotators, and whose arrows are numbered by 1, 2, 3, . . . as above. This means that the diagram schemes are these symbols, and that the evaluation of the diagram scheme yields objects and morphisms in E. Forms have, by hypothesis, uniquely determined values under these four maps: f ormN ame : F → D, identif ier : F → M ono(E), diagram : F → Diagrams(D, E), type : F → T . This means that a form can be written as F N : Id.T (Dg), where F N is the form’s name denotator, Id its identifier, T its type, and Dg its diagram. We impose a small number of axioms for these structures. To this end, we call the domain dom(Id) of a form F N : Id.T (Dg) the form’s space, whereas the codomain cod(Id) is called its frame space. Accordingly, for a denotator N : A@F (C), the composition Id ◦ C with its form’s identifier Id is called the frame coordinate, it uniquely determines the denotator’s coordinate. Axiom 8 Here are the conditions for this setup: (i) The map formName is injective, i.e., the form’s name is a key. (ii) For all form diagrams, except for simple type, the vertex denotators of the diagram schemes are form names, and their values are the spaces of the respective forms.
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(iii) If the form’s type is Limit or Colimit, its frame space is the limit or colimit of the diagram. (iv) If the form’s type is Power, the diagram has just one vertex and no arrows, and the frame space is ΩS , where S is the space of the vertex form. (v) There is a denotator ∅ which is not a form name, and for a simple type form, the diagram has exactly the vertex ∅, no arrow, and the value is a ‘representable’ object X in R. In other words, the simple type frame space is just a representable object in disguise. Such a diagram is represented by ∅X. The language of forms and denotators has been encoded in an ASCII-based textual form, like TEX, which is therefore called Denotex and is available in BNF14 . In RUBATOr , a Denotex parser is available for communication with Denotex files. Our present notation in this section, such as F N : Id.T (Dg) for forms and N : A@F (C) for denotators, is an illustration of the Denotex notation. Example 100 An elementary form for names can be set up as follows: The form N F represented by f n : Id.Simple(Dg) is simple with the diagram ∅ZhU N ICODEi. The identifier Id is the identity on the representable presheaf @ZhU N ICODEi, and the name f n is a denotator f n : 0@F N (C), whose coordinate C is the zero-addressed homomorphism C : 0 → ZhU N ICODEi with value C(0) = “NameForm” with denotatorN ame(f n : 0@N F (C)) = f n, i.e., it is its proper name denotator. So its identification resides on its coordinate value “NameForm” and the form named f n. This identifies the entire N F form. Then, general U N ICODE names n may be defined by n : 0@N F (Cn), where the value Cn(0) = “anyName” is any U N ICODE string combination, such as “3.Violin+4.Piano”, and which are their proper name denotators, i.e., denotatorN ame(n : 0@N F (C)) = n.
14 Denotex
was developed in collaboration with Thomas Noll, J¨ org Garbers, Stefan G¨ oller, and Stefan M¨ uller.
Appendix H
Complements on General and Algebraic Topology H.1
Topology
Refer to [261] for general topology, and to [498] for algebraic topology.
H.1.1
General
A topological space is a pair (X, OpenX ) of a set X andSa set OpenX of open subsets of X such that X is open, U ∩ V is open if U, V are so, and i Ui is open for any family (Ui ) of open sets, in particular the union of the empty family, the empty set, is open. The complement X − U of an open set U is called closed. Therefore the collection ClosedX of closed sets fits with the corresponding axioms: the union of any two closed sets is closed, the intersection of any family of closed sets is closed1 , and the empty set is closed. If we define the closure Y of any subset of X as the intersection of all closed sets containing Y , then the topology is again defined by the axioms for the closure operator : 2X → 2X , i.e., ∅ = ∅, is idempotent, Y ⊆ Y , and Y ∪ Z = Y ∪ Z. Given two topologies OpenX , Open0X , one says that OpenX is coarser than Open0X or that Open0X is finer than OpenX iff OpenX ⊆ Open0X . On any set X, the coarsest topology consists just of X and of the empty set, it is called the indiscrete topology, whereas the finest topology is the powerset of X, it is called the discrete topology. The intersection of any family of topologies on X is the finest topology which is coarser than each member of the family. Every set of subsets S of X is contained in the intersection of all topologies containing this subset, a family containing at least the discrete topology. It consists of all unions of finite intersections (the empty intersection gives X) of members of S and is denoted by Open(S). A neighborhood W of x ∈ X is a subset containing an open set U which contains x. Finite intersections of neighborhoods of x are neighborhoods, supersets of neighborhoods are neighborhoods. An accumulation point of a subset Y of X is a point not in Y which intersects 1 The
intersection of the empty family being defined as the total space X.
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APPENDIX H. COMPLEMENTS ON GENERAL AND ALGEBRAIC TOPOLOGY
Y in each of its neighborhoods. The closure of a subset Y of X is the union of Y and of its accumulation points. The interior U o of a subset is the union of all open subsets of U . It is also the complement of the closure of its complement. The interior operator has an evident set of axioms corresponding to the closure axioms which also characterize the topology. The boundary ∂U of a subset is the difference U − U o . A subset B of open sets is called a base for the topology iff any open set is the union of a family of B members, or, equivalently, every neighborhood of a point contains a neighborhood S from B. The axioms for a set of subsets B of X to be a base for a topology is that X = B, and that for any two U, B ∈ B, U ∩ V is the union of members of B. A subbase for a topology OpenX on X is a set S of subsets of X such that OpenX = Open(S).
H.1.2
The Category of Topological Spaces
Suppose that (X, OpenX ), (Y, OpenY ) are topological spaces. A set map X → Y is continuous iff the inverse map 2Y → 2X induces a map OpenY → OpenX . The set-theoretic composition of continuous maps is continuous, the identity map is so, and therefore, we have the category Top of topological spaces and continuous maps. An isomorphism of topological spaces is called a homeomorphism. Any subset W of a topological space becomes a topological space by the coarsest topology OpenW = OpenX |W such that the inclusion W ⊂ X is continuous; its open sets are just the intersections of open sets of X with W , this topology is called the relative topology on W . More generally, given any set map f : X → Y into a topological space (Y, OpenY ) the coarsest topology OpenY |f (smallest set of open sets) on X such that f becomes continuous is given by the set of inverse images of open sets of Y , we also call it the relative topology with respect to f . Conversely, for a set map f : X → Y , where (X, OpenX ) is a topological space, we have a finest topology such that f becomes continuous, it is given by the set of all subsets of Y such that their inverse image is open in X. This is the Qquotient topology OpenX /f . If (Xi ) is a family of topological spaces, the cartesian product i Xi has the coarsest topology such that the projections Q to all factors become continuous. A base of this product topology is given by the products i Ui of open sets Ui ⊆ Xi with Ui = Xi except for a finite number of indices. This is a limit in the category Top. The coarsest topology on the set-theoretical limit lim(D) of a diagram of continuous maps is the limit in Top, a similar construction (this time with the finest topology) yields the colimit of a diagram of continuous maps. If we are given a family fi : Xi → X of set maps whose domains are topological spaces, there is a finest topology which makes these maps continuous. Its universal property is that with this topology on X, a map g : X → Y into a topological space Y is continuous iff all compositions g ◦ fi are so. This is a particular case of a quotient topology for the situation ` i Xi → X. This topology is called the coinduced topology. If the maps fi are inclusions of subspaces Xi of a topological space X, the topology of X is called coherent or weak if it is coinduced from the relative topologies on the spaces Xi . If we are given a set X, together with a collection of subsets Ci of X which are topological spaces such that for all indexes i, j, the intersections Ci ∩ Cj have the same relative topology as inherited from Ci or from Cj , and that these intersections are closed in both, Ci and Cj . Then the coinduced topology is coherent with this family, in other words, the coinduced topology relativizes to the given topologies on all Ci .
H.1. TOPOLOGY
H.1.3
1147
Uniform Spaces
Topologies are often defined by relations that stem from metrical distance functions. The axiomatics is as follows: Definition 166 A uniformity on a set X is a set U of uniform sets U ⊆ X 2 such that: (i) Each uniform set contains the diagonal ∆. (ii) If U is uniform, so is U −1 . (iii) If U is uniform, then there is a uniform V such that V ◦ V . (iv) If U, V are uniform, then so is U ∩ V . (v) If U is uniform, then so is every superset in X 2 . The prototype of a uniformity is given by a distance function, i.e., a pseudo-metric d : X × X → R as defined in definition 171 in appendix I.1.1. The uniformity contains all U ⊆ X 2 which contain a set of type U = {(x, y)|d(x, y) < }, > 0. Each uniformity U gives rise to a uniform topology Open(U) whose open sets are those V such that for each x ∈ V , there is a uniform set U with U [x] ⊂ V , where U [x] = {y|(x, y) ∈ U }. So the uniform topology imitates metrical neighborhoods.
H.1.4
Special Issues
Definition 167 A topological space X is said to be: (i) T0 iff for any two different points x, y ∈ X, at least one of them is not the specialization of the other; (ii) T1 iff every point is closed, i.e., no other point dominates it; (iii) T2 (Hausdorff) iff every two different points have disjoint neighborhoods. Definition 168 A subset L ⊂ X of a topological space X is said to be locally closed iff one of the equivalent properties holds: (i) L = O ∩ C, O open, C closed. (ii) Every point l ∈ L has an open neighborhood Ul such that Ul ∩ L is closed in Ul . (iii) L is open in its closure in X. See [65, I,§3.3] for a proof. Definition 169 A topological space X is called quasi-compact iff every covering of X by open sets admits a finite subcovering. A Hausdorff quasi-compact space is called compact. Typically, prime spectra of commutative rings are quasi-compact but not compact.
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APPENDIX H. COMPLEMENTS ON GENERAL AND ALGEBRAIC TOPOLOGY
Algebraic Topology
Refer to [498] for this section.
H.2.1
Simplicial Complexes
A simplicial complex K is a set V of vertexes, together with a subset K of 2V whose elements are called simplexes such that (1) each singleton {v}, v ∈ V is a simplex, (2) each non-empty subset of a simplex is a simplex. If for a simplex s of K, card(s) = q + 1, one says that s is a q-simplex or a q-dimensional simplex. A subsimplex s0 ⊆ s of a simplex s is called a face of s; it is called a q-face if it is a q-dimensional simplex, we also write s0 ≤ s instead of s0 ⊆ s. Evidently, a simplicial complex is completely determined by its simplex set K and may be identified with it. Example 101 Let U be a covering of a set X by non-empty subsets. The nerve n(U ) of U is the simplicial complex with V = U , and the simplexes T s being those finite sets s = {u0 , u1 , . . . up } in U which have non-empty intersection ∩s = i ui . The dimension dim(K) of a simplicial complex K is the maximal dimension of its simplexes, including the special cases dim(∅) = −1 dim(K) = ∞ if no maximal dimension exists. A simplicial map f : K1 → K2 is a set map f : V1 → V2 on the underlying vertex sets such that the induced map 2f : 2V1 → 2V2 carries simplexes to simplexes, i.e., restricts to a map f : K1 → K2 , meaning that if s ∈ K1 , then f (s) ∈ K2 . One may also say that it is a set map F : K1 → K2 which is induced by a map f on the underlying vertex sets. The simplicial complexes and their simplicial maps define the category Simpl of simplicial complexes. A subcomplex L of a simplicial complex K is a subset of simplexes which is also a simplicial complex. L is full iff a simplex of K whose vertexes belong to L is also in L. For example, given a simplicial complex K and a natural number k, the k-dimensional skeleton K|k is the subcomplex of all simplexes of dimension ≤ k. For a covering U , the k-dimensional skeleton of its nerve is denoted by nk (U ). Example 102 Let Covens be the category of set coverings, whose objects are pairs (X, I) of sets X and coverings I of X by non-empty subsets. The morphisms are pairs (f, φ) : (X, I) → (Y, J) with f : X → Y, φ : I → J two maps such that for all i ∈ I, f (i) ⊂ φ(i). We then have the nerve functor n : Covens → Simpl : (X, I) 7→ n(I).
H.2.2
Geometric Realization of a Simplicial Complex
We have a functorially defined geometric representation of simplicial complexes K by topological spaces |K| as follows. The set |K| is the subset of those functions α : V (K) → I = [0, 1] into the real unit interval I such that 1. the support supp(α) = {v ∈ V (K)|α(v) 6= 0} of α is a simplex, P 2. v∈V (K) α(v) = 1.
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The value α(v) is called the v th barycentric coordinate of α. On the set I (V (K)) of functions with finite support, one has the Euclidean metric d(α, β) = kα − βk2 . We induce this metric and its associated topology (see section H.1.3) on |K| and denote it by |K|d . For a simplex s ∈ K, the closed simplex |s| is defined by |s| = {α ∈ K|supp((α) ⊂ s}. P ∼ Evidently, if dim(s) = q, there is a homeomorphism |s|d → ∆q = {x ∈ | I q+1 | xi = 1} onto the “standard closed q-simplex”. If s, t ∈ K, either s ∩ t = ∅ or a common face, and then |s ∩ t| = |s| ∩ |t|, so |s|d ∩ |t|d is closed in both, |s|d , |t|d , and the relative topologies from |s|d , |t|d coincide on the intersection. By the remarks on coinduced topologies in section H.1.2, we have the coherent topology on |K| which is coinduced from the topologies on the closed simplexes. This means that Fact 19 A subset E ⊆ |K| is closed/open iff each intersection E ∩ |s|d is closed/open. Therefore, a function f : |K| → X into a topological space X is continuous iff its restrictions f ||s| are so for all simplexes s of K. In particular, the identity |K| → |K|d is continuous, therefore, |K| is Hausdorff, it is also normal, see [498, 3.1, Th.17]. Also, |K| is compact iff K is finite. Call K locally finite, iff every vertex belongs to a finite number of simplexes. Then Theorem 73 For a simplicial complex K the following statements are equivalent: (i) K is locally finite. (ii) The identity |K| → |K|d is a homeomorphism. (iii) |K| is metrizable, i.e., there is a metric whose topology is the coherent topology. See [498, 3.2, Th.8] for a proof. If f : K1 → K2 is a simplicial map, we have the continuous map X |f |(α)(v) = α(w) f (w)=v
which is continuous for both topologies on |K|. We are therefore given two functors |?|, |?|d : Simpl → Top and a natural transformation Id : |?| → |?|d . P A continuous map f : |K| → X ⊂ Rn is said to be linear iff f (α) = v∈V (K) α(v)f (v) for all α ∈ |K|. Any function on the vertexes may uniquely be extended to a continuous linear map, this is the universal property of affine pointsets in general position. In particular, the maps |f | associated with a simplicial map f is linear. Definition 170 A geometric realization of a simplicial complex K in Rn is a linear embedding (injection) of |K| in Rn . Theorem 74 If a simplicial complex K has a geometric realization in Rn , then it is countable, locally finite and has dimension ≤ n. Conversely, if it is countable, locally finite, and has dimension ≤ n, then it has a geometric realization as a closed subset of R2n+1 . Example 103 For the nerve n(U ) of a finite covering U , we write N (U ) for the geometric realization |n(U )|, we also write Nk (U ) for |nk (U )|.
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H.2.3
APPENDIX H. COMPLEMENTS ON GENERAL AND ALGEBRAIC TOPOLOGY
Contiguity
A simplicial pair is a couple (K, L), where K is a subcomplex of L. A simplicial map of pairs f : (K1 , L1 ) → (K2 , L2 ) is a simplicial map f : L1 → L2 which induces a simplicial map on the respective subcomplexes. Two simplicial maps f, f 0 : (K1 , L1 ) → (K2 , L2 ) are called contiguous if for every simplex s in K1 or L1 , the union f (s) ∪ f 0 (s) is a simplex in K2 or Ls . Contiguity is an equivalence relation and defines contiguity classes of simplicial maps. Two continuous maps f, g : X ⇒ Y of topological spaces are called homotopic iff there is a continuous map (a homotopy) F : X × I → Y such that f = F (?, 0), g = F (?, 1); the homotopy relation is an equivalence relation. If X 0 ⊆ X is a subspace, and if f |X 0 = g|X 0 , a homotopy is called relative to this subspace, iff F |X 0 × t = f |X 0 = g|X 0 , all t ∈ I. Lemma 96 ([498, lemma 2, p.130]) Contiguous simplicial maps which agree on a subcomplex define contiguous maps which are homotopic relative to the space of the subcomplex.
H.3
Simplicial Coefficient Systems
A simplicial complex K can be viewed as a category whose objects are the simplexes s of K, and whose morphisms are the inclusions s ⊆ t of simplexes. For a commutative ring R, a coefficient system of R-modules is a covariant functor M : K → R Mod with values in the category R Mod of R-modules and affine homomorphisms. Let ∆q = {0, 1, 2, . . . q} be the standard simplex of dimension q. A singular simplex of dimension q is a simplicial map s : ∆q → K, i.e., a sequence s0 , s1 , . . . sq of points in K which define a simplex. If we have any set map f : ∆p → ∆q , we have the singular p-simplex f (s) = s ◦ f : ∆p → K. For a singular simplex s, we denote M (s) = M (Im(s)). Clearly Im(f (s)) ⊆ Im(s). Therefore we have an affine homomorphism fs : M (f (s)) → M (s). Denote by Q Sn (K) the set of singular simplexes of dimension n in K. Then we have a module C n (K; M ) = s∈Sn (K) M (s), whose elements are called the singular cochains of dimension n. For a map f : ∆p → ∆q , we have an affine map f : C p (K; M ) → C q (K; M )
(H.1)
which has f ((as )s∈Sp (K) ) = (bt )t∈Sq (K) and bt = ft (af (t) ). In other words, C ∗ (K; M ) = (C n (K; M ))n is a simplicial cochain complex.
H.3.1
Cohomology
Suppose now that the simplicial cochain complex stems from a system of coefficients with linear maps. Then all the transition maps of equation (H.1) are linear. Consider now the strictly increasing ith -face maps Fni : ∆n−1 → ∆n leaving aside index i in ∆n , i.e., mapping ∆n−1 onto the subset {0, 1, 2, . . . ˆi . . . n}. Then we have the coboundary map dn : C n (K; M ) → C n+1 (K; M ), dn (a) =
n+1 X j=0
j (−1)j Fn+1 (a),
(H.2)
H.3. SIMPLICIAL COEFFICIENT SYSTEMS
1151
and dn+1 ◦ dn = 0. This means that Im(dn ) ⊆ Ker(dn+1 ), and we may consider the cohomology groups H n (K; M ) = Ker(dn )/Im(dn−1 ) (H.3) for n ≥ 0, with the trivial extension to C −1 (K; M ) = 0.
Appendix I
Complements on Calculus I.1
Abstract on Calculus
I.1.1
Norms and Metrics
Definition 171 A pseudo-metric on a set V is a (pseudo-distance) function d : V × V → R such that: 1. (Positivity) 0 ≤ d(x, y), and d(x, x) = 0 for all (x, y) ∈ V × V ; 2. (Symmetry) d(x, y) = d(y, x) for all (x, y) ∈ V × V ; 3. (Triangle inequality) d(x, z) ≤ d(x, y) + d(y, z) for all (x, y, z) ∈ V × V × V . If conversely d(x, y) = 0 implies x = y, the pseudo-metric (pseudo-distance function) is called a metric (distance function). Definition 172 For a pseudo-metric space (X, d), if 0 < r, x ∈ X, the open ball of radius r around x is Br (x) = {y|d(y, x) < r}. The system of open balls {Br (x)|0 < r, x ∈ X} 1
is a base of a topology , the (uniform) topology associated with the pseudo-metric d. Evidently, this topology is Hausdorff iff the pseudo-metric is a metric. A map f : V → V of a pseudo-metric space V is called an isometry iff d(f (x), f (y)) = d(x, y), for all (x, y) ∈ V × V . Lemma 97 ([73, Lemma 4]) Given an action µ : G × V → V of a group G on a pseudo-metric space (V, d) by isometries, then inf d(g.x, y) = inf d(g.x0 , y)
whenever G.x= G.x0 ,
(I.1)
inf d(g.x, y) = inf d(g.x, y 0 )
whenever G.y = G.y 0 .
(I.2)
g∈G g∈G 1 In
g∈G
g∈G
fact, the system {Br = {(x, y) ∈ V 2 |d(x, y) < r}|0 < r} is a base of a uniformity.
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APPENDIX I. COMPLEMENTS ON CALCULUS With the above notation, we may define d∗ (G.x, G.y) = inf d(g.x, y), g∈G
(I.3)
and lemma 97 guarantees that this is a well-defined function d∗ : G\V × G\V → R. Definition 173 If d(x, y) is a pseudo-metric on a set V , and if we have a group action G × V → V , we say that g acts by isometries, iff each map g. : V → V is an isometry, i.e., iff d(g.x, g.y) = d(x, y), for all x, y ∈ V, g ∈ G. Lemma 98 ([73, Lemma 5]) Let d be a pseudo-metric on V , and µ : G × V → V a group action by isometries. Then the function d∗ defined in (I.3) is a pseudo-distance on the orbit space G\V .
I.1.2
Completeness
A Cauchy sequence in a uniform space (X, U) is a sequence (xi )i=0,1,2,... of elements in X such that for every uniform set U ∈ U, there is an index t such that (xi , xj ) ∈ U for all i, j > t. A uniform space is (sequentially) complete iff every Cauchy sequence converges. Lemma 99 A closed subspace of a complete uniform (in particular: a metric space) space is complete. Definition 174 A norm on real vector space X is a function k k : X → R such that for all (x, y) ∈ V × V : 1. (Positivity) 0 ≤ kxk, and kxk = 0 iff x = 0; 2. (Homogeneity) kλ.xk = |λ|.kxk; 3. (Triangle inequality) kx + yk ≤ kxk + kyk. Every norm gives rise to an associated metric d(x, y) = kx − yk, and therefore to an associated topology. A normed vector space with a complete associated (uniform) topology is called a Banach space. Example 104 On Rn , we have three well-known norms. If x = (x1 , . . . xn ) ∈ Rn , then P 1. the absolute or 1-norm is kxk1 = i |xi |, pP 2 2. the Euclidean norm is kxk2 = i xi , 3. the uniform norm is kxk∞ = max{|xi || i = 1, . . . n}. For real numbers a < b, we have the vector space C 0 [a, b] of continuous real-valued functions on the interval [a, b]. On C 0 [a, b], we have three well-known norms (corresponding to the above three norms). For f ∈ C 0 [a, b], we have:
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Rb 1. the absolute or 1-norm is kf k1 = a |f |, Rb 2. the Euclidean norm is kf k2 = ( a f 2 )1/2 , 3. the uniform norm is kf k∞ = M ax[a,b] |f |. Two norms k k1 , k k2 on a real vector space X are called equivalent iff there are two positive constants a, b such that k k1 ≤ a.k k2 , k k2 ≤ b.k k1 . Equivalent norms give rise to the same associated uniformities and topologies, so they have the same Cauchy sequences. Theorem 75 Any two norms on a finite-dimensional real vector space are equivalent. See [307, Th.3.4.1] for a proof. The theorem implies that every finite-dimensional normed real vector space is Banach, since the standard Rn is so under the Euclidean norm. We shall therefore mainly work in Rn .
I.1.3
Differentiation
We say that two functions f, g : U → Rm that are defined in a neighborhood U of 0 ∈ Rn define the same germ iff they coincide on a common neighborhood of 0. (We are in fact considering the colimit of function spaces on the neighborhood system of 0.) The set of germs in 0 of functions f with f (0) = 0 is denoted by F0 . Within this vector space, we have the vector subspace DF0 of those f with f (0) = 0 and kf (z)k/kzk → 0 if z → 0. We evidently have LinR (Rn , Rm ) ∩ DF0 = {0}. Definition 175 A function f : U → Rm which is defined in a neighborhood U ⊆ Rn of a point x is differentiable in x iff there is a linear map D ∈ LinR (Rn , Rm ) such that ∆x f − D ∈ DF0 , where ∆x f (z) = f (x+z)−f (x). By the above, D is uniquely determined and is denoted by Dfx . The coefficient of row i and column j of the matrix of Dfx in the canonical basis is denoted by ∂fi /∂xj , whereas the matrix is called the Jacobian of f in x. A function f : O → V on an open set O ⊆ Rn with values in an open set V ⊆ Rm is differentiable if it is differentiable in each point of its domain O. ∼
A differentiable function on O defines its derivative Df : O → LinR (Rn , Rm ) → Rnm , which may again be differentiated according to the norm on the space of linear maps. Inductively we define Dt+1 f = D(Dt f ), if it exists. The function f is C r iff all derivatives Df, D2 f, . . . Dr f exist and are continuous, C 0 denotes just the set of continuous functions. This definition is however not in the right shape for functorial behavior. One therefore adds the linear behavior to the function as follows: Let T O = O × Rn be the tangent bundle of the open set O. Then we define T f : T O → T Rn by T f (x, u) = (f (x), Dfx (u)). This implies that if g : U → Rl is a second differentiable function on an open set U ⊆ Rm , then g ◦ f is differentiable and T (g ◦ f ) = T g ◦ T f. So we have a functor T : f 7→ T f and the natural transformation pr1 : T → Id of first projection. More generally, defining T r+1 f = T (T r f ), we also have T r (g ◦ f ) = T r g ◦ T r f.
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APPENDIX I. COMPLEMENTS ON CALCULUS
Moreover, if we identify LinR (Rn , LinR (Rn , Rm )) with BilR (Rn , Rm )), etc. for higher multilinear maps, the higher derivatives Dr fx identify to r-linear maps (Rn )r → Rm . Proposition 115 If f is C r , then Dr fx is a symmetric r-linear matrix. The category of r times differentiable or C r functions has the property that the linear parts of the tangent maps compose as normal linear maps do, and this means that the Jacobians of isomorphisms are invertible quadratic matrices. A curve in Rn is a C 1 -map y : U → Rn . Its derivative Dyt in a point t ∈ U is a linear map R → Rn which identifies to the image of 1 in Dyt (1) ∈ Rn , meaning that the derivative can be identified with a continuous map y 0 : U → Rn : t 7→ y 0 (t) = Dyt (1).
I.2
Ordinary Differential Equations (ODEs)
Throughout this section, D denotes an open set in Rn , and f : D → Rn denotes a continuous vector field (a function) with components fi , i = 1, . . . n. Definition 176 Let ζ ∈ R, η ∈ D, J(ζ) an open interval containing ζ, and U (η) ⊂ D an open neighborhood of η in D. Denote by A(f, ζ, η, J(ζ), U (η)) the set of all C 1 -functions y : J(ζ) → U (η) such that y 0 = f ◦ y and y(ζ) = η. (I.4) Denote by B(f, ζ, η, J(ζ), U (η)) the set of all C 1 -functions y : J(ζ) → U (η) such that Z
?
f ◦ y.
y=η+
(I.5)
ζ
Lemma 100 With the above definitions, we have A(f, ζ, η, J(ζ), U (η)) = B(f, ζ, η, , J(ζ), U (η)). The easy proof is left to the reader.
I.2.1
The Fundamental Theorem: Local Case
The following theorem is called the local case of the fundamental theorem of ordinary differential equations. Theorem 76 With the preceding notation and definitions, suppose that f is locally Lipschitz, i.e., for every x ∈ D, there is a neighborhood U (x) ⊂ D and a positive number L such that x1 , x2 ∈ U (x) implies |f (x1 ) − f (x2 )| ≤ L.|x1 − x2 |. Then for any “initial condition” ζ ∈ R, η ∈ D, there is an open interval J(ζ) containing ζ, and an open neighborhood U (η) of η such that A(f, ζ, η, J(ζ), U (η)) is a singleton. The element of A is called the local solution of the differential equation y 0 = f ◦ y at J(ζ), U (η).
I.2. ORDINARY DIFFERENTIAL EQUATIONS (ODES)
1157
The proof uses lemma 100 and refers to the set B. In fact, it is shown that the operator Z ? Tζ,η,f = η + f ◦y ζ
is a contraction, and contractions have a unique fixpoint. Proposition 116 Let T : X → X be a contraction on a complete metric space2 (X, d), i.e., there is a constant 0 < c < 1 such that d(T (x), T (y)) ≤ c.d(x, y) for all x, y ∈ X. Then, T has a unique fixpoint z = T (z). Proof. It suffices to show that the sequence (xn = T n (x)) is Cauchy. In fact, setting k = |n−m|, we have d(xn , xm ) = d(T n (x), T m (x)) = cM in(n,m) .d(x, T k ). But d(x, T k ) ≤ d(x, T (x)) + d(T (x), T 2 (x)) + . . . d(xk−1 , T k ) 1 d(x, T (x)). ≤ (1 + c + . . . ck−1 )d(x, T (x)) ≤ 1−c So this term is limited, while cM in(n,m) tends to zero as n, m tend to infinity, QED. Corollary 30 Let X be a complete metric space, and B = Br (x) the closed ball of radius r > 0 around x. Let T : B → X a contraction with d(T (x), x) ≤ (1 − c)r, 0 < c < 1. Then T has a unique fixpoint in B. Proof. We know from lemma 99 that B is complete. Further, for y ∈ B, we have d(T (y), x) ≤ d(T (y), T (x)) + d(T (x), x) ≤ c.d(y, x) + (1 − c)r ≤ r. Therefore, T leaves B invariant and the claim follows from proposition 116, QED. Corollary 31 With the notation of corollary 30, suppose that T : Br (x) → X is a contraction with d(T (x), x) < (1 − c)r, 0 < c < 1. Then there is a unique fixpoint of T in Br (x). Next, we need some auxiliary results concerning uniform convergence of continuous functions. Let W be a Banach space (in our case W = Rn ), A a set, then we set B(A, W ) = {f : A → W |kf k∞ < ∞}.
Proposition 117 The set B(A, W ) with the usual scalar multiplication and addition of functions is a Banach space. It is clear that B(A, W ) is a vector space. Let (fn )n be a Cauchy sequence in B(A, W ). Since for any x ∈ A, kfn (x) − fm (x)k ≤ kfn − fm k, and the right term converges to zero, the left term is also a Cauchy sequence in W and converges to limn→∞ fn (x) = f (x). We first show that limn→∞ fn = f . For 0 < , let N be such that n, m > N implies kfm − fn k > . Then by definition, for all x ∈ A, kfn (x)−f (x)k = kfn (x)−limm>N fm (x)k = limm>N kfn (x)−fm (x)k ≤ . Therefore kfn − f k ≤ , and f = (f − fn ) + fn is a sum of two elements of B(A, W ) and therefore lives in B(A, W ), whereas limn→∞ fn = f , QED. 2 See
this appendix, section I.1.1.
1158
APPENDIX I. COMPLEMENTS ON CALCULUS
Theorem 77 Let A be a metric space, W a Banach space, and let BC(A, W ) = B(A, W ) ∩ C 0 (A, W ) be the set of continuous functions with limited norm. Then BC(A, W ) ⊂ B(A, W ) is a closed sub-vector space, and therefore also Banach. Proof. It is clearly a sub-vector space. Let (fn ) be a Cauchy sequence in BC(A, W ). It converges to f in B(A, W ). We have to show that it is also continuous. In fact, given 0 < select n such that kf − fn k < /3. Let a ∈ A. Take 0 < δ such that d(x, a) < δ implies kfn (x) − fn (a)k < /3. Then kf (x) − f (a)k ≤ kf (x) − fn (x)k + kfn (x) − fn (a)k + kfn (a) − f (a)k < /3 + /3 + /3, QED. We are now ready for the proof of the local theorem. Recall that we are given a locally Lipschitz vector field function f : D → Rn . Consider the Banach space BC = BC(J(ζ), Rn ) for an interval J(ζ) whose length δ will be determined in the course of the proof. Select 0 < r such that (1) the closed ball Br (η)− ⊂ D, and (2) f |Br (η)− is Lipschitz with a constant L. Then f is evidently limited on Br (η)− , let m be an upper bound. Let η¯ : J(ζ) → Br (η)− : t 7→ η be the constant map. Consider the closed ball Br (¯ η )− ⊂ BC around η¯. For every g ∈ Br (¯ η )− , n f ◦ g : J(ζ) → R lives in BC. R? We now show that the operator T (g) = η + ζ f ◦ g defines a contraction T : Br (¯ η )− → BC with contraction constant c such that d(T (¯ η ), η¯) < (1 − c)r. According to corollary 30, this will imply that T has a unique fixpoint in Br (¯ η )− and we are done. Rx Evidently, T (g) is continuous. Further, for any x ∈ J(ζ), we have |T (g)(x)| ≤ |η| + | ζ f ◦ g| ≤ |η| + |x − ζ|.kf ◦ gk∞ which evidently is finite. We are left with the contraction claims. We have Z t kT (¯ η ) − η¯k∞ = lubJ(ζ) kT (¯ η )(t) − η¯(t)k = lubJ(ζ) k f (η)k ζ
= lubJ(ζ) |t − ζ|.|f (η)| ≤ δ.|f (η)| ≤ δ.m. For two functions g1 , g2 ∈ Br (¯ η )− , we have Z t kT (g1 ) − T (g2 )k = lubJ(ζ) | f (g1 ) − f (g2 )| ≤ δ.kf ◦ g1 − f ◦ g2 k ζ
= δ.lubJ(ζ) |f (g1 (s)) − f (g2 (s))| ≤ δ.L.lubJ(ζ) |g1 (s) − g2 (s)| = δ.L.kg1 − g2 k. This means that T is a contraction with c = δ.L if δ is such that δ.L < 1. Further, we need r δ.m < (1 − c)r = (1 − δ.L)r, i.e., δ < m+Lr solves the problem, QED.
I.2.2
The Fundamental Theorem: Global Case
The global fundamental theorem deals with maximal integral curves y : J → D for the differential equation y 0 = f ◦ y.
I.2. ORDINARY DIFFERENTIAL EQUATIONS (ODES)
1159
Definition 177 We say that u ∼ v for x, y ∈ D iff there is a curve y : J → D, defined on an open interval J for the differential equation y 0 = f ◦ y, and such that {u, v} ⊂ y(J). Lemma 101 The relation ∼ is an equivalence relation. The equivalence class of an element x ∈ D is denoted by [x]. It is clearly reflexive and symmetric. It is transitive for the following reason. Let yi : Ji → D, i = 1, 2 be two integral curves such that y1 (t1 ) = x, y1 (t2 ) = y, y2 (t3 ) = y, y2 (t4 ) = z. By an evident parameter shift, we may suppose t2 = t3 . We claim that y1 |J1 ∩ J2 = y2 |J1 ∩ J2 . Suppose that y1 (t) 6= y2 (t) for a t > t2 . Let t2 ≤ t0 be the infimum of these t. Since our curves are continuous, we have y1 (t0 ) = y2 (t0 ). But then, according to the local theorem 76, there is an -ball U (t0 ) around t0 and a neighborhood U (y1 (t0 ) = y2 (t0 )) such that there is a unique integral curve y : U (t0 ) → U (y1 (t0 )). But we may suppose WLOG that is so small that both y1 |U (t0 ), y2 |U (t0 ) have their codomains in U (y1 (t0 )). Evidently, these solutions must then coincide with the unique solution on the open interval U (t0 ), but this contradicts the choice of t0 . A symmetric argument holds for the supremum s0 ≤ t2 of those arguments with y1 (t) 6= y2 (t). Therefore, y1 |J1 ∩ J2 = y2 |J1 ∩ J2 ,and we may extend the integral curves y1 , y2 to the domain J1 ∪ J2 , whence the transitivity of the ∼-relation, QED. Theorem 78 Let x ∈ D. Then there is a unique integral curve y : J → D with y(0) = x, y 0 = f ◦ y, and such that J contains all domains of any integral curve z, z(0) = x, z 0 = f ◦ z. We R have y(J) = [x] and write x f for this curve; it is called the global solution through x. Proof. Let Γ = {Γyi ⊂ R × D|Γyi = graph of solution yi of yi0 = yiS◦ f, x = yi (0)}. Since two solutions coincide on the intersection of their domains, the union Γ is functional, and the union of the domains is an open interval J. Further, the function y of this graph is a solution of the differential equation y 0 = f ◦ f which reaches all elements equivalent to x, QED. Corollary 32 Let x1 ∼ x2 and
R x1
f (t2 ) = x2 . Then
R x2
f=
R x1
f ◦ et2 and J2 = e−t2 J1 .
R Definition 178 The quotient D/ ∼= { x f |x ∈ D} is called the phase portrait of the vector field f and denoted by D/f . An integral curve which is not an injective function of its parameter is called a cycle of the field f . R R R Proposition 118 Let w f be a cycle with w f (t1 ) = w f (t1 + T ). Then the cycle’s domain R is R and w f is T -periodic. R Proof. Let y : J → D be the cycle w f with y(t1 ) = z. Consider the function yˆ = y ◦ eT : J − T → D. Evidently, yˆ(t1 ) = y(t1 ), and yˆ also solves the differential equation since yˆ0 (t) = y 0 (t + T ) = f ◦ y(t + T ) = f ◦ yˆ(t). So, since yˆ has a common value with y att1 , by maximality of y, we have yˆ = y|J − T , and J − T ⊂ J, whence J = [−∞, b[. Symmetrically, exchanging t1 with t1 + T , and T with −T , we obtain J = [a, ∞[, i.e., J = R. Now, for any t ∈ R, with yˆ = y ◦ eT , uniqueness guarantees yˆ = y, whence the periodicity of y, QED.
1160
APPENDIX I. COMPLEMENTS ON CALCULUS
Proposition 119 Suppose that D− is compact (e.g.: D is bounded), and that f is locally LipR schitz on D− .RIf the domain J =]a, b[ of a maximal curve x f has finite upper bound b, then t → b implies x f (t) → ∂D R Sketch of proof: Write y = x f , and suppose that the closure of y(J) were in D. Then, since y(J)− is compact, there is a convergent sequence tn → b with a convergent image sequence y(tn ) → q, q ∈ D. It can be shown that q is uniquely determined, i.e., another such sequence yields the same limit. We then set y(b) = limt→b y(t) = q, and y may be extended to a local solution containing b, a contradiction, QED.
I.2.3
Flows and Differential Equations
On an open set O ⊆ Rn , a vector field (a C 1 -map) f : O → Rn can also be viewed by its graph as a section O → T O : x 7→ F (x) = (x, f (x)). If x ∈ O, an integral curve of F in x is a curve y : U (0) → O defined on an open neighborhood U (0) of 0 such that y(0) = Rx and y 0 = f ◦ y. By the main theorem 78 of ODEs, there is a unique maximal integral curve x f for every point x ∈ O. For a vector field F on O ⊆ Rn , a flow box is a triple (U, a, W ) where U ⊆ O is open, a is a positive real number of ∞, and W : U ×] − a, a[→ O is C 1 such that for all x ∈ U , Wx : ] − a, a[→ O : t 7→ W (x, t) is an integral curve of F at x. Two flow boxes (U, a, W ), (U 0 , a0 , W 0 ) always coincide in their maps W, W 0 on the intersection (U ∩ U 0 ) × (] − a, a[∩] − a0 , a0 [) of their domains. For each point x ∈ O, there is a flow box (U, a, W ) with x ∈ U . Let DF = {(x, t) ∈ R R×O| there is an integral curve x whose domain contains t}. Then (1) DF is open in R×Rn ; (2) there is a unique map WF : DF → O such that t 7→ WF (t, x) is an integral curve at x for all x ∈ O.
I.2.4
Vector Fields and Derivations
For a C 1 -function f : O → R, we have the derivative T f : T O → T R, whose second component evaluates to linear forms on Rn . This map df = pr2 ◦ T f = Df is called the differential of f . If F : O → T O is a vector field, the composition LF f = df ◦ F : O → R is called the Lie derivative of f with respect to F . If we denote by grad(f ) the differential of f as a tangent vector (∂x1 f, . . . ∂xn f ) (the old-fashioned gradient of f ), the Lie derivative is just the scalar product of grad(f ) with the vector field. If F(O) denotes the real algebra3 of C 1 -function on O, the map LF : F(O) → F(O) is a derivation in the sense that: (i) LF is linear; (ii) for f, g ∈ F(O), we have LF (f.g) = f.LF (g) + LF (f ).g; (iii) If c ∈ F(O) is constant, then LF c = 0. Therefore, we also have d(f.g) = df.g + f.dg and dc = 0 for a constant c. Denote by V F (O) the vector space of all C 1 -vector fields on O. Then: 3 Multiplication
goes pointwise.
I.3. PARTIAL DIFFERENTIAL EQUATIONS
1161
Theorem 79 The Lie map L? : V F (O) → Der(F(O)) : F 7→ LF is an isomorphism of vector spaces. See [2, Th.8.10] for a proof. In particular, the Lie bracket [LF , LG ] = LF ◦ LG − LG ◦ LF which is a derivation, must be the Lie derivative of a unique vector field which is denoted by [F, G], the Lie bracket of the vector fields F and G. The Lie bracket makes the vector space V F (O) into a real Lie algebra, see section E.4.4.
I.3
Partial Differential Equations
For this section, refer to [252]. We only need a short review of quasi-linear first order partial differential equations (PDE). Recall that a PDE is an equation of type E(x1 , x2 , . . . u, ux1 , ux2 , . . . ux1 x1 , ux1 x2 , . . .) = 0 where u is a function of the n real variables x1 , x2 , . . . xn , with its partial derivatives ux1 , . . ., the higher partial derivatives ux1 x1 . . . etc. A solution is meant to be such a function u which is defined in an open set O of Rn . Its order m is the highest number of iterated partial derivatives, whereas E is called quasi-linear iff it is an affine function of the derivatives of u of highest order m, with coefficients that are functions of the variables x1 , x2 , . . . u, ux1 . . . until derivatives of order m − 1. A first-order quasi-linear PDE has the shape X ai (x1 , x2 , . . . u)uxi = c(x1 , x2 , . . . u) i
and can be solved by a system of ODEs, this is the method of characteristics. We illustrate the method for two variables, i.e., for the equation a(x, y, u)ux + b(x, y, u)uy = c(x, y, u).
(I.6)
The solution u(x, y) is represented as a surface z = u(x, y) in R3 . Such a surface is called an integral surface of the equation (I.6). We have a vector field F (x, y, z) = (a(x, y, z), b(x, y, z), c(x, y, z)) on the common domain U of the three functions a, b, c. The tangent space of an integral curve at x, y, u(x, y) is spanned by the vectors X = (1, 0, ux ) and Y = (0, 1, uy ). Their vector product Y ∧ X = (ux , uy , −1) is the normal vector to the integral surface. Therefore equation (I.6) just means that the scalar product (F (x, y, z), Y ∧ X) vanishes identically, i.e., the vector field F is tangent to the integral surface. Clearly, only the direction of the vectors of the vector field F , the characteristic directions matter for the equation (I.6). It is easily seen that an integral curve of F , if it crosses a point of an integral surface, is entirely contained in this surface. Therefore an integral surface is the union of integral curves of the directional vector field F . An integral surface can be constructed by finding a curve Γ which lies in an integral surface, and which
1162
APPENDIX I. COMPLEMENTS ON CALCULUS
is never parallel to an integral curve of F . This is the Cauchy problem for the equation (I.6). Then, the parameter of Γ and the curve parameter of a flow box (see section I.2.3) around Γ describe the integral surface. Technically, the existence condition for Γ(t) = (Γx (t), Γy (t), Γz (t)) to generate a surface is that the projection Γxy (t) = (Γx (t), Γy (t)) is never parallel to the projection Fxy of the directional field on the xy plane. The existence of a curve Γ is again guaranteed by the main theorem of ODEs, and we are done.
Part XVII
Appendix: Tables
1163
Appendix J
Euler’s Gradus Function This table lists the rational numbers x/y with Euler’s gradus suavitatis Γ(x/y) ≤ 10, see also [71]. Γ
Intervals
2
1/2
3
1/3, 1/4
4
1/6,2/3,1/8
5
1/5,1/9,1/12,3/4,1/16
6
1/10,2/5,1/18,2/9,1/24,3/8,1/32
7
1/7,1/15,3/5,1/20,4/5,1/27,1/36,4/9,1/48,3/16,1/64
8
1/14,2/7,1/30,2/15,3/10,5/6,1/40,5/8,1/54,2/27,1/72,8/9,1/96,3/32,1/128
9
1/21,3/7,1/25,1/28,4/7,1/45,5/9,1/60,3/20,4/15,5/12,1/80,5/16,1/81,1/108, 4/27,1/144,9/16,1/192,3/64,1/256
10
1/42,2/21,3/14,6/7,1/50,2/25,1/56,7/8,1/90,2/45,5/18,9/10,1/120,3/40,5/24, 8/15,1/160,5/32,1/162,2/81,1/216,8/27,1/288,9/32,1/384,3/128,1/512
1165
Appendix K
Just and Well-Tempered Tuning This table lists the just coordinates of the just tuning intervals (with respect to c, second tone in first column) according to Vogel [547], see subsubsection 7.2.1.2, together with the value in Cents, and the deviation in % from the tempered tuning with 100, 200, 300, etc. Cents. Tone name c d[
Frequency ratio
Octave coord.
Fifth coord.
Third coord.
Pitch (Ct)
% deviation
1
0
0
0
0
0
16/15
4
-1
-1
111.73
+11.73
d
9/8
-3
2
0
203.91
+1.96
e[
6/5
1
1
-1
315.65
+5.22
e
5/4
-2
0
1
386.31
-3.42
f
4/3
2
-1
0
498.05
-0.39
45/32
-5
2
1
590.22
-1.63
g
3/2
-1
1
0
701.96
+0.28
a[
8/5
3
0
-1
813.69
+1.71
a
5/3
0
-1
1
884.36
-1.74
b[
16/9
4
-2
0
996.09
-0.39
b
15/8
-3
1
1
1088.27
-1.07
f]
1167
Appendix L
Chord and Third Chain Classes L.1
Chord Classes
This section contains the list of all isomorphism classes of zero-addressed chords in P iM od12 . The meanings of the column items are explained in subsection 11.3.7; here we give a short definition. • Class Nr. is the number of the isomorphism class, numbers with extension “.1” indicate the class number for classification under symmetries from Z (no fifth or fourth transformations). Autocomplementary classes have a star after the number. • Representative of Nr. without hat is the number’s representative in full circles, the one with hat is the complementary chord. • Group of symmetries is Sym(N r.). To keep notation readable, we use the notation with linear factor to the left. • Conj. Class denotes the conjugacy class symbol of Sym(N r.) and refers to the numbering 1, 2, . . . 19 from [402]. d • Card. End. Cl. N r.|N r. is the pair of numbers of conjugacy classes of endomorphisms in d Nr. and in its complement N r., respectively.
1169
1170
APPENDIX L. CHORD AND THIRD CHAIN CLASSES
Class Nr. 1
Chord Classes Representative Group of d N r. = •, N r. = ◦ Symmetries −→ • • • • • • • • • • •• GL(Z12 )
Conj. Class
] End. d N r.|N r.
19
28|28
8
1|31
3
3|23
8
3|25
8
3|19
One/Eleven Element
2
• ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦
Z× 12 Two/Ten Elements
3
• • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦
3.1
• ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦◦
4
• ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦
5 6 7
• ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦◦
h−1e−1 i {1, 7, −1e−2 , 5e−2 } −3
{1, 5, 7e
−3
, −1e
8
8
}
• ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦◦
{1, 7, 5e , −1e }
8
3|31
• ◦ ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦◦
Z× 12
13
3|28
h−1e−2 i
2
4|14
{1}
1
4|30
{1}
1
8|36
6Z12
ne
Three/Nine Elements
8
• • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦
8.1
• ◦ • ◦ ◦ ◦ ◦ • ◦ ◦ ◦◦
9
• • ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦◦
9.1
• ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦
10
• • ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦◦
10.1
• ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦◦
11
• • ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦◦
h5i
4
4|20
12
• • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦◦
h7e6 i
6
5|29
8
8
6
6
13
• ◦ • ◦ • ◦ ◦ ◦ ◦ ◦ ◦◦
{1, 7, −1e , 5e }
8
4|18
14
• ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ ◦◦
h7i
6
8|31
• ◦ ◦ • ◦ ◦ • ◦ ◦ ◦ ◦◦
{1, 5, −1e , 7e }
8
5|32
16
• ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦◦
Z× 12
15
4|20
17
• • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦◦
h−1e−3 i
3
4|8
17.1
• ◦ • ◦ ◦ • ◦ • ◦ ◦ ◦◦
18
• • • ◦ • ◦ ◦ ◦ ◦ ◦ ◦◦
{1}
1
5|19
18.1
• ◦ • ◦ • ◦ ◦ • ◦ ◦ ◦◦
19
• • • ◦ ◦ • ◦ ◦ ◦ ◦ ◦◦
{1}
1
5|19
19.1
• • ◦ • ◦ ◦ ◦ ◦ • ◦ ◦◦
20
• • • ◦ ◦ ◦ • ◦ ◦ ◦ ◦◦
{1}
1
7|23
15
4Z12
ne
Four/Eight Elements
L.1. CHORD CLASSES
1171
Class Nr.
Chord Classes—Continued Representative Group of d N r. = •, N r. = ◦ Symmetries
20.1
• • ◦ ◦ ◦ • ◦ • ◦ ◦ ◦◦
21
• • • ◦ ◦ ◦ ◦ • ◦ ◦ ◦◦
{1, 7, −1e−2 , 5e−2 } −4
9
7|9
2
6|20
• • ◦ • • ◦ ◦ ◦ ◦ ◦ ◦◦
22.1
• ◦ • ◦ ◦ • ◦ ◦ ◦ • ◦◦
23
• • ◦ • ◦ • ◦ ◦ ◦ ◦ ◦◦
h5i
4
5|13
24
• • ◦ • ◦ ◦ • ◦ ◦ ◦ ◦◦
h7e6 i
6
6|17
25
• • ◦ • ◦ ◦ ◦ • ◦ ◦ ◦◦
{1}
1
0|3
25.1
• • ◦ • ◦ • ◦ ◦ ◦ ◦ ◦◦
26
• • ◦ • ◦ ◦ ◦ ◦ ◦ • ◦◦
{1}
1
12|31
26.1
• ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦◦
27
• • ◦ • ◦ ◦ ◦ ◦ ◦ ◦ •◦
11
5|13
h−1e i
3
6|14
h7i
6
10|23
4
11|23
10
9|19
8
7|15
• • ◦ ◦ • • ◦ ◦ ◦ ◦ ◦◦
28.1
• • ◦ ◦ ◦ • ◦ ◦ • ◦ ◦◦
29
• • ◦ ◦ • ◦ ◦ • ◦ ◦ ◦◦
30 31 32
• • ◦ ◦ • ◦ ◦ ◦ • ◦ ◦◦ • • ◦ ◦ • ◦ ◦ ◦ ◦ • ◦◦ • • ◦ ◦ ◦ • • ◦ ◦ ◦ ◦◦
i
] End. d N r.|N r.
22
28
h−1e
Conj. Class
{1, 7e−3 , 5e2 , −1e−1 } 7
4
h5e i −1
{1, −1e
−4
, 5e
6
, 7e }
6
{1, 5, −1e , 7e } −1
3
−1
33
• • ◦ ◦ ◦ ◦ • • ◦ ◦ ◦◦
{1, 7, −1e , 5e , e6 , 7e6 , 5e5 , −1e5 }
14
7|14
34
• ◦ • ◦ • ◦ • ◦ ◦ ◦ ◦◦
{1, 7, −1e6 , 5e6 }
9
6|17
8
11|19
35
• ◦ • ◦ • ◦ ◦ ◦ • ◦ ◦◦
4
4
{1, 7, −1e , 5e } −2
−2
36
• ◦ • ◦ ◦ ◦ • ◦ • ◦ ◦◦
{1, 7, −1e , 5e , e6 , 7e6 , 5e4 , −1e4 }
13
9|28
37
• ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦◦
3Z12 Z× 12 n e
17
7|21
h−1e−4 i
2
5|7
{1}
1
6|10
{1}
1
8|12
{1}
1
8|12
Five/Seven Elements
38
• • • • • ◦ ◦ ◦ ◦ ◦ ◦◦
38.1
• ◦ • ◦ • ◦ ◦ • ◦ • ◦◦
39
• • • • ◦ • ◦ ◦ ◦ ◦ ◦◦
39.1
• • ◦ • ◦ • ◦ ◦ ◦ ◦ •◦
40
• • • • ◦ ◦ • ◦ ◦ ◦ ◦◦
40.1
• • ◦ • ◦ ◦ • ◦ • ◦ ◦◦
41
• • • • ◦ ◦ ◦ • ◦ ◦ ◦◦
1172
APPENDIX L. CHORD AND THIRD CHAIN CLASSES
Class Nr.
Chord Classes—Continued Representative Group of d N r. = •, N r. = ◦ Symmetries
41.1
• • • ◦ ◦ • ◦ • ◦ ◦ ◦◦
42
• • • ◦ • • ◦ ◦ ◦ ◦ ◦◦
42.1
• • ◦ • ◦ • ◦ ◦ • ◦ ◦◦
43
• • • ◦ • ◦ • ◦ ◦ ◦ ◦◦
43.1
• • ◦ • ◦ • ◦ • ◦ ◦ ◦◦
44
Conj. Class
] End. d N r.|N r.
{1}
1
6|16
{1}
1
8|20
• • • ◦ • ◦ ◦ • ◦ ◦ ◦◦
h7i
6
7|9
45
• • • ◦ • ◦ ◦ ◦ • ◦ ◦◦
{1}
1
16|22
45.1
• • ◦ ◦ • ◦ • ◦ • ◦ ◦◦
46
• • • ◦ • ◦ ◦ ◦ ◦ • ◦◦
4
5|12
2
8|14
{1}
1
8|18
{1}
1
10|18
h−1e−2 i
2
9|13
h5e−4 i −2
47
• • • ◦ • ◦ ◦ ◦ ◦ ◦ •◦
47.1
• ◦ • ◦ • ◦ • ◦ ◦ • ◦◦
48
• • • ◦ ◦ • • ◦ ◦ ◦ ◦◦
48.1
• • ◦ • ◦ ◦ ◦ • • ◦ ◦◦
49
• • • ◦ ◦ • ◦ ◦ • ◦ ◦◦
49.1
• • ◦ • ◦ ◦ ◦ ◦ • • ◦◦
50
• • • ◦ ◦ • ◦ ◦ ◦ • ◦◦
50.1
• • ◦ • • ◦ ◦ ◦ • ◦ ◦◦
51
• • • ◦ ◦ ◦ • • ◦ ◦ ◦◦
h7i
6
9|11
52
• • • ◦ ◦ ◦ • ◦ • ◦ ◦◦
{1, −1e−2 , 7e6 , 5e4 }
8
7|17
53
• • ◦ • • ◦ • ◦ ◦ ◦ ◦◦
{1}
1
10|20
53.1
• • ◦ • ◦ ◦ • ◦ ◦ ◦ •◦
54
• • ◦ • • ◦ ◦ • ◦ ◦ ◦◦
{1}
1
14|26
54.1
• • ◦ ◦ • ◦ • ◦ ◦ • ◦◦
55
• • ◦ • ◦ • • ◦ ◦ ◦ ◦◦
{1, 5, −1e6 , 7e6 }
8
8|8
56
• • ◦ • ◦ • ◦ ◦ ◦ • ◦◦
h5i
4
16|16
6
6
12|16
6
57
• • ◦ • ◦ ◦ • • ◦ ◦ ◦◦
h−1e
i
h7e i
58
• • ◦ • ◦ ◦ • ◦ ◦ • ◦◦
h7e i
6
18|23
59
• • ◦ • ◦ ◦ ◦ • ◦ • ◦◦
h7i
6
13|29
60
• • ◦ ◦ • • ◦ ◦ • ◦ ◦◦
h5i
4
11|19
4
4
61
• • ◦ ◦ • ◦ ◦ • • ◦ ◦◦
{1, 7, −1e , 5e }
8
14|14
62
• ◦ • ◦ • ◦ • ◦ • ◦ ◦◦
{1, 7, −1e4 , 7e4 }
8
11|19
L.1. CHORD CLASSES
Class Nr.
1173
Chord Classes—Continued Representative Group of d N r. = •, N r. = ◦ Symmetries
Conj. Class
] End. d N r.|N r.
h−1e−5 i
3
5|5
{1}
1
9|9
{1}
1
9|9
h−1e−4 i
2
12|6
Six/Six Elements
63*
• • • • • • ◦ ◦ ◦ ◦ ◦◦
63.1*
• • ◦ • ◦ • ◦ ◦ • ◦ •◦
64*
• • • • • ◦ • ◦ ◦ ◦ ◦◦
64.1*
• • ◦ • ◦ • ◦ • ◦ ◦ •◦
65
• • • • • ◦ ◦ • ◦ ◦ ◦◦
65.1
• • • ◦ • ◦ ◦ • ◦ • ◦◦
66
• • • • • ◦ ◦ ◦ • ◦ ◦◦
66.1
• • • ◦ ◦ • ◦ • ◦ • ◦◦
67*
• • • • ◦ • ◦ • ◦ ◦ ◦◦
h5e−2 i
5
6|6
68*
• • • • ◦ • ◦ ◦ • ◦ ◦◦
{1}
1
9|9
69
• • • • ◦ • ◦ ◦ ◦ • ◦◦
{1}
1
15|11
69.1
• • ◦ • • ◦ • ◦ • ◦ ◦◦
70*
• • • • ◦ • ◦ ◦ ◦ ◦ •◦
{1, 5, −1e−3 , 7e−3 }
10
6|6
71*
• • • • ◦ ◦ • • ◦ ◦ ◦◦
{1}
1
11|11
71.1*
• • • ◦ ◦ • ◦ • • ◦ ◦◦
72
• • • • ◦ ◦ • ◦ • ◦ ◦◦
6
8|10
i
3
13|9
h7e6 i −3
73
• • • • ◦ ◦ • ◦ ◦ • ◦◦
73.1
• • ◦ • ◦ ◦ • ◦ • • ◦◦
h−1e
74*
• • • • ◦ ◦ ◦ • • ◦ ◦◦
h−1e−3 i
3
7|7
75*
• • • ◦ • • ◦ ◦ • ◦ ◦◦
{1}
1
17|17
75.1*
• • ◦ ◦ • • ◦ ◦ • ◦ •◦
76*
• • • ◦ • • ◦ ◦ ◦ • ◦◦
h5e−5 i
4
10|10
77*
• • • ◦ • ◦ • ◦ • ◦ ◦◦
4
h5e i
4
14|14
78*
• • • ◦ • ◦ • ◦ ◦ ◦ •◦
{1}
1
23|23
78.1*
• • ◦ • ◦ • ◦ • ◦ • ◦◦
79
• • • ◦ • ◦ • • ◦ ◦ ◦◦
6
18|10
9
15|11
4
11|11
h7i −2
−2
• • • ◦ • ◦ ◦ • ◦ ◦ •◦
{1, 7, 5e
81*
• • • ◦ • ◦ ◦ ◦ • • ◦◦
−4
h5e
82*
• • • ◦ ◦ • • ◦ ◦ • ◦◦
{1}
1
17|17
82.1*
• • ◦ • • ◦ ◦ • • ◦ ◦◦
83*
• • • ◦ ◦ ◦ • • • ◦ ◦◦
{1, −1e−2 , 5e−2 , 7}
13
12|12
80
, −1e
}
i
1174
APPENDIX L. CHORD AND THIRD CHAIN CLASSES
Class Nr.
Chord Classes—Continued Representative Group of d N r. = •, N r. = ◦ Symmetries
Conj. Class
] End. d N r.|N r.
7
14|14
8
15|23
12
20|20
ne6Z12 84* 85 86*
• • ◦ • • ◦ • ◦ ◦ • ◦◦ • • ◦ • • ◦ ◦ • ◦ • ◦◦ • • ◦ • ◦ ◦ • • ◦ • ◦◦
h7e3 i −4
{1, −1e
−4
, 5e
, 7}
6Z12
h7i n e 4
8
4
8
87*
• • ◦ • • ◦ ◦ • • ◦ ◦◦
{1, e , e , 5, 5e , 5e , −1e−1 , −1e3 , −1e7 , 7e−1 , 7e3 , 7e7 }
16
12|12
88*
• ◦ • ◦ • ◦ • ◦ • ◦ •◦
Z12 Z× 12 n e
18
12|12
L.2. THIRD CHAIN CLASSES
L.2
1175
Third Chain Classes
The following list of third chain translation classes shows the class number in the first column, where equivalence (∼) means that the same pc set is generated. The second column shows the pitch classes in the order of appearance along the third chain. The third column shows the third chain, the fourth column shows the chord class of the pc set, and the fifth column shows lead-sheet symbols as systematically derived in subsection 25.2.1. Third Chains Chain Nr.
Pitch Classes
Third
Chord
Lead-Sheet
∼ equiv.
from 0
Chain
Class
Symbols
1
0,3
2
0,4
Two Pitch Classes 3
5
trd
4
6
T rd
Three Pitch Classes 3
0,3,6
33
15
C0, Cm5-
4
0,3,7
34
10.1
Cm
5
0,4,7
43
10.1
C
6
0,4,8
44
16
C+, C5+
Four Pitch Classes 7
0,3,6,9
333
37
C07-
8
0,3,6,10
334
26.1
C07
9
0,3,7,10
343
22.1
Cm7
10
0,3,7,11
344
30
Cm7+
11
0,4,7,10
433
26.1
C7
12
0,4,7,11
434
28.1
C7+
13
0,4,8,11
443
30
C+7+
Five Pitch Classes 14
0,3,6,9,1
3334
58
C07-/9-
15
0,3,6,10,1
3343
53.1
C09-
16
0,3,6,10,2
3344
56
C09
17
0,3,7,10,1
3433
53.1
Cm9-
18
0,3,7,10,2
3434
42.1
Cm9
19
0,3,7,11,2
3443
59
Cm7+/9, Cmmaj7/9
20
0,4,7,10,1
4333
58
C9-
21
0,4,7,10,2
4334
47.1
C9
22
0,4,7,11,2
4343
42.1
C7+/9, Cmaj7/9
23
0,4,7,11,3
4344
60
C7+/9+
24
0,4,8,11,2
4433
56
C+7+/9
1176
APPENDIX L. CHORD AND THIRD CHAIN CLASSES Third Chains—Continued
Chain Nr.
Pitch Classes
Third
Chord
Lead-Sheet
∼ equiv.
from 0
Chain
Class
Symbols
25
0,4,8,11,3
4434
60
C+7+/9+
Six Pitch Classes 26
0,3,6,9,1,4
33343
84*
C07-/9-/11-
27
0,3,6,9,1,5
33344
85b
C07-/9-/11
28
0,3,6,10,1,4
33433
79
C09-/11-
29
0,3,6,10,1,5
33434
65.1b
C09-/11
30
0,3,6,10,2,5
33443
69.1
C011
31
0,3,7,10,1,4
34333
84*
Cm9-/11-
32
0,3,7,10,1,5
34334
64.1*
Cm9-/11
33
0,3,7,10,2,5
34343
63.1*
Cm11
34
0,3,7,10,2,6
34344
75.1*
Cm11+
35
0,3,7,11,2,5
34433
69.1
Cm7+/11
36
0,3,7,11,2,6
34434
82*
Cm7+/11+
37
0,4,7,10,1,5
43334
73.1
C9-/11
38
0,4,7,10,2,5
43343
64.1*
C11
39
0,4,7,10,2,6
43344
78.1*
C11+
40
0,4,7,11,2,5
43433
65.1b
C7+/11
41
0,4,7,11,2,6
43434
66.1b
C7+/11+
42
0,4,7,11,3,6
43443
82*
C7+/9+/11+
43
0,4,8,11,2,5
44333
85b
C+7+/11
44
0,4,8,11,2,6
44334
78.1*
C+7+/11+
45
0,4,8,11,3,6
44343
75.1*
C+7+/9+/11+
46
0,4,8,11,3,7
44344
87*
C+7+/9+/(11)/13-
47
0,3,6,9,1,4,7
333433
58b
C07-/9-/11-/13-
48
0,3,6,9,1,4,8
333434
54.1b
C07-/9-/11-/13
49
0,3,6,9,1,5,8
333443
54.1b
C07-/9-/13
50
0,3,6,10,1,4,7
334333
58b
C09-/11-/13-
51
0,3,6,10,1,4,8
334334
47.1
C09-/11-/13
52
0,3,6,10,1,5,8
334343
38.1
C09-/13
53
0,3,6,10,1,5,9
334344
54.1b
C09-/13+
54
0,3,6,10,2,5,8
334433
47.1b
C013
55
0,3,6,10,2,5,9
334434
54.1b
C013+
56
0,3,7,10,1,4,8
343334
54.1b
Cm9-/11-/13
57
0,3,7,10,1,5,8
343343
38.1b
Cm9-/13
Seven Pitch Classes
L.2. THIRD CHAIN CLASSES
1177 Third Chains—Continued
Chain Nr.
Pitch Classes
Third
Chord
Lead-Sheet
∼ equiv.
from 0
Chain
Class
Symbols
58
0,3,7,10,1,5,9
343344
47.1b
Cm9-/13+
59
0,3,7,10,2,5,8
343433
38.1b
Cm13
60
0,3,7,10,2,5,9
343434
38.1b
Cm13+
61
0,3,7,10,2,6,9
343443
54.1b
Cm11+/13+
62
0,3,7,11,2,5,8
344333
54.1b
Cm7+/13
63
0,3,7,11,2,5,9
344334
47.1b
Cm7+/13+
64
0,3,7,11,2,6,9
344343
54.1b
Cm7+/11+/13+
65
0,3,7,11,2,6,10
344344
60b
Cm7+/11+/(13)/15-
66
0,4,7,10,1,5,8
433343
54.1b
C9-/13
67
0,4,7,10,1,5,9
433344
54.1b
C9-/13+
68
0,4,7,10,2,5,8
433433
47.1b
C13
69
0,4,7,10,2,5,9
433434
38.1b
C13+
70
0,4,7,10,2,6,9
433443
47.1b
C11+/13+
71
0,4,7,11,2,5,8
434333
54.1b
C7+/13
72
0,4,7,11,2,5,9
434334
38.1b
C7+/13+
73
0,4,7,11,2,6,9
434343
38.1b
C7+/11+/13+
74
0,4,7,11,2,6,10
434344
45.1b
C7+/11+/(13)/15-
75
0,4,7,11,3,6,9
434433
54.1b
C7+/9+/11+/13+
76
0,4,7,11,3,6,10
434434
55b
C7+/9+/11+/(13)/15-
77
0,4,8,11,2,5,9
443334
54.1b
C+7+/13+
78
0,4,8,11,2,6,9
443343
47.1b
C+7+/11+/13+
79
0,4,8,11,2,6,10
443344
62b
C+7+/11+/(13)/15-
80
0,4,8,11,3,6,9
443433
54.1b
C+7+/9+/11+/13+
81
0,4,8,11,3,6,10
443434
45.1b
C+7+/9+/11+/(13)/15-
82
0,4,8,11,3,7,10
443443
60b
C+7+/9+/(11)/13-/15-
83
0,3,6,9,1,4,7,10
3334333
37b
C07-/9-/11-/13- . . .
84
0,3,6,9,1,4,7,11
3334334
26.1b
C07-/9-/11-/13- . . .
85
0,3,6,9,1,4,8,11
3334343
22.1b
C07-/9-/11-/13 . . .
86
0,3,6,9,1,5,8,11
3334433
26.1b
C07-/9-/13 . . .
87
0,3,6,10,1,4,7,11
3343334
29b
C09-/11-/13- . . .
88
0,3,6,10,1,4,8,11
3343343
18.1b
C09-/11-/13 . . .
89
0,3,6,10,1,5,8,11
3343433
17.1b
C09-/13 . . .
90
0,3,6,10,2,5,8,11
3344333
26.1b
C013 . . .
91
0,3,6,10,2,5,9,1
3344344
31b
C013+ . . .
Eight Pitch Classes
1178
APPENDIX L. CHORD AND THIRD CHAIN CLASSES Third Chains—Continued
Chain Nr.
Pitch Classes
Third
Chord
Lead-Sheet
∼ equiv.
from 0
Chain
Class
Symbols
92
0,3,7,10,1,4,8,11
3433343
31b
Cm9-/11-/13 . . .
93
0,3,7,10,1,5,8,11
3433433
18.1b
Cm9-/13 . . .
94
0,3,7,10,2,5,8,11
3434333
22.1b
Cm13 . . .
95
0,3,7,10,2,5,9,1
3434344
18.1b
Cm13+ . . .
96
0,3,7,10,2,6,9,1
3434434
29b
Cm11+/13+ . . .
97
0,3,7,11,2,5,9,1
3443344
34b
Cm7+/13+ . . .
98
0,3,7,11,2,6,9,1
3443434
29b
Cm7+/11+/13+ . . .
99
0,3,7,11,2,6,10,1
3443443
28b
Cm7+/11+/(13)/15- . . .
100
0,4,7,10,1,5,8,11
4333433
29b
C9-/13 . . .
101
0,4,7,10,2,5,8,11
4334333
18.1b
C15
102
0,4,7,10,2,5,9,1
4334344
22.1b
C13+ . . .
103
0,4,7,10,2,6,9,1
4334434
26.1b
C11+/13+ . . .
104
0,4,7,11,2,5,9,1
4343344
18.1b
C7+/13+ . . .
105
0,4,7,11,2,6,9,1
4343434
17.1b
C7+/11+/13+ . . .
106
0,4,7,11,2,6,10,1
4343443
25.1b
C7+/11+/(13)/15- . . .
107 ∼ 84
0,4,7,11,3,6,9,1
4344334
26.1b
C7+/9+/11+/13+ . . .
108 ∼ 87
0,4,7,11,3,6,10,1
4344343
29b
C7+/9+/11+/(13)/15- . . .
109
0,4,7,11,3,6,10,2
4344344
30b
C7+/9+/11+/(13)/15- . . .
110
0,4,8,11,2,5,9,1
4433344
31b
C+7+/13+ . . .
111
0,4,8,11,2,6,9,1
4433434
18.1b
C+7+/11+/13+ . . .
112
0,4,8,11,2,6,10,1
4433443
34b
C+7+/11+/(13)/15- . . .
113 ∼ 85
0,4,8,11,3,6,9,1
4434334
22.1b
C+7+/9+/11+/13+ . . .
114 ∼ 88
0,4,8,11,3,6,10,1
4434343
18.1b
C+7+/9+/11+/(13)/15- . . .
115
0,4,8,11,3,6,10,2
4434344
35b
C+7+/9+/11+/(13)/15- . . .
116 ∼ 92
0,4,8,11,3,7,10,1
4434433
31b
C+7+/9+/(11)/13-/15- . . .
117
0,4,8,11,3,7,10,2
4434434
30b
C+7+/9+/(11)/13-/15- . . .
Nine Pitch Classes 118
0,3,6,9,1,4,7,10,2
33343334
15b
C07-/9-/11-/13- . . .
119
0,3,6,9,1,4,7,11,2
33343343
9.1b
C07-/9-/11-/13- . . .
120
0,3,6,9,1,4,8,11,2
33343433
9.1b
C07-/9-/11-/13 . . .
121
0,3,6,9,1,5,8,11,2
33344333
15b
C07-/9-/13 . . .
122
0,3,6,10,1,4,7,11,2
33433343
10b
C09-/11-/13- . . .
123
0,3,6,10,1,4,8,11,2
33433433
13b
C09-/11-/13 . . .
124
0,3,6,10,1,5,8,11,2
33434333
9.1b
C09-/13 . . .
125
0,3,6,10,2,5,9,1,4
33443443
10b
C013+ . . .
L.2. THIRD CHAIN CLASSES
1179 Third Chains—Continued
Chain Nr.
Pitch Classes
Third
Chord
Lead-Sheet
∼ equiv.
from 0
Chain
Class
Symbols
126
0,3,7,10,1,4,8,11,2
34333433
10b
Cm9-/11-/13 . . .
127
0,3,7,10,1,5,8,11,2
34334333
9.1b
Cm9-/13 . . .
128
0,3,7,10,2,5,9,1,4
34343443
9.1b
Cm13+ . . .
129
0,3,7,10,2,6,9,1,4
34344343
15b
Cm11+/13+ . . .
130
0,3,7,11,2,5,9,1,4
34433443
13b
Cm7+/13+ . . .
131 ∼ 119
0,3,7,11,2,6,9,1,4
34434343
9.1b
Cm7+/11+/13+ . . .
132
0,3,7,11,2,6,9,1,5
34434344
14b
Cm7+/11+/13+ . . .
133 ∼ 122
0,3,7,11,2,6,10,1,4
34434433
10b
Cm7+/11+/(13)/15- . . .
134
0,3,7,11,2,6,10,1,5
34434434
11b
Cm7+/11+/(13)/15- . . .
135
0,4,7,10,1,5,8,11,2
43334333
15b
C9-/13 . . .
136
0,4,7,10,1,5,8,11,3
43334334
10.1b
C9-/13 . . .
137
0,4,7,10,2,5,8,11,3
43343334
10.1b
C17
138
0,4,7,10,2,6,9,1,5
43344344
10.1b
C11+/13+ . . .
139
0,4,7,11,2,6,9,1,5
43434344
8.1b
C7+/11+/13+ . . .
140
0,4,7,11,2,6,10,1,5
43434434
12b
C7+/11+/(13)/15- . . .
141
0,4,7,11,3,6,9,1,5
43443344
14b
C7+/9+/11+/13+ . . .
142
0,4,7,11,3,6,10,1,5
43443434
12b
C7+/9+/11+/(13)/15- . . .
143
0,4,7,11,3,6,10,2,5
43443443
11b
C7+/9+/11+/(13)/15- . . .
144
0,4,8,11,2,6,9,1,5
44334344
10.1b
C+7+/11+/13+ . . .
145
0,4,8,11,2,6,10,1,5
44334434
14b
C+7+/11+/(13)/15- . . .
146
0,4,8,11,3,6,9,1,5
44343344
10.1b
C+7+/9+/11+/13+ . . .
147
0,4,8,11,3,6,10,1,5
44343434
8.1b
C+7+/9+/11+/(13)/15- . . .
148
0,4,8,11,3,6,10,2,5
44343443
14b
C+7+/9+/11+/(13)/15- . . .
149 ∼ 136
0,4,8,11,3,7,10,1,5
44344334
10.1b
C+7+/9+/(11)/13-/15- . . .
150 ∼ 137
0,4,8,11,3,7,10,2,5
44344343
10.1b
C+7+/9+/(11)/13-/15- . . .
151
0,4,8,11,3,7,10,2,6
44344344
16b
C+7+/9+/(11)/13-/15- . . .
Ten Pitch Classes 152
0,3,6,9,1,4,7,10,2,5
333433343
5b
C07-/9-/11-/13- . . .
153
0,3,6,9,1,4,7,11,2,5
333433433
4b
C07-/9-/11-/13- . . .
154
0,3,6,9,1,4,8,11,2,5
333434333
5b
C07-/9-/11-/13 . . .
155
0,3,6,10,1,4,7,11,2,5
334333433
3b
C09-/11-/13- . . .
156
0,3,6,10,1,4,8,11,2,5
334334333
4b
C09-/11-/13 . . .
157 ∼ 152
0,3,6,10,2,5,9,1,4,7
334434433
5b
C013+ . . .
158
0,3,6,10,2,5,9,1,4,8
334434434
6b
C013+ . . .
159
0,3,7,10,1,4,8,11,2,5
343334333
5b
Cm9-/11-/13 . . .
1180
APPENDIX L. CHORD AND THIRD CHAIN CLASSES Third Chains—Continued
Chain Nr.
Pitch Classes
Third
Chord
Lead-Sheet
∼ equiv.
from 0
Chain
Class
Symbols
160
0,3,7,10,1,4,8,11,2,6
343334334
6b
Cm9-/11-/13 . . .
161
0,3,7,10,1,5,8,11,2,6
343343334
3.1b
Cm9-/13 . . .
162
0,3,7,10,2,5,9,1,4,8
343434434
3.1b
Cm13+ . . .
163
0,3,7,10,2,6,9,1,4,8
343443434
7b
Cm11+/13+ . . .
164
0,3,7,11,2,5,9,1,4,8
344334434
6b
Cm7+/13+ . . .
165
0,3,7,11,2,6,9,1,4,8
344343434
3.1b
Cm7+/11+/13+ . . .
166
0,3,7,11,2,6,9,1,5,8
344343443
7b
Cm7+/11+/13+ . . .
167 ∼ 160
0,3,7,11,2,6,10,1,4,8
344344334
6b
Cm7+/11+/(13)/15- . . .
168 ∼ 161
0,3,7,11,2,6,10,1,5,8
344344343
3.1b
Cm7+/11+/(13)/15- . . .
169
0,3,7,11,2,6,10,1,5,9
344344344
6b
Cm7+/11+/(13)/15- . . .
170
0,4,7,10,1,5,8,11,2,6
433343334
7b
C9-/13 . . .
171
0,4,7,10,1,5,8,11,3,6
433343343
3.1b
C9-/13 . . .
172
0,4,7,10,2,5,8,11,3,6
433433343
6b
C19
173
0,4,7,10,2,6,9,1,5,8
433443443
6b
C11+/13+ . . .
174
0,4,7,11,2,6,9,1,5,8
434343443
3.1b
C7+/11+/13+ . . .
175 ∼ 170
0,4,7,11,2,6,10,1,5,8
434344343
7b
C7+/11+/(13)/15- . . .
176
0,4,7,11,2,6,10,1,5,9
434344344
3.1b
C7+/11+/(13)/15- . . .
177 ∼ 171
0,4,7,11,3,6,9,1,5,8
434433443
3.1b
C7+/9+/11+/13+ . . .
178
0,4,7,11,3,6,10,1,5,8
434434343
6b
C7+/9+/11+/(13)/15- . . .
179
0,4,7,11,3,6,10,1,5,9
434434344
7b
C7+/9+/11+/(13)/15- . . .
180 ∼ 172
0,4,7,11,3,6,10,2,5,8
434434433
6b
C7+/9+/11+/(13)/15- . . .
181
0,4,7,11,3,6,10,2,5,9
434434434
3.1b
C7+/9+/11+/(13)/15- . . .
182
0,4,8,11,2,6,10,1,5,9
443344344
6b
C+7+/11+/(13)/15- . . .
183
0,4,8,11,3,6,10,1,5,9
443434344
3.1b
C+7+/9+/11+/(13)/15- . . .
184
0,4,8,11,3,6,10,2,5,9
443434434
7b
C+7+/9+/(11)/13-/15- . . .
185
0,4,8,11,3,7,10,1,5,9
443443344
6b
C+7+/9+/(11)/13-/15- . . .
186
0,4,8,11,3,7,10,2,5,9
443443434
3.1b
C+7+/9+/(11)/13-/15- . . .
187
0,4,8,11,3,7,10,2,6,9
443443443
6b
C+7+/9+/(11)/13-/15- . . .
Eleven Pitch Classes 188
0,3,6,9,1,4,7,10,2,5,8
3334333433
2b
C07-/9-/11-/13- . . .
189
0,3,6,9,1,4,7,11,2,5,8
3334334333
2b
C07-/9-/11-/13- . . .
190
0,3,6,10,1,4,7,11,2,5,8
3343334333
2b
C09-/11-/13- . . .
191
0,3,6,10,1,4,7,11,2,5,9
3343334334
2b
C09-/11-/13- . . .
192
0,3,6,10,1,4,8,11,2,5,9
3343343334
2b
C09-/11-/13 . . .
193 ∼ 191
0,3,6,10,2,5,9,1,4,7,11
3344344334
2b
C013+ . . .
L.2. THIRD CHAIN CLASSES
1181 Third Chains—Continued
Chain Nr.
Pitch Classes
Third
Chord
Lead-Sheet
∼ equiv.
from 0
Chain
Class
Symbols
194 ∼ 192
0,3,6,10,2,5,9,1,4,8,11
3344344343
2b
C013+ . . .
195
0,3,7,10,1,4,8,11,2,5,9
3433343334
2b
Cm9-/11-/13 . . .
196
0,3,7,10,1,4,8,11,2,6,9
3433343343
2b
Cm9-/11-/13 . . .
197
0,3,7,10,1,5,8,11,2,6,9
3433433343
2b
Cm9-/13 . . .
198 ∼ 195
0,3,7,10,2,5,9,1,4,8,11
3434344343
2b
Cm13+ . . .
199
0,4,7,10,1,5,8,11,2,6,9
4333433343
2b
C9-/13 . . .
200
0,4,7,10,1,5,8,11,3,6,9
4333433433
2b
C9-/13 . . .
201
0,4,7,10,2,5,8,11,3,6,9
4334333433
2b
C21
202 ∼ 199
0,4,7,10,2,6,9,1,5,8,11
4334434433
2b
C11+/13+ . . .
203 ∼ 191
0,4,7,11,3,6,10,2,5,9,1
4344344344
2b
C7+/9+/11+/(13)/15- . . .
204 ∼ 192
0,4,8,11,3,6,10,2,5,9,1
4434344344
2b
C+7+/9+/(11)/13-/15- . . .
205 ∼ 195
0,4,8,11,3,7,10,2,5,9,1
4434434344
2b
C+7+/9+/(11)/13-/15- . . .
206 ∼ 196
0,4,8,11,3,7,10,2,6,9,1
4434434434
2b
C+7+/9+/(11)/13-/15- . . .
207
0,3,6,9,1,4,7,10,2,5,8,11
33343334333
1b
C07-/9-/11-/13- . . .
208 ∼ 207
0,4,7,10,2,5,8,11,3,6,9,1
43343334334
1b
C23
209 ∼ 207
0,4,7,10,2,6,9,1,5,8,11,3
43344344334
1b
C11+/13+ . . .
210 ∼ 207
0,4,8,11,3,7,10,2,6,9,1,5
44344344344
1b
C+7+/9+/(11)/13-/15- . . .
Twelve Pitch Classes
Appendix M
Two, Three, and Four Tone Motif Classes
M.1
Two Tone Motifs in OnP iM od12,12
ClassN r.
Representative
1
(0, 0), (0, 1)
2
(0, 0), (0, 2)
3
(0, 0), (0, 3)
4
(0, 0), (0, 4)
5
(0, 0), (0, 6) 1183
1184
M.2
APPENDIX M. TWO, THREE, AND FOUR TONE MOTIF CLASSES
Two Tone Motifs in OnP iM od5,12 ClassN r.
Representative
1
(0, 0), (0, 1)
2
(0, 0), (0, 2)
3
(0, 0), (0, 3)
4
(0, 0), (0, 4)
5
(0, 0), (0, 6)
6
(0, 0), (1, 0)
7
(0, 0), (1, 1)
8
(0, 0), (1, 2)
9
(0, 0), (1, 2)
10
(0, 0), (1, 4)
11
(0, 0), (1, 6)
M.3. THREE TONE MOTIFS IN ON P IM OD12,12
M.3
1185
Three Tone Motifs in OnP iM od12,12
Refer to the discussion in subsection 11.3.8 for the entries of this table. The order of these representatives is a historical one. After this table, the representatives are also visualized on a 12 × 12 square in list M.1.
Class Nr.
Three-Element Motif Classes in OnP iM od12,12 Representative Kernel Class Weight
1
(0, 0), (1, 0), (2, 0)
Z.(1, 2) × Z.(1, 1)
(1, 1, 2)
0
2
(0, 0), (1, 0), (3, 0)
Z.(1, 2) × Z.(0, 1)
(1, 2, 3)
0
3
(0, 0), (1, 0), (4, 0)
Z.(1, 0) × Z.(0, 1)
(1, 3, 4)
0
4
(0, 0), (1, 0), (5, 0)
Z.(1, 2) × Z.(1, 1)
(1, 1, 4)
0
5
(0, 0), (1, 0), (6, 0)
Z.(1, 2) × Z.(1, 0)
(1, 1, 6)
0
6
(0, 0), (2, 0), (4, 0)
(Z4 × 2Z4 ) × Z.(1, 1)
(2, 2, 4)
0
7
(0, 0), (2, 0), (6, 0)
(Z4 × 2Z4 ) × Z.(0, 1)
(2, 4, 6)
0
(3, 3, 6)
0
(4, 4, 4)
0
Z23
Volume
(0, 0), (3, 0), (6, 0)
Z.(1, 2) ×
9
(0, 0), (4, 0), (8, 0)
(Z24 )
10
(0, 0), (1, 0), (0, 1)
0×0
(1, 1, 1)
1
11
(0, 0), (2, 0), (0, 1)
Z.(2, 0) × 0
(1, 1, 2)
2
12
(0, 0), (3, 0), (0, 1)
0 × Z.(1, 0)
(1, 1, 3)
3
13
(0, 1), (0, 2), (3, 0)
0 × Z.(1, 1)
(1, 1, 1)
3
14
(0, 0), (0, 1), (4, 0)
Z.(1, 0) × 0
(1, 1, 4)
4
15
(0, 0), (1, 2), (2, 0)
Z.(1, 2) × 0
(1, 1, 2)
4
16
(0, 0), (2, 0), (0, 2)
2Z24
(2, 2, 2)
4
17
(0, 0), (6, 0), (0, 1)
Z.(2, 0) × Z.(1, 0)
(1, 1, 6)
6
18
(0, 0), (3, 0), (0, 2)
Z.(2, 0) × Z.(0, 1)
(1, 2, 3)
6
19
(0, 0), (0, 2), (3, 1)
Z.(2, 0) × Z.(1, 1)
(1, 1, 2)
6
20
(0, 0), (4, 0), (0, 2)
(Z4 × 2Z4 ) × 0
(2, 2, 4)
4
(3, 3, 3)
3
(2, 2, 6)
0
(2, 2, 2)
0
(4, 4, 4)
4
(3, 3, 6)
6
(6, 6, 6)
0
8
× Z.(1, 1)
×0
Z23
(0, 0), (4, 0), (0, 4)
0×
22
(0, 0), (6, 0), (0, 2)
23
(0, 2), (0, 4), (6, 0)
24
(0, 0), (4, 0), (0, 4)
25
(0, 0), (6, 0), (0, 3)
26
(0, 0), (6, 0), (0, 6)
2Z24 × Z.(1, 0) 2Z24 × Z.(1, 1) Z24 × 0 Z.(2, 0) × Z23 2Z24 × Z23
21
1186
APPENDIX M. TWO, THREE, AND FOUR TONE MOTIF CLASSES
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Figure M.1: Representatives of the 26 isomorphism classes of three-element motives in OnP iM od12,12 .
M.3. THREE TONE MOTIFS IN ON P IM OD12,12
1187
10 11
12 13 21 19
17
18
14 15 25
16
3 2
20 24
5 23 22 1
4 6
26
8
7
9 Figure M.2: Hasse diagram of dominance/specialization among the 26 isomorphism classes of motives in OnP iM od12,12 .
1188
M.4
APPENDIX M. TWO, THREE, AND FOUR TONE MOTIF CLASSES
Four Tone Motifs in OnP iM od12,12
This list was calculated by Straub in [513], refer to subsection 11.3.8 for details. The list’s numbering follows Straub’s algorithm; * denotes classes which are not determined by volume and class weight.
Class Nr.
Four-Element Motif Classes Representative Class Weight
Volume
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
(0,0),(0,1),(0,2),(0,7) (0,0),(0,1),(0,2),(0,3) (0,0),(0,1),(0,2),(0,6) (0,0),(0,1),(0,2),(0,5) (0,0),(0,1),(0,2),(0,4) (0,0),(1,0),(0,5),(0,6) (0,0),(0,1),(0,4),(0,5) (0,0),(0,1),(0,3),(0,5) (0,0),(0,1),(0,4),(0,8) (0,0),(0,1),(0,6),(0,7) (0,0),(0,1),(0,3),(0,6) (0,0),(0,1),(0,3),(0,7) (0,0),(0,1),(0,4),(0,7) (0,0),(0,1),(0,3),(0,10) (0,0),(0,1),(0,3),(0,4) (0,0),(0,1),(0,3),(0,9) (0,0),(0,1),(0,4),(0,9) (0,0),(0,2),(6,0),(6,10) (0,0),(0,2),(0,4),(6,0) (0,0),(0,2),(0,4),(6,2) (0,0),(0,2),(0,4),(0,6) (0,0),(0,2),0,4),(0,8) (0,0),(0,2),(6,0),(6,2) (0,0),(0,2),(6,0),(6,6) (0,0),(0,2),(0,6),(6,2) (0,0),(0,2),(0,6),(6,0) (0,0),(0,2),(0,6),(0,8) (0,0),(0,3),(0,6),(0,9) (0,0),(0,6),(6,0),(6,6)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
(1,1,5,5) (1,1,2,2) (1,4,5,7) (1,4,2,3) (1,2,3,6) (4,4,5,5) (4,4,3,3) (4,2,2,6) (4,3,3,9) (5,5,5,5) (5,2,2,8) (5,2,3,7) (5,3,3,8) (2,2,2,2) (2,2,3,3) (2,3,7,8) (3,3,3,3) (23,23,22,22) (23,6,22,7) (6,6,22,22) (6,6,7,7) (6,7,7,9) (22,22,22,22) (22,22,22,26) (22,22,7,7) (22,7,7,26) (7,7,7,7) (8,8,8,8) (26,26,26,26)
M.4. FOUR TONE MOTIFS IN ON P IM OD12,12
1189
Four-Element Motif Classes—Continued Class Nr. Representative Class Weight Volume 29 30 31 32 33 34* 35* 36* 37* 38* 39* 40* 41* 42* 43 44* 45* 46* 47 48* 49* 50 51 52 53 54 55 56 57 58 59 60 61 62
(0,0),(0,1),(0,2),(1,0) (0,0),(0,1),(0,5),(1,0) (0,0),(0,1),(0,6),(1,0) (0,0),(0,1),(0,3),(1,0) (0,0),(0,1),(0,4),(1,0) (0,0),(0,1),(1,0),(1,5) (0,0),(0,1),(1,0),(7,7) (0,0),(0,1),(1,0),(1,1) (0,0),(0,1),(1,0),(3,5) (0,0),(0,1),(1,0),(3,11) (0,0),(0,1),(1,0),(1,2) (0,0),(0,1),(1,0),(5,10) (0,0),(0,1),(1,0),(4,10) (0,0),(0,1),(1,0),(2,4) (0,0),(0,1),(1,0),(2,5) (0,0),(0,1),(1,0),(1,3) (0,0),(0,1),(1,0),(7,9) (0,0),(0,1),(1,0),(3,3) (0,0),(0,1),(1,0),(6,8) (0,0),(0,1),(1,0),(1,4) (0,0),(0,1),(1,0),(4,4) (0,0),(0,1),(1,0),(1,6) (0,0),(0,1),(1,0),(2,2) (0,0),(0,1),(1,0),(2,3) (0,0),(0,1),(1,0),(6,9) (0,0),(0,1),(1,0),(8,8) (0,0),(0,1),(1,0),(3,4) (0,0),(0,1),(2,0),(3,1) (0,0),(0,1),(2,0),(3,4) (0,0),(0,1),(3,0),(4,1) (0,0),(0,1),(0,2),(2,1) (0,0),(0,1),(0,2),(2,0) (0,0),(0,1),(0,5),(2,0) (0,0),(0,1),(0,5),(2,1)
(1,10,10,11) (4,10,10,14) (5,10,10,17) (2,10,11,12) (3,10,12,14) (10,10,10,10) (10,10,10,10) (10,10,10,10) (10,10,10,13) (10,10,10,13) (10,10,11,11) (10,10,11,11) (10,10,11,15) (10,10,11,15) (10,10,11,19) (10,10,12,12) (10,10,12,12) (10,10,12,12) (10,10,15,19) (10,10,14,14) (10,10,14,14) (10,10,17,17) (10,11,11,13) (10,11,13,15) (10,11,12,18) (10,13,14,14) (10,12,15,18) (11,11,12,12) (11,12,12,15) (12,12,14,14) (1,11,11,15) (1,11,11,16) (4,11,11,14) (4,11,11,20)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2
1190
APPENDIX M. TWO, THREE, AND FOUR TONE MOTIF CLASSES Four-Element Motif Classes—Continued Class Nr. Representative Class Weight Volume 63 64 65 66 67 68 69 70 71 72 73* 74* 75* 76* 77* 78* 79* 80* 81* 82* 83* 84* 85* 86* 87* 88* 89* 90* 91 92 93 94 95 96
(0,0),(0,1),(0,6),(2,1) (0,0),(0,1),(0,6),(2,0) (0,0),(0,1),(0,3),(2,0) (0,0),(0,1),(0,3),(2,1) (0,0),(0,1),(0,4),(2,1) (0,0),(0,1),(0,4),(2,0) (0,0),(0,1),(2,0),(4,6) (0,0),(0,1),(2,0),(4,0) (0,0),(0,1),(2,0),(6,6) (0,0),(0,1),(2,0),(6,0) (0,0),(0,1),(2,0),(2,1) (0,0),(0,1),(2,0),(2,5) (0,0),(0,1),(2,0),(2,7) (0,0),(0,1),(2,0),(2,11) (0,0),(0,1),(2,0),(6,5) (0,0),(0,1),(2,0),(6,11) (0,0),(0,1),(2,0),(4,7) (0,0),(0,1),(2,0),(8,11) (0,0),(0,1),(2,0),(2,2) (0,0),(0,1),(2,0),(8,10) (0,0),(0,1),(2,0),(4,1) (0,0),(0,1),(2,0),(8,5) (0,0),(0,1),(2,0),(8,4) (0,0),(0,1),(2,0),(2,4) (0,0),(0,1),(2,0),(6,7) (0,0),(0,1),(2,0),(6,1) (0,0),(0,1),(2,0),(2,9) (0,0),(0,1),(2,0),(2,3) (0,0),(0,1),(2,0),(4,3) (0,0),(0,1),(2,0),(4,2) (0,0),(0,1),(2,0),(4,9) (0,0),(0,1),(2,0),(4,8) (0,0),(0,1),(4,2),(6,1) (0,0),(0,1),(4,2),(6,4)
(5,5,11,11) (5,22,11,11) (2,11,15,18) (2,11,16,18) (3,11,14,18) (3,11,20,18) (23,11,11,15) (6,11,11,14) (22,11,15,17) (7,11,14,17) (11,11,11,11) (11,11,11,11) (11,11,11,11) (11,11,11,11) (11,11,11,19) (11,11,11,19) (11,11,15,15) (11,11,15,15) (11,11,15,16) (11,11,15,16) (11,11,14,14) (11,11,14,14) (11,11,14,20) (11,11,14,20) (11,11,17,17) (11,11,17,17) (11,11,18,18) (11,11,18,18) (11,15,15,19) (11,15,16,19) (11,14,14,19) (11,14,20,19) (15,15,17,17) (15,16,18,18)
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
M.4. FOUR TONE MOTIFS IN ON P IM OD12,12
1191
Four-Element Motif Classes—Continued Class Nr. Representative Class Weight Volume 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113* 114* 115* 116 117 118* 119* 120* 121* 122* 123 124 125 126 127 128* 129* 130
(0,0),(0,1),(4,0),(6,1) (0,0),(0,1),(4,0),(6,4) (0,0),(0,1),(0,2),(3,0) (0,0),(0,1),(0,2),(3,1) (0,0),(0,1),(0,5),(3,0) (0,0),(0,1),(0,4),(3,2) (0,0),(0,1),(0,6),(3,2) (0,0),(0,1),(0,6),(3,1) (0,0),(0,1),(0,6),(3,0) (0,0),(0,1),(0,3),(3,2) (0,0),(0,1),(0,3),(3,1) (0,0),(0,1),(0,3),(3,0) (0,0),(0,1),(0,4),(3,0) (0,0),(0,1),(0,4),(3,1) (0,0),(0,1),(3,0),(6,0) (0,0),(0,3),(0,6),(3,0) (0,0),(0,1),(3,0),(3,5) (0,0),(0,1),(3,0),(3,11) (0,0),(0,1),(3,0),(9,11) (0,0),(0,1),(3,2),(3,8) (0,0),(0,1),(3,0),(3,2) (0,0),(0,1),(3,0),(9,7) (0,0),(0,1),(3,0),(3,7) (0,0),(0,1),(3,0),(3,1) (0,0),(0,1),(3,0),(9,3) (0,0),(0,1),(3,0),(3,3) (0,0),(0,1),(3,0),(6,5) (0,0),(0,1),(3,0),(6,1) (0,0),(0,1),(3,0),(3,6) (0,0),(0,1),(3,0),(3,10) (0,0),(0,1),(3,0),(6,9) (0,0),(0,3),(3,0),(3,3) (0,0),(0,3),(3,0),(9,9) (0,0),(0,3),(3,0),(3,6)
(14,14,17,17) (14,20,18,18) (1,13,12,18) (1,12,12,19) (4,4,12,12) (4,3,13,12) (5,13,13,17) (5,12,12,17) (5,12,12,25) (2,13,12,19) (2,12,12,18) (2,12,21,18) (3,3,12,12) (3,3,12,21) (8,12,12,17) (8,21,21,25) (13,13,12,12) (13,13,12,12) (13,13,12,12) (13,13,17,17) (13,12,19,18) (12,12,12,12) (12,12,12,12) (12,12,12,12) (12,12,12,21) (12,12,12,21) (12,12,19,19) (12,12,17,17) (12,12,17,25) (12,12,18,18) (12,21,18,18) (21,21,21,21) (21,21,21,21) (21,21,25,25)
2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
1192
APPENDIX M. TWO, THREE, AND FOUR TONE MOTIF CLASSES Four-Element Motif Classes—Continued Class Nr. Representative Class Weight Volume 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161* 162* 163* 164*
(0,0),(0,1),(0,2),(4,3) (0,0),(0,1),(0,2),(4,1) (0,0),(0,1),(0,2),(4,0) (0,0),(0,1),(0,5),(4,2) (0,0),(0,1),(0,5),(4,3) (0,0),(0,1),(0,5),(4,0) (0,0),(0,1),(0,5),(4,1) (0,0),(0,1),(0,6),(4,3) (0,0),(0,1),(0,6),(4,1) (0,0),(0,1),(0,6),(4,0) (0,0),(0,1),(0,3),(4,2) (0,0),(0,1),(0,3),(4,1) (0,0),(0,1),(0,3),(4,0) (0,0),(0,1),(0,3),(4,3) (0,0),(0,1),(0,4),(4,1) (0,0),(0,1),(0,4),(4,0) (0,0),(0,2),(2,0),(6,10) (0,0),(0,2),(2,0),(4,4) (0,0),(0,1),(4,0),(8,6) (0,0),(0,2),(0,4),(2,0) (0,0),(0,2),(0,4),(4,2) (0,0),(0,2),(0,4),(4,0) (0,0),(0,2),(2,0),(2,6) (0,0),(0,2),(4,0),(6,2) (0,0),(0,2),(0,6),(2,0) (0,0),(0,2),(0,6),(4,0) (0,0),(0,2),(0,6),(4,2) (0,0),(0,1),(4,0),(8,0) (0,0),(0,2),(4,0),(8,0) (0,0),(0,4),(0,8),(4,0) (0,0),(0,1),(4,2),(4,7) (0,0),(0,1),(4,2),(4,3) (0,0),(0,1),(4,0),(4,7) (0,0),(0,1),(4,0),(4,11)
(1,15,15,15) (1,15,14,14) (1,15,14,20) (4,15,15,14) (4,15,15,20) (4,14,14,14) (4,14,14,24) (5,5,15,15) (5,5,14,14) (5,7,15,14) (2,2,15,15) (2,2,14,20) (2,3,15,14) (2,3,15,20) (3,3,14,14) (3,3,14,24) (23,16,16,16) (23,16,20,20) (6,15,15,14) (6,16,16,20) (6,20,20,20) (6,20,20,24) (22,22,16,16) (22,22,20,20) (22,7,16,20) (7,7,20,20) (7,7,20,24) (9,14,14,14) (9,20,20,20) (9,24,24,24) (15,15,15,15) (15,15,15,15) (15,15,14,14) (15,15,14,14)
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
M.4. FOUR TONE MOTIFS IN ON P IM OD12,12
1193
Four-Element Motif Classes—Continued Class Nr. Representative Class Weight Volume 165* 166* 167* 168* 169 170 171* 172* 173* 174* 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198
(0,0),(0,1),(4,0),(4,2) (0,0),(0,1),(4,0),(4,10) (0,0),(0,1),(4,0),(4,1) (0,0),(0,1),(4,0),(4,5) (0,0),(0,1),(4,0),(4,4) (0,0),(0,2),(2,0),(2,2) (0,0),(0,2),(2,0),(8,8) (0,0),(0,2),(2,0),(2,4) (0,0),(0,2),(4,0),(4,2) (0,0),(0,2),(4,0),(4,10) (0,0),(0,2),(4,0),(4,4) (0,0),(0,4),(4,0),(4,4) (0,0),(0,1),(0,2),(6,1) (0,0),(0,1),(0,2),(6,3) (0,0),(0,1),(0,2),(6,4) (0,0),(0,1),(0,2),(6,0) (0,0),(0,1),(0,5),(6,0) (0,0),(0,1),(0,4),(6,5) (0,0),(0,1),(0,5),(6,3) (0,0),(0,1),(0,5),(6,1) (0,0),(0,1),(0,6),(6,5) (0,0),(0,1),(0,6),(6,1) (0,0),(0,1),(0,6),(6,2) (0,0),(0,1),(0,6),(6,4) (0,0),(0,1),(0,6),(6,3) (0,0),(0,1),(0,6),(6,0) (0,0),(0,1),(0,3),(6,0) (0,0),(0,1),(0,3),(6,4) (0,0),(0,1),(0,3),(6,5) (0,0),(0,1),(0,3),(6,1) (0,0),(0,1),(0,3),(6,3) (0,0),(0,1),(0,4),(6,1) (0,0),(0,1),(0,4),(6,3) (0,0),(0,1),(0,4),(6,2)
(15,15,14,20) (15,15,14,20) (14,14,14,14) (14,14,14,14) (14,14,14,24) (16,16,16,16) (16,16,20,20) (16,16,20,20) (20,20,20,20) (20,20,20,20) (20,20,20,24) (24,24,24,24) (1,1,17,17) (1,2,19,18) (1,23,18,18) (1,22,19,17) (4,4,17,17) (4,3,19,18) (4,6,18,18) (4,7,19,17) (5,5,19,19) (5,5,17,17) (5,22,19,19) (5,22,18,18) (5,8,18,18) (5,26,17,17) (2,2,17,25) (2,2,18,18) (2,23,19,18) (2,22,17,18) (2,22,18,25) (3,3,17,25) (3,3,18,18) (3,6,19,18)
4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
1194
APPENDIX M. TWO, THREE, AND FOUR TONE MOTIF CLASSES Four-Element Motif Classes—Continued Class Nr. Representative Class Weight Volume 199 200 201 202 203 204* 205* 206* 207* 208* 209* 210* 211* 212* 213* 214* 215*
(0,0),(0,1),(0,4),(6,0) (0,0),(0,1),(0,4),(6,4) (0,0),(0,1),(6,3),(6,9) (0,0),(0,3),(0,6),(6,3) (0,0),(0,3),(0,6),(6,0) (0,0),(0,1),(6,0),(6,5) (0,0),(0,1),(6,0),(6,11) (0,0),(0,1),(6,2),(6,3) (0,0),(0,1),(6,2),(6,5) (0,0),(0,1),(6,0),(6,1) (0,0),(0,1),(6,0),(6,7) (0,0),(0,1),(6,0),(6,3) (0,0),(0,1),(6,0),(6,9) (0,0),(0,1),(6,3),(6,4) (0,0),(0,1),(6,3),(6,10) (0,0),(0,3),(6,0),(6,3) (0,0),(0,3),(6,0),(6,9)
(3,7,17,18) (3,7,18,25) (22,8,18,18) (8,8,25,25) (8,26,25,25) (19,19,17,17) (19,19,17,17) (19,19,18,18) (19,19,18,18) (17,17,17,17) (17,17,17,17) (17,18,18,25) (17,18,18,25) (18,18,18,18) (18,18,18,18) (25,25,25,25) (25,25,25,25)
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
M.5. THREE TONE MOTIFS IN ON P IM OD5,12
M.5
1195
Three Tone Motifs in OnP iM od5,12
Refer to the discussion in subsection 11.3.8 for the entries of this table. The order of these representatives is a historical one.
Cl. Nr.
Three-Element Motif Classes in OnP iM od5,12 Representative Kernel in Z5 × Z12
1
(0,0),(1,0),(0,4)
Z24 × (Z3 × 0) × (0 × Z5 )
2
(0,0),(1,0),(4,4)
Z24 × (Z3 × 0) × Z5 .(1, 1)
3
(0,0),(1,0),(3,4)
Z24 × (Z3 × 0) × Z5 .(2, 1)
4
(0,0),(0,4),(1,8)
Z24 × Z3 .(1, 1) × (Z5 × 0)
5
(0,0),(1,4),(4,8)
Z24 × Z3 .(1, 1) × Z5 .(1, 1)
6
(0,0),(1,0),(0,10)
(Z4 × 2Z4 ) × (Z3 × 0) × (0 × Z5 )
7
(0,0),(1,0),(4,10)
(Z4 × 2Z4 ) × (Z3 × 0) × Z5 .(1, 1)
8
(0,0),(1,0),(3,10)
(Z4 × 2Z4 ) × (Z3 × 0) × Z5 .(2, 1)
9
(0,0),(0,4),(1,2)
(Z4 × 2Z4 ) × Z3 .(1, 1) × (Z5 × 0)
10
(0,0),(1,4),(0,2)
(Z4 × 2Z4 ) × Z3 .(1, 1) × (0 × Z5 )
11
(0,0),(1,4),(4,2)
(Z4 × 2Z4 ) × Z3 .(1, 1) × Z5 .(1, 1)
12
(0,0),(1,4),(3,2)
(Z4 × 2Z4 ) × Z3 .(1, 1) × Z5 .(2, 1)
13
(0,0),(0,4),(1,6)
(Z4 × 2Z4 ) × (0 × Z3 ) × (Z5 × 0)
14
(0,0),(1,4),(0,6)
(Z4 × 2Z4 ) × (0 × Z3 ) × (0 × Z5 )
15
(0,0),(1,4),(4,6)
(Z4 × 2Z4 ) × (0 × Z3 ) × Z5 .(1, 1)
16
(0,0),(1,4),(3,6)
(Z4 × 2Z4 ) × (0 × Z3 ) × Z5 .(2, 1)
17
(0,0),(3,4),(1,6)
(Z4 × 2Z4 ) × (0 × Z3 ) × Z5 .(1, 2)
18
(0,0),(1,4),(1,6)
(Z4 × 2Z4 ) × (0 × Z3 ) × Z5 .(4, 1)
19
(0,0),(1,0),(0,1)
(Z4 × 0) × (Z3 × 0) × (0 × Z5 )
20
(0,0),(1,0),(4,1)
(Z4 × 0) × (Z3 × 0) × Z5 .(1, 1)
21
(0,0),(1,0),(3,1)
(Z4 × 0) × (Z3 × 0) × Z5 .(2, 1)
22
(0,0),(0,4),(1,5)
(Z4 × 0) × Z3 .(1, 1) × (Z5 × 0)
23
(0,0),(1,4),(0,5)
(Z4 × 0) × Z3 .(1, 1) × (0 × Z5 )
24
(0,0),(1,4),(4,5)
(Z4 × 0) × Z3 .(1, 1) × Z5 .(1, 1)
25
(0,0),(1,4),(3,5)
(Z4 × 0) × Z3 .(1, 1) × Z5 .(2, 1)
26
(0,0),(0,4),(1,9)
(Z4 × 0) × (0 × Z3 ) × (Z5 × 0)
27
(0,0),(1,4),(0,9)
(Z4 × 0) × (0 × Z3 ) × (0 × Z5 )
1196
APPENDIX M. TWO, THREE, AND FOUR TONE MOTIF CLASSES Three-Element Motif Classes in OnP iM od5,12 —continued Cl. Representative Kernel Nr. in Z5 × Z12 28
(0,0),(1,4),(4,9)
(Z4 × 0) × (0 × Z3 ) × Z5 .(1, 1)
29
(0,0),(1,4),(3,9)
(Z4 × 0) × (0 × Z3 ) × Z5 .(2, 1)
30
(0,0),(3,4),(1,9)
(Z4 × 0) × (0 × Z3 ) × Z5 .(1, 2)
31
(0,0),(1,4),(1,9)
(Z4 × 0) × (0 × Z3 ) × Z5 .(4, 1)
32
(0,0),(0,6),(1,1)
Z4 .(1, 2) × (Z3 × 0) × (Z5 × 0)
33
(0,0),(1,6),(0,1)
Z4 .(1, 2) × (Z3 × 0) × (0 × Z5 )
34
(0,0),(1,6),(4,1)
Z4 .(1, 2) × (Z3 × 0) × Z5 .(1, 1)
35
(0,0),(1,6),(3,1)
Z4 .(1, 2) × (Z3 × 0) × Z5 .(2, 1)
36
(0,0),(0,10),(1,5)
Z4 .(1, 2) × Z3 .(1, 1) × (Z5 × 0)
37
(0,0),(1,10),(0,5)
Z4 .(1, 2) × Z3 .(1, 1) × (0 × Z5 )
38
(0,0),(1,10),(4,5)
Z4 .(1, 2) × Z3 .(1, 1) × Z5 .(1, 1)
39
(0,0),(1,10),(3,5)
Z4 .(1, 2) × Z3 .(1, 1) × Z5 .(2, 1)
40
(0,0),(0,10),(1,9)
Z4 .(1, 2) × (0 × Z3 ) × (Z5 × 0)
41
(0,0),(1,10),(0,9)
Z4 .(1, 2) × (0 × Z3 ) × (0 × Z5 )
42
(0,0),(4,10),(1,9)
Z4 .(1, 2) × (0 × Z3 ) × Z5 .(1, 1)
43
(0,0),(3,10),(1,9)
Z4 .(1, 2) × (0 × Z3 ) × Z5 .(1, 2)
44
(0,0),(1,10),(3,9)
Z4 .(1, 2) × (0 × Z3 ) × Z5 .(2, 1)
45
(0,0),(1,10),(1,9)
Z4 .(1, 2) × (0 × Z3 ) × Z5 .(4, 1)
Appendix N
Well-Tempered and Just Modulation Steps N.1
12-Tempered Modulation Steps
N.1.1
Scale Orbits and Number of Quantized Modulations
In the following table, the exclamation sign (!) in column 6 means that quantization is not possible for every translation quantity p in the notation of theorem 30. Orbits and Number of Quantized Modulations Class # Min. Cadence Sets # Quanta # Quant. Mod. 38
9
42
54 (!)
38.1
5
20
26
47
6
28
30
47.1
15
66
114
50
7
34
42
50.1
6
36
46
52
5
24
24 (!)
55
6
30
32 (!)
61
10
38
62
62
5
24
24 (!)
39
9
29
93
39.1
6
23
55
40
10
24
108
40.1
7
26
72
1197
1198
APPENDIX N. WELL-TEMPERED AND JUST MODULATION STEPS Orbits and Number of Quant. Mod.—Continued Class # Min. Cadence Sets # Quanta # Quant. Mod. 41
7
25
75
41.1
6
21
53
42
6
22
54
42.1
7
28
74
43
6
22
57
43.1
7
26
72
44
9
23
89
45
7
21
63
45.1
10
21
105
46
6
26
56
48
10
23
109
48.1
7
28
68
49
7
21
71
49.1
7
26
74
51
9
13
86
53
7
27
67
53.1
9
25
91
54
7
32
71
54.1
21
32
226
56
7
24
70
57
8
21
71
58
18
17
185
59
11
22
101
60
6
21
60
N.1. 12-TEMPERED MODULATION STEPS
N.1.2
1199
Quanta and Pivots for the Modulations Between Diatonic Major Scales (No.38.1)
Quanta and Pivots for the Modulations Between Diatonic Major Scales Transl. p Cadence Quantum Modulator Pivots 1 1 2 2 2 3
{II, V } {II, III} {V II} {II, V } {IV, V } {II, V }
• ◦ • • ◦ • • • • • •• • ◦ • • ◦ • • • • • •• ◦ • • ◦ • • ◦ • ◦ ◦ ◦• ◦ • • ◦ • • ◦ • ◦ • ◦• ◦ • • ◦ • • ◦ • ◦ • ◦• • ◦ • ◦ ◦ • ◦ • • • ••
e5 11
{II, III, V, V II}
5
{II, III, V, V II}
6
{II, IV, V II}
6
{II, IV, V, V II}
6
{II, IV, V, V II}
7
{II, III, V, V II}
7
e 11 e 11 e 11 e 11 e 11
3
{II, III}
• ◦ • ◦ ◦ • ◦ • • • ••
e 11
{II, III, V, V II}
4
{V II}
◦ ◦ • • ◦ • • ◦ ◦ • ◦•
e8 11
{II, IV, V, V II}
4
{IV, V }
◦ • • • • • • • ◦ • ◦•
e8 11
{II, III, V, V II}
8
4
{II, III}
• • • • ◦ • • • • • ◦•
e 11
{V, V II}
5
{V II}
◦ ◦ • ◦ • • ◦ • ◦ ◦ ••
e9 11
{II, IV, V II}
6 6 6 6
{II, III} {IV, V } {IV, V } {II, III}
◦ • • • • • ◦ • • • •• ◦ • • • • • • • • • ◦• • • • • ◦ • • • • • ◦• • • • • ◦ • ◦ • • • ••
6
{II, III, V, V II}
10
{II, IV, V, V II}
6
{II, IV, V, V II}
10
{II, III, V, V II}
11
e
e 11 e
e 11
7
{V II}
• ◦ • ◦ ◦ • • ◦ ◦ • ◦•
e 11
{III, V, V II}
8
{V II}
◦ • • ◦ ◦ • ◦ • ◦ ◦ ••
e0 11
{II, V II}
8
{IV, V }
◦ • • • • • ◦ • • • ••
e0 11
{II, IV, V, V II}
0
8
{II, III}
• • • • ◦ • ◦ • ◦ • ••
e 11
{II, III, V, V II}
9
{II, V }
◦ ◦ • ◦ • • • • • • ◦•
e1 11
{II, IV, V, V II}
9
{IV, V }
◦ ◦ • ◦ • • • • • • ◦•
e1 11
{II, IV, V, V II}
10
{V II}
• ◦ • • ◦ • ◦ ◦ ◦ • ◦•
e2 11
{III, V, V II}
10 10 11 11
{II, V } {II, III} {II, V } {IV, V }
• ◦ • • ◦ • ◦ • ◦ • ◦• • ◦ • • ◦ • ◦ • ◦ • ◦• ◦ • • ◦ • • • • • • •• ◦ • • ◦ • • • • • • ••
2
{II, III, V, V II}
2
{II, III, V, V II}
3
{II, IV, V, V II}
3
{II, IV, V, V II}
e 11 e 11 e 11 e 11
1200
N.1.3
APPENDIX N. WELL-TEMPERED AND JUST MODULATION STEPS
Quanta and Pivots for the Modulations Between Melodic Minor Scales (No.47.1)
Transl. p
Quanta and Pivots for the Modulations Between Melodic Minor Scales Cadence Quantum Modulator Pivots
1
{II, IV }, {IV, V II}
1 1
{III, V I}, {V, V I} {III, V II}
• • • • ◦ • • ◦ ◦ • •◦ • • • • • • ◦ • • ◦ •• • ◦ ◦ • • ◦ • • • • ◦•
e3 11
{II, IV, V II}
3
{I, III, V, V I}
3
{III, V, V II}
3
e 11 e 11
1
{IV, V }
• • • • ◦ • • • • • •◦
e 11
{II, IV, V, V II}
1
{II, III}
• ◦ ◦ • • • • • • • ••
e3 11
{II, III, V, V II}
3
1
{I, V II}
• • • • • ◦ • • • • ◦•
e 11
{I, III, V, V II}
2
{III, V }, {III, V II}, {II, III}
• • ◦ • • • ◦ • ◦ • ◦•
e4 11
{II, III, V, V II}
2
{II, IV }, {II, V I}, {I, II}
• ◦ • ◦ • • ◦ • ◦ • ◦•
e4 11
{I, II, IV, V I}
4
2
{I, III}, {III, V I}, {III, IV }
◦ • • • ◦ • ◦ • ◦ • ◦•
e 11
{I, III, IV, V I}
3
{III, V }, {III, V I}, {V, V I}, {I, V }
• ◦ • • ◦ • • • ◦ ◦ ••
e5 11
{I, III, V, V I}
5
3
{III, V }, {III, V I}, {V, V I}, {I, V }
• ◦ • • ◦ • • • ◦ ◦ ••
e 11
{I, III, V, V I}
3
{III, V }, {III, V I}, {V, V I}, {I, V }
• ◦ • • ◦ • • • ◦ ◦ ••
e5 11
{I, III, V, V I}
5
3
{III, V }, {III, V I}, {V, V I}, {I, V }
• ◦ • • ◦ • • • ◦ ◦ ••
e 11
{I, III, V, V I}
4
{III, V }
◦ • • ◦ • • ◦ ◦ • • •◦
e6 11
{III, V }
4
{I, III}
◦ ◦ • • • ◦ ◦ • ◦ ◦ ◦•
e6 11
{I, III}
4 4 4 4 4
{II, V I}, {I, II} {IV, V II} {III, V I} {III, V II} {V, V I}
• • • ◦ • • • • ◦ • ◦• • • • • • • • ◦ ◦ • ◦◦ ◦ • • • • • ◦ • ◦ ◦ ◦• • ◦ ◦ • ◦ ◦ • • ◦ • ◦• • • • • • • • • ◦ ◦ ◦•
6
{I, II, IV, V I}
6
{II, IV, V II}
6
{I, III, V I}
6
{III, V, V II}
6
{I, III, V, V I}
6
e 11 e 11 e 11 e 11 e 11
4
{III, IV }
◦ • • • • • ◦ • ◦ • ◦•
e 11
{I, III, IV, V I}
4
{II, III}
• • ◦ • ◦ • • • ◦ • ◦•
e6 11
{II, III, V, V II}
4 4 4 4 4 5 5
{I, V II} {I, III} {III, V } {III, V II}, {II, III} {III, V I}, {III, IV } {I, II}, {I, V }, {III, V I}, {V, V I} {II, V I}, {I, II}
• ◦ • • • ◦ • • ◦ • ◦• • ◦ ◦ • • ◦ ◦ • • ◦ ◦• ◦ ◦ • • ◦ ◦ • • ◦ ◦ •• ◦ • • • ◦ • • • ◦ • •• • • ◦ • • • ◦ • • • ◦• • ◦ • • • • ◦ • • ◦ ◦• • ◦ • ◦ ◦ • ◦ • • • ••
6
{I, III, V, V II}
4
{I, III}
4
{III, V }
4
{II, III, V, V II}
4
{I, III, IV, V I}
7
{I, III, V, V I}
7
{I, II, IV, V I}
e 11 e e e e
e 11 e 11
N.1. 12-TEMPERED MODULATION STEPS
1201
Transl. p
Quanta and Pivots for Melodic Minor Scales—Continued Cadence Quantum Modulator
5
{IV, V II}, {IV, V }
• ◦ • • • • ◦ • ◦ • •◦
e7 11
{II, IV, V, V II}
5
{III, V II}
• ◦ ◦ • • ◦ ◦ • • • ••
e7 11
{III, V, V II}
6
{III, V }, {III, V II}, {II, III}
• • ◦ • ◦ • ◦ • • • ◦•
e8 11
{II, III, V, V II}
6
{I, III}, {III, V I}, {III, IV }
◦ • • • ◦ • • • ◦ • ◦•
Pivots
8
{I, III, IV, V I}
6
e 11
6
{I, III}, {III, V I}, {III, IV }
• • ◦ • ◦ • • • ◦ • ◦•
e
{I, III, IV, V I}
6
{III, V }, {III, V II}, {II, III}
◦ • • • ◦ • ◦ • • • ◦•
e6
{II, III, V, V II}
7
{III, V }, {I, V }, {III, V II}, {I, V II}
• ◦ • • ◦ ◦ • • ◦ • ••
e9 11
{I, III, V, V II}
9
7
{II, V I}, {I, II}
• ◦ • ◦ • • ◦ • ◦ • ••
e 11
{I, II, IV, V I}
7
{IV, V II}, {IV, V }
• ◦ • • • • • • ◦ • ◦◦
e9 11
{II, IV, V, V II}
9
7
{III, V I}
◦ ◦ • • • • • • ◦ ◦ ••
e 11
{I, III, V I}
8
{III, V }
• ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ••
e10 11
{III, V }
10
8
{I, III}
◦ ◦ • • ◦ ◦ ◦ • • ◦ ◦•
e 11
{I, III}
8
{II, V I}
• • • ◦ ◦ • ◦ ◦ • • ••
e10 11
{II, IV, V I}
8 8 8 8
{IV, V II}, {IV, V } {III, V I} {III, V II} {V, V I}
• • • • ◦ • ◦ • • • •◦ ◦ ◦ • • ◦ • ◦ • • ◦ ◦• • • ◦ • ◦ ◦ ◦ • ◦ • •• • ◦ • • ◦ • ◦ • • ◦ ••
10
{II, IV, V, V II}
10
{I, III, V I}
10
{III, V, V II}
10
{I, III, V, V I}
10
e 11 e 11 e 11 e 11
8
{III, IV }
◦ • • • ◦ • ◦ • • • ◦•
e 11
{I, III, IV, V I}
8
{II, III}
• • ◦ • ◦ • ◦ • ◦ • ••
e10 11
{II, III, V, V II}
8 8 8 8 8 9 9 9 9 10 10 10
{I, V II} {I, III} {III, V } {III, V II}, {II, III} {III, V I}, {II, IV } {III, V } {II, IV }, {II, V I} {I, III}, {I, V }, {III, V II}, {I, V II} {I, II} {II, V }, {III, V II}, {II, III} {II, IV }, {IV, V II}, {IV, V } {I, III}, {III, V I}, {III, IV }
• • • • ◦ ◦ ◦ • • • •• • ◦ ◦ • • ◦ ◦ • • ◦ ◦• ◦ ◦ • • ◦ ◦ • • ◦ ◦ •• ◦ • • • ◦ • • • ◦ • •• • • ◦ • • • ◦ • • • ◦• • ◦ ◦ • • ◦ ◦ • • ◦ ◦• • ◦ • ◦ ◦ • • ◦ ◦ • ◦• • ◦ • • • ◦ ◦ • • • ◦• • ◦ • ◦ • • • • ◦ • ◦• • • ◦ • ◦ • ◦ • ◦ • ◦• • ◦ • • ◦ • ◦ • ◦ • •◦ ◦ • • • ◦ • ◦ • ◦ • ••
10
{I, III, V, V II}
8
{I, III}
8
{III, V }
8
{II, III, V, V II}
8
{I, III, IV, V I}
11
{III, V }
11
{II, IV, V I}
11
{I, III, V, V II}
11
{I, III, IV, V I}
0
{II, III, V, V II}
0
{II, IV, V, V II}
0
{I, III, IV, V I}
e 11 e e e e
e 11 e 11 e 11 e 11 e 11 e 11 e 11
1202
APPENDIX N. WELL-TEMPERED AND JUST MODULATION STEPS
Transl. p
Quanta and Pivots for Melodic Minor Scales—Continued Cadence Quantum Modulator
11
{II, IV }, {II, V I} {III, V I}
11
◦ ◦ • • ◦ • • • • ◦ ••
{III, V II}, {I, V II}
11
{I, III, V I}
1
{I, III, V, V II}
1
{I, III, V, V I}
1
e 11
• • • • ◦ • • • • ◦ ••
{I, IV, V I}
1
e 11
• • • • ◦ • • ◦ • • ••
{V, V I}
11
e1 11
• • • ◦ • • ◦ ◦ • • ◦•
Pivots
e 11
11
{III, IV }
◦ ◦ • • • • • • • • ••
e 11
{I, III, IV, V I}
11
{I, II}
• • • ◦ • • • • • • ◦•
e1 11
{I, II, IV, V I}
N.1.4
Quanta and Pivots for the Modulations Between Harmonic Minor Scales (No.54.1)
For this table, we need a numbering of the 21 minimal cadence sets: 0 = {II, V II} 5 = {V, V II} 10 = {III, V II} 15 = {I, V II} 20 = {V, V I}
1 = {I, III} 6 = {I, V I} 11 = {I, IV } 16 = {I, II}
2 = {II, IV } 7 = {IV, V II} 12 = {II, V } 17 = {II, III}
3 = {III, V } 8 = {I, V } 13 = {III, V I} 18 = {III, IV }
4 = {IV, V I} 9 = {II, V I} 14 = {V I, V II} 19 = {IV, V }
Quanta and Pivots for the Modulations Between Harmonic Minor Scales Transl. p Cadence Nr. Quantum Pivots 3/9
1,3,6,8,10,11,15-20
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦
{I, II, III, IV, V, V I, V II}
3/9
2,4,7,9,14
• ◦ • • ◦ • • ◦ • • ◦•
{II, IV, V I, V II}
3/9
5,12
◦ • • ◦ • • ◦ • • ◦ ••
{II, V, V II}
4/8
0,7,12,14-17,19
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦
{I, II, III, IV, V, V I, V II}
4/8
1,6,13
• ◦ ◦ • • ◦ ◦ • • ◦ ◦•
{I, II, V I}
4/8
2
• • • ◦ • • • ◦ • • •◦
{II, IV }
4/8
3
◦ ◦ • • ◦ ◦ • • ◦ ◦ ••
{III, V }
4/8
4,11,18
• • ◦ • • • ◦ • • • ◦•
{I, III, IV, V I}
4/8
5,10
◦ • • • ◦ • • • ◦ • ••
{III, V, V II}
4/8
8,20
• ◦ • • • ◦ • • • ◦ ••
{I, III, V, V I}
6
2,7
• ◦ • ◦ ◦ • • ◦ • ◦ ◦•
{II, IV, V II}
6
3,10,17
◦ • • • ◦ • ◦ • • • ◦•
{II, III, V, V II}
6
4,9,14
• ◦ • • ◦ • • ◦ • • ◦•
{II, IV, V I, V II}
N.2. 2-3-5-JUST MODULATION STEPS
1203
Quanta and Pivots for Harmonic Minor Scales—Continued Transl. p Cadence Nr. Quantum Pivots 6
5,12
◦ • • ◦ ◦ • ◦ • • ◦ ◦•
{II, V, V II}
6
8,11,13,15,16,18
• • • • ◦ • • • • • ◦•
{I, II, II, IV, V, V I, V II}
6
19
• • • ◦ ◦ • • • • ◦ ◦•
{II, IV, V, V II}
1,2,5,7,10,11
any cadence set
• • • • • • • • • • ••
{I, II, II, IV, V, V I, V II}
N.1.5
Examples of 12-Tempered Modulations for all Fourth Relations
Examples of Modulations for all Fourth Relations Start → Target Neutral Pivots Cadence C
F
IC
V IIF ∪ IIF
IF , IVF , VF , IF
C
B[
IC , VC
IIIB[ , VB[ ∪ V IIB[
V IIB[ , IB[
C
E[
IC , VC
IIE[ , VE[ ∪ V IIE[
VE[ ∪ V IIE[ , IE[
C
A[
IC , IVC
IIA[ ∪ V IIA[
IVA[ , VA[ , IA[
C
D[
IC
IIID[ , IID[ ∪ V IID[
IID[ , VD[ , ID[
C
G[
IC , IVC ∪ V IC , V IC ∪ V IIC
IIG[ ∪ V IIG[ , IIG[
VG[ ∪ V IIG[ , IG[
C
B
IC , VC , IIC
IVB , IIB ∪ V IIB
IVB , VB , IB
C
E
IC , V IC
VE ∪ V IIE
IVE , VE , IE
C
A
IC , VC
IVA
IVA ∪ V IIA , VA , IA
C
D
IC , V IC , VC
V IID ∪ VD
IID , VD , ID
C
G
IC , VC
IIIG
IIG , VG , IG
N.2
2-3-5-Just Modulation Steps
The following tables show data for modulations from C-tonic.
N.2.1
Modulation Steps between Just Major Scales
Here, we have the two modulators Φ1 Φ2
= eb , = eb .A.
(N.1) (N.2)
The numbering of minimal cadence sets is the one used in formula (26.2). The tonics D∗ and B[∗ are the usual third comma shifted representatives of D and B-flat.
1204
APPENDIX N. WELL-TEMPERED AND JUST MODULATION STEPS Pivots for the Modulations Between Just Major Scales Translation Target Tonic Modulator Cadence Pivots
N.2.2
(1, 0)
G
Φ2
5
{V, V II}
(−1, 0)
F
Φ2
1
{II, IV }
(2, 0)
D
Φ2
1
{II, IV }
(−2, 0)
B[
Φ2
5
{V, V II}
(0, 1)
E
Φ2
1
{II, V, V II}
(0, 1)
E
Φ1
5
{II, V, V II}
(0, −1)
A[
Φ2
5
{II, IV, V II}
(0, −1)
A[
Φ1
5
{II, V, V II}
(1, 1)
B
Φ2
1
{II, V, V I}
(−1, −1)
D[
Φ2
5
{II, V, V II}
(1, −1)
E[
Φ2
1
{II, V, V II}
(1, −1)
E[
Φ1
1
{II, IV, V II}
(−1, 1)
A
Φ2
5
{II, IV, V II}
(−1, 1)
A
Φ1
1
{II, IV, V II}
(−2, 1)
D∗
Φ2
5
{III, V, V II}
(2, −1)
B[∗
Φ2
1
{II, IV, V I}
Modulation Steps between Natural Minor Scales
We have the two modulators Φ1 Φ2
= eb , = eb .A.
The minimal cadence sets are these: J1 = {V II}, J2 = {III, V I}, J3 = {V, V I}, J4 = {IV, V }, J5 = {II}, J6 = {III, IV }.
Pivots for the Modulations Between Natural Minor Scales Translation Target Tonic Modulator Cadence Pivots (−1, 0)
F
Φ2
5
{II, IV }
(1, 0)
G
Φ2
1
{V, V II}
(−2, 0)
B[
Φ2
1
{V, V II}
(2, 0)
D
Φ2
5
{II, IV }
N.2. 2-3-5-JUST MODULATION STEPS
1205
Pivots for the Modulations Between Natural Minor Scales—Continued Translation Target Tonic Modulator Cadence Pivots (0, −1)
A[
Φ2
1
{II, IV, V II}
(0, −1)
A[
Φ1
5
{II, IV, V II}
(0, 1)
E
Φ1
5
{II, IV, V II}
(0, 1)
E
Φ2
5
{II, V, V II}
(−1, −1)
D[
Φ2
1
{III, V, V II}
(1, 1)
B
Φ2
5
{II, IV, V I}
(1, −1)
E[
Φ1
1
{II, V, V II}
(1, −1)
E[
Φ2
5
{II, V, V II}
(−1, 1)
A
Φ1
1
{II, V, V II}
(−1, 1)
A
Φ2
1
{II, IV, V II}
(2, −1)
B[∗ ∗
Φ2
5
{II, IV, V I}
Φ2
1
{III, V, V II}
(−2, 1)
N.2.3
D
Modulation Steps From Natural Minor to Major Scales
We have the two modulators Φ1 Φ2
= eb .A, = eb .B, B =
! 1 1 . 0 −1
The minimal cadence sets are the same as for major scales. Pivots for the Modulations From Natural Minor to Major Scales Translation Target Tonic Modulator Cadence Pivots (−2, 0)
B[
Φ1
5
{V, V II}
(−1, 0)
F
Φ1
1
{II, IV }
(1, 0)
G
Φ1
5
{V, V II}
(2, 0)
D
Φ1
1
{II, IV }
(−1, −1)
D[
Φ1
5
{II, V, V II}
(0, −1)
A[
Φ1
5
{II, IV, V II}
(1, −1)
E[
Φ1
1
{II, V, V II}
(2, −1)
B[∗
Φ1
1
{II, IV, V I}
1206
APPENDIX N. WELL-TEMPERED AND JUST MODULATION STEPS
N.2.4
Modulation Steps From Major to Natural Minor Scales
We have the two modulators (same as above). Φ1 Φ2
= eb .A, b
= e .B, B =
! 1 1 . 0 −1
The minimal cadence sets are those of minor scales. Pivots for the Modulations From Major to Natural Minor Scales Translation Target Tonic Modulator Cadence Pivots (−2, 0)
B[
Φ1
1
{V, V II}
(−1, 0)
F
Φ1
5
{II, IV }
(1, 0)
G
Φ1
1
{V, V II}
(2, 0)
D
Φ1
5
{II, IV }
(−2, 1)
D
Φ1
1
{III, V, V II}
(−1, 1)
A
Φ1
1
{II, IV, V II}
(0, 1)
E
Φ1
5
{II, V, V II}
(1, 1)
B
Φ1
5
{II, IV, V I}
N.2.5
Modulation Steps Between Harmonic Minor Scales
We have the unique translation modulator Φ = eb . The minimal cadence sets are these: J1 = {III}, J2 = {II}, J3 = {V II}, J4 = {I, IV }, J5 = {I, V }, J6 = {I, V I}, J7 = {IV, V }, J8 = {IV, V I}, J9 = {V, V I}.
Pivots for the Modulations Between Harmonic Minor Scales Translation Target Tonic Modulator Cadence Pivots (3, 0)
A∗
Φ
8
{II, IV, V I}
(−3, 0)
E[∗
Φ
8
{II, IV, V I}
(0, 1)
E
Φ
2
{II, IV }
(0, 1)
E
Φ
4
{I, III, IV, V I}
(0, 1)
E
Φ
8
{I, III, IV, V I}
N.2. 2-3-5-JUST MODULATION STEPS
1207
Pivots for the Modulations Between Harmonic Minor Scales—Continued Translation Target Tonic Modulator Cadence Pivots (0, −1)
A[
Φ
2
{II, IV }
(0, −1)
A[
Φ
4
{I, III, IV, V I}
(0, −1)
A[
Φ
8
{I, III, IV, V I}
(2, 1)
f]
Φ
8
{II, IV, V I, V II}
(−2, −1)
G[
Φ
8
{II, IV, V I, V II}
(0, 2)
G]
Φ
4
{I, III, IV, V I}
(0, −2)
F[
Φ
4
{I, III, IV, V I}
(1, 2)
D]
Φ
4
{I, III, IV, V I}
(−1, −2)
B[,[
Φ
4
{I, III, IV, V I}
(1, 0)
G
Φ
1,5,7,9
{I . . . V II}
(−1, 0)
F
Φ
1,5,7,9
{I . . . V II}
(2, 0)
D
Φ
9
{I . . . V II}
(−2, 0)
B[
Φ
9
{I . . . V II}
(3, 0)
A∗
Φ
9
{I . . . V II}
(−3, 0)
E[∗
Φ
9
{I . . . V II}
(−2 . . . 1, 1)
Φ
7
{I . . . V II}
(−2 . . . − 1, 1)
Φ
9
{I . . . V II}
Φ
5,9
{I . . . V II}
Φ
4
{I . . . V II}
Φ
8
{I . . . V II}
(−1 . . . 2, 1)
Φ
7
{I . . . V II}
(1 . . . 2, −1)
Φ
9
{I . . . V II}
Φ
5,9
{I . . . V II}
Φ
4
{I . . . V II}
(2, 1)
F]
(1 . . . 2, 1) (1, 1)
(−2, −1)
B
G[
(−2 . . . − 1, −1) (−1, −1)
D[
Φ
8
{II, IV, V I}
(0, 2)
G]
Φ
7
{II, IV, V I}
(0, −2)
F[
Φ
7
{I . . . V II}
N.2.6
Modulation Steps Between Melodic Minor Scales
We have the two modulators Φ1 Φ2
= eb , = eb .A.
1208
APPENDIX N. WELL-TEMPERED AND JUST MODULATION STEPS The minimal cadence sets are these: J1 = {I}, J2 = {II}, J3 = {III}, J4 = {I, IV }, J5 = {V I}, J6 = {V II}.
Pivots for the Modulations Between Melodic Minor Scales Translation Target Tonic Modulator Cadence Pivots (1, 0)
G
Φ2
6
{V, V II}
(−1, 0)
F
Φ2
2
{II, IV }
(2, 0)
D
Φ2
2
{II, IV }
(2, 0)
D
Φ1
5
{II, IV, V II}
(−2, 0)
B[
Φ1
6
{V, V II}
(−2, 0)
B[
Φ1
5
{II, IV, V II}
(0, 1)
E
Φ2
2
{II, V, V II}
(0, 1)
E
Φ2
3
{I, III, IV, V I}
(0, 1)
E
Φ1
5
{I, III, IV, V I}
(0, 1)
E
Φ1
6
{II, III, V, V II}
(0, −1)
A[
Φ1
5
{I, III, IV, V I}
(0, −1)
A[
Φ2
5
{I, III, V, V I}
(0, −1)
A[
Φ1
6
{II, III, V, V II}
(0, −1)
A[
Φ2
6
{II, IV, V II}
(−1, 1)
A
Φ1
2
{II, IV, V I, V II}
(−1, 1)
A
Φ1
3
{I, III, V, V I}
(−1, 1)
A
Φ2
5
{I, III, V, V I}
(−1, 1)
A
Φ2
6
{II, IV, V II}
(1, −1)
E[
Φ1
2
{II, IV, V I, V II}
(1, −1)
E[
Φ1
3
{I, III, V, V I}
(−1, 1)
E[
Φ2
3
{I, III, IV, V I}
(−1, 1)
E[
Φ2
2
{II, V, V II}
(−2, 1)
D
∗
Φ2
5
{III, V, V II}
(2, −1)
B[∗
Φ2
1
{II, IV, V I}
N.2.7
General Modulation Behaviour for 32 Alterated Scales
The following list refers to the 32 scales as defined in 27.1.6.1. Following Radl [429], we say that a scale type
N.2. 2-3-5-JUST MODULATION STEPS
1209
• has no modulations if its modulation domain is empty (always excluding the start tonality!), • has infinite modulations if its modulation domain is infinite, • has modulations if its modulation domain is not empty (always excluding the start tonality!), has limited modulations if the transitive closure (all tonics which can be reached by successive modulations from relative modulation domains) of its modulation domain is not total space.
No.
Modulation Behaviour in 32 Alterated Scale Types Scale Type Behaviour
1
c, d, e, f, g, a, b
has modulations: see table N.2.1
2
c, d, e, f, g, a, b[
has modulations: ±(1, 0) with III, ±(0, 1) with V, V I
3
c, d, e, f, g, a[ , b
has modulations: corresponds to No. 11
4
c, d, e, f, g, a[ , b[
has modulations: corresponds to No. 9
5
c, d, e, f] , g, a, b
has modulations: see table N.2.2
6
c, d, e, f] , g, a, b[
has modulations (special table in [429])
7
c, d, e, f] , g, a[ , b
has modulations: ±(1, 0) with II, ±(0, 1) with III, V II
8
c, d, e, f] , g, a[ , b[
has limited modulations: see No. 25
9
c, d, e[ , f, g, a, b
has modulations: see table N.2.6
10
c, d, e[ , f, g, a, b[
has modulations: corresponds to No. 2
11
c, d, e[ , f, g, a[ , b
has modulations: see table N.2.5
12
c, d, e[ , f, g, a[ , b[
has modulations: see table N.2.2
13
c, d, e[ , f] , g, a, b
has modulations: ±(1, 0) with II, ±(−1, 1) with I, V II
14
c, d, e[ , f] , g, a, b[
has modulations: ±(1, 1) with I, V , ±(0, 1) with I, V
15
c, d, e[ , f] , g, a[ , b
has no modulations
16
c, d, e[ , f] , g, a[ , b[
has modulations: corresponds to No. 7
17
c, d[ , e, f, g, a, b
has modulations: see No. 7
18
c, d[ , e, f, g, a, b[
has modulations: corresponds to No. 14
19
c, d[ , e, f, g, a[ , b
has no modulations: see No. 15
20
c, d[ , e, f, g, a[ , b[
has modulations: corresponds to No. 13
21
c, d[ , e, f] , g, a, b
has infinite modulations
22
c, d[ , e, f] , g, a, b[
has infinite modulations
23
c, d[ , e, f] , g, a[ , b
has modulations: ±(1, 0) with V , ±(0, 1) with II
24
c, d[ , e, f] , g, a[ , b[
has infinite modulations
25
c, d[ , e[ , f, g, a, b
has limited modulations
1210
APPENDIX N. WELL-TEMPERED AND JUST MODULATION STEPS
No.
Modulation Behaviour in 32 Alterated Scale Types—Continued Scale Type Behaviour
26
c, d[ , e[ , f, g, a, b[
has modulations: corresponds to No. 6
27
c, d[ , e[ , f, g, a[ , b
has modulations: see No. 16
28
c, d[ , e[ , f, g, a[ , b[
has modulations: see table N.2.1
29
c, d[ , e[ , f] , g, a, b
has infinite modulations: corresponds to No. 24
30
c, d[ , e[ , f] , g, a, b[
has infinite modulations: corresponds to No. 22
31
c, d[ , e[ , f] , g, a[ , b
has modulations: corresponds to No. 23
32
c, d[ , e[ , f] , g, a[ , b[
has infinite modulations: corresponds to No. 21
Appendix O
Counterpoint Steps O.1
Contrapuntal Symmetries
All the following tables relate to representatives of strong dichotomies (X/Y ) which are indicated in the table after the counterpoint theorem 33 in subsection 31.3.3.
O.1.1 k 2
4 5 7
Class Nr. 64 g
g.X[ε]
6
6
g.X[ε] ∩ X[ε]
card(g.X[ε] ∩ X[ε]) 48
e (1 + ε.6)
(1 + ε.6)Z12 + ε.e X
eε.8 5
Z12 + ε.e8 5X
z even: z + ε.{5, 11} z odd: z + ε.X Z12 + ε.{4, 5, 7, 9}
e6 (1 + ε.6)
see k = 2
see k = 2
48
ε.11
e
11
11
Z12 + ε.e 11X
Z12 + ε.{2, 4, 7, 9}
48
6
see k = 2
see k = 2
48
6
e (1 + ε.6)
9
e (1 + ε.6) eε.3 7
see k = 2 Z12 + ε.e3 7X
see k = 2 Z12 + ε.{2, 4, 5, 7}
48
11
eε.3 7 eε.11 11 eε.8 5
see k = 9 see k = 5 see k = 2
see k = 9 see k = 5 see k = 2
48
1211
1212
APPENDIX O. COUNTERPOINT STEPS
O.1.2 k 0 1
2 3
Class Nr. 68 g ε.6
e
g.X[ε] ∩ X[ε]
card(g.X[ε] ∩ X[ε])
6
Z12 + ε.(X − {0})
60
3
Z12 + ε.{0, 2, 3, 5} z even: z + ε.{2, 8} z odd: z + ε.X z even: z + ε.{0, 2, 8} z odd: z + ε.(X − {0})
48
Z12 + ε.{0, 1, 3, 5}
48
z z z z
48
g.X[ε] 7
ε.3
Z12 + ε.e 7X
e 7 eε.6 (1 + ε.6)
Z12 + ε.e 7X (1 + ε.6)Z12 + ε.e6 X
7 + ε.6
(1 + ε.6)Z12 + ε.7X
5
Z12 + ε.5X
ε.6
e
(1 + ε.6)
6
(1 + ε.6)Z12 + ε.e X
even: z + ε.{2, 8} odd: z + ε.X even: z + ε.{0, 2, 8} odd: z + ε.(X − {0})
7 + ε.6
(1 + ε.6)Z12 + ε.7X
5
eε.3 11 eε.6 (1 + ε.6) 7 + ε.6
Z12 + ε.e3 X see k = 1 see k = 3
Z12 + ε.{0, 1, 2, 3} see k = 1 see k = 3
48
8
eε.3 11 eε.3 7 5
see k = 5 see k = 1 see k = 2
see k = 5 see k = 1 see k = 2
48
O.1. CONTRAPUNTAL SYMMETRIES
O.1.3 k
1213
Class Nr. 71
g ε.8
g.X[ε] 8
g.X[ε] ∩ X[ε]
card(g.X[ε] ∩ X[ε])
0
e 5 eε.8 (5 + ε.6)
Z12 + ε.e 5X (1 + ε.6)Z12 + ε.e8 5X
Z12 + ε.{1, 2, 6, 7} z even: z + ε.{1, 2, 6, 7} z odd: z + ε.{0, 1, 2, 7}
48
1
eε.3 (7 + ε.3)
(1 + ε.3)Z12 + ε.e3 7X
42
eε.3 (7 − ε.3)
(1 − ε.3)Z12 + ε.e3 7X
eε.9 (7 + ε.3)
(1 + ε.3)Z12 + ε.e9 7X
eε.9 (7 − ε.3)
(1 − ε.3)Z12 + ε.e9 7X
z z z z z z z z z z z z z z z z
2
eε.6 (1 + ε.6)
(1 + ε.6)Z12 + ε.e6 X
z even: z + ε.{0, 1, 6, 7} z odd: z + ε.X
60
3
eε.6 (1 + ε.6)
see k = 2
see k = 2
60
ε.2
2
= 0, 4, 8 : z + ε.{0, 3} = 1, 5, 9 : z + ε.{0, 1, 3, 6, 7} = 2, 6, 10 : z + ε.{3, 6} = 3, 7, 11 : z + ε.{0, 1, 2, 6, 7} = 0, 4, 8 : z + ε.{0, 3} = 1, 5, 9 : z + ε.{0, 1, 2, 6, 7} = 2, 6, 10 : z + ε.{3, 6} = 3, 7, 11 : z + ε.{0, 1, 3, 6, 7} = 0, 4, 8 : z + ε.{3, 6} = 1, 5, 9 : z + ε.{0, 1, 2, 6, 7} = 2, 6, 10 : z + ε.{0, 3} = 3, 7, 11 : z + ε.{0, 1, 3, 6, 7} = 0, 4, 8 : z + ε.{3, 6} = 1, 5, 9 : z + ε.{0, 1, 3, 6, 7} = 2, 6, 10 : z + ε.{0, 3} = 3, 7, 11 : z + ε.{0, 1, 2, 6, 7}
6
e 5 eε.2 (5 + ε.6)
Z12 + ε.e 5X (1 + ε.6)Z12 + ε.e2 5X
Z12 + ε.{0, 1, 2, 7} z even: z + ε.{0, 1, 2, 7} z odd: z + ε.{1, 2, 6, 7}
48
7
eε.9 (7 + ε.3) eε.3 (7 + ε.3)
see k = 1 see k = 1
see k = 1 see k = 1
42
1214
APPENDIX O. COUNTERPOINT STEPS
O.1.4 k 0
1 2
4 5 8
Class Nr. 75
g
g.X[ε]
ε.9
9
e 7 eε.9 (7 + ε.4)
Z12 + ε.e 7X (1 + ε.4)Z12 + ε.e9 7X
eε.9 (7 − ε.4)
(1 − ε.4)Z12 + ε.e9 7X
eε.6 (1 + ε.6)
(1 + ε.6)Z12 + ε.e6 X
eε.6 (1 + ε.6)
see k = 0
ε.8
e
8
g.X[ε] ∩ X[ε]
card(g.X[ε] ∩ X[ε])
Z12 + ε.{1, 4, 5, 8} z = 0, 3, 6, 9 : z + ε.{1, 4, 5, 8} z = 1, 4, 7, 10 : z + ε.{0, 1, 5, 8} z = 2, 5, 8, 11 : z + ε.{0, 1, 4, 5} z = 0, 3, 6, 9 : z + ε.{1, 4, 5, 8} z = 1, 4, 7, 10 : z + ε.{0, 1, 4, 5} z = 2, 5, 8, 11 : z + ε.{0, 1, 5, 8} z even: z + ε.{2, 8} z odd: z + ε.X
48
see k = 0
48
z z z z z z
56
(5 + ε.4)
(1 + ε.4)Z12 + ε.e 5X
eε.8 (5 − ε.4)
(1 − ε.4)Z12 + ε.e8 5X
eε.6 (1 + ε.6)
see k = 1
see k = 1
48
see k = 2
see k = 2
56
Z12 + ε.{0, 1, 4, 5} z = 0, 3, 6, 9 : z + ε.{0, 1, 4, 5} z = 1, 4, 7, 10 : z + ε.{1, 4, 5, 8} z = 2, 5, 8, 11 : z + ε.{0, 1, 5, 8} z = 0, 3, 6, 9 : z + ε.{0, 1, 4, 5} z = 1, 4, 7, 10 : z + ε.{0, 1, 5, 8} z = 2, 5, 8, 11 : z + ε.{1, 4, 5, 8}
48
ε.8
e
(1 ± ε.4)
ε.5
5
e 11 eε.5 (11 + ε.4)
Z12 + ε.e 11X (1 + ε.4)Z12 + ε.e5 11X
eε.5 (11 − ε.4)
(1 − ε.4)Z12 + ε.e5 11X
= 0, 3, 6, 9 : z + ε.{0, 1, 4, 8} = 1, 4, 7, 10 : z + ε.{0, 1, 4, 5, 8} = 2, 5, 8, 11 : z + ε.{0, 2, 4, 5, 8} = 0, 3, 6, 9 : z + ε.{0, 1, 4, 8} = 1, 4, 7, 10 : z + ε.{0, 2, 4, 5, 8} = 2, 5, 8, 11 : z + ε.{0, 1, 4, 5, 8}
O.1. CONTRAPUNTAL SYMMETRIES
1215
1216
APPENDIX O. COUNTERPOINT STEPS
O.1.5 k 0
Class Nr. 78
g ε.9
e
g.X[ε] 9
g.X[ε] ∩ X[ε]
card(g.X[ε] ∩ X[ε])
z z z z z z z z z z z z z z z z
42
(7 + ε.3)
(1 + ε.3)Z12 + ε.e 7X
= 0, 4, 8 : z + ε.{1, 4} = 1, 5, 9 : z + ε.{0, 2, 4, 6, 10} = 2, 6, 10 : z + ε.{1, 10} = 3, 7, 11 : z + ε.{0, 1, 4, 6, 10} = 0, 4, 8 : z + ε.{1, 4} = 1, 5, 9 : z + ε.{0, 1, 4, 6, 10} = 2, 6, 10 : z + ε.{1, 10} = 3, 7, 11 : z + ε.{0, 2, 4, 6, 10} = 0, 4, 8 : z + ε.{1, 10} = 1, 5, 9 : z + ε.{0, 1, 4, 6, 10} = 2, 6, 10 : z + ε.{1, 4} = 3, 7, 11 : z + ε.{0, 2, 4, 6, 10} = 0, 4, 8 : z + ε.{1, 10} = 1, 5, 9 : z + ε.{0, 2, 4, 6, 10} = 2, 6, 10 : z + ε.{1, 4} = 3, 7, 11 : z + ε.{0, 1, 4, 6, 10}
eε.9 (7 − ε.3)
(1 − ε.3)Z12 + ε.e9 7X
eε.3 (7 + ε.3)
(1 + ε.3)Z12 + ε.e3 7X
eε.3 (7 − ε.3)
(1 − ε.3)Z12 + ε.e3 7X
1
eε.6 (1 + ε.6)
(1 + ε.6)Z12 + ε.e6 X
z even: z + ε.{0, 4, 6, 10} z odd: z + ε.X
60
2
eε.6 (1 + ε.6)
see k = 1
see k = 1
60
4
5 + ε.4
(1 + ε.4)Z12 + ε.5X
56
5 − ε.4
(1 − ε.4)Z12 + ε.5X
z z z z z z
6
eε.9 (7 ± ε.3) eε.3 (7 ± ε.3)
see k = 0 see k = 0
see k = 0 see k = 0
42
10
eε.6 (5 + ε.2)
(1 + ε.2)Z12 + ε.e6 5X
52
eε.6 (5 − ε.2)
(1 − ε.2)Z12 + ε.e6 5X
z z z z z z z z z z z z
= 0, 3, 6, 9 : z + ε.{0, 2, 6, 10} = 1, 4, 7, 10 : z + ε.{0, 2, 4, 6, 10} = 2, 5, 8, 11 : z + ε.{1, 2, 4, 6, 10} = 0, 3, 6, 9 : z + ε.{0, 2, 6, 10} = 1, 4, 7, 10 : z + ε.{1, 2, 4, 6, 10} = 2, 5, 8, 11 : z + ε.{0, 2, 4, 6, 10}
= 0, 6 : z + ε.{0, 2, 4, 6} = 1, 7 : z + ε.{1, 2, 4, 6, 10} = 2, 8 : z + ε.{0, 4, 6, 10} = 3, 9 : z + ε.{0, 2, 6, 10} = 4, 10 : z + ε.{0, 2, 4, 10} = 5, 11 : z + ε.{0, 2, 4, 6, 10} = 0, 6 : z + ε.{0, 2, 4, 6} = 1, 7 : z + ε.{0, 2, 4, 6, 10} = 2, 8 : z + ε.{0, 2, 4, 10} = 3, 9 : z + ε.{0, 2, 6, 10} = 4, 10 : z + ε.{0, 4, 6, 10} = 5, 11 : z + ε.{1, 2, 4, 6, 10}
O.1. CONTRAPUNTAL SYMMETRIES
O.1.6 k 0
Class Nr. 82
g ε.6
e
g.X[ε] (1 + ε.6)
6
(1 + ε.6)Z12 + ε.e X
eε.6 (7 + ε.6)
(1 + ε.6)Z12 + ε.e6 7X
eε.11 (11 − ε.4)
(1 + ε.4)Z12 + ε.e11 11X
eε.11 (11 + ε.4)
(1 − ε.4)Z12 + ε.e11 11X
eε.11 11
Z12 + ε.e11 11X
g.X[ε] ∩ X[ε]
card(g.X[ε] ∩ X[ε])
z even: z + ε.{3, 9} z odd: z + ε.X z even: z + ε.{3, 7, 9} z odd: z + ε.(X − {7}) z = 0, 3, 6, 9 : z + ε.{3, 4, 7, 8} z = 1, 4, 7, 10 : z + ε.{0, 3, 7, 8} z = 2, 5, 8, 11 : z + ε.{0, 3, 4, 7} z = 0, 3, 6, 9 : z + ε.{3, 4, 7, 8} z = 1, 4, 7, 10 : z + ε.{0, 3, 4, 7} z = 2, 5, 8, 11 : z + ε.{0, 3, 7, 8} Z12 + ε.{3, 4, 7, 8}
48
z z z z z z
56
(5 − ε.4)
(1 + ε.4)Z12 + ε.e8 5X
eε.8 (5 + ε.4)
(1 − ε.4)Z12 + ε.e8 5X
4
eε.6 (1 + ε.6) eε.6 (7 + ε.6)
(1 + ε.6)Z12 + ε.e6 X (1 + ε.6)Z12 + ε.e6 7X
see k = 0 see k = 0
48
7
7
Z12 + ε.7X
Z12 + ε.(X − {7})
56
Z12 + ε.{0, 3, 4, 7} see k = 0 see k = 0 z = 0, 3, 6, 9 : z + ε.{0, 3, 4, 7} z = 1, 4, 7, 10 : z + ε.{3, 4, 7, 8} z = 2, 5, 8, 11 : z + ε.{0, 3, 7, 8} z = 0, 3, 6, 9 : z + ε.{0, 3, 4, 7} z = 1, 4, 7, 10 : z + ε.{0, 3, 7, 8} z = 2, 5, 8, 11 : z + ε.{3, 4, 7, 8}
48
3
8
ε.8
1217
e
ε.3
3
e 7 eε.6 (1 + ε.6) eε.6 (7 + ε.6) eε.3 (7 + ε.4)
Z12 + ε.e 7X (1 + ε.6)Z12 + ε.e6 X (1 + ε.6)Z12 + ε.e6 7X (1 + ε.4)Z12 + ε.e3 7X
eε.3 (7 − ε.4)
(1 − ε.4)Z12 + ε.e3 7X
= 0, 3, 6, 9 : z + ε.{0, 4, 7, 8} = 1, 4, 7, 10 : z + ε.(X − {7}) = 2, 5, 8, 11 : z + ε.(X − {9}) = 0, 3, 6, 9 : z + ε.{0, 4, 7, 8} = 1, 4, 7, 10 : z + ε.(X − {9}) = 2, 5, 8, 11 : z + ε.(X − {7})
1218
APPENDIX O. COUNTERPOINT STEPS
O.2
Permitted Successors for the Major Scale
For the sweeping orientation, given a cantus firmus step CF : x 7→ y, one is allowed to move from a consonance c (i.e., x + ε.c) in the top row to a consonance d (i.e., y + ε.d) in the right column iff there is a ∗ in the corresponding matrix entry.
1. Oblique Motion in Cantus Firmus CF : 0 7→ 0
CF : 2 7→ 2
CF : 4 7→ 4
CF : 5 7→ 5
CF : 7 7→ 7
CF : 9 7→ 9
CF : 11 7→ 11
0 4 7 ∗ ∗ ∗ ∗∗ ∗∗∗
0 3 ∗ ∗ ∗∗ ∗
0 3 ∗ ∗ ∗∗ ∗∗
0 4 7 ∗ ∗ ∗ ∗∗ ∗∗∗
0 4 7 ∗ ∗ ∗ ∗∗ ∗∗∗
0 3 ∗ ∗ ∗∗ ∗∗
0 3 8 ∗∗0 ∗ ∗3 ∗∗ 8
9 ∗0 ∗4 ∗7 9
7 9 ∗∗0 ∗ 3 ∗7 ∗ 9
7 8 ∗∗0 ∗∗3 ∗7 ∗ 8
9 ∗0 ∗4 ∗7 9
9 ∗0 ∗4 ∗7 9
2. Minor Ascending Second in Cantus Firmus CF : 4 7→ 5
CF : 11 7→ 0
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
3 ∗ ∗ ∗ ∗
7 8 ∗∗0 ∗∗4 ∗7 ∗∗9
3 ∗ ∗ ∗ ∗
8 ∗ ∗ ∗ ∗
0 4 7 9
3. Minor Descending Second in Cantus Firmus CF : 5 7→ 4
CF : 0 7→ 11
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗
4 ∗ ∗ ∗ ∗
7 9 ∗∗0 ∗∗3 ∗7 ∗∗8
4 ∗ ∗ ∗
7 ∗ ∗ ∗
9 ∗0 ∗3 ∗8
4. Major Ascending Second in Cantus Firmus CF : 0 7→ 2
CF : 2 7→ 4
CF : 5 7→ 7
CF : 7 7→ 9
CF : 9 7→ 11
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
0 4 7 8 ∗ ∗∗0 ∗ ∗∗4 ∗∗ ∗7 ∗∗∗∗9
0 4 7 9 ∗ ∗∗0 ∗∗∗∗3 ∗∗ ∗7 ∗ ∗∗8
0 ∗ ∗ ∗
4 7 ∗ ∗∗ ∗ ∗∗
9 ∗ ∗ ∗ ∗
0 3 7 9
3 ∗ ∗ ∗ ∗
7 9 ∗∗0 ∗∗3 ∗7 ∗∗8
3 ∗ ∗ ∗
7 ∗ ∗ ∗
8 ∗0 ∗3 ∗8
5. Major Descending Second in Cantus Firmus CF : 2 7→ 0
CF : 4 7→ 2
CF : 7 7→ 5
CF : 9 7→ 7
CF : 11 7→ 9
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
0 4 7 9 ∗ ∗∗0 ∗ ∗∗4 ∗∗ ∗7 ∗∗∗∗9
0 3 7 8 ∗ ∗∗0 ∗∗∗∗4 ∗∗ ∗7 ∗ ∗∗9
0 ∗ ∗ ∗ ∗
3 7 ∗ ∗∗ ∗ ∗∗
9 ∗ ∗ ∗ ∗
0 4 7 9
3 ∗ ∗ ∗ ∗
7 8 ∗∗0 ∗∗3 ∗7 ∗∗9
3 ∗ ∗ ∗ ∗
8 ∗ ∗ ∗ ∗
0 3 7 8
7 8 ∗∗0 ∗∗3 ∗7 ∗ 8
O.2. PERMITTED SUCCESSORS FOR THE MAJOR SCALE
1219
6. Minor Ascending Third in Cantus Firmus CF : 2 7→ 5
CF : 4 7→ 7
CF : 9 7→ 0
CF : 11 7→ 2
0 3 7 9 ∗ ∗∗0 ∗∗∗∗4 ∗∗ ∗7 ∗ ∗ 9
0 ∗ ∗ ∗ ∗
0 3 7 8 ∗ ∗∗0 ∗ ∗∗4 ∗∗ ∗7 ∗ ∗∗9
0 3 8 ∗∗∗0 ∗ ∗3 ∗∗∗7 ∗ ∗9
3 7 8 ∗∗∗0 ∗∗∗4 ∗ ∗7 ∗∗9
7. Minor Descending Third in Cantus Firmus CF : 5 7→ 2
CF : 7 7→ 4
CF : 0 7→ 9
CF : 2 7→ 11
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗
4 ∗ ∗ ∗ ∗
7 9 ∗∗0 ∗ 3 ∗7 ∗ 9
4 ∗ ∗ ∗ ∗
7 9 ∗∗0 ∗ 3 ∗7 ∗∗8
4 ∗ ∗ ∗ ∗
7 9 ∗∗0 ∗ 3 ∗7 ∗∗8
3 ∗ ∗ ∗
7 ∗ ∗ ∗
9 ∗0 ∗3 ∗8
8. Major Ascending Third in Cantus Firmus CF : 0 7→ 4
CF : 5 7→ 9
CF : 7 7→ 11
0 4 7 9 ∗ ∗∗0 ∗∗∗∗3 ∗∗ ∗7 ∗ ∗∗8
0 4 7 9 ∗ ∗∗0 ∗∗∗∗3 ∗∗ ∗7 ∗ ∗∗8
0 4 7 9 ∗ ∗∗0 ∗∗∗∗3 ∗ ∗∗8
9. Major Descending Third in Cantus Firmus CF : 4 7→ 0
CF : 9 7→ 5
CF : 11 7→ 7
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
3 ∗ ∗ ∗ ∗
7 8 ∗∗0 ∗∗4 ∗7 ∗∗9
3 ∗ ∗ ∗ ∗
7 8 ∗∗0 ∗∗4 ∗7 ∗∗9
3 ∗ ∗ ∗ ∗
8 ∗ ∗ ∗ ∗
0 4 7 9
10. Ascending Fourth in Cantus Firmus CF : 0 7→ 5
CF : 2 7→ 7
CF : 4 7→ 9
CF : 7 7→ 0
CF : 9 7→ 2
CF : 11 7→ 4
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
4 ∗ ∗ ∗ ∗
7 9 ∗∗0 ∗∗4 ∗7 ∗∗9
3 ∗ ∗ ∗ ∗
7 9 ∗∗0 ∗∗4 ∗7 ∗∗9
3 ∗ ∗ ∗ ∗
7 8 ∗∗0 ∗∗3 ∗7 ∗∗8
4 ∗ ∗ ∗ ∗
7 9 ∗∗0 ∗∗4 ∗7 ∗∗9
3 ∗ ∗ ∗ ∗
7 8 ∗∗0 ∗∗3 ∗7 ∗∗8
3 ∗ ∗ ∗ ∗
8 ∗ ∗ ∗ ∗
0 3 7 8
11. Descending Fourth in Cantus Firmus CF : 5 7→ 0
CF : 7 7→ 2
CF : 9 7→ 4
CF : 0 7→ 7
CF : 2 7→ 9
CF : 4 7→ 11
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗
4 ∗ ∗ ∗ ∗
7 9 ∗∗0 ∗∗4 ∗7 ∗∗9
4 ∗ ∗ ∗ ∗
7 9 ∗∗0 ∗∗3 ∗7 ∗∗9
3 ∗ ∗ ∗ ∗
7 8 ∗∗0 ∗∗3 ∗7 ∗∗8
12. Ascending Tritone in Cantus Firmus CF : 5 7→ 11
CF : 11 7→ 5
0 4 7 9 ∗∗0 ∗∗∗ 3 ∗ ∗∗8
0 3 8 ∗∗0 ∗∗∗4 ∗∗∗7 ∗ ∗9
4 ∗ ∗ ∗ ∗
7 9 ∗∗0 ∗∗4 ∗7 ∗∗9
3 ∗ ∗ ∗ ∗
7 9 ∗∗0 ∗∗3 ∗7 ∗∗8
3 ∗ ∗ ∗
7 ∗ ∗ ∗
8 ∗0 ∗3 ∗8
1220
APPENDIX O. COUNTERPOINT STEPS
Part XVIII
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1221
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Index Symbols 2F u , 66 B x, 280 Dia(N ames/E), 1138 H i (∆/GI ), 433 M ono(E), 1137 ˙ @M ˜ , 206 Rn @B T `2 (R), 226 , 50 C(3) , 325 A Q2 (U, V ; W ), 1093 , 50 {}, 50 !, 1121, 1131 !/α, 402 (G), 1076 (G : H), 1067 (GI )∨ , 336 (I1 /J1 ), 631 (I7 /J7 ), 631 (K1 /D1 ), 632 (K7 /D7 ), 632 (X/C(X)), 631 (X[ε]/Y [ε]), 634 (X|C(X)), 631 (f /α), 405 (ri,j ), 1085 (x, y), 1058 /N , 360 0R , 1075 12-T emp, 512 1R , 1075 2, 1062 3CH, 546 3Chains, 319 < N , 413
<Mod , 90, 93 ?s , 1110 ?e , 521 ?βσµ , 146 A(f, ζ, η, J(ζ), U (η)), 1156 ALLSERn , 237 ASCII, 406 AX, 1133 A@F , 1120 A@M OT , 466 A@M OTF , 466 A@M OTF,n , 466 A@ <M , 94 A × B, 1058 AB , 1062 AF , 1091 Aq , 1102 As , 1102 At,y,G , 951 Athreshold , 1029 AbsDyn, 81, 778 AbsDynamicEvents, 778 AbsT po, 414 AbsT poEvt, 415 AbsT poInComp, 415 AccentU,S , 769 Ad, 170 AddT o(c1, c2, at), 974 AllExtB (m), 539 AllExtB (n, m), 539 AllSet, 1058 Alphabet, 1063 AltM aj(i), 577 AltM aj(i)(3) , 577 An(B, M ), 1095 ArchaicF orm, 101 1255
1256 Arg, 748 ArtiSlurS,U , 769 Aut, 1066 Aut(K), 174 Aut(c), 1116 AutR (M ), 1083 B(∆), 1096 B(f, ζ, η, J(ζ), U (η)), 1156 BA α, 1133 BDVmean (w), 787 BDV (w), 787 BIT , 51, 71 ˜ , 205 B @F BN , 353 Br (x), 1153 B/M , 1095 BarLine, 81 Basic, 417 BeetM otChordF iber, 423 BeetM otChordF iberObject, 424 BetterF orm, 101 C, 80, 534, 1030 C(F ), 64 C(G, F, P, w), 233 C(X), 630 C(i), 577 C+, 534 C/OnBeetSon, 422 C0, 534 CD, 415, 821 CIN T (S), 254 CIN T (Ser), 253 CIN Tm (S), 254 CN (GI ), 346 COM (Cont), 251 CON Tn,k (X), 251 CP , 475 CT (D), 67 C ∗ (K; M ), 1150 C 0 [a, b], 1154 C 5 , 509 C i (X), 630 C n (K; M ), 1150 C ? (GI ), 374 CGI , 351
INDEX Cad, 552 Cc, 631 CcF ourier, 86 CcM , 630, 631 CellHierarchyBP , 725 CellU , 725 CellBP , 725 Ch/P ianoChord, 413 Chrono − F ourier, 288 ChronoF ourierSound, 288 Ci, 630 CiM , 630 ClH (U ), 715 Cln , 215 ClassZChord, 509 ClosedX , 1145 Cm, 534 ColimCirc, 77 CommaZM odule, 511 Consonant, 286 Count, 417 Covens, 1148 Crescendo, 79 D, 80, 323, 721, 1028 DEG, 556 DF0 , 1155 DN R, 256 Dr f , 1155 Dµ (M ), 486 DK , 1112 Df , 1108 Dx (T0 ), 802 D (M ), 482 Dk,n , 150 Daughters, 725 Dc, 630 DcM , 630 DenOrb(R, n, B, M ), 207 Der(L), 1105 Df , 1155 Dfx , 1155 Dg, 557 Dg ∗ , 557 Di, 630 DiM , 630
INDEX Dia, 470 Dia(N ames), 1138 Diaπ, 476 Diak , 470 Dil, 1084 Dir, 797 Dir(Q), 1064 Dom(X), 461 Dp, 557 Dup(T ), 407 Duration, 79 DurationV alue, 977 DynSymb, 81 E, 80, 1028 E2M , 73 EHLD, 768, 950 EHLDGC, 799 EM B(A, B), 254 EN H, 511 EX, 1131 E F , 129 EdRg , 472 Ed∆Rg , 472 El(G), 489 El(M ), 489 Elast, 471 End(A), 170 End(c), 1116 EndR (M ), 1083 Envelope, 84 EulerChord, 109 EulerChordEvent, 110 EulerM odule, 73, 106, 109 EulerM oduleq [ε], 618 EulerP lane, 110, 113 EulerZChord, 509 EulerRM odule, 684 EvtS (U ), 766 Ex, 511 Ex(D, Lα,β ), 373 ExT opA (F ), 521 ExtA (M ), 518 F a G, 1120 F (D), 67 F CM (Alphabet), 1064
1257 F D, 821 F G(Alphabet), 1067 F M (Alphabet), 1063 F M -Object, 87 F O(P EX), 1135 F ORM S, 1138 F SH, 417 F0 , 1155 Fx , 1107 Fπ , 672 Fsym , 672 F ermata, 781 F ermataE,S , 769 F ib(e. ), 914 F ieldU , 726 F ieldU,R , 726 F in, 1124 F in(Onset), 97 F in(P itch), 97 F in(S), 1061 F latten, 88 F lattenn , 89 F ormList(S), 71 F ourier, 84, 86 F ourierSound, 84 F rameU , 726 F unc(C, D), 1119 F und(H), 715 F ushi, 417 F ushiOrnament, 417 F ushiP ic, 417 F ushiST RG, 417 G, 80, 1032 G.x, 315 G/H, 1067 G/Ω t Ω∗ , 770 GL(n, p), 1070 GLn (R), 1070 G o H, 1069 GI , 309 GI∗ , 351 Gopp , 1066 Gm , 1066 Gv,k,l , 226 GesP t (M ), 474
1258 Gesµt (M ), 486 Gest (M ), 474 GlC t , 676 Glissando, 79 GlobP erf ScoreBP. , 729 GlobP erf ScoreBP , 728 Grassr,n , 1113 H, 80, 715, 1018, 1032 HA α, 1133 H \ G, 1067 H / G, 1067 i H∆ (GI ), 432 n H (K; M ), 1151 H ? (GI ), 374 Hl? (X M axM et ), 460 Hprime , 1033 HarM in, 577 HarM in(3) , 577 Hom, 1066, 1075, 1115 Homloc , 343 HomC , 1115 I, 696, 721 I(F ), 64 I(H), 700 ICV (A, B), 253 ID, 721 IIIX , 321 IIX , 321 IN T (Ser), 252 IN Tm (Ser), 252 IV (A, B), 253 IVX , 321 I|i, 329 IU , 725 IX , 321 Ix , 621 IdA , 170, 1059 Ide , 1116 Idempot(M ), 1064 Im(f ), 1062 Importance, 415 InT op(F ), 520 InitP erfU , 726 InitSetU , 725 InstruN ame, 82
INDEX Int(E), 518 Int(M ), 518 IntM od12,q [ε], 621 IntM od12 , 621 IntT hirds3,4,q [ε], 621 IntT hirds3,4 , 621 Inte (M ), 518 IntA , 522 Intg , 1067 Intonation, 684 Item(F ), 976 J(TsE , w), 794 JCK, 416 JCKF U , 418 JK (X), 1129 JKt , 325 K(X), 1129 KUs,t , 144 K1 f K2 , 217 K4 , 251 KE , 697 Kb , 857 KT opS , 767 Ker(f ), 1067 KernelU , 726 Knot, 87 KnotBasic , 88, 330 KnotP erOns12 , 612 KnotP erOns , 612 Kq, 74 Kt, 74 K|k, 1148 L, 80, 1030 LB, 971 LCP , 475 BP,k LP SInstrument , 728 BP. LP SInstrument , 728 BP LP SInstrument , 728 LF f , 1160 Lψ f (a, b), 289, 1025 Label, 415 LegatoSlurU , 768 LegatoSlurU,S , 769 Lev, 329 LimCirc, 77
INDEX Limint(F ), 317 LinR , 1083 LinR (GI , M ), 351 List(F ), 976 ListEntryF , 417 ListF , 417 LocC t , 670 LocP erf ScoreBP , 724 LocF , 108 Loudness, 79 M/N , 351 M OT , 466 M OTF , 466 M OTn , 466 M OTF,n , 466 M [ε], 127 M @F , 1137 M @, 1091, 1119 ι
M |f /IdA , 352 M n , 209 M ? , 1063 Mn , 253 Mq , 1102 Ms , 1102 Mt , 546 Mx (T0 ), 802 M12 , 318 MOP , 466 MO , 466 M[ϕ] , 1084 Mjust , 318 Md,a , 540 Mi,j , 950 M aelzel, 414 M aj, 576 M akroBasic , 88, 330 M akroP erOns12 , 612 M akroP erOns , 612 M arcatoU,S , 769 M atchP , 1130 M athP itch, 72 M ax(X), 329 M axM et(X), 329 M edia, 415
1259 M elM in, 577 M elM in(3) , 577 M in(L, S), 705 M in(X, S), 705 M odelF orm, 101 M onEnd(F ), 518 M or(C), 1116 M osG n , 378 M osG n,λ , 379 M osG n,k , 378 M other, 725 M t/M otif , 413 N (D), 67 N (F ), 63 N (U ), 1149 N Comp, 415 N F , 255 N oφ H, 1068 Ne , 1063 Nk (U ), 1149 Nred , 365 N at(F, G), 1119 N atM in, 576 N atM in(3) , 576 N ewContainer, 974 N oteGroup, 413 N umber, 415 OP D, 145 OP LDZ , 532 OS, 670 Ob(C), 1116 ObExT opB (F ), 526 OnM odm , 75 OnP iM odm,n , 75 Onset, 79, 110 Onset??S , 80 OnsetP roj, 422 Op, 749 Open(G), 407 Open(S), 1145 OpenX , 1107, 1145 Openf , 1107 OpenX,x , 1107 Openf,x , 1107 OrchSet, 82
1260 P (E), 639 P (Q), 1079 P (Q)/ ∼, 1117 P EX, 1135 P F , 256 P O, 612 P ∧ Q, 421 P + , 1130 PT r¨aumerei , 873 P ara, 117, 119 P art(I), 378 P artial, 86 P ause, 81 P er(R), 380 P erOns, 612 P erOns12 , 612 P ercussion, 612 P eriodsw , 1018 P hysCrescendo, 81 P hysDuration, 81 P hysGlissando, 81 P hysInstri , 673 P hysLoudness, 81 P hysOnset, 81 P hysOrchestra, 673 P hysP itch, 81 P hysicalBruteF orceOperator, 791 P iM od12,(7) , 618 P iM odn , 75 P iT hirds3,4 , 620 P ianoSelector, 54 P iano-N ote, 51 P itch, 79 P itchChange, 417 P itchSymb, 80 P owerCirc, 77 P tch, 140 P ythagorasLine, 560 QN ormalize(c), 974 QReduce(c, [n]), 974 Qw , 793 Qβσµ , 146 Qθ (E1 , E2 ), 640 Qθ (K/D), 640 R(P ara), 327
INDEX R.S, 126 R.f , 172 REd, 472 RT , 977 RT C, 449 RU LES, 1133 R[ε], 1077 RhM i, 1077 RhQi, 1079 R∗ , 612 ∗ ROns , 612 R2 , 882 RC , 1085 R× , 1075 RR , 1084 RGI , 351 RV uza , 380 Rmax , 726 Rmin , 726 Rad(M ), 1088 Rat, 748 Rate, 414 RelDyn, 81, 779 RelDynamicEvents, 779 RemainderSplit ∝ µ, 789 RemoveF rom(c1, c2), 974 RepA,n? , 363 Review, 415 Rg, 469 Rgo (M ), 469 Rgp (M ), 469 Rhythm(P ara), 327 S!, 419 S, 323 S(E), 639 S(EX), 1131 S(p, u.), 1070 S/ ∼, 1060 SEGk [M 1/n ], 252 SERMn , 150 SERMk,n , 150 SERn , 150 SERk,n , 150 SO2 (71), 949 SP E, 286
INDEX S[Alphabet], 1077 ShAlphabeti, 1077 S⊗R ?, 1091 S −1 A, 1101 S −1 M , 1102 S , 121 Sn (K), 1150 Sq , 618 Sq [ε], 618 StMt , 546 SA,n? , 363 Sat, 449 SatF B , 526 Satellite, 811 ScalarOperatorw , 795 ScoreF orm, 724 ScoreInstri , 673 ScoreOrchestra, 673 SemInT op(F ), 520 Sema(E), 1140 SemiEnd(F ), 518 Semitone, 1032 Sg, 557 Sh(C, J), 1130 SimplexU , 725 SimplexesU , 725 Sound, 146 Sp, 557 Sp(X), 459 Sp(x), 329 Spµ (x), 467 Spec, 1109 Spec(A), 1108 Spec(R), 179 Spec(f ), 1109 Special, 417 Split ∝ µ, 789 SplitU,ν , 788 StaccatissimoU,S , 769 StaccatoU,S , 769 StepT une, 782 Sub(X), 1062 Sub(BP), 715 SubM ∗ (N ), 480 SubC , 1126
1261 Support, 1015 SupportF orm(E), 411 Switch, 71 Syllabic, 286 Sym(K), 174 Symi (A, B), 254 SymbolicBrueF orceOperator, 790 SynCirc, 76 T , 323, 721, 1017 T (E), 697 T (F ), 64 T (G), 1070 T (p., u.. ), 1070 T F , 544 T Ff,t , 544 T GI , 677 T ID, 721 T In , 150 T K, 669 T O, 1155 T RU T H(F ), 530 T RU T H(I), 407 T RU T H(h), 409 T T O, 253 T r f , 1155 T t K, 669 T0 , 801, 1147 T1 , 1147 T2 , 1147 Tk K, 669 T2,R , 475 TX,x , 1112 TΛ , 801 TTON , 589 TVALmode , 589 TVALtype , 589 Tε , 128 Tζ,η,f , 1157 Tred , 128 T empo, 682 T empoOperatorw , 793 T ension(λ. , µ. , ω, φmin ), 590 T enutoU,S , 769 T erminal, 976 T f , 669, 1155
1262 T fk , 669 T g, 557 T imeSig, 82 T imeSig(p/q)S , 769 T itle, 415 T op(H), 715 T opS , 767 T orSeq, 471 T oroidm,l γ , 471 T p, 557 T r(D, T ), 636 T rans, 380 T rans(D, T ), 636 T riv(K), 174 T une, 782 T ypes, 1137 U N ICODE, 406 U V , 1094 U n , 727 U o , 1146 Ug , 141 Un , 253 Us , 141 Ux , 321 Un+1 , 727 Ux/x+1 , 321 U tai, 418 V (E), 1108 V F (O), 1160 V IIX , 321 V IX , 321 VX , 321 Vn,x , 213 V al(I), 407 V alCh(p/q = r/s)S , 769 V erbAbsT po, 414 V owel, 286 W Pn (U, TR ), 727 W ∗ , 605 R Wn,i , 768 W eight(U ), 727 W eightListBP , 726 W eightn (U ), 727 W eightBP , 727 X(R, n, B, M ), 207
INDEX X(harmo), 879 X(melod), 879 X(metric), 879 X@F , 1091 X-chromatic, 509 X-harm, 509 X-major, 509 X-mel, 509 X (3) , 321 X M axM et , 329 X M etLg[L] , 329 X M etP er[P ] , 329 X ? , 1091 X.cyc(g) , 1071 X6 , 559 Xnr , 213 Xadd , 878 YAff , 1111 YComRings , 1111 Y ear, 415 Z(R), 1075 Z(harmo), 879 Z(metric), 879 ZN F , 256 [C/C(X)], 631 [C|C(X)], 631 [K), 283 [K], 235 [M ), 489 [M 1/n ], 252 [R], 407 [S], 1119 [ ni ], 378 [a, b], 556 [b, p, g], 328 [l|x|k], 130 [s], 1060 [x), 489 [x], 1159 [xy], 541 &, 1131 An , 1112 Ab, 1107 Aff, 1109 kf k1 , 1155
INDEX kf k2 , 1155 kf k∞ , 1155 kxk1 , 1154 kxk2 , 1154 kxk∞ , 1154 k k, 1154 @M , 1091, 1119 @, 62, 1090 @red R M , 1095 @R , 1090 @loc X, 342 C, 1076 CHR, 51 Colimit, 1137 ComGlobA , 347 ComLoClassgen,M n+1,End(B) , 207 ComLoClassgen,lf,sp n+1,0R , 215 ComLoClassgen n+1,OR , 210 ComLoClassn+1,0R , 215 ComLocA , 172 ComLocemb A , 172 ComLocgen A , 172 ComLocin A , 172 ComMod, 1108 ComRings, 1107 ComRings@ , 1112 ∆Rg, 469 ∆nΓ(GI ), 358 ∆, 686, 763, 1131 ∆n , 211, 357 ∆GI , 358 ∆i,j , 1098 Den, 402 Den(x, y), 402 DenColimit , 402 DenLimit , 402 DenPower , 402 DenSimple , 402 DenSyn , 402 Den∞ , 406 Den∞ /sig, 410 ForSem, 1141 ΓBf , 354 ΓΛnf , 355
1263 Γ, 353, 1049, 1109 Γ(GI ), 351 Γ(U, F ), 1109 ι
Γ(f /IdA ), 353 Γ(myF M Object), 87 Γw , 292 Γt , k468 Γt , 468 Γt (M, j), 377 ΓDir , 803 ΓRedIndia,c , 470 Γt,n , 468 µ Glob, 428 Gr, 1107, 1117 H, 1076 Λ ↓, 763 Λ ↑, 763 Λ l, 764 Λ∞ , 764 LieR , 1104 Limit, 1137 LinMod, 1117 LinModR , 1117 Loc, 164 LocEnd(A) , 170 Loc@A , 165 µ Loc, 428 LocRgSpaces, 1108 MR , 1117 Mm,n (R), 1084 Mod@ , 1091 Mod, 1091, 1107, 1117 ModR , 1091, 1117 Mon, 1107, 1117 N, 1058, 1116 µ ObGlob, 428 ObLoc, 158 µ ObLoc, 428 ObLocEnd(A) , 170 ObLoc@A , 165 Ω ∝ µ, 770 Ω, 727, 763, 872, 1126 Ω(p), 588 ΩSh , 1130
1264 ΩX ω , 876 Φ, 1131 Π, 743, 1135 Πw , 232 Power, 1137 Ψ, 742 Q, 72, 73, 1058, 1076 R, 51, 72, 1058, 1076 R[Q] , 72 ⇒, 1022, 1132 }R, 100 Rings, 1107, 1117 BC(A, W ), 1158 BP, 715 B(A, W ), 1157 C1 × C2 |K, 714 GlDifft , 676 Glob, 335 LocDifft , 670 ObGlob, 335 TantR , 669 TantCat , 670 Schemes, 1111 SetsU , 1116 ΣI Mi , 1084 Simpl, 1148 SinLoc, 169 SinLocEnd(A) , 170 SinLoc@A , 170 Sob, 1110 Simple, 1137 Syn, 1137 TON, 544, 545, 588, 590 T(h), 409 TI , 407 TA I , 407 Tex, 406 Texig(Den)(h), 409 Texig(Den)I , 406 Top, 1117, 1146 Υ, 1135 VAL, 544, 545, 588, 590 VALmode , 589 VALtype , 589 Ξ, 1131, 1135
INDEX ΞD , 508 ΞK , 508 Ts, 690 Z, 51, 1058, 1076 Zn , 1069, 1076 Z2 , 71 Z471 , 947 TsΛ,Dir , 797 TsΛ,U , 801 TsΛ , 797 Tsex , 720 α, 638 α+ , 618, 689 α− , 618 α+ , 128 α− , 128 ¯ 489 X, z¯, 1076 β, 638 β(x), 354 T V , 1058 S LV , 1058 I Mi , 1084 ⊥, 408, 1132 , 1094 •, 971 ∩, 346 ˇ?@A , 165 ˇ 121 S, χ, 506, 508 χ(Y ), 1062 χ? , 507 χ `σ , 1126 I Mi , 1062 δY , 324 δ, 638, 743 δ(X/Y ), 633 δ(Y |X), 633 δ[X|Y ], 633 δ@A , 358 δij , 1085 f˙/IdA , 359 s, ˙ 359 ∅, 176, 1057 ∅R , 1091
INDEX ∃x, 1136 ∃x P (x, y), 421 ∀x, 1136 ∀x P (x, y), 421 D, 1108 H, 722 MusGen, 159 M, 1136 M α[x], 1136 M0 , 559 M1 , 559 M2 , 559 SM , 1066 T, 484 T/Ges, 484 Tsp T oroid , 489 Tsp T oroid /Ges, 489 Tµ/Ges , 484 Tµ , 484, 486 Tµ /Ges, 486 Tt,P,d , 483 Tt,P,d /Gest , 483 W, 727 gl(L), 1105 gl(n, R), 1105 h, 716, 763 h|U , 718 sl(L), 1105 sl(n, R), 1105 γ, 743 −→ GL(n, p), 1070 ˆ?, 66, 157 ˆb, 874 fˆ, 157, 1025 −>, 1131 ∈, R 1057 RC F , 1122 f , 1159 x ι, 763 ιj , 1062, 1084 κ, 552 κ(x), 1108 κJ , 553 κRelDyn , 779 κorb , 560
1265 hSie , 1063 hSi, 513, 1063 hSi, 1118 |—— α, 1134 CL
B, 1146 C r , 1155 E, 1137 F(O), 1160 M, 210 Mr , 213 R, 1137 µ, 763 ¬, 1132 ω, 424 Y , 1145 f , 1150 ∂Dynamics, 689 ∂Intonation, 689 ∂T empo, 689 ∂U , 718, 1146 ∂Z, 689 ∂Tsλ,ex , 720 ∂fi /∂xj , 1155 ∂ Y R, 704 ∂Y R, 704 φ(n), 1069 πY , 324 πj , 1084 π Qm,l , 477 I Mi , 1062 ψa,b , 289 ρD , 1016 σ(X/Y ), 633 σ(X|Y ), 633 σ[X|Y ], 633 ∼, 1060 ∼P s, 1046 √ x, 1064 @, 483, 1079 i @λi , 587 i @λi , 1079 , 1094 ⊂, 1057 ⊆, 1057
1266 τ (X, D, S), 705 |Ser|1 , 280 |Ser|2 , 280 θ, 638 ˜ 1108 A, ˜ nr , 213 X ×, 216 >, 408, 1132 → − T 2 (R), 226 U , 719 Λ, 764 Ts, 719 α, 1132 ϕx , 803 ∨, 1132 ∧, 1132 ∧r , 1103 c, 526 M ℘C , 712 ]x[, 325 + τx , 323 0 τx , 323 R Glob, 431 R GlobA , 431 loc Loc@ , 343 0 C, 1116 1 C, 1116 A Covn? , 357 A ∆n , 357 A ∆n? /N , 360 R R, 1084 F R SatA , 527 R µ Glob, 431 R µ GlobA , 431 a, alpha525 a tc b, 1121 a ×c b, 1121 abcardt (M ), 468 abcardt (m), 468 ad(x), 1105 ad, 1062 add, 407 at, 545
INDEX b f , 874 book, 78, 85 bottom(x), 1081 c, 1029 c-spacen (X), 251 c5 , 509 cm , 925 c(3) , 561 (3) ch , 561 (3) cm , 561 char , 577 cmel , 577 cnat , 576 card(A), 1059 causalEnd, 926 causalStart, 926 char(F ), 1076 codom(f ), 1115 colim(∆), 1121 coord(F ), 1139 ctµ, (M ), 497 cyc(g), 1071 d, 472, 1028 dB, 1029 d∗ , 1154 d2E x, 874 d∗P , 478 dn , 1150 dt , 472 d1,c , 473 d2,c , 473 dE x, 874 d∗P,n , 478 d∞,c , 473 def ormLiGr, 925 den(E), 412 df , 1160 dim(K), 1148 dim(M ), 1085 dom(f ), 1115 drap(k), 207 e, 1028 eM , 1090 em , 1090
INDEX ez , 913 enh, 515 evp , 179 exz , 523 exp(F, G), 405 exp(ad(x)), 1105 ext(E), 410 f /α, 155 f : a → b, 1115 f @S, 121 f ◦ g, 1115 ι
f /α, 335 ι
f /α ? M , 350 f −1 (C), 1062 f∗ F , 1107 fS,T , 1102 f inalEnd, 926 f inalStart, 926 f rame(F ), 1139 f un(F ), 1139 g, 556, 1032 g ◦ f , 1059 glb, 422 grad(f ), 1160 groundclass(C I ), 381 h, 1031, 1032 hx , 1108 i, 1130 i < x, y >, 253 i{x, y}, 253 ic < k >, 253 ic{k}, 253 id(F ), 1139 intexz,B,G (Ch), 559 intexz,B , 524 ip < a, b >, 252 ip{a, b}, 252 isX , 459 iso∩, 346 j, 1110 jak, 261 jakπ , 266 k-Contra, 470 ks , 143
1267 key, 1032 l, 1063 l(Cont), 251 l(M ), 328, 1089 lambdaw (x), 317 lev, 1061 lev(x), 329 levi , 329 lim(∆), 1121 lim(D), 1078 loc, 342 lub, 422 m−1 , 1066 mx , 1108 m12 , 318 mjust , 318 modo , 508 modp , 112 monexz , 523 myF M w , 292 myF Mw , 292 n(U ), 1148 nW , 498 nW (M ), 497 nΓ(σ), 351 nk (U ), 1148 n∞ (K I ), 317 newset, 176 o, 73 o-T empClass, 110 obintexz,B , 526 p, 556 p(M ), 328 p-ClassChord, 110 p-EulerClass, 110 p-Scale, 113 pQ x , 1013 pBP Instrument , 728 pB , 689 pj , 1062 pOnset , 119 pU,V , 715 px∆ , 637 pβσµ , 145 pcf , 618
1268 pint , 618 pmeter , 327 pct, 780 prµ, (M ), 497 prof , 460 pv, 73 q, 73 qX , 1111 r(TsE , w), 794 rH , 1016 rt , 253 rel(x, ai ), 786 resGI /IdA , 358 res@A , 358 res ι , 358 f /IdA
resclass(C I ), 381 ret, 363 revk , 150 round(x), 1081 s0 ≤ s, 1148 s(TsE , w, t), 794 s(x), 1108 sG, 557 sG∗ , 557 sP , 557 scalemodp , 113 sem, 1138 set, 176 sigDen , 406 sing, 1124 sp(x), 329 span, 467 sti (x), 317 supp(χ), 410 sym(A), 254 t, 73, 468 t(F ), 1139 ,tk ttpoopt , 786 tn , 468 top(x), 1081 true, 1126 w, 763 w-P itch, 160 w-P itchClass, 111
INDEX w-T emperedScale, 113 S wharmo , 786 S wmetro , 785 S wmotif , 785, 786 wEvt.,RelEvt.. , 780 wGrpArti , 783 wloc , 921 worth , 921 xhmax , 873 xhmean , 873 xmelodic , 873 xmetric , 873 x/E, 410 x < y, 1060 x > y, 1110 ˆ 315 x@G, ∼ x , 403 x z, 278 x y, 488 xh , 320 xm , 320 xU,V , 542 xalt , 128 xj,s , 859 xred , 128 y(ti , j), 877 C/opp b, 1121 C/b, 1121 C α, 1133 C@ , 1119 Copp , 1117 Cspaces , 1107 Cspaces , 1107 ? Cspaces , 1107 X aP , 1131 |, 1131 |?|, 1149 |?|d , 1149 |K|, 1148 |f |, 1149 |m|α , 525 |s|, 1149 Colimit, 69 Limit, 69 Loc, 105
INDEX Mod@ , 63 Power, 68, 107 PerCell, 713 Simple, 68 Syn, 68 head, 1063 tail, 1063 Colimit, 64 Limit, 64 Power, 64 Simple, 64 Syn, 64 A A-addressed function, 351 abelian, 1066 abelian group finitely generated -, 373 absolute dynamical sign, 777 dynamics, 831 logic, 176 music, 934 pitch, 700 symbolic dynamics, 81 tempo, 682, 780 absorbing point, 525 absorption, 1015 coefficient, 1015 abstract cardinality, 468 complement, 254 gestalt, 474 specialization, 488 identity, 16 inclusion, 254 motif, 468 onset, 150 specialization, 488 abstraction, 492, 494, 515 concept framework, 468 textual -, 440
1269 accelerando, 739, 768, 782 accelerated motion, 738 accentuation, 720 accessory parameter, 999 accumulation point, 1145 acoustics virtual -, 850 action complement -, 630 faithful -, 1066 free -, 1066 left -, 1066 motor -, 739 right -, 1066 transitive -, 1067 activities fundamental -, 4, 7 activity combinatorial -, 242 construction -, 197 instinctive -, 757 interpretative -, 300, 307, 308 acuteness, 290 ad-hoc polymorphism, 968 adapted tempo curve, 699 Add-Element, 983 address, 61, 63, 1091, 1120 change, 63, 83 technique, 83 faithful, 523 fixed vs. variable, 106 for a chord, 111 full -, 523 fully faithful -, 523 functor, 170 killing, 204 navigation, 169 variable, 61 zero -, 61, 62 addressed adjointness, 166 comma category, 165 adic representation, 1080
1270 adjoint left -, 1120 right -, 1120 adjointness addressed -, 166 adjunction, 1062 admitted successor, 647 tonalities, 566 Adorno, Theodor Wiesengrund, 186, 300, 302, 665, 691, 696, 741, 757, 792 Adrien, Jean-Marie, 1027 affine counterpoint group, 475 dual, 1091 Lie bracket, 541 tensor product, 1094 transformation modular -, 948 affine functions complex of -, 351 on functorial global compositions, 431 after qualifier, 983 Agawu, Kofi, 400 Age of Enlightenment, 41 aggregate, 253 Agmon, Eytan, 248, 250 agogical architecture, 963 operator, 872 agogics, 304, 780 global -, 764 primavista -, 764 AgoLogic, 699, 758, 953, 963 Agon, Carlos, viii, 257, 382 Alain, 175 aleatoric component, 242 aleatorics, 70 algebra, 1075 Boolean -, 123, 1132 general linear -, 1105
INDEX Heyting -, 123, 1132 Lie -, 1104 logical -, 1132 monoid -, 1077 quiver -, 1079 Riemann -, 586 algebraic geometry, 668 topology, 200 algorithm, 1022 Euclidean -, 1076, 1080 in FM synthesis, 87 off-line -, 919 real-time -, 919 TX802, 289 algorithmic extraction of performance fields, 916 Alighieri, Dante, 138, 196 aliquid pro aliquo, 16 all-interval n-phonic series, 237 series, 244 allomorph, 539 allomorphic extension, 539 allowed successor pairing, 646 α-restriction, 525 alphabet of creativity, 242 of music, 106 alphabetic ordering, 40, 43, 58 alteration, 127, 129, 196, 198, 276, 567, 618 as tangent, 128 direction of -, 951 elementary -, 129 force field, 952 in pitch, 62 pitch -, 952 successively increased -, 952 two-dimensional -, 950 altered note, 127
INDEX scale, 585 ambient space, 107 ambient space, 107 coproduct -, 124 dual -, 128 product -, 124 ambiguity, 300, 307 theory of -, 300 tonal -, 601 ambitus, 320 American (musical) set theory, 139, 219, 247–258 contour theory, 467 jazz, 538 notation, 533 theory, 534 amplitude, 291, 1020 modulation, 288, 1003 spectrum, 1020 Amuedo’s decimal normal rotation, 256 Amuedo, John, 255, 258, 534 analysis, 1018 -by-synthesis, 741, 755 chord -, 533 coherent -, 772 comparative -, 333 FM -, 289 immanent -, 458, 465 metrical -, 835 motivic -, 262, 491 musical -, 744 neutral -, 272, 305 normative -, 457 principal component -, 898 regression -, 860, 877, 880 situs, 199, 277 sonic -, 842 spectral -, 638 text -, 741 analytical discourse, 12 vector, 876
1271 weight, 666, 671, 785 anchor note, 760 Andreatta, Moreno, viii, 257, 382 ANSI-C, 945 anthropic principle, 565, 567, 658 anthropology computer-aided -, 925 anti-homomorphism ring -, 1076 antisymmetric, 1059 antiworld, 560, 605, 933 anvil, 1037 Appassionata, 667, 907 application framework, 808 apposition, 18 approach bigeneric -, 540 categorical -, 967 historical -, 565, 574 nonparametric -, 856 statistical -, 855 systematic -, 574 transformational -, 249 approximate, 829 arbitrary, 18 archicortex, 642 architectural principle, 869 architecture agogical -, 963 modulatory -, 603 Argerich, Martha, 894, 895, 927 argument, 748 Aristotle, v, 30, 31, 43, 50, 934 arpeggio, 88, 161, 697, 720, 760 field, 698 arrow, 618, 1063 self-addressed -, 626 articulated listening, 304 articulation, 304, 682, 769, 783, 832 double -, 19 field, 687–689 initial -, 702 operator, 720 artistic fantasy, 692
1272 artistry combinatorial -, 243 arts, 5 Ashkenazy, Vladimir, 884, 893, 895, 897 Assayag, G´erard, viii, 955 associated metric, 1154 metrical rhythm, 327 topology, 1154 AST, 247, 332, 470, 498, 534 global -, 382–385 software for -, 255 asymmetries of communication, 910 Atarir , 758 Mega ST4, 955 atlas, 308, 676 A-addressed, 309 projective -, 360 standard -, 357 atlases equivalent, 314 atom semantic -, 538 atomic formula, 1135 atomism ontological -, 27 atonal music, 248 attack, 1018 auditory cortex, 639, 641 gestalt, 481 nerve, 1037 representation, 240 augmentation, 161 augmented, 540 Augustinus, 564, 611 Auroux, Sylvain, 41 auto, 1116 autocomplementarity, 220, 517 function, 508, 632 autocomplementary marked dichotomy, 631 autocorrelation, 925 autocorrelogram, 864 automorphism, 1066, 1075, 1116 group, 174, 1083
INDEX of interpretable compositions, 372 autonomy, 7 Avison, Charles, 303 axiom, 1133 of choice, 1060 axis third -, 113 diachronic -, 399, 575 fifth -, 113 of combination, 138, 260 of selection, 138, 260 paradigmatic -, 194 synchronic -, 399, 575 syntagmatic -, 18, 194 B B¨atschmann’s Bezugssystem, 12 B¨atschmann, Oskar, 12, 187 B´ek´esy, Georg von, 1041 Babbitt, Milton, 247–249 Bach, Johann Sebastian, 137, 141, 144, 196, 231, 243, 248, 303, 304, 394, 595, 693, 740, 835, 857, 860, 907 background, 503 Bacon, Francis, 5 ball open -, 1153 Banach space, 1154 Banach, Stefan, 1154 band frequency -, 640 bandwidth, 856, 874 bankruptcy scientific -, 24 bar grouping, 864 bar-line, 720, 768 bar-lines, 81 barline meter, 115 Barlow, Klarenz, 1053 Barthes, Roland, 17, 195 barycentric coordinate, 1149 base, 1085 for a topology, 1146 sheaf on -, 1108
INDEX Basic, 88 basic extension, 518 intension, 518 series, 150 theme, 246 basilar membrane, 1038 basis, 1129 calculation, 918 coordinate, 1028 deformation, 797 of a tangent composition, 669 of disciplines, 6 parameter, 79, 795 space, 689, 715, 763 specialization, 797 basis-pianola operator, 795 Baudelaire, Charles, 268, 601, 963 Bauer, Moritz, 262 beat, 457, 1051 frequency, 1051 meter, 115 strong -, 457 weak -, 457 beauty, 419 Beethoven, Ludwig van, 118, 145, 161, 245, 303, 327, 337, 394, 422, 492, 495, 559, 560, 563, 567, 594, 603, 693, 935, 941, 1035 before qualifier, 983 behave well, 483 Benjamin, Walter, 665, 691, 792 Beran operator, 876 Beran, Jan, viii, 245, 246, 594, 745, 855, 876 Berg, Alban, 150, 248, 301 Berger, Hans, 640 Bernard, Jonathan W., 249 Beschler, Edwin, ix Bessel function, 1024 Bezugssystem, B¨ atschmann’s, 12 Bezuoli, Giuseppe, 30 biaffine, 1093 bidirectional dialog, 34
1273 big bang, 399 big science, 239 bigeneric approach, 540 major tonality, 547 bigeneric morphemes construction of -, 541 bijective, 1059 bilinear form, 354 Binnenstruktur, 301 biological inheritance principle, 763 bipolar recording, 638 Bissonanz, 514 Blake, William, 505, 911 block, 716, 947 type, 379 Boccherini, Luigi, 295, 994 body performance -, 712 B¨osendorfer, 700, 764, 765, 833, 849 boiling down method, 787 book concept, 56 BOOLE, 50 Boole, George, 1132 Boolean algebra, 123, 1132 combination of (class) chords, 111 operation, 947 topos, 1134 bottle M¨obius -, 677 bottom wall, 768 Boulez, Pierre, 33, 39, 40, 45, 105, 106, 137, 152, 299, 301, 309, 349, 369, 939, 941, 999, 1007, 1048 bound Chopin rubato, 760 variable -, 1136 boundary, 1146 bow angle, 288 application, 288
1274 parameter, 1003 pressure, 288, 1002 velocity, 288, 1002 box, 969 factory -, 972 flow -, 1160 temporal -, 979 Box-Value, 983 bracket Lie -, 1104, 1161 brain emotional -, 641 breaking symmetry -, 936 Brendel, Alfred, 883, 884 brilliance, 290 Brinkman, Alexander, 255 Bruijn, Nicolass Govert de, 232, 236, 379 bundle of ontologies, 171 tangent -, 1155 Bunin, Stanislav, 891 Buteau, Chantal, viii, 456, 490 C C-major, 556 inner symmetry of -, 147 c-motif, 119 C-scale frame, 577 C-scheme, 1112 C-space, 1107 c-space, 251 CAC, 967 cadence, 502, 551–562, 566, 986, 1008 parameter, 552 Rameau’s -, 554 cadential, 552, 565 family minimal -, 554 formula, 551 set, 554 Cage, John, 70, 306, 694 calculation basis -, 918 field -, 918
INDEX precision, 775 calculus, 692 camera obscura, 734 canon, 140, 194, 328 canon cancricans, 857, 860 canonical curve, 872, 890, 899 operator, 253 P ara-meter, 329 program, 394 canons classification of -, 380 cantus durus, 320 firmus, 619 mollis, 320 Capova, Sylvia, 894 cardinality, 817, 1059 abstract, 468 of a gestalt, 475 of a local composition, 107 carrier, 87, 1022 Cartesian product, 1058 cartesian closed, 1127 product, 1121 case linear -, 250 cyclic -, 250 Casella, Alfredo, 223 Castine, Peter, 255 catastrophe, 605, 606, 608, 610 theory, 277, 604 categorical approach, 967 categories equivalent -, 1119 category cocomma -, 1121 cocomplete -, 1122 comma -, 1121 complete -, 1122 finitely cocomplete -, 1122 complete -, 1122 isomorphic -, 1118
INDEX matrix -, 1117 of cellular hierarchies, 722 of commutative global composition, 347 of coverings of sets, 357 of denotators, 402–406 of elements, 158, 1122 of forms, 67 of functorial global compositions, 335 of local compositions, 105 of objective global compositions, 335 of performance cells, 713 of sheaves, 1130 of textual semioses, 409 opposite -, 1117 path -, 1079 product -, 1118 quotient -, 1117 skeleton -, 1117 Cauchy problem, 1162 sequence, 1154 Cauchy, Augustin, 1154, 1162 causal coherence, 929 depth, 821 relation, 985 causal-final variable, 927 causality, 925 CDC Cyber, 639 ˇ Cech cohomology, 431 Celibidache, Sergiu, 140 cell, 373 complex, 394 Deiters’ -, 1039 hair -, 1038 outer hair -, 1039 pillar -, 1039 performance -, 711 cellular hierarchies category of, 722 classification of -, 718 hierarchy, 716, 725
1275 product -, 718 restriction -, 718 type of a -, 716 organism, 394 Cent, 1031 center, 1075 central pitch detector, 1045 CERN, 239 chain, 1089 proof -, 1133 third -, 820 chamber pitch, 684, 1031 change of material, 982 of orientation, 619, 626, 646 of perspective, 393 program -, 947 value -, 769 CHANT, 291 chant Gregorian -, 620 character string, 71 characteristic, 1076 function, 407, 1062 map, 1126 characteristics method of -, 1161 charge semantic -, 490 chart, 307, 309 chart of level j, 329 Chim Chim Cheree, 219 choice axiom of -, 1060 Chomsky, Noam, 286 Chopin rubato, 667, 682, 698, 759, 924 bound -, 760 free -, 760 Chopin, Fr´ed´eric, 98, 760 chord, 109, 219–227, 304, 502 structural -, 503 a scale’s -, 112 addresses, 111 analysis, 533
1276 circle -, 514 class -, 111 classification, 194, 502 closure, 523 complement, 111 core -, 590 dictionary, 219 difference, 111 diminished seventh -, 563, 604, 608, 610 event, 110 foundation -, 535 fundamental -, 534 inspector, 820 intersection, 111 inversion, 509 isomorphism classes, 219 just class -, 111 n-, 109 pivotal -, 563 progression, 502 prolongational -, 503 self-addressed -, 225 sequence, 591 coherent -, 591 standard -, 531 symbol, 533, 535 tempered class -, 111 tesselating -, 377 union, 111 CHORD-CLASSIFIER, 256, 535 Chowning, John, 1022 chromatic (tempered) class chord, 111 Michel -, 582 octave just -, 114 Roederer -, 582 scale, 506 Vogel -, 582 chronospectrum, 287 circle chord, 514 of fifths, 513
INDEX of fourths, 513 circular colimit, 77 definition, 55, 176 denotator, 85–89 denotators folding -, 448 form, 76 limit, 77 set, 79 synonymy, 76 circularity conceptual -, 176 of forms, 56 CL, 1134 class, 188, 968 contour -, 252 resultant -, 381 chord, 111 just -, 111 tempered -, 111 contiguity -, 1150 counterpoint dichotomy -, 631 dichotomy -, 630 equivalence -, 1060 ground -, 381 marked counterpoint dichotomy -, 630 dichotomy -, 630 meta-object -, 982 nerve, 346, 376, 390 number, 219 precedence list, 971 segment -, 252, 255 set -, 253, 254 third comma -, 325 Vuza -, 380 weight, 230, 346 classical logic, 1134 classification, 997 epistemological -, 6 semiotics of sound -, 294
INDEX chord -, 502 geometric -, 216 in musicology, 192 local musical interpretation of -, 211 local theory of -, 191 of canons, 380 of cellular hierarchies, 718 of chords, 194 of motives, 228–231 of music-related activities, 4 of rhythms, 380 of sounds, 11 recursive -, 216 sound -, 284 technique, 205 theory, 999 classifier subobject -, 1126 CLOS, 256, 967, 968 closed cartesian -, 1127 locally -, 1147 path, 1063 point, 279 set, 1145 sieve, 1130 simplex, 1149 closure, 1145 hierarchy -, 715 objective -, 524 Clough, John, 248 cluster Cortot -, 894, 898 Horowitz -, 894, 898 Clynes, Manfred, 734, 738 CMAP, 255 coarser, 1145 coboundary map, 1150 cochain complex of a global composition, 374 singular -, 1150 cochlear Fourier analysis, 1051
1277 cocomma category, 1121 cocomplete category, 1122 cocone, 1119 coda, 304, 603 code, 259 codification of a symmetry, 154 codomain, 1059, 1115 coefficient absorption -, 1015 largest -, 890 system, 1150 coefficients signs of -, 887 cognitive dimension, 219 effort, 218 independence, 219 musicology, 23 psychology, 218, 276 science, 743 coherence, 503, 667, 772, 834 causal -, 929 final -, 929 harmonic -, 544 inter-period -, 929 coherent analysis, 772 chord sequence, 591 topology, 1146 Cohn, Richard, 384 cohomology, 374 ˇ Cech -, 431 group, 1151 l-adic -, 460 module of a global composition, 374 resolution -, 432 coinduced topology, 1146 Coleman, Ornette, 959 Coleman, Steve, 458 colimit, 308, 1121 circular -, 77 form, 67 topology, 1146 colinear, 281
1278 collaborative environment, 240 collaboratory, 35, 809 collective responsibility, 770 color coordinate, 1021 encoding, 923 parameter, 1004 sound -, 194 space, 1000 coloring, 947 Coltrane, John, 694, 733 COM matrix, 470 combination axis of -, 138, 260 linear -, 1084 weight -, 827 combinatorial activity, 242 artistry, 243 topology, 310 combinatoriality, 257 combinatorics creative -, 301 comes, 194, 243, 835 comma category, 1121 addressed -, 165 fifth -, 74 Pythagorean -, 74 syntonic -, 74, 115 third -, 74, 325 common language, 25 taste, 907 common-note function, 249 communication, 4, 5, 10, 12, 27 asymmetries of -, 910 coordinates, 16 process, 15 communicative dimension, 25 commutative, 1063 local composition module of a -, 125 diagram, 1118
INDEX global compositions, 347 local composition, 125 polynomials, 1077 commutativity relation, 1117 compact, 1147 comparative analysis, 333 discourse, 601 comparative criticism, 912 comparison matrix, 251 competence, 401 historical, 424 stylistic, 424 complement, 1058 abstract -, 254, 257 action, 630 literal -, 257 theorem, 254 complete category, 1122 harmony, 995 quiver, 1063 uniform space, 1154 completeness, 41, 48, 57 finite -, 166 completion semantic -, 57 complex cell -, 394 module -, 350 numbers, 1076 of affine functions, 351 quotient -, 351 set -, 382 simplicial -, 940, 1148 simplicial cochain -, 1150 complexity, 201 degree of -, 197 formal -, 465 measure, 311 of performance, 664 component aleatoric -, 242 alteration, 128
INDEX idempotent -, 1064 irreducible -, 330 metrical -, 327 reduced -, 128 composed frame, 968 composer, 13 perspective of the -, 301 composition, 198, 1059 t-fold tangent -, 669 global standard -, 357 commutative local -, 125 computer assisted -, 967 computer-aided -, 935, 955 concept, 694 dodecaphonic -, 149 functorial local -, 121 generic, 212 global -, 47, 169, 999 N -formed -, 354 oriented -, 355 resolution of a -, 393 global functorial -, 314 global objective -, 309 interpretable -, 370 local -, 47, 89, 105, 107 embedded -, 126 dimension of a -, 217 generating -, 126 projecting -, 216 standard -, 357 local objective -, 107 locally free local -, 213 modular -, 307 musical -, 33 non-interpretable -, 371, 376 tangent -, 669 tools fractal -, 137 compositional design, 255 idea, 391 space, 249
1279 compositions commutative global -, 347 local category of -, 105 computation symbolic -, 967 computational musicology, 23 computer assisted composition, 967 performance research, 764, 850 science, 7, 188 computer-aided anthropology, 925 composition, 935, 955 conativity, 259 concatenation, 159, 1059 principle, 160, 624 concept, 10, 39 architecture, 184 composition -, 694 construction history, 55 denotator -, 808 form -, 808 format, 48 framework, 3, 5, 9 abstraction -, 468 dynamic -, 399 fuzzy -, 200 grouping -, 305 human - construction, 55 leafing, 58, 60 of a book, 56 of instantaneous velocity, 30 of music, 23 paradigmatic -, 280 poietical -, 15 point -, 175 RUBATOr -, 807 score -, 307, 693, 909, 978 set -, 176 space, 23, 34, 36 stable -, 276 surgery, 99–102, 770 concepts
1280 standard of basic musicological -, 108 void pointer -, 35 conceptual circularity, 176 explicitness, 23 failure, 26 genealogy, 75 identification, 280 laboratory, 33 navigation, 39 precision, 35 profoundness, 109 universality, 109 zoom-in, 21 conceptualization dynamic -, 79 fuzzy -, 455 human -, 175 precise -, 258 process, 245 concert form, 957 master, 761 pitch, 699 concert for piano and orchestra, 307 condition initial -, 1156 instrumental -, 850 conductor, 668, 683, 761 cone, 1119 configuration counting series, 233 conjugation, 1067, 1076 conjugation class of endomorphisms, 220 of symmetry group, 220 conjunction, 421, 1131 connective predicate -, 1135 connotation, 19 connotator, 398, 1142 consonance, 564, 571, 619, 1049 deformed -, 646 imperfect -, 635, 646, 657 perfect -, 635, 646, 657 consonance-dissonance, 1035
INDEX dichotomy, 508, 632, 657 consonant, 286 interval, 503, 640 mode, 547 constant functor, 1119 module complex, 350 part, 525 shift, 129 structural -, 1105 constraint gestural -, 751 programming, 935, 967 constraints semiotic -, 284 construction activity, 197 of bigeneric morphemes, 541 recursive -, 49 construction history of concept, 55 contact, 259 point, 288 container, 973, 978 content, 17, 410, 497, 786, 817 interval -, 252 mathematical -, 17 maximal structure -, 1047 musical -, 17 context, 259, 895 problem, 819 real-time -, 917 contiguity, 18 class, 1150 contiguous simplicial maps, 1150 continuous, 1146 gesture, 986 method, 776 stemma, 803 weight, 775 contour, 193, 470 class, 252 space, 251 theory, 332
INDEX American -, 467 contra, 646 contraction, 1157 contrapunctus III, 835 contrapuntal form, 304 group, 137 interval oriented -, 619 meaning of Z-addressed motives, 120 motion shape type, 470 sequence, 646 symmetry, 647 local character of a -, 647 technique, 194 tension, 646 tradition, 243, 1052 contravariant functor, 1118 contravariant-covariant rule, 972 control group, 936 interactive -, 982 of transformation, 244 conversation topos of -, 995 coordinate barycentric -, 1149 basis -, 1028 color -, 1021 fifth -, 1032 function, 212 geometric -, 1021 octave -, 1032 ontological -, 10 pianola -, 1028 third -, 1032 coordinates, 52 of existence, 701 coordinator, 50, 1139 form -, 64 of a form, 65 coproduct, 1062 ambient space, 124 of local compositions, 124
1281 type, 53 core chord, 590 correlate electrophysiological -, 637 cortex auditory -, 639, 641 Corti organ of -, 1038 Cortot cluster, 894, 898 Cortot, Alfred, 891, 894, 897 coset left -, 1067 right -, 1067 cosmology, 565 counterpoint, 161, 508, 618, 637, 995 dichotomy, 630 class, 631 double -, 624 theorem, 649, 653 theory, 936, 1008 countersubject, 836 counting series configuration -, 233 coupling monogamic -, 769 polygamic -, 769 covariant functor, 1118 covering, 308 equivalent -, 309 family, 1129 motif, 467 sieve, 1129 cp, 251 CPL, 971 cpset, 251 creative combinatorics, 301 extension, 245 creativity, 242, 399 alphabet of -, 242 creator, 12, 13 crescendo, 79, 668, 722, 738, 1029 wedge, 778 critical
1282 distance, 1015, 1016 fiber, 911, 915 criticism comparative -, 912 journalistic -, 885 music -, 772 critique, 911 music -, 905 cross-correlation stemmatic -, 771 cross-semantical relation, 745 cube topographic -, 19, 36 cul-de-sac, 657 interval, 653 culture of performance, 757 curve, 282, 1156 canonical -, 872, 890, 899 integral -, 1158 intonation -, 684 tempo -, 682, 738, 758, 877, 947 curvilinear reduction, 937 CX5M Yamaha -, 639 cycle, 1063, 1071, 1159 index, 233, 1071 of variations, 956 pitch -, 252 cyclic case, 250 extension, 253 group, 1069 interval succession, 253, 254 Czerny, Carl, 758, 924 D d’Alembert, Jean Le Rond, 5, 40, 58 da capo, 140 dactylus, 260 grid, 265, 266 Dahlhaus, Carl, 26, 147, 300, 323, 324, 544, 574, 594, 819, 994, 1053 dance, 735 Dannenberg, Roger, 918
INDEX data ethnomusicological -, 99 dataglove, 738 daughter, 752 tempo, 682 daughters, 725 Davin, Patrick, 986 DBMS, 808 de la Motte, Helga, 25, 694 Debussy, Claude, 223, 600, 756 decay, 1018 decomposition hierarchical -, 858, 872 natural -, 855 orthonormal -, 11 spectral -, 856 Sylow -, 542, 620 Deep Purple, 231 default weight function, 587 definition circular, 55, 176 of music, 6 of musical concepts, 114 deformation, 276, 720 basis -, 797 degree of -, 951 hierarchy -, 799 non-linear, 827 non-linear -, 776, 889 of a tempo curve, 699 pianola -, 720, 797 deformed consonance, 646 dichotomy, 646 dissonance, 646 degree, 304, 321, 535, 537, 566 different -, 323 modulation -, 566 of complexity, 197 of deformation, 951 of freedom, 219 of organization, 869 of symmetry, 254 parallel -, 324 system
INDEX irreducible -, 556 theory, 531 Deiters’ cell, 1039 Delalande, Francois, 738, 740 delay, 1002 relative -, 288 Deligne, Pierre, 427 delta Kronecker -, 1085 Dennett, Daniel, 181 denotator, 47, 67–69 attributes, 48 circular -, 85–89 concept, 808 genealogy, 47 flow chart, 49 image, 69 language, 723 morphism, 108 name, 52 non-zero-addressed -, 82 ontology, 398 orchestra instrumentation -, 82 philosophy, 185 reference -, 403 regular -, 79–85 self-addressed -, 82 truth -, 407 Z-addressed -, 62 denotators circular - folding, 448 linear ordering among -, 58 ordering on -, 89–99 ordering principle on -, 57 relations among -, 105 Denotex, 811, 1143 DenotexRUBETTEr , 811 dense, 280 densification, 986 depth, 23, 25, 240 causal -, 821 EEG, 637 stereotactic -, 638 electrode, 639 final -, 821
1283 in mathematics, 25 in musicology, 26 in the humanities, 591 semantic -, 465 derivation, 1105, 1160 inner -, 1105 outer -, 1105 derivative, 1155 Lie -, 1160 derived serial motif, 237 Desain, Peter, 664 Descartes, Ren´e, 12, 178, 1049 description object -, 244 verbal -, 756 design compositional -, 255 matrix, 877 Desmond, Paul, 218 development, 304, 603 history, 745 software -, 723 syn- and diachronic of music, 242 Dezibel, 1029 di-alteration, 129 Diabelli Variations, 394 diachronic, 17 axis, 399, 575 index, 273 normalization, 909 diaffine homomorphism, 1090 diagonal embedding, 1098 field, 686 diagram, 1118 scheme, 1117 commutative -, 1118 filtered -, 1107 Hasse -, 267, 1061 of forms, 67 dialog, 996 bidirectional, 34 experimental navigation -, 35 dialogical principle, 997 diameter, 633
1284 diastematic, 816 index shape type, 470 shape type, 470 diatonic scale, 658 dichotomy class, 630 marked -, 630 consonance-dissonance -, 508, 632, 657 counterpoint -, 630 deformed -, 646 interval -, 630 major -, 631, 657 marked counterpoint -, 630 marked interval -, 630 Riemann -, 636 Saussurean -, 17 dictionary of expressive rules, 747 Diderot, Denis, 5, 40, 58 difference, 1058 genealogical -, 912 phenomenological -, 912 different degree, 323 differentiable, 1155 differential, 1160 equation, 1156 semantic -, 198 differentiation rules, 742 digital age, 40 diinjective, 1101 dilatation, 160, 1097 time -, 83 dilinear homomorphism, 1084 part, 1090 dimension, 1085, 1148 cognitive -, 219 communicative -, 25 of a local composition, 217 of a simplex, 1148 ontological -, 19 diminished, 540 diminished seventh chord, 563, 604, 608, 610
INDEX Ding an sich, 23 direct image, 1107 sum module, 1084 directed graph, 1063 direction of alteration, 951 directional endomorphism, 797 Director Musices, 742 discantus, 619 disciplinarity dynamic -, 809 discipline basic -, 6 discourse analytical -, 12 comparative -, 601 esthesic -, 15 discoursivity, 41, 48, 57 discrete, 774 interpretation, 311 field, 917 gesture, 986 nerve, 311 topology, 1145 disjoint, 1058 sum, 1121 disjunction, 421, 1131 dissonance, 564, 571, 619, 1049 deformed -, 646 emancipation of -, 33 dissonant interval, 640 mode, 547 distance, 276, 279 critical -, 1015, 1016 Euclidean - for diastematic types, 472 Euclidean - for rigid types, 472 for toroidal types, 473 function, 472 natural -, 441 on toroidal sequences, 473 relative Euclidean - for rigid types, 472 third -, 622 to an initial set, 704
INDEX Distributed RUBATOr , 922 distributed laboratory, 35 distributive, 1132 distributor, 835 divertimenti, 994 division of pitch distances, 72 of time regular -, 456 divisor resulting -, 382 documentation, 4, 5, 7 dodecaphonic composition, 149 composition principle, 137 method, 936, 940 paradigm, 150 series, 149, 197, 236, 301, 309, 394 vocabulary, 243 dodecaphonism, 162, 251 communicative problem of -, 162 esthetic principles of -, 162 domain, 1059, 1115 fundamental scientific -, 6 modulation -, 580 dominance, 267, 329 topology, 283, 488 dominant, 323, 502, 541, 545 role of major scale, 657 seventh, 508 dominate, 278, 1110 double articulation, 19 counterpoint, 624 drama musical -, 908 Dreiding, Andr´e, 355 Dress, Andreas, 355 driving grid, 951 drum ear -, 1037 dual affine -, 1091
1285 ambient space, 128 linear -, 1091 numbers, 127, 618, 1077 dualism between major and minor, 147 Dufourt, Hugues, 967 duration, 51, 79 period, 115 dux, 194, 243 DX7, 1022 dynamic concept framework, 399 conceptualization, 79 disciplinarity, 809 navigation, 45 dynamical initialization, 701 knowledge management, 399 modularity -, 809 sign absolute -, 777 relative local -, 778 relative punctual -, 777 dynamically loadable module, 808 dynamics, 303, 304, 682, 685 absolute -, 831 historical -, 271, 273 mechanical -, 739 of performance, 800 primavista -, 764 relative -, 831 symbolic absolute, 81 relative, 81 E -ball, 280 -neighborhood, 482 -paradigm, 280 ear drum, 1037 inner -, 1037
1286 middle -, 1037 outer -, 1036 ecclesiastical mode, 319, 655, 657 editing geometric -, 946 editor, 968 EEG depth -, 637 response, 637 semantic charge of -, 638, 640 test, 638 effect groove -, 952 effective, 1066 Eggebrecht, Hans Heinrich, 23, 24, 26 Ego poetic -, 262, 268 Ehrenfels transpositional invariance criterion, 108 Ehrenfels, Christian von, 108, 203, 276, 301, 332, 334, 465 Eimert, Herbert, 152, 258 Eitz, Carl, 1032 elastic, 816 shape type, 471 electrode depth -, 639 electrophysiological correlate, 637 element, 1057 neutral -, 1063 elementary alteration, 129 gesture, 986 neighborhood, 489 shift, 129 elements category of -, 158, 1122 emancipation of dissonance, 33 embedded local composition, 126 embedding, 1118 diagonal -, 1098 number, 254 Yoneda -, 1091, 1120 emotion, 642, 734–737 emotional
INDEX brain, 641 function of music, 642 landscape, 295 emotivity, 259 empty form name, 55 set, 176 string, 52 encapsulated history, 675 encapsulation, 26, 188, 973 speculative -, 30 encoding color -, 923 formula rubato -, 751 Encore, 986 Encyclop´edie, 5, 41, 43, 58 encyclopedia, 40, 440 encyclopedic ordering, 58 encyclopedism, 56 encyclospace, 41, 43, 58 endo, 1116 endolymph, 1038 endomorphism, 1116 directional -, 797 enharmonic -, 516 right-absorbing -, 524 ring, 1083 energy, 739 spectrum, 1020 enharmonic, 515 endomorphism, 516 group, 517 identification, 515 ensemble rules, 742 Ensemble Intercontemporain, 986 enumeration musical - theory, 232 of motives, 238 theory global -, 376 envelope, 84, 1018 environment collaborative -, 240
INDEX experimental -, 827 epi, 1116 epilepsy therapy surgical -, 638 epileptiform potential, 638 epimorphism, 1116 epistemology of musicology, 29 epsilon gestalt topology, 484 topology, 483 Epstein, David, 739 equation differential -, 1156 spring -, 1020 equivalence phonological -, 263 class, 1060 paradigmatic transformation -, 259 perceptual -, 280 relation, 305, 1060 syntagmatic -, 263 equivalent atlases, 314 categories, 1119 covering, 309 norms, 1155 equivariant, 1067 Erwartung, 223 Escher, Cornelis Maurits, 196 EspressoRUBETTEr , 916, 922 essential parameter, 999 esthesic, 12, 1021 identification, 303 esthesis, 12, 15, 258 esthetic, 259 esthetics, 15, 259 of music, 393 ethnological form, 57 ethnology inverse -, 909 ethnomusicological data, 99 ethnomusicology, 909 Euclid, 178, 617 Euclidean
1287 algorithm, 1076, 1080 geometry, 353 metric, 279 Euler function, 1069 module, 73, 218 plane, 110 point, 73, 1031 space, 1031 Euler’s identity, 1020 Euler, Leonhard, 73, 581, 619, 1032, 1049, 1165 European score notation, 79 Eustachian tube, 1037 evaluation, 359, 1132 event percussion -, 612 time -, 674 evolution, 763 exact sequence split -, 1069 exchange of pitch and onset, 152 parameter -, 160, 161 existence, 67, 397 mathematical -, 175, 398, 413 musical -, 413 experiment mental -, 666 musicological -, 33, 34 physical -, 32 experimental environment, 827 humanities, 29 material, 401 natural sciences, 29 strategy, 841, 851 experimentation, 982 experiments of the mind, 34 explanatory variable, 877 explicitness conceptual -, 23 exponentiable, 1127 exposition, 304, 603, 959 expression, 406, 733, 916
1288 human -, 692 instrumental -, 994 rhetorical -, 692 expressive rules dictionary of -, 747 expressivity pure -, 737 rhetorical -, 674 extension, 373, 401, 410, 670 allomorphic -, 539 basic -, 518 creative -, 245 cyclic -, 253 strict -, 539 topology, 521 exterior score, 694 extraterritorial part, 720 extroversive semiosis, 400 F f -morphism, 1107 F-to-enter level, 881 face, 1148 facticity, 397, 420, 565 finite - support, 411 full -, 410 factor pressure decrease -, 1016 strength -, 742 factory box, 972 faithful action, 1066 address, 523 functor, 1118 point, 523 False, 1132 family, 293 covering -, 1129 minimal cadential -, 554 of violins, 997 violin -, 295, 1009 fantasy artistic -, 692 faster uphill, 742
INDEX father, 752 Feldman, Jacob, 739 Feldman, Morton, 306 Fermat, Pierre de, 26 fermata, 668, 766, 769, 781 Ferretti, Roberto, viii feuilleton, 772 feuilletonism, 905 FFT, 638 fiber, 743, 1062 critical -, 911, 915 group, 936 product, 1078, 1121 of local compositions, 167 structure, 913 sum, 1121 of local compositions, 169 Fibonacci sequence, 413 Fibonacci, Leonardo, 70, 413 fibration linear -, 914 fiction, 397, 565 fictitious performance history, 763 field, 726 arpeggio -, 698 calculation, 918 diagonal -, 686 discrete -, 917 finite -, 949 fundamental -, 720 interpolation, 918, 922 intonation -, 684 of equivalence, 191 of fractions, 1101 operator, 792 paradigmatic -, 150 parallel articulation -, 689 parallel crescendo -, 689 parallel glissando -, 689 performance -, 685, 690, 712 prime -, 1076 selection, 969 skew -, 1075
INDEX tempo -, 683 tempo-intonation -, 686 vector -, 1156 writing, 969 fifth, 73, 1031 axis, 113 coordinate, 1032 sequence, 321 Fifth symphony, 303 film music, 733 filtered diagram, 1107 filtering input -, 918 final coherence, 929 depth, 821 retard, 738 vertex, 802 finale, 956 finalis, 319 finality, 925 fine arts, 14, 186 finer, 1145 fingering, 303, 738, 757 finite, 1057 completeness, 166 cover topology, 430 field, 949 locally -, 1149 monoid, 1063 multigraph, 1062 finitely cocomplete category, 1122 complete category, 1122 generated, 1069, 1084 finitely generated abelian group, 373 Finscher, Ludwig, 993, 994 Finsler’s principle, 175 Finsler, Paul, vi, 175 first representative, 220 FIS, 249 Fitting’s lemma, 1089
1289 Fitting, Hans, 1089 fixpoint, 1157 group, 1066 flasque module complex, 370 flat, 130 flatten, 88 flattening operation, 88, 331 Fleischer, Anja, viii, 590 FLOAT, 50 flow box, 1160 interpolation, 706 flying carpet, 927 FM, 289, 1022 -object generalized, 292 analysis, 289 synthesis, 86 folding, 442 circular denotators, 448 colimit denotators, 446 limit denotators, 446 foramen ovale recording, 638 force field alteration -, 952 modulation -, 567, 571 forces in physics, 649 foreground, 503 form, 50, 61–67 bilinear -, 354 circular -, 76 circularity, 56 colimit -, 67 concept, 808 concert -, 957 contrapuntal -, 304 coordinator, 64, 65 ethnological -, 57 Forte’s prime -, 256 functor, 64 identifier, 64, 65 limit -, 67 list -, 976 morphisms
1290 wrap -, 402 musical -, 6 name, 50, 51 empty -, 55 names, see Symbols normal -, 255 of a symmetry, 135 pointer character, 55 powerset -, 66 prime -, 257 Rahn’s normal -, 255 regular -, 76 semiotic global -, 1141 simple -, 66 simplify to a -, 75 sonata -, 304, 603, 956 space, 64 Straus’ zero normal -, 256 synonym -, 66 type, 64 typology, 65 form semiotics morphism of -, 1141 formal complexity, 465 structure, 967 formalism Lie -, 800 formant, 291 manifold, 291 open - set, 291 forms category of -, 67 diagram of -, 67 ordering on -, 89–99 formula, 1135 atomic -, 1135 cadential -, 551 propositional -, 1136 quantifier -, 1136 Forte’s prime form, 256 Forte, Allen, 247–249, 255, 383 foundation chord, 535 four
INDEX part texture, 995 Fourier analysis cochlear -, 1051 decomposition, 84 ideology, 286 paradigm, 284 representation, 899, 1000 theorem, 10 transform, 1025 Fourier’s theorem, 1019 Fourier, Jean-Baptiste, 512 fractal, 70, 196, 198, 943 composition tools, 137 principle, 964 fractions field of -, 1101 frame, 712, 726, 968, 1026, 1139 composed -, 968 simple -, 968 space, 64 structure, 718 wavelet -, 290 framework, 973 application, 808 concept -, 3, 5, 9 hermeneutical -, 12 free action, 1066 Chopin rubato, 760 commutative monoid, 1064 group, 1067 jazz, 665 locally -, 1110 module, 1085 monoid, 1064 variable -, 1136 free jazz, 14 freedom of choice, 658 frequency, 72, 84, 1018, 1021 band, 640 beat -, 1051
INDEX fundamental -, 1019 modulation, 289, 1022 modulation -, 288, 1003 of variable inclusion, 888 Freud, Sigmund, 643 Friberg, Anders, 742 Fripertinger, Harald, viii, 203, 231, 257, 376, 378, 1071 Frost, Robert, 333 Fryd´en, Lars, 741, 755 fugue, 194, 243 full address, 523 functor, 1118 model, 880 point, 523 subcategory, 1118 subcomplex, 1148 fully faithful address, 523 functor, 1118 point, 523 function, 1059 A-addressed -, 351 autocomplementarity -, 508, 632 Bessel -, 1024 characteristic -, 407, 1062 common-note -, 249 Euler -, 1069 generic -, 971 gradus suavitatis -, 1049, 1165 horizontal poetical -, 942 index -, 1081 interval -, 249 inverse -, 1059 level -, 329, 1061 of a symmetry, 136 poetical -, 18, 138, 259, 295, 303, 934, 942 theory, 324, 531 tonal -, 304, 323, 544 value tonal -, 544 vertical poetical -, 942 function harmony, 35
1291 functional, 1059 programming, 967 semantics, 541 functor, 1117 address -, 170 constant -, 1119 contravariant -, 1118 covariant -, 1118 faithful -, 1118 form -, 64 full -, 1118 fully faithful -, 1118 global section -, 1109 module -, 172 nerve -, 1148 of orbits, 1114 open -, 1113 open covering of -, 1113 representable -, 1120 resolution -, 358 support -, 314 functorial global composition, 314 local composition, 121 fundamental activities, 4, 7 chord, 534 field, 720 note, 535 period, 1019 pitch, 532 scientific domain, 6 series, 137 space, 715 fushi, 14, 416 Fux rule, 657 Fux, Johann Joseph, 636, 656, 1008, 1053 fuzziness, 531 fuzzy concept, 200 conceptualization, 455 logic, 409 set, 198 theory, 194
1292 G G-prime form, 254 G¨ otterd¨ ammerung, 814 Gabriel, Peter, 185 Gabrielsson, Alf, 734, 738 Galilei, Galileo, 29, 30, 32, 35, 664 Galois, Evariste, vi Garbers, J¨org, viii, 807, 1143 Garbusow, Nikolai, 221 gate function hippocampal -, 642 Gegenklang, 324, 556 Gell-Mann, Murray, 176 genealogical difference, 912 genealogy conceptual, 75 of denotator concept, 47 poietic, 154 general linear algebra, 1105 pause, 782 position, 391 General Midi, 287 general position, 126, 212 musical meaning of -, 127 generated finitely -, 1069, 1084 generating local composition, 126 Generative Theory of Tonal Music (=GTTM), 312, 457 generator sound -, 849 time -, 936 generic composition, 212 function, 971 linear visualization, 440 point, 279, 330, 1110 score, 665 genotype, 943 geodesic, 295 geographic information system, 809
INDEX orientation, 43 geometric classification, 216 coordinate, 1021 editing, 946 parameter, 1000 realization, 1149 representation, 946 geometry analytical -, 178 algebraic -, 178, 201, 668 Euclidean -, 353 germ, 326, 1155 rhythmic -, 152, 326 germinal melody, 269, 270, 956 gestalt, 106, 203, 332, 465, 492 abstract -, 474 auditory -, 481 cardinality of a -, 475 global -, 307 musical -, 106, 152 paradigm, 816 psychology, 106 small -, 483 specialization, 488 category, 490 stability, 276 gestural constraint, 751 rationale, 908 semantics, 908 gesture, 735, 738–741 continuous -, 986 discrete -, 986 elementary -, 986 instrumental -, 986 orchestral -, 986 Get-Editor, 983 Get-View, 983 Giannitrapani, Duilio, 640 Gianoli, Reine, 891 Gigue Nr. 32, 231 Gilels, Emil, 756 Gilson, Etienne, 13 GIS, 249, 384
INDEX structure, 248 Glarean, 320 glide reflection, 1097 glissando, 79, 668, 689, 722, 986, 1032 global, 299 affine functions module of -, 432 agogics, 764 AST, 382–385 composition, 169, 999 cochain complex of a -, 374 enumeration theory, 376 form semiotic, 1141 functorial composition, 314 morphism, 335 gestalt, 307 molecule, 355 molecules morphism of -, 355 morphisms, 300 object, 299 objective composition, 309 objective composition morphism, 335 performance score, 728 predicate, 552 score, 307, 946 section, 350, 1121 functor, 1109 slope, 822 solution, 1159 standard composition, 357 tangent composition, 675 technical parameter, 1005 tension, 822 theory, 269 threshold, 819 globalization
1293 metrical -, 116 orchestral -, 673 G¨oller, Stefan, viii, 441, 1143 Goethe, Johann Wolfgang von, 147, 198, 394, 996 Goldbach conjecture, 32 Goldbach, Christian, 32 Goldberg Variations, 394 golden section, 70 Goldstein, Julius, 1045 Gottschewski, Hermann, 31 Gould, Glenn, 667, 740, 841, 851, 907 GPL, vii GPS, 728 gradus suavitatis function, 1049, 1165 Graeser, Wolfgang, 135, 137, 248, 304 Gram identity, 355 grammar locally linear -, 802 performance -, 747 rule-based -, 748 grand unification, 564 granddaughter, 802 grandmother, 802 graph, 1058, 1062 directed -, 1063 of a FM-denotator, 87 Riemann -, 821 weighted -, 292 graphical interface design, 439 MOP, 982 Grassmann scheme, 1113 greeking, 441, 463 Gregorian chant, 620 Greimas, Algirdas Julien, 934, 936, 937 grid dactylus -, 265, 266 driving -, 951 vector
1294 horizontal -, 951 vertical -, 951 groove effect, 952 Grothendieck topology, 180, 430, 1129 topos, 1130 Grothendieck, Alexander, vi, ix, 175, 180, 185, 427, 430, 436, 1129 ground class, 381 group, 1066 affine counterpoint -, 475 automorphism -, 174, 1083 cohomology -, 1151 contrapuntal -, 137 control -, 936 cyclic -, 1069 enharmonic -, 517 fiber -, 936 fixpoint -, 1066 free -, 1067 homomorphism, 1066 isomorphism, 1066 isotropy -, 1066 Klein -, 251, 548 linear counterpoint -, 475 opposite -, 1066 p-Sylow -, 1069 paradigmatic of isometries, 478 paradigmatic -, 474 product -, 1068 quotient -, 1067 rhythmical -, 977 simple -, 1067 Sylow -, 218 symmetric -, 1066 symmetry -, 174, 220, 571, 816 theory, 191, 259 torsion -, 1070 group-theoretical method, 241, 250 grouping, 456, 503, 739, 764 bar -, 864 concept, 305 hierarchical -, 743
INDEX instrumental -, 728 metrical -, 303 of sounds, 88 rules, 742 stemmatic -, 770 structure (=G), 457 time -, 118 GTTM, 312, 457, 752 Guarneri del Ges` u, 1000 gyri Heschl’s -, 1044 H H¨olderlin, Friedrich, 9, 138 H¨ ullakkord, 523 Haegi, Hans, 355 hair cell, 1038 Haj´os group, 377, 382 Haj´os, Gy¨orgy, 377 Halle, Morris, 286 Halsey, George, 376, 381 Hamilton, William, 1076 Hamiltonian, 800 hammer, 1037 Hammerklavier-Sonate, 118, 245, 327, 337, 559, 560, 563, 567, 594, 603, 667, 693, 907, 941 hanging orientation, 128, 619 Hanslick, Eduard, 17, 303, 307, 935, 997 harmolodic, 959 harmonic coherence, 544 knowledge, 591 logic, 546 minor, 584 morpheme, 546 motion, 503 path, 586 progression, 154 semantics, 531 strip, 310, 321, 538 tension, 586, 587 topology, 538 weight, 587, 786
INDEX harmonic minor scale, 575, 577 tonality, 560 harmonical-rhythmical scale, 959 harmony, 106, 221, 637 complete -, 995 jazz -, 337 Keplerian -, 33 Riemann -, 322 HarmoRUBETTEr , 546, 586, 787, 819, 866 Harnoncourt, Nicolas, 909 Harris, Craig, 255 Haschemann, 947 Hashimoto, Shuji, 738, 745 Hasse diagram, 267, 1061 specialization -, 269 hat Mexican -, 1025 Hauptmann, Moritz, 531 Hausdorff topology, 1147 Hausdorff, Felix, 1147 hayashi, 14 Haydn, Joseph, 295, 994, 995 Hazlitt, William, 831 heartbeat, 738 Hebb, Donald O, 871 Hegel, Georg Wilhelm Friedrich, 39, 942 Heijink, Hank, 918 helicotrema, 1038 Helmholtz, Hermann von, 619, 1049, 1051 Hemmert, Werner, viii Hempfling, Thomas, ix Hentoff, Nat, 733 hermeneutics unicorn of -, 14 Hertz, 1018 Herv´e, Jean-Luc, 986 Heschl’s gyri, 639, 1044 Hess system, 638 Hesse, Hermann, 200, 695 Hewitt, Edwin, 376, 381 hexameter, 261 Heyting algebra, 123, 407, 1132 logic, 530
1295 Heyting, Arend, 407, 530, 1132 Hichert, Jens, 629, 649, 651, 653, 657 hidden symmetry, 136 hierarchical decomposition, 858, 872 grouping, 743 organism, 304 smoothing, 857 hierarchy, 306, 674, 767 cellular -, 716, 725 closure, 715 deformation, 799 metrical -, 455 of performance development, 757 parallel -, 718 performance -, 674 piano -, 722 space -, 715 standard -, 717 tempo -, 758 violin -, 722 Himmelfahrtsoratorium, 595 Hindemith, Paul, 147, 503, 512 Hintergrund, 503 hippocampal gate function, 642 memory function, 642 hippocampus, 639, 642 histogram, 865 historical dimension of music, 108 approach, 565, 574 dynamics, 271, 273 instrumentation, 393 localization, 273 musicology, 399 process, 763, 771 rationale, 994 reality, 594 historicity in music, 271 history, 5 development -, 745 encapsulated -, 675 of music, 6
1296 hit point, 706 problem, 704 Hjelmslev, Louis, 16, 19, 398 Hofmann, Ernst Theodor Amadeus, 303 homeomorphism, 1146 homomorphism diaffine -, 1090 dilinear -, 1084 group -, 1066 Lie algebra -, 1104 linear module -, 1083 monoid -, 1063 ring -, 1075 structural -, 1075 homotopy, 1150 relative -, 1150 Honing, Henkian, 664 Horace, 755 horizontal grid vector, 951 poetical function, 942 poeticity, 261 Horowitz cluster, 894, 898 Horowitz, Vladimir, 884, 891, 895, 897, 927 human conceptualization, 175 expression, 692 precision, 757 humanism, 997 humanities, 200, 275 experience in the -, 34 experimental -, 29 Husmann, Heinrich, 1052 hypermedia, 43 hyperouranios topos, 23, 42 I I, 24 Ib´erie, 223 ICMC, 240, 744 icon
INDEX instrumental -, 947 idea compositional -, 391 musical -, 934, 935 ideal, 1076 left -, 1076 right -, 1076 idempotent, 1064 component, 1064 identification conceptual -, 280 enharmonic -, 515 esthesic -, 303 identifier, 1139 form -, 64, 65 identity, 492, 496, 1115 of a point, 178 abstract, 16 Euler’s -, 1020 Jacobi -, 1104 of a work, 16 slice, 336 ideology Fourier -, 286 IL, 1134 image, 1062 denotator -, 69 direct -, 1107 inverse -, 1062 imagination, 5 imitation, 492, 494 immanent analysis, 465 imperfect consonance, 635, 646, 657 implementation, 763 implication, 421, 1131, 1132 importance relative -, 786 Impromptu, 760 improvisation, 45 jazz -, 218 in absentia, 18, 953 in praesentia, 18, 952 in-time music, 986 inbuilt performance
INDEX grammar, 907 included literally -, 255 inclusion abstract -, 254, 257 literal -, 257 incomplete semiosis, 401 incorrect politically -, 907 indecomposable, 1089 space, 715 independence cognitive -, 219 index cycle -, 1071 diachronic -, 273 function, 1081 set, 1060 indiscrete interpretation, 312 topology, 1145 individual variable, 1135 ineffability, 25, 693 infinite, 1057 interpretation, 317 message, 905 performance, 666 infinitely small, 692 infinitesimal, 774 information, 39 paratextual -, 831 system geographic -, 809 InfoRUBETTEr , 810 inharmonic, 819 inharmonicity, 290 inheritance, 763, 968 principle biological -, 763 property, 479 initial, 1121 articulation, 702 condition, 1156 design matrix, 878
1297 moment, 697 performance, 712, 726 set, 696, 712, 725 polyhedral -, 704 value, 683 initial set distance to an -, 704 initialization dynamical -, 701 injective, 1059 module, 1103 inlet, 972 inner derivation, 1105 ear, 1037 logic, 757 score, 665, 694 input filtering, 918 real-time -, 946 inspector chord -, 820 instance, 968 instantiation, 968 instinctive activity, 757 instrument name, 82 space, 726 instrumental condition, 850 expression, 994 gesture, 986 grouping, 728 icon, 947 parameter, 1002, 1004 technique, 1002 variety, 673 vector, 1005 voice, 269 instrumentation historical -, 393 orchestra - denotator, 82 instrumentum, 42 Int´egrales, 394 INTEGER, 50
1298 integer, 1076 integral curve, 1158 surface, 1161 integrated serial motif, 237 integration method, 829 intensification, 964 intension, 401, 519 basic -, 518 topology, 520 intensity, 739 inter-period coherence, 929 interaction interpretative -, 873 matrix, 857 interactive control, 982 interface design graphical -, 439 interictal period, 638 interior, 1146 interlude, 836 intermediate performance, 756 internal structure, 311 interpolation, 692, 917 field -, 918, 922 flow -, 706 interpretable composition, 370 automorphism group of -, 372 molecule, 356, 385 interpretation, 269, 272, 665 discrete -, 311 indiscrete -, 312 infinite -, 317 iterated -, 317 just triadic degree -, 325 metrical, 328 motivic -, 332, 467 of a local composition, 316 of weights, 800 rhythmical -, 328 semantic -, 598 silly -, 312
INDEX singleton -, 336 sketchy -, 757 tangent -, 677 tetradic -, 337 third chain -, 319 triadic -, 337, 548, 553, 566 triadic degree -, 320 interpretative activity, 300, 307, 308 interaction, 873 interspace, 692 sequence, 234 structure, 234 interval unordered p-space -, 252 unordered pc -, 253 class content vector, 253 consonant -, 503, 640 content, 252 contrapuntal oriented -, 619 cul-de-sac -, 653 cyclic - succession, 253 dichotomy, 630 dissonant -, 640 function, 249 multiplication, 623 ordered p-space -, 252 ordered pc -, 253 succession, 252 cyclic -, 254 mth -, 254 successive -, 640 time -, 83 vector, 253, 257 interval-class vector, 249 intonation, 682, 683 curve, 684 field, 684 intratextual, 400 introversive semiosis, 400 intuition musical -, 246 intuitionistic logic, 539, 1134
INDEX invariance transformational -, 276, 332 vector, 254 invariant pcset, 254 inversa, 839 inverse, 1066 ethnology, 909 function, 1059 image, 1062 left -, 1066 performance theory, 743, 790, 913 right -, 1066 inversion, 140, 302, 321 chord -, 509 real, 148 retrograde -, 73, 144 tonal -, 148, 952 inverted weight, 827 invertible, 1075 IRCAM, 291, 967 irreducible, 716 component, 330 degree system, 556 topological space, 1110 iso, 1116 isometry, 305, 1153 isomorphic, 1066 category, 1118 isomorphism, 1116 group -, 1066 monoid -, 1063 ring -, 1075 isomorphism classes of local rhythms, 221 of chords, 219 isotropy group, 1066 isotypic tesselation, 376 ISPW, 918 istesso tempo, 673 iterated interpretation, 317 J Jackendoff, Ray, 305, 311, 457, 461, 752, 873
1299 Jacobi identity, 1104 Jacobi, Carl, 1104 Jacobian, 1155 Jacobson, Nathan, 1088 Jakobson, Roman, 18, 138, 191, 259, 272, 286, 295, 305, 400, 934, 942 Jandl, Ernst, 963 Jauss, Hans Robert, 187 Java, 808 Java2D, 922 jazz, 13, 14, 45, 218, 694 American -, 538 CD review, 415 free -, 14, 665 harmony, 337 improvisation, 218 lead-sheet notation, 533 JCK, 416 jnd, 280 Johnson, Tom, 594, 953 join, 1132 journalistic criticism, 885 J´ozef Marja Hoene-Wro´ nski, 387 Julia set, 198 Julia, Gaston, 198 Jupiter Symphony, 458 just, 111 chromatic octave, 114 class chord, 111 modulation, 577 scale, 113 triadic degree interpretation, 325 tuning, 1032 just-tempered tuning, 1033 justest scale, 325 tuning, 560 juxtaposition, 72 K k-partition, 378 K¨ohler, Egmont, 230
1300 Kagel, Maurizio, 152, 394 kairos, 994 Kaiser, Joachim, 303, 603, 907 kansei, 738, 745 Kant, Immanuel, v, 10, 23, 32, 43, 175 Karajan, Herbert von, vi, 700, 740, 945 Karg-Elert, Sigfrid, 137, 504 Kepler, Johannes, 136 Keplerian harmony, 33 kernel, 726, 1067 Naradaya–Watson -, 857 smoothing, 856, 874 smoothing -, 857 symbolic -, 712 view, 826 key, 948 function of music, 643 musicogenic, 644 signature, 768 killing address -, 204 Kinderszenen, 495 kindred, 293 Kircher, Athanasius, 242 Klangrede, 19, 996 Klavierst¨ uck III, 385 Klein group, 251, 475, 548 knot in FM synthesis, 88 knowledge, 39, 440 crash, 412 harmonic -, 591 hiding, 240, 440 human -, 5 management dynamical -, 399 ontology, 420 private -, 29 space, 10, 29 Koenig, Thomas, 833 Kollmann, August, 995 Kontra-Punkte, 152 Kopiez, Reinhard, 291, 734, 736 KORG, 1027 Kronecker delta, 1085
INDEX Kronecker, Leopold, 1085 Kronmann, Ulf, 738, 739 Krull, Wolfgang, 1090 KTH school, 750, 755 Kubalek, Antonin, 894 Kunst der Fuge, 137, 248, 304, 740, 835, 849 Kuriose Geschichte, 764, 765, 771, 849, 860 Kurzweil, 850 KV 449, 231 L l, 1029 L’essence du bleu, 941 L’isle joyeuse, 223 λ-abstraction, 970 l-adic cohomology, 460 λ&-calculus, 968 λ-function, 969 L-system, 943 L¨ udi, Werner, 664, 955 L´evi-Strauss, Claude, 593 La mort des artistes, 268, 963 laboratory conceptual -, 33 distributed -, 35 Lagrangian, 800 landscape emotional -, 295 Langer, Susan, 734 Langner, J¨org, 736 language, 19 common -, 25 denotator -, 723 langue, 19 large orchestra performance of a -, 761 largest coefficient, 890 lattice, 1132 law Weber-Fechner -, 1029 Lawrence, David Herbert, 905 Lawvere, William, 180, 435 layer RUBATOr -, 810 layers of reality, 10
INDEX lazy path, 1063 LCA, 639 Le sacre du printemps, 223 lead-sheet notation, 535, 694 learning process, 674 learning by doing, 36 leaves of a stemma, 764 left action, 1066 adjoint, 1120 coset, 1067 ideal, 1076 inverse, 1066 legato, 783 LEGO, 943 Leibnitz, Gottfried Wilhelm, 565 λεκτ oν, 17 lemma Fitting’s -, 1089 length, 328, 1089 minimal -, 835 of a local meter, 115 path -, 1063 LEP, 239 Lerdahl, Fred, 290, 291, 294, 305, 311, 457, 461, 752, 873 Les fleurs du mal, 268, 963 level, 329 connotative -, 19 denotative -, 19 F-to-enter -, 881 function, 329, 1061 meta -, 19 metrical -, 457 neutral -, 258 object-, 19 sound pressure -, 1029 levels of reality, 10 Levelt, Wilhelm, 1049, 1052 Lewin, David, 83, 247–250, 376, 384, 498 Lewis, Clarence Irving, 693 lexical, 418 lexicographic ordering, 58, 90, 1060 Leyton, Michael, viii, 933, 935
1301 LH, 764 library, 60, 446 Lie algebra, 1104 homomorphism, 1104 linear -, 1105 bracket, 1104, 1161 affine -, 541 derivative, 1160 formalism, 800 operator, 774 Lie, Sophus, 1104, 1160 Lied auf dem Wasser zu singen..., 262 Ligeti, Gy¨orgy, 33 limbic structure, 638 system, 642, 737, 1045 limit, 1121 circular -, 77 form, 67 ring, 1078 topology, 1146 limited modulations, 585 transposition, 151 mode with -, 151 line, 1062 linear (in)dependence, 1085 algebra special -, 1105 case, 250 combination, 1084 counterpoint group, 475 dual, 1091 fibration, 914 Lie algebra, 1105 module homomorphism, 1083 ordering, 1060 on a colimit, 92 on a limit, 92 on finite subsets, 92
1302 representation, 1086 visualization generic -, 440 metrical -, 441 linear ordering among denotators, 58 linguistics, 194 structuralist -, 305 Lipschitz locally -, 1156 Lipschitz, Rudolf, 1156 LISP, 534 list form, 976 listener, 12, 14 listening articulated -, 304 music -, 1035 procedure, 743 Liszt, Franz, 18, 20, 603 literally included, 255 Lluis Puebla, Emilio, viii local, 299 character of a contrapuntal symmetry, 647 composition, 89, 105, 107 commutative -, 125 embedded -, 126 functorial -, 121 generating -, 126 morphism, 124 objective -, 107 sequence of a -, 234 wrapped as -, 108 compositions coproduct of -, 124 fiber sum of -, 169 product of -, 124 meter, 115 length of a -, 115 period of a -, 115 meters simultaneous -, 609 morphism, 1108 optimization, 821 orientation, 324 P ara-meter, 327
INDEX performance score, 724 rhythm, 116, 127 ring, 1089 score, 307, 946 solution, 1156 standard composition, 357 symmetry, 648, 649 technical parameter, 1005 threshold, 819 local topography, 19 local-global patchwork, 307 locality principle, 920 localization, 1101 historical -, 273 of epilepsy focus, 638 of musical existence, 24 locally closed, 1147 finite, 1149 free, 1110 linear grammar, 802 Lipschitz, 1156 ringed space, 1108 trivial structure, 307 locus Riemann -, 820 logarithmic perception, 668 LoGeoRUBETTEr , 811 logic, 419 absolute, 176 classical -, 1134 fuzzy, 409 harmonic -, 546 Heyting -, 530 inner -, 757 intuitionistic -, 539, 1134 musical -, 323 of orbits, 243 of toposes, 277
INDEX performance -, 674 performing -, 934 predicate -, 530 logical, 1128 algebra, 1132 connective symbol, 1131 motivation, 776 switch operator, 71 time, 611 loop, 1063 Lord, John, 231 loudness, 51, 79, 739, 1029 LPS, 724, 755 Luening, Otto, 306 Lussy, Mathis, 747 M mth interval succession, 254 M.M., 670, 682 M¨alzel’s metronome, 31, 1028 M¨alzel, Johannn Nepomuk, 414, 670, 682, 693 Ma m`ere l’oye, 223 Mac OS X, 807, 813 machine performance -, 852 precision, 757 Turing -, 670 MacLean, Paul, 642 macro, 331 -event, 89 germ, 331 Maiguashca, Mesias, 70, 137 major, 146, 657 dichotomy, 631, 657 mode, 545 scale, 575, 576 dominant role of -, 657 third, 1031 tonality, 560, 582 bigeneric -, 547 major-minor problem, 146 making music, 24
1303 Malt, Mikhail, 979 manifold, 307 formant -, 291 musical -, 295 of opinions, 997 semantic -, 295 map, 1059 characteristic -, 1126 coboundary -, 1150 performance -, 712 refinement -, 336 simplicial -, 1148 maquette, 978 Marek, Ceslav, 303 marked counterpoint dichotomy, 630 class, 630 dichotomy autocomplementary -, 631 class, 630 rigid -, 631 strong -, 631 interval dichotomy, 630 Martinet, Andr´e, 17 Marx, Adolf Bernhard, 603 Maschke, Heinrich, 1088 Mason’s theorem, 130 Mason, Robert, 129, 567 Mason-Mazzola theorem, 130 mass-spring, 1027 Massinger, Philip, 824 master concert -, 761 matching, 918 of structures, 869 score-performance -, 918 material change of -, 982 experimental -, 401 musical -, 978 of music, 106 time, 611 Math-motif, 494 Mathematicar , 929
1304 mathematical existence, 175, 398 model, 565 morphism, 344 overhead, 623 mathematically equivalent morphisms, 344 mathematics, 6, 195 matrilineal, 764 scheme, 762 matrix, 1085 category, 1117 comparison -, 251 design -, 877 initial design -, 878 interaction -, 857 product, 1085 Riemann -, 544, 586, 819 value -, 925 verse -, 261 Matterhorn, 183 Mattheson, Johann, 303, 996, 1050 MAX, 137, 256, 535, 953 maximal, 381 meter nerve topology, 460 topology, 329, 459 structure content, 1047 mayamalavagaula, 658 Mayer, G¨ unther, 271 Mazzola, Christina, ix Mazzola, Guerino, 268, 611, 613, 745, 873, 945 Mazzola, Silvio, ix MDZ71, 293 mean performance, 881 tempo, 881 meaning of sound, 295 paratextual -, 400 textual -, 400
INDEX topological -, 192 transformational -, 193 measure for complexity, 311 measurement, 30 mechanical dynamics, 739 mechanism modulation -, 566 mediante tuning, 1033 mediation, 935 meet, 1132 mela, 658 melakarta, 658 melodic charge, 742 minor, 584, 657 variation, 959 melodic minor scale, 575, 578 tonality, 560 melody, 276, 331 germinal -, 269, 270, 956 retrograde of a -, 137 MeloRUBETTEr , 467, 497, 785, 816 membrane basilar -, 1038 Reissner’s -, 1038 tectorial -, 1039 memory, 641, 642 function hippocampal -, 642 mental experiment, 666 organization, 39 time, 664 tone parameters, 79 Mersenne, Marin, 1049 message, 13, 27 infinite -, 905 passing, 968 messaging, 188 Messiaen mode, 151 scale, 959 Messiaen, Olivier, 150, 152, 161, 959
INDEX meta-object, 968, 978, 982 class, 982 protocol, 982 meta-programming, 967, 982 meta-vocabulary, 243 metalanguage, 259 metalevel, 19 metamere, 1046 metaphor, 26 metasystem, 19 meter, 114, 455–463, 1029 beat -, 115 barline -, 115 local -, 115 method, 188, 968 boiling down -, 787 continuous -, 776 dodecaphonic -, 936, 940 group-theoretical -, 241, 250 integration -, 829 of characteristics, 1161 operational -, 31 selection, 971 statistical -, 745, 818 metric, 1153 associated -, 1154 Euclidean -, 279 metrical analysis, 835 component, 327 globalization, 116 grouping, 303 hierarchy, 455 level, 457 linear visualization, 441 profile, 835 quality, 456 rhythm associated -, 327 similarity, 199, 472 structure (=M), 457 weight, 455, 456, 785 metronome, 118 M¨alzel’s -, 31, 1028
1305 MetroRUBETTEr , 457, 814 Mexican hat, 289, 1025 Meyer wavelet, 1026 Meyer-Eppler, Werner, 1035, 1046 mezzoforte, 1030 Michel chromatic, 582 micro -motif, 784 timing, 270 micrologic, 692 microstructure timing -, 871 middle ear, 1037 middleground, 503 MIDI, 287, 946 Mikaleszewski, Kacper, 756 minimal cadential set, 554 length, 835 Minkowski, Hermann, 377, 1070 minor, 146 harmonic -, 584 melodic -, 584, 657 mode, 545 natural -, 582 tonality, 582 Mittelgrund, 503 Mitzler, Laurentz, 1050 mixed weight, 815 M¨obius bottle, 677 strip, 549, 579, 941 M¨obius strip, 307, 322, 538 modal structure, 383 synthesis, 1027 mode, 152 aeolian -, 320 authentic -, 320 consonant -, 547 dissonant -, 547 dorian -, 320 ecclesiastical -, 319, 655, 657 hypoaeolian -, 320
1306 hypodorian -, 320 hypoionian -, 320 hypolocrian -, 320 hypolydian -, 320 hypomixolydian -, 320 hypophrygian -, 320 ionian -, 320 locrian -, 320 lydian -, 320 Messiaen -, 151 mixolydian -, 320 phrygian -, 320 plagal -, 320 rhythmic -, 611 with limited transpositions, 151 model, 1136 mathematical, 565 physical -, 29 template fitting -, 1045 modeling physical -, 289, 850 modification of functional relations, 982 syntax -, 982 modular affine transformation, 948 composition, 307 modularity dynamical -, 809 modulatio, 564 modulation, 559, 563–592, 1008 amplitude -, 288, 1003 degree, 566 domain, 580 force, 567, 571 frequency, 288, 1003 frequency -, 1022 just -, 577 mechanism, 566 path, 600 pedal -, 608 pitch -, 288, 1003 plan, 607
INDEX quantized -, 572 quantum, 567, 568, 572, 573 rhythmical -, 576, 610, 613, 959 theorem, 572 topos-theoretic background of -, 568 well-tempered -, 571 modulations limited -, 585 modulator, 87, 289, 572, 596, 600, 602, 1022 modulatory architecture, 603 region, 592 module, 1083 as basic space type, 69 complex, 350 constant -, 350 flasque -, 370 of A-addressed forms, 351 representative -, 363 retracted -, 352 direct sum -, 1084 dynamically loadable -, 808 free -, 1085 functor, 172 injective -, 1103 of a commutative local composition, 125 of global affine functions, 432 product -, 1084 projective -, 1102 semi-simple -, 1087 shaping -, 807 simple -, 1087 structuring -, 807 modules in music, 70 modus ponens, 1134 molecule, 355 global -, 355 interpretable -, 356, 385 Molino, Jean, 12, 14, 696 moment initial -, 697 mono, 1116 monochord, 24 monogamic coupling, 769 monoid, 1063
INDEX algebra, 71, 1077 finite -, 1063 free -, 1064 free commutative -, 1064 homomorphism, 1063 isomorphism, 1063 morpheme -, 540 multigeneric -, 543 trigeneric -, 540 word -, 1064 monomorphism, 1116 Monteverdi, Claudio, 909 Montiel Hernandez, Mariana, viii, 334 mood, 736 MOP, 982 graphical -, 982 Morlet wavelet, 1025 morpheme harmonic -, 546 monoid, 540 Morphemfeld, 506 morphic, 749, 1140 morphing, 952 morphism, 196, 1115 t-fold differentiable tangent -, 669 t-fold tangent -, 669 global -, 300 local -, 1108 mathematical -, 344 mathematically equivalent -, 344 of denotators, 108 of form semiotics, 1141 of formed compositions, 355 of functorial global compositions, 335 of functorial local compositions, 156 of global molecules, 355 of local compositions, 124, 154–158 of objective global compositions, 335 of objective local compositions, 154 of performance cells, 713 tangent -, 669, 676 Morris, Robert, 247, 249, 250, 258, 383, 385, 498
1307 Morrison, Joseph, 1027 MOSAIC, 1027 mosaic, 378 mother, 724, 752 primary -, 764, 765 prime -, 765 tempo, 682 motif, 118, 193, 279, 331 abstract -, 468 classification, 228–231 covering, 467 Reti’s definition of a -, 491 rhythmic -, 613 serial, 149 space, 467 Z-addressed -, 120 motion, 734, 738 accelerated -, 738 harmonic -, 503 sense of -, 739 trigger, 738 motivated, 18 motivation, 419 geometric -, 420, 422 logical -, 420, 776 motives enumeration of -, 238 motivic analysis, 262, 491 interpretation, 332, 467 nerve, 467 simplex, 467 weight, 496, 785 work, 338 zig-zag, 339, 941 motor action, 739 movement tensed -, 646 Mozart, Wolfgang Amadeus, 231, 458, 598 M¨ uller, Stefan, viii, 912, 1143 multigeneric monoid, 543 multigraph, 1062 finite -, 1062 multimedia object, 441, 449 multiple-dispatching, 968
1308 multiplication interval -, 623 scalar -, 1083 multiplicity, 254 Mumford, David, 366 Murenz wavelet, 1025 music, 3, 8, 9, 14, 25 absolute -, 934 alphabet of -, 106 atonal -, 248 composition technology, 564 concept of -, 23 critic, 304 role of -, 906 criticism, 303, 772 critique, 905 definition of -, 6 deixis, 18 emotional function of -, 642 esthetics of -, 393 fact of -, 10 film -, 733 historical dimension of -, 108 history, 6 in-time -, 986 key function of -, 643 listening, 1035 material of -, 106 psychology, 291, 305 research, 8 semiotic perspective of -, 16 software, 307 syn- and diachronic development of -, 242 tape -, 306 theory, 813 thinking -, 25 music theory professional -, 247 musical concepts definition of -, 114 analysis, 744 composition, 33
INDEX drama, 908 gestalt, 106, 152 idea, 934, 935 intuition, 246 logic, 323 manifold, 295 material, 978 onset, 1028 ontology, 23 process, 978 prosody, 270 reality, 171 semantics, 162 taste, 643 tempo, 31 topography, 19 unit, 106 musicological experiment, 33, 34 ontology, 398 musicology, 3, 14, 813, 871 cognitive -, 23 computational -, 23 historical -, 399 systematic -, 399 traditional -, 24, 31 Musikalisches Opfer, 144 musique concr`ete, 306 Muzzulini, Daniel, 537, 554, 571, 585 Mystery Child, 952 N n-circle, 513 n-cube, 671 N -formed global composition, 354 n-modular pitch, 250 n-phonic series all-interval -, 237 N -quotient, 360 name, 76 instrument, 82 of a denotator, 52 of a form, 51 names
INDEX ordering on -, 90 naming policy, 51, 52, 68 Naradaya–Watson kernel, 857 narration, 933 narrativity theory of -, 934 Nattiez, Jean-Jacques, 272, 304, 473, 940 natural, 1119 decomposition, 855 distance, 441 minor, 582 transformation, 1118 natural minor tonality, 560 natural sciences experience in the -, 34 nature exterior -, 32 interior -, 32 nature’s performance, 925 navigation, 34, 43 address -, 169 conceptual -, 39 dynamic -, 45 productive -, 44, 45 receptive -, 44, 89 topographical -, 21 trajectory, 35 visual -, 439 negation, 421, 1131, 1132 neighborhood, 199, 276, 817, 1145 elementary -, 489 nerve, 937, 1148 auditory -, 1037 class -, 346, 376, 390 discrete -, 311 functor, 1148 motivic -, 467 of a global functorial composition, 344 of a global objective composition, 310 weight, 460 induced -, 460 Neuhaus, Harry, 756 νε˜ υ µα, 193 neumes, 16, 193, 693
1309 neural pitch processing, 1045 neuronal oscillator, 737 neutral, 12, 1021 analysis, 272, 305 element, 1063 level, 258 neutral level, 12, 14 neutralization, 565 Newton, Isaac, 399 NeXT, 808, 833 NEXTSTEP, 807, 813 nexus, 382 nihil ex nihilo, 28 nilpotent, 1089 Ninth Symphony, 495 Noether, Emmy, 138 Noh, 14, 416, 768 Noll, Thomas, viii, 82, 221, 506, 510, 512, 515, 519, 524, 529, 538, 540, 546, 564, 571, 633, 636, 744, 820, 1064, 1143 non-commutative polynomials, 1077 non-interpretable composition, 371, 376 non-invertible symmetry, 153 non-lexical, 418 non-linear deformation, 776, 827, 889 non-linearity, 1043 non-parametric approach, 856 norm, 19, 1154 normal form, 255 Rahn’s, 255 order, 257 subgroup, 1067 normalization diachronic -, 909 synchronic -, 909 normative analysis, 457 norms equivalent -, 1155 not parallel, 795 notation American jazz -, 533 European score -, 79
1310 lead-sheet -, 533, 535, 694 notched tone space, 1048 note alterated, 127 anchor -, 760 satellite -, 760 note against note, 619, 646 number embedding -, 254 prime -, 278 numbers complex -, 1076 dual -, 618, 1077 rational -, 1076 real -, 1076 O object, 188, 968, 1116 description, 244 global -, 299 multimedia -, 441, 449 prototypical -, 280 visualization principle, 441 object-oriented programming, 55, 723, 763, 766, 770, 967, 968 objective closure, 524 global - composition, 309 local - composition, 107 trace, 121 Objective C, 808, 825 objectlevel, 19 objectystem, 19 observation, 30 OCR, 767 octave, 73, 1031 coordinate, 1032 period, 110 octave class, 139 ODE, 792, 829, 1156 Ode an die Freude, 495 Oettingen, Arthur von, 137, 147, 504, 512, 514, 1032 off-line algorithm, 919
INDEX ON-OFF, 71 ondeggiando, 720 onomatopoiesis, 18, 938 onset, 51, 79 abstract -, 150 musical -, 1028 origin, 115 physical -, 1028 self-addressed -, 83 time, 1013 weight, 116 ontological atomism, 27 coordinate, 10 dimension, 19 perspective, 6 shift, 171 ontology, 9, 171, 180, 184, 398 musicological -, 398 denotator -, 398 knowledge -, 420 musical -, 23 time -, 936 open ball, 1153 covering of a functor, 1113 formant set, 291 functor, 1113 semiosis, 401 set, 278, 1145 source, 808 Open-Editor, 983 OpenMusic, 256, 384, 935, 943, 967–990 openness, 290 operation Boolean -, 947 flattening -, 88, 331 operationalization, 245 operationalized thinking, 196 operator, 727, 749, 752 agogical -, 872 articulation -, 720 basis-pianola -, 795 Beran -, 876 canonical -, 253
INDEX field -, 792 Lie -, 774 performance -, 727, 744, 773–803 physical -, 749, 791 pianola -, 801 prima vista -, 749 smoothing -, 874 splitting -, 788 sub-path -, 588, 1079 support -, 1015 symbolic -, 749, 789 tempo -, 793 test -, 791 Todd -, 752 T T O -, 253 validation -, 424 opinions manifold of -, 997 opposite category, 1117 group -, 1066 opposition, 18 optimal path, 820 optimization local -, 821 orbit, 1066 set-theoretic -, 1114 space, 1066 orbits functor of -, 1114 Orchestervariationen, 137 orchestra instrumentation denotator, 82 orchestral gesture, 986 globalization, 673 orchestration, 948 order, 1067, 1069 normal -, 257 of a PDE, 1161 ordered p-space interval, 252 pair, 1058 pc interval, 253 ordering, 440 alphabetic -, 40, 43, 58
1311 encyclopedic -, 58 lexicographic -, 58, 90, 1060 linear -, 1060 on a colimit, 92 on a limit, 92 on finite subsets, 92 on coefficient rings, 94 compound (naive) denotators, 59 compound (naive) forms, 59 coordinators, 91 denotators, 89–99 diagrams, 91 direct sums, 94, 95 forms, 89–99 identifiers, 91 matrix modules, 95 Mod, 93–96 morphisms, 95 names, 90 simple forms, 90 types, 90 universal construction functors, 92 ZhASCIIi, 95 partial -, 1060 powerset -, 60 principle, 440 on denotators, 57 universal -, 44 ordinal, 1116 Oresme, Nicholas, 30, 664 organ of Corti, 1038 organic composition principle, 868 principle, 198 organism cellular, 394 hierarchical -, 304 organization degree of -, 869 mental -, 39 orientation, 8, 322, 619 hanging -, 128 change of -, 619, 626, 646
1312
INDEX
geographic -, 43 hanging -, 619 local -, 324 ontological -, 9 recursive -, 21 sweeping -, 128, 619 oriented contrapuntal interval, 619 global composition, 355 origin, 328 of onset, 115 OrnaMagic, 941, 950, 952, 953 ornament, 720, 949 pattern, 246 OrnamentOperator, 784 orthogonality principle, 920 orthonormal decomposition, 11 Orthonormalization, 879 oscillator, 736 neuronal -, 737 oscillogram, 736, 737 Osgood, Charles, 198 ostinato, 979 ottava battuta, 657 outer derivation, 1105 ear, 1036 hair cell, 1039 pillar cell, 1039 outlet, 972 output prestor -, 946 oval window, 1037, 1041 overhead mathematical -, 623 overloading, 968 P p-group, 1069 p-pitch, 252 p-scale, 112 p-space, 252
p-Sylow group, 1069 Paganini, Niccol`o, 996 painting, 183, 946 pair ordered -, 1058 polarized -, 646 simplicial -, 1150 Yoneda -, 1137 Palestrina–Fux theory, 655 paper science, 176 Papez, James, 642 Par´e, Ambroise, 32 P ara-rhythm, 327 paradigm, 18 dodecaphonic -, 150 Fourier -, 284 general affine -, 161 gestalt -, 816 phonological -, 269 παρ` αδειγµα, 192 paradigmatic concept, 280 field, 150 group, 474 strategy, 940 theme, 272, 473 tool, 953 transformation equivalence, 259 paradigmatics uncontrolled -, 201 parallel, 795 articulation field, 689 crescendo field, 689 degree, 324 glissando field, 689 hierarchy, 718 not -, 795 performance field, 689 map, 689 space, 718 Parallelklang, 556
INDEX parameter accessory -, 999 basis -, 79, 795 bow -, 1003 cadence -, 552 color -, 1004 essential -, 999 exchange, 160, 161 geometric -, 1000 global technical -, 1005 instrumental -, 1002, 1004 local technical -, 1005 pianola -, 79, 795 primavista -, 722 space, 434 system -, 575 technical -, 289 vibrato -, 1003 parametric polymorphism, 968 paratextual, 769 information, 831 meaning, 400 paratextuality, 424 Parncutt, Richard, 738 parole, 19 part, 301, 307, 334 dilinear -, 1090 extraterritorial -, 720 translation -, 1090 partial, 86, 513 ordering, 1060 partials, 10 participation value, 639 particle physics, 567 partition, 1058 int´erieure, 14 partitioning, 257 passing message -, 968 patch, 969, 978 patchwork local-global -, 307 path, 1063
1313 category, 1079 closed -, 1063 harmonic -, 586 lazy -, 1063 length, 1063 modulation -, 600 optimal -, 820 patrilineal, 764 pattern, 246 pause, 81, 768 general -, 782 pc, 253 pc-space, 253 pcseg, 253 pcset, 253 invariant -, 254 PDE, 798, 1161 quasi-linear -, 1161 Peano axioms, 32 pedal modulation, 608 voice, 608 peer, 810 perception logarithmic -, 668 perceptional pitch concept, 1047 perceptual equivalence, 280 percussion, 269 event, 612 perfect consonance, 635, 646, 657 performance body, 712 cell, 711 cells category of -, 713 morphism of -, 713 complexity of -, 664 culture of -, 757 development hierarchy of -, 757 dynamics of -, 800 field, 685, 690, 712 parallel -, 689 prime mother -, 768 fields
1314 algorithmic extraction of -, 916 grammar, 747 inbuilt -, 907 hierarchy, 674 history fictitious -, 763 real -, 763 infinite -, 666 initial -, 712, 726 intermediate -, 756 logic, 674 machine, 852 map, 712 parallel -, 689 mean -, 881 nature’s -, 925 of a large orchestra, 761 operator, 727, 744, 773–803 plan, 757 primavista -, 766 procedure, 743 real-time -, 738 research computer-assisted -, 764, 850 score global -, 728 local -, 724 structural rationale of -, 395 synthetic -, 741 theory, 387, 393 inverse -, 743, 790, 913 tradition, 907 PerformanceRUBETTEr , 708, 792, 794, 824, 889 performer, 27 performing logic, 934 perilymph, 1038 period, 503 fundamental -, 1019 in the Euler module, 110, 112 interictal -, 638 octave -, 110 of a local meter, 115 of a Vuza canon, 381
INDEX of a Vuza rhythm, 380 of duration, 115 temporal -, 456 periodicity, 856 higher level -, 117 Perle, George, 248 permutation, 1059 perspective, 27, 181, 184, 393, 566 change of -, 393 f -, 336 of the composer, 301 ontological -, 6 variation of -, 182 perspectives sum of -, 394 Petsche, Hellmuth, 638, 640 phase, 1020 portrait, 1159 spectrum, 291, 1020 phaticity, 259 phenomenological difference, 912 phenotype, 943 philosophy, 5 denotator -, 185 Yoneda -, 997 phoneme, 287 phonological equivalence, 263 paradigm, 269 poeticity, 263 photography, 183 phrasing, 303 physical model, 29 modeling, 289, 850 onset, 1028 operator, 749, 791 pitch, 1031 sound, 84 time, 664 tone parameters, 81 PhysicalOperator, 829 physics, 6 particle -, 567 pi -rank, 1070
INDEX pianissimo, 738 piano hierarchy, 722 Piano concert No.1, 245, 246 Pianoforte Schule, 758 pianola coordinate, 1028 deformation, 720, 797 operator, 801 parameter, 79, 795 space, 689, 715, 763 specialization, 801 piecewise smooth, 1018 Pinocchio, 450 pitch, 51, 79 -class self-addressed -, 83 -class set, 248, 253 absolute -, 700 alteration, 62, 952 chamber -, 684, 1031 class, 111, 139 segment, 253 set, 253 concept perceptional -, 1047 concert -, 699 cycle, 252 detector central -, 1045 difference, 73 distance, 72 fundamental -, 532 mathematical -, 72 modulation, 288, 1003 physical -, 1031 processing neural -, 1045 segment, 252 spaces, 250 symbolic -, 80 pivot, 572 pivotal chord, 563 pixel, 417 plan performance -, 757
1315 plane transformation, 949 Plato, 29, 43, 202 platonic ideas, 23 playing, 24 Plomp, Reiner, 1049, 1052 Podrazik, Janusz, 257 Poe, Edgar Allan, 933 Poem of Wind, 268 poetic Ego, 268 poetical function, 18, 138, 259, 295, 303, 934, 942 functions spectrum of -, 266 poeticity, 138, 259 vertical -, 261 horizontal -, 261 phonological, 263 poetics timbral -, 295 verse -, 303 poetology, 258 poiesis, 12, 13, 258 retrograde -, 15 poietic, 12, 1021 genealogy, 154 point, 177–181, 1057 generic -, 279 absorbing -, 525 accumulation -, 1145 closed -, 279 concept, 175 etymology, 178 Euler -, 1031 faithful -, 523 full, 523 fully faithful -, 523 generic -, 330, 1110 identity, 178 turning -, 565 pointer, 25, 26, 43, 177 scheme, 55 polarity, 631, 640 at x, 637 in musical cultures, 658
1316 profile, 279 polarized pair, 646 politically incorrect, 907 P´olya enumeration theory main theorems of -, 233 theory, 232 weight function, 232 P´olya, George, 232, 378 polygamic coupling, 769 polygon, 948 polyhedral initial set, 704 polymorphism ad-hoc -, 968 parametric -, 968 polynomials commutative -, 1077 non-commutative -, 1077 polyphony, 995 polyrhythm, 965 polysemy, 129, 200 Popper, Karl, 997 Porphyrean tree, 191 portrait phase -, 1159 position general -, 391 privileged -, 633 Posner, Roland, 261, 942 post-serialism, 245 potential epileptiform -, 638 sink -, 739 power spectral -, 640 window, 638 powerset, 1062 form, 66 ordering, 60 type, 54 PR, 457 Pr´eludes, 600 practising, 769, 771 pre-Hilbert space, 1020 pre-morphism, 410
INDEX pre-object, 410 precise conceptualization, 258 precision calculation -, 775 conceptual -, 35 human -, 757 machine -, 757 PrediBase, 808 predicate, 410 atomic -, 412 connective, 1135 deictic, 420 global -, 552 logic, 530 mathematical -, 412, 420 morphic -, 410 objective, 410 primavista -, 414, 769 European -, 414 non-European -, 414 punctual -, 410 PV -, 414 relational -, 410 shifter -, 418 textual -, 544, 552 variable, 1135 preferences, 820 prehistory of the string quartet, 994 presence, 497, 785, 817 presheaf, 1119 pressure bow -, 1002 decrease factor, 1016 variation, 1013 prestor , 47, 137, 245, 246, 268, 269, 293, 532, 639, 699, 758, 941, 943, 945– 953, 955 output, 946 prestor , primary mother, 764, 765 primavista, 674, 766 agogics, 764 dynamics, 764
INDEX operator, 749 parameter, 722 performance, 751, 766 predicate, 769 PrimavistaOperator, 766 PrimavistaRUBETTEr , 831 prime, 1080 field, 1076 form, 257 mother, 765 performance field, 768 number, 278 spectrum, 278, 293, 1108 stemma, 764 vector, 73, 1033 principal component analysis, 898 principal homogeneous G-set, 249 principle anthropic -, 565, 567, 658 architectural -, 869 concatenation -, 160, 624 dialogical -, 997 fractal -, 964 locality -, 920 normative -, 458 object visualization -, 441 of relevance, 17 ordering -, 440 organic composition -, 868 organic -, 198 orthogonality -, 920 packing -, 441 sonata -, 163 variation -, 394 priority, 879, 891 privileged position, 633 problem Cauchy -, 1162 context -, 819 hit point -, 704 wild -, 913 procedure listening -, 743 performance -, 743
1317 rule based -, 747 rule learning -, 747 statistical -, 242 process, 19, 401 historical -, 763, 771 learning -, 674 musical -, 978 of conceptualization, 245 product, 1062 ambient space, 124 Cartesian -, 1058 cartesian -, 1121 category, 1118 cellular hierarchy, 718 fiber -, 1078, 1121 group, 1068 matrix -, 1085 module, 1084 of local compositions, 124 of the cells, 714 ring, 1077 semidirect -, 1068 tensor -, 1078 topology, 1146 type, 52 weight function, 232 wreath -, 1069 production, 4 of a musical work, 13 profile metrical -, 835 program canonical -, 394 change, 947 programme narratif, 935 programming constraint -, 935, 967 functional -, 967 language visual -, 967 object-oriented -, 55, 188, 723, 763, 766, 770, 967, 968 progression harmonic, 304 chord -, 502
1318 contrapuntal, 304 harmonic -, 154 Project of Music for Magnetic Tape, 306 projecting local composition, 216 projection, 153, 1062 projective atlas, 360 functions, 362 module, 1102 Prokofiev, Serge, 223 prolongational reduction (=PR), 457 proof chain, 1133 propagation sexual -, 763, 773 property inheritance -, 479 propositional formula, 1136 variable, 1131 prosody musical -, 270 protocol meta-object -, 982 prototype, 241 prototypical object, 280 pseg, 252 pseudo-metric, 1153 on abstract gestalt space, 478 psychological reality, 665 psychology, 6 cognitive -, 218, 276 gestalt -, 106 music -, 291, 305 psychometrics, 199, 279 Puckette, Miller, 918 pullback, 1121 pure expressivity, 737 pushout, 1121 PVBrowserRUBETTEr , 811 Pythagoras, 530 Pythagorean, 33 school, 114, 413 tonality, 561 tradition, 24, 186, 1049 tuning, 325, 581, 1032
INDEX Pythagoreans, 12 Q quale, 693 qualifier after -, 983 before -, 983 quality metrical -, 456 quantifier existence -, 421 formula, 1136 universal -, 421 quantization, 835 quantized modulation, 572 quantum modulation -, 567, 568, 572, 573 quantum mechanics, 505, 516 quartet string -, 934, 993 quasi-coherent, 1109 quasi-compact, 1147 quasi-homeomorphism, 1111 quasi-linear PDE, 1161 quaternions, 1076 quatuor concertant, 994 quatuor dialogu´e, 996 quiver, 1063 algebra, 1079 complete -, 1063 Riemann -, 586 Riemann index -, 589 stemma -, 801 quotient category, 1117 complex, 351 dominance topology, 283 group, 1067 ring, 1076 topology, 1146 R radical, 1064 Radl, Hildegard, 560, 576, 581, 585 Raffael, 186 Raffman, Diana, 25, 693
INDEX raga, 658 Rahn, John, 247–250, 498 Rameau’s cadence, 554 Rameau, Jean-Philippe, 502, 512, 531, 554, 1050 ramification mode, 48 random, 15 rank, 1085 torsion-free -, 1070 Raphael, 201 rational numbers, 1076 rationale, 748 gestural -, 908 historical -, 994 Ratner, Leonard, 400 Ratz, Erwin, 567, 604 Ravel, Maurice, 223 RCA, 639 real, 829 inversion, 148 numbers, 1076 performance history, 763 real-time algorithm, 919 context, 917 input, 946 performance, 738 reality, 10 historical -, 594 levels of -, 10 mental -, 12 musical -, 171 physical -, 11 psychological -, 12, 665 realization geometric -, 1149 reason, 5 recapitulation, 304, 603 receiver, 259 reception, 4 receptive navigation, 89 recitation tone, 319 recombination, 982 weight -, 776
1319 reconstruction, 493 recording bipolar -, 638 foramen ovale -, 638 recta, 839 recursive classification, 216 construction, 49 orientation, 21 typology, 48, 56 reduced diastematic shape type, 470 strict style, 657 reduction, 1095 curvilinear -, 937 reductionism, 7 Reeves, Hubert, 203 reference denotator, 403 tonality, 546 referentiality, 259 refinement map, 336 reflection, 982 glide -, 1097 reflexive, 1059 reflexivity, 967 Regener, Eric, 249 region modulatory -, 592 register, 947, 969 regression analysis, 860, 877, 880 regular denotator, 79–85 division of time, 456 form, 76 representation, 1086 structure, 856 regularity time -, 116 rehearsal, 674, 741, 745, 769, 771 Reichhardt, Johann Friedrich, 996 reification, 982 Reissner’s membrane, 1038
1320 relation causal -, 985 commutativity -, 1117 cross-semantical -, 745 equivalence -, 305, 1060 K -, 382 Kh -, 382 KI -, 383 temporal -, 985 relative delay, 288 dynamics, 831 homotopy, 1150 importance, 786 motivic topology, 486 symbolic dynamics, 81 tempo, 682, 832 topology, 1146 relative local dynamical sign, 778 tempo, 780 relative punctual dynamical sign, 777 tempo, 780 relevance, principle of -, 17 Rellstab, Ludwig, 20 Remak, Robert, 1090 Remove-Element, 983 renaming, 55 repetition, 140 replay, 140 Repp, Bruno, 871, 872, 876, 898, 927 representable functor, 1120 representation adic -, 1080 auditory -, 240 Fourier -, 899, 1000 geometric -, 946 linear -, 1086 regular -, 1086 score -, 742 textual -, 937
INDEX representative first -, 220 module complex, 363 reprise, 964 reset, 71 resolution, 434, 999, 1009 cohomology, 432 functor, 358 of a global composition, 358, 393 response EEG -, 637 responsibility collective -, 770 restriction cellular hierarchy -, 718 of modulators, 606 scalar -, 127, 1084 resultant class, 381 resulting divisor, 382 retard final -, 738 Reti, Rudolph, 201, 275, 456, 465, 490, 816, 873 Reti-motif, 493 retracted module complex, 352 retraction, 1116 retrograde, 15, 25, 142, 152, 160, 254, 302 address involution, 150 inversion, 73, 144, 161 of a melody, 137 retrogression, 253 reverberation time, 1016 reversed order score played in -, 143 tape played in -, 145 sound, 145 revolution experimental -, 32 RH, 639, 764 rhetorical expression, 692 expressivity, 674 shaping, 674 rhetorics, 996
INDEX rhythm, 140, 152, 455–463, 974 local -, 116, 127 Vuza -, 380 rhythmic germ, 152, 326 mode, 611 motif, 613 scale, 613 rhythmical group, 977 modulation, 576, 610, 613, 959 theory, 612 structure, 958 rhythms, 114 classification of -, 380 local isomorphism classes of -, 221 Richards, Whitman, 739 richness semantic -, 692 Richter, Sviatoslav, 756 Riemann algebra, 586 dichotomy, 636 graph, 821 harmony, 322 index quiver, 589 locus, 820 matrix, 544, 586, 819 quiver, 586 transformation, 384 Riemann, Bernhard, 307 Riemann, Hugo, 116, 147, 194, 245, 250, 307, 455, 502, 506, 531, 543, 546, 564, 571, 586, 590, 619, 636, 814, 819, 841, 866, 873 Ries, Ferdinand, 993, 995 right action, 1066 adjoint, 1120 coset, 1067 ideal, 1076 inverse, 1066 right-absorbing endomorphism, 524
1321 rigid, 321, 340, 567, 571, 576, 816 difference shape type, 469 marked dichotomy, 631 shape type, 469 ring, 1075 anti-homomorphism, 1076 endomorphism -, 1083 homomorphism, 1075 isomorphism, 1075 limit -, 1078 local -, 1089 product -, 1077 quotient -, 1076 self-injective -, 1103 simple -, 1077 ringed space, 1107 ritardando, 739, 782 RMI, 810 Roederer chromatic, 582 Roland R-8M, 269, 955 role exchange, 72 of a music critic, 906 rotation, 246, 253, 254, 949 Amuedo’s decimal normal -, 256 roughness, 1051 round window, 1041 Rousseau, Jean-Jacques, 303 row-class, 255 RUBATOr , 47, 457, 764, 788, 789, 807–811, 871, 895, 916 concept, 807 Distributed -, 922 layer, 810 rubato Chopin -, 667, 682, 698, 759, 924 encoding formula, 751 RUBETTEr , 808, 813–832 Rufer, Joseph, 150, 162 rule -based procedure, 747
1322 grammar, 748 contravariant-covariant -, 972 Fux -, 657 learning procedure, 747 preference - (=PR), 457 well-formedness - (=WFR), 457 rules differentiation -, 742 ensemble -, 742 grouping -, 742 Rulle, 742 Runge-Kutta-Fehlberg, 792, 829 Russian Quartets, 995 Ruwet, Nicolas, 272, 304, 940 S Sabine’s formula, 1017 Sachs, Klaus-J¨ urgen, 645, 646 Salzer, Friedrich, 503 Sands’ algorithm, 377 Sands, Arthur, 377 Sarabande Nr. 52, 231 Sarcasmes, 223 satellite, 449 note, 760 saturation, 526, 1102 sheaf, 526, 540 Saussure, Ferdinand de, 17, 194, 242, 272, 305, 574 Sawada, Hideyuki, 738 SC, 253 scala media, 1038 tympani, 1037 vestibuli, 1038 scalar, 1083 multiplication, 72, 1083 restriction, 127, 1084 ScalarOperator, 829 scale, 112, 538 12-tempered -, 318 major -, 575 melodic minor -, 575, 578 altered -, 585
INDEX chromatic -, 506 diatonic -, 658 harmonic minor -, 575, 577 harmonical-rhythmical -, 959 just -, 113, 318 justest -, 325 major -, 321, 576 Messiaen -, 959 minor harmonic -s, 321 melodic -, 321 rhythmic -, 613 whole-tone -, 657 SCALE-FINDER, 256, 535 SCALE-MONITOR, 256, 535 scales common 12-tempered -, 113 scatterplot, 862 Sch¨onberg, Arnold, 33, 106, 137, 150, 162, 223, 243, 245, 248, 249, 301, 310, 321, 394, 501, 512, 563, 565, 567, 611, 936, 940 Sch¨afer, Sabine, 833 Schaeffer, Pierre, 306 scheme, 1111 diagram -, 1117 Grassmann -, 1113 matrilineal -, 762 mental -, 14 Molino’s -, 12 sonata -, 613 Schenker, Heinrich, 331, 400, 503 scherzo, 956 Schmidt, Erhard, 1090 school KTH -, 750, 755 Pythagorean -, 114, 413 Zurich -, 744 School of Athens, 186, 201 Schubert, Franz, 262, 283, 956 Schumann, Robert, 495, 764, 765, 818, 860, 947, 996 Schweizer, Albert, 834 science cognitive -, 743
152, 259, 538,
849,
INDEX computer -, 188 contemplative -, 29 doing -, 30 experimental -, 32 paper -, 176 scientific bankruptcy, 24 score, 12, 14, 71, 414, 946 concept, 307, 693, 909, 978 European - notation, 79 exterior -, 694 generic -, 665 global -, 307, 946 inner -, 665, 694 interior -, 14 local -, 307, 946 played in reversed order, 143 representation, 742 semantics, 696 transformation -, 948 score-following, 918 score-performance matching, 918 Scriabin, Alexander, 222, 587, 964 SEA, 467 Second Book of Pr´eludes, 756 section, 1109, 1116 global -, 350, 1121 segment class, 252, 255 pitch -, 252 selection axis of -, 138, 260 field -, 969 method -, 971 stepwise forward -, 881 self-addressed arrow, 626 chord, 225 contrapuntal intervals, 626 denotator, 82 onset, 83 pitch-class, 83 self-injective ring, 1103 self-modulating, 1022 self-referential, 22, 176
1323 self-similar time structure, 964 Selibidache, Sergiu, 696 semantic atom, 538 charge, 490 of EEG, 638, 640 completion, 57 depth, 465 differential, 198 interpretation, 598 loading, 48 manifold, 295 richness, 692 semantics functional -, 541 gestural -, 908 harmonic -, 531 incomplete, 99 musical -, 162 of weights, 497 score -, 696 semi-simple module, 1087 semidirect product, 1068 semigroup, 1063 semiosis, 10 extroversive -, 400 incomplete -, 401 introversive -, 400 open -, 401 paratextual -, 424 textual -, 406, 424 semiotic constraints, 284 marker visual -, 981 of E-forms, 1138 semiotical symmetry, 161 semiotics, 6, 16 of sound classification, 294 semitone, 74 sender, 13, 259 sense of motion, 739 sentence, 1131, 1136 valid -, 1132 sentic state, 734
1324 separating module complex, 360 sequence Cauchy -, 1154 chord -, 591 contrapuntal -, 646 Fibonacci -, 413 interspace -, 234 of a local composition, 234 sequencer, 953 sequentialization, 937 serial motif, 149 integrated -, 237 derived -, 237 technique, 152–154 serialism, 245 series all-interval -, 237, 244 basic -, 150 dodecaphonic -, 149, 197, 236, 301, 309, 394 fundamental -, 137 (k, n)-, 149, 236 n-phonic -, 149, 236 time -, 856 set, 1057 cadential -, 554 circular -, 79 class, 253, 254 closed -, 1145 complex, 249, 382 theory, 248 concept, 176 empty -, 176 fuzzy -, 198 in AST, 248 index -, 1060 initial -, 696, 712, 725 minimal cadential -, 554 of operations, 255 open -, 278, 1145 pitch-class -, 248 small -, 1116 source -, 248
INDEX support -, 309 theory, 305 SET-SLAVE, 255, 535 set-theoretic orbit, 1114 seventh dominant -, 508 natural -, 513 subdominant -, 508 tonic -, 508 sexual propagation, 763, 773 SGC, 255 Shakespeare, William, 773 shape, 492 shape type, 468 contrapuntal motion -, 470 diastematic index -, 470 diastematic -, 470 elastic -, 471 reduced diastematic -, 470 rigid difference -, 469 rigid -, 469 toroidal sequence -, 471 toroidal -, 471 shaping module, 807 rhetorical -, 674 vector, 876 sharp, 130 sheaf, 1111, 1130 on a base, 1108 saturation -, 526 sheafification, 1130 shearing, 144, 147, 160, 246, 1096 sheaves category of -, 1130 Sherman, Robert, 218 shift, 129 constant -, 129 elementary -, 129 ontological -, 171 shifter, 696, 701
INDEX esthesic -, 419 poietic -, 418 sieve, 1126 closed -, 1130 covering -, 1129 sight-reading, 674, 875 sign, 16 deictic -, 18 lexical -, 18 shifter -, 18 signature, 720 key -, 768 time -, 768 significant, 16 significate, 16 signification, 16, 17, 410 process, 16 signs of coefficients, 887 system of -, 6 similarity, 194, 198, 276, 279 metrical -, 199, 472 simple form, 66 simplify to a -, 75 frame, 968 group, 1067 module, 1087 ring, 1077 simple forms ordering on -, 90 simplex, 725, 1148 closed -, 1149 dimension of -, 1148 motivic -, 467 singular -, 1150 standard -, 1150 simplicial cochain complex, 1150 complex, 940, 1148 map, 1148 metrical weight, 329 pair, 1150
1325 weight, 346 simplify to a simple form, 75 Simula, 968 simultaneous local meters, 609 singleton interpretation, 336 singular cochain, 1150 simplex, 1150 sink potential, 739 Siron, Jacques, 694 sister, 802 site, 1129 Zariski -, 1112 skeleton, 1148 category, 1117 sketchy interpretation, 757 skew field, 1075 slave tempo -, 759 Slawson, Wayne, 290 slice, 121 identity -, 336 f -slice, 336 Sloboda, John, 807 slope global -, 822 slot, 968 slur, 768 SMAC, 744 small gestalt, 483 infinitely -, 692 set, 1116 smallness, 290 Smith III, Julius O, 1027 smooth piecewise -, 1018 smoothing hierarchical -, 857 kernel, 857 kernel -, 856, 874 operator, 874 SMPTE, 946
1326 SNSF, 744, 807 sober, 1110 weight, 460 socle, 1089 software development, 723 engineering, 184 for AST, 255 music -, 307 solution global -, 1159 local -, 1156 sonata form, 304, 603, 956 principle, 163 scheme, 613 theory, 603 Sonatine, 223 sound, 1013 classification, 284 color, 194 colors space of -, 290 conceptualization of -, 15 generator, 849 grouping, 88 meaning of -, 295 natural -, 11 physical -, 84 pressure level, 1029 reversed -, 145 speech, 996 transformation, 145 Sound Pattern of English (=SPE), 286 sounding analysis, 842 source open -, 808 set, 248 space, 1139 ambient -, 107 Banach -, 1154 basis -, 689, 715, 763 color -, 1000 compositional -, 249
INDEX contour -, 251 Euler -, 1031 form -, 64 fundamental -, 715 hierarchy, 715 indecomposable -, 715 instrument -, 726 locally ringed -, 1108 motif -, 467 of sound colors, 290 orbit -, 1066 parallel -, 718 parameter -, 434 pianola -, 689, 715, 763 pre-Hilbert -, 1020 ringed -, 1107 tangent -, 669 top -, 715 topological -, 1145 vector -, 1085 span, 633, 817 time -, 83 SPE, 286 special linear algebra, 1105 specialization, 196, 267, 282, 488, 719 abstract -, 488 abstract gestalt -, 488 basis -, 797 co-inherited -, 489 gestalt -, 488 Hasse diagram, 269 inherited -, 489 pianola -, 801 topology, 489 specialize, 278 species, 293 spectral analysis, 638 decomposition, 856 participation vector, 638, 639 power, 640 vector, 1001 spectrum, 874
INDEX amplitude -, 1020 energy -, 1020 of poetical functions, 266 phase -, 291, 1020 prime -, 278, 293, 1108 speculum mundi, 41 speech, 19 sound -, 996 SPL, 1029 split exact sequence, 1069 local commutative composition, 215 SplitOperator, 827 splitting, 764 operator, 788 spring equation, 1020 SQL, 811 Staatliche Hochschule f¨ ur Musik, 764, 765, 833 stability gestalt -, 276 stabilizer, 1066 stable concept, 276 staccato, 783 stalk, 1107 standard global - composition, 357 atlas, 357 chord, 531 hierarchy, 717 local - composition, 357 of basic musicological concepts, 108 simplex, 1150 composition, 211 standardized tempo, 877 Stange-Elbe, Joachim, viii, 764, 833 state sentic -, 734 stationary voice, 608 statistical approach, 855 method, 745, 818 procedure, 242 Steibelt, Daniel, 145, 161 Steinway, 955
1327 stemma, 674, 745, 752, 755–772 continuous -, 803 leaves of a -, 764 prime -, 764 quiver, 801 tempo -, 758 theory, 895, 911 tree, 802 stemmatic cross-correlation, 771 grouping, 770 Stengers, Isabelle, 30, 664 stepwise forward selection, 881 stereocilia, 1038 stereotactic depth EEG, 638 stirrup, 1037 Stockhausen, Karlheinz, 70, 152, 286, 385 Stolberg, Leopold, 262, 283, 956 Stone, Peter, 137 Stopper, Bernhard, 110 strategy experimental -, 841, 851 paradigmatic -, 940 target-driven -, 841, 851 Straub, Hans, viii, 230, 347, 537, 553, 555, 1188 Straus’ zero normal form, 256 strength factor, 742 stretching, 246 time -, 789 strict extension, 539 style, 656 reduced -, 657 STRING, 50 string empty, 52 of operations, 255 quartet, 82, 295, 934, 993 prehistory of the -, 994 theory, 993 strip harmonic -, 310, 321, 538 M¨obius -, 549, 579, 941 strong marked dichotomy, 631
1328 structural constant, 1105 homomorphism, 1075 rationale of performance, 395 structuralist linguistics, 305 structure formal -, 967 frame -, 718 internal -, 311 interspace -, 234 limbic -, 638 local vs. global -, 106 locally trivial -, 307 modal -, 383 of fibers, 913 regular -, 856 rhythmical -, 958 transitional -, 564 structures matching of -, 869 Structures pour piano, 152 structuring module, 807 Stucki, Peter, ix style, 869 strict -, 656 sub-complex Kh, 257 sub-path operator, 588, 1079 subbase for a topology, 1146 subcategory, 1118 full -, 1118 Yoneda -, 1137 subclass, 968 subcomplex, 1148 full -, 1148 subconscious, 643 subdivision, 757 subdominant, 323, 502, 541, 545 seventh, 508 subgroup normal -, 1067 subject, 24 subjectivity, 32 subobject, 1126 classifier, 1126
INDEX relation, 91 substance, 50 substitution theory, 1048 subtyping, 971 succession, 935 interval -, 252 successive interval, 640 successively increased alteration, 952 successor admitted -, 647 pairing allowed -, 646 sum direct -, 74 disjoint -, 1121 fiber -, 1121 of perspectives, 394 SUN, 810 Sundberg, Johan, 671, 738, 739, 741, 747 super-summativity, 203, 276, 332 superclass, 968 supersensitivity, 834 support, 410 functor, 314 of a local composition, 107 operator, 1015 set, 309 supporting valence, 1047 surface integral -, 1161 surgery concept -, 770 surgical epilepsy therapy, 638 surjective, 1059 suspension, 875 sweeping orientation, 128, 619 Swing, 922 switch vocabulary -, 293 Sylow decomposition, 95, 542, 620 group, 218 Sylow, Ludwig, 1069 symbol logical connective -, 1131
INDEX symbolic absolute dynamics, 81 computation, 967 kernel, 712 operator, 749, 789 pitch, 80 relative dynamics, 81 Symbolic Composer, 137 SymbolicOperator, 829 symmetric, 1059 group -, 1066 SYMMETRICA, 379 symmetries in music, 15, 137–154 musical meaning of -, 159 semantical paradigm for -, 159 symmetry, 108, 116, 135, 196, 1096 of parameter roles, 152 breaking, 936 codification of a -, 154 contrapuntal -, 647 degree of -, 254 form of a -, 135 function of a -, 136 group, 174, 220, 571, 816 conjugation class of the -, 220 hidden -, 136 inner of C-major, 147 local -, 648, 649 non-invertible, 153 semantical function of -, 135 semiotical -, 161 transformation, 305, 306 underlying -, 155 synchronic, 17 axis, 399, 575 normalization, 909 synonym form, 66 synonymy circular -, 76 type, 54 syntagm, 18
1329 syntagmatic equivalence, 263 syntax modification, 982 Synthesis, 268, 576, 610, 613, 940, 941, 950, 955–964 synthesis, 1018 modal -, 1027 synthetic performance, 741 syntonic comma, 115 system auditory -, 11 coefficient -, 1150 Hess -, 638 limbic -, 642, 737, 1045 meta -, 19 object-, 19 of signs, 6 non-linguistic -, 16 parameter, 575 vestibular -, 739 weight -, 768 systematic approach, 574 musicology, 399 understanding, 994 T t-fold tangent composition, 669 morphism, 669 t-fold differentiable tangent morphism, 669 t-gestalt, 474 t¨onend bewegte Formen, 307, 935 tactus, 457 Take Five, 218 tangent, 128, 621 bundle, 1155 composition, 669 basis of a -, 669 global -, 675 interpretation, 677 morphism, 669, 676 space, 669 Zariski -, 1112 torus, 621 tape music, 306
1330 target-driven strategy, 841, 851 taste common -, 907 musical -, 643 tautology, 1132 Taylor, Cecil, 664, 963 technical parameter, 289 technique instrumental -, 1002 tectorial membrane, 1039 telling time, 934 tempered, 111 class chord, 111 scale space, 113 tuning, 1032 template fitting model, 1045 tempo, 664, 668, 670, 682 absolute -, 414, 682, 780 curve, 247, 270, 682, 738, 758, 877, 947 adapted -, 699 deformation of -, 699 daughter -, 682 discrete -, 31 field, 683 hierarchy, 758 istesso -, 673 mean -, 881 mother -, 682 musical -, 30, 31 operator, 793 relative -, 682, 832 relative local -, 780 relative punctual -, 780 slave, 759 standardized -, 877 stemma, 758 weight -, 794 tempo-intonation field, 686 TempoOperator, 829 temporal box, 979 period, 456 relation, 985 tenor tone, 319
INDEX tensed movement, 646 tension, 503, 786 contrapuntal -, 646 global -, 822 harmonic -, 586, 587 tensor product, 1078 affine -, 1094 Terhardt, Ernst, 1053 terminal, 1121 terminology, 248 territory, 719 tesselating chord, 377 tesselation isotypic -, 376 test EEG -, 638 operator, 791 Turing -, 955 Wilcoxon -, 640 tetractys, 33, 1049 tetradic interpretation, 337 tetrahedron, 4, 7 text analysis, 741 textual abstraction, 440 meaning, 400 predicate, 544, 552 representation, 937 semioses category of -, 409 textuality, 406–424 texture four part -, 995 The Sonic Language of Myth, 458 theme, 331, 503 basic -, 246 paradigmatic -, 272, 473 Reti’s definition of a -, 491 theorem, 1133 complement -, 254 counterpoint -, 649, 653 Fourier’s -, 1019 Mason’s -, 130 Mason-Mazzola -, 130
INDEX modulation -, 572 theory American jazz -, 534 catastrophe -, 277, 604 classification -, 999 contour -, 332 counterpoint -, 936, 1008 degree -, 531 function -, 531 global -, 269 group -, 259 music -, 813 of ambiguity, 300 of narrativity, 934 Palestrina–Fux -, 655 performance -, 387, 393 rhythmical modulation -, 612 set -, 305 sonata -, 603 stemma -, 895, 911 string quartet -, 993 substitution -, 1048 valence -, 1035 wavelet -, 1025 thesis world-antiworld -, 604 Thiele, Bob, 733 thinking, 24 by doing, 31, 33 music, 24 operationalized -, 196 thinking music, 25 third, 502 axis, 113 chain, 318, 532, 820 closure, 319 interpretation, 319 minimal -, 319 weak -, 319 comma, 325 class, 325 coordinate, 1032
1331 degree tonality, 548 distance, 622 major -, 73, 1031 weight, 820 Thom, Ren´e, 196, 277 3D vision, 439 threshold global -, 819 local -, 819 tie, 720 Tierny, Miles, 180, 435 tiling lattice, 517 timbral poetics, 295 time, 5, 411, 1029 -slice, 307 -span reduction (=TSR), 457 dilatation, 83 event, 674 generator, 936 grouping, 118 interval, 83 logical -, 611 material -, 611 mental -, 664 onset -, 1013 ontology, 936 physical -, 664 regularity, 116 reverberation -, 1016 series, 856 signature, 82, 768 span, 83 reduction, 752 stretching, 789 structure self-similar -, 964 telling -, 934 told -, 934 timing micro -, 270 microstructure, 871 Tinctoris, Johannes, 629 Todd operator, 752 Todd, Neil McAgnus, 674, 739, 742, 744, 755
1332 told time, 934 tolerance, 779, 827 Ton, 618 tonal ambiguity, 601 function, 304, 323, 544 value, 544 inversion, 148, 952 tonalities admitted -, 566 tonality, 304, 323, 502, 531, 544, 551 harmonic minor -, 560 major -, 560, 582 melodic minor -, 560 minor -, 582 natural minor -, 560 Pythagorean -, 561 reference -, 546 third degree -, 548 tone recitation -, 319 space notched -, 1048 tenor -, 319 tone parameters mental -, 79 physical -, 81 tonic, 319, 502, 541, 545 seventh, 508 tonical, 323 Tonort, 618 tonotopy, 1045 tool paradigmatic -, 953 top space, 715 top-down, 757 topic, 43, 400 topographic cube, 19, 36 topographical navigation, 21 topography, 9, 137 local -, 19 local character of -, 27 musical -, 19 topological
INDEX meaning, 192 space, 1145 irreducible -, 1110 topology, 43, 191, 199, 275 algebraic -, 200 associated -, 1154 base for a -, 1146 coherent -, 1146 coinduced -, 1146 colimit -, 1146 combinatorial -, 310 discrete -, 1145 dominance -, 283, 488 epsilon -, 483 epsilon gestalt -, 484 extension -, 521 finite cover -, 430 Grothendieck -, 180, 430, 1129 harmonic -, 538 Hausdorff -, 1147 indiscrete -, 1145 Lawvere–Tierny -, 435 limit -, 1146 maximal meter nerve -, 460 maximal meter -, 329, 459 on gestalt spaces, 479 on motif spaces, 479 product -, 1146 quotient -, 1146 quotient dominance -, 283 relative -, 1146 relative motivic -, 486 specialization -, 489 subbase for a -, 1146 uniform -, 1147 weak -, 1146 Zariski -, 199, 293 topor, 1139 topos, 3, 10, 23, 1128 Boolean -, 1134 Grothendieck -, 1130 hyperouranios -, 23 logic, 530 of conversation, 995
INDEX Platonic -, 178 topos-theoretic background of modulation, 568 toroidal sequence shape type, 471 shape type, 471 torsion group, 1070 torsion-free rank, 1070 torus tangent -, 621 TOS, 736 total, 1059 Tr¨ aumerei, 818, 857, 860, 899, 927 trace objective -, 121 track, 307 tradition, 401 contrapuntal -, 243, 1052 performance -, 907 Pythagorean -, 24, 1049 traditional musicology, 24 transcendence, 23 transform Fourier -, 1025 TransforMaster, 953 transformation, 492, 495, 935 control of -, 244 natural -, 1118 of sound, 145 plane -, 949 Riemann -, 384 score, 948 symmetry -, 305, 306 transformational approach, 249 invariance, 276, 332 meaning, 193 Transici´ on II, 152, 394 transitional structure, 564 transitive, 1059 action, 1067 transitivity, 280 translation, 159, 1090
1333 part, 1090 transposability, 203 transposition, 139, 160, 276, 624 limited -, 151 transvection, 144, 160 tree, 407 stemma -, 802 triad, 106, 502 augmented -, 321 diminished -, 321 major -, 321 minor, 321 triadic degree interpretation, 320 interpretation, 337, 548, 553, 566 trigeneric monoid, 540 trigger motion -, 738 trill, 88, 760 True, 1132 truth denotator, 407 TTO operator, 253 tube Eustachian -, 1037 Tudor, David, 306 tuning, 304 just -, 1032 just-tempered -, 1033 justest -, 560 mediante -, 1033 Pythagorean -, 325, 581, 1032 tempered -, 1032 12-tempered -, 106 well-tempered -, 1032 turbidity, 147 Turing machine, 670 test, 955 turning point, 565 12-tempered scales common -, 113 tuning, 106
1334 two-dimensional alteration, 950 TX7 Yamaha -, 639 type, 50, 1139 casting, 405 change, 402 coproduct -, 53 form -, 64 of a cellular hierarchy, 716 powerset -, 54 product -, 52 shape -, 468 synonymy -, 54 types ordering on -, 90 typology of forms, 65 recursive, 56 recursive -, 48 U Uhde, J¨ urgen, 302, 567, 604 Unbewusstes, 643 uncertainty relation, 299, 516 uncontrolled paradigmatics, 201 underlying symmetry, 155 understanding, 395, 997 musical works, 393 systematic -, 994 unfolding, 937 unicorned view, 906 uniform topology, 1147 uniformity, 1147 union, 1058 unit musical -, 106 unity, 41, 48, 56 universal ordering, 44 universe, 1116 of structure, 400 of topics, 400 unordered p-space interval, 252 pc interval, 253
INDEX Ursatz, 400 Ussachevsky, Vladimir, 306 Utai, 416 utai, 14 V Val´ery, Paul, 15, 47, 187, 663, 670, 681, 696, 711, 774 valence, 1046 supporting -, 1047 theory, 1035 valid sentence, 1132 validation operator, 424 valuation interpretative -, 15 value change, 769 initial -, 683 matrix, 925 participation -, 639 Var`ese, Edgar, 392, 394 variable bound, 1136 causal-final -, 927 explanatory -, 877 free, 1136 inclusion frequency of -, 888 individual -, 1135 predicate -, 1135 propositional -, 1131 variable address, 61 variation, 492, 495, 618, 950 melodic -, 959 of the perspective, 182 pressure -, 1013 principle, 394 Variationen f¨ ur Klavier, 394, 860 variations cycle of -, 956 varieties of sounds, 284 variety instrumental -, 673 vector, 1083 analytical -, 876
INDEX field, 1156 instrumental -, 1005 interval -, 253, 257 interval-class -, 249 invariance -, 254 prime -, 1033 shaping -, 876 space, 1085 spectral participation -, 639 spectral -, 1001 spectral participation -, 638 velocity, 739, 948, 1030 instantaneous -, 30 bow -, 1002 concept of instantaneous -, 30 physical -, 30 verbal description, 756 Vercoe, Barray, 918 Verdier, Jean-Louis, 431 Verillo, Ronald, 738 Vers la flamme, 587, 964 verse matrix, 261 poetics, 303 vertex, 1062, 1148 final -, 802 vertical grid vector, 951 poetical function, 942 poeticity, 261 vestibular system, 739 vibrato, 288, 290, 738, 1002 parameter, 1003 Vieru, Anatol, 257, 383 view, 968 kernel -, 826 unicorned -, 906 Villon, Fran¸cois, 138, 242, 243 viola, 993 violin, 993 family, 295, 997, 1009 hierarchy, 722
1335 violoncello, 993 virtual acoustics, 850 visual navigation, 439 programming language, 967 semiotic marker, 981 visualization, 917, 918 vocabulary dodecaphonic -, 243 extension, 45 switch, 250, 293 Vogel chromatic, 582 Vogel, Martin, 115, 506, 512, 517, 576, 582, 1032, 1167 voice, 619 crossing, 619 instrumental -, 269 leading, 304 pedal -, 608 stationary -, 608 Voisin, Fr´ed´eric, 986 Volkswagen Foundation, 807 volume, 230 Vordergrund, 503 vowel, 286 Vuza class, 380 rhythm, 380 Vuza, Dan Tudor, 83, 257, 328, 376, 380 W W, 605 Wagner, Richard, 259, 814 walking, 738 wall bottom -, 768 Ward, Artemus, 747 wave, 1018 waveguide, 1027 wavelet, 289, 1025 frame, 290 Meyer -, 1026
1336 Morlet -, 1025 Murenzi -, 1025 theory, 1025 wavelet-transformed, 1025 weak topology, 1146 Weber-Fechner law, 1029 Webern, Anton von, 149, 150, 152, 198, 248, 301, 394, 860, 907 Wedderburn, Joesph, 1088 wedge crescendo -, 778 Wegner, Peter, 29 weight, 726, 742, 744, 752, 775, 818 analytical -, 666, 671, 785 class -, 346 combination, 827 continuous -, 775 function default -, 587 P´olya -, 232 product -, 232 harmonic -, 587, 786 induced nerve -, 460 inverted -, 827 metrical -, 455, 456, 785 mixed -, 815 motivic -, 496, 785 nerve -, 460 onset -, 116 profile, 267 recombination, 776 simplicial -, 346 simplicial metrical -, 329 sober -, 460 system, 768 tempo, 794 third -, 820 watcher, 827 weighted graph, 292 well-ordered, 1060 well-tempered modulation, 571 tuning, 1032 Well-Tempered Piano, 303 Weyl, Hermann, 196
INDEX WFR, 457 whatness, 23 whereness, 23 White, Andrew, 694 whole, 301, 334 whole-tone scale, 657 Whymper, Edward, 183 Wicinski, A.A., 756 Widmer, Gerhard, 744 Wieland, Renate, 302 Wieser, Heinz-Gregor, 637 Wilcoxon test, 640 wild problem, 913 Wille, Rudolf, 3, 135, 551 window oval -, 1037, 1041 power -, 638 round -, 1041 Winson, Jonathan, 642 Wittgenstein, Ludwig, 43, 397 WLOG, 131 Wohltemperiertes Klavier, 141 Wolff, Christian, 306 word, 71, 1064 monoid, 1064 work, 12, 14 identity of a -, 16 motivic -, 338 production of a -, 13 world, 560, 605 world-antiworld thesis, 604 wrap form morphisms, 402 wrapped as local composition, 108 wreath product, 1069 writing field -, 969 Wulf, Bill, 35, 809 Wyschnegradsky, Ivan, 110 X Xenakis, Iannis, 33, 258 Y YAMAHA, 1027
INDEX Yamaha, 834, 849, 1022 CX5M, 639 RX5, 269, 955 TX7, 639 TX802, 269, 289, 293, 955 Yoneda embedding, 1091, 1120 lemma, 171, 341, 393 pair, 1137 philosophy, 109, 175, 184, 566, 997 subcategory, 1137 Yoneda, Nobuo, 175, 299, 392, 997 Z Z-addressed motives contrapuntal meaning of -, 120 Z-relation, 257 Zahorka, Oliver, 764, 807, 833 Zariski site, 1112 tangent, 128 space, 1112 topology, 199, 293 Zariski, Oskar, 199 Zarlino, Gioseffo, 147 Zauberfl¨ ote, 598 Zermelo, Ernst, 1061 zero address, 61, 62 zig-zag motivic -, 339, 941 Zurechth¨oren, 1035 Zurich school, 744
1337
