Mathematics and Computation in Music: First International Conference, MCM 2007, Berlin, Germany, May 18-20, 2007. Revised Selected Papers (Communications in Computer and Information Science)

Communications in Computer and Information Science 37 Timour Klouche Thomas Noll (Eds.) Mathematics and Computation...

Author: Timour Klouche | Thomas Noll

207 downloads 1177 Views 15MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Communications in Computer and Information Science

37

Timour Klouche Thomas Noll (Eds.)

Mathematics and Computation in Music First International Conference, MCM 2007 Berlin, Germany, May 18-20, 2007 Revised Selected Papers

13

Volume Editors Timour Klouche Staatliches Institut für Musikforschung Preußischer Kulturbesitz Abteilung III Akustik und Musiktechnologie Tiergartenstr. 1, 10785 Berlin, Germany E-mail: [email protected] Thomas Noll Escola Superior de Musica de Catalunya Departament de Teoria i Composicio i Direccio C. Padilla 155-Edifici L’Auditori 08013 Barcelona, Spain E-mail: [email protected]

Library of Congress Control Number: Applied for CR Subject Classification (1998): H.5.5, J.5, I.1, G.2, I.6 ISSN ISBN-10 ISBN-13

1865-0929 3-642-04578-2 Springer Berlin Heidelberg New York 978-3-642-04578-3 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. springer.com © Springer-Verlag Berlin Heidelberg 2009 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper SPIN: 12699315 06/3180 543210

Preface

This volume comprises a selection of papers presented at the first International Conference on Mathematics and Computation in Music – mcm2007. The conference took place at the Staatliches Institut für Musikforschung PK – National Institute for Music Research in Berlin during May 18–20, 2007 and was jointly organized by the National Institute for Music Research Berlin and the Society of Mathematics and Computation in Music. The papers were selected for the conference by the program committee and classfied into talks and posters. All papers underwent further selection, revision and elaboration for this book publication. The articles cover a research field which is heterogeneous with respect to content, scientific language and methodology. On one hand, this reflects the heterogeneity and richness of the musical subject domain itself. On the other hand, it exemplifies a tension which has been explicitly intended by both the organizers and the founders of the society, namely to support the integration of mathematical and computational approaches to music theory, composition, analysis and performance. The subdivision into three parts reflects the original structure of the program. These parts are opened by invited papers and followed by talks and posters. We are grateful to the Deutsche Forschungsgemeinschaft for sponsoring the conference, to the publishing team at Springer, to the collaborators at the National Institute for Music Research and the Museum of Musical Instruments, to Hans-Reinhard Wirth, head of the Department of Music Technology and Acoustics, to Conny Restle, director of the Musikinstrumenten-Museum (Museum of Musical Instruments), and to Thomas Ertelt, director of the Institute. Without his foresight and commitment, you would not hold this book in your hands.

November 2008

Timour Klouche Thomas Noll

Table of Contents

I-Mathematical Modeling and Computation in Music Invited Lectures Rhythm and Transforms, Perception and Mathematics . . . . . . . . . . . . . . . . William A. Sethares Visible Humour – Seeing P.D.Q. Bach’s Musical Humour Devices in The Short-Tempered Clavier on the Spiral Array Space . . . . . . . . . . . . . . . Elaine Chew and Alexandre Fran¸cois

1

11

Metalanguage and Representation Category-Theoretic Consequences of Denotators as a Universal Data Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gérard Milmeister

19

Normal Form, Successive Interval Arrays, Transformations and Set Classes: A Re-evaluation and Reintegration . . . . . . . . . . . . . . . . . . . . . . . . . Ciro Scotto

25

Melodic, Motivic and Metric Levels of Description A Model of Musical Motifs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Torsten Anders Melodic Clustering within Motivic Spaces: Visualization in OpenMusic and Application to Schumann’s Tr¨ aumerei . . . . . . . . . . . . . . . . . . . . . . . . . . Chantal Buteau and John Vipperman Topological Features of the Two-Voice Inventions . . . . . . . . . . . . . . . . . . . . Kamil Adilo˘glu and Klaus Obermayer Comparing Computational Approaches to Rhythmic and Melodic Similarity in Folksong Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anja Volk, J¨ org Garbers, Peter van Kranenburg, Frans Wiering, Louis Grijp, and Remco C. Veltkamp

52

59 67

78

Harmonic Levels of Description Automatic Modulation Finding Using Convex Sets of Notes . . . . . . . . . . . Aline Honingh

88

VIII

Table of Contents

On Pitch and Chord Stability in Folk Song Variation Retrieval . . . . . . . . J¨ org Garbers, Anja Volk, Peter van Kranenburg, Frans Wiering, Louis Grijp, and Remco C. Veltkamp

97

Bayesian Model Selection for Harmonic Labelling . . . . . . . . . . . . . . . . . . . . Christophe Rhodes, David Lewis, and Daniel M¨ ullensiefen

107

The Flow of Harmony as a Dynamical System . . . . . . . . . . . . . . . . . . . . . . . Peter Giesl

117

Tonal Implications of Harmonic and Melodic Tn -Types . . . . . . . . . . . . . . . Richard Parncutt

124

Computational Models in Music Psychology Calculating Tonal Fusion by the Generalized Coincidence Function . . . . . Martin Ebeling Predicting Music Therapy Clients’ Type of Mental Disorder Using Computational Feature Extraction and Statistical Modelling Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geoff Luck, Olivier Lartillot, Jaakko Erkkil¨ a, Petri Toiviainen, and Kari Riikkil¨ a Nonlinear Dynamics, the Missing Fundamental, and Harmony . . . . . . . . . Julyan H.E. Cartwright, Diego L. Gonz´ alez, and Oreste Piro

140

156

168

Computational Models for Musical Instruments Dynamic Excitation Impulse Modification as a Foundation of a Synthesis and Analysis System for Wind Instrument Sounds . . . . . . . . . . . Michael Oehler and Christoph Reuter Non-linear Circles and the Triple Harp: Creating a Microtonal Harp . . . . Eleri Angharad Pound

189

198

Comparative Computational Analysis Applying Inner Metric Analysis to 20th Century Compositions . . . . . . . . Anja Volk

204

Tracking Features with Comparison Sets in Scriabin’s Study op. 65/3 . . . Atte Tenkanen

211

Computer Aided Analysis of Xenakis-Keren . . . . . . . . . . . . . . . . . . . . . . . . . Kamil Adilo˘glu and G. Ada Tanir

220

Table of Contents

IX

Automated Extraction of Motivic Patterns and Application to the Analysis of Debussy’s Syrinx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Olivier Lartillot

230

Pitch Symmetry and Invariants in Webern’s Sehr Schnell from Variations Op.27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elaine Chew

240

Computational Analysis Workshop: Comparing Four Approaches to Melodic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chantal Buteau, Kamil Adilo˘glu, Olivier Lartillot, and Christina Anagnostopoulou

247

Posters Computer-Aided Investigation of Chord Vocabularies: Statistical Fingerprints of Mozart and Schubert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eva Ferkov´ a, Milan Zd´ımal, and Peter Sidl´ık The Irrelative System in Tonal Harmony . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miroslaw Majchrzak

250 257

II-Mathematical Approaches to Music Analysis and Composition Invited Lectures Mathematics and the Twelve-Tone System: Past, Present, and Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robert Morris Approaching Musical Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John Rahn A Transformational Space for Elliott Carter’s Recent Complement-Union Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John Roeder

266 289

303

Mathematical Approaches to Composition Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tom Johnson

311

From Mathematica to Live Performance: Mapping Simple Programs to Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Katarina Miljkovic

318

Nonlinear Dynamics of Networks: Applications to Mathematical Music Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jonathan Owen Clark

330

X

Table of Contents

Mathematical Approaches to Musical Analysis and Performance Form, Transformation and Climax in Ruth Crawford Seeger’s String Quartet, Mvmt. 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Edward Gollin

340

A Local Maximum Phrase Detection Method for Analyzing Phrasing Strategies in Expressive Performances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eric Cheng and Elaine Chew

347

Klumpenhouwer-Networks Subgroup Relations among Pitch-Class Sets within Tetrachordal K-Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jerry G. Ianni and Lawrence B. Shuster K-Net Recursion in Perlean Hierarchical Structure . . . . . . . . . . . . . . . . . . . Gretchen C. Foley Webern’s Twelve-Tone Rows through the Medium of Klumpenhouwer Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Catherine Nolan Isographies of Pitch-Class Sets and Set Classes . . . . . . . . . . . . . . . . . . . . . . Tuukka Ilom¨ aki

354 365

375 386

Leonard Euler at the Crossroads of Music Theory The Transmission of Pythagorean Arithmetic in the Context of the Ancient Musical Tradition from the Greek to the Latin Orbits During the Renaissance: A Computational Approach of Identifying and Analyzing the Formation of Scales in the De Harmonia Musicorum Instrumentorum Opus (Milan, 1518) of Franchino Gaffurio (1451–1522) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Herbert Kreyszig and Walter Kreyszig Combinatorial and Transformational Aspects of Euler’s Speculum Musicum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Edward Gollin

392

406

Posters Structures Ia Pour Deux Pianos by Boulez: Towards Creative Analysis Using OpenMusic and Rubato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yun-Kang Ahn, Carlos Agon, and Moreno Andreatta

412

Table of Contents

XI

The Sieves of Iannis Xenakis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimitris Exarchos

419

Tonal, Atonal and Microtonal Pitch-Class Categories . . . . . . . . . . . . . . . . . Fernando Gualda

430

Using Mathematica to Compose Music and Analyze Music with Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christopher W. Kulp and Dirk Schlingmann

441

III-Mathematical Approaches to Music Theory Invited Lectures A Diatonic Chord with Unusual Voice-Leading Capabilities . . . . . . . . . . . . Norman Carey

449

Mathematical and Musical Properties of Pairwise Well-Formed Scales . . . David Clampitt

464

Eine Kleine Fourier Musik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emmanuel Amiot

469

Towards New Music-Theoretical Concepts WF Scales, ME Sets, and Christoffel Words . . . . . . . . . . . . . . . . . . . . . . . . . Manuel Dom´ınguez, David Clampitt, and Thomas Noll Interval Preservation in Group- and Graph- Theoretical Music Theories: A Comparative Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robert Peck Pseudo-diatonic Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Franck Jedrzejewski

477

489 493

Dasian, Diatonic and Dodecaphonic Set Theory Affinity Spaces and Their Host Set Classes . . . . . . . . . . . . . . . . . . . . . . . . . . José Oliveira Martins

499

The Step-Class Automorphism Group in Tonal Analysis . . . . . . . . . . . . . . Jason Yust

512

A Linear Algebraic Approach to Pitch-Class Set Genera . . . . . . . . . . . . . . Atte Tenkanen

521

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

531

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

533

Rhythm and Transforms, Perception and Mathematics William A. Sethares University of Wisconsin, Department of Electrical and Computer Engineering, 1415 Engineering Drive, Madison WI 53706 [email protected]

Abstract. People commonly respond to music by keeping time, tapping to the beat or swaying to the pulse. Underlying such ordinary motions is an act of perception that is not easily reproduced in a computer program or automated by machine. This paper outlines the flow of ideas in Rhythm and Transforms (Sethares 2007), which creates a device that can “tap its foot” along with the music. Such a “beat finding machine” (illustrated in Fig. 1) has implication for music theory, on the design of sound processing electronics such as musical synthesizers, on the uses of drum machines in recording and performance, and on special effects devices. The beat finder provides a concrete basis for a discussion of the relationship between the mind’s processing of temporal information and the mathematical techniques used to describe and understand regularities in data. Extensive sound examples (Sethares 2008) demonstrate beatbased signal processing techniques, methods of musical (re)composition, and new kinds of musicological analysis.

1

What Is Rhythm?

How can rhythm be described mathematically? How can it be detected automatically? People spontaneously clap in time with a piece of music, and can effortlessly internalize and understand rhythmic phenomena – but it is tricky to create a computer program that can keep time to the beat. Teaching the computer to synchronize to music requires both interesting mathematics and unusual kinds of signal processing. There are many different ways to think about and notate rhythmic patterns. A variety of different notations, tablatures, conventions, and illustrations are used throughout Rhythm and Transforms to emphasize the distinction between symbolic notations (which accentuate high level information about a sound) and acoustical notations (which allow the sound to be recreated). Surveying the musics of the world shows many different ways of conceptualizing the use of rhythmic sound: for instance, the timelines of West Africa, the clave of Latin America (illustrated in Fig. 2), and the tala of India. T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 1–10, 2009. c Springer-Verlag Berlin Heidelberg 2009

2

W.A. Sethares

Fig. 1. A foot-tapping machine designed to mimic people’s ability to synchronize to complex rhythmic sound must “listen” to the sound, locate the underlying rhythmic pulse, anticipate when the next beat timepoint will occur, and then provide an output

2

time 3

1

L

H H

2

time

time 3

1

L

H (a)

4

4

(b)

H

H

(c)

4

4

H 3 2

1

L

L H

3 2

1

Fig. 2. The son clave rhythm is arranged in necklace notation; the 3-2 clave begins at the larger arrow while the 2-3 clave begins at the smaller arrow. (a) The beats of the two 4 measures are indicated inside the circle along with the 16 timepoints that represent 4 the tatum (short for “temporal atom,” the fastest pulsation present in the music) (b) repeats the basic clave in the outer circle and shows how various other rhythmic parts complement, augment, and can substitute for the straight clave pattern. The middle circle shows the c´ ascara. The inner circle shows a bell pattern with low (L) and high (H) bells. (c) shows the guanguanc´ o (rumba) clave.

2

Auditory Perception

The auditory system is not simple. Underlying the awareness of rhythmic sounds are basic perceptual laws that govern the recognition of auditory boundaries, events, and successions. Research into the mechanisms of perception sheds light on the physical cues that inspire rhythmic patterns in the mind of the listener. These cues help distinguish features of the sound that are properties of the signal (such as amplitude and frequency) from those that are properties of the perceiving mind (such as loudness and pitch). Just as pitch is a perceptual correlate of frequency and loudness is a perceptual correlate of amplitude, the “beat” is a perceptual correlate. A major part of Rhythm and Transforms is the search for physically measurable correlates of the beat perception. Fig. 3 illustrates this idea.

Rhythm and Transforms, Perception and Mathematics

memory stream of incoming stimuli pressure wave in air

selection and filtering processes

3

long term

short organized patterns term and perceptions memory

expectation and attention

Fig. 3. Perception of sound is not a simple process; it begins with a physical waveform and may end with a high level cognitive insight (for example, understanding the meaning of a sound). There are constant interactions between long term memory, attention and expectation, and the kinds of patterns formed. There are also constant interactions between memory, attention, expectation, and the ways that the raw information is selected and filtered. The time span over which the short term memory organizes perceptions is called the perceptual present.

3

Transforms

Transforms model a signal as a collection of waveforms of a particular form: e.g., sinusoids for the Fourier transform, mother wavelets for the wavelet transforms, periodic basis functions for the periodicity transforms. All of these methods are united in their use of inner products as a basic measure of the similarity and dissimilarity between signals, and all may be applied (with suitable care) to problems of rhythmic identification. A transform must ultimately be judged by the high frequencies = blue light middle frequencies = yellow light complex light wave

complex sound wave

low frequencies = red light

prism

Digitize Waveform in Computer

Fourier Transform

high low frequencies frequencies middle = treble = bass frequencies = midrange

Fig. 4. Just as a prism separates light into its simple constituent elements (the colors of the rainbow), the Fourier Transform separates sound waves into simpler sine waves in the low (bass), middle (midrange), and high (treble) frequencies. Similarly, the auditory system transforms a pressure wave into a spatial array that corresponds to the various frequencies contained in the wave.

4

W.A. Sethares

insight it provides and not solely by the elegance of its mathematics. Transforms and the various algorithms derived from them (for instance, the phase vocoder and short time Fourier transform) are mathematical operations that have no understanding of psychoacoustics or of the human perceptual apparatus. Thus a square wave may be decomposed into its appropriate harmonics by the Fourier transform irrespective of the time axis. It makes no difference whether the time scale is milliseconds (in which case we would hear pitch) or on the order of seconds (in which case we would hear rhythm). It is, therefore, necessary to explicitly embed psychoacoustical insights into the mathematics (Terhardt (1982) and Parncutt (1994) provide two well known examples) in order to make more practical and effective models. Mathematics is perceptually agnostic – it is only the interpretation of the mathematics that makes a psychoacoustic model. Fig. 4 presents one such interpretation.

4

Adaptive Oscillators

One way to model biological clocks is with oscillators that can adapt their period and phase to synchronize to external events. To be useful in the beat tracking problem, the oscillators must be able to synchronize to a large variety of possible input signals and they must be resilient to noises and disturbances. Clock models can be used to help understand how people process temporal information and the models are consistent with the importance of regular successions in perception. One simple situation is shown in Fig. 5. θ1 swing left

f1

swing right coupling

swing left

f2 swing right

θ2

synchronization achieved

Fig. 5. When two oscillators are coupled together, their frequencies may influence each other. When the outputs synchronize in frequency and lock in phase, they are said to be entrained. The musicians represent one “oscillator” and the beat finding machine represents a second. When they synchronize, the machine has “found the beat.”

5

Statistical Models

The search for rhythmic patterns can take many forms. Models of statistical periodicity do not presume that the signal itself is periodic; rather, they assume that there is a periodicity in the underlying statistical distributions. In some cases, the randomness is locked to a known periodic grid on which the statistics


5

Fig. 6. The simplest useful model is a generalization of the “ball and urn” problem where a collection of urns are mounted on a carousel. Each time a ball is removed from one of the N urns (indicated by the arrow), the platform rotates, bringing a new urn into position. When N is unknown, it is necessary to infer both the percentage of balls in each urn and the number of urns (the periodicity) from the experiments. In terms of the periodicity-finding goals of beat tracking, inferring N is often more important than inferring the individual percentages of black or white balls.

are defined. In other cases, the random fluctuations may be synchronized to a grid with unknown period. In still other cases, the underlying rate or period of the repetition may itself change over time. The statistical methods relate the signal (for example, a musical performance) to the probability distribution of useful parameters such as the period and phase of a repetitive phenomenon. One simple model is shown in Fig. 6.

6

Automated Rhythm Analysis

Just as there are two kinds of notations for rhythmic phenomenon (the symbolic and the acoustical), there are two ways to approach the detection of rhythms; from a high level symbolic representation (such as an event list, musical score, or standard MIDI file) or from a acoustical representation such as a direct encoding in a .wav file. Both aspire to understand and decompose rhythmic phenomena, and both exploit a variety of technologies such as the transforms, adaptive oscillators, and statistical techniques. A preliminary discussion of the rhythmic parsing of symbolic sequences is then generalized by incorporating perceptually motivated feature vectors to create viable beat detection algorithms for audio. The performance of the various methods is compared in a variety of musical passages. A visual representation is shown in Fig. 7.

6

W.A. Sethares

(a)

(c)

τ

ω

σS2

σL2 σt

(b)

T

(d)

3

4 seconds

5

3

4 seconds

5

Fig. 7. A few seconds of four feature vectors of Pieces of Africa by the Kronos Quartet are shown. The estimated beat times (which correctly locate the pulse in cases (a), (c), and (d)) are indicated by the bumps in the curve σt that are superimposed over each vector. The three timing parameters T (period), τ (phase), and δT (change in period, not shown) are estimated from the feature vectors.

7

Beat-Based Signal Processing

There is an old adage in signal processing: if something is known about a signal, use the knowledge. The ability to detect beat timepoints is information about the naturally occurring points of division within a musical signal and it makes sense to exploit these points when manipulating the sound. Signal processing techniques can be applied on a beat-by-beat basis or the beat can be used to control the parameters of a continuous process. Applications include beatsynchronized special effects, spectral mappings with harmonic and/or inharmonic destinations (as illustrated in Fig. 8), and a variety of sound manipulations that exploit the beat structure. Illustrative sound examples can be heard online (Sethares 2008). There are two ways to exploit beat information. First, each beat interval may be manipulated individually and then the processed sounds may be rejoined. To the extent that the waveform between two beat locations represents a complete unit of sound, this is an ideal application for the Fourier transform since the beat interval is analogous to a single “period” of a repetitious wave. The processing may be any kind of filtering, modulation, or signal manipulation in either the time or frequency domain. For example, Fig. 9 shows the waveform of a song partitioned into beat-length segments by a series of envelopes. Each of the segments can be processed separately and then rejoined. Using envelopes that decay to zero at the start and end helps to smooth any discontinuities that may be introduced.

magnitude


spectrum of original (source) sound

f1

f2

f3

f4

g2 g3

g4

f5

f6

f7

f9 ...

f8

spectral mapping

source

7

magnitude

destination g1

0

g5 g6

g7

g8 g9 ...

spectrum of destination sound

500

1000 frequency

1500

Fig. 8. In this schematic representation of a spectral mapping, a source spectrum with peaks at f1 , f2 , f3 , . . . is mapped into a destination spectrum with peaks specified at g1 , g2 , g3 , . . .. The spectrum of the original sound (the plot is taken from the G string of a guitar with fundamental at 194 Hz) is transformed by the spectral mapping for compatibility with the destination spectrum. The mapping changes the frequencies of the partials while preserving the energy in each partial, leaving the magnitudes approximately the same.

The second method uses beat locations to control a continuous process. For example, a resonant filter might sweep from low to high over each beat interval. The depth of a chorusing (or flanging) effect might change with each beat. The cutoff frequency of a lowpass filter might move at each beat boundary. There are several commercially available software plug-ins (for example, Camelspace and SFXMachine) that implement such tasks using the tempo specified by the audio sequencer; the performer implicitly implements the beat tracking. Since certain portions of the beat interval may be more perceptually salient than others, these may be marked for special treatment. For example, time stretching by a large factor often smears the attack transients. Since the beat locations are known, so are the likely positions of these attacks. The stretching can be done nonuniformly: to stretch only a small amount in the vicinity of the start of the beat and to stretch a larger amount in the steady state portions between beat locations.

8

W.A. Sethares

waveform beat locations

beat interval

envelopes fade in

fade out

enveloped beat interval

Processing

Fig. 9. A collection of windows separates the waveform into beat intervals, which can be processed independently. After processing, the intervals are windowed again to help reduce clicks and edge discontinuities. The final step (not shown) is to sum the intervals to create a continuous output.

8

Musical Composition and Recomposition

The beats of a single piece may be rearranged and reorganized to create new structures and rhythmic patterns including the creation of beat-based “variations

Fig. 10. This mosaic of Scot Joplin (created from many smaller pictures) presents a visual analog of an audio collage: a piece is deconstructed into beats, and then reconstructed by reordering the beats. A series of sound examples available on the website (Sethares 2008) demonstrate this.


9

on a theme.” For example, it is easy to remove every fourth beat. The effect is to change a piece in 4/4 time into 3/4, as is demonstrated by transforming Scott Joplin’s “Maple Leaf Rag” into the “Maple Leaf Waltz,” which can be heard on the author’s website (Sethares 2008). Similarly, two pieces may be merged in a time-synchronous manner to create hybrid rhythmic textures that inherit tonal qualities from both. See Fig. 10.

9

Musical Analysis via Feature Scores

Traditional musical analysis often focuses on the use of note-based musical scores. Since scores only exist for a small subset of the world’s music, it is helpful to be able to analyze performances directly, to probe both the symbolic and the acoustical levels. For example, Figure 11 displays a skeletal tempo score that shows how time evolves in several different performances of the Maple Leaf Rag. More generally, Banuelos (2005) details several psychoacoustically motivated feature scores that are particularly useful in an analysis of Alban Berg’s Violin Concerto, subtitled Dem Andenken eines Engels, that merges standard analytical techniques with new feature scores in an elegant and insightful way. By conducting analyses in a beat-synchronous manner, it is possible to track changes in a number of psychoacoustically significant musical variables. This allows the automatic extraction of new kinds of symbolic feature scores directly from the performances.

Beat Interval T (sec)

0.4 0.35 0.3 0.25 0.2 100

200

300 400 Beat Number

500

600

700

Fig. 11. A tempo score is a plot of the duration of each beat vs. the beat number; it shows how the tempo changes over time. In this plot, 29 performances of the Maple Leaf Rag are played in a variety of tempos ranging from T = 0.22 to T = 0.4 sec per beat. The plot shows how the tempo of each performance varies over time.

10

Conclusions

The ability to decompose a piece into its primitive beat-elements is a surprisingly powerful technique for musical analysis, for musical composition (such as

10

W.A. Sethares

beat-synchronous sound collages) and for audio signal processing (where the beat boundaries provide a natural partitioning of the signal). Rhythm and Transforms (Sethares 2007) contrasts two ways of understanding temporal regularities in the world around us: directly via perception and indirectly via mathematical analysis. “Rhythm” alludes to the perceptual apparatus that allows people to effortlessly observe and understand rhythmic phenomena while “transforms” evokes the mathematical tools used to detect regularities and to study patterns. The book develops a variety of such applications and provides a wealth of sound examples (Sethares 2008) that concretely demonstrate the efficacy and the limitations of the techniques.

References Banuelos, D.: Beyond the Spectrum of Music, DMA Thesis, University of Wisconsin (2005) Parncutt, R.: A perceptual model of pulse salience and metrical accent in musical rhythms. Music Perception 11(4) (1994) Sethares, W.: Rhythm and Transforms. Springer, Heidelberg (2007) Sethares, W.: Sound examples accompanying this article can be heard (2008), http://eceserv0.ece.wisc.edu/~ sethares/RT.html Terhardt, E., Stoll, G., Seewann, M.: Algorithm for extraction of pitch and pitch salience from complex tonal signals. Journal of the Acoustical Society of America 71, 679–688 (1982)

Visible Humour – Seeing P.D.Q. Bach's Musical Humour Devices in The Short-Tempered Clavier on the Spiral Array Space Elaine Chew1 and Alexandre François2 1

University of Southern California Viterbi School of Engineering Epstein Dep of Industrial and Systems Engineering, Hsieh Dep of Electrical Engineering 2 Computer Science Department Los Angeles, California, USA

Abstract. We explore the use of the Spiral Array spatial visualization of tonal evolution through time for the visual analysis of P.D.Q. Bach's The ShortTempered Clavier. In particular, we analyze situations in which we can see some of the humour devices abstracted in an earlier study by David Huron. We conclude that although we can see a good number of Schickele's humour devices – such as incongruent styles, musically improbable tonality and harmony shifts, and excessive repetition – we do not yet have sufficient information to form a robust computer-based method for detecting musical humour.

1 Introduction In this paper, we explore the use of the Spiral Array spatial visualization of pitches, chords, and keys, and the tracking of their dynamic evolution through time for the visual analysis of musical humour. We visually analyze, using the Music on the Spiral Array . Real-Time (MuSA.RT) system (Chew & François 2005), pieces from The Short-Tempered Clavier – Preludes and Fugues in All the Major and Minor Keys Except the Really Hard Ones (S. 3.14159, easy as), composed by P.D.Q. Bach, a.k.a. Peter Schickele. In particular, we focus on the points of departure from musical expectation (and good taste), and determine if these strategies for violating expectation and thus eliciting laughter can be traced visibly in MuSA.RT. This paper is inspired by Huron's work on music-engendered laughter (Huron 2004), where he systematically identifies and classifies laughter-eliciting musical devices in P.D.Q. Bach's music – each involving expectation violation. P.D.Q. Bach is Peter Schickele, a classically trained contemporary composer who has built a career on writing laughter-eliciting musical compositions under his pseudonym. Huron's article presents the results of a study of 629 instances of laughter-eliciting music by Schickele. Huron states that, unlike most musical humourists, "beyond the visual gags and language-based humor, most of Schickele's humour devices are to be found in the core musical domains of pitch, time, and timbre.'' Excluding visual and language-based comedy, he proceeds to identify nine categories of musical devices that elicit laughter: (1,2) incongruity of instrumental sounds and of musical genres, (3,4) musically improbable tonality and metric shifts, (5,6) implausible delays and excessive repetition, (7) incompetence cues, and (8,9) incongruous quotations and misquotations. T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 11–18, 2009. © Springer-Verlag Berlin Heidelberg 2009

12

E. Chew and A. François

In this paper, we will show the visual counterparts of some of these humourevoking devices, particularly the ones based on pitch and time, using examples from The Short-Tempered Clavier.

2 MuSA.RT and Visualization The tool we use for our explorations is MuSA.RT, an interactive music analysis and visualization system, based on Chew's Spiral Array mathematical model for tonality (Chew 2000), that is capable of tracking and displaying the trajectory of the tonal content and context of music in real-time during a live performance. The Spiral Array is a geometric representation of tonal entities (pitches, chords, and keys) in threedimensional space. The chords and keys are determined via the centers of effect (c.e.'s) generated by the notes in the piece, which are mapped to their respective pitch class representations in the Spiral Array using a pitch spelling algorithm. In MuSA.RT, each c.e. is located at the aggregate position of the notes present and in the recent past, where the effect of notes past are decreasing according to a linear decay function. MuSA.RT stands for Music on the Spiral Array . Real-Time. The system takes as input MIDI from, say, a piano keyboard. The tonal structures are computed and mapped to the display instantaneously in response to the MIDI input. The virtual viewpoint can be manipulated via a game controller or an automatic pilot. To coordinate the concurrent processing of the multiple data streams, the system is implemented in the style of François’ Software Architecture for Immersipresence (François 2004). MuSA.RT maps musical performances to distinctive spatio-temporal paths and patterns inside the Spiral Array that are characteristic of the piece, and of the particular interpretation. It provides a means of visualizing not only pitch clusters, and key finding and chord tracking algorithms and their results, but also the unfolding and history of the patterns over time. When in autopilot, the movement of the spiral is in synchrony with the rhythm of the music. 2.1 Seeing Style Differences The visualizations in MuSA.RT can reveal pitch structures, and to some extent, time structures as well. However, timbre is presently not a feature that is visualized in the system. We show, by the example of the Prelude No. 5 in D minor from The ShortTempered Clavier the differences visible when the same theme is repeated in multiple styles (or genres). In MuSA.RT Opus 2.6, when shown in color, a silver helix shows the outermost spiral on which the pitch classes reside, and the inner and intertwined blue and red helices denote the spirals on which the minor and major keys are represented, respectively. The closest major or minor triad (corresponding to an upward- or a downwardpointing triangle) is highlighted in pink or blue, and its name colored and indicated by a sphere of the same color. The closest major or minor key appears as a sphere on the corresponding major or minor key spiral, and a sphere of the appropriate color marks the letter name of the key. When the chord and key are of the same name, the sphere denoting the chord appears inside the sphere for the key. A violet trail traces the history of the key c.e., and a purple trail that of the chord c.e. Figure 1 shows the complete history of Prelude No. 5 in D minor.

Visible Humour – Seeing P.D.Q. Bach's Musical Humour Devices

13

Fig. 1. Prelude No.5 in D minor (views: center, right)

Fig. 2. Prelude No. 5, first part - arpeggiated chords (views: center, right)

Fig. 3. Prelude No. 5, middle part - eighth note walking bass, two against three (views: center, right)

14


Next, we segment the piece into its three stylistic sections, and visualize each part independently. The initial arpeggiated chords that present the theme create relatively smooth trajectories around D minor, as shown in Figure 2. The middle section with its staccato walking bass and interplay of groupings of two against three traces a more jagged version of the initial trails outlined by the first section – see Figure 3. The final section with the boogie bass, in a faster triplet rhythm, again outlines the same tonal regions, but results in a tangled mess, as shown in Figure 4.

Fig. 4. Prelude No. 5, last part - boogie bass (views: center, right)

3 Expectations Violated In this section, we explore the visualizations of incongruous musical genres, musically improbable tonality and harmonic shifts, and excessive repetitions. 3.1 The Jazz Ending The Preludes and Fugues in Schickele's The Short-Tempered Clavier by and large begin in a classical or baroque style. In the category of incongruous musical genres, Schickele frequently switches from baroque/classical to jazz sporadically, and particularly at the ends of pieces. For example, the Prelude No. 1 in C, a parody of J. S. Bach's Prelude No. 1 in C, occasionally slips from the classical triadic treatment to a jazzy minor seventh chord, with a definitive next-to-last measure switch to the C minor seventh chord, before ending on the C major triad, evoking a tierce picarde-like effect. This shift towards the jazz genre toward the end is visible as a southward swing of the trajectory of the piece, a departure from the tightly wound main cluster, shown in Figure 5. Schickele employs a similar strategy in Prelude No. 8 in G minor, where the entry into the last measure veers sharply from G minor, using the F# major triad to step chromatically into the G major ending. Again, the resulting and unexpected tonal shift is visible in the views of the trajectories of the piece, shown in Figure 6; the trajectories travel north sharply to end in the G major triad, giving this tierce picarde a jazzy twist.


15

Fig. 5. Prelude No.1 in C major (views: left, center)

Fig. 6. Prelude No.8 in G minor (views: center, right)

3.2 Improbable Harmonies Another technique employed by Schickele is the insertion of musically improbable harmonies (chords) and tonality shifts, which typically map to distant locations in the Spiral Array space. In the Fugue No. 2 in C minor, beginning in measure 26, Schickele begins building towards, and preparing for, a climax in C minor, only to land on a cheerful and triumphant C major in measure 34. This is visible in the c.e. trails for the piece shown in Figure 7, where the trajectories meander around the B region, only to shoot up toward the E pitch class in a rather acrobatic fashion at measure 34.

䅋

䅊㩷

16


Fig. 7. Fugue No.2 in C minor (views: center, right)

䅊䅋䅋

In Fugue No. 6 in E major, the composer inserts a jarring chord based on stacked Perfect fourths (E , A , D) in the last four measures of the piece. This departure from expectation is visible as the large upward swoop in the c.e. trails for the piece shown in Figure 8.

䅊

Fig. 8. Fugue No.6 in E major (views: center, right)

3.3 Excessive Repetition One of the Schickele's comedic strategies most easy to identify is that of excessive repetition. Typically, a figure is repeated three times, once to introduce it, the second


17

time as confirmation, and the third to re-affirm the figure. Anything beyond this level of repetition, unless it is an intentional ostinato, can be deemed to be in bad taste. Figure 9 shows an example of a ludicrous and awkward amount of repetition in Prelude No. 10 in A major.

Fig. 9. Prelude No.10 in A major (views: center, right)

4 Conclusions In conclusion, we are able to see stylistic changes, improbable tonal and harmonic shifts, and incessant repeats, some of the laughter-eliciting devices uncovered by Huron in Schickele's music. These strategies are distilled from examples of laughtereliciting music, thus implying that Schickele's humour derives from subsets of these devices. However, it is unclear that the converse is true; that is to say, whether all such expectation-violating events would result in humourous situations. For example, Pachelbel's Canon in D, and Ravel's Bolero all exhibit fairly large numbers of repeats, but most listeners do not find either of these pieces funny. In much post-tonal music, tonal (key) and harmonic (chord) progressions that may have been previously deemed to be improbable in earlier genres, are frequently invoked in practice. Thus, although we can see a good number of Schickele's humour devices, we concede that musical humour cannot be reduced to simply the effect of a few musical features. There are aspects of musical humour that may not be visible in the pitch and time structures of the music, such as the cultural context. One could create a humorous situation that does not employ unexpected musical gestures; one could also create a musical situation that traces an extravagant c.e. curve, i.e. having unexpected pitches and chords, that may not elicit laughter. Humour results from a complex combination of stimulants, and we do not yet have a robust way of automatically detecting musical humour.

Acknowledgements This material is based upon work, and made use of Integrated Media Systems Center Shared Facilities, supported by the National Science Foundation under grant

18


No. 0347988 and Cooperative Agreement No. EEC-9529152. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors, and do not necessarily reflect the views of the National Science Foundation.

References Chew, E.: Towards A Mathematical Model of Tonality. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA (2000) Chew, E., François, A.R.J.: Interactive multi-scale visualizations of tonal evolution in MuSA.RT Opus 2. Computers in Entertainment 3(4), 1–16 (2005) François, A.R.J.: A hybrid architectural style for distributed parallel processing of generic data streams. In: Proc. International Conference on Software Engineering, Edinburgh, Scotland, pp. 367–376 (2004) Huron, D.: Music-engendered laughter: An analysis of humor devices in PDQ Bach. In: Proc. Of the Intl. Conf. on Music Perception and Cognition, pp. 93–95 (2004) Schickele, P.: The Short-Tempered Clavier – Preludes and Fugues in All the Major and Minor Keys Except for the Really Hard Ones, S. 3.14159, easy as edn., pp. 1807–1742. Theodore Pressor Company, Bryn Mawr

Category-Theoretic Consequences of Denotators as a Universal Data Format Gérard Milmeister Institut für Informatik Universität Zürich [email protected]

Abstract. The RUBATO C OMPOSER Music Software uses denotators as the universal data format. Denotators are objects (or points) in general spaces called forms. Denotators and forms constitute a general architecture of concepts, which itself is embedded in category theory. More precisely, the category in question is the category of set-valued presheaves over the modules. This embedding allows some new and important properties which are useful for modeling musical objects. In this paper we discuss the consequences of using the limit and colimit constructions of category theory in the implementation of denotators in RUBATO C OMPOSER.

1 Introduction The RUBATO C OMPOSER music software [4] uses the theory of forms and denotators [2] to model its universe of objects. Denotators play the role of objects and forms the role of data types in this scheme. In fact, the various kinds of constructions available with forms correspond mostly to the kinds traditionally found with structured data types. Apart from simple forms, which match the basic data types, such as numbers and strings, but comprise the whole theory of mathematical modules, there are limit and colimit forms, which correspond to product and disjoint union data types, such as they are found in complete analogy in functional programming languages, such as Haskell or SML. Additionally powerset forms describe sets of denotators. Such a set data type construction is usually not available natively in programming languages but is generally provided as part of the standard library (for example Set in Java). Thus, forms and denotators allow the usual data type constructions found in modern programming languages. There would be nothing especially interesting about this, except for the fact that the theory is embedded in category theory. This fact could already have been anticipated by the use of the category-theoretic terms limit and colimit instead of the more familiar product and coproduct. Apart from the fact that denotators live in a functorial space based on the category of modules (to be exact: the set-valued presheaves over the modules), the main difference between the categorical view and the traditional conception is the presence of diagrams. T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 19–24, 2009. c Springer-Verlag Berlin Heidelberg 2009

20

G. Milmeister

2 Diagrams in Category Theory To keep things simple, concepts from category theory will be used henceforth in an intuitive way and in terms of forms and denotators. In the category of forms, the objects are the forms and morphisms (or arrows) are functions between forms (i.e., a function f : F → G that maps a denotator in the form F to a denotator in the form G). A diagram in the category is a directed graph with forms as vertexes and functions as arrows. It may be required that any paths in the diagram commute. A very basic example shall illustrate the concept of a diagram: e B

A

g

f ? C

h

? - D

In this diagram, there are four objects (forms) A, B, C and D and four morphisms (functions between forms) e : A → B, f : A → C, g : B → D and h : C → D. Commutativity of this diagram requires that following any of the paths (as the composition of functions) always leads to the same result. With a little imagination it is obvious that such a diagram is nothing else than a graphical (and category-theoretical) expression for an equation! Indeed, the commutativity of the above diagram can be rendered as an equation (the left and the right side of the equation corresponding to the two possible paths in the diagram): ∀x ∈ A : g(e(x)) = h(f (x))

3 Limits The common Cartesian product P of n factors F1 ,. . . Fn is drawn as the diagram in figure 1 (left). The arrows in such a product diagram are the projections pi to the respective factors. There are no arrows between the factor objects, i.e., the diagrams consisting of the factors is discrete. P p1

F1

p2

P

@ @ pn @ R @ ?

F2

···

p1

Fn

F1

f

p2

@ @ pn @ R @ ?

- F2

···

Fig. 1. Product diagram (left), limit diagram (right)

Fn

Category-Theoretic Consequences of Denotators

21

In the general case of a product, the limit, a diagram may be non-empty. A simple example is shown in figure 1 (right). Here the diagram features an arrow f from F1 to F2 . An object P is a solution whenever, for a given x ∈ P , we have f (p1 (x)) = p2 (x) or, in words, the second factor of x is equal to f applied to the first factor of x. More generally, such a diagram embodies constraints on the factors of a traditional product. Example. In music, triads provide a straightforward example where limits can be put to good use. A triad form can be defined1 as Triad:.Limit(Note, Note, Note) Note:.Limit(Onset, Pitch, Duration, Loudness) This simple definition does not guarantee that all notes have the same onset (a condition which is arguably sensible). This condition can be accomplished by extending the definition Triad:.Limit(Note, Note, Note, Onset). and adding an arrow f : Note → Onset from each of the three Note factors to the Onset factor and define f as the projection from a Note to its Onset factor. It must be noted that diagrams will not provide solutions for every constraint programming problem that may ever arise. However, many standard situations can be modeled in a quite natural way. The next examples shall illustrate two particular cases. The case shown above involves equations of the type f (X) = Y where X and Y are factors of the limit. A more general type has the form f (X, Y ) = g(Z). Two additional features are illustrated here: functions with more than one argument and multiple arrows. For the two-argument function f we need to add the product space X × Y to factors of the limit. For the equality involving f : X × Y → W and g : Z × W another factor W is needed. With the addition of the projections to enforce the consistency between X and Y factors on the one hand and the X × Y factor on the other, we have the following: P:.Limit(X, Y , Z, X × Y , W ) with the morphisms f :X ×Y →W g:Z→W pX : X × Y → X pY : X × Y → Y. Until now, only equalities have been considered. One possibility to implement inequalities is by means of predicates. A predicate is a morphism of the type F → 1

The denoteX notation from [2] is used here.

22

G. Milmeister

Boolean, where F is a form and Boolean is a predefined form for representing the two Boolean values TRUE and FALSE. An inequality X < Y will result in a morphism h : X × Y → Boolean. Of course, a Boolean factor must be added to the factors of the limit as well. Many problems in mathematical music theory can be formulated as the solution of so-called K-nets (Klumpenhouwer networks). These networks are another application of limit diagrams. The following diagram is an example of such a network. The vertexes are modules, in particular the integers modulo 12 representing pitch classes, the arrows are affine morphisms known from music, such as inversions or transpositions. Z12 6

f M 6

I3

g

Z12

T6

- Z12

Solving such a network means finding those values that satisfy the diagram. This requires the use of methods from linear algebra and finite domain constraint solving algorithms. An overview of some more complex applications of constraint programming in music composition and analysis is given in [5].

4 Colimits To a certain extent the definitions of coproducts and colimits are analogous to the definitions of products and limits, respectively. The coproduct C of n cofactors F1 ,. . . Fn is drawn as the diagram in figure 2 (left). The arrows in such a coproduct diagram are the injections ij from the respective cofactors into the coproduct. There are no arrows between the cofactor objects, i.e., the diagrams consisting of the cofactors is discrete. An example of a colimit with a non-empty cofactor diagram is shown in figure 2 (right). The equation defined by this diagram is i1 (x) = i2 (g(x)) F1

F2

@ @ i1 @ R @

i2

? C

···

in

Fn

F1

g F2

@ @ i1 @ R @

i2

?

···

in

C

Fig. 2. Coproduct diagram (left), colimit diagram (right)

Fn

Category-Theoretic Consequences of Denotators

23

for every x ∈ F1 . The interpretation of this condition is a little less obvious than for the limit case. The equation says that the injection of x ∈ F1 into the colimit form C is equal to the injection g(x) ∈ F2 into C. In more operational terms, consider a denotator c ∈ C which is the injection of a denotator x ∈ F1 . It now makes sense to reinterpret c by applying g to x and considering c as the injection of g(x). Example. Consider a general music object as an input to an analytic software component, which may be either a score, an harmonic analysis of a score, or a formal analysis of a score. The design of the analytic component and the exact shape of the score and its analyses are not important in this context. Such a form may be defined as follows: Analysis:.Colimit(Score, HarmonicAnalysis, FormalAnalysis). If we define f : Score → HarmonicAnalysis and g : Score → FormalAnalysis as the functions which carry out the actual analyses and add them as arrows to the diagram of the colimit form Analysis, the analytic procedures are embedded into the form! Then, whenever a denotator of form Analysis is encountered and it contains a score, both analyses can be retrieved without further ado.

5 Integration in RUBATO C OMPOSER The implementation of forms and denotators in RUBATO C OMPOSER already supports diagrams for types limit and colimit. The implementation of the various morphisms between forms has not yet progressed very far, currently only morphisms of simple forms are available. The consequences of having arbitrary diagrams have to be explored more deeply, since various interesting situations may arise. There are essentially two cases that may happen during the construction of a denotator of type limit. First, all the factors required by the form are supplied. In this case the constructor must check all morphisms of diagrams for any inconsistency. If there is an inconsistency, the construction fails. Second, only some of the factors are supplied. The constructors must then check the diagram for all the given factors, just as in the first case. In addition, the constructor must try to infer the missing factors from the diagram. This is far from obvious: If the diagram contains a loop on one of the factors, the solution for this factor would be a fixed point for the morphism associated with the loop. A diagram may contain one or more cycles. In either case, it may be impossible to find a solution (either positive or negative) computationally. In the case where all morphisms are affine mappings, linear algebra provides the environment for solving such equations (see [3] for example). For situations that are well known in constraint programming research, existing implementations (for example a Java library such as Choco [1]) should be used. As shown above, adding constraints may result in enriching the diagram of a limit with auxiliary factors. It may be convenient to hide these from the user. The Rubato framework needs to provide the tools for managing such forms of increasing complexity and expose only the essential information.

24

G. Milmeister

The colimit situation does not pose such difficulties, since the morphisms are only applied on demand. The reinterpretation semantics of colimit diagrams can be put to good use for the reforming operation of RUBATO C OMPOSER. To reform a denotator of a given form means to convert it into a denotator of another given form. This is akin to casting or conversion in programming languages. For the above colimit example this gives us the possibility to convert a denotator of form C which is the injection of a denotator of form F1 to a denotator of form F2 by simply applying the transformation g. If the colimit diagram contains a loop or a cycle, we may consider the orbit of all denotators induced by the diagram. Thus a colimit form is a method of specifying in a compact way a collection of denotators by giving a set of generators. Of course, it must be explored how to expose the effects of diagrams to the user of forms and denotators, especially in the context of graphical systems such as the RU BATO C OMPOSER GUI. Situations such as the dynamic2 failure of limit constructions must be made transparent and manageable. The graceful handling of constraints and their failures are part of the further research and development of the RUBATO C OM POSER framework. A set of specialized rubettes may help to approach this goal. One possibility to handle constraints and their failures is not to enforce them automatically, but only on demand. A CheckConstraints rubette could be developed that may be placed at strategic points in a RUBATO C OMPOSER network that actually enforces constraints at execution time. The problem of the solution of diagrams could be delegated to a SolveLimit rubette that may be configured with a limit form and generates all solutions. A more sophisticated version may propose to the user the option of removing arrows in order to increase the solution set. The case of an infinite set of solutions must be considered as well. Specific to colimit forms, an OrbitGenerator rubette could produce all denotators on the orbit induced by a given colimit form.

References [1] Choco constraint programming system, http://choco.sourceforge.net [2] Mazzola, G.: The Topos of Music. Birkhäuser, Basel (2002) [3] Mazzola, G., Andreatta, M.: From a Categorical Point of View: K-nets as Limit Denotators. In: Perspectives of New Music, vol. 44(2). Princeton University Press, Princeton (2006) [4] Milmeister, G.: The Rubato Composer Music Software: Component-Based Implementation of a Functorial Concept Architecture, Zürich (2006) [5] Truchet, C.: Some Constraint Satisfaction Problems in Computer Assisted Composition and Analysis. In: Mazzola, G., Noll, T., Lluis-Puebla, E. (eds.) Perspectives in Mathematical Music Theory, epOs Music, Osnabrück, pp. 330–342 (2004)

2

Dynamic, since limit constructions may fail because of the values of the factors, and not only because of their forms.

Normal Form, Successive Interval Arrays, Transformations and Set Classes: A Re-evaluation and Reintegration Ciro Scotto Eastman School of Music, University of Rochester [email protected]

Normal form is an ordering standard for pitch-class sets that facilitates finding structural relationships and properties through comparative analysis. However, the analytical process often fails to correctly identify important relationships or properties, because the normal form algorithm generates misaligned orderings for many pitchclass sets. For example, a comparative analysis of normal form INT1 (Morris 1987, 40-41, 107-109) relationships often indicates pitch-class sets are not inversionally related even though they are members of the same Tn/TnI type.1 The normal form ordering also obscures important structural properties, such as symmetry, for many pitch-class sets. In these problematic cases, a comparative analysis of normal form orderings cannot produce the relevant structural information without the aid of supplemental ad hoc operations. Creating jury-rigged add-on procedures seems to be the accepted solution to working around these informational inconsistencies, whereas modifying the algorithm to eliminate them is an approach that has not been pursued in the literature. In this paper, I will introduce a new normal form conceptualization and a new algorithm that corrects the problems inherent in John Rahn’s (Rahn 1980, 3139) normal form algorithm.2 1

2

The INT1 of an ordered pitch-class set is the series of ordered pitch-class intervals between adjacent pitch-classes of the ordered pitch-class set. The CINT1 of an ordered pitch-class set is the series of ordered pitch-class intervals between adjacent pitch-classes of the ordered pitch-class set that includes the ordered pitch-class interval between the first and last pitch class, or includes the ordered pitch-class interval between the pitch-classes in the last and first order positions of the pitch-class set. As Robert Morris has demonstrated, the CINT can be generalized to calculate intervals for pitch-classes in non-adjacent order positions as well, so the CINT for adjacent pitch-class is technically CINT1. Although the INT of a pitch or pitchclass segment is generally understood to be the succession of intervals between adjacent pitches or pitch-classes, it can also be generalized to list the interval between pitches or pitchclasses n order position apart. In spite of the informal practice of naming the interval succession between adjacent pitch or pitch-classes an INT, the author will adopt Morris’s practice of indicating the interval succession between adjacent pitches or pitch-classes as INT1 for consistency with CINTn notation. Although a review of the procedure will be part of this article, readers may wish to review this section of Rahn’s text. Rahn defines normal form on p. 38: “The “normal form” of a set is that ordering of its members which is increasing within an octave and most packed to the left; if there is more than one such ordering, it is the remaining ordering with the smallest initial pc number.”

T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 25–51, 2009. © Springer-Verlag Berlin Heidelberg 2009

26

C. Scotto

The new conceptualization and normal form algorithm will provide analysts, composer, teachers, and students with a consistent and efficient method of obtaining structural information about pitch-class sets. Correcting the ordering misalignments, for example, allows the new procedure to eliminate many steps in the process of generating a pitch-class set’s normal form and identifying its set class. The new algorithm will also reveal the symmetrical properties of pitch-class sets without having to create multi-sets, and it will reveal a pitch-class set’s degree of symmetry. However, the primary function of normal form comparisons and the most important information obtained by them is identifying the transformational type and the specific transformation relating pitch-class sets. To reinforce this purpose, the new algorithm will become the foundation of a normal form function, which will be the basis of an error proof system for identifying transformational relationships. The new function is analogous to and incorporates aspects of Richard Chrisman’s (Chrisman 1971) use of successive interval arrays to identify transformational types and set classes.3 The algorithm generates misaligned orderings that produce informational inconsistencies, because it contains an asymmetrical bias that only allows it to scan the interval series of a pitch-class set in one direction, left-to-right, for the cyclic permutation producing the smallest perimeter interval. The algorithm, therefore, does not examine every cyclic permutation for the ordering producing the smallest perimeter interval. The new algorithm scans the interval series of a pitch-class set in two directions, right-to-left and the left-to-right, examining every possible cyclic permutation for the one producing the smallest perimeter interval. It also creates left and right normal form classes to distinguish a normal form ordering generated by a right-to-left or a left-to-right cyclic permutation. The classes also help distinguish the normal form of a pitch-class set from its set-class representative or prime form with greater precision. A set class representative, for example 5-20[0,1,5,8], is a normal form ordering, but it is not a pitch-class set. The normal form of a pitch-class set can be either a member of the left or right normal form class, while the set class representative can only be a member of the left normal form class. Comparing a pitch-class set in normal form to its set-class representative and comparing it to other normal form members of a set class reveals different structural information. The former comparison identifies the set class, and the latter comparison indicates the specific transformations relating pitchclass sets. The normal form classes also produce INT1 equivalence classes that facilitate identifying the Tn and/or Tn/TnI types of pitch-class sets. If the normal form of a pitch-class set is a member of either the left or right normal form classes, its INT1 demonstrates a transpositional relationship to at most eleven other members of the same class. The INT1s from different normal form classes, however, will not be related by transposition; they will be related by inversion. The INT 1s in the left and right normal form classes will be equivalence classes called INT1-types that partition the domain of all pitch-class sets creating a partition isomorphic to the Tn equivalence classes. Each INT1-type of one normal form class is inversionally related to at most one INT1-type from the other normal form class. The left normal form INT1-types from an inversionally related pair of INT1-types will be equivalence classes that partition the domain of all possible pitch-class sets creating a partition 3

The successive interval array of a pitch-class set is equivalent to either a pitch-class set’s INT1 or CINT1.

Normal Form, Successive Interval Arrays, Transformations and Set Classes

27

isomorphic to Tn/TnI equivalence classes. Since INT1-types from the left normal form class uniquely identify the set class of any pitch-class set, a pitch-class set’s intervals will ultimately determine its set class membership. INT1 types from the left normal form class will also discriminate between Z-related pitch-class sets, because there are no Z-related INT1-types. The 48 set-classes that contain the wrap-around interval as an internal interval manifest the problems produced by the asymmetrical bias in the algorithm. Of the 48, 31 are inversionally symmetrical, such as set-class 4-20[0,1,5,8]. The members of these set classes do not generate R-invariant INT1s, and they do not generate right normal forms, which further conceals their symmetrical structure. The new algorithm generates both a right and left normal form for members of these set classes, which reveals their symmetrical structure. Since the right normal forms generate independent Tn-classes for these set classes, the total number of Tn classes will increase to 384 from the current number of 352. For the 17 remaining non-inversionally symmetrical set classes listed in Table 1, the asymmetrical bias in the algorithm produces ordering misalignments that generate INT1 relationships indicating the members of a set class are not inversionally related.

Example 1. Transformational relationships in piano part of Columbine from Pierrot Lunaire

Let us begin investigating and correcting the source of the asymmetrical bias in Rahn’s formulation of the algorithm by examining some transformational relationships and issues relating to their derivation in measures 3-5 in the piano part of Columbine

28

C. Scotto

from Pierrot Lunaire (see Example 1a). The vertical pitch-class set labeled X and the one at the end of measure 4 marked with a single asterisk are related at T1.4 What process produced this analytical information? If the pitch-class sets are ordered pitch or pitch-class transpositions, then the specific transposition relating them is easily identified. A pair of pitch-classes in identical order positions will reveal the transposition operation, because identifying the specific transposition only requires identifying the interval relating the pair of pitch classes (see example 1b). For example, the asterisk chord could become an ordered pitch transposition of chord X one half step higher by transferring its pitch Ab up two octaves, so the T1 transposition relating the E of chord X to the F the asterisk chord could easily be heard as the operation relating both chords. When the pitch ordering of pitch-class sets does not does not reveal a transformational relationship, the operation cannot be inferred from a single pair of pitches or pitchclasses in identical order positions. The pitch ordering of pitch-class sets X and Z on beat 1.5 in measure 5, for example, does not indicate they are transformationally related. In these cases, either every possible ordered pair must be examined to find the one-toone correspondences producing identical transformations, or pitch-class sets can be abstractly reordered to discover identically ordered or retrograde related INT1s or CINT1s. Generating and comparing the normal forms of pitch-class sets X and Z demonstrates how the original algorithm produces informational inconsistencies from misaligned orderings and lays the groundwork for correcting the problem (see Example 2a). First, place the set’s pitch-classes in ascending numerical order, then add the first pitch class to the end of the series and calculate the ordered pitch-class interval between adjacent pitch classes to find the largest interval. Rotate the ordering to begin with the right hand member of the ordered pair forming the largest interval. Of course when rotated, the largest interval becomes the smallest perimeter interval (formed by the pitch-classes in order positions 0 and n-1, where n equals the cardinality of the pitch-class set) enclosing the other members of set. When two or more cyclic orderings produce the smallest perimeter interval, the permutation containing the next smallest perimeter interval determines the normal form ordering. The next smallest perimeter interval produced by the pitch classes in order positions 0 and n-2 of the cyclic permutations is 4. The normal form for pitch-class sets X is , and for pitch-class set Z it is <e,2,3,7>.5 4 5

C = 0, t = 10, and e = 11 throughout this paper. Before proceeding further, it is necessary to address one issue for the purpose of consistency and clarity. The normal form procedure requires placing the pitch classes of a pitch-class set in a particular order determined by a rule. Consequently, the normal form of a pitch-class set is an ordered pitch-class segment. Most theorists do not directly indicate this property in the notation used for a set in normal form. For example, throughout Basic Atonal Theory Rahn uses the same curly braces for both unordered pitch-class sets and pitch-class sets in normal form. Straus (Straus 2005, 39) uses square brackets to indicate normal form. Since square brackets are also used to indicate the set class representative in both Rahn and Morris, I will follow their lead and not use them to indicate normal form to avoid further confusion. Robert Morris uses angle brackets for normal form (Morris 199, 40) indicating the ordering property of the pitch-class segment. The convention of using angle brackets for pitch-class sets in normal form will be followed in this paper. However, the use of thick angle brackets will help to further differentiate an ordered pitch-class set in normal form from a pitch-class set that is simply an ordered segment.


2a)

29

2b)

X = {4,e,3,7}

*’ = {e,3,2,7} = Z

X =

*’ = <e,3,2,7> = Z

INT1(X) =

INT1(Z) =

3 4 7 e 3 1 3 4 3 3 4 7 e = 8/4 e 3 4 7 = 8/5 NF =

3 4 7 e 3 1 3 4 4

2 3 7 e 2 1 4 4 3

r1 (1-4-4-3)→(4-4-3-1) 2 3 7 e 2 1 4 4 3 Pitch Classes 7 e 2 3 = 8/7 r1 (2,3,7,e,2)→(3,7,e,2,3) e 2 3 7 = 8/4 NF = <e,2,3,7>

INT1(NFX) =

INT1(NFZ) = (0,3,4,8)Tn

→ 4-19[0,1,4,8] 2c)

2d)

3 4 7 e 3—3 7 e 2 3 1 3 4 4 4 4 3 1 T 6I Must find normal form to generate set class.

New NF Algorithm 3 4 7 e 3 1 3 4 4

2 3 7 e 2 1 4 4 3

3 4 7 e—7 e 2 3 1 3 4 4 3 1 T 6I

LNF

RNF

INT1(LNFX) =

INT1(RNFX) = New Tn Type (0,4,7,8)Tn

Examples 2a-d. Comparing the normal forms of pitch-class sets X and Z

30

C. Scotto

Looking a little further down the road, the new normal form algorithm eliminates the extra step of calculating the next smallest perimeter interval. The INT1 generated by the normal form of pitch-class set X is .6 The INT1 also generates the set class for pitch-class set X. Starting with pitch-class 0 add pitch-classes that replicate the series of intervals between adjacent pitch-classes. In the context of the new normal form function, will become an INT1 equivalence class signifying a pitch-class set is a member of set class 4-19[0,1,4,8]. The normal form ordering of pitch-class set Z is <e,2,3,7> producing the INT1 . The INT1 reveals a BIP1 relationship (Forte 1973, 64-72) between pitch-class sets X and Z, but it does not indicate a transformational relationship.7 Comparing CINT1 permutations, however, reveals the normal form comparison produced erroneous results (see Example 2b). First, place the set’s pitch-classes in ascending numerical order adding the first pitch class to the end of the series, and generate the CINT1. This method cyclically permutes the CINT1 instead of the pitch classes. All cyclic permutations of the CINT1 (reading in one direction, left-to-right) of one pitch-class set must be compared to one CINT1 of the other set to find a rotation indicating a transformational relationship. The CINT1s of pitch-class sets X and Z are and , respectively. Applying r1 to Z’s CINT1 produces a retrograde related pair of CINT1s indicating X and Z are inversionally related.8 However, since CINT1 permutations do not produce different normal form orderings of the pitch-classes for each member of a set-class, they only show the transformational type not the specific transformations relating set class members. Finding the TnI operation requires the extra step of also applying the operation r1 to the pitch-class ordering that generated the original CINT1. Furthermore, neither the CINT1s nor generates the set class for the pitch-class sets. The CINT1 must be put in normal order to reveal the set class identity of the pitch-class set. 6

7

8

Chrisman’s notational convention for successive interval arrays use dashes to separate the linear succession of ordered pitch-class intervals. This same format will be used for INT1 notation. However, to indicate that the interval succession is ordered, the INT1 will also be enclosed in angle brackets. According to Forte, the BIP is the reduction of the interval succession for an ordered segment to a normalized pattern. Morris discusses generalized BIP relations (Morris 1991, 44-46). Pitchclass sets in a BIP1 relationship do not guarantee the pitch-class sets are transformationally related. For example, the ordered pitch-class set A= generates the INT1 , while the ordered pitch-class set B= generates the INT1 . Even though the ordered pitch-class sets produce a BIP1 relationship, their interval vectors are not identical. The interval vector for pitch-class set A is , while the interval vector for pitch-class set B is , so the pitch-class sets are not related to each other by Tn or TnI. Chrisman uses the notation Pn(A) to identify each of the cyclic ordering of a CINT1. If P0(A) is 1-1-1-9, then P1(A) will be 1-1-9-1 (Chrisman 1971, 64). The operation rn is a rotation operation that shifts the intervals of the CINT1 n places to the left, so the interval in order position z is now in order position z-n mod #CINT1. The maximum value of n is the #CINT1-1. Rotation can also be formalized as transposition of order numbers. The operation rn is equivalent to the operation Tn mod #CINT1 where the n of Tn equals #CINT1 – rn. For a CINT1 whose cardinality is 5, the operation r2 shifts the interval in order position 2 to order position 0, and the equivalent transposition operation, T3 mod 5, shifts the interval in order position 2 to order position 0.


31

The new normal form algorithm refines the CINT1 system in several ways. Examining all the cyclic permutations of the CINT1 to find the pair indicating a transformational relationship is not necessary, because the new algorithm generates a single ordering that always indicates the presence or absence of a transformational relationship. The new algorithm simultaneously reorders the pitch-classes and the CINT1 of a pitch-class set, so it generates both the transformational type and the specific transformation relating pitch-class sets. The INT1s of normal form orderings will be equivalence classes. Therefore, they immediately identify the set class membership of any pitch-class set, eliminating the extra step of placing a CINT1 in normal order to reveal the set class identity of a pitch-class set. Furthermore, the pitch ordering of a pitch-class set may not obviously correspond to its set-representative ordering or normal form. In comparisons of the pitch ordering of a pitch-class set to its normal form, CINT1s are really of no help. Reexamining pitch-class sets X and Z using the new algorithm demonstrates these points and corrects the misaligned orderings. Place the set’s pitch-classes in ascending numerical order adding the first pitch class to the end of the series, and generate the CINT1. The new algorithm examines and interprets the CINT1 by reading it in both the right-to-left and the left-to-right directions to determine the cyclic permutation producing the smallest perimeter interval. The intervals in the series reveal every perimeter interval in a cyclic permutation of the pitch classes. That is, the intervals produced by pitch-class 0 and the pitch-classes in order positions n-1, n-2…n-(n-1), where n is the cardinality of the pitch-class set. The order of the intervals in relation to their size in the series determines the cyclic permutation producing the smallest perimeter intervals. When the CINT1 reveals two or more cyclic permutations producing a smallest perimeter interval, the order of the other perimeter intervals in relation to their size is especially important. The cyclic permutation, therefore, must meet an interval succession criterion to be the normal form: the series of decreasing interval sizes from largest to smallest where, in general, a smaller interval does not precede a larger interval will be the permutation producing the normal form ordering of the pitch classes. The CINT1 of pitch-class set X is 1-3-4-4 (see Example 2c). The interval four appears twice in the CINT1, so pitch-class set X produces four (two reading leftward and two reading rightward) cyclic permutations producing the smallest perimeter interval. However, the interval succession criterion necessitates only examining the permutation beginning with the interval four between pitch-class adjacency <e,3> reading right-to-left and the permutation beginning with the interval four between pitch-class adjacency reading left-to-right, since all other permutations violate it. The right-to-left reading produces the interval permutations 4-4-3-1, while the left-to-right reading produces permutation: 4-4-1-3. Only the permutation 4-4-3-1 meets the interval succession criterion, so it represents the cyclic permutation of the pitch-classes that will be the normal form. Reading the CINT1 in the right-to-left direction means the pitch-class on the right of the ordered pair <e, 3> will be the first pitch-class of the cyclic permutation. The ordering is a member of the left normal form class, because the CINT1 was read in a right-to-left direction. It is labeled LNF to signify its class membership.

32

C. Scotto

The CINT1 of pitch-class set Z is 1-4-4-3 (see Example 2d). Once again, the interval succession criterion necessitates only examining the permutation beginning with interval four between pitch-class adjacency reading right-to-left and the permutation beginning with interval four between pitch-class adjacency reading leftto-right. The right-to-left reading produces the interval permutation 4-4-1-3, while the left-to-right reading produces the interval permutation 4-4-3-1. Only the permutation 4-4-3-1 meets the interval succession criterion, so it represents the cyclic permutation of the pitch-classes that is the normal form. Reading the CINT1 in left-to-right direction means the pitch-class on the left of the ordered pair will be the last pitchclass of the cyclic permutation. The ordering is a member of the right normal form class, because the CINT1 was read in a left-to-right direction. It is labeled RNF is to signify its class membership. The LNF and RNF of pitch-class sets X and Z generate INT1s and , respectively. The INT1s are retrograde related, so the ordered segments are related by retrograde inversion, specifically RT 6I(X)→Z. In general, if RTnI is the transformation relating two pitch-class sets in normal form, then the transformation TnI relates the unordered pitch-class sets. The normal form ordering produced by the original algorithm generates the T n type (0,3,4,8)Tn for pitch-class set Z, but the RNF generates a new Tn type, (0,4,7,8)Tn. The INT1 of the original T n type representative (0,3,4,8) obscures the inversional relationship linking the Tn types, while the INT1 generated by the new Tn type representative (0,4,7,8)Tn clarifies the inversional relationship linking it to Tn type (0,1,4,8).9 The obvious difference between the algorithms is their choice of <e,2,3,7> or as the normal form ordering. Since both of these orderings produce the smallest perimeter interval, they were both potential normal form orderings in the original algorithm (see Example 2a). However, the ordering was rejected, because its next perimeter interval was larger than the next perimeter interval produced by the ordering <e,2,3,7>. A jury-rigged workaround for the original algorithm adds an exception rule: if the INT1s of two normal form orderings indicate the pitchclass sets are not inversionally related, and if there are at least two cyclic permutations producing the smallest perimeter interval, the INT1 of the rejected cyclic permutation will demonstrate the pitch-class sets are actually inversionally related. Jury-rigging the original algorithm prohibits the creation of a normal from function, because the INT1s of normal forms will not create equivalence classes. Furthermore, although the modified algorithm appears to choose an ordering that violates the criteria determining the normal form, it is the original algorithm’s ordering that does not conform to the criteria. The work of the next section produces a formal version of the new algorithm and demonstrates why the ordering produces the smallest perimeter interval. 9

The different INT1/Tn types in each of the lists is analogous to the differences in the Forte (Forte 1973, 179-181) and Rahn (Rahn 1980, 140-143) set-class lists produced by Forte’s normal order procedure and Rahn’s normal form algorithm. Each list contains different Tn/TnI type representatives.


33

Example 3. Clockwise and counterclockwise permutations producing the smallest perimeter interval

Rahn’s definition says “the “normal form” of a set is that ordering of its members which is increasing within an octave and most packed to the left; if there is more than one such ordering, it is the remaining ordering with the smallest initial pc number” (Rahn 1980, 38). The definition only specifies a single criterion for determining the correct ordering: the cyclic permutation of the pitch classes producing the smallest perimeter interval is the normal form. The other conditions are part of a subroutine for choosing between cyclic permutations producing identical smallest perimeter intervals: 1) eliminate the cyclic permutations that do not produce the smallest perimeter interval; 2) remove the last pitch class from the remaining cyclic permutations and

34

C. Scotto

recalculate the perimeter intervals; 3) if a cyclic permutation produces the smallest perimeter interval, it is the normal form, or else go to the next step; 4) if the new perimeter intervals are identical, repeat steps 1 through 3, unless the perimeter interval is formed by the first and second pitch places, then go to the next step; 5) if the final perimeter intervals are identical, the ordering beginning with the smallest pitch-class number is the normal form. An abstract representation of the pruning procedure is illustrated in Example 3. It shows five cyclic permutations of an ordered set containing the elements with point distances between elements of 1, 2, 3, and 3, respectively. Permutations A2 and A3 are immediately eliminated, since their perimeter intervals are eleven and ten points, respectively. Eliminating the final elements from permutations A1, A4, and A5 leaves two permutations with an identical perimeter interval of six points (see Example 3b). Repeating steps 1 through 3 reveals A1 as the normal form, since the interval between elements , which is three points, is smallest perimeter interval. Running the normal form orderings of pitch-class set Z generated by the original and new algorithms through the pruning process demonstrates the RNF ordering produces the smallest perimeter interval (see Example 3d). Pairing the orderings with abstract representations facilitates neutral interval comparisons. Create an ordered set containing the elements with point distances between elements of 3, 1, and 4, respectively. Associate the alphabetic elements with the pitch-classes of the ordering <e, 2, 3, 7>, so the ordered pitch-class intervals produced by adjacent pitch classes correspond to the point distances between alphabetic labels. The number of points traversed moving from a starting to goal element determines interval size. Generate the cyclic permutation corresponding to the RNF ordering in the same way, and normalize it to read in a left-to-right direction simply to facilitate point count comparisons. Cyclic permutations A1 and A2 produce identical perimeter intervals of eight and four points, respectively. The final perimeter interval of A1 and A2 is 3 points and 1 point, respectively. A2, the RNF , ultimately produces the smallest perimeter interval, so it is the normal form.

Example 4. Uninterpreted Mod 12 space


35

Examining an un-interpreted version of the mod-12 pitch-class space reveals the asymmetrical bias in the original algorithm is the result of its interpretation of the space.10 A circle containing twelve equally spaced points represents the more general symmetrical structure of an un-interpreted mod-12 pitch-class space (see Example 4). Two rules determine movement around the space: 1) the move from one point to any other point must be made through points adjacent to the current point, and 2) once a direction is chosen, subsequent moves continue in the chosen direction until the goal point is reached.11 The important and relevant feature of this model is movement can proceed in two directions, clockwise or counterclockwise. Calculating ordered pitchclass intervals in the original algorithm limits movement to the clockwise direction in its comparisons of perimeter intervals. Therefore, it does not examine the perimeter intervals of cyclic permutations generated by moving around the circle counterclockwise. a) 0135=5 1 3 5 0 = (–1) 11 3 5 0 1 = (–2) 10 5 0 1 3 = (-2) 10

i i i i

b) 5310=7 3105=2 1053=2 0531=1

i i i i

The asymmetrical bias of ordered pitch-class intervals is the source of the ordering misalignments. The first step in the original algorithm of placing the pitch classes in ascending numerical order is really a convention determined by the formula for calculating ordered pitch-class intervals, because the normal form for the pitch-class set {0, 1, 3, 5}, for example, could be calculated ordering the pitch classes as either an ascending or descending series (see Example 5). The smallest perimeter interval can be calculated from within the permutation as the smallest interval between its first and last pitch classes, or it could be calculated from outside the permutation as the largest interval between its first and last pitch classes. Each method produces equivalent results, and the normal form in each case is identical to within retrogression.12 The ascending series corresponds to the clockwise motion produced by calculating ordered pitch-class intervals. The ordered pitch-class interval i is 5, so with pitch-class 0 as the first element of the ordered pair, the only direction to pitch-class 5 that traverses only 5 points is clockwise.

10

11

12

Another way of stating this is that the circle in Example 4 is similar to an uninterrupted model of the group Ζ12. That is, the representation models the structure of the group, but does not have any particular tokens associated with the elements in the representation. If moves along the diameter or radii through the center of the circle were allowed, the familiar concept of interval would be lost. We calculate our intervals by the number of steps it takes to move from one point to another through intervening points. In the latter method, the smallest perimeter interval is calculated indirectly, since the complement of the largest interval outside the permutation will be the smallest perimeter interval inside the permutation. Calculating the normal form by the outside interval appears in Rahn’s shortcut for normal form.

36

C. Scotto

Example 5. The asymmetrical bias of ordered pitch-class intervals

The descending series, however, reveals the clockwise bias inherent in calculating ordered pitch-class intervals. A descending series of pitch classes may intuitively imply a counterclockwise motion, but it actually produces a complementary clockwise motion. The ordered pitch-class interval i is the perimeter interval of the descending series and, subtracting pitch-class 5 from 0 initially produces the value –5. Interpreting the value literally would mean moving from pitch-class 5 counterclockwise to pitch-class 0, since counterclockwise is the only direction in which pitch-class 0 is five points from pitch-class 5. Negative intervals are, of course, converted to their mod 12 equivalents, so the -5 counterclockwise motion is converted to the clockwise complementary path between pitch-classes 5 and 0 traversing 7 points. Obviously, the normalization preserves many important concepts, such as complementary intervals. However, it also limits other concepts, such as normal form, since the bias those concepts inherit produces the inconsistencies in normal form comparisons. If mod 12 normalizes movement around the space to the clockwise direction, then mod -12 normalizes movement around the space to the counterclockwise direction (see Example 6a). The mod -12 space is just a reinterpretation of the group “Z12” with


37

a)

{2, 3, 7, e} mod –12→{-10, –9, –5, –1} b) (-10, –9, –5, –1} Step 1 Step 2

NF

Step 3 ← -9 = –5/–8 = -4/–8 = –11=

Step 2a Remove Last Pitch-Class to Create Next Container Interval < –10, –9. -5> < –1, –10, –-9>

i i i i

-10 –(1) = -9 -1 –(-5) = 4 mod -12 = -8 -5 –(-9) = 4 mod -12 = -8 -9 –(-10) = 1 mod -12 -11

Step 3 Repeated i -10 –(-5) = -5 i -1 –(-9) = 8 mod -12 = -4

Step 4 LNF <e, 2, 3, 7>

RNF

i<e, 7> 7—11 = -4 mod 12 = 8 i<e, 3> 3—11= -8 mod 12 = 4 i<e, 2> 2-11= -9 = 3

i -5 –(-9) = 4 mod -12 = -8 ABS = 8 i -1 –(-9) = 8 mod -12 = -4 ABS = 4 i -10 –(-9) = -1 ABS = 1

Example 6. a) Mod -12 and mod 12 spaces, b) algorithmic steps for generating a right and left normal form

38

C. Scotto

Step 5 RNF mod 12 → RNF NF generated by original algorithm = <e, 2, 3, 7>

Example 6. (continued)

different tokens.13 Calculating the ordered interval i, for example, by subtracting –4 from –1, [–1 –(–4)] produces a value of 3. Applying mod –12 produces a value of –9 normalizing the movement to the counterclockwise direction. The full computational version of the new algorithm examines the intervals of a pitch-class set from both the clockwise and counterclockwise perspectives to determine the smallest perimeter interval. It generates the additional counterclockwise cyclic permutations by applying mod –12 to the integers of a pitch-class set. The intervals of the counterclockwise cyclic permutations are calculated in the “Z–12” space. The clockwise cyclic permutation producing the smallest “Z12” perimeter interval is compared to the counterclockwise cyclic permutation producing the smallest “Z–12” perimeter interval, and the clockwise or counterclockwise cyclic permutation producing the smallest perimeter interval is the normal form. Clockwise and counterclockwise cyclic permutations are members of the left or LNF or right or RNF normal classes, respectively. To generate the counterclockwise cyclic permutations for pitch-class set Z, transform the pitch-classes {2,3,7,e} into their “Z–12” counterparts, {-10, -9, -5, -1} (see Examples 6a and b).14 Place the pitch classes in descending order from right to left starting with -1, and produce all the cyclic permutations. Calculate the first container interval for each permutation to find the smallest perimeter interval. Two orderings emerge in step 3: and . Step 2a removes the

13

14

John Fraleigh describes the isomorphism with regard to token exchange: “Suppose that a set has three elements. As before, we may as well let the set be {e, a, b}. For e to be an identity, a binary operation * on this set has to have a table [where] …each row and each column are to contain each element exactly once…so * does give a group structure on G = {e, a, b}. Now suppose that G’ is any other group of three elements and imagine a table for G’ with identity element appearing first. Since our filling out of the table for G = {e, a, b} could be done in only one way, we see that if we take the table for G’ and rename the identity e, the next element listed a, and the last element b, the resulting table for G’ must be the same as the one we had for G. As explained…this renaming gives an isomorphism of the group G’ with the group G” (Fraleigh 1999, 60). The pitch-class set is transformed by mod –12 rather than simply taking the inverse of each pitch-class (i.e., {-2, –3, –7, –11}), because transforming the pitch class by mod –12 preserves the spatial relationships on the circle of their mod 12 counterparts. Taking the inverse of each pitch class would flip the spatial relationships 180 degrees, which would yield results in the calculations for normal form equivalent those produced by calculating the LNF.


39

last pitch class from each permutation, and repeating step 3 calculates the new perimeter interval. Since the interval –4 is smaller than –5, the ordering produces the small perimeter interval. Step 4 determines whether the clockwise or counterclockwise cyclic permutation produces the smallest perimeter interval. Comparing positive and negative intervals from the spaces is not a problem, since they are indicators of direction not size. Nevertheless, taking the absolute value of the intervals facilitates comparisons. The first two intervals in both the clockwise and counterclockwise permutations tie with values of 8 and 4, respectively. The next ordered pitch-class intervals for the clockwise and counterclockwise permutations is 3,–1 with an absolute value of 1, respectively. The permutation produces the smallest perimeter interval. Step 5 transforms the pitch classes back to their positive counterparts producing the RNF . The complete computational algorithm presented in Figure 1 is an adaptation of Morris’s algorithm (Morris 1991, 40). Although it is computationally intensive, the shortcut version demonstrated earlier produces equivalent results with a minimum of computational overhead in about three easy steps. The complete algorithm is presented as the theoretical counterpart of the pragmatic shortcut in the interest of completeness. Spans represented by the formula SK1(X) are another name for perimeter intervals, and span calculation is the formal version of the pruning procedure. Calculating spans for RNFs requires another set of ordered pitch-class intervals calculated from right-to-left or moving counterclockwise around the circle. The combination of clockwise and counterclockwise intervals will be called bidirectional ordered pitchclass intervals: DEF

ic = b–a mod 12 and icc = -a–(-b) mod –12 (for any two pitch classes a, b, the ic bidirectional ordered pitch-class interval between a and b in that order equals the number b–a (mod 12) or the bidirectional ordered pitchclass interval icc between -b and -a in that order equals the number -a–(-b) (mod –12))15

Some pitch-class sets produce both an LNF and an RNF, so conditions 1, 2, or 3 in step 12 of the algorithm determine the normal form in these cases. The steps in the procedure are illustrated by running pitch-class set {0,1,3} through the algorithm (see Example 7a). Calculating the counterclockwise intervals from the pitch-class set DL by applying mod —12 to the resulting intervals visually simplifies the illustration (see Example 7b). Lr has one member and Rr has no members in step 11, so LNF is the normal form.

15

This definition is an expanded version of Rahn’s definition of ordered pitch-class interval (Rahn 1980, 25). The interval type icc allows the calculation of Z-12 intervals without having to first flip the pitch-classes into the “Z–12” space. Although it might appear as if these new intervals no longer have complements because ic and icc might appear as if they are complements of each other, this is not the case. The complement of ic is still ic, because these intervals belong to the Z12 group, and in that group they are still complements, since they are both taken mod 12.

40

C. Scotto

a) 1)

DL={0, 1, 3}, DR={0, -11, -9}

2)

DL=, DR=

3)

k = 3–1 = 2

4-5) Lr ={ A0 S2 (DL) = ic 3 – 0 =3 A1 S2 (DL) = ic 0 – 1 = –1 mod 12 = 11 A3 S2 (DL) = ic 1 – 3 =–2 mod 12 =10} Rr={ A3 S2 (DR) = icc -11 – 0 =-11 A4 S2 (DR) = icc -9 –(-11) = 2 mod -12 = -10 A5 S2 (DR) = icc 0 –(-9) = 9 mod -12 = -3} 6) {A0=3, A1=11, A2=10, A3=11, A4=10, A5=3} 7) m=3 8) Delete A1, A2, A3, A4 from Lr and Rr. 9) k = 2-1 = 1 10) 1 > 0, go to step 5 52) Lr ={ A0 013 S1 (DL) = ic 1 – 0 =1 } Rr={ A5 S1 (DR) = icc -11 –(-9) = -2} 6 2)

{A0 = 1, A5 = 2}

7 2)

m=1

8 2)

Delete A5

9 2)

k=1–1=0

Example 7. a) Illustration of steps in the algorithm, b) calculating the counterclockwise intervals from the pitch-class set DL


102)

41

If k = 0, go to step 11

11) If k = 0 and either Lr or Rr, but not both, has only one member, it is the NF, and it will be either a LNF or RNF: NF = LNF b) 4-5) Rr={ A3 S2 (DR) = icc -11 – 0 =-11 A4 S2 (DR) = icc -9 –(-11) = 2 mod -12 = -10 A5 S2 (DR) = icc 0 –(-9) = 9 mod -12 = -3} Rr={ A3 S2 (DR) = icc =1-0 =1 mod -12 = -11 A3 S2 (DR) = icc =3-1 =2 mod -12 = -10 A3 S2 (DR) = icc =0-3 = -3} Example 7. (continued)

Producing normal form INT1 equivalence classes that partition the domain of all possible pitch-class sets creating a partition isomorphic to Tn/TnI equivalence classes is only possible with the new algorithm, because it corrects the misaligned orderings that misidentify inversional relationships among pitch-class sets. Identifying a pitchclass set’s Tn/TnI type usually involves generating the Tn types for a pitch-class set and its inversion, and the representative in “most normal form” becomes the set’s Tn/TnI type (Rahn 1980, 81-2). The representative in most normal form has its intervals “most packed from the right,” which means the larger intervals are on the right proceeding to smaller intervals at the left.16 Interval distribution relative to size and direction distinguishes the representative in “most normal form.” The Tn representative that is not in “most normal form” has the larger intervals on the left proceeding to the smaller intervals on the right, which means it is “most packed from the left.” Interval distribution also identifies the class membership of a normal form orderings. The intervals in the INT1s of all LNFs are “most packed from the right,” which distinguishes a LNF from a RNF where the intervals in the INT1s are “most packed from the left.” For example, the normal forms of pitch-class sets {0,e,9} and {0,1,3} are RNF and LNF, generating INT1s and , respectively. Since the INT1 of the representative in “most normal form,” always has the smaller intervals on the left proceeding to larger intervals at the right, the INT1 of the Tn/TnI type representative is always an LNF. The new algorithm guarantees the INT1s of the normal

16

The original phrase that Rahn (1980, 38) uses is “most packed to the left.” It is a little unfortunate, since it often causes confusion with Forte’s normal form algorithm that breaks ties by choosing the cyclic permutation that has the smallest initial interval. Straus (2005) uses the phrase most packed from the right, which avoids confusion and is closer in meaning to the normal form criterion.

42

C. Scotto

forms for all the members of a set class will be identical to within retrogression, so the INT1 of an LNF can always be generated from an RNF simply by retrograding the INT1 of an RNF. For example, the normal form for the pitch-class set {0,t,4,7,e} is , and the INT1 of the normal form ordering is . Since the smaller intervals are on the right proceeding to the larger intervals on the left, the interval series is a member of the RNF class, and retrograding it produces the INT1 type . Using the intervals of the INT1 type and starting with pitch-class 0 generate the remaining pitch classes of the set class representative. Pitch-class set {0,t,4,7,e} is a member of set-class 5-Z38[0,1,2,5,8]. Since the INT1 of an LNF always uniquely identifies the set class of any pitch-class set, and an LNF can always be generated from an RNF simply by retrograding the INT1 of an RNF, the INT1s of LNFs are equivalence classes that exhaustively partition all pitch-class sets into exclusive classes. Furthermore, INT1 types easily distinguish between Z-related pitch-class sets. The new algorithm generates the LNF for pitch-class set {3,4,7,8,t}, so the INT1 of the LNF is also an INT1 type. Comparing the INT1s of pitch-class sets {0,t,4,7,e} and {3,4,7,8,t} reveals they are neither transpositions nor inversions of each other nor are they members of the same set class, since their INT1s are not identical to within retrogression. The INT1 type indicates pitch-class set {3,4,7,8,t} is a member of set class 5-Z18[0,1,4,5,7], which is the Z partner of pitchclass set {0,t,4,7,e}.17 Although interval vectors cannot reliably identify set class membership and distinguish between Z-related pitch-class sets, INT1 types can perform both functions. Therefore, intervals can partition the domain of all possible pitch-class sets into equivalence classes, if a specific specialized subset of intervals instead of the total intervallic content of pitch-class sets generates the equivalence classes. The interval succession criterion determines a pitch-class set’s normal form in the shortcut algorithm. The interval distribution relative to size and direction that distinguishes the LNF class from the RNF class is the basis for the criterion. The order of interval sizes in an INT1 of an LNF is from smaller intervals on the left proceeding to the larger intervals on the right, while the order of interval sizes is reversed for the INT1 of an RNF. When reading the interval series produced by a pitch-class set in 17

Rahn (1979-80) demonstrates that Lewin’s EMB(A, B) and his extension of embed, MEMBn(X, A, B), both distinguish between Z-related sets 483-498). The pitch-class sets generated by the INT1 types of a pair of Z-related sets are in a relationship determined by a special case of EMB(A, B): if #A = #B and /A/ /B/ then EMB(A, B) = EMB(B, A) = 0. In other words, if the cardinality of sets A and B are equal and if A and B are not members of the same set class, then pitch-class set A will not be embedded in B and pitch-class set B will not be embedded in A. This works whether pitch-class sets A and B are ordered or unordered. With ordered pitch-class sets taken from a pair of Z-related sets, a corollary about interval follows. If the ordered pitch-class set A is not embedded in B, and if the ordered pitch-class set B is not embedded in A, then the INT1s of either A or B cannot be reproduced in the other pitch-class set, even though the interval vectors of pitch-class sets A and B are identical. Reproducing the INT1 of pitch-class set A in B or reproducing the INT1 of B in A would be equivalent to embedding pitch-class set A in B or embedding pitch-class set B in A, or it would be equivalent to demonstrating the pitch-class sets are transformationally related.


43

ascending order in the direction opposite the normal form class (i.e., reading an interval series from right-to-left for an LNF), the potential perimeter intervals in the series become anti-perimeter intervals. The permutation with a series of decreasing interval sizes from largest to smallest produces a permutation with the smallest perimeter intervals, because a large anti-perimeter interval decreases the size of its complement, a perimeter interval. This is why the intervals in the INT1s of all LNFs are “most packed from the right,” and the intervals in the INT1s of all RNFs are “most packed from the left.” When more than one cyclic permutation produces the largest anti-perimeter interval, the remaining anti-perimeter intervals determine the normal form. This is why the other anti-perimeter intervals in the series must be ordered from largest to smallest. For example, the interval series for the pitch class set {3, 4, 6, 8} contains two cyclic permutations producing the largest anti-perimeter interval: The cyclic permutation reading right-to-left for the LNF is , and reading left-to-right for the RNF it is (see Example 8).18 The next anti-perimeter interval for the

Example 8. Left and right cyclic permutations 18

Reading the interval series from left-to-right for the RNF is equivalent to reading the pitch classes that produced the series backwards. Reading the pitch classes in the reverse order produces the mod 12 complements of the intervals in the series, . However, the intervals of the RNF space are taken mod -12, so the intervals in the series become , and the shortcut algorithm is simply taking the absolute values of the intervals in the series.

44

C. Scotto

RNF is 1, while the next anti-perimeter interval for the LNF is 2. Since interval 2 is larger than interval 1, pitch-class 6 will be closer to pitch-class 3 than pitch-class 4 is to pitch-class 8.19 Therefore, only the interval series meets the interval succession criterion, and the normal form is LNF. The interval succession criterion makes generating the normal form of a pitch-class easier and more efficient. For example, pitch-class set {1, 0, t, 7, 4} is one of the problematic pitch-class sets in Table 1 (see Example 9). The interval series generated by the initial steps of the shortcut contains three 3s, which means the pitch-class set has six cyclic permutations, three from the right and three from the left, capable of producing the smallest perimeter interval. However, the interval succession criterion necessitates only examining the permutation reading right-to-left beginning with interval three between pitch-class adjacency and the permutation reading left-toright beginning with interval three between pitch-class adjacency . Of the two remaining permutations, only RNF whose INT1 is meets the interval succession criterion. The set class is always generated from an LNF, so retrograding the INT1 produces the INT1 type indicating pitch-class set {1, 0, t, 7, 4}, is a member of set class 5-31[0, 1, 3, 6, 9]. A = {1, 0, t, 7, 4} 1) 2) 0 1 1

4 3

7 3

t 3

(0)

2

3) 1←33321 t→33312 12) RNF INT 13) Set class: a)

Retrograde RNF INT

b) Begin with pitch-class 0 5-31[0, 1, 3, 6, 9] Example 9. Shortcut algorithm for normal form 19

Another way to verify the smallest perimeter interval is to add the two intervals between pitch-classes 3 and 6 and compare the value to the sum of adding the two intervals between pitch-classes 8 and 4. Adding the intervals for pitch-classes 3 and 6 produces a value 3, while adding the intervals for pitch-classes 8 and 4 produces a value of 4. The interval 3 is smaller than 4, so the LNF produces the smallest perimeter interval, and it is the normal form for the pitch-class set.


45

The full algorithm demonstrates how LNFs and RNFs reveal hidden symmetries, when the INT1 of the normal form is not R-invariant (see Example 10). For pitchclass set {0, 1, 2, 7}, the fourth pass sets k to 0 and m to 1 leaving Lr and Rr each with one member. Since all perimeter intervals are identical and the orderings of the LNF and RNF are rotations of each other, the pitch-class produces both an LNF and an RNF. In these cases condition 2 applies: the normal form is determined by context, or the LNF is chosen by convention. The INT1s generated by equivalent LNFs and RNFs are retrograde related indicating the pitch-class set is symmetrical, and it will map into itself under inversion. The set Lr and Rr will also illustrate transpositional symmetry. The multiplicity of identical normal form types, LNFs or RNFs, represents the number of operations that map the pitch-class set into itself under Tn, so the cardinality of Lr or Rr equals the degree of transpositional symmetry. The number of LNF/RNF pairs represents the number of operations that map the pitch-class set into itself under TnI and equals the degree of inversional symmetry. The sum of both numbers equals the degree of symmetry. For example, when k=0 for the pitch-class set {0, 1, 2, 7} the cardinality of Lr is one and the number of LNF/RNF pairs is also one, so the degree of symmetry is 2. Since the pitch-class set {0,1,2,7} generates two normal form orderings, LNF and RNF that are rotations of each other, it is also a member of two Tn types. The LNF is a member of Tn type (0,1,2,7)Tn, while the RNF is a member of new Tn type (0,5,6,7)Tn. The addition of another Tn type for this pitch-class set better reflects its symmetrical structure. 1)

LNF

A0 0127 ic 7 – 0 = 7 A1 1270 ic 0 – 1 = –1 = 11 A2 2701 ic 1 – 2 = –1 = 11 A3 7012 ic 2 – 7 = –5 = 7

RNF A4 0127 icc 0 – 7 = –7 A5 7012 icc 7 – 2 = 5 = –7 A6 2701 icc 2 – 1 = 1 = –11 A7 0127 icc 1 – 0 = 1 –11

2) A0 0127 ic 2 – 0 = 2 A3 7012 ic 1 – 7 = –6 = 6

A4 0127 icc 1 – 7 = –6 A5 7012 icc 0 – 2 = –2

3) A0 0127 ic 1 – 0 = 1

A5 7012 icc 1 – 2 = –1

LNF INT

RNF INT

Example 10. LNFs and RNFs reveal hidden symmetries

46

C. Scotto

Fortunately, the shortcut algorithm quickly and easily reveals hidden symmetries and a pitch-class set’s degree of symmetry (see Example 11). The interval series generated by pitch-class set {4,5,9,0} contains two fours. The interval succession criterion necessitates only examining the permutation reading right-to-left beginning with interval four between pitch-class adjacency and the permutation reading left-to-right beginning with interval four between pitch-class adjacency . All perimeter intervals in the LNF and RNF are identical and the orderings are rotations of each other, so the INT1s generated by the equivalent LNF and RNF will be retrograde related indicating the pitch-class set will map into itself under inversion. The LNF is a member of Tn type (0,1,5,8)Tn, while the RNF is a member of the new Tn Type (0,3,7,8)Tn. The degree of symmetry for the pitch-class set is 2, since the algorithm only produces one LNF/RNF pair, and the multiplicity of LNFs is also one. In general, the number of identical normal form permutations generated by a pitch-class set equals its degree of symmetry.20 The normal form function, NFTR[], has at its foundation four possible INT1s relationships and transformational definitions (see Example 12). Since the INT1s of the normal forms generated by the new algorithm always correctly indicate the presence or absence of a transformational relationship in comparative analysis, the function uniquely associates a pair of INT1s with a transformational type. The elements of the domain are sets containing the INT1s generated by a pair of pitchclass sets in normal form. The codomain is the set containing the values 1, 2, 3, or 4, and the range is sets whose elements are a pair of INT1s and a value. The values indicate the transformational relationship: 1 = the pitch-class sets generating INT1<X> and INT1 from normal form orderings are Tn-related; 2 = the pitch-class sets generating INT1<X> and INT1 from normal form orderings are TnI-related; 20

The presence or absence of the cyclic interval creates some interesting disparities between INT1 and CINT1 systems. In the INT1 system, most set classes with a degree of symmetry of 2 or higher will produce an INT1that is R-invariant. Some set classes, however, need to repeat a pitch-class in order to reveal they are capable producing an R-invariant INT1, when the unmodified algorithm generated their normal form. These pitch-class sets generate two equivalent normal forms using the modified algorithm, and the INT1s of the normal forms reflect the pitch-class set’s symmetrical structure. The inclusion of the cyclic interval in the CINT1 system essentially reverses the situation. Pitch-class sets capable of generating an Rinvariant CINT1 will be the pitch-class sets that generate two equivalent normal forms. In the INT1 system a pitch-class set with a degree of symmetry of 2 producing an R-invariant INT1 will no longer produce an R-invariant CINT1 due to the inclusion of the cyclic interval. For example, the pitch-class set {0,1,6,7} in normal form is producing the INT1 1-5-1, while placing the same pitch-class set in ascending order and including the cyclic interval produces the pitch-class set generating a CINT1 of 1-5-1-5, which is not Rinvariant. Although the CINT1 1-5-1-5 is not R-invariant, there is a cyclic permutation of the CINT1 that is related by retrogression to the original order of the CINT1, 5-1-5-1. In the shortcut version of the modified algorithm, the interval series produces four equivalent permutations indicating the degree of symmetry of pitch-class set {0,1,6,7} is 4. The smallest initial pitch-class condition eliminates two of the orderings, and the remaining ordering fall under condition 1: if there is one LNF and RNF, and if the RNF is the LNF read backwards, the NF is the LNF, by convention.


47

A = {4, 5, 9, 0} 1) 2) (0,1,5,8)Tn/(0,3,7,8)Tn 0 4 5 9 (0) 4 1 4 3

3)

44341 54341

4) LNF INT Condition 2 from step 12

RNF INT

5) Set class—4-20[0, 1, 5, 8]

4 5 9 0—LNF RNF—9 0 4 5 3 4 1 4 3 Example 11. Revealing hidden symmetries with the shortcut algorithm

3 = the unordered pitch-class sets generating INT1<X> and INT1 from normal form orderings are Tn and/or TnI related; and 4 = the pitch-class sets generating INT1<X> and INT1 from normal form orderings are not Tn or TnI –related. Comparing Sets/ Normal Form and Normal Form Comparisons The INT1the two sets in normal form can be in 1 of 4 relationships. 1) The normal forms of pitch-class sets A and B generate INT1s whose intervals are in the same order: INT1(NF(A))=, INT1(NF(B))=. 2) The normal forms of pitch-class sets A and B generate INT1s whose intervals are retrograde related: INT1(NF(A))=, INT1(NF(B))=. 3) The normal forms of pitch-class sets A and B generate INT1s whose intervals are in the same order and retrograde related: INT1(NF(A))=, INT1(NF(B))=.

48

C. Scotto

4) The normal forms of pitch-class sets A and B generate INT1s whose intervallic contents are not identical or the intervallic content is identical but the order of the intervals is neither the same nor retrograde related: INT1(NF(A))=, INT1(NF(B))= or INT1(NF(A))=, INT1(NF(B))=. Definitions and Transformational Relationships Following from the Five Normal Form INT1 Relationships: 1)

2)

3)

4)

Relationship 1: Definition 1—unordered pitch-class sets A and B are related by the operation of transposition if and only if the INT1 generated by pitch-class set A in normal form contains the same ordered pitch-class intervals in the same order as the INT1 generated by pitchclass set B in normal form. Relationship 2: Definition 2—unordered pitch-class sets A and B are related by the operation of inversion if and only if the INT1 generated by pitch-class set A in normal form contains the same ordered pitchclass intervals in the reverse order of the INT1 generated by the pitchclass set B in normal form. Relationship 3: Definition 3—unordered pitch-class sets A and B are related by the operation of transposition and inversion if and only if the INT1 generated by pitch-class set A in normal form is R-invariant with the INT1 generated by pitch-class set B in normal form. (a) Definition 3a—unordered pitch-class set A will map into itself under inversion if and only if the INT1 generated by pitch-class set A in normal form is R-invariant. Relationship 4: Definition 4—unordered pitch-class sets A and B are not related by the operation of inversion or transposition if and only if the INT1 generated by pitch-class set A in normal form does not contain the same ordered pitch-class intervals as the INT1 generated by the pitch-class set B in normal form or the INT1 generated by pitchclass set A in normal form contains the same ordered pitch-class intervals as the INT1 generated by the pitch-class set B in normal form but the order of the intervals is neither the same or reversed. Example 12. INT relationships and the normal form function

In this paper, I have demonstrated that creating a normal form function produces a consistent system for determining transformational relationships. Simply placing a pitch-class set in normal form is sufficient to determine its transformational type, the specific transformation relating it to any other pitch-class set, its degree of symmetry, and its set class membership. Rahn writes “the ability to take a set and quickly, almost automatically, list it in normal form is absolutely crucial to all subsequent use of nontonal theory” (Rahn 1980, 31). It is my hope that there are now many more reasons to work towards this goal.


49

References Chrisman, R.: Identification and Correlation of Pitch-Sets. Journal of Music Theory 15(1-2), 58–83 (1971) Forte, A.: The Structure of Atonal Music. Yale University Press, New Haven (1973) Fraleigh, J.: A First Course in Abstract Algebra. Addison-Wesley Publishing Company, Inc., Reading (1999) Morris, R.: Composition with Pitch-Classes. Yale University Press, New Haven (1987) Morris, R.: Class Notes for Atonal Music Theory. Frog Peak Music, Lebanon (1991) Rahn, J.: Basic Atonal Theory. Longman Inc., New York (1980) Rahn, J.: Relating Sets. Perspectives of New Music 18(1-2), 483–498 (1979-1980) Straus, J.N.: Introduction to Post-Tonal Theory. Pearson Prentice Hall, Upper Saddle River (2005)

Appendix Rahn/Morris/Scotto Normal Form Algorithm

Definition: span (sub-k) of ordered set X: Sk1 (X) = xk – x0 mod 12 where k ≤ #x – 1 Sk2 (X) = xk – x0 mod –12 where k ≤ #x – 1 Examples Sk1 X = ; S2 (X) = x2 – x0 = 3 – 0 = 3 S1 (X) = x1 – x0 = 1 – 0 = 1

Sk2 X = >; S2 (X) = x2 – x0 = -11 – 0 = –11 S1 (X) = x1 – x0 = -9 – 0 = –9

50

C. Scotto

1)

DL is a pitch-class set in mod 12 space; DR is a pitch-class set in mod -12 space.

2)

Write DL as an ordered set by placing the pitch classes in ascending numerical order. Call this L. Write DR as an ordered set by placing the pitch classes in descending numerical order. Call this R.

3)

Let k = #D – 1

4)

Construct the set Lr consisting of all rotations of L. Construct the set Rr consisting of all rotations of R.

5)

Find all the values Sk for each member of Lr and Rr.

6)

Take the absolute value of each Sk

7)

Find the smallest values of Sk from the members of Lr and Rr. Call it m.

8)

Delete all members of Lr and Rr with Sk greater than m.

9)

k=k–1

10)

If k = 0, go to step 11; if k >0, go to step 5

11)

12)

a)

If k = 0 and either Lr or Rr, but not both, has only one member, it is the NF, and it will be either a LNF or RNF.

b)

If k = 0 and Lr and Rr each have one members, go to step 12.

Apply conditions 1, 2, or 3. Condition 1: if there is LNF and RNF, and if the RNF is the LNF read backwards, the NF is the LNF, by convention. Condition 2: If there is one LNF and one RNF, and if the RNF is a rotation of the LNF, the choice of NF is context dependent. Condition 3: If all the members of Lr and Rr produce the smallest perimeter interval, choose the LNF that begins on the smallest pitch-class integer by convention. Fig. 1. Rahn/Morris/Scotto Normal Form Algorithm


51

Table 1. Set Classes where the INTs of a Pitch-class Set and its Inversion in NF are not Retrogrades of Each Other

4-19[0, 1, 4, 8] 5-13[0, 1, 2, 4, 8] 5-31[0, 1, 3, 6, 9] 5-32[0, 1, 4, 6, 9] 6-Z46[0, 1, 2, 4, 6, 9] 6-Z47[0, 1, 2, 4, 7, 9] 6-Z-44[0, 1, 2, 5, 6, 9] 6-27[0, 1, 3, 4, 6, 9] 7-10[0, 1, 2, 3, 4, 6, 9] 7-16[0, 1, 2, 3, 5, 6, 9] 7-29[0, 1, 2, 3, 6, 7, 9] 7-21[0, 1, 2, 4, 5, 8, 9] 7-22[0, 1, 2, 5, 6, 8, 9] 9-7[0, 1, 2, 3, 4, 5, 7, 8, t] 9-8[0, 1, 2, 3, 4, 6, 7, 8, t] 9-10[0, 1, 2, 3, 4, 6, 7, 9, t] 9-11[0, 1, 2, 3, 5, 6, 7, 9, t]

A Model of Musical Motifs Torsten Anders Interdisciplinary Centre for Computer Music Research University of Plymouth

Abstract. This paper presents a model of musical motifs for composition. It defines the relation between a motif’s music representation, its distinctive features, and how these features may be varied. Motifs can also depend on non-motivic musical conditions (e.g., harmonic, melodic, or rhythmic rules). The model was implemented as a constraint satisfaction problem.

1

Introduction

Compositional aspects such as harmony and counterpoint have often been formalised and implemented successfully. For example, Pachet and Roy (2001) provide a survey of constrained-based harmonisation systems. A key aspect of such systems is the introduction of formal models of established musical concepts such as note pitches, pitch classes, scale degrees, chord roots and so forth. At the end of their survey, Pachet and Roy (2001) point out: “However, what remains unsolved is the problem of producing musically nice or interesting melodies.” I believe, in order to formalise melody composition we need to model important melodic concepts such motifs and their relations. A crucial aspect of the motif concept is the diversity of possible motifs and their variations. The motif definition of the New Grove clearly points out this diversity. A short musical idea, melodic, harmonic, rhythmic, or any combination of these three. A motif may be of any size, and is most commonly regarded as the shortest subdivision of a theme or phrase that still maintains its identity as an idea. [Drabkin] Motifs have been modelled for music analysis. For example, Buteau and Mazzola (2000) model the similarity of motifs, including motifs of different lengths. However, a motif model for composition is missing (to my knowledge). L¨ othe (1999) proposes a system creating minuet melodies over a given harmonic progression. The author discusses the importance of motif variations, but does not present a formalisation. The constraint-based composition system OMRC (Sandred, 2003) and its successor PWMC1 support the composition of pieces from pre-composed motifs. These systems allow the user to apply further constraints on the music (e.g., rhythmic and harmonic rules). However, motif variations are severely restricted (only pitch transpositions are permitted). 1

Personal communication, PRISMA meeting, January 2007 in Montbéliard, France.

T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 52–58, 2009. c Springer-Verlag Berlin Heidelberg 2009

A Model of Musical Motifs

53

This research presents a model of musical motifs for composition. The model expresses the relation between a motif’s music representation, its identity (often notated a vs. b, cf. (Schoenberg, 1967)), and how it is varied (a1 vs. a2 ). Various musical aspects (e.g., the rhythm, melody, or harmony) can define the identity of a motif. The model distinguishes between a motif as an abstract concept and instances of this motif (actual occurances in the music). The identity of a motif (the abstract concept) is described formally by a set of features. Motif instances can vary these features in many ways, while retaining the motif’s identity. In general, the term ‘motif variation’ is often used to indicate how a given prototypical instance of a motif is transformed in order to obtain another instance. On the contrary, the present model formalises motif variations by defining the relation between a symbolic description of the motif (the abstract concept) and its instances. The user defines which relations are regarded as variations, and which are not (compare changing the melodic contour with a mere transposition). The model is implemented as part of the constraint-based composition system Strasheela (Anders, 2007).2 Users define a set of motifs (by features characterising their identity), and a set of variations on these motifs. Rules on motific identity and variation can be applied. For example, a rule may constrain that a certain phrase consists of variations of the same motif, where the motif’s identity is unknown in the definition. Additionally, users can constrain other aspects of the music. For example, harmonic, rhythmic, and formal rules are defined independently of the motif definition, but directly affect the motifs in the solution. For efficiency, Strasheela uses state-of-the-art constraint programming techniques: a constraint model based on the notion of computational spaces (Schulte, 2002) makes search strategies programmable. Paper Outline The rest of the paper is organised as follows. The motif model formalism is explained in Sec. 2. Section 3 demonstrates the model with two motifs from Beethoven’s 5th symphony. The text concludes with a discussion (Sec. 4).

2

The Formal Model

The proposed motif model is stated as a constraint satisfaction problem (CSP). A CSP closely resembles a mathematical specification. A CSP imposes constraints (relations) between variables (unknowns), where each variable has a domain (a set of possible values). However, a CSP is also executable: modern constraint solvers efficiently find solutions for a CSP (i.e., determine each variable to a value of its domain which is consistent with all its constraints). In this model, a motif is a tuple of the three variables representation, description, and variation (Fig. 1). The following paragraphs outline how single domain values of these variables are constructed. Formally, this text notates variables by a disjunction (∨) of its domain values. 2

Strasheela is available for download at http://strasheela.sourceforge.net/

54

T. Anders motif ::= representation , description , variation representation ::=

some hierarchic music representation ∨...

description ::=

feature 1 : variable list 1 , feature 2 : variable list 2 , . . . ∨...

variation ::=

motif → (0 ∨ 1) ∨...

makeVariation ::= feature 1 : f1 : motif → variable list 1 , feature 2 : f2 : motif → variable list 2 , . . . → variation domain value (a function) Fig. 1. A motif consists of its music representation, a symbolic description, and a variation function

The variable representation basically stores the information recorded in a music notation of an instance of the motif. For example, representation expresses the temporal organisation of notes in the motif and their pitches. Its domain is the set of all motif representation candidates. In an efficient implementation of the model, the representation is not a variable itself but it contains variables (e.g., all note pitches and durations in the representation may be variables). The model abstracts away from the actual music representation format: this information can be encoded in any hierarchic representation format which supports variables and an interface for accessing score information (e.g., a variant of CHARM (Harris et al., 1991), or Smoke (Pope, 1992) supporting variables). The model was implemented using the Strasheela music representation (Anders, 2007). The variable description symbolically states distinctive motif features. Each domain value of this variable describes the features of a motif (an abstract concept, see above) with its own identity (e.g., one domain value describes motif a and another motif b). Because we have no agreed feature set which distinguishes the identity of a motif (cf. the Grove motif definition above), description can contain any information (e.g., the motif’s note durations and its melodic intervals). description can have an arbitrary format, but a consistent format of its domain values simplifies the CSP definition. The following format combines flexibility with convenience: description is a tuple of feature-value pairs (Fig. 1). A feature is a descriptive label (e.g., durations) and its value is a list of (often determined) variables (e.g., the note durations for motif a). The variable variation denotes a specific motif variation. The variation domain consists of functions which map a motif to a Boolean variable (Fig. 1). These functions formalise how motif instances vary the description of a motif (e.g., whether the note pitches defined in the description are followed literally by


55

the representation or in reverse order). When the constraint solver decides for a variation domain value (a function), then and only then this function returns 1 (i.e., true) and it constrains the relation between the motif’s representation and its description. This approach is highly generic because arbitrary constraints can be applied by the variation functions. However, these functions can be complex to define. In a still flexible but more convenient approach, variation functions are created by the function makeVariation . makeVariation expects a tuple of feature-value pairs, where the features correspond to the features of the description, and their values are functions mapping a motif instance to a list of variables (e.g., a function returning the note durations of a motif). Please note that makeVariation unifies this list with the corresponding list in the selected motif description. For example, a model instance may constrain the note durations in the motif’s representation to be equal to the durations in the description. This affects which domain values are selected for these variables.3 Figure 2 summarises the relations between all variables of the model. myMotif is any motif instance in the score (a subsection or a whole piece). The model’s essence is highlighted in bold font.4 For brevity, the definition of makeVariation is omitted. ∀ myMotif ∈ score : ∃ representation , description , variation : representation = representation 1 , . . . , representation n ∧ description = description 1 , . . . , description n ∧ variation = variation 1 , . . . , variation n ∧ myMotif = representation, description, variation ∧1= map(getInitialDomain(variation), f : f (v ) := v (myMotif )) ∧ variation(myMotif ) = 1 Fig. 2. Relations between the motif model variables (essence in bold font)

3

An Example

This section models well-known motifs from the first movement of Beethoven’s Fifth Symphony as an example. Figure 3 classifies some motif instances according to motif identity and variation. The presented classification allows for 3

4

In the implementation, description and variation are encoded by finite domain integers. They point as indices in the respective domains. Selection constraints (Duchier et al., 1998) care for efficient constraint propagation. The function map applies the given function f to every element of the variation’s domain and returns the collected results.

56

T. Anders

considerable mutability of the first variation of motif a. Note that other classifications can be expressed with this model as well. A set of motifs and their classification is modelled by defining domains for the three variables representation, description, and variation. The set of solutions for a single motif instance includes all motifs shown in Fig. 3 – among similar motifs. However, additional rules can further restrict the music (e.g., rhythmic, harmonic, and contrapuntal rules), and many motif instances can be part of a CSP.

Fig. 3. Motifs from Beethoven’s Symphony No. 5 (one possible classification)

The representation domain consists of note sequences, where each note in a sequence has parameter values for its duration and pitch (Fig. 4).5 As these parameters can have any value, all shown Beethoven motifs are members of this domain. Please note that instances of motif a and b differ in length: the motif length is not fixed in representation.6 Similarily, different variations of the same motif identity can differ in length (e.g., a motif can be reduced or embellished). Because a motif’s variation defines the relation between its description and representation, a variation can reduce the representation to, say, the highest or longest note of the description (this is of course an extreme case). The description domain characterises rhythmic and melodic features which distinguish the two Beethoven motifs a and b. Please note that the feature sets differ between motifs: description a specifies the pitchContour (the sequence of pitch interval directions), whereas description b specifies scaleDegreeIntervals (the sequence of distances between note pitches measured in scale degrees). Also, note that description a makes use of variables (e.g., the last note duration is not fixed). 5 6

The pause is not modelled for simplicity. It can be addressed by a note offset parameter (Anders, 2007). The implementation encodes all motif instances with the same – maximum – length internally. Notes are marked as ‘non-existing’ by setting their duration to 0 (Anders, 2007).


representation :=

57

sequence 1 with notes of specific duration and pitch, sequence 2 with notes of specific duration and pitch, ...

description a := durations : (, , , (

pitchContour : (→, (

, . . . , )) →, ), )

description b := durations : (♩, ♩, ♩, ♩, ♩, ♩, ♩, ♩), scaleDegreeIntervals : (3, −1, 1, 1, −3, 0, −1) description := descriptiona , descriptionb , . . . variation 1 := durations : getNoteDurations , pitchContour : getPitchContour , scaleDegreeIntervals : getScaleDegreeIntervals variation 2 := durations : getNoteDurations , pitchContour : f : f (myMotif ) := getDescription (myMotif ) = description a variation :=

∧ inverse(getPitchContour (myMotif )) makeVariation(variation 1 ), makeVariation(variation 2 ), . . .

Fig. 4. Definition of the three variables representation, description, and variation which model the Beethoven motifs (results in classification of Fig. 3)

Finally, the functions in the variation domain constrain the relation between the representation and the description of a motif instance. The functions getNoteDurations, getPitchContour, and getScaleDegreeIntervals access the motif’s representation. For example, getNoteDurations can be implemented as shown in (1), where getNotes returns the notes in the motif’s representation, and getDuration returns the duration of a note. Please remember that makeVariation unifies the variable list returned by these functions with the corresponding variable list in the description. description values can differ in their set of features (see above): variations only constrain those motif aspects specified by the description of a motif (e.g., variation 1 does not constrain the pitch contour in case the motif’s description is motif b ). variation 2 inverses the pitch contour of a motif (cf. Fig. 3), but variation 2 is only permitted for motif a. getNoteDurations(myMotif ) := map(getNotes(myMotif ), getDuration)

4

(1)

Discussion

This paper presented a motif model as a CSP which specifies the relation between the motif’s music representation, a description of distinctive motif features, and

58

T. Anders

motif variation definitions. The model was designed for computer-aided composition, but it can also be used as an executable representation of a motivic analysis. This research does not propose a new concept of motivic similarity, but allows for the application of various similarity models (e.g., the pitch contour). The model does not express a degree or genealogy of variations. However, it supports various additional cases. Non-motivic sections can be modelled by a variation function which does not apply any constraint at all.7 Contrapuntal motif combinations (e.g., a fugue subject) can be search for by constraining multiple motif instances to the same description, but leaving feature values in the description itself undetermined in the definition. Overlapping motifs are possible if the music representation supports such nesting. Finally, higher-level formal relations can be expressed by nesting ‘motif’ instances (e.g., a theme may contain a motif sequence, and is specified by the theme’s description and constrained by its variation).

References Anders, T.: Composing Music by Composing Rules: Design and Usage of a Generic Music Constraint System. Ph. D. thesis, School of Music & Sonic Arts, Queen’s University Belfast (2007) Buteau, C., Mazzola, G.: From Contour Similarity to Motivic Topologies. Musicae Scientiae 4(2), 125–149 (2000) Drabkin, W.M., Macy, L.(ed.): Grove Music Online. Oxford Music Online, http://www.oxfordmusiconline.com/subscriber/article/grove/music/19221 (accessed September 24, 2008) Duchier, D., Gardent, C., Niehren, J.: Concurrent Constraint Programming in Oz for Natural Language Processing. Programming Systems Lab, Universit¨ at des Saarlandes, Germany (1998) Harris, M., Smaill, A., Wiggins, G.: Representing Music Symbolically. In: IX Colloquio di Informatica Musicale, Genoa, Italy (1991) L¨ othe, M.: Knowledge Based Automatic Composition and Variation of Melodies for Minuets in Early Classical Style. In: Burgard, W., Christaller, T., Cremers, A.B. (eds.) KI 1999. LNCS, vol. 1701, pp. 159–170. Springer, Heidelberg (1999) Pachet, F., Roy, P.: Musical Harmonization with Constraints: A Survey. Constraints Journal 6(1), 7–19 (2001) Pope, S.T.: The Smoke Music Representation, Description Language, and Interchange Format. In: Proceedings of the International Computer Music Conference, San Jose (1992) ¨ Searching for a Rhythmical Language. In: PRISMA 01. EuresisEdizioni, Sandred, O.: Milano (2003) Schoenberg, A.: Fundamentals of Musical Composition. Faber and Faber, London (1967) Schulte, C.: Programming Constraint Services. Springer, Heidelberg (2002)

7

To eliminate symmetries (i.e., different solutions which are equivalent), this nonmotivic variation should determine the motif description to some domain value.

Melodic Clustering within Motivic Spaces: Visualization in OpenMusic and Application to Schumann’s Träumerei Chantal Buteau and John Vipperman Brock University [email protected], [email protected]

Abstract. Based on the concepts of motive contour, gestalt and motive similarity, our model of motivic structure yields topological motivic spaces of a composition in which open neighborhoods correspond to groupings of similar motives. In Buteau 2006 we presented a model extension of an earlier approach in order to integrate the concept of melodic clustering in motivic spaces, demonstrated an application to the soprano voice of Schumann’s Träumerei, and provided a comparison with a human-made segmentation (clustering) analysis (Repp 1992) and a machine learning approach (Cambouropoulos and Widmer 2000). In this short paper, we present our novel dynamic visualization of melodic clustering in OpenMusic software, and extend our initial analysis of Träumerei to multi-voice clustering.

1 Introduction As shown in recent works (such as Cahill and Maid´ın 2005 and Cambouropoulos and Tsougras 2004), computer-aided analysis and content-based music retrieval are promising research domains that contribute to the development of a better understanding of the concept of melodic similarity. In computer-aided analysis, any reasonable model of a germinal motif, i.e. those short melodies having a germinal function such as the opening motif in Beethoven’s Fifth symphony, necessitates the inclusion of melodies of different lengths into the method. Our topological approach (Buteau 2003; Mazzola 2002) to the modeling of motivic structure includes the concept of contour similarity for different lengths. It is an immanent approach that formalizes Rudolph Réti’s (1951) method in which melodic segments are compared with one another in order to determine which melodic segments are germinal motives. Melodic clustering, that is an organization of melodic segments into ‘significant’ categories, is another important analytical structure offering insight into melodic similarity. Our approach builds on work introduced in Buteau 2006, wherein we presented a model extension to motivic spaces that includes the concept of melodic clustering, its application to the soprano voice of Schumann’s Träumerei, the seventh piece of Kinderszenen, op.15, and a comparison with human-made clustering analysis (Repp 1992) and machine learning approach (Cambouropoulos and Widmer 2000). The results were very close to these reference clusterings. In this paper, we briefly present our novel

We would like to express our gratitude to Carlos Agon (IRCAM, Paris) for his continuous support in the design and implementation of OM-Melos Clustering Tool in OpenMusic.


60

C. Buteau and J. Vipperman

dynamic visualization of melodic clustering in OpenMusic software, called OM-Melos Clustering Tool, and extend our initial analysis of Träumerei to multi-voice clustering1 . Like our results on the soprano voice, our clustering analysis on the ’primary’ motives (Repp 1992) of Träumerei, appearing in all 4 voices, yield a melodic segmentation very close to the human-made clustering reference. The results on the complete four-voice segmentation are not all as close to the reference as the soprano only results, but there are significant similarities, and overall the complete segmentations are simply coarser. As a consequence, the resulting melodic clusterings within motivic spaces overall strongly contribute to the validation of our topological model of motivic structure.

2 Topological Model of Motivic Structure This section will briefly review motivic spaces; for details and examples, see Buteau 2001; Buteau 2003; Mazzola 2002. Tones are parameterized by at least onset and pitch values. Motives M are non-empty finite sets of tones: M = {m1 , ..., mn } such that all onset values in M are different. We set card(M ) = n. Given a music composition S we consider a (finite) collection of motives in S that we denote M OT (S). We impose that M OT (S) satisfies the Submotif Existence Axiom (SEA), that is every sub-motives of a motif in M OT (S), down to a minimal cardinality, is also in M OT (S). The shape of a motif M is the image of M by a set mapping2 t : M OT (S) → Γt ; for example, Com(M ) = the COM matrix of M , Rg(M ) = projection of M on the onsetpitch plane, and Dia(M ) = vector of consecutive pitch intervals. These 3 examples are respectively called COM-matrix, Rigid, and Diastematic types. We consider a group P action on Γt induced by a group action on M OT (S), e.g., the affine counterpoint paradigmatic group P = CP or the group P = T r of transpositions −1 and translations in time. We introduce the gestalt of a motif M as GesP (P · t (M ) := t t(M )). We consider pseudo-metrics dn for shapes of cardinality n that we retract to motives: the distance between motives M and N with same cardinality n is dt (M, N ) := dn (t(M ), t(N )), and their gestalt distance is gdP t (M, N ) := inf p,q∈P dn (p · t(M ), q · t(N )). For example, the Euclidean distance or relative Euclidean distance REdt (Buteau 2003) for t = Com, Rg, and Dia. If P is a group of isometries, then gdP t is also a pseudo-metric. Given a > 0, we introduce the -neighborhood of a motif M ∗ as Vt,d,P (M ) := {N ∈ M OT (S)|N ∗ ⊂ N s.t. gdP t (N , M ) < }, or simply denoted V (M ). If our setup (defined by t, P , and d) fulfills the inheritance property (Buteau 2001; Mazzola 2002), corresponding to impose that similar motives have similar associated sub-motives, then these neighborhoods form a basis for a topology Tt,P,d on3 M OT (S) (Buteau 2001; Mazzola 2002). The topological space is called motivic space of S. The topology Tt,P,d being only of type T0 (not Hausdorff) with no intuitive 1 2

3

Note that our method does not restrict to monophonic music. The exact construction of the model is on the set M OT of all possible motives from which we take a finite collection M OT (S) of motives in S; see Buteau 2001 for details. In the exact construction the space of a composition is defined as the relativization to M OT (S) of the topology on M OT .

Melodic Clustering within Motivic Spaces

61

geometrical representation, we introduce4 the following functions: pres (M ) := 1 ∗ · #{N ∗ ⊂ N |gdP t (N , M ) < }, where m = card(M ) and n = N ∈ M OT (S) 2n−m 1 ∗ card(N ); con (M ) := N ∈ M OT (S) 2m−n · #{M ∗ ⊂ M |gdP t (M , N ) < }; and the weight of motif M at radius as weight (M ) := pres (M ) · con (N ). The motivic topology for S corresponds to the motivic structure of S (Buteau 2003; Mazzola 2002). The formalization of the germinal function of a motif, i.e. of being omnipresent in a composition given a similarity threshold , is formalized by the motives with largest weights at radius (Buteau 2003; Mazzola 2002). 2.1 Melodic Clustering within Motivic Spaces We now introduce the definition of a melodic cluster in our topological spaces. Given a set X ⊂ M OT (S) of motives and > 0, the -variation set of motif M in X X X is ∈ X|N ∈ V (M ) or M ∈ V (N )}, or equivalently V ar (M ) = V ar (M ) := {N V (M )∪W (M ) ∩X, where W (M ) = {N ∈ M OT (S)|M ∈ V (N )} is a closed set in M OT (S). The intersection with set X corresponds to consider the relative subspace to X. In order to model clustering approaches, such as r Cambouropoulos and Widmer 2000, we introduce an additional set CM intersection with the variations that depends on the cardinality of the motif M . For example, we require that motives in the variation set of M should have a cardinality of at least 70% of the cardinality of M . This can be formalized by the cardinality restriction min(card(M),card(N )) function r : X × X → {0, 1} with r(M, N ) := 1 if max(card(M),card(N )) ≥ 70%, and r 0 otherwise; together with the set CM = {N ∈ X|r(M, N ) = 1}. We call X a clustering set and we introduce the -cluster ClusterX (M ) of motif M r in X (with respect r) as ClusterX (M ) := V arX (M ) ∩ CM . Given a set X of motives and a cardinality restriction function r, clustering the set X of motives corresponds to construct all the -clusters ClusterX (M ), i.e. for all motives M ∈ X and all similarity threshold > 0. Note that the introduction of the -variation sets of motives involves some kind of new ’distance’-function between any two motives in X, which satisfies the reflexivity and symmetry properties of a pseudo-metric, but in general does not satisfy the triangular inequality; take for example two motives M and N of same cardinality with a common sub-motif but with distance gdt (M, N ) = 0. Since weight functions are global functions, it is of interest to introduce local weight functions to a relative space of M OT (S), in particular to the clustering set X, and to compare it with the global weight section on X, i.e. weight|X . We define X locW eightX (M ) as being the product of locP resX (M ) with locCon (M ) whose definitions remain the same as pres and con functions, except for the sum index that changes to ’N ∈ X’.

3 Model Implementation and Visualization in OpenMusic The motivic model was first partially implemented by Mazzola and Zahorka (1994) as a module of the software RUBATOc . It was completely reimplemented by Buteau (2004) 4

For computational efficiency purposes, these functions can be redefined on corresponding quotient spaces of gestalts and for analysis purposes, they can be generalized (Buteau 2001).

62


in JAVA where the major improvements are the rich diversity of the outputs unveiling all details of the topological spaces and a significant enhancement of computational efficiency. The clustering extension was designed, in 2006, based on an algorithm to find maximal cliques in a dynamic graph (Stix 2004). It is implemented in line with the core program efficiency: calculations are reduced to motif classes. In addition to our core JAVA implement, called Melos, we designed a visualization tool in OpenMusic (Agon and Assayag 2002a), called OM-Melos Clustering Tool and implemented in 2007 by Vipperman. Figure 1 shows the overall flowchart of our implementation: the program input is a score file (MIDI format or text file5 for specification of the analysis segmentation), a clustering set file (MIDI or text file), and analysis parameter settings, e.g. topological parameters t, P , and dt . The output6 is a text file to be passed to our implementation in OpenMusic for the detailed visualization of (topological) melodic clustering.

Fig. 1. The overall flowchart of the implementation of our topological melodic clustering model and its visualization (in OpenMusic)

The OM-Melos Clustering Tool automatically exhibits the melodic clustering as a dynamic table (a OM-Maquette interface (Agon and Assayag 2002b)) of labeled colored boxes. It shows the initial state at which motives (represented by boxes) in the clustering set are linked (same color) to one another with same gestalt; see Figure 2. With a key command, it exhibits the melodic clustering by labeling the boxes with the -cluster numbers (possibly more than one number) at each similarity threshold . Additionally, the Clustering Motif Info Window (see Figure 3) displays, for each clustering 5

6

The OM-Melos Score Tool, implemented in OpenMusic, reads and displays a music piece from a MIDI file, allows the user to easily segment the piece for the analysis, and saves it in a text file. More details about the complete OM-Melos tool can be found in Buteau and Vipperman 2008. Note that the program returns two files, one of which is passed to computer algebra system MAPLE for three-dimensional weight graphs and Motivic Evolution Trees (Buteau 2003) and the other file is passed to OpenMusic. For more details on OM-Melos, see Buteau and Vipperman 2008.


63

Fig. 2. The dynamic clustering maquette of OM-Melos Clustering Tool displays the resulting melodic clustering. This example shows the clustering of the soprano voice of Schumann’s Träumerei (see Figure 4) constructed in the motivic space with t = Com, P = T r, and dt = REdt , in which we added a score line to exemplify how this representation relates to the score. The figure shows the initial state at which clustering motives (boxes) are linked to one another (represented by same color) if they share same gestalt. For example, the ‘salient’ (Repp 1992) ascending motives (motives 1, 6, 10, ... in Figure 4) all have same COM -matrix (shown in brown). The number of lines (6 in this example) in the table is determined by the user.

Fig. 3. The Clustering Motif Info Window of OM-Melos Clustering Tool displays, for a motif in the clustering set X local and global weight function graphs of the motif (two bottom left boxes), the motif’s shape (third box from the bottom left), and the notes forming the motif (bottom right box - the motif is displayed in the upper key). This example shows topological information about the salient ascending motif 1 (see Figure 4) in the soprano voice melodic segmentation of Schumann’s Träumerei (t = Rg, P = T r, dt = REdt ).

motif box, local and global weight function graphs of the motif and its shape, and shows and plays the notes forming the motif. Two additional visualization functionalities are implemented: the dynamic motif clusters (an OM-Maquette interface) and clustering motif set displays reveal other important details for the melodic clustering analysis.

64


4 Application to Schumann’s Träumerei In Buteau 2006 our melodic clustering analysis of the soprano voice (28 motives) of Träumerei (see Figure 4) compared well with the melodic/rhythmic segmentation suggested by music theorist Repp (1992) and with a computer-generated clustering (Cambouropoulos and Widmer 2000). In this section we briefly discuss the extension of our clustering analysis to the ’primary’ (Repp 1992) motives (36 in total) “which represent the leading voice(s) in the polyphonic quartet” (Repp 1992) and involve all 4 voices. We also address the complete 4 voice segmentation (70 motives), and compare it to the melodic segmentation reference (Repp 1992)7 .

Fig. 4. The soprano voice of Schumann’s Träumerei with the melodic/rhythmic segmentation proposed by Repp (1992)

We constructed the melodic clustering within topological spaces with shape types rigid, COM-matrix, diastematic, and elastic (Mazzola 2002), the paradigmatic groups T r and CP , and the relative Euclidean distance function dt = REdt , for both the primary motives and complete 4 voice segmentations, with the 70%-cardinality ratio restriction function. Using OM-Melos Clustering Tool, we visualized, dynamically, the melodic clustering and compared it with the segmentation reference. Figure 5 shows the melodic segmentation for the primary motives proposed by Repp and by our topological model at a fixed similarity threshold. The tables should read as follows: each cell in the table corresponds, in a chronologically consistent manner, to a primary motif in the score. The very left column indicates the phrase structure of the Träumerei containing two main phrases, A and B, that appear in some variations (Ai and Bi). The small letter symbols correspond to clustering labels (accordingly to Cambouropoulos and Widmer 2000 with the soprano voice clustering). An empty cell in a table corresponds to a monadic category. The primary motives cluster set shares most of its motives with the soprano voice segmentation (motif clusters ’a’ to ’g’ in Table 1). This contributes to our resulting clusterings of primary motives that concord well with the segmentation reference. 7

Cambouropoulos and Widmer (2000) confined their melodic clustering approach to the soprano voice.


65

Table 1 Table 2 Melodic Segmentation Melodic Segmentation According to Repp Within Motivic Spaces A1 a b c d e i A1 a b c d e i B1 a b f h g j B1 a b’ f i B2 a b f h g j B2 a b f h g B3 a b f h g j B3 a b f h g A1 a b c d e i A1 a b c d e i A2 a b c d e A2 a b’ c d e Fig. 5. The melodic/rhythmic segmentation of primary motives of Schumann’s Träumerei according to Repp (1992) in Table 1 and according to our topological approach in Table 2 with parameters t = Dia, P = T r, dt = REdt , and similarity threshold = 0.4714. Analysis with other topological parameters yield similar results.

Fig. 6. Motives in Schumann’s Träumerei (bars 10 - 12): 2-note motives B and E are identified together in motivic spaces with the COM -matrix shape type, whereas Repp (1992) distinguishes them, possibly with their onset but identifies them with simultaneous (super-)motives A and C, respectively, in other voices. Motives F, G and H are similarly identified.

Our resulting 4-voice clusterings8 do not compare as closely to the segmentation reference as the soprano and the primary motives segmentations do. For instance, many of the 2-note motives are all identified together in our motivic spaces, e.g. with COM matrix shape type, whereas Repp distinguishes them possibly with their onset but identifies them with simultaneous (super-)motives in other voices. For example, motives B and E in Figure 6 are not the same ’melodic gesture’ according to Repp 1992, whereas they are identified together in motivic spaces with t = Com. Furthermore, motives A and B are regrouped as same melodic gesture according to Repp 1992, as are motives F, G, and H. Motives C, D, and E are regrouped as related gestures (i.e. primary gesture C and secondary gesture D-E). In motivic spaces with t = Com and P = T r, the motif onset information is lost and motives B, D,and E are identified together in the same gestalt, as are motives G and H. However, when using the 70%-cardinality ratio restriction function, they cannot, for any > 0, be in the same -cluster as their super-motives A, C, and F respectively. This was not an issue when we dealt with only one voice clustering set. In fact, when considering each of the 4 voices separately, our resulting clusterings compare well with 8

Our Java implement computes multi-voice clusterings, but their visualization implementation in OM-Melos Clustering Tool is still in progress.

66


the segmentation reference. In general, for each constructed motivic space on the 4voice composition, some ’Repp’s clusters’ are inevitably regrouped in the same cluster in the motivic spaces, especially when weakening the 70%-ratio restriction function. This makes our clustering results coarser but not contradictory to Repp’s segmentation. Finally, for the motivic space on the soprano voice with rigid or elastic type (with P = T r and dt = REdt ) and generalized weight function9, the so-called ‘salient’ ascending motif ‘6 - 1’ (see Figure 3) is prominent both locally and globally. Further systematic investigations on local and global weight functions on this composition are ongoing.

References Agon, C., Assayag, G.: Object-Oriented Programming in OpenMusic. In: Mazzola, G., et al. (eds.) Topos of Music, pp. 967–990. Birkhäuser, Basel (2002) Agon, C., Assayag, G.: Programmation Visuelle et Editeurs Musicaux pour la Composition Assistée par Ordinateur. In: IHM 2002. ACM Computer, Poitiers (2002) Buteau, C.: Reciprocity between Presence and Content Functions on a Motivic Composition Space. Tatra Mt. Math. Publ. 23, 17–45 (2001) Buteau, C.: A Topological Model of Motivic Structure and Analysis of Music: Theory and Operationalization. Ph.D Thesis, Universität Zürich, Zürich (2003) Buteau, C.: Motivic Spaces of Scores through RUBATO’s MeloTopRUBETTE. In: Lluis-Puebla, E., Mazzola, G., Noll, T. (eds.) Perspectives in Mathematical and Computational Music Theory, pp. 330–342. Verlag epOs-Music, Osnabrück (2004) Buteau, C.: Melodic Clustering Within Topological Spaces of Schumann’s Träumerei’. In: Proceedings of ICMC, New Orleans, pp. 104–110 (2006) Buteau, C., Vipperman, J.: Representations of Motivic Spaces of a Score in OpenMusic. Journal of Mathematics and Music 2(2) (2008) ´ Melodic Similarity Algorithms - Using Similarity Ratings for DevelCahill, M., Maid´ın, D.O.: opment And Early Evaluation. In: Proceedings of ISMIR 2005, London, pp. 450–453 (2005) Cambouropoulos, E., Costas Tsougras, C.: Influence of Musical Similarity on Melodic Segmentation: Representations and Algorithms. In: Proceedings of the International Conference on Sound and Music Computing (SMC), Paris, France (2004) Cambouropoulos, E., Widmer, G.: Melodic Clustering: Motivic Analysis of Schumann’s Träumerei. In: Proceedings of JIM, France (2000) Mazzola, G., et al.: The Topos of Music. Birkhäuser, Basel (2002) Mazzola, G., Zahorka, O.: The RUBATO Performance Workstation on NEXTSTEP. In: Proceedings of ICMC 1994, San Francisco (1994) Repp, B.: Diversity and commonality in music performance: An analysis of timing microstructure in Schumann’s Träumerei. Journal of Acoustical Society of America 92(5), 2546–2568 (1992) Réti, R.: The Thematic Process in Music. Greenwood Press, Connecticut (1951) Stix, V.: Finding All Maximal Cliques in Dynamic Graphs. Computational Optimization and Applications 27, 173–186 (2004)

9

We used the weight function defined with squaring the content (to make more weight on larger motives as opposed to short ones) and having a factor of 0.8 for the cardinality difference between motives.

Topological Features of the Two-Voice Inventions Kamil Adilo˘glu and Klaus Obermayer Berlin University of Technology [email protected], [email protected]

Abstract. The similarity neighbourhood model is a mathematical model making use of statistical, semiotical and computational approaches to perform melodic analysis of given music pieces. This paper is dedicated to the investigation of topological features and conditions in connection with the model on the one hand and concrete analyses on the other. Therefore, checking the topological features of the model as well as the analysis results is a good practice not only for theoretical but also for practical reasons. The topological features of the similarity neighbourhood model are investigated from a theoretical viewpoint, in order to figure out under which conditions the collection of the results yielded by the model define a topology. These topological features are then tested practically on the two-voice inventions. These investigations and tests have shown that the similarity neighbourhood model defines a topology not for all cases, but depending on the analysed musical piece.

1 Introduction The similarity neighbourhood model has been designed to extract the melodic structure of a given musical piece. The model is based on the similarities between melodies of equal length. However, on the basis of the detected similarities, the model further identifies sub- and super-segment relationships within the given piece. The similarity neighbourhood model is inspired by topology. For this reason, a topological terminology is used in the exposition of the model to explain some musical and/or music-theoretical relationships. However, we did not mathematically investigate the family of neighbourhoods as a whole and therefore missed the potential of topological methodology. This applies to the entire model on a theoretical level as well as to the interpretation of individual analytical results. In the field of mathematical music theory, there are several research studies, in which melodic similarities in music pieces are investigated by using topology in a more strict theoretical sense. Buteau (2001, 2003) defined a topological model, in which the - neighbourhoods are defined for each motif based on the similarity degree of them. The inheritance property guarantees that the similarity of two motives is passed on their sub-motives. This ensures that the set of all - neighbourhoods forms a base for a topology. Adiloglu et al. (2006a) defined a correlation based model to identify similarities between melodies. This approach differs also in another respect from Buteaus’s approach: Only sequences of consecutive notes are considered as melodies. Based on the similarities, the neighbourhood sets are defined for each melody. These neighbourhood sets only contain equal length melodies. T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 67–77, 2009. c Springer-Verlag Berlin Heidelberg 2009

68

K. Adilo˘glu and K. Obermayer

Mazzola (2002) and Buteau (2001, 2003) quantified the presence of a melody m within a longer melody m in terms of an intensity number which measures the cardinality of those submotives of m which are similar to m. The total presence of m is a weighted sum of all these intensity numbers. The content of a melody is defined in a reciprocal manner accordingly. Adiloglu and Obermayer (2005) redefined presence and content in terms of sets of melodies rather than in terms of numbers. The presence neighbourhood set of a given melody is the set of melodies that contain the given melody. The content neighbourhood set of a given melody consists of the sub-segments of the given melody. Ruwet (1987) claims that the main criterium, which governs the process of segmentation is repetition. Therefore he suggests to start the segmentation of a given piece with the longest repeated passages. He extends this idea into a quite flexible segmentationalgorithm for musical pieces. Adiloglu and Obermayer Adiloglu and Obermayer (2006c) made use of his method to perform a reduction of the analysis results of their model. In this paper, we present the topological investigation of the similarity neighbourhood model (for the theoretical details see Adiloglu and Obermayer 2005, 2006b, 2006c, and Adiloglu et al. 2006a).The topological features of the model are presented on a practical test scenario on the Two-Voice Inventions of J.S. Bach.

2 The Similarity Neighbourhood Model In the similarity neighbourhood model, we use only the chromatic pitch height values of the notes and ignore their durations and inter-onset intervals between neighbouring notes, i.e. we ignore the parameters that constitute rhythm or other parameters that are related to the articulation of the piece. Temporal and articulation information is simply reduced to the sequential order of the notes in time. Hence, we define a melody m of length n as a sequence of n integers (t1 , t2 , . . . , tn ) ∈ Zn , whose coordinates ti denote chromatic pitches. A monophonic piece M in itself is regarded as a sequence of integers M = (τ1 , ..., τN ). To simplify the situation in a polyphonic piece we disregard the vertical contrapuntal incidences, and consider the piece from a syntagmatic point of view merely as a list of voices (Mk )k=1,...,ν , where ν is the number of voices within the piece M . The basic information of a polyphonic analysis is then a ν × ν-upper-triangular-matrix of comparative analyses for every pair (Mk , Ml ) of voices from which further global information can be extracted. The voices are supposed to be disjoint, i.e. the total νnumber of pitch-occurrences of a polyphonic piece is supposed to be the sum N = k=1 Nk , where Nk is the length of the voice Mk . In analogy to the monophonic case, we designate the Nk pitch-coordinates of the voice Mk by using the same symbols as follows: Mk = (τk,1 , ..., τk,Nk ). We need to distinguish between abstract submelodies of a given melody M and concrete occurrences of such sub-melodies within M , which shall be called sub-segments. A sub-melody of M of length n is a sequence (t1 , ..., tn ) ⊂ Zn such that there exists an index i ≤ Nk − n within the k th voice of the piece M with (t1 , t2 , ..., tn ) = (τk,i , τk,i+1 , ...τk,i+n−1 ).

Topological Features of the Two-Voice Inventions

69

n The sub-segment Mk,i = (i, k, (τk,i , ..., τk,i+n−1 )) is the concrete occurrence of the sub-melody (τk,i , ..., τk,i+n−1 ), which starts at index i within the voice k. Thus, it is modeled as a 3-tuple, which consists of the index i, the voice index k and the submelody of length n, which starts at index i within the voice k. In order to denote only n the sub-melody of length n, we write M(i) (putting subscript i in parantheses), i.e. n n Mk,i = (i, k, Mk,(i) ). We use the same terms and notation for the identification of abstract sub-melodies (s1 , ..., sn ). which start at index j relative to any given melody m = (t1 , ..., tn ), s. n t. (j + n − 1 ≤ n) (e.g. m = Mk,i being a sub-segment of M ) and concrete ocn currences mj = (j, (tj , ..., tj+n −1 )) of such sub-melodies as sub-segments of m. Similarly, mn(j) denotes the sub-melody, whereas mnj is the concrete occurrence of this

sub-melody within the given melody m, s. t. mnj = (i, mn(j) ). Note that the voice index k is not necessary for a sub-segment of a given melodic segment m, since the voice index of m is already set. In order to have a transformation invariant representation, the shape of the melodic n segment m = Mk,i is calculated based on the chromatic distances between the conn n secutive pitch-coordinates of the melody Mk,(i) . The shape is defined as μ(Mk,(i) )= (t2 − t1 , t3 − t2 , . . . , tn − tn−1 ). So, the shape of a given melody is the sequence of intervals between the consecutive pitches. We use of the correlation coefficient d : Rn−1 × Rn−1 → [−1, 1] for calculating the absolute value |d(μ(m1 ), μ(m2 ))| of the correlation between the shapes μ(m1 ) and μ(m2 ) of two melodies m1 and m2 of the same length n in order to decide upon their similarity with the help of some threshold. Similarity includes — as a special case of maximal similarity — the musical transformations of chromatic pitch transposition and inversion, because μ(m + t) = μ(m) and d(x, −x) = −1. Note, that the retrograde of a melody cannot be identified in this way. Those segments, which are similar to a given melodic segment m are stored in the similarity neighbourhood of the given melodic segment. Definition 1. The similarity neighbourhood URn (m, M ) of a given melodic segment n m = Mk,i within the voice k of a given piece M is defined as: n n n : |d(μ(Mk,(i) ), μ(Ml,(j) ))| > R}, where URn (m, M ) = {Ml,j

R=2

1 n c1

−1

.

(1) (2)

The distance is calculated between the given melody and all other equal length melodies within the given piece. The similarity neighbourhood set of the melodic segment m thus contains equal length n melodic segments m = Ml,j similar to m. The members m of a similarity neighbourhood set of a melodic segment m are be said to be first order similar with respect to R. Likewise, two melodic segments m1 and m2 are be said to be second order similar, if the intersection of their similarity neighbourhood sets is non-empty, and contains neither m1 nor m2 . A re-iteration of

70


this principle leads to an equivalence relation, whose equivalence class Ecn (M ) shall be called the connectivity component with the index number c, which contains all equal length melodic segments being first- or higher-order similar to each other. Hereby, every connectivity component can be indexed also by its corresponding representative melody m∗ , which is defined to be the melodic segment with the similarity neighbourhood set of largest cardinality. The connectivity component obtained by this process contains all melodic segments of the same length, which are related to each other in the following sense: For any two different melodic segments ma1 and maj , of a connectivity component, there exists a chain of first-order similarity connection between melodic segments ma1 ma2 ma3 · · · maj . In fact, often the similarity neighbourhood sets of two similar melodic segments of the same length contain a lot of common segments and the connectivity components Ecn (M ) of melodies of length n are the fixed points of the iterative procedure of unification. Constructing the connectivity components simplifies the control of the results, and indicates the second, third and higher order similarities of melodies in a better way.

3 Inheritance Property Intuitively, the inheritance property says that similar melodies have similar submelodies. The inheritance property defined by Mazzola and Buteau addresses a one-directional relationship between motives and sub-motives. The similarity of two motives implies the similarity of their corresponding submotives. However, the inheritance property, which we define below in terms of two conditions, considers a bidirectional relationship, namely, from segments to sub-segments, as well as from sub-segments to segments: n Definition 2. Suppose we are given a melodic segment m = Mk,i of length n and a n sub-segment m = mj of length n (with j + n − 1 ≤ n). n n 1. Further suppose we are given a melodic segment m = Ml,i , s.t. m ∈ UR (m, M ), which is similar to m. In association with m we consider its sub-segment m = n m j . We say that the similarity between the segments m and m is inherited by their sub-segments m and m if these sub-segments are similar as well, i.e. if m ∈ URn (m, M ). n ∈ 2. Further suppose we are given a melodic segment m = Ml,i +j−1 , s.t. m n UR (m, M ), which is similar to m. In association with m we consider the melodic n n segment m = Ml,i within the ambient melody. If m exists, m contains m = mj at the same relative location as m contains m. We say that that the similarity between the sub-segments m and m is inherited by the segments m and m if these segments are similar as well, i.e. if m ∈ URn (m, M ).

The first part of this definition requires that the similarity between two melodic segments implies that their corresponding sub-segments are similar as well. The second part requires the converse — which is not as intuitive as the first part. It says that if


71

Fig. 1. Bidirectional Inheritance Property

there are two similar sub-segments, their corresponding supersegments are also similar, if they exist within M . These conditions are not fulfilled in general. But if they are fulfilled, they help to remove redundant information from the analytical results. An instance of such a bidirectional relationship is shown in Figure 1.

4 Redundant Melodies Ruwet (1987) presents a method for paradigmatic partitioning starting with the identification of the longest repeated segments within a given piece. In the following steps, Ruwet identifies shorter segments as well as sub-segments of the previously identified longer segments, which partition the given piece any further. That is to say, the shorter segments or the sub-segments of the longer segments help to decrease the amount of the unpartitioned parts of the given piece by defining new partitions in those areas. The following strategy employs these ideas of Ruwet in a more general situation, where the segments are not strictly partitioning. From the similarity point of view, all of the sub-melodies of two similar melodies appear within, where these similar melodies appear. Therefore, pursuing Ruwet ideas, segments, which contribute to the partitioning of the given piece should be distinguished from the ones, which only appear redundantly within these partitioning segments. Hence, a melody is called redundant, if there is a longer melody containing the given melody, up to similarity, wherever the given melody appears. Due to the exhaustive nature, the similarity neighbourhood model identifies the redundant melodic segments as well. However these melodic segments can be removed from the set of the results. Definition 3. By weak reduction we mean the following reduction procedure, which is applied to the entire collection of neighbourhoods in order to yield an analogous family n n of neighbourhoods. Given two melodic segments m = Mk,i and m = Mk,j of length n n and n respectively, where m = mj (with j + n − 1 ≤ n). The melody m is removed from the results, if card(URn (m , M )) ≤ card(URn (m, M )). According to Definition 3, the melodic segments are removed, if the number of occurrences is less than or equal to the number of occurrences of the melodic segments containing them. This definition assumes that the shorter melodic segment only appears within the longer melodic segment containing it. However this does not have to be true for all cases.

72


From music-theoretical point of view, melodies appearing not only within their super-segments but also independently within a given musical piece are important for this particular musical piece. These kinds of melodies should not be removed. On the contrary, they should be kept in the results and investigated further. Therefore, before reducing a melody, the first part of the inheritance property should be checked as well. For this reason, we define the strong reduction method that uses the inheritance property as a criterion for reduction. Definition 4. By strong reduction we mean the following reduction procedure, which is applied to the entire collection of neighbourhoods in order to yield an analogous n n family of neighbourhoods: Given two melodic segments m = Mk,i and m = Mk,j of n length n and n respectively, where m = mj (with j + n − 1 ≤ n). The melody m is removed from results, if card(URn (m , M )) ≤ card(URn (m, M )), and if for all melodic segments m ∈ URn (m , M ), there exists a melodic segment m ∈ URn (m, M ) such that m = mnj . The employed condition guarantees that just those melodies are removed from the analysis, which appear only within similar ambient melodies at analogous positions. Therefore, we call this reduction method strong reduction.

5 Finding Subsequences A music-theoretical melodic analysis explains how the melodic material is introduced and used throughout the given piece. Therefore the segment relationships between melodies should be identified as well. Adiloglu and Obermayer (2005) defined presence and content of melodies to investigate these relationships. The inheritance property is utilised as a condition to define these sets. The presence of a given melody is the appearance of the melody within other ambient melodies of the same piece and within similar melodies to those ambient melodies. The content of a given melody is the sub-segments of the given melody. These neighbourhood sets are called weak, if they do not satisfy the inheritance property. However, for the strong neighbourhood sets the inheritance property is enforced. Here we define the presence neighbourhood sets only for the connectivity components, in order to simplify the interpretation of the presence of two similar melodic segments within their super-segments. Definition 5. The presence n’-neighbourhood for the connectivity component Ecn (M ) (consisting of melodies of length n) is defined to be the union of those connectivity compo nents Edn (M ) (each consisting of melodies of length n ) containing a melodic segment m , such that a melodic segment m in the connectivity component Ecn (M ) is a subsegment of m . The presence neighbourhood for the connectivity component Ecn (M ) is the union of all presence n’-neighbourhood sets where n ∈]length(m), length(M )]:


PEq n (Ecn (M )) =

73

n

Edn (M )|m = m i ,

m ∈Edn (M) m∈Ecn (M)

PEq (Ecn (M )) =

PEq n (Ecn (M )).

The presence n’-neighbourhood sets for a connectivity component eventually consists of several connectivity components, due to the fact that two melodies containing the same melody can possibly belong to separate connectivity components. In this case, all of these connectivity components are included into the presence n’-neighbourhood set of the given connectivity component. However, the following result helps two make inferences about the construction of the presence neighbourhood sets of the connectivity components. Corollary 1. Suppose that there are two melodic segments m = Mkn1 ,i+j and m = Mkn2 ,i +j ∈ Ecn (M ) such that they are contained by the sub-segments Mkn1 ,i and Mkn2 ,i of length n at analogous positions i + j and i + j respectively. These two super segments Mkn1 ,i and Mkn2 ,i belong to the same connectivity component Edn (M ), if the second part of the inheritance property holds. The similarity relation between two melodic segments does not necessarily imply a similarity between their super-segments. The inheritance property is necessary in order this statement to be true. Hence, Corollary 1 mathematically formulates the necessity of the inheritance property to imply a similarity relation between the super-segments of two similar melodic segments. In order to ease the interpretation of the relationships between the sub-segments of two similar melodic segments we define the content neighbourhood sets for the connectivity components as well. Definition 6. The content n’-neighbourhood for the connectivity component Ecn (M ) of melodies of length n is defined to be the connectivity components Edn (M ) of melodies of length n ( with n < n), each of which containing at least one melodic segment m , such that m is a sub-segment of a melodic segment m in the connectivity component Ecn (M ). The content neighbourhood for the connectivity component Ecn (M ) is the union of the content n’-neighbourhood sets for n ∈ [minimum melody length, length(m∗)[: CEq n (Ecn (M )) = Edn (M )|m = mni ,

m ∈Edn (M) m∈Ecn (M)

CEq (Ecn (M )) =

CEq n (Ecn (M )).

For the same reason as in the presence neighbourhood sets, the content n’neighbourhood sets of a given connectivity component can contain more than one connectivity component. The following corollary explains the role of the inheritance property in the construction of these sets.

74


Corollary 2. Suppose that there are two melodic segments Mkn1 ,i and Mkn2 ,i of length n which belong to the same connectivity component Ecn (M ), such that they contain the sub-segments Mkn1 ,i+j and Mkn2 ,i +j of length n at analogous positions i + j and i + j respectively. These two sub-segments Mkn1 ,i+j and Mkn2 ,i +j belong to the same connectivity component Edn (M ), if the first part of the inheritance property holds. During the construction of the connectivity components, all of the first- and higherorder similar melodies are collected in the same connectivity component. However, the inheritance property does not have an influence on constructing the connectivity components. It does not have an influence on the relationships between the connectivity components either. Hence it is not possible to enforce the inheritance property in this case. However the condition is true that if the second part of the inheritance property holds, for two similar melodies their super-melodies should be contained in the same connectivity component.

6 Melodic Topologies In the similarity neighbourhood model, we do not aim at defining a topological space at first place. However investigating the topological features of this model adds new valuable information to the analytical results and it helps to understand the commonalities and differences between strict topological approaches and ours. Three aspects of the model have been investigated in order to find out whether they define topologies Adiloglu and Obermayer (2006b). It is obvious that the connectivity components define a topological base, because the intersection of the them is always empty. What deserves attention is the intersection behaviour of the presence as well as of the content neighbourhood sets for the connectivity components. Theorem 1. The collection of presence neighbourhood sets for the connectivity components for a given a musical piece M define a base for a topology τ on the set {m|m ∈ M }, if and only if the inheritance property is satisfied. Proof. It is enough to show that given the connectivity components Ecn (M ) and Edn (M ) and a melodic segment m ∈ Edn (M ) such that m ∈ PEq (Ecn (M )), the following is true: PEq (Edn (M )) ⊂ PEq (Ecn (M )). If the inheritance property holds, there exists a melodic segment m such that m = and due to Corollary 2, m ∈ Ecn (M ). Suppose that there exists a melodic segment m ∈ Eun (M ) such that m ∈ PEq (Edn (M )). This means that there exists a melodic segment m such that m = n m j and due to Corollary 2, m ∈ Edn (M ). In the same way, due to Corollary 2, there should exist another melodic segment n o = m i such that o ∈ Ecn (M ). So n m i

PEq n (Ecn (M )) ⊃ Eun (M ) ⊂ PEq n (Edn (M )).


75

This statement is true for the content neighbourhood sets for connectivity components as well. However, in order to prove it for the content neighbourhood case, Corollary 1 together with the inheritance property is used. 6.1 Melodic Topologies on the Syntagms The experiments have shown that the reduction process decreases the number of results quite efficiently. This step makes the obtained results more clear by simply eliminating the irrelevant repetitions. However, the reduction process considered from the topological point of view does not make the results obey the inheritance property. For both of the reduction methods, the remaining similarity neighbourhood sets contain at least one melodic segment, whose super-segment is not contained in the similarity neighbourhood set like the other super-segments. Otherwise the similarity neighbourhood of the sub-segments would be reduced. The following expresses this statement in a mathematical way: ∃m such that m ∈ URn (m, M ). ∀URn (m , M ), where n > n, there is no m ∈ URn (m , M ), for which the following is true: m ⊂ m . Hence, the similarity neighbourhood set URn (m, M ) will not be reduced. The fact will in turn cause that the weak as well as the strong presence neighbourhood set (P resn (m, M ) or P resn (m, M ) R

R

respectively) of the similarity neighbourhood set URn (m, M ) will not contain a melody m , which is a super-segment of the melody m. Because of this fact, it cannot be guarann teed that the presence neighbourhood set PEq (Ecn (M )) for the connectivity component n m ∈ Ec (M ) contain a melody m , which is a super-segment of the given melody m. The melodic segment m, which secured the corresponding similarity neighbourhood set from reduction causes that the similarity neighbourhood set in relation to other similarity neighbourhood sets violates the inheritance property. In the previous section, We considered the cases, where the inheritance property does not hold. Therefore we will not repeat them here. As a consequence, the reduction process does not reform the results so that the inheritance property holds after the reduction has been applied. Therefore, theoretically it is not possible to prove that the inheritance property is satisfied for the similarity neighbourhood sets as well as for the connectivity components of the reduced results. Hence the same theorem proven in the previous section are also valid for the reduced results concerning the melodic topologies. Even-though, it is not possible to prove theoretically, that the reduced results define a topological base, experimentally the topological investigations can differ for the reduced and non-reduced case. In the following section, we investigated the Two-Voice Inventions from a topological viewpoint. 6.2 Investigation of the Inventions The analyses of the two-voice inventions have been tested for their topological features before and after the reduction process. For each case, the collection of the similarity neighbourhood sets defines a topological base. However, the collections of the presence and content neighbourhood sets for the connectivity components do not always define a topological base. This depends upon the concrete inventions.

76

K. Adilo˘glu and K. Obermayer Table 1. Topological Investigation of the Two-Voice Inventions Inv’s Inv 01 Inv 02 Inv 03 Inv 04 Inv 05 Inv 06 Inv 07 Inv 08 Inv 09 Inv 10 Inv 11 Inv 12 Inv 13 Inv 14 Inv 15

Prototypes Pres Cont 19 − 58 5 − 7 42 − 108 5 − 7 21 − 38 5 − 7 36 − 61 5 − 7 48 − 95 5 − 7 19 − 45 5 − 7 15 − 37 5 − 7 12 − 97 5 − 7 30 − 51 5 − 7 25 − 40 5 − 7 26 − 73 5 − 7 32 − 57 5 − 7 18 − 37 5 − 7 51 − 73 5 − 7 17 − 37 5 − 7

Syntagms Pres Cont 58 58 108 108 38 38 61 61 95 95 45 45 37 37 97 97 51 51 40 40 73 73 57 57 37 37 73 73 37 37

Table 1 summarises the topological investigation of the whole corpus. The larger of these two numbers indicate that the longest melodies within the connectivity components are 58 notes long. The longer melodies are singletons. The smaller number means that from this length on, the connectivity components would define a topology. However, connectivity components of the shorter melodies break the rules to define a topological base. Single numbers mean that the whole collection defines a topology. Table 1 shows that the connectivity components constructed without performing the reduction, which are shown in the “Prototypes” column, do not define topologies. The larger values in the “Prototypes” columns indicate the maximum length melodies identified within the piece. The smaller ones indicate the shortest melodies, for which the complete set of melodies up to the longest melodies defines topological base. However, shorter melodies exist, which do not satisfy the topological requirements. On the other hand, the connectivity components constructed after the reduction (“Syntamgs” column) define topological bases not only for the presence neighbourhood sets but also for the content neighbourhood sets.

7 Conclusion The similarity neighbourhood model is a simple but effective model to perform paradigmatic melodic analysis of pieces. The aim of the model is to help identifying the melodic structure of a given piece. These results can be used as input to perform further analysis, such as syntagmatic analysis by considering the interaction of the similarity relations between melodies with their environment or harmonic analysis by studying the solidarity of the melodic variation with harmonic progression. Since the distance measure can measure only the similarities between equal length melodies, the relationships between different length melodies cannot be measured in


77

the same way. Therefore the presence- and content neighbourhood sets were defined to explain the sub-segment relationships of melodies. From the music-theoretical viewpoint, the relations between the melodies of different length are explained to understand how the melodic material is introduced and used throughout the musical piece. The development of the similarity neighbourhood model does not aim to define a topology. Buteau 2001 (2001), Buteau 2003 (2003), and Mazzola (2002) actually make use of the inheritance property in order to construct topological bases. Such a basis can be actually defined by exhausting the limits of inheritance. Some of the distance measures tested on these models obey the inheritance property, such that it can be proven mathematically, without considering the concrete musical piece analysed. Hence the generation of a topological base is guaranteed for these distance measures. For the similarity neighbourhood set, the collection of the connectivity components define a topological base simply because the intersections of the connectivity components is always empty. This topology, however, does not yield any valuable information to the analysis process. On the other hand, the presence as well as the content neighbourhood sets for connectivity components of the similarity neighbourhood model can define a topological base depending on the given musical piece as long as the inheritance property is satisfied by these sets. The tests have shown that the collection of the reduced results satisfy inheritance property for the whole corpus of Two-Voice Inventions. The reduction process inspired by the ideas of Ruwet produces music-theoretically relevant results Adiloglu and Obermayer (2006c). From a mathematical point of view, results obtained after the reduction process have a mathematically stable structure. Even-though a mathematical proof is not possible that the inheritance property is satisfied, practical results are promising in the sense that reduction yields a music-theoretically relevant results as well as brings to the results a mathematical structure.

References Adiloglu, K., Obermayer, K.: Finding Subsequences of Melodies in Musical Pieces. In: Proceedings of ICMC. Pompeu Fabra University, Barcelona (2005) Adiloglu, K., Noll, T., Obermayer, K.: A Paradigmatic Approach to Extract the Melodic Structure of a Musical Piece. Journal of New Music Research 35(3), 221–236 (2006) Adiloglu, K., Obermayer, K.: Melodic Topologies. In: Proceedings of ICMC. Tulane University, New Orleans (2006) Adiloglu, K., Obermayer, K.: A Reduction Method for the Paradigmatic Melodic Analysis. In: Mathematics and Computation in Music, Berlin (2002) (in print) Chantal, B.: Reciprocity between Presence and Content Functions on a Motivic Composition Space. Tatra Mt. Math. Publications 23, 17–45 (2001) Buteau, C.: A Topological Model of Motivic Structure and Analysis of Music: Theory and Operationalization. PhD Thesis. University Zurich, Zurich (2003) Mazzola, G.: The Topos of Music. Birkhäuser, Basel (2002) Ruwet, N.: Methods of Analysis in Musicology. Music Analysis 6, 11–36 (1987)

Comparing Computational Approaches to Rhythmic and Melodic Similarity in Folksong Research Anja Volk1 , J¨ org Garbers1 , Peter van Kranenburg1, Frans Wiering1 , Louis Grijp2 , and Remco C. Veltkamp1 1

Department of Information and Computing Sciences, Utrecht University 2 Meertens Institute Amsterdam

Abstract. In this paper we compare computational approaches to rhythmic and melodic similarity in order to find relevant features characterizing similarity in a large collection of Dutch folksongs. Similarity rankings based on Transportation Distances are compared to an approach of rhythmic similarity based on Inner Metric Analysis proposed in this paper. The comparison between the two models demonstrates the important impact of rhythmic organization on melodic similarity.

1

Introduction

Computational approaches to melodic similarity such as proposed by Ahlb¨ ack (2004), M¨ ullensiefen (2004) and Typke (2007) contribute to the study of melodies in the different areas of music cognition, ethnomusicology and music information retrieval. In this paper we study rhythmic similarity in the context of melodic similarity as a first step within the interdisciplinary enterprise of the WITCHCRAFT project (Utrecht University and Meertens Institute). The project makes use of and contributes to methods in these three areas in order to develop a content based retrieval system for a large collection of Dutch folksongs. The retrieval system will give access to the collection Onder de groene linde hosted by the Meertens Institute to both the general public and musical scholars. For the latter it is of special interest to be able to classify, identify and trace melodic variants with the help of the retrieval system to be designed. The similarity between different variants of a folksong melody is based on a variety of musical dimensions, such as rhythm, contour or cadence notes. According to cognitive studies metric and rhythmic structures play a central role in the perception of melodic similarity. For instance, in Ahlb¨ ack 2004 an exact repetition of a pitch sequence has not be recognized if it was not congruent with the fundamental metrical structure.In the immediate recall of a simple melody studied in Sloboda 1985 metrical structure was the most accurately remembered structural feature. In this paper we focus on rhythmic similarity by comparing similarity rankings based on Inner Metric Analysis (IMA) to Transportation Distances (see Typke 2007 and Bosma et al. 2006). T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 78–87, 2009. c Springer-Verlag Berlin Heidelberg 2009

Comparing Computational Approaches to Rhythmic and Melodic Similarity

79

Transportation Distances have been successfully applied to the measurement of melodic similarity of, for instance, RISM incipits or karaoke pieces (see Typke 2007). By excluding the pitch factor we apply the Transportation Distances in this paper to rhythm only in order to study the impact of rhythm on melodic similarity. Inner Metric Analysis has been successfully applied to the study of metric structures of musical pieces in the context of music analysis (Fleischer 2003), music cognition (Volk 2003) and classification (Chew et al. 2005). We therefore propose in this paper an approach to the measurement of rhythmic-metric similarity based on IMA and compare the results to those of the Transportation Distances.

2 2.1

Two Computational Approaches to Rhythmic Similarity Transportation Distances

Transportation distances consider melodies as weighted point sets. A similarity (or distance) measure between two melodies is defined on the basis of weight flows between these point sets that have to be minimized. As a metaphor the point set of one melody is regarded as heaps of sand and the point set of the second melody as holes in the ground. Transportation distances compute the minimum amount of work needed to fill the holes with sand. We compare two instances of these distances, namely the Earth Mover’s Distance (EMD) and the Proportional Transportation Distance (PTD). Both distance measures and their application to melodies are described in detail in Typke 2007. In the application to melodies, every note is a point in the Euclidean space with the two coordinates pitch and onset time, the duration of the note determines its weight. In this article we apply these distances to rhythms instead of melodies, hence the coordinates of the points are determined by the onset time only. Figure 1 gives an example for two short rhythms with the minimal flow of weights according to the EMD. The arrows indicate which amount of weight from the first rhythm is transported to which note in the second rhythm. For instance, the first rhythm is being described with the point set r1 = {(0.0, 1.0), (1.0, 0.5), (1.5, 0.25), (1.75, 0.25), (2.0, 0.5), (2.5, 0.5), (3.0, 1.0)}. The first coordinate of each point is the onset time of the note, the second is its weight (which equals its duration). The difference between EMD and PTD becomes evident in the similarity comparison between pieces of different total weight, which is in this case the total length. The EMD realizes partial matching, hence ignores the extra notes in the longer piece. Within the PTD approach the total weight of each piece is normalized in order to prevent partial matching, hence the existence of extra notes that can not be matched in the second melody is effectively penalized. 2.2

Inner Metric Analysis

Inner Metric Analysis (see Mazzola 2002, Fleischer 2003) describes the inner metric structure of a piece of music generated by the actual notes inside the

80

A. Volk et al.

Fig. 1. Example of minimal weight flow between two rhythms. The point size depicts the weight.

bars as opposed to the outer metric structure associated with a given abstract grid such as the bar lines. The model assigns a metric weight to each note of the piece. Figure 2 gives an example for the song OGL 199141 (from the collection Onder de groene linde) belonging to the melody group Deze morgen. The notes of the first phrase are shown in the top example of figure 4. For each note a line depicts the metric weight such that the higher the line, the higher the corresponding weight. The background gives the bar lines for orientation. The metric weight profile corresponds to the typical accent hierarchy of a 6/8 bar.

Fig. 2. Metric weight of OGL 19914, melody group Deze morgen in 6/8

The details of the model have been described in Fleischer 2003 and Chew et al. 2005. The general idea is to search for all pulses (chains of equally spaced events) of a given piece and then to assign a metric weight to each note. The pulses are chains of equally spaced onsets of the notes of the piece called local meters. Let On denote the set of all onsets of notes in a given piece. We consider every subset m ⊂ On of equally spaced onsets as a local meter if it contains at least three onsets and is not a subset of any other subset of equally spaced onsets. Let k denote the number of onsets a local meter consists of minus 1. Hence k counts the number of repetition of the period (distance between consecutive onsets of the local meter) within the local meter. The metric weight of an onset is then calculated as the weighted sum of the length k of all local meters mk that coincide at this onset.

1

OGL as the abbreviation of Onder der groene linde.


81

Let M () be the set of all local meters of the piece of length at least . The general metric weight of an onset, o ∈ On, is as follows: W,p (o) = kp . {m∈M():o∈mk }

In all examples of this paper we have set the parameter = 2, hence we consider all local meters that exist in the piece. In order to obtain stable layers in the metric weights of the folksongs we have chosen p = 3. 2.3

Defining Similarity Based on Inner Metric Analysis

Metric weights of short fragments of musical pieces have been used in Chew et al. 2005 to classify dance rhythms of the same meter and tempo using the Pearson correlation coefficient. In this article we want to modify this approach to measure the similarity between the rhythms of two complete melodies in terms of the metric structure implied by these rhythms. The similarity measurement is hence carried out on the analytical information given by the metric weights. Since the metric weight is defined only on note onsets, we define in a first step for each of the two pieces the metric weight of all silence events as zero and hence obtain the metric grid weight. The silence events are inserted along the finest grid of the piece which is determined by the shortest existing interval between two consecutive onsets of the piece. Thus we obtain a weight for all events e of the piece along the finest onset grid. We want to compare the consecutive weights within cells of equal total duration (for instance 4 quarter notes in length) of the two pieces. Therefore in cases where the finest onset grids of the two pieces differ, we adapt the grids of the pieces to a common finer grid by adding events e with the weight zero along the finer grid. In the second step, the metric grid weight is split into consecutive segments that cover an area of equal duration in the piece. These segments contain the

Fig. 3. The first three correlation windows of two metric weights to be compared

82

A. Volk et al.

weights to be compared with the Pearson correlation coefficient, we therefore call them correlation windows. The first correlation window of each piece starts with the first full bar, hence the weights of an upbeat are disregarded. For all examples of this article we have set the size of the correlation window to one bar. Figure 3 shows an example for two metric grid weights with the first 3 correlation windows. For the computation of the similarity measure both grid weights are completely covered with disjoined correlations windows. Let wi, i=1,...,n denote the consecutive correlation windows of the first piece and vj, j=1,...,m those of the second piece. Let ck, k=1,...,min(n,m) denote the correlation coefficient between the grid weights that are covered by the windows wk and vk . Then we define the similarity IM Ac,s that is defined on the subsets of the two pieces from the beginning until the end of the shorter piece as the mean of all correlation coefficients: min(n,m) 1 IM Ac,s = ck min(n, m) k=1

The partial similarity IM Ac,s disregards all extra notes at the end of the longer piece. However, in many contexts it might be important to add a penalty for these extra notes that have no counterpart in the shorter piece. Therefore we define the correlation coefficient value between the additional correlation windows of the longer piece to the empty correlation windows of the shorter piece as zero ck, k=min(n,m)+1,...max(n,m) = 0. Hence we obtain the similarity measure IM Ac,e taking into consideration the entire pieces: IM Ac,e =

1 max(n, m)

max(n,m)

ck

k=1

We will use the latter measure for the application to rhythmic similarity ranking in section 3.

3

Evaluation of the Rhythmic Similarity Approaches

In this section we compare the similarity measurements based on IMA, EMD and PTD in a first and simple approach to rhythmic similarity of melodies. The application of these measurements is simple in so far as it does not contain a segmentation procedure and the search for similar segments that are shifted in time. Since the pieces contain musically meaningful segments (phrases) we applied IMA, EMD and PTD to both single phrases and complete pieces. The evaluation of the similarity measurements is based on melody groups of the collection Onder de groene linde (OGL) of related songs. The melodies belonging to one group are being considered as musically similar. The current test corpus of digitized melodies contains 141 songs which are segmented into 567 phrases in total. One melody (or melody phrase) of such a group is selected as the query and the similarity measure to all other melodies (or melody phrases) in the test corpus is calculated and ordered (the ordered list starts with the most similar melody).


83

A good similarity measurement should therefore list the members of the group the query belongs to among the top hits of the list, if the members are more similar to each other than to members of other melody groups. The melody groups in our test corpus that have been constructed by musicologists fulfill this condition to a certain extent. However, sometimes a very similar song was assigned to a different group based on other than musical reasons - for instance, because of the text. A typical comparison of those ranking lists include the number of melodies that should have been found within the top hits of the list (because they are group members), but get a very low rank (”false negatives”) and the number of melodies that end up high in the ranking list but do not belong to the same melody group as the query (”false positives”). Since our melody groups have not been tested to always contain the most similar melodies, we will in our comparison not only count the false positives but check whether they are nevertheless musically similar. In the following section 3.1 we discuss one example in detail using the melody group Deze morgen, in section 3.2 we summarize very briefly further results of the comparison. 3.1

A Detailed Comparison on the Melody Group Deze Morgen

The melody group called Deze morgen contains 12 melodies which are very similar to each other. However, two songs have one phrase less than the others. First we want to compare the results of the ranking lists for a single phrase, in the second step we will use the entire piece. As the query for the single phrase we used the first phrase of OGL 19914 (the top melody from figure 4). For the evaluation of the ranking list we focus on the ranks that have been assigned to the other first phrases of melodies in this group, since they are all rhythmically very similar to the query. The ranking list according to IMA contains among the first 19 elements 11 members of the group and misses among the top 20 hits only one phrase at rank 29 (see figure 5). All false positives with a better rank than 29 are musically very similar to the query (for instance, most of them are second phrases from melodies of the same group). Figure 4 lists the best hits from the ranking list according to IMA with the exclusion of melodies that duplicate the rhythmic structure of melodies that are displayed. Hence the displayed 9 melodies stem from the best 19 matches. PTD ranks 10 group members among the first 22 matches. Thus it misses the first phrase of OGL 37511 in figure 5 which is placed on rank 68 and a very similar phrase to the latter one is placed on rank 73. The first phrase of OGL 37511 was ranked lower than all other members of the group (rank 29) by IMA as well, indicating that this rhythm is somewhat less similar to the query. However, the low rank of 68 according to PTD is very drastic. For instance, figure 6 gives three examples of phrases that are assigned a higher similarity to the query according to PTD. These are rhythmically less similar to the query than the missed phrase from figure 5.

84

A. Volk et al.

86

Ik ben d'r van de

86

Ik

ben er van de

86

Ik ben van de

ze

mor

ze

86

Ik ben van de

86

En ik ben van de

86

Ik

86

Ik

ben van de

86 Ik

ben van de

ben van de

mor

ze

mor

En ik ben er van de

86

mor

ze mor

ze mor

mor

ze

mor

ge

staan,

gen en vroeg op

mor

ze

gen d'r vroeg op

gen vroeg op

gen vroeg op

ze

ze

staan.

staan

ge

staan

re gen vroeg op

ge

staan

gen vroeg op

ge

gaan

gen vroeg op

ge

staan

ge

staan

gen vroeg op

ge

staan

re gen vroeg op

ge staan,

Fig. 4. Excerpt from the top hits from the list according to IMA (melodies of same rhythm excluded, the listed rhythms cover the first 19 matches)

86

Ik

ben van de

ze

mor

9 8

re gen en vroeg

op

ge staan,

Fig. 5. IMA assigns OGL 37511 rank 29

One of the reasons for the low rank of OGL 37511 are two long notes near the end of the query (the notes of the syllables ”op” and ”staan” in the top melody of figure 4). Both of them do not have a counterpart in OGL 37511 and therefore their weight is distributed over 5 different notes each that are located much earlier in the piece. In contrast to this, the first phrase shown in figure 6 contains many notes that are located in roughly the same area as the long end notes of the query. Also here the weight is distributed among 4 to 6 notes, but this weight has to be transported only locally and not to notes far apart as in OGL 37511. This results in a much higher similarity ranking. The false positives within the first 24 matches of the PTD list are all rhythmically similar except rank 11 which is shown in figure figure 6. On the other hand the ranking list according to IMA contains up to 29 similar elements in the beginning, hence the last elements are being missed by PTD.


86

Hij vroeg me

en o

op

de

zen dag of ik zijn huisvrouw we zen mag.

wee, en o wee en ach had den we maar een

46 Waar

zo

85

ve

le

he

ren

lo

ge

vree

ren

Fig. 6. PTD assings rank 11 (top melody), rank 34 (middle melody) and rank 50 (bottom melody)

86 Een

juf

fer

tje

fijn

Fig. 7. Melodic phrase at rank 3 according to EMD

86

Ik

ben van de

ze mor

re gen vroeg op

ge

staan

Fig. 8. Melodic phrase at rank 79 according to EMD

EMD ranks 10 group members among the first 24 matches and misses two phrases at the ranks 58 and 79. The false positive on rank 3 (see figure 7) demonstrates the partial matching of the EMD: since the 5 notes of that very short phrase can be matched with the first 5 notes in the query, this phrase gets a rather high similarity measurement. Among the false positives within the first 24 matches are in sum 4 examples of such shorter melodies that are rhythmically not very similar to the query. The low rank 79 for the first phrase of OGL 25904 (see figure 8) has its main reason in the existence of many shorter melody phrases in the test corpus. These phrases match a part of the query with a lower total weight than the weight of this phrase leading to a higher similarity value. In summary all three methods miss only very few group members by assigning them a low rank within the list. While the list ordered according to IMA contains 29 rhythmically similar melodies at the top and covers in this range also all first phrases from the group Deze morgen, both PTD and EMD miss 2 phrases. For the comparison of the entire melodies using OGL 19914 as the query we will give only a very short overview. The ranking list according to IMA contains within the first 11 hits 1 false positive and misses 2 melodies that have one phrase less then all other group members (rank 27, 57). The ranking list according to PTD contains within the first 17 hits 7 false positives at the beginning of the list. The two melodies being missed at the beginning (rank 45 and 107) are again the songs that have one phrase less. The ranking list according to EMD contains

86

A. Volk et al.

within the first 17 hits 9 false positives, three melodies have a much lower rank, two of them are the same being missed by IMA and PTD. Hence the melody group of Deze morgen is an example that demonstrates how much the rhythmic structure alone determines similarity. The comparison of the three models reveals the best results for the IMA, while the EMD has the most false positives due to its partial matching. 3.2

Summary of Further Results

A problem in the application of the PTD occurred in the comparison of last phrases. For instance, in the ranking list for the last phrases of the melody group called Heer Halewijn A both IMA and EMD find 9 similar phrases among the first 10 respective 11 hits. In contrast to this, PTD ranks only 2 of them at the top of the list, the others are ranked lower than rank 31. The reason for this is a difference in the duration of the last note (due to different transcription strategies of the recorded melodies). If the last note in all the examples is adjusted to the same duration, PTD lists 8 of the phrases among the top 11 melodies. Similar effects are observed in the ranking lists for the entire melodies. In most of the examples IMA gains the best result. However, if the query is rhythmically only very little differentiated (such as a quasi continuous eighth note chain), then the results of both Transportation Distances are more convincing. In general rhythmic similarity seems to be an important component of the similarity of melodies in the current test corpus of melodies from Onder de groene linde.

4

Conclusion

The aim of the comparison of the computational approaches to rhythmic similarity in this paper is a first test in how far the different methods are suited for finding rhythmically similar melodies. For the application of the PTD a solution concerning the length of the last note of a phrase has to be found. For the application of the EMD it might be necessary to filter out hits that are much shorter then the query if one is not interested in partial matching. The use of the metric weights obtained by IMA in their role as weights in the Transportational Distances instead of the duration could be a promising merge of the two models. The application of the Transportational Distances to pitches only while ignoring the rhythm information and a comparison to the results obtained in this paper is a further step towards the investigation of the importance of rhythmic similarity in the context of melodic similarity.

References Ahlb¨ ack, S.: Melody beyond notes. PhD thesis. G¨ oteborgs Universitet (2004) Bosma, M., Veltkamp, R.C., Wiering, F.: Muugle: A framework for the comparison of Music Information Retrieval methods. In: Proceedings of the ICMPC 2006, pp. 1297–1303 (2006)


87

Chew, E., Volk, A., Lee, C.-Y.: Dance Music Classification Using Inner Metric Analysis. In: Proceedings of the 9th INFORMS Computer Society Conference, pp. 355–370. Kluwer (2005) Fleischer (Volk), A.: Die analytische Interpretation. Schritte zur Erschließung eines Forschungsfeldes am Beispiel der Metrik. dissertation. de - Verlag im Internet Gmbh, Berlin (2003) Mazzola, G.: The Topos of Music. Birkh¨ auser, Basel (2002) M¨ ullensiefen, D.: Variabilit¨ at und Konstanz von Melodien in der Erinnerung. PhD Thesis, Hamburg (2004) van Dijk, M.B.G., Kuijer, H.J., Dekker, A.J. (eds.): Onder de groene linde. Verhalende liederen uit de mondelinge overlevering. Uitgeverij Uniepers, Amsterdam (19871991) Sloboda, J.A., Parker, D.H.H.: Immediate recall of melodies. In: Hower, P., Cross, I., West, R. (eds.) Musical structure and cognition, pp. 143–167. Academic Press, London (1985) Typke, R.: Music Retrieval Based on Melodic Similarity. PhD thesis, Utrecht University (2007) Volk, A.: The Empirical Evaluation of a Mathematical Model for Inner Metric Analysis. In: Proceedings of the 5th Triennal ESCOM Conference, Hanover (2003) Wiering, F., Veltkamp, R.C., Typke, R.: Transportation Distances in Music Notation Retrieval. Computing in Musicology 13, 113–128 (2004)

Automatic Modulation Finding Using Convex Sets of Notes Aline Honingh Music Informatics Research Group, Department of Computing, City University, London [email protected] Abstract. Key finding algorithms, designed to determine the local key of segments in a piece of music, usually have difficulties at the locations where modulations occur. A specifically designed program to indicate modulations in a piece of music is presented in this paper. It was previously shown that the major and minor diatonic scale, as well as the diatonic chords, form convex sets when represented in the Euler lattice (Honingh and Bod 2005). Therefore, a non-convex set within a piece of music may indicate that this specific set is not part of a diatonic scale, which could indicate a modulation in the music. A program has been developed that finds modulations in a piece of music by localizing nonconvex sets. The program was tested on the first five preludes and fugues in a major key from the first book of Bach’s Well-tempered Clavier. It has been shown that the algorithm works best for modulations that involve many chromatic notes.

1

Introduction

When a piece of music is said to be in a specific key, we usually mean that the piece starts and ends in this key. It sometimes happens that the piece is entirely in the same key, however often, other keys occur at several instances in the music. A modulation is the act or process of changing from one key to another. In the research of key finding (see for example Krumhansl 1990; Temperley 2001; Longuet-Higgins and Steedman 1971; Chew 2002, 2006), the most difficult part of the analysis is usually formed by the modulations. Vos and van Geenen (1996) developed a key-finding model and detected only two of the six modulations that were analyzed by Keller (1976), when the model was tested on the 48 fugues of Bach’s Well-Tempered Clavier. Furthermore, it also found modulations in 10 other cases in which there was no modulation according to Keller (1976). Temperley (2001) tested his model on the same corpus and found only two of the modulations correctly. Therefore, a specially designed program to indicate the modulations in a piece of music would be a helpful tool to implement in several key finding models.

2

Probability of Convex Sets in Music

It has been observed that the major and minor diatonic scales as well as the diatonic chords form so-called convex sets if they are represented in the Euler T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 88–96, 2009. c Springer-Verlag Berlin Heidelberg 2009

Automatic Modulation Finding Using Convex Sets of Notes Bbb

Db

F

A

Ebb

Gb

Bb

D

F#

Abb

Cb

Eb

G

B

D#

Dbb

Fb

Ab

C

E

G#

B#

Bbb

Db

F

A

C#

E#

Gb

Bb

D

F#

A#

Eb

G

B

D#

89

Fig. 1. Representation of the Euler lattice with the major and minor diatonic scale indicated by a region in lines and dashed lines respectively

lattice1 (Honingh and Bod 2005). This means that the regions or shapes that are described by these scales in the Euler lattice do not have holes or inlets, see figure 1. Let us briefly go into this notion of convexity. Each note can be found at more than one location in the Euler lattice2 (for example, we can find two instances of the note D in fig. 1), which means that each chord can be represented by several configurations of notes in the Euler lattice (see fig. 2). A set of notes defined by its note names (like C, E, G) is said to be convex if (at least) one of its configurations in the Euler lattice constitutes a shape without any holes or inlets (A more formal definition and detailed explanation is given in Honingh 2006). An example is given in figure 2. The finding that the diatonic scales and chords form convex sets, might suggest that non convex subsets of the diatonic scales are not so common. If this is indeed the case, a non-convex set within a piece of music may indicate that this specific set is not part of a diatonic scale which could in turn indicate a modulation in the music. To verify the correctness of this reasoning, we need to investigate the convexity of all possible subsets of the diatonic scales. Hence, we will address the question “what is the chance for a set of n randomly chosen notes from a piece of music to be convex?”. Assuming a certain piece is in one and the same key, this means calculating the chance that a subset of n notes from a scale is convex. We calculate for each possible n-note set that is a subset of the major diatonic scale, whether it is convex or not. This results in a percentage of convex sets. A Matlab program was written for this purpose. The results are displayed in table 1. The values 1 and 7 are left out because the convexity of one note does not mean anything, and there is only one configuration for 7 notes within one scale which is the whole scale and which is necessarily convex. The Euler lattice 1 2

The ‘Euler lattice’ and minor variants of it are known under various names such as ‘Tonnetz’, ‘Oettingen lattice’, and ‘harmonic network’. The different positions of a note in the Euler-lattice are connected to their frequency ratios. See Honingh (2006) for more information.

90

A. Honingh Fb

Dbb

Ab

Bbb

Db

F

Gb

Bb

D

F#

A#

Cb

Eb

G

B

D#

F##

Fb

Ab

C

E

G#

B#

D##

Db

F

A

C#

E#

Bb

D

F#

A#

C##

Bbb

C#

Ebb

Gb

Dbb

A

G

B

D#

G##

F## Fb

Ab

Bbb

Db

F

Ebb

Gb

Bb

D

F#

A#

Abb

Cb

Eb

G

B

D#

F##

Fb

Ab

C

E

G#

B#

D##

Db

F

A

C#

E#

G##

C##

Gb

Bb

D

F#

A#

Eb

G

B

D#

F##

C A

Fb

Ab

Bbb

Db

F

E

Abb

Bbb

Eb

C

Dbb

C A

C#

Ebb

Gb

Bb

D

F#

A#

Abb

Cb

Eb

G

B

D#

F## D##

Fb

Ab

C

E

G#

B#

Bbb

Db

F

A

C#

E#

G##

Gb

Bb

D

F#

A#

C##

Eb

G

B

D#

F##

E C#

Dbb

E

Fb

Ab

Bbb

Db

F

C A

E C#

Ebb

Gb

Bb

D

F#

A#

Abb

Cb

Eb

G

B

D#

F## D##

Fb

Ab

C

E

G#

B#

Bbb

Db

F

A

C#

E#

G##

Gb

Bb

D

F#

A#

C##

Eb

G

B

D#

F##

Fig. 2. Possible configurations of the triad C, E, G. The set C, E, G is said to be a convex set since the first configuration in this figure constitutes a shape without any holes or inlets. Table 1. Percentage of n-note sets that are convex if chosen from a major scale number of notes in percentage convex the set 2 100 % 3 94.29 % 4 94.29 % 5 100 % 6 100 %

is an infinitely large 2-dimensional lattice, however for reasons of computing, we consider here a 9 × 9 lattice. This is big enough to contain all sets that we want to consider and it contains also enough configurations of a set to calculate whether it is a convex set or not. Given a set of note names to the computer program, the program computes every configuration in the 9 × 9 lattice. In the 9 × 9 plane, every note name has 2 or 3 possible configurations. Therefore, if a set consists of n notes, the number of possible configurations lies between 2n and 3n . One could argue that the notes from a piece of music in one key do not only come from one scale, even if the piece of music is in one and the same key. Often more notes appear in a piece of music than only the notes from the scale. For example, in the first fugue from the Well-tempered Clavier of Bach, which is written in C major, the notes that appear throughout the piece are the notes from the major scale in C plus the additional notes F , B, C and G. The idea that the key contains more notes than the scale of the tonic has been formalized

Automatic Modulation Finding Using Convex Sets of Notes

91

Table 2. Percentage of n-note sets that are convex if chosen from the set of notes representing the C major scale with an additional F number of notes percentage in the set convex 2 100 % 3 92.86 % 4 88.57 % 5 92.86 % 6 100 % 7 100 % Table 3. Percentage of n-note sets that are convex if chosen from the set of notes representing the C neutral minor scale with additional D, A, E, B, F number of notes percentage in the set convex 2 100 % 3 80.91 % 4 59.80 % 5 52.02 % 6 49.03 % 7 51.01 % 8 58.79% 9 71.82% 10 89.39 % 11 100%

by, among others, Van de Craats (1989). He claims that in a major key, the augmented fourth is often used and should therefore be included in the scale. This means that in C major, the scale would contain the notes (given in a fifth sequence): F, C, G, D, A, E, B, F . A piece of music in C minor can contain the notes (given in a sequence of fifths): D, A, E, B, F, C, G, D, A, E, B, F , according to Van de Craats. In accordance with the latter claim, Longuet-Higgins (1987) states that “a note is regarded as belonging to a given key if its sharpness3 relative to the tonic lies in the range -5 to +6 inclusive”. Results by other researchers (Youngblood 1985; Knopoff and Hutchinson 1983; Krumhansl and Kessler 1982) are in agreement with Longuet-Higgins’ and Van de Craats’ suggestions. These ‘scales’ of 8 and 12 notes respectively can be used as new key-contents for our Matlab program, to calculate the percentages of sets that are convex. The results can be found in tables 2 and 3. The bigger the total set of notes is to choose from, the higher are the percentages of non-convex subsets. Therefore, in table 3 the percentages of convex sets decrease to a minimum of 49.03% at n = 6, meaning that there is a reasonable chance of finding a 6 note set that is non-convex in a piece of music written in a minor key. From both tables 2 and 3 we see that the highest 3

Sharpness is understood here as the position of pitch name on the line of fifths.

92

A. Honingh

percentages of convex sets appear for the smallest and biggest possible sets in the key. This suggests that the smallest and the biggest non-convex sets are the best indicators of modulations. From the above results we learn that if we choose randomly a set of notes from one key, there is a high chance for the set to be convex. Therefore, we hypothesize that, if we analyze a piece of music by dividing it into sets of n notes, most of the sets are convex. It is thus more special in a piece of music for a set to be non-convex than convex. And because we have seen that sets from one key tend to be convex, a non-convex set within a piece could point to a change of key or modulation. 2.1

Finding Modulations by Means of Convexity

A Matlab program is written that finds modulations in a piece of music by localizing non-convex sets. The more sets that are not convex around a certain location, the stronger is the indication of a change of key. To be able to judge all n-tone sets on convexity, we introduce a sliding window of width n moving over the piece. We start with a window of width 2 after which we enlarge it to 3, etc. We stop at a width of 7 notes, since non-convex n-tone sets with n > 7 rarely occur for a major key. Furthermore, for n > 7, the computation gets highly intensive since all possible configurations (which is a number between 2n and 3n ) should be checked. For each non-convex set a vertical bar is plotted at the position of the notes in the piece that it affects. The hight of the bar represents the number of notes in the set. For each n, a sliding window is moving over the piece resulting in a histogram. These histograms belonging to n = 2 to 7 are plotted in the same figure such that the result is one histogram presenting all non-convex sets in a piece of music. If a set of notes contains several identical notes, the set will be reduced to the set of notes that contains of each note only one. For example, the set of notes {D, E, D, F } gets reduced to the set {D, E, F }. If such a set turns out to be non-convex, it will be indicated in the histogram with a bar of which the hight is associated with the number of notes in the reduced set. The music that we tested the model on is from the Well-tempered Clavier of Bach. Data files containing the notes and other information from all preludes and fugues in the first book of J. S. Bach’s Well-tempered Clavier (BWV 846869) was made available by Meredith (2003). We used these files as input for our program. The only input used by our model are the note names, so no rhythm, meter, note length, key information etc. was involved. As an example we consider the third prelude from book I of the Well-tempered Clavier. The bars in figure 3 show the position of the non-convex sets in the piece. The xaxis represents number of bars in the piece of music, the prelude consists of 104 bars. The values on the y-axis indicate the number of notes in a set. Looking at figure 3 we see three regions in the music in which a lot of non-convex sets appear. We will now see how these regions relate to the structure of the piece. In Bruhn (1993), an analysis of the third prelude can be found. The analysis states that from bar 31 to 35 there is a modulation from A minor to D minor,


93

Bach WTC, BWV848a

number of notes in non−convex set

7

6

5

4

3

2

1

10

20

30

40

50

60

70

80

90

100

measure no.

Fig. 3. Histogram of non-convex sets in the third prelude from the Well-tempered clavier. On the x-axis, the number of bars in the piece is indicated, the y-axis indicates the number of notes in the non-convex set.

from bar 35 to 39 a modulation from D minor to G major, from bar 39 to 43 a modulation from G major to C major and from bar 43 to 47 a modulation from C major to F major. One can see from figure 3 that this region of modulations in bars 31 to 46 is precisely indicated by the first cluster of bars. Looking to the second cluster of bars (bars 63 to 72) in figure 3, one can see that this pattern is repeated a bit later in bars 87 to 96 . These two regions correspond to two (similar) pieces in G having a pedal on the tonic. There are no modulations involved but the notes of the seventh chords are melodically laid out in a way that in forming sets often the fifth is omitted and therefore some sets are nonconvex. The last region of bars in figure 3 is from bar 97 to 102. This region represents a melodic line in which a lot of chromatic notes are involved. One can not become aware of one specific key until the last two bars where the piece again resolves in C major. In the regions in between the marked parts (white space in fig 3) no modulations are present. In those regions the music is in a certain key, which can vary over time, i.e. there can be (sudden) key changes from one bar to another. This method of looking at non-convex sets is therefore only suitable for longer modulation processes. We have learned that sets consisting of 6 and 7 notes give a stronger indication to a change of key than sets consisting of less notes. Therefore regions 1 (bars 31 to 46) and 4 (bars 97 to 102) have strong(-er) indications of a modulation than regions 2 (bars 63 to 72) and 3 (bars 87 to 96) which is in accordance with the analysis of the piece. Thus, this third prelude serves as an indication that the modulation finding program works well. Unfortunately, the method did not work well for all pieces. For pieces in a minor key it was not sufficient to calculate the non-convex sets up to 7 notes, since, according to table 3 we can learn most about these pieces if we look at

94

A. Honingh

10 or 11 note sets. Since the number of possible configurations of a set of notes is a number between 2n and 3n , the analysis of configurations in a minor key requires too much computational time.

3

Results

It is difficult to test the performance of the algorithm for a number of reasons. From the side of looking at the histograms it is difficult to decide which instances of plotted bars to count as an indication of a modulation. We would have to decide on an offset regarding both the number of notes in the non-convex set and the number of non-convex sets. To give an example, an instance of a nonconvex set of 4 notes would not indicate a modulation, but perhaps a nonconvex set of 7 notes would; and one instance of a non-convex set would not be a strong indication of a modulation, but three instances are. From the side of the music, it is difficult from the point of a definition of a modulation. Many types of modulations exist and our algorithm works for some better than for others. For example, a modulation can be made by using a common chord of two (closely related) keys, in which case the algorithm would have difficulty indicating the modulation. The algorithm performs best on modulations that involve some chromatism in between one key and the other, like for example a sequential modulation. Since we do want to be able to make a general judgment on the performance of the algorithm, we decide on the following. We count only the instances that include non-convex sets of 6 or 7 notes in the histogram as indications of modulations. Furthermore, we decide on counting every peak in the histogram as one modulation if the peaks are widely separated, which is, more than three bars apart. More than one peak within three bars therefore merges to one instance of a modulation. We checked the results using the analysis of Bruhn (1993). This work gives a thorough analysis of the Well tempered Clavier, however it was sometimes difficult to interpret the true points of modulation from it. Therefore, we have chosen to only count a modulation when either Bruhn uses the word ’modulation’, or a clear key change is pointed out, or a region of chromatism is pointed out. Herewith, note that we also count the pieces of chromatism that can occur within one key, which therefore do not indicate a real modulation. The algorithm was tested on the first five preludes and fugues in major keys from the first book of Bach’s Well-tempered Clavier. The results are given in table 4. The number of correctly indicated modulations represents the total number of modulations noted by Bruhn (1993) that are correctly identified by the algorithm. Table 4. Results of modulation searching process on the five first preludes and fugues in major keys from book I of the Well-tempered Clavier no. of correctly indicated no. of false no. of false modulations positives negatives 11 7 12


95

The number of false positives represents the number of modulations marked by the program which are not modulations according to Bruhn (1993). The number of false negatives represents the instances that are modulations according to Bruhn (1993) which have however not been marked as modulations by the program. The false positives appear due to regions in the music where many non-diatonic notes are present, such as complicated extended cadences, pedal notes, and chromatic ornamentation. The false negatives appear mostly due to modulations to closely related keys.

4

Conclusions

We have seen that studying non-convex sets can give a rough analysis of the modulations in a piece. The pieces in a major key are easier to analyze than pieces in a minor key, since in the latter some ‘background noise’ of non-convex sets is present. This analysis method uses only little information of the music (only the note names under octave equivalence) which indicates that the method can still be improved. Furthermore, it can perhaps be integrated in other modulation finding theories to optimize the results. Since the method can also be used to visualize repeated (and closely related) passages in a piece of music (of which an example was given in fig. 3), the method can possibly contribute to a structural analysis of music.

Acknowledgments A substantial part of this research was carried out at the University of Amsterdam in the context of the NWO project ’Towards a unifying model for linguistic, musical and visual processing’. The author wants to thank Rend Bod, Henk Barendregt, Elaine Chew and Timour Klouche for helpful comments and suggestions.

References Bruhn, S.: J.S. Bach’s Well-Tempered Clavier: In-depth Analysis and Interpretation. Mainer International Ltd., Hong Kong (1993); Transcription for the Web published (2002-2003) Chew, E.: The spiral array: An algorithm for determining key boundaries. In: Anagnostopoulou, C., Ferrand, M., Smaill, A. (eds.) ICMAI 2002. LNCS, vol. 2445, pp. 18–31. Springer, Heidelberg (2002) Chew, E.: Slicing it all ways: Mathematical models for tonal induction, approximation and segmentation using the spiral array. INFORMS Journal on Computing 18(3) (2006) Honingh, A.K.: The Origin and Well-Formedness of Tonal Pitch Structures. Ph. D. thesis, University of Amsterdam, The Netherlands (2006) Honingh, A.K., Bod, R.: Convexity and the well-formedness of musical objects. Journal of New Music Research 34(3), 293–303 (2005)

96

A. Honingh

Keller, H.: The Well-Tempered Clavier by Johann Sebastian Bach. Norton. Translated by Leigh Gerdine, London (1976) Knopoff, L., Hutchinson, W.: Entropy as a measure of style: The influence of sample length. Journal of Music Theory 27, 75–97 (1983) Krumhansl, C.L.: Cognitive Foundations of Musical Pitch. Oxford Psychology Series, vol. 17. Oxford University Press, Oxford (1990) Krumhansl, C.L., Kessler, E.J.: Tracing the dynamic changes in perceived tonal organization in a spatial representation of musical keys. Psychological Review 89, 334–386 (1982) Christopher Longuet-Higgins, H.: The perception of melodies. Mental Processes: Studies in Cognitive Science, pp. 105–129. British Psychological Society/ MIT Press, London (1987/1976); published earlier as Longuet-Higgins (1976) Christopher Longuet-Higgins, H., Steedman, M.: On interpreting Bach. In: Christopher Longuet-Higgins, H. (ed.) Mental Processes: Studies in Cognitive Science, pp. 82– 104. British Psychological Society/ MIT Press, London (1987/1971); published earlier as Longuet-Higgins and Steedman (1971) Meredith, D.: Pitch spelling algorithms. In: Proceedings of the Fifth Triennial ESCOM Conference, pp. 204–207. Hanover University of Music and Drama, Germany (2003) Temperley, D.: The Cognition of Basic Musical Structures. MIT Press, Cambridge (2001) Van de Craats, J.: De fis van Euler: Een nieuwe visie op de muziek van Schubert, Beethoven, Mozart en Bach. Aramith Uitgevers, Bloemendaal (1989) Vos, P.G., van Geenen, E.W.: A parallel-processing key-finding model. Music Perception 14(2), 185–224 (1996) Youngblood, J.E.: Style as information. Journal of Music Theory 2, 24–35 (1985)

On Pitch and Chord Stability in Folk Song Variation Retrieval J¨ org Garbers1 , Anja Volk1 , Peter van Kranenburg1, Frans Wiering1 , Louis P. Grijp2 , and Remco C. Veltkamp1 1

Department of Information and Computing Sciences, Utrecht University 2 Meertens Institute Amsterdam [email protected]

Abstract. In this paper we develop methods for computer aided folk song variation research. We examine notions and examples of stability for pitches and implied chords for a group of melodic variants. To do this we employ metrical accent levels, simple alignment techniques and visualization techniques. We explore how one can use insight into stability of a known set of variants to query for additional variants.

1

Introduction

The goal of the WITCHCRAFT project (What Is Topical in Cultural Heritage: Content-based Retrieval Among Folksong Tunes) is to develop a content-based retrieval system for a large number of folk song melodies stored as audio and notation. Its purpose is on the one hand to aid folk song researchers in tracing and classifying variants of folk songs and on the other hand to allow the general public to search for melodies with a simple Query by Humming or Keyboard interface. Representing melodies and melodic queries as weighted point sets in the onsetpitch domain, as done in the Muugle system [1], proved to perform well in combination with a couple of pre- and post-processing methods in the general public query task [10]. In the initial part of our project we have tested Muugle’s fitness on a test corpus of 141 symbolically encoded Dutch folk songs for the purpose of the folk song research task: variant classification. Although the results were quite promising, it became clear, that an extended system, which uses more information from the user-query, from the data and from additional feature extractors, would enable researchers to retrieve and classify folk songs in more informed ways. The present paper is about this topic. Overview We assume that classification and retrieval of melodic variants can benefit from the investigation into stable features across melodies which are known to be T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 97–106, 2009. c Springer-Verlag Berlin Heidelberg 2009

98

J. Garbers et al.

related to each other. To know what typically remains stable from one variation of a melody to another allows folk song researchers to decide, if any given melody belongs to a variant group or not. To support this kind of classification with a search engine, we must find ways to formulate queries that specify what should be matched and how strictly. For convenience we prefer to automatically derive such queries from a set of melodies, that are known to belong to the same variant group. (See [5].) Note that the examples and figures, included in this article, are only given to exemplify our computer aided methods. All figures (except for the first) were automatically generated from Humdrum-**kern sources with the help of the Humdrum toolkit, the Guido note-viewer, Rubato and additional scripts that are to be executed once. [7, 6, 8] In section 2 we develop modifications to our present computational framework in order to allow to search for variants in a database, when a group of variant melodies that exemplify the stability and variability of a melody class is given as a query. In section 3 we examine the stability of pitch across variants and how to use this information for querying and in section 4 we do the same for chords that can be implied from the variants.

2

Modifications of the Retrieval System

The unmodified Muugle system compares a melody query, given as a sequence of events in the onset-pitch-duration domain, with melodies from a database and comes up with a ranked list of close matches. It does so (in principle) by computing the so called “Earth-Movers-Distance” (EMD) between the query melody and every database melody, represented as weighted point sets. (See [11] in this volume.) By interpreting the note durations as weights, Muugle assures that it always matches similar amounts of musical duration. Besides of this the EMD requires the definition of a “ground distance”, which in the unmodified Muugle system is realized in terms of the euclidean distance in the onset-pitch domain, with pitch measured in semitones and onset in seconds. A scaling factor in the onset dimension is used to balance the influence of pitch and time in the computation of the ground distance. In preparation of the following sections we need a generalization of our initial ‘melody matches melody’ - approach. In the generalization we want to match a single pitch q at some onset in a melody either a) against a set P of alternative pitches for that same onset or b) against a distribution P of such alternative pitches. The idea is to use event sequences consisting of such alternatives as queries in order to find matching melodies in the database. To formulate these kinds of query with respect to the EMD, we simply have to redefine the pitch distance component of the ground distance for pitch sets and for pitch distributions: a) Let P be a set of (alternative) pitches and q a fixed pitch. Then the minimum pitch distance between q and P is the minimum of the distances between q and any of the pitches in P .

On Pitch and Chord Stability in Folk Song Variation Retrieval

99

b) Let P be a distribution of pitches and q a fixed pitch. Then the average pitch distance between q and P is the weighted average of the distances between q and the pitches in P . Technically, we leave the pre- and post-processing features of the Muugle system intact, as they allow us to compute and combine partial matches and gain transposition and tempo invariance. Several effects of these modifications on the retrieval performance are studied in [3].

Fig. 1. Two EMD matching options for F and C with the refined pitch distance a)

Figure 1 illustrates the effects of the refined ground distance (a). There are four pitch sets in the upper sequence consisting of one pitch each (a melody) and two pitch sets in the lower sequence (a chord sequence). All note duration must flow (in terms of the EMD; see [9] for the formal definition) from the melody to the chords. The G clearly flows to the C major chord and the A to the F major chord. The rest depends on the onset scaling factor: If it is large then F selects a ‘close mismatch’ both in time and pitch and matches with the remaining duration from C major chord and C matches perfectly. If it is low, then F matches with the F from F major chord and the C satisfies the remaining duration of the C major chord.

3

Pitch Stability

In this section we develop methods that help to investigate the pitch variability of a given group of melodic variants. 3.1

Metrical Levels

Metrical symbols such as time signatures (4/4, 6/8) and barlines are used in common music notation to encode metrical accents structures on the time axis and can be used to infer note accents. As a working hypothesis we assume that metrically more accented notes are more stable across folk song variations than less accented notes. We expect smaller amounts of pitch variation on accented onsets in comparison to less accented onsets.

100

J. Garbers et al.

In order to test this hypothesis we visually explore our data by using the Humdrum metpos command to mark the notes of each folk song in a variant group with its position within a metric hierarchy (Levels: 1=bar, 2=half-bar, 3=eights, 4=sixteenth). Then we align the songs in each set by dropping upbeats and unmatched verses. For each metrical level we extract all notes above that level and produce views to compare the projection behaviors on the different levels. When looking for the characteristics of an aligned variant group, we start methodically with very abstract views and proceed to detail views, if necessary. Figures 2-4 show some automatically derived views for the manually aligned variant group ‘Frankrijk B1’ of Onder de groene Linde. [2] 3.2

Evaluation of Pitch Stability

Figure 3 gives us a quick view on the pitch material used per bar at the different metrical levels. By definition we get less or an equal number of pitches at higher metrical levels. But it is interesting to see that there are quite different ranges, both in pitch number and ambitus: While the variation in pitch in bar 3 is reflected on all metrical levels, the variation vanishes on higher metrical levels in bar 7. This might lead us to different matching strategies for different segments (e.g. contour search vs. chord search) when looking for additional members of the variant class that this variant group is a subset of in the database. Figure 4 provides us with slightly more detailed snapshots across all variants taken at different metrically motivated grid positions. The note stability increases from the sixteenth up to the half-bar-level but not up to the bar level. In bar 6, second beat, we have even more stability than on the first beat. To investigate further, where the remaining instability comes from, we look at figure 2 and check the onset positions where the bottom staff contains more than one note. By looking at the other staffs we check, how many variants are responsible for each pitch. In some cases all variants agree except for one outlier (often the first line, e.g. in bar 4), in other cases we find corresponding subgroups within the variants (e.g. last beat in bar 4). Such subgrouping can be interpreted as local pitch alternatives or might lead to the insight that the group actually consists of different coherent subgroups, if the pitches are more often stable within such subgroups. 3.3

Query Formulation

Assume that there are still unclassified melodies in the database and some partial variant groups are already established. To present the user good additional candidates for a given variant group, we can proceed as follows: We first either manually or automatically align the given melodies. Then we compute the pitch sets or pitch distributions for each onset at every metrical level. We construct a query with all alternative pitches or pitch distributions for every onset. We use this for searching in the Muugle database, as described in section 2.


101

Before actually querying the database, one might also want to refine the query (i.e. the pitch distributions) by hand, to get closer to the melodic model that one believes the variants stem from. For this refinement one can use harmonic information (see following Section 4). Matching for example any (new) melody candidate’s first and last bar against the G major chord, seems to be a good generalization (see figure 4).

4

Implied Chord Stability

In this section we develop methods that help to investigate the harmonic variability of a given group of melodic variants and that help to automatically find good candidates for new members of such a group. 4.1

Harmonization

Not all melodies follow harmonic building principles or have implied harmonizations. However, many melodies do allow genre specific harmonizations or already follow associated harmonic constraints. This allows even less trained singers to sing an additional voice, in folk songs typically one third or fourth apart. While melody proceeds at beat level or faster, harmony typically changes more slowly at bar or half-bar level. Melodies that strongly suggest specific harmonizations often contain chord notes as long notes and/or contain them on metrically strong beat positions, and have non-chord tones such as passing tones on metrically weaker beat positions. (Suspensions are an exception of this rule.) When locating actual notes within hypothesized triads, we enter the domain of interpretation and ambiguity. We interpret the given tones in the light of a harmonic model to aid the understanding of the music or to generate accompaniments. In our evaluation we follow the approach described in [4] because we have a harmonic analyzer (HarmoRubette) available as a tool within Rubato, that produces the desired harmonic information for the best harmonic path of a given sequence of pitch sets. We can use this information later to generate prototypical chords that best represent the harmonic information. In a successful analysis those chords would be much the same as those a musician would use in an accompaniment of that melody. 4.2

Evaluation of Implied Chord Stability

We have tried different harmonic analytic models, i.e. different music-theoretical parameterizations of the HarmoRubette. But since the models and their analytic results were still far from optimal we do not go into their details, but show the preliminary results for the different metrical levels. The HarmoRubette generates for each onset a function symbol and a key, such as S(G), the subdominant in G. When running the automatic analysis on the melodies at different metrical levels, we find visually irritating results. For

102

J. Garbers et al.

some variants the results are completely in G major, for some in C major. That makes T(C) and S(G) look quite dissimilar and symbol sequences difficult to compare. An option to cope with this ambiguity is to constrain the analysis to a single key (e.g. G major). Otherwise one would need to invest more knowledge about the harmonic structures behind the melodies and reevaluate them. We did not follow such an approach yet. Two other options are to listen to the represented chords or to compare chord roots instead of the function symbols. (See figure 5 for illustration.) We get more symbol stability, but not surprisingly still different interpretations for ’A’ pitches in G and in C. Already now, the resulting set of chords and — in this case — the extended set of involved chord notes can be used in the role of note distributions in the extended distance measure of section 2. However, this will only sort out harmonically distant melodies but will not result in a fine ranking, because it allows very many melodic alternatives. 4.3

Contextualization

If a melody is naively (mis)-interpreted as a sequence of one-note-chords, the HarmoRubette naturally yields strange results. It chooses fluctuating tonalities, as the harmony sequences are so much under-specified. Analytic ambiguity is inherent to this naive harmonic analysis. But it is nevertheless an interesting point of departure. What we prefer is to analyze more constraining chord sequences. In the following we present several ideas where these more constraining additional notes may come from in the environment. First, the additional notes can come from notes that belong to lower metrical levels of the time span that an accentuated pitch represents. Figure 2 shows many examples, where often from three 8th notes two can be considered chord tones. In practice the harmonic analyzer can be left alone to figure out, which notes make sense as chord notes in the larger context. We simply have to feed it all notes at once. Another option is to try to derive a common chord scheme from the whole set of variations. We tested this by running the analysis on the chords of figure 4 and found the consistent key G major with some very short deviations to ’ii’ and ’vi’ on lower metrical levels. (See figure 5.) From this we might conclude, that to test, if a new candidate melody belongs to the variation group, we just have to build the common chord set and see, if the new notes do not ‘disturb’ the analysis. The HarmoRubette also comes up with note weights that express the conformance of the notes with the analyzed harmonic loci. However, this might not be possible in general for manually unprepared melodies, that can differ in slightly shifted onsets. In such a case the note distribution matching strategy within the EMD seems to be more promising.


5

103

Excerpts from the Variation Group ‘Frankrijk B1’

Fig. 2. The melodies at metrical levels 3 (eights) and above and chords resulting from projecting all notes. (Bars 3-5)

Fig. 3. A collected view of all pitches per bar from figure 4. The staffs refer to reductions to different metrical grids.

104

J. Garbers et al.

Fig. 4. Four views of all notes of all variations. Each staff shows the notes that fall on the grid of a particular metrical level.


105

Fig. 5. Automatic root analysis of the sequence of alternative pitches at metrical level 2 (half-bar). The left column shows the pitches, the right columns show their shared functional analysis and root chords.

6

Summary

For a group of folk song variations we have looked into the note stability and the stability of the ’best harmonic symbol sequence’ on the onset, tactus and bar levels. Therefore we developed a set of tools and views that allow us to get a quick impression about the stability of features for a set of variants at different metrical levels. We found them quite useful to visually examine the pitch stability and found our hypothesis verified, that melody tones at strong positions are more stable among variants. We have also presented the idea to use this information in a refined transportation distance measure, that can match pitch distributions with pitches.

106

J. Garbers et al.

In the follow up paper [3] we show that this leads to better retrieval performance and in [5] we elaborate on making automatic alignments rather than manual alignments. We will further study these methods within the WITCHCRAFT project to improve our general public search engine.

Acknowledgements This work was kindly supported by the Netherlands Organization for Scientific Research within the WITCHCRAFT project NWO 640-003-501, which is part of the program Continuous Access to Cultural Heritage. Further we want to thank the developers of the Humdrum and Rubato toolkits and the encoders of the Dutch folk songs, who made this investigation possible.

References [1] Bosma, M., Veltkamp, R.C., Wiering, F.: Muugle: A framework for the comparison of music information retrieval methods. In: Proceedings of the ICMPC 2006, pp. 1297–1303 (2006) [2] van Dijk, M.B.G., Kuijer, H.J., Dekker, A.J. (eds.): Onder de groene linde. Verhalende liederen uit de mondelinge overlevering. Uitgeverij Uniepers, Amsterdam (1987-1991) [3] Garbers, J., van Kranenburg, P., Volk, A., Wiering, F., Grijp, L., Veltkamp, R.C.: Using pitch stability among a group of aligned query melodies to retrieve unidentified variant melodies. In: Dixon, S., Bainbridge, D., Typke, R. (eds.) Proceedings of the Eighth International Conference on Music Information Retrieval, pp. 451–456. Austrian Computer Society (2007) [4] Garbers, J., Noll, T.: New perspectives of the harmorubette. In: Lluis-Puebla, E., Mazzola, G., Noll, T. (eds.) Perspectives in Mathematical and Computer-Aided Music Theory, Verlag epOs-Music, Osnabr¨ uck (2003) [5] Garbers, J., Wiering, F.: Towards structural alignment of folk songs. In: Bello, J.P., Chew, E. (eds.) Proceedings of the Nineth International Conference on Music Information Retrieval (2008) [6] Hoos, H.H., et al.: Guido, http://guidolib.sourceforge.net/ [7] Huron, D., et al.: Humdrum, http://music-cog.ohio-state.edu/Humdrum/ [8] Mazzola, G., Zahorka, O., Garbers, J.: Rubato, http://www.rubato.org [9] Rubner, Y., Tomasi, C., Guibas, L.J.: The earth mover’s distance as a metric for image retrieval. International Journal of Computer Vision 40(2), 99–121 (2000) [10] Typke, R.: Music Retrieval Based on Melodic Similarity. PhD thesis, Utrecht University (2007) [11] Volk, A., Garbers, J., van Kranenburg, P., Wiering, F., Grijp, L., Veltkamp, R.C.: Music retrieval based on melodic similarity. In: Noll, T., Klouche, T. (eds.) Mathematics and Computation in Music: Proceedings of the MCM 2007 (2008)

Bayesian Model Selection for Harmonic Labelling Christophe Rhodes , David Lewis, and Daniel M¨ ullensiefen Department of Computing, Goldsmiths, University of London, SE14 6NW, United Kingdom [email protected]

Abstract. We present a simple model based on Dirichlet distributions for pitch-class proportions within chords, motivated by the task of generating ‘lead sheets’ (sequences of chord labels) from symbolic musical data. Using this chord model, we demonstrate the use of Bayesian Model Selection to choose an appropriate span of musical time for labelling at all points in time throughout a song. We show how to infer parameters for our models from labelled ground-truth data, use these parameters to elicit details of the ground truth labelling procedure itself, and examine the performance of our system on a test corpus (giving 75% correct windowing decisions from optimal parameters). The performance characteristics of our system suggest that pitch class proportions alone do not capture all the information used in generating the ground-truth labels. We demonstrate that additional features can be seamlessly incorporated into our framework, and suggest particular features which would be likely to improve performance of our system for this task.

1

Introduction

This paper introduces a straightforward model for labelling chords based on pitch-class proportions within windows, and using this model not only to generate chord labels given a symbolic representation of a musical work, but also to infer the relevant level of temporal granularity for which a single label is justified. The generation of these chord labels was initially motivated by the desire to perform automated musical analysis over a large database of high-quality MIDI transcriptions of musical performances, as part of a larger study investigating musical memory. While the MIDI transcriptions are of high-fidelity with respect to the performances they represent, they do not include any analytic annotations, such as song segmentation, principal melody indications, or significant rhythmic or harmonic motifs; all of these must be generated if desired, but it is not practical to do so manually over the collection of some 14,000 pop song transcriptions.

Corresponding author.


108

C. Rhodes, D. Lewis, and D. M¨ ullensiefen

A time sequence of chord labels, as a compact representation of the harmony of the musical work, can not only be used as the basis for the detection of largerscale harmonic features (such as cadences, clichés and formulae), but can also inform a structural segmentation of the music, since harmony is an indicator of structure in many popular music styles. Such segmentation is a necessary first step for other feature extraction tools – it is, for example, a prerequisite for the melody similarity algorithms presented in M¨ ullensiefen and Frieler (2004). A second use for these chord labels is the automatic generation of lead sheets. A lead sheet is a document “displaying the basic information necessary for performance and interpretation of a piece of popular music” (Tagg 2003b). The lead sheet usually gives the melody, lyrics and a sequence of short chord labels, usually aligned with the melody, allowing musicians to accompany the singer or main melody instrument without having a part written out for them. An advantage of the model we present in this paper is that the overall framework is independent of the type of harmony scheme that it is used with: for example, it can be adapted to generate labels based on tertial or quartal harmonic classification (Tagg 2003a). Furthermore, a similar model selection stage can be used to choose which harmonic classification is most appropriate for a given work, a decision which can be guided by information not present in the observed musical data (such as a genre label) by incorporating that information into a prior probability model. The rest of this paper is organized as follows: after a discussion of previous related work in section 2, we present our model for the dependency of pitchclass content on the prevailing chord, forming the core of our simple model, and discuss its use in window size selection in section 3. We discuss implementation of parameter inference and empirical results in section 4, and draw conclusions and suggest further work in section 5.

2

Previous Work

Most previous work on chord label assignment from symbolic data is implemented without an explicit model for chords: instead, preference rules, template matching and neural network approaches have been considered (Temperley 2001, Chapter 6 and references therein); an alternative approach involving knowledge representation and forward-chaining inference has also been applied to certain styles of music (Pachet 1991; Scholz et al. 2005). One attempt to use probabilistic reasoning to assign chord labels uses a Hidden Markov Model approach with unsupervised learning (Raphael and Stoddard 2004) of chord models; however, the authors note that they do not provide for a way of making decisions about the appropriate granularity for labelling: i.e. how to choose the time-windows for which to compute a chord label. There has been substantial work in the symbolic domain on the related task of keyfinding. For instance, Krumhansl (1990, Chapter 4) presents a decision procedure based on Pearson correlation values of observed pitch-class profiles with profiles generated from probe-tone experiments. Another class of algorithms used

Bayesian Model Selection for Harmonic Labelling

109

for keyfinding is a geometric representation of keys and tones, attempting to capture the perceived distances between keys by embedding them in a suitable space (Chuan and Chew 2005). The profile-based model has been refined (Temperley 2001, Chapter 7) by making several modifications: altering details of the chord prototype profiles; dividing the piece into shorter segments; adjusting the pitchclass observation vector to indicate merely presence or absence of that pitch class within a segment, rather than the proportion of the segment’s sounding tones, and thus avoiding any attempt at weighting pitches based on their salience; and imposing a change penalty for changing key label between successive segments. There are existing explicit models for keys and pitch-class profiles: one such (Temperley 2004) is defined such that for each key, the presence or absence of an individual pitch class is a Bernoulli distribution (so that the pitch-class profile is the product of twelve independent Bernoulli distributions); in this model, there are also transition probabilities between successive chords. This model was further refined in Temperley (2007) by considering not just pitch classes but the interval between successive notes. These models are based on the notion of a fixed-size ‘segment’, which has two effects: first, the key models are not easily generalized to windows of different sizes, as the occurrence of a particular scale degree (i.e. pitch relative to a reference key) is not likely to be independent in successive segments; second, unless the segment length is close to the right level of granularity, a postprocessing stage will be necessary to smooth over fragmented labels. There has been more work towards chord recognition in the audio domain, where the usual paradigm is to model the chord labels as the hidden states in a Hidden Markov Model generating the audio as observation vectors (Bello and Pickens 2005; Sheh and Ellis 2003). One problem in training these models is in the lack of ground truth, of music for which valid chord labels are known (by ‘valid’ here, we mean sufficient for the purposes for which automated chord labelling is intended, though of course these may vary between users); approaches have been made to generate ground truth automatically (Lee and Slaney 2006), but such automatic ground truth generation depends on a reliable method of generating labels from the symbolic data or from something that can be mapped trivially onto it; without such a reliable method, hand-annotated ground truth must be generated, as for example in Harte et al. (2005). One feature of the method presented in this paper in contrast to most existing harmony or key identification techniques is that it has an explicit musicallymotivated yet flexible model for observable content (i.e. pitch-class distributions) at its core, rather than performing some ad-hoc matching to empirical prototypes. This flexibility confers two modelling advantages: first, the parameters of the model can be interpreted as a reflection of musical knowledge (and adjusted, if necessary, in a principled way); second, if evidence for additional factors influencing chord labels surfaces, in general or perhaps for a specific style of music under consideration, these additional factors can be incorporated into the model framework without disruption.

110

3


Model

The repertoire of chords that we represent is triad-based (though an extension to include other bases is possible with some care over the dimensionality of the relevant spaces); motivated by their prevalence in western popular music, we aim to distinguish between major, minor, augmented, diminished and suspended (sus4 and sus9) triads with any of the twelve pitch classes as the root, and we will infer probability distributions over these chord labels given the musical content of a window. Of the six, it should be noted that augmented and diminished chords are much rarer in popular music, and that suspended chords, despite their names, are frequently treated in popular music as stable and not as needing to resolve, and so require categories of their own – e.g. in soul or country music where they form independent sonorities; see Tagg (2003a). We introduce the Dirichlet distribution on which our chord model is based, give our explicit model for the dependence of pitch-class proportions on the chord, and then explain how we can use this to perform selection of window size in a Bayesian manner. 3.1

Dirichlet Distributions

The Dirichlet distribution is a model for proportions of entities within a whole. Its density function is 1 αi −1 p(x|α) = x (1) B(α) i i with support on the simplex i xi = 1. The normalizing constant B(α) is defined as Γ (αi ) (2) B(α) = i Γ ( i αi )

∞ where Γ is the gamma function Γ (x) = 0 tx−1 e−t dt. Note that for each individual component of the whole, represented by an individual random variable xi , the corresponding αi controls the dependence of the density (1) for small values of this component: if αi > 1, the probability density tends towards zero in the limit xi → 0; if αi < 1, the density increases without limit as xi → 0. 3.2

The Chord Model

Our introductory chord model is triad-based, in that for each chord we consider the tones making up the triad separately from the other, non-triadic tones. The proportion of a region made up of triad tones is modelled as a Beta distribution (a Dirichlet distribution with only two variables), and the triad tone proportion is then further divided into a Dirichlet distribution over the three tones in the triad. Denoting the proportion of non-triadic tones as t¯, and that of triadic tones as t, where the latter is made up of root r, middle m and upper u, we can write our chord model as for tone proportions given a chord label c as p(rmutt¯|c) = p(tt¯|c)p(rmu|tt¯c)

(3)


111

with support on the simplexes t + t¯ = 1, r + m + u = 1; each of the terms on the right-hand side is a Dirichlet distribution. We simplify the second term on the right-hand side by asserting that the division of the harmonic tones is independent of the amount of harmonic tones in a chord, so that p(rmu|tt¯c) = p(rmu|c). In principle, each chord model has two sets of independent Dirichlet parameters α; in practice we will consider many chords to be fundamentally similar, effectively tying those parameters. This simple chord model does not allow for certain common harmonic labels, such as seventh chords or open fifths (as these are not triadic); we leave this extension for further work. Additionally, there is a possible confusion even in the absence of noise between the suspended chords, as the tones present in a sus4 chord are the same as those in a sus9 chord four scale degrees higher. 3.3

Bayesian Model Selection

We start with a set of possible models for explaining some data, where each individual model is in general parameterized by multiple parameters. Given this set of distinct models, and some observed data, we can make Bayesian decisions between models in an analogous fashion to selecting a particular set of parameters for a specific model; in general, we can generate probability distributions over models (given data) in a similar way to the straightforward Bayesian way of generating probability distributions over the parameter values of a given model. For a full exposition of Bayesian Model Selection, see e.g. MacKay (2003, Chapter 28). In the context of our problem, of chord labelling and window size selection, we choose a metrical region of a structural size: in our investigation for popular music, we choose this region to be one bar, the basic metrical unit in that style. The different models for explaining the musical content of that bar, from which we will aim to select the best, are different divisions of that bar into independently-labelled sections. For example, one possible division of the bar is that there is no segmentation at all; it is all one piece, with one chord label for the whole bar. Another possible division is that the bar is made up of two halves, with a chord label for each half bar. These divisions of the bar play the rˆ ole of distinct models, each of which has Dirichlet parameters for each independentlylabelled section of the bar. In our experiment described in section 4, the corpus under consideration only contains works in common time, with four quarter beats in each bar, and we consider all eight possible divisions of the bar that do not subdivide the quarter beat (i.e. , 1+1+1+1, 1+1+2, 1+2+1, 2+1+1, 2+2, 1+3, 3+1, 4). The Bayesian Model Selection framework naturally incorporates the Occam factors in a quantitative manner: if there is evidence for two different chord labels, then the whole-bar model will not be a good fit to the data; if there is no evidence for two distinct chord labels, then there are many more different poor fits for a more fine-grained model than for the whole-bar model. To be more precise, we can write the inference over models M given observed data D as p(D|M )p(M ) p(M |D) = (4) p(D)

112


where p(D|M ) =

p(D|cM )p(c|M )

(5)

c

is the normalizing constant for the inference over chord labels c for a given model M . Note that there is an effective marginalization over chord labels for each model – when considering the evidence for a particular model, we add together contributions from all of its possible parameters, not simply the most likely. We can use the resulting probability distribution (4) to select the most appropriate window size for labelling. The flexibility of this approach is evident in equation (5): the chord models p(D|cM ) can differ in parameter values or even in their identity between window sizes, and that the prior probabilities for their generation p(c|M ) can also be different for different models of the bar M .

4 4.1

Experiment Parameter Estimation

In order to test our chord model, (see equation 3), we must choose values for the α parameters of the Dirichlet distributions. We summarize the maximumlikelihood approach (from a labelled ‘training set’) below, noting also the form of the prior for the parameters in the conjugate family for the Dirichlet distribution); in addition, we performed a large search over the parameter space for the training set, attempting to maximize performance of our model at the labelling task with a binary loss function. i ) log xi ]−log B(α) We can rewrite the Dirichlet density function (1) as e− i [(1−α , demonstrating that it is in the exponential family, and that i log xi is a sufficient statistic for this distribution; additionally, there is a conjugate prior for the parameters of the form π(α|A0 , B 0 ) ∝ e−

i

[(1−αi )A0i ]−B 0 log B(α)

(6)

with support αi ∈ R+ 0. Given N observations x(k) , the posterior density is given by p(α|x(k) ) ∝ p(x(k) |α)π(α), which is e

−

i

(k) (1−αi ) A0i + N −(B 0 +N ) log B(α) k log xi

;

(7)

that is, of the same form as the prior in equation (6), but with the hyperparameters A0 and B 0 replaced by A = A0 + k log x(k) (with the logarithm operating componentwise) and B = B 0 + N . The likelihood is of the form of equation (7), with A0 and B 0 set to 0. The maximum likelihood estimate for parameters is then obtained by equating the first derivatives of the log likelihood to zero; from equation (2), we see that ∂ log B(α) ∂ = Ψ (αi ) − Ψ = log Γ (αk ) − log Γ αk αk , ∂αi ∂αi k k k (8)


113

where Ψ is the digamma function; therefore, ∂ log L ∂ log B(α) , = Ai − B = Ai − B Ψ (αi ) − Ψ αk ∂αi ∂αi

(9)

k

giving Ψ ( k αk ) = Ψ (αi ) − ABi for the maximum point, which we solve numerically for αi using the bounds discussed in Minka (2003). In addition, performing a quadratic (Gaussian) approximation around the maximum, we can obtain estimates for the error bars on the maximum likelihood estimate from ∂ 2 log L = −σα−2 , giving 2 i ∂α i

max

σαi =

B Ψ (αi ) − Ψ

− 12 αk

;

(10)

k

for the purpose of the confidence interval estimates in this paper, we disregard 2 L covariance terms arising from ∂∂αlog . i ∂αj We defer detailed discussion of a suitable form of the prior on these chord parameters to future work. We have derived an approximate noninformative prior (Jaynes 2003, Chapter 12) within the conjugate family, but its use is inappropriate in this setting, where we can bring considerable musical experience to bear (and indeed the maximum a posteriori estimates generated using this noninformative prior give inferior performance than the maximum likelihood estimates in our experiment). 4.2

Results

Our corpus of MIDI transcriptions is made up of files each with thousands of MIDI events, with typically over five instruments playing at any given time; each bar typically contains several dozen notes. We selected 16 songs in broadly contrasting styles, and ground-truth chord labels for those transcriptions of performances were generated by a human expert, informed by chord labels as assigned by song books to original audio recordings. We then divided our corpus of 640 labelled bars into “training” and “testing” sets of 233 and 407 bars respectively. Based on an initial inspection of the training set, we performed maximum likelihood parameter estimation for the chord models for three different sets of labels: major or minor chord labels for an entire bar; major or minor labels for windows shorter than a bar; and all other labels. From the inferred parameters for major and minor chords at different window sizes in table 1, there was clear evidence for qualitatively different label generation at sub-bar window sizes from the behaviour of labelling whole bars: the sub-bar window sizes have high probability density for small non-triadic tones, while whole-bar windows have a vanishing probability density near a zero proportion of non-triadic tones (from the different qualitative behaviour of distributions with Dirichlet parameters below and above 1.0: 0.72 and 1.45 in our case). We interpret this as showing that the ground-truth labels were generated

114


such that a sub-bar window is only labelled with a distinct chord if there is strong evidence for such a chord – i.e. only small quantities of non-triadic tones. If no sub-bar window is clearly indicated, then a closest-match chord label is applied to the whole bar, explaining the only slight preference for chord notes in the whole-bar distribution. There was insufficient ground-truth data to investigate this issue over the other families of chords (indeed, there was only one example of an augmented chord in the training data set). Table 1. Maximum likelihood estimates and 1σ error bars for Dirichlet distributions, based on labelled ground truth Chord, win Maj/Min, bar Maj/Min, sub-bar other

αtt¯ {6.28, 1.45} ± {0.49, 0.099} {3.26, 0.72} ± {0.32, 0.054} {5.83, 1.04} ± {0.82, 0.12}

αrmu {3.91, 1.62, 2.50} ± {0.23, 0.11, 0.15} {4.04, 2.66, 2.29} ± {0.21, 0.15, 0.13} {4.08, 2.35, 1.49} ± {0.38, 0.23, 0.16}

Using the maximum likelihood estimates of table 1, we performed inference over window sizes and chord labels over the testing set, obtaining 53% of correct windows and 75% of correct labels given the window. Additionally, we performed a large (but by no means exhaustive) search over the parameter space on the training data, and obtained parameter values which performed better than these maximum likelihood estimates on the testing set, giving 75% windows and 76% chords correctly. It should be noted that the training and testing sets are quite similar in character, being individual bars drawn from the same pieces; it would be difficult to justify claims of independence between the sets. Validation on an independent testset (i.e. music excerpts drawn from different pieces) is currently being undertaken. We interpret these trends as suggesting that the model for chords based simply on tone proportions is insufficiently detailed to capture successfully enough of the process by which ground-truth labels are assigned. The fact that maximum likelihood estimates perform noticeably worse than a set of parameters from training indicates that there is structure in the data not captured by the model; we conjecture that inclusion of a model for the chord label conditioned on the functional bass note in a window would significantly improve the performance of the model. Another musically-motivated refinement to the model would be to include an awareness of context, for instance by including transition probabilities between successive chord labels (in addition to the implicit ones from the musical surface). This corresponds to removing the assumption that the labels are conditionally independent given the musical observations: an assumption that is reasonable as a first approximation, but in actuality there will be short-term dependence between labels as, for instance, common chord transitions (such as IV-V-I) might be favoured over alternatives in cases where the observations are ambiguous; similarly, enharmonic decisions will be consistent over a region rather than having an independent choice made at the generation of each label.


115

The performance of our approach, without any of the above refinements, is at least comparable to techniques which do relax the assumption of conditional independence between labels; for example, the algorithm of Temperley (2001), which infers chord labels over the entire sequence (using dynamic programming to perform this inference efficiently), achieves a comparable level of accuracy (around 77%) on those pieces from our dataset for which it correctly computes the metrical structure.

5

Conclusions

We have presented a simple description of the dependence of chord labels and pitch-class profile, with an explicit statistical model at its core; this statistical model can be used not only to infer chord labels given musical data, but also to infer the appropriate granularity for those labels. Our empirical results demonstrate that adequate performance can be achieved, while suggesting that refinements to the statistical description could yield significant improvements. The model presented ignores all context apart from the bar-long window in question, and operates only on pitch-class profile data; incorporation of such extra information can simply be achieved by extending the statistical model. Similarly, we can incorporate available metadata into our model, for instance by defining a genre-specific chord label prior; and we can change the repertoire of chords under consideration without alteration of the framework, simply by replacing one component of the observation model.

Acknowledgments C.R. is supported by EPSRC grant GR/S84750/01; D.L. and D.M. by EPSRC grant EP/D038855/1.

References Bello, J.P., Pickens, J.: A Robust Mid-Level Representation for Harmonic Content in Musical Signals. In: Proc. ISMIR, pp. 304–311 (2005) Chuan, C.-H., Chew, E.: Polyphonic Audio Key Finding Using the Spiral Array CEG Algorithm. In: Proc. ICME, pp. 21–24 (2005) Harte, C., Sandler, M., Abdallah, S., G´ omez, E.: Symbolic Representation of Musical Chords: A Proposed Syntax for Text Annotations. In: Proc. ISMIR, pp. 66–71 (2005) Jaynes, E.T.: Probability Theory: The Logic of Science. Cambridge University Press, Cambridge (2003) Krumhansl, C.L.: Cognitive Foundations of Musical Pitch. Oxford University Press, Oxford (1990) Lee, K., Slaney, M.: Automatic Chord Recognition from Audio Using an HMM with Supervised Learning. In: Proc. ISMIR (2006) MacKay, D.J.C.: Information Theory, Inference, and Learning Algorithms. Cambridge University Press, Cambridge (2003)

116


Minka, T.: Estimating a Dirichlet Distribution (2003), http://research.microsoft.com/~ minka/papers/dirichlet/ M¨ ullensiefen, D., Frieler, K.: Cognitive Adequacy in the Measurement of Melodic Similarity: Algorithmic vs. Human Judgments. Computing in Musicology 13, 147–176 (2004) Pachet, F.: A meta-level architecture applied to the analysis of Jazz chord sequences. In: Proc. ICMC (1991) Raphael, C., Stoddard, J.: Functional Harmonic Analysis Using Probabilistic Models. Computer Music Journal 28(3), 45–52 (2004) Scholz, R., Dantas, V., Ramalho, G.: Funchal: a System for Automatic Functional Harmonic Analysis. In: Proc. SBCM (2005) Sheh, A., Ellis, D.P.W.: Chord Segmentation and Recognition using EM-trained Hidden Markov Models. In: Proc. ISMIR, pp. 185–191 (2003) Tagg, P.: Harmony entry. In: Shepherd, J., Horn, D., Laing, D. (eds.) Continuum Encyclopedia of Popular Music of the World. Continuum, New York (2003a) Tagg, P.: Lead sheet entry. In: Shepherd, J., Horn, D., Laing, D. (eds.) Continuum Encyclopedia of Popular Music of the World. Continuum, New York (2003b) Temperley, D.: The Cognition of Basic Musical Structures. MIT Press, Cambridge (2001) Temperley, D.: Bayesian Models of Musical Structure and Cognition. Musicae Scientiae 8, 175–205 (2004) Temperley, D.: Music and Probability. MIT Press, Cambridge (2007)

The Flow of Harmony as a Dynamical System Peter Giesl Department of Mathematics, University of Sussex, Mantell Building, Falmer, BN1 9RF, UK [email protected]

Abstract. When analysing the evolution of harmony within a composition, one can distinguish between two parts: on the one hand a dynamical system and on the other hand the composer. The dynamical system summarises the rules of harmony, whereas the composer intervenes at certain points, e.g. by choosing new initial values. This viewpoint is helpful for the musical analysis of a composition and will be exemplified by analysing the first movement of Beethoven’s first symphony.

1 Dynamical Systems Applied to Harmony In this section we recall definitions and concepts from dynamical systems and apply them to harmony. By harmony we mean chords which we often denote by its functional class according to Riemann’s theory of functional harmony (Riemann 1893). A discrete dynamical system (X, St ) consists of a metric space X, called the phase space, which in our case is the space of all harmonies or chords. The flow operator St : X → X maps the initial harmony x to the harmony St x after the time t. The time is assumed to be discrete, i.e. we measure the time in steps. Moreover, S0 = id is the identity operator and St is a semi-group, i.e. St+s = St ◦ Ss for all t, s ≥ 0. The metric on the space of all harmonies or chords can be determined by the degree of relationship between two chords. All chords in a close functional relation to the tonic, e.g. dominant or subdominant, have a smaller distance than parallel chords or chords with no direct functional relation1 . Each discrete dynamical system is given by the iteration of a map g: X → X; g = S1 maps a chord to its successor. The sequence of chords x0 , x1 , . . . is thus given by the iteration xn+1 = g(xn ). For example, starting with the dominant D, the next chord is the tonic T which then stays there, i.e. g(D) = T, g(T) = T or in a different notation D → T, T → T. This results in St D = T for all t ≥ 1 and St T = T for all t ≥ 0. The map g and the initial chord x0 determine the sequence of chords, the flow of harmony. This is certainly a simplified viewpoint and one could extend the class of dynamical systems under consideration by allowing a multi-valued map g, possibly with certain probabilities. Then a harmony x could be followed by any harmony of the set g(x) = {y1 , . . . , yk } of harmonies y1 , . . . , yk . Another possibility would be a timedependent map g(xn , n) or a map including the history, i.e. g(xn , xn−1 , . . . , xn−k ). 1

Note that some difficulties occur when considering a discrete phase space with a metric in the definitions of stability. However, we will not go into further detail since we are only interested in the general concept.


118

P. Giesl

We, however, restrict ourselves mostly to the simplest version described above. The dynamical system is thus the frame, the underlying system of rules, which can be altered by the composer, cf. Section 3. Hence, the composer can alter the function g or, in the multi-valued setting, he can choose one of the harmonies in the set g(x). In the following we will transfer and interpret more concepts and definitions from dynamical systems to harmony. The rules of each voice in counterpoint, cf. Section 2, will help to explain these concepts. A fixed point x0 is a fixed point of the map g, i.e. g(x0 ) = x0 , and thus defines a constant solution. An example is the tonic T. Furthermore, one is interested in the behaviour near a fixed point. An asymptotically stable fixed point (in the following denoted by stable) is a fixed point, such that all solutions with initial chords in a neighbourhood2 of the fixed point finally tend to the fixed point while staying near. The set of all initial chords tending to the fixed point is called the basin of attraction of the fixed point. The tonic T is a stable fixed point and the subdominant is part of its basin of attraction since S → D → T. Hence, all chords with a functional relation to the tonic belong to the basin of attraction. There can be several stable fixed points, and each has its basin of attraction, which can be regarded as its region of influence. The basins of attraction are disjoint and are often separated by an unstable (fixed) point. A fixed point is called unstable, if in each nontrivial neighbourhood of the fixed point there are chords such that the corresponding solutions tend away from the unstable fixed point. The change from one stable fixed point to another is a modulation. Let us consider two fixed points T and T’ where T is A major and T’ is E major, cf. Figure 1. An unstable point could be the diminished seventh chord f − g − b − d. This is an unstable situation which is at the boundary of both basins of attraction. Depending on the next chord, it can proceed to either of the basins of attraction. One could imagine a ball on the top of a hill with two valleys, one to each side: a light wind could let the ball roll to the one or the other direction and it will come to a halt in either of the two valleys. Hence, the unstable chord f − g − b − d could tend to A major or, understood as f − a − c − d, to E major. Coming back to the image of the ball, we define a Lyapunov function for a stable fixed point. A Lyapunov function is a function V : X → R which decreases along solutions, i.e. V (xn+1 ) ≤ V (xn ), and which has a strict minimum at the fixed point. We generalize the concept slightly and only claim the first property, i.e. V decreases along solutions. Then it can have several minima at different fixed points and it has a maximum at an unstable fixed point. In dissipative physical systems the energy is an example of a Lyapunov function. In our case the harmonic energy would be such a generalized Lyapunov function, since it decreases along solutions: a tense chord with high harmonic energy seeks for resolution and leads to a chord which has less energy until it tends to a minimum. The harmonic energy is like gravitation which moves the ball downwards. If we start with a Lyapunov function, we can define a dynamical system in the following way: the multi-valued map g maps x to the set g(x) := {y ∈ X | V (y) ≤ V (x)} 2

Here we assume the neighbourhood to be nontrivial, i.e. it should not only consist of the fixed point.

The Flow of Harmony as a Dynamical System

119

V

A

f - g# - b - d

Eb

Fig. 1. Above: The figure shows two stable fixed points (A in white and E in black) with their respective basin of attraction (dashed and black line). The basins of attraction are separated by an unstable point (diminished seventh in grey). Below: The Lyapunov function V is sketched for the three chords: the two stable fixed points are minima of V while the unstable point is a maximum.

of all harmonies with lower harmonic energy than x. This leaves some freedom for the sequence of chords, since we have several choices. As an example let us define the following Lyapunov function: V (T) = 0, V (D) = 1, V (S) = 2, V (Sp) = 3. Then for all the following sequences the Lyapunov function decreases: Sp → D → T, D → T, Sp → S → D → T.3

2 Dynamical Systems Applied to Counterpoint The intention of this section is twofold: on the one hand we want to show that similar mechanisms as on the global level of harmony are valid on the local level of voices and counterpoint. On the other hand these local mechanisms can explain some of the global features of the preceding section. On this local level the phase space X = Y n is given by the pitch Y for each of the n voices. The map g = (g1 , . . . , gn ): Y n → Y n determines the next note of each voice. The definition of g depends on the energy of all vectors in Y n – some have a high harmonic energy (dissonant) and some a low one (consonant), and the map g seeks to decrease this energy. Let us illustrate these ideas by two voices (n = 2): an interval f − e1 would relax to f − d1 , i.e. g(f, e1 ) = (f, d1 ). The interval f − d1 has a lower energy, but it could still relax to e − e1 , i.e. g(f, d1 ) = (e, e1 ), cf. Figure 2. The function g for more than two voices is partly a composition of the corresponding functions of two voices, however, other aspects must also be taken into account. If we consider again the example of the diminished seventh chord, then again f − d1 leads to e − e1 , or g − b leads to a − a. Thus, a choice has to be made, which pair of voices 3

Note the following equivalences of the diatonic functions: T=I, Sp=ii, S=IV and D=V.

120

P. Giesl

e g

e f

d f

e e

Harmonic Energy V

Fig. 2. A phrygian clausula together with the harmonic energy. The highest energy appears when the lower voice is moved to f , resulting in the dissonant interval f − d1 . From then on the energy decreases until a resolution is obtained.

forms the clausula4 and which voices have to follow. This choice explains the instability of the chord: the decision of the role of each voice determines the future development5. Another example is the deceptive cadence6 : here, some voices follow the rules of the dynamical system, and only the bass is set differently. On a microlocal level, the same concept applies to only one voice, defining a dynamical system on X = Y . Here, the function g: Y → Y depends on the potential energy given by the musical scale or, in Gregorian melodic, by other laws7 : in the latter case a Finalis is a stable fixed point, a Ténor would be an unstable fixed point as small changes will lead the melody away from it.

3 The Composer In summary, the dynamical system is responsible for the general rules, the resolution of tensions and the automatism within the harmonic flow, whereas the composer can introduce a new starting point, change the system or push it into an ambiguous situation. 4

5

6 7

A clausula in this context consists of two voices, one of which goes a tone upwards (Discant Clausula), the other one at the same time going a tone downwards (Tenor Clausula) to end in an octave, cf. (Schwind and Polth 1996) . Usually one of the steps is a semitone (minor second) and the other a whole tone (major second); if the Discant Clausula has the semitone, then the clausula is called authentic, otherwise phrygian, cf. (Jans 1992). In fact, there are even more possibilities and combinations, cf. (Giesl 1999). These clausulae can even lead to a sequence of chords by changing the roles of each voice, cf. (Giesl 2001). A deceptive cadence is D followed by any chord which is not the tonic, often D→Tp or V→vi. The melodic theory of Christoph Hohlfeld, for example, could provide an appropriate function g, cf. (Giesl 2002).


121

A dynamical system cannot explain the flow of harmony nor the counterpoint in all details; on the contrary, an external influence, the composer, is needed to take decisions and to start with an initial situation. The composer can reset the system at any time by starting with a new chord or situation, he is a control to the dynamical system. He can act against the gravitation and put the ball on a hill, or push it into some direction. The composer mostly acts only at particular, singular times and seldom over a longer period. As an example consider the cadence: S → D → T is the determined sequence by the dynamical system. Then, since T is a fixed point, the system would stay there. Now the composer sets a new initial chord, e.g. the subdominant S, and the rest follows by the rules of the dynamical system. However, combinations which are used very often, can become new rules. Therefore, we will not regard or mention the subdominant after the tonic as an action of the composer. In fact, all harmonies that stay within the basin of attraction of a stable fixed point (cadence harmony) are governed by the dynamical system. Only new initial chords outside the basin of attraction or a new stable fixed point are important and notable events for the analysis. As an example we will provide a short harmonic analysis of the first movement of Beethoven’s first symphony by highlighting the decisions of the composer in the appendix.

4 Summary We have described the flow of harmony as a dynamical system on different macroscopic and microscopic levels. The dynamical system summarises the rules, which depend on the cultural and historical situation. The flow seeks to decrease the harmonic energy and enhances the resolution of tensions. The composer is not restricted to the dynamical system: he chooses a new initial chord which then evolves following the rules of the dynamical system. He leads the system to a point with high musical energy, then the dynamical system takes over. This viewpoint is helpful for the musical analysis of compositions as well as for improvisation: one distinguishes between the internal rules (dynamical system) and the external control (individual decisions). Finding the dynamical system for a whole epoch summarises its theory on harmony. Identifying the important harmonic decisions in a specific composition is a more appropriate analysis of the harmonic flow than just listing all harmonies. This analysis, which is based on the harmonic energy as a parameter which is accessible for the audience, thus highlights the important points which make the composition unique and special.

References Giesl, P.: Von Stimmführungsvorgängen zur Harmonik. Eine Anwendung der Clausellehre auf Wagners Tristan und Isolde. Die Musikforschung 52, 403–436 (1999) Giesl, P.: Von Stimmführungsvorgängen zu Kleinterzzirkeln. Eine Deutung der Teufelsmühle durch die Clausellehre. Die Musikforschung 54, 378–399 (2001)

122

P. Giesl

Giesl, P.: Zur melodischen Verwendung des Zweiten Modus in Messiaens Subtilité des Corps Glorieux. In: Edler, A., Meine, S. (eds.) Musik, Wissenschaft und ihre Vermittlung; Bericht u¨ ber die Internationale Musikwissenschaftliche Tagung Hannover 2001, pp. 259–264. Wißner, Augsburg (2002) Jans, M.: Modale Harmonik. Beobachtungen und Fragen zur Logik der Klangverbindungen im 16. und frühen 17. Jahrhundert. Basler Jahrbuch für Historische Musikpraxis 16, 185 (1992) Riemann, H.: Vereinfachte Harmonielehre oder die Lehre von den tonalen Funktionen der Harmonie, London (1893) Schwind, E., Polth, M.: Article Klausel und Kadenz, 2nd edn. MGG 5, pp. 256–282. Bärenreiter, Kassel (1996)

A Harmonic Analysis of Beethoven’s 1st Symphony, 1st Movement Exposition 18 Starting with a dominant seventh chord, the resolution follows. 2 Another dominant seventh chord is set, another resolution follows... 3 ...and yet another one leading to the basin of attraction of the stable fixed point C which, however, the audience only realises after some time. 9 g, leading to a deceptive cadence. 18 A7 , leading to d minor. 24 f 56 , leading back9 to C. 41 A, leading to d minor. 42 B, leading to e minor and followed by a sequence... 44→45 the unstable situation is clarified to G. 59 Within the basin of attraction of G its dominant D is stabilised by c, 60 leads back with c to G. 64 Within the basin of attraction of G its subdominant C is stabilised by G7 . Note that a is resolved to g in 65/66, and then g → a in 66/67, leading back to G. 77 G – now g minor – is no longer a stable fixed point. We are in a sequence of falling fifths, a free fall without any gravitation. B major seems to be stabilised in 79-81... 82 ...but the action D7 moves us into the basin of attraction of G – note the phrygian and authentic clausulae10 to d in 84. 94–98 For the sequence of diminished seventh chords with resolutions, cf. the following figure: d → e e ← d c → d c b → a a ← b g←a g → a a ← g g ← f → e e ← f e →d 8 9

10

The numbers denote the bars. This is the phrygian version of the Tenor and Discant Clausula (a, f → g) in contrast to the authentic one corresponding to a dominant-tonic relation (a, f → g), cf. footnote 4 and (Giesl 1999). See footnote 4 for explanation.


123

93 The diminished seventh chord is an unstable situation, coming from g → a/f , the resolution is e = d/f → e. 95 The diminished seventh chord is an unstable situation, coming from e → d/f , the resolution is g/b → a. 97 The diminished seventh chord is an unstable situation, coming from a → g/b, the resolution is c/e → d and leads back to the basin of attracion of G. 107 Adding the seventh to G, we are led back to the basin of attraction of C. Development We again have a sequence of falling fifths in 112-124 starting with the newly introduced A7 and again in 124-132, but this time each chord is more stabilised and the decision to move on is taken just before the next harmony. More decisions are made in 149, 153, 157, and 159, finally leading to a minor. The melodic move from e to f (175 to 176) leads us back to C. Recapitulation The same decisions as in the Exposition are taken. In the new part 191-200 an accelerated sequence of decisions causes an unstable situation. Coda 262 C 7 leads to F in 265. 266 A is introduced, leading to d minor in 269. 270 G leads back to the basin of attraction of C. 274 and 276 g leads to a deceptive cadence which is finally resolved in 279 From 279 up to the end we stay at the stable fixed point C.

Tonal Implications of Harmonic and Melodic Tn-Types Richard Parncutt University of Graz

Music composed of tones (in the psychoacoustical sense of sounds that have pitch) can never be completely atonal (Reti 1958). Consider any quasi-random selection of tones from the chromatic scale, played either simultaneously or successively. Most such sets generate associations with musically familiar pitch-time patterns and corresponding tonal stability relationships (Auhagen 1994). A pattern of pitch can imply a tonal centre simply because it reminds us of a tonal passage: it has tonal implications that depend on the intervals among the pitch classes (pcs) in the set.1 The only clear exceptions to this rule are trivial: the null set (cardinality = 0)2 and the entire chromatic aggregate (cardinality = 12). Since every interval, sonority and melodic fragment has tonal implications, even the so-called “atonal” music of composers such as Ferneyhough, Ligeti and Nono is full of fleeting tonal references: at any given moment during a performance, some pitches are more likely than other pitches to function as psychological points of reference. In the following, I will use the terms “tonal” and “atonal” in this broad, psychological sense. A number of terms have been coined in an attempt to map out the diverse terrain that separates (major-minor or harmonic) tonality from (extreme) atonality, including pantonal, extratonal, atonical, neotonal and polytonal. Tonality is multidimensional in the sense that there are many different ways of bridging that gap that manifest as different styles (such as impressionism, bebop and minimalism). The present analysis attempts to map this complexity onto a single dimension. Instead of dividing music into “tonal” and “atonal”, I conceive of degrees of tonality or atonality and imagine the possibility of evaluating the degree of tonality of a passage of music as a positive whole number between, say, 0 for completely atonal and 1 for completely tonal. That number should generally be higher for music that has clearer or longer-lasting tonal anchors. The “atonal” repertoire avoids tonal references by favoring pc-sets with relatively weak tonal implications. A well-known exception is Berg’s violin concerto, a work that is usually regarded as 12-tone but can barely be regarded as “atonal”, because the row at the beginning of the first movement begins with a minor triad. It follows from this exceptional example that, as a rule, “atonal” composers deliberately avoid consonant intervals between successive notes and prefer tonally weak or ambiguous pc-sets. From a logical viewpoint, they may find “atonal” pc-sets in two main ways: 1

The term “pitch” in “pitch-class set” is misleading, because a pc-set is primarily a configuration of intervals. Each set is defined by the number of times each interval class (of which there are six: 1, 2, 3, 4, 5 or 6 semitones) occurs in the set. This set of 6 numbers is called the interval vector (Forte 1973). For example, a major or minor triad contains no semitone, no tone, one minor third, one major third, one fourth (fifth) and no tritone, so its interval vector is [001110]. 2 The cardinality of a pc-set is simply the number of members in the set. T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 124–139, 2009. © Springer-Verlag Berlin Heidelberg 2009

Tonal Implications of Harmonic and Melodic Tn-Types

125

either by borrowing them - consciously or unconsciously - from the existing “atonal” literature, or by discovering them by aurally guided exploration and trial and error, exploring the various possibilities creatively and listening carefully. Since the “atonal” repertoire presumably includes all possible pc-sets, it is no longer possible to find “new” ones, suggesting that these two strategies cannot be separated. How might a composer best find pc-sets corresponding to a given desired degree of tonality or atonality? Composers in atonal idioms (including serial approaches) have intuitively favored pc-sets that avoid perfect intervals (fifths, fourths) and favor tritones and semitones. This paper presents a new method by which composers can systematically and quickly seek and find pc-sets of any specified cardinality and strength of tonal implication. This aim is appropriate given the large number of possible pc-sets from which a composer can choose and the long history of constructive interaction between composition and music theory. My approach is intended to shed light on three areas simultaneously: perception, analysis and compositional practice. The tonal implications of a sounding musical fragment depend not only on the underlying pc-set but also, of course, on its musical realization. The realization of a pc-set has several aspects: properties of individual tones (duration, loudness, timbre, temporal envelope); whether melodic (successive) or harmonic (simultaneous); if melodic, the order of the tones (especially important in 12-tone music) and which tones are repeated; if harmonic, voicing (octave register of each tone, spacing between the tones, doubling in different octaves) and onset synchronicity. A tone is more likely to be perceived as a tonal center if it is repeated (or doubled), has a longer duration, or is simply louder than other tones (Oram and Cuddy 1995; Parncutt 1988; 1997). Here, I assume that it is possible to separate effects associated with the intervals within a set from effects of the set’s specific realization, and focus only on the former. While this assumption may not be entirely valid, it is a good first approximation and a useful starting point. In the following analysis of the tonal implications of pc-sets, I make use of Forte’s convenient and well-known method of enumerating all possible pc-sets within given constraints. While Forte’s method is often referred to as pc-set theory, in the present approach it is no more than a systematic classification system or taxonomy, because the taxonomy itself does not generate predictions that can be empirically tested – unlike the perceptual theory with which it is combined in this paper. While the musicanalytical application of Forte’s taxonomy is usually confined to “atonal” music, there is no reason why it should not be applied to any music composed within the confines of the 12-tone chromatic scale. Tn-types of cardinality 3 Rahn (1980) broke down Forte’s pc-sets into types. One such type is the transpositional type or Tn-type. A Tn-type is a pc-set that is invariant under transposition but not inversion. The subscript n refers to the size of a transposition in semitones, and Tn-type refers to all 12 possible transpositions of a given collection of pcs. The mathematical jargon sounds complicated, but the concept is fundamentally simple. The major and minor triads are both examples of Tn-types. A major triad is a set of three pcs: a root and two further tones, 4 and 7 semitones above the root – 047 for short. The intervallic inversion of the major triad is the minor triad 037, and both

126

R. Parncutt

belong to the same pc-set, whose prime form (Forte 1973) is 037. Because 037 is the 11th in Forte’s list of pc-sets of cardinality 3, it is also referred to as 3-11. When the two Tn-types corresponding to this pc-set are separated, the minor triad is labelled 311A and the major 3-11B. Table 1. All Tn-types of cardinality 3 (after Rahn 1980) set name

3-1

3-2

3-3

3-4

3-4

3-6

3-7

3-8

3-9

3-10 3-11 3-12

prime form

012

013

014

015

016

024

025

026

027

036

023

034

045

056

035

046

inversion

037

048

047

For purpose of argument, let us begin by enumerating all possible Tn-types of cardinality 3. There are 19 of them, and they are presented in Table 1. In the table, “set name” corresponds to “name” in Appendix 1 of Forte (1973); the number before the dash is the cardinality, and the number after the dash is the set’s position in a list of all possible sets of that cardinality. The prime form “012” corresponds to C-C#-D in all chromatic transpositions, “013” to C-C#-D#, and so on. Some of the pc-sets (prime forms) in Table 1 are symmetrical and some are not. For example, 012 is symmetrical, but 013 is asymmetrical. An asymmetrical set may be broken down into two Tn-types, which are labelled A and B: e.g. 013 is labelled 32A, and 023 is labelled 3-2B. The tonal implications of the 19 Tn-types of cardinality 3 vary markedly. At one extreme, the major and minor triads have strong tonal implications; the major is the more strongly tonal, since its root is perceived more clearly (Parncutt 1988). At the other extreme, 012 has almost no tonal implications – by which I mean that the tones sound about equally important and no other (virtual) tones are strongly implied. (Even this is not quite true: when 012 is presented harmonically, 0 and 2 are more audible than 1, due to masking.) All other Tn-types of cardinality 3 have tonal implications with various degrees of strength. For example, 023 may be heard as either the 1st, 2nd and 3rd degrees of a minor scale or the 6th, 7th and 8th degrees of a major scale, suggesting that either the 0 or the 3 in 023 may be heard as a point of reference. The major-third (4-semitone) interval embedded within 014 suggests that its reference pitch is 0, regardless of whether the pattern is heard as Neapolitan, Arabic or Flamenco; in Terhardt’s approach, both pitch-salience patterns and cultural associations are learned, but since pitch-salience patterns are ultimately based on universal aspects of pitch perception in speech, they are expected to vary less than cultural associations across listeners and musical contexts. Given the wide range of tonal implications within Tn-types of cardinality 3 (and any other cardinality for that matter), it is surprising that many pc-set theorists tacitly consider all pc-sets a priori to be equivalent or value-free, as if they had no tonal implications – or as if tonal implications did not exist. Can the tonal implications that we learn from music simply disappear (which is psychologically implausible), or are they arbitrary (which is psychoacoustically and ethnomusicologically implausible)? It may be possible to make tonal implications disappear in a magical, ideal world of mathematics located in a far-off galaxy and inhabited by aliens, but in real music


127

heard by real human beings, pc-sets will always have tonal implications. Moreover, the appeal of so-called atonal music may be due not to an absence of tonal implications, but to their multiplicity, fluctuation and intangibility. The tonal implications of a Tn-type may be understood and quantified by first evaluating the perceptual salience of each chromatic scale degree in the context of that Tn-type. By “salience” I mean the (subjective) importance of something for a listener, or the (objective) probability that a listener will notice or become consciously aware of something - in this case, a tone at a given chromatic scale degree. The perceptual profile of a Tn-type is a set of 12 values, one for each of the 12 chromatic scale degrees. Each value reflects the perceptual salience of that scale degree in the context of (i.e. during or following presentation of) that Tn-type. In the following, I will distinguish between two kinds of perceptual profile, harmonic and tonal, and present separate algorithms for calculating these profiles that are based on contrasting empirical data and perceptual-cognitive3 theory. The harmonic profile The harmonic profile of a Tn-type is a vector of twelve values, each of which is an estimate of the perceptual salience of a pc. In Parncutt (1988; 1989), I assumed the salience of pitches in chords to be proportional to the probability that a pitch will function as the root when the tones are sounded simultaneously (i.e., as a sonority). I assumed that when a given Tn-type is heard repeatedly in different voicings and contexts, the probability increases that a certain pitch will be heard as a reference – a long process that involves learning, history and culture (Parncutt 2005). I then developed a simple algorithm for pc-salience within harmonically presented Tn-types that was based on the virtual pitch algorithm of Terhardt et al. (1982) and the chordroot model of Terhardt (1982). The model was tested by presenting chords of octavecomplex tones (OCTs, Shepard tones) followed by individual OCTs and asking listeners how well the single OCT fits with the chord (Parncutt 1993). In that experiment, and many other experiments reported for example by Krumhansl (1990), OCTs are used to operationalize the music-theoretical concept of a pc (which is equivalent to the music-psychological concept of chroma). Terhardt assumed that the root of a chord is a virtual pitch. By that, he meant that the root corresponds to the fundamental of an approximately harmonic series of audible pure-tone components (partials). Those components, which are a subset of all the chord’s audible4 partials, generally include harmonics of different chord tones. There are usually several possible candidates for the root of a chord; “the” root may be the one corresponding to the most salient virtual pitch, but may also depend on the music with which a listener is familiar, and thus indirectly on the history of musical 3

There is no clear boundary between “perceptual” and “cognitive”. Terhardt’s theory tends to be regarded as perceptual or psychoacoustical, but it is also cognitive in the sense that it involves information processing (or better: his algorithm to predict the pitch salience profile of a complex sound involves information processing). Krumhansl’s approach is explicitly cognitive, but it is based on empirical data obtained from perceptual or psychoacoustical experiments. 4 By “audible” I mean present in the running spectral analysis of the sound which is performed physiologically by the basilar membrane and transmitted to the brain along the auditory nerve. The initial masking stage of Terhardt’s algorithm predicts what is “audible” in this sense and what is not, and assigns spectral pitches to all audible partials.

128

R. Parncutt

syntax and implicitly learned conventions of music theory. Whichever way you look at it, the root is assumed to be learned and enters culture when listeners are repeatedly exposed to consistent patterns of pitch relationships within musical sonorities. Table 2. Root-support intervals (after Parncutt (1988)) Rootsupport interval

diatonic notation size in semitones

root-support weight

P1, P8…

P5, P12…

M3, M10…

m7, m14… M2, M9…

0

7

4

10

2

10

5

3

2

1

Abbreviations: P=perfect, M=major, m=minor

According to Terhardt (1982), the virtual pitch at the root of a chord is generated by the chord’s tones and that the intervals octave/unison, perfect fifth, major third, minor seventh and major second/ninth determine the root. I call these intervals root supports (see Table 2; Parncutt 1988). They are octave generalizations of the intervals between spectral and virtual pitches in typical harmonic complex tones such as voiced speech sounds (i.e., between harmonic overtones and the fundamental). The chord-root model includes free parameters called root-support weights. These are quantitative estimates of the influence of each root-support interval on the salience of the virtual pitch at the lower tone of the interval, and hence on the perceived root of a chord. The weights used in the present calculations are presented in Table 2 and Figure 1 (b, c). They are assumed to depend on the position of the corresponding element in the harmonic series: intervals that occur early in the series are assumed in Terhardt’s approach to be more familiar to the ear and therefore to play a more important role in the determination of virtual pitches and chord roots. The values in Table 2 have been tested by studying the predictions of the model and comparing them with both music-theoretic intuition and various published sources of empirical data. The predictions of the chord-root algorithm (including an additional masking procedure) were tested experimentally in Parncutt (1993) for a limited set of chords of octave-complex tones; when Krumhansl and Kessler (1982) asked how well octavecomplex probe tones follow single chords (rather than short progressions), they obtained essentially the same results (that is, the correlation coefficients between the two sets of profiles are highly significant). The predictions of the chord-root algorithm may also be considered to apply to a typical or average voicing5 of a given Tn-type when it is realized as regular musical tones (harmonic complex tones). Note the absence of the minor third from the root-support intervals presented in Table 2 and Figure 1 (b, c). In this approach, the m3 is not assumed to have any direct influence on the root. First, it is not found in the lower reaches of the harmonic series between an element of the series and the fundamental. Second, the root of the minor

5

The idea of a “typical or average voicing” could be quantified by documenting all voicings of a given Tn-type in a given musical repertoire using software such as David Huron’s Humdrum.


129

poids (1/n)

1

0 40

36

32

28

24

20

16

12

8

4

0

interval (semitones)

weight

(a) The template assumed by Parncutt (1989), in which weights are set to the reciprocal of harmonic number.

10 8 6 4 2 0 0

1

2

3

4

5

6

7

8

9

10 11

interval class (semitones) (b) The octave-generalized template used in the present calculations (similar to that of Parncutt (1988), but without the m3 interval).

0 11

1

10

2

9

3 8

4 7

5 6

(c) A circular representation of the same template; the numbers are intervals above the root in semitones. Fig. 1. The harmonic series template for calculating virtual pitch salience

130

R. Parncutt

triad can be explained solely in terms of the P5 interval between the fifth and root of the chord. Figure 2 (below) shows how the theory correctly predicts the root of the minor triad without explicitly including the minor third as a root-support interval. Incidentally, the omission of the minor-third interval from the root supports does not contradict the relatively high salience of the third degree of the minor scale in the K-K profiles (see Figure 4 below). On the contrary: this model can explain why Krumhansl found the third degree of the minor scale to be more salient than the fifth: the strong minor third is also present in the pc-salience profile of the minor triad (Parncutt, in preparation). Table 3. Matrix used to calculate the harmonic profiles of Tn-types

10 0 2 0 0 5 0 0 3 0 1 0 0 10 0 2 0 0 5 0 0 3 0 1 1 0 10 0 2 0 0 5 0 0 3 0 0 1 0 10 0 2 0 0 5 0 0 3 3 0 1 0 10 0 2 0 0 5 0 0 0 3 0 1 0 10 0 2 0 0 5 0 0 0 3 0 1 0 10 0 2 0 0 5 5 0 0 3 0 1 0 10 0 2 0 0 0 5 0 0 3 0 1 0 10 0 2 0 0 0 5 0 0 3 0 1 0 10 0 2 2 0 0 5 0 0 3 0 1 0 10 0 0 2 0 0 5 0 0 3 0 1 0 10 The harmonic profile of a Tn-type is calculated by a simple pattern-matching routine in which the octave-generalized template illustrated in Figure 1 (b) is compared with the pcs of the set, in all 12 transpositions around the pc cycle. One way to represent this routine is by matrix multiplication. The first column of the matrix in Table 3 corresponds to the template in Figure 1 (b). Successive columns are generated by rotating the template down, one element at a time. The Tn-type is expressed as a vector (1 row and 12 columns) of 12 numbers corresponding to 12 pcs (0 to 11), with the value 1 for each pc that is present and 0 for pcs that are absent. For example, the major triad (047) is denoted (1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0). This vector is then multiplied by the matrix in Table 3. The result after matrix multiplication is the Tn-type’s harmonic profile. The calculated harmonic profile of the major triad is (18, 0, 3, 3, 10, 6, 2, 10, 3, 7, 1, 0). According to this profile, the most salient pitches evoked by a C major triad are C (pc 0, predicted salience = 18), followed by E and G (pcs 4 and 7, salience = 10 in each case). Tones F (pc 5, salience = 6) and A (pc 9, salience = 7) are predicted to be strongly implied although they are not among the chord’s notes. Similar results are obtained for the minor triad; one striking difference is that the difference in salience between the root and the third is smaller for the minor triad, which can explain why the minor triad is tonally more ambiguous and - in that sense - less consonant than the


Minor triad 037

Major triad 047 notes

notes

pitches

pitches

0

0

0

11

1 2

8

4

9

3

7

2

8

4

9

7

10

2

8

4

3

5

7

10

2

9

8

4

7

5 6

1

11

2

9

5 6

pitches 0

8

4

5

11

1

10

3

7

3

0

10

3

4

7

notes

0 11

8

9

Augmented triad 048

pitches

1

2

6

Diminished triad 036

0

1

10

5

6

notes

11

9

3

6

11

1

1

10

5

0

11

11

10

131

2

9

3

8

2

9

4

7

1

10

3

8

4

5 7

6

5

6 6

Fig. 2. Calculated harmonic profiles of four common triads

major. That can in turn explain why minor triads and tonalities are less prevalent and less stable than major triads and tonalities (Eberlein 1994).6 The implied pitches at the 4th and 6th scale degree above the root (M6 for the major triad, m6 above the minor) can explain why chord progressions in which roots fall through fifth or third intervals are more prevalent in tonal music than progressions in the other direction (Eberlein 1994): the pitches that are implied by the first triad (the 4th and 6th) are realized as tones in the second (root and 3rd; Parncutt 2005). The predictions of the model for major, minor, diminished and augmented triads are shown in Figure 2. Corresponding experimental data are presented in Figure 3. In Parncutt (1993), 27 listeners (mainly musicians) rated how well a probe tone went with a preceding chord. Both chords and probe tones were constructed from octavecomplex (Shepard) tones. Trials were shuffled and rotated randomly around the chroma cycle. Filled diamonds in Figure 3 are mean experimental ratings; bars are 95% confidence intervals about those means.

6

Eberlein calculated the frequency of occurrence of different sonorities including major and minor triads, and presented the results in an appendix. He consistently found more major than minor triads in the music of the 18th and 19th centuries - even in a sample that included equal numbers of pieces in major and minor tonalities. The reason is evidently that the dominant triad tends to be major in both modes.

Probe tone’s perceived goodness of fit

132

R. Parncutt

major triad 047

minor triad 037

major-minor seventh 047Q

half-diminished seventh 036Q

Interval class relative to conventional root (semitones) Fig. 3. Experimental data on the salience of pitches evoked by common musical sonorities composed of octave-complex tones (after Parncutt 1993). The diamonds denote the mean responses of 27 listeners; the error bars are the 95% confidence intervals. In each trial, listeners heard a chord of octave-complex (Shepard) tones followed by a single such tone. They were asked to rate how well the tone went with the chord on a scale from 0 (very badly) to 3 (very well). In the chord labels, the letter Q means 10.

The tonal profile The tonal profile of a Tn-type is similar to the harmonic profile, but it is calculated in a quite different way. Each value is an estimate of the probability that a chromatic scale degree will be perceived as the tonic when the Tn-type is realized melodically (successively) in random order(s) and register(s). The calculation involves the major and minor key profiles of Krumhansl and Kessler (K-K) (1982), which are reproduced in Figure 4. They comprise 24 values that may be regarded as measures of the stability of chromatic scale steps in the context of major and minor keys (12 for each). In the following, I will not consider Krumhansl’s well-known explanation of the psychological distances between musical keys based on correlation coefficients between key profiles, nor will I develop the mathematical procedures based on the KK profiles proposed by Temperley (e.g. 2007). Instead, I propose a new algorithm for the pc-salience profile of a Tn-type that is based on the assumption that listeners are familiar with the tonal stability relations within major and minor keys, as represented by K-K profiles. I begin by subtracting a constant (2.23) from all values in the profiles so that the minimum value becomes zero. I then estimate the probability that a given set of tones will occur in a given key by adding up the stability, according to the K-K profiles, of those tones in that key. For example, the probability that the set CEF# will

133

mean goodness-of-fit rating


pitch class Fig. 4. The key profiles of Krumhansl and Kessler (1982). The full line denotes the (i.e. any) minor key (or tonality), the dotted line the major key.

occur in the key of C major is estimated by adding up the stability of C, of E and of F# in the C-major key profile. The novel aspect of this procedure is as follows: I then calculate the tonal profile of the Tn-type as a weighted mean of all 24 K-K profiles (one for each major and minor key), where the weights are the probabilities calculated in the previous step (i.e. how often we expect the Tn-type in question to occur in each key). The underlying idea is that any Tn-type can be heard in any key, but with different probabilities; the tonal profile of a Tn-type is therefore a weighted mean of all 24 key profiles, where each weight is the probability that a key will be cognitively instantiated when the Tn-type is heard. Finally, I normalize that weighted-mean profile so that its mean is 10; individual values are rounded to the nearest whole number. This new algorithm is conceptually simple, but the weighted mean of all 24 keys would be very time-consuming to perform by hand. Although the algorithm is based on the culturespecific assumption that tonality is limited to the Western major and minor modes, it yields intuitively reasonable results for all Tn-types (see appendix). Table 4. Calculated harmonic and tonal profiles for C major and minor triads pitch class

in semitones

0

as letter

C

major triad

harmonic profile

34

3-11B (047)

tonal profile

minor triad

harmonic profile

3-11A (037)

tonal profile

1

2

3

4

5

E

F

G

6

19

11

4 19

D

6

7

0

6

22

0

13

5

17

10

29

2

4

25

0

15

14

7

10

12

8

11

7 14

8

9

10

G

11 B

6

13

2

0

0 22

4

13

4

9

0 19

15

4

2

6

10

8

11

8

134

R. Parncutt

Table 4 compares the two kinds of perceptual profile, harmonic and tonal, for the C-major and minor triads according to these procedures. Both profiles have been normalized so that their mean is 10, and all entries have been rounded to the nearest whole number. The correlation coefficient between the harmonic and tonal profiles is quite high (r = 0.84 for both major and minor), although the two profiles have been calculated on the basis of quite different assumptions and using quite different procedures and numerical values. The results for the major and minor triads are also similar in the following ways. In both harmonic profiles, the most likely root is the conventional root, the third and fifth have relatively high salience, and the fourth and sixth (M6 in the major triad, m6 in the minor) are strongly implied. The approximately equal stability of root and fifth in the tonal profiles is consistent with the idea that the root of a chord does not generally (or even often) coincide with the tonic; for example, a repeated chord near the end of a classical development section is often perceived as a dominant rather than a tonic. The perceptual profile of a Tn-type of cardinality between 1 and 11 always has peaks, which means that it is always to some extent tonal: the clearer the peaks, the clearer the tonality.7 In Parncutt (1988), I developed a simple mathematical formulation of the “peakedness” of a pc-set’s perceptual profile and called it root ambiguity. It was calculated by dividing the sum of the 12 values by their maximum and taking the square root of the result; the square root came from a model developed to account for empirical data on the number of tones simultaneously perceived in a set of musical and non-musical sonorities (their multiplicity) in Parncutt (1989). According to this procedure, the calculated harmonic ambiguity of the major triad is 1.87, which makes it the least ambiguous of all 19 Tn-types of cardinality 3 and is consistent with its ability to blend (to fuse perceptually). Perceptual profiles, consonance and prevalence The appendix presents the calculated perceptual profiles of all Tn-types of cardinality 3.8 These data, when extended to include cardinalities, have interesting compositional and music-analytical applications: they can help composers to find Tn-types of any given degree of tonal strength and analysts to analyze the tonal strength of Tn-types found in the repertoire. However, things are not quite that simple, because the tonal strength of a Tn-type depends not only on the intervals in the set, but also on the prevalence of a set in the tonal literature and the contexts in which it normally appears. And that depends in turn on its consonance, or lack of dissonance.9 This theory is not circular: the prevalence of a Tn-type is assumed originally to depend causally on just two factors, its (lack of) roughness and the peakedness of its tonal 7

My basic assumption is that the flatter the profile, the more ambiguous the tonal implications. I have not considered bitonality, that is, the possibility that a single profile can imply more than one root/chord or tonic/tonality. Bitonality may be regarded as an example of tonal ambiguity. I also deliberately fail to distinguish between ambiguity and multiplicity. A profile with two main peaks may cause a listener to perceive one peak or the other at different times (ambiguity), or both at once (multiplicity). That distinction is beyond the present scope. 8 Profiles for Tn-types of larger cardinality may be obtained directly from the author. 9 This idea applies regardless of how the term “consonance” is defined or understood. The rank order of consonance of common triads is presumably the same as their rank order of prevalence in tonal music: major, minor, diminished, augmented (Parncutt 2006).


135

profile (cf. Terhardt 1976). But the theory is complicated by the gradual historical evolution of tonal syntax (Parncutt, in preparation). Pitch patterns may be perceived as consonant because they are often heard in tonal music and are therefore familiar. A pitch pattern may also be performed and therefore heard more often because it is a subset of commonly-used scales. Since rough sonorities are generally less prevalent in tonal music, they may also have fewer or weaker tonal implications. The roughness of a Tn-type may be predicted on the basis of the average roughness of the six interval classes (cf. Huron 1994). In a first approximation, the roughest interval is the minor second, followed by the major second and tritone (Plomp and Levelt 1965). These may be combined with the interval vector of each pc-set, which shows how often each interval class occurs in the set. In the absence of a comprehensive table of such calculations, consider the interaction between roughness and the calculated ambiguity of the harmonic profiles in the appendix. The least ambiguous sets according to the appendix are 047 (major), 035 (part of a seventh chord), 027 (suspended), and 037 (minor), in that order. The reason why 037 is more prevalent in tonal music than 027 or 035 evidently involves the roughness of the major second interval within 035 and 027. The most ambiguous Tn-types of cardinality 3 are predicted to be 036, followed by 012, 013 and 023, then by 014, 034, 046 and 048. The model predicts that 036 (the diminished triad) has four root candidates of approximately equal salience, making it highly ambiguous. None of its three tones is reinforced by a root-support interval (see Table 2), so all have approximately equal salience, and a non-chord tone - 8 relative to 036, or Ab relative to CEbGb - is reinforced by all three tones, which gives it the character of a “pitch at the missing fundamental”. Why is 036 so prevalent in tonal music in spite of its tonal ambiguity? First, it is relatively smooth because it contains no major or minor seconds. Second, it is a subset of the prevalent major-minor (dominant) seventh chord (4-27B or 0368), which is the least ambiguous Tn-type of cardinality 4. Third, it is a subset of the standard major and minor scale sets (Parncutt 2006). Thus, it is both relatively smooth and relatively prevalent. The other listed sets are less prevalent because they contain rough second intervals. These sets may therefore be considered suitable for composition of “atonal” music. Conclusion In this paper, I have sketched a new, systematic approach to the enumeration and perceptual analysis of Tn-types. I have attempted to explain the relative tonalness, consonance and prevalence of Tn-types on the basis of the pitch-salience profiles and the roughness of corresponding musical sonorities. The preliminary findings are promising and the approach shows potential for future application in music analysis and composition. This is not a new investigation in the sense that an answer is sought to a new question. Rather, I have considered the implications of existing empirical and theoretical work for music theory, analysis and composition. The novel aspects of this paper include the systematic application of the algorithm presented in Parncutt (1988) to all possible Tn-types, and consideration of the implications of that procedure for both the history of tonal-harmonic syntax and contemporary composition. Another

136

R. Parncutt

original element is the development of a new algorithm for the pitch-salience of a Tntype based on the K-K profiles. The models that I have presented are incomplete in that they do not account for differences in the musical realization of Tn-types. It would be possible, but beyond the present scope, to account quantitatively in the presented models for parameters such as register, doubling, loudness, doubling and repetition. Acknowledgments. I am grateful to Helga de la Motte-Haber, Timour Klouche and an anonymous reviewer for their insightful questions, criticism and suggestions.

References Auhagen, W.: Experimentelle Untersuchungen zur auditiven Tonalitätsbestimmung in Melodien. Kölner Beiträge zur Musikforschung, vol. 180. Bosse, Kassel (1994) Eberlein, R.: Die Entstehung der tonalen Klangsyntax. Peter Lang, Frankfurt (1994) Forte, A.: The structure of atonal music. Yale University Press, New Haven (1973) Huron, D.: Interval-class content in equally tempered pitch-class sets: Common scales exhibit optimum tonal consonance. Music Perception 11, 289–305 (1994) Krumhansl, C.L., Kessler, E.J.: Tracing the dynamic changes in perceived tonal organization in a spatial representation of musical keys. Psychological Review 89(4), 334–368 (1982) Krumhansl, C.L.: Cognitive Foundations of Musical Pitch. Oxford University Press, New York (1990) Oram, N., Cuddy, L.L.: Responsiveness of Western adults to pitch distributional information in melodic sequences. Psychological Research 57(2), 103–118 (1995) Parncutt, R.: Revision of Terhardt’s psychoacoustical model of the root(s) of a musical chord. Music Perception 6, 65–94 (1988) Parncutt, R.: Harmony: A psychoacoustical approach. Springer, Berlin (1989) Parncutt, R.: Pitch properties of chords of octave-spaced tones. Contemporary Music Review 9, 35–50 (1993) Parncutt, R.: A model of the perceptual root(s) of a chord accounting for voicing and prevailing tonality. In: Leman, M. (ed.) Music, gestalt, and computing - Studies in cognitive and systematic musicology, pp. 181–199. Springer, Berlin (1997) Parncutt, R.: Perception of musical patterns: Ambiguity, emotion, culture. In: Auhagen, W., Ruf, W., Smilansky, U., Weidenmüller, H. (eds.) Music and science - The impact of music. Nova Acta Leopoldina, vol. 92(341), pp. 33–47. Deutsche Akademie der Naturforscher Leopoldina, Halle (2005) Parncutt, R.: Peer commentary on N. D. Cook & T. X. Fujisawa, The psychophysics of harmony perception: Harmony is a three-tone phenomenon. Empirical Musicology Review 1(4) (2006), http://emusicology.org/ Parncutt, R.: Key profiles as pitch salience profiles of tonic triads (in preparation) Plomp, R., Levelt, W.J.M.: Tonal consonance and critical bandwidth. Journal of the Acoustical Society of America 38, 548–560 (1965) Rahn, J.: Basic atonal theory. Schirmer, New York (1980) Reti, R.: Tonality, atonality, pantonality: A study of some trends in twentieth century music. Greenwood Press, Westport (1958) Temperley, D.: Music and probability. MIT Press, Cambridge (2007)


137

Terhardt, E.: Ein psychoakustisch begründetes Konzept der musikalischen Konsonanz. Acustica 36, 121–137 (1976) Terhardt, E.: Die psychoakustischen Grundlagen der musikalischen Akkordgrundtöne und deren algorithmische Bestimmung. In: der Musik, T. (ed.) Carl Dahlhaus and Manfred Krause, pp. 23–50. Technical University of Berlin, Berlin (1982) Terhardt, E., Stoll, G., Seewann, M.: Algorithm for extraction of pitch and pitch salience from complex tonal signals. Journal of the Acoustical Society of America 71, 679–688 (1982)

138

R. Parncutt

Appendix: Calculated Perceptual Profiles of All Tn-Types of Cardinality 3 Row 1: Tn-type labels, harmonic profile (12 values), ambiguity of harmonic profile (a) Row 2: correlation between harmonic and tonal profiles (r), tonal profile (12 values), ambiguity of tonal profile (a)

3- 1

(012)

21 19 23

4

r = 0.72

11 10 11

9 10 11

3- 2A (013) r = 0.75

3- 2B (023) r = 0.75

3- 3A (014) r = 0.75

3- 3B (034) r = 0.75

3- 4A (015) r = 0.83

3- 4B (045) r = 0.83

3- 5A (016) r = 0.73

3- 5B (056) r = 0.73

4 10 10 10 9 11

6

6

8

2

a=2.29

9 11 10

9

a=3.26

19 21

4 23

0 13 10

0 15

6

2

8

a=2.29

11 11

8 12

8 11

9

9 12

8 11

8

a=3.11

21

2 23 19

4 13

0 10 15

0

8

6

a=2.29

12

8 11 11

8 11

8 12

9 11

8

a=3.11

2

a=2.20

9

25 19

6

4 19 10 13

0

11 11

9

9 12 10

9

9 11 11

8 10

a=3.13

25

2

6 19 19 13

4

0 15 10

2

6

a=2.20

12

9

9 11 11 10

8 11 11

9 10

a=3.13

19 25

4

13 11

9 10

25

6

14

4

6

6 11

2

a=2.05

8 14

8

9 11

9 11

7

a=2.97

6

2 19 29

4

4

6 10 11

0

a=2.05

8 10

9 11 13

7 11

9 11

8

a=2.97

10 13

19

9

2

0 29 10

19 19 10

6

6 15

4

2 10 29

8 10 10 10 12

6 10

2

12 10 10 10

2 29 19

0 10

6

9

2 11

a=2.05

8 11 10 10 10

a=3.08

4 10

a=2.05

0 11 10

8 13 10 10 10 10 11

8

a=3.08


3- 6

(024)

27

0 25

0 23 10

4 10

6 10

8

0

a=2.12

r = 0.75

12

8 12

8 12 10

8 12

8 12

8 10

a=3.11

21

6 23

2

4 29

0 13

6

0 17

0

a=2.05

14

7 12

9

8 14

7 12

8 11 11

7

a=2.95

19

8

4 21

0 32

0

4 15

0 11

6

a=1.93

14

8

9 12

7 14

7 11 11

8 12

7

a=2.95

21

0 29

0

6 10 19 10 10

0

8 10

a=2.05

11

9 12

9

9 10 11 11

8 11

9 10

a=3.14

25

0 11

0 21 10 23

0 10 10

2 10

a=2.20

9 11

8 11

a=3.14

3- 7A (025) r = 0.85

3- 7B (035) r = 0.85

3- 8A (026) r = 0.59

3- 8B (046) r = 0.59

3- 9

3-10

11 10

9

9 12

9 11 10

(027)

30

0 23

6

4 11

0 29

6

8

0

a=1.98

r = 0.90

14

6 14

8 10 11

7 15

7 11 10

8

a=2.82

(036)

19

2 10 19

2 15

a=2.51

r = 0.62

3-11A (037) r = 0.84

3-11B (047) r = 0.84

3-12

139

2 13 19

0 19

4

0

11 10

8 12

8 10 11 10 11

8 11 10

a=3.11

29

2

4 25

0 15

0 19 15

4

2

6

a=2.05

14

7 10 12

8 11

7 14 10

8 11

8

a=2.95

34

0

6

6 19 11

4 19

6 13

2

0

a=1.87

14

7 11

8 12 10

7 14

8 11

8 10

a=2.95

0 25 10

6

6

0

a=2.20

8 10

a=3.15

(048)

25 10

6

r = 0.74

12 10

8 10 12 10

0 25 10

8 10 12 10

Calculating Tonal Fusion by the Generalized Coincidence Function Martin Ebeling Peter-Cornelius-Conservatory of music, Mainz [email protected]

Abstract. Models of pitch perception in the time domain suggest that the perception of pitch is extracted from neuronal pulse series by networks for periodicity detection. A neuronal mechanism for periodicity detection in the auditory system has been found in the inferior colliculus (Langner 1983). The present paper proposes a mathematical model to compute the degree of coincidence in the periodicity detection mechanism for musical intervals represented by pulse series. The purpose of this model is to study the logical structure of coincidence and to define a measure value for the degree of coincidence. The model is purely mathematical but has a strong relation to physiological data presented by Langner. As the sensation of consonance depends mostly on pitch, frequency is the only parameter to be regarded in the model. The integration of other parameters and the adaptation to further physiological data should be easy but still lies ahead. The model is a mathematical basis for a concept of consonance based on pitch perception models in the time domain. In contrast to the concept of the sensory consonance it does not refer to the percept of roughness, which nevertheless is important for the perceived pleasantness of consonances.

1 Background 1.1 Tonal Fusion and Roughness Carl Stumpf observed that consonant intervals show a tendency to cohere into a single sound image. He called this phenomenon Tonverschmelzung – tonal fusion. Consonant intervals show a stronger tendency to fuse than less consonant or dissonant intervals. From extensive hearing experiments Stumpf deduced a system of rules which he termed Stufen der Tonverschmelzung and illustrated it in a curve which he called System der Verschmelzungsstufen in einer Curve (Stumpf 1890/1965). The curve shows the degree of fusion for all intervals over a range of an octave. Not only the consonant intervals which are the prime (1:1), minor third (5:6), major third (4:5), pure fourth (3:4), pure fifth (2:3), minor sixth (5:8), major sixth (3:5) and the pure octave (1:2) - indicated by their frequency ratio - have a higher degree of fusion, but also slightly mistuned intervals nearby the consonant intervals. Stumpf’s concept of tonal fusion is an attempt to define consonance and dissonance psychologically. He emphasizes consciousness and focuses on the mental acts of attending to and referring to sound. Thus, his approach is essentially different from T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 140–155, 2009. © Springer-Verlag Berlin Heidelberg 2009

Calculating Tonal Fusion by the Generalized Coincidence Function

141

Helmholtz’s (1877) idea to explain consonance and dissonance as a result of the sensation of roughness, a theory based on the physiology of hearing. While Helmholtz deals with the perception of tone, Stumpf, by contrast focuses on the apperception of tone. Without discussing roughness in this paper it should be mentioned that the important phenomenon of roughness (Zwicker and Fastl 1999) is widely accepted for the explanation of consonance and dissonance as it is closely combined with the psychophysical model of critical bandwidth filters on the basilar membrane (Plomp and Levelt 1965). Several methods for the calculation of roughness have been proposed (Aures 1985; Kameoka and Kuriyagawa 1969). Considerations of the underlying logic of neuronal processing in the auditory system reveal that tonal fusion is besides roughness a quite considerable concept to explain consonance. This paper is based on the descriptions of a periodicity detection mechanism in the inferior colliculus (IC) found by Langner (1983; 2007). The logic of its operation is studied mathematically by the definition and calculation of the Generalized Coincidence Function (Ebeling 2007).

Fig. 1. Stumpf’s “System der Verschmelzungsstufen in einer Curve” (Stumpf 1890/1965)

1.2 Interspike Interval Distributions, Pitch Estimates and Harmony 1.2.1 Neuronal Code and Pitch The inner ear provides a frequency analysis mechanism which transforms incoming sound into a neuronal code (Zhang, Heinz, Bruce and Carney 2001). Due to the mechanics of the basilar membrane and the frequency selectivity of the hair cells, sound induced pressure waves travelling through the cochlea are converted into neural impulses, representing the differentiatedly resolved frequency components of the sound (Goldstein and Scrulovic 1977). In the case of a single pure tone, the travelling pressure wave maximally activates hair cells at a certain place on the basilar membrane and makes them basically react with a periodic firing pattern. According to the volley principle of Wever (1949) this results in a running spike train in the auditory nerve with a period corresponding to the reciprocal of the frequency of the tone. Two pure tones falling into the same critical bandwidth stimulate the same group of hair cells and cannot be resolved in the auditory system. They produce a pitch

142

M. Ebeling

percept, corresponding to a frequency between the frequencies of the two pure tones and an amplitude modulation with a frequency equal to the difference frequency. All partials of an idealized harmonic sound are multiples of a fundamental frequency. As a result of superposition the period of the fundamental is equal to the period of the envelope of the harmonic sound. The period of the fundamental is also encoded in the cochlea in amplitude modulations resulting from superposition of frequency components above the third harmonic. As a consequence the period of the fundamental is coded temporally in spike intervals in the auditory nerve and can be analysed by neurons in the auditory brain stem (cochlear nucleus: CN) and midbrain (inferior colliculus: IC) (Langner 2005). 1.2.2 Interspike Intervals The time between neural spikes, called interspike interval (ISI), can be measured either between successive discharges (1st- order ISI) or between both successive and non-successive spikes (all- order ISI). Counting all ISI in a discharge pattern leads to histograms that show the interspike interval distributions for the entire auditory nerve. Cariani and Delgutte (1996) have shown that ISI- histograms (autocorrelograms) computed from all-order ISIs show high peaks for periods corresponding to the pitch. This demonstrates that the most frequent all-order interspike interval corresponds to the perceived pitch (Cariani and Delgutte 1996, 1698). 1.2.3 Coinciding Periodicity Patterns for Intervals Tramo, Cariani, Delgutte and Braida (2001) analysed the neuronal responses to harmonic intervals. They used stimuli of isolated, harmonic intervals (minor second, perfect fourth, triton, perfect fifth) formed by complex tones. Each of the two complex tones contained the first six harmonics with equal amplitude and equal phase. They found that in addition to the pitches of notes actually present in the interval, for consonant intervals, the fine timing of auditory nerve fiber responses contain strong representations of harmonically related pitches implied by the interval (subharmonics, e.g. Rameau’s fundamental bass; for the perceptual root of a chord, see Parncutt 1989; 1997). Moreover, all or most of the six partials can be resolved by finely tuned neurons throughout the auditory system. By contrast, dissonant intervals evoked auditory nerve fiber activity that does not contain strong representations of constituent notes or related bass notes. As in the case of dissonant intervals the two complex tones contain many partials too close together to be resolved, these partials interfere with one another, and thus cause coarse fluctuations in the firing of peripheral and central auditory neurons. This gives rise to the perception of roughness and dissonance. (Tramo, Cariani, Delgutte and Braida 2001, 92). Tramo, Cariani, Delgutte and Braida (2001) determined the ISI distributions embedded in the responses of axons throughout the auditory nerve during stimulation with musical intervals. Comparing these ISI histograms with the graphs of the computed autocorrelation functions (with primaries, consisting of six equally strong harmonics) they found the same periodicity patterns in both, the autocorrelation and the ISI distributions.


143

1.2 Autocorrelation 1.2.1 Autocorrelation versus Fourier-Analysis Measuring and counting the all-order ISI is an analysis in the time domain which is, from a logical point of view equivalent to the computation of an autocorrelation function. The autocorrelation function shows peaks for all periods of a signal. As periods are distances in time, the autocorrelation function has to be regarded as an analysis in the time domain. From the investigations of Cariani and Delgutte (1996) it becomes probable that a neuronal autocorrelation mechanism for the detection of the periods of running spike trains in the auditory system provides the sensation of pitch. It must be pointed out that the autocorrelation function is as powerful a means to sound analysis as the Fourier-transform. The famous theorem of Wiener - Khintchine (Wiener 1930; Hartmann 2000) says that the autocorrelation function is the Fourier-transform of the power spectrum (energy spectral density) (see also Papoulis 1962, 246). As a consequence, the autocorrelation analysis is equivalent to a Fourier-analysis of a signal. The Fourier-analysis is used for spectral analysis in the frequency domain; the autocorrelation analysis is a periodicity analysis in the time domain. The power spectrum shows all frequencies inherent in the signal but no phase-shifts, the autocorrelation function shows all periods inherent in the signal including all subharmonic periods, but also no phase-shifts. 1.2.2 Hearing Theories and Autocorrelation Neuronal spike patterns in the auditory system can mathematically be represented by pulse sequences. Forming their autocorrelation functions (Papoulis 1962, 249) provides all information about their periodicity and ISIs. Thus, the “existence of a central processor capable of analyzing these interval patterns could provide a unified explanation for many different aspects of pitch perception” (Cariani and Delgutte 1996, 1698). Since Licklider (1951), a lot of auditory theories operating in the time-domain presume an autocorrelation mechanism or a related model to detect the periodicity of the stimuli (overviews: Hartmann 2000; Cheveigné 2005). These models have been tested psychoacoutically or with computer simulations (e.g. Meddis and Hewitt 1991; Patterson and Allerhand 1995) using different stimuli. Those tests give evidence for (e.g. Yost, Patterson and Sheft 1996) and sometimes against (e.g. Kaernbach 1998) an autocorrelation mechanism in the auditory system. Few models are based on physiological data using properties of neuronal circuits in the auditory pathway (Langner 1983; Meddis and Hewitt 1991). The present paper refers to properties of Langner’s model of periodicity detection in the inferior colliculus (IC) so that this model should briefly be presented. 1.3 Langner’s Neuronal Correlator Langner (1983) measured the responses of neurons in the cochlear nucleus (CN) and inferior nucleus (IC) to amplitude modulated signals (Hartmann 2000, 399) and proposed a model that performs a correlation between signal fine-structure and modulation envelope: The model of Langner is based on neuronal delay and coincidence mechanisms. Its processing elements are a trigger, an oscillator, a reducer, and a coincidence neuron. These elements have their counterparts in well-described on-type, which discharge only to stimulus onset, chopper neurons, and pauser neurons in the

144

M. Ebeling

CN and disc cells in the IC. The oscillator responds with short bursts of regular intrinsic oscillations to each modulation period. The integrator collects the energy of the signal, thus generating intervals precisely related to the signal fine structure. By integrating synchronized activity of many nerve fibres the reducer is able to code frequencies to the upper limit of phase coupling. The trigger unit synchronizes the responses of oscillator and reducer cycle to the modulation. The coincidence unit is activated by simultaneous inputs from oscillator and reducer. It responds best when signal fine-structure (detected by the integrator) and signal envelope (which the oscillator is synchronized to) are correlated and the envelope period matches the reducer delay. Thereby each of such a neuronal circuit responds best to a periodically modulated sound (BMF) and is simultaneously representing a certain frequency and a certain pitch (Langner 2005). Three different periods are crucial for coincidence detection:

τ m - The period of the envelope (Hartmann 2000, 412-426) τ c - The period of a carrier frequency, that means the fine- structure of the sound, and

τ o - The period of intrinsic oscillation. Langner (2005; 2007) assumes that the detection of the envelope period yields to the sensation of pitch, whereas the timbre of the sound corresponds to the finestructure of the sound, represented by τ c . The inner oscillation provides a time slot of coincidence, with periods of

τ o = 0.8

ms; τ o = 1.2 ms…up to τ o = 2.4 ms or generally: τ o = 0.8 ms + k ⋅ 0.4 ms . (Langner and Schreiner 1988, 1813). Mathematically described, these three periods correlate if there are small integers n, m, so that the “periodicity equation” is valid (see Langner and Schreiner 1988, 1818):

m ⋅ τm + n ⋅ τc + τo = 0 .

(1)

The intrinsic oscillation with period τ o contributes a fuzziness to the coincidencedetection as intrinsic oscillation raises the coincidence neuron onto an excitation level closely under threshold. At the beginning of stimulation each coincidence neuron shows a responds characteristic of a comb filter. But after about 30 ms inhibition from the onset neuron the coincidence neuron in functionally converted to a bandpass filter (Voutsas, Langner, Adamy and Ochse 2005; Langner 2007). This grants that the whole bank of coincidence circuits acts like an autocorrelator onto the modulation frequencies.

2 Mathematical Model of Generalized Coincidence 2.1 Correlation Functions Applying autocorrelation functions makes it necessary to classify functions (signals) according to their average power that is defined by:

Calculating Tonal Fusion by the Generalized Coincidence Function ____ 2

f

(t ) = Tlim →∞

1 2T

³

T

−T

145

2

f (t ) dt .

(2)

The proposed model of generalized coincidence makes exclusive use of functions ___ 2

(t ) = 0 . Nevertheless, it can easily be ex___ which have the property that 0 < f 2 (t ) < ∞

with finite energy, which means that tended to finite power functions,

f

(Papoulis 1962). In the case of functions with finite energy, the correlation functions of two functions f 1 (t ), f 2 (t ) are defined by: autocorrelation function ∞

ρ i (τ ) = ³ f i (t ) f i (t + τ )dt

(3)

−∞

cross correlation functions ∞

ρ12 (τ ) = ³ f 1 (t ) f 2 (t + τ )dt

(4)

−∞

∞

ρ 21 (τ) = ³ f 2 (t ) f 1 (t + τ )dt

(5)

−∞

Substituting t ′ = t + τ in (4) shows that

ρ12 (τ ) = ρ 21 (− τ ) .

(6)

Let S (t ) = f 1 (t ) + f 2 (t ) be the sum of two functions f 1 (t ), f 2 (t ) . Using definition (3) immediately leads to the sum formula of autocorrelation functions: ∞

∞

−∞

−∞

³ ³ [ f (t ) + f (t )]⋅ [ f (t + τ ) + f (t + τ )]dt = f (t ) f (t + τ )dt + f (t ) f (t + τ )dt + f (t ) f (t + τ )dt + f (t ) f (t + τ )dt ³ ³ ³ ³

ρ S (τ ) =

S (t )S (t + τ )dt =

∞

1 −∞

1

2

1

∞

1

−∞

2

∞

2

2

1 −∞

= ρ1 (τ ) + ρ 2 (τ ) + ρ12 (τ ) + ρ 21 (τ )

∞

2

−∞

2

1

(7) 2.2 Sequence Representation of a Tone In the auditory system, a pitch is represented by a periodic pulse train which mathematically can be represented by a sequence of equally spaced pulses (M positive integer or ∞ ):

xμ (t ) =

M

¦ I (t − mT )

m=− M

μ

(8)

The constant T is the period of the pulse train and it is the reciprocal of the frequency corresponding to the perceived pitch. The function I μ (t ) describes the pulse

146

M. Ebeling

form. A neuronal pulse is built up by a lot of neuronal discharges randomly distributed around time mT and coincidences randomly occur in a time window. Therefore,

I μ (t ) should be a density function, that means: i.

I μ (t ) ≥ 0 for every t;

ii.

³

∞

−∞

(9)

I μ (t )dt = 1 .

(10)

Furthermore, the spread of all single discharges is determined by parameter scribing the “width” of the pulse

I μ (t ) . Taking μ as a real number, I μ (t ) becomes

a family of functions with the generalized limit third property of

I μ (t ) follows:

iii.

μ de-

δ(t ) (Papoulis 1962, 277). Thus, a

lim I μ (t ) = δ(t )

(11)

μ →0

The limit

lim xμ (t ) = μ→0

M

¦ δ(t − mT )

(12)

m =− M

is the idealized case of all neuronal discharges occurring exactly at time

mT . Ex-

I μ (t ) may be the gaussian pulse with μ to determine the variance or the rectangular pulse with the width μ .

amples for

If two pulses fulfil the properties (i)-(iii), their cross correlation functions also fulfil the properties (i)-(iii). Considering definition (3), property (i) and (ii) become obvious for the cross correlation function. Property (iii) can be proved by applying the definition of the generalized limit: If ∞

χ μν (τ ) = ³ I μ (t )Jν (t + τ )dt

(13)

−∞

is the cross correlation function of two pulses

lim lim ³

∞

()()

I μ (t ) and J ν (t ) , it can be shown, that

()

()

χ μν τ φ τ dτ = φ 0 for every continuous test function φ t . By μ →0 ν →0 − ∞ definition of the generalized limit this is equivalent to property (iii). As by definition the autocorrelation function is a special case of a cross correlation function thus follows that the properties (i)-(iii) are also valid for the autocorrelation functions of

I μ (t ) and J ν (t ) .

2.3 Sequence Representation of an Interval In the model the sum of two simultaneously running pulse trains is the mathematical representation of neural spike trains corresponding to an interval. If ν 1 , ν 2 are the


frequencies of the two tones constituting the interval and ing periods, the frequency ratio of the interval is

s= Let

147

T1 , T2 are the correspond-

ν 2 T1 −1 or T2 = s T1 = ν 1 T2

(14)

I μ (t ) and J ν (t ) be two families of pulse functions with properties (i)-(iii) as

above. The two tones of the interval shall be represented by the two sequences

xμ (t ) =

M

¦ I μ (t − mT1 ) ; xν (t ) =

m= − M

N

¦ J ν (t − nT2 ) =

n= − N

¦ J (t − ns N

n=− N

ν

T1 ) (15)

−1

Their sum

S (t ) = xμ (t ) + x ν (t )

(16)

is the mathematical representation of the interval with the frequency ratio s. Furthermore, let:

α μ (τ ) be the autocorrelation function of the pulse I μ (t ) ,

α ν (τ ) be the autocorrelation function of the pulse J ν (t ) , χ μν (τ ) , χ νμ (τ ) be the cross correlation functions of the pulses I μ (t ) , J ν (t ) .

From (6) follows:

χ μν (− τ ) = χ νμ (τ )

(17).

By induction on N the equation for N , k , m, n integers, N ≥ 0, k ≤ N:

{

}

# (n,m) n ≤ N ∧ m ≤ N ∧ (n − m) = k = 2 N + 1 − k

(18) can be proved. Together with the definitions (3)-(5) and the sum formula (7) (the linearity of integration respectively), it follows that: The autocorrelation function of xμ (t ) is the sequence: 2M

ρ μ (τ ) = ¦ (2 M + 1 − n )α μ (τ − nT1 ) ;

(19)

n = −2 M

The autocorrelation function of x ν (t ) is the sequence

ρν (τ ) =

¦ (2 N + 1 − n )αν (τ − ns −1T1 ) ; 2N

n = −2 N

(20)

148

M. Ebeling

The cross correlation functions of xμ (t ) and x ν (t ) are (21)

(22) As the parameters n and m are as well positive as negative and as

χ μν (− τ ) = χ νμ (τ ) from (17), both cross correlation functions are equal: ρ μν (τ ) = ρ νμ (τ )

(23)

2.4 Autocorrelation Function of an Interval Applying the sum formula (7) to S (t ) = xμ (t ) + x ν (t ) , it follows from (19)-(21) that

ρ S (τ, s ) = ρ μ (τ ) + ρ ν (τ ) + ρ μν (τ ) + ρ νμ (τ ) = ρ μ (τ ) + ρ ν (τ) + 2ρ μν (τ) = =

¦ (2M + 1 − m )α (t − mT ) + ¦ (2 N + 1 − n )α (t − ns 2M

2N

μ

m = −2 M ∞

+2¦

∞

¦ χ (τ − (ns

m = −∞ n = −∞

μν

1

n = −2 N

−1

ν

−1

T1

)

(24)

) )

− m T1

is the autocorrelation function of the interval with the frequency ratio s. As the autocorrelation function also depends on the frequency ratio s, it is introduced as second variable. 2.5 Definition of the Generalized Coincidence Function We define the generalized coincidence function as the integral

Κ (s ) := ³ ρ 2S (τ, s )dτ D

0

(25)

As we are only interested in positive periods up to a certain length, the integration is performed over the interval 0, D with D>0. For each frequency ratio s, Κ (s ) is a measure value of overall coincidence between the two pulse trains representing the two tones of the interval with regard to pulse forms and pulse widths. This becomes clear from the example of rectangular pulses (see Ebeling 2007).

[

]

3 Application of the Model to Rectangular Pulse Sequences 3.1 Correlation Functions of Rectangular Pulses 3.1.1 Autocorrelation Function of the Rectangular Pulse To apply the model, the degree of coincidence shall be calculated for all frequency ratios s within an octave, that means 1 ≤ s ≤ 2 . As a pulse function to fulfil the properties (i)-(iii) we take the rectangular pulse


149

μ 1 ° μ if t < 2 ° I μ (t ) := ® °0 otherwise ° ¯

(26)

Its autocorrelation function is the triangle pulse. (see also Papoulis 1962, 243).

1 § τ · ° ¨¨1 − ¸¸ if τ < μ μ¹ °° μ © α μ (τ ) = Δ μ (τ ) := ® ° 0 otherwise ° °¯

(27)

Fig. 2. The autocorrelation function of the rectangular pulse is the triangle pulse

3.1.2 Cross Correlation Function of the Rectangular Pulse Consider two real functions f1 (t ) and f 2 (t ) with the Fourier transforms

F1 (ω )

and F2 (ω ) . It can be shown (see Papoulis 1962, 244) that the definition of the cross correlation function given in (4) is equivalent to

ρ12 (τ ) =

1 2π

∞

³ F1 (ω )F2 (ω )e

iωτ

dω

(28)

−∞

Thus, the cross-correlation function is the Fourier transform of the cross-energy

spectrum E12 (ω ) := F1 (ω )F2 (ω ) . Recall that the Fourier transform of the rectangular pulse I ε (t ) is (see Papoulis 1962, 20)

150

M. Ebeling

§ ε· 2 sin¨ ω ¸ 1 © 2 ¹ =: F (ω ) I ε (t ) ↔ ε

ε

ω

Considering two rectangular pulses

E μν (ω ) =

(29)

I μ (t ) and Iν (t ) the cross-energy spectrum

§ § μ −ν · § μ +ν · · cos cos ω ω ¸ ¸¸ ¨ − ¨ ¸ ¨ ¨ © 2 ¹¹ ¹ μν ω 2 © © 2 2

(30)

is obtained. As this function is even, equation (28) becomes

1 ρ12 (τ ) = 2π This leads to integrals of the form

∞

∞

³ E μν (ω )cos(ωτ )dω

(31)

−∞

³−∞

cos(ax ) dx = −π a . This evaluation can be x2

shown using the calculus of residue. As a result, the cross correlation function of the two pulses I μ (τ ), I ν (τ ) is the function:

1 § μ +ν · if ° μν ¨ 2 − τ ¸ © ¹ ° °° 1 χ μν (τ ) := ® if ° μ ° ° °¯ 0 otherwise

μ −ν 2

. This representation might be quite complex; I have written and used csound instruments that take over 40 parameters. However, for easiest mathematical treatment we would want to make sure that all the parameters were independent of one another. The case of parameters that are partly dependent on each other, for example the usual treatment of attack, duration, sustain and decay controls in computer music, is more complicated. Suppose then we have a note with n independent parameters pi, represented as a list or n-tuple. In general, each parameter may have a different perceptual space, which may require a different algebraic object – let’s just say, group – to act on the objects in that space. If the parameters are independent, we can define an action on the notes using the direct product group of the respective groups for each parameter in the list. (See Example 11.) The action in Example 11 is non-uniform because each parameter is acted on by a different group element, but in addition, the groups from which the group elements are taken also vary from parameter to parameter. Instead of being an exponentiation of some one group G as in Example 9, this is the direct product group of a number of different groups. Yet this action remains homogeneous in the sense that the same action is applied at all its levels (one transition). We call it a “levelhomogeneous group-heterogeneous non-uniform action.”

( ( &! " # H

4 5 - - - V ! 4 5 - - - &" H 4 4 5 5 - - - * 4 5 - - - + ! 4*4+ 5*5+ - - - * + Example 11. Level-homogeneous group-heterogeneous non-uniform action on orderings

So actions can be: level-homo or level-hetero, group-homo or group-hetero. If group hetero, then non-uniform. If group-homo, then either uniform or non-uniform. Ignoring “hard” dependencies such as “if group-hetero then non-uniform,” there would be a total of 8 such classifications, a sort of Eight-Fold Way. If we bring other entities such as rings, fields, and modules into the picture, the scheme exfoliates too madly to bother with. Admittedly, this classification scheme for actions is rather complicated. However, this is one of its virtues, leading us away from any simplistic tendencies and toward ideas of mathematical actions that are complex enough perhaps to begin to model Isobel’s musical acts.

Approaching Musical Actions

299

Finally, let’s take a different approach to Isobel’s representations.14 We informally tie together three notions: relation, net, and space. Define an n-ary relation R in the usual way as any subset of the Cartesian product of n sets. If all the sets in the Cartesian product are the same, the product is homogeneous; if not, heterogeneous. If it is homogeneous, we call it a relation “on” its unique underlying set. Considering R as a relation, we can ask if it is reflexive, symmetric, transitive, connected, and so on. This can also be considered an n-dimensional net, or digraph, when a net is defined as Lewin did originally, identifying arrows with ordered pairs (tuples, in the general case) of objects. This is also interpretable under certain conditions as the total note-space, with each point in it a note as defined above. Each subspace, and in particular each dimension, would have its own metric, defining its interval structure, and so on in the familiar development of transpositions and other isometries and isomorphisms. We can color this net or relation in more or less elaborate ways by giving a formal interpretation of it, as defined in Example 12 for homogeneous relations on a set S (the definition for heterogeneous relations follows mutatis mutandis).

" $ &$" , " $ $ $"(

"(" . *$ + / $#""# $() $ " $" .( / "

) %& 0 " () $ %$! )$ ,! 7&

$ " , * $"#"# 4 + ) (" "# 7& $# . * $ + " &$" ,0 %) ,0 ) as its identity relation. Monoidal acts are defined for it (see the definition of S-act). So we could define a Net whose mode-contents are binary relations, and whose arrow-labels are also binary relations that act on the node content! This would open up quite a new field of inquiry about Nets. 14

What follows is endebted to Rahn (1994).

300

J. Rahn

L "$! &$" , " $ 2 )

#) N&O (" , $" , Y H

T*$ 1+ " 2 2 X ) / & " 2 %) $ & $" &,1U Example 13. Composition of binary relations

Finally, consider phrase-structure grammars, which have been used as models of Schenkerian-type theories of tonal music and therefore are one kind of plausible candidate for external representations of Isobel’s internal representations of piece-tilnow and piece-to-come. Any such grammar is a kind of finite-state machine and therefore can be formalized as an S-act.

Example 14. Grammar and metagrammar

Approaching Musical Actions

301

These grammars produce structures that are trees, which are a kind of graph or net (or relation!) that is partially ordered in a particular way. Note that the grammar does tie together in this way the ideas of mathematical action, relation, and net. Example 14 is taken from my 1994 article, “Network Models.” It shows a little invented phrase-structure grammar for a kind of non-tonal music, purely as a methodological illustration, formalized as a formal theory with axioms and with inference rules in the form of Emil-Post productions, along with one particular derivation sequence modelling the structure in this grammar of one of the musical pieces, that is, theorems or sentences, producible by this grammar. Imagine Isobel imagining her piece in this way, that is, a basic vocabulary of pc sets, a secondary vocabulary emphasizing pc sets of Tn-type (0 3 7) and (0 2 5) and emphasizing transformation of any pcset by T1. She realizes that one stretch of her music, perhaps even the music-til-now, is the particular production within this grammar shown in Example 14 as A Derivation. Clearly, many other stretches of music can be represented as sentences within this grammar, but not just any stretch of any music. So the grammar as a whole lends a certain flavor to the piece, which from Isobel’s standpoint, is good. But what does Isobel do next? She would like to move to some closely related new stretch of music, but it should not be so closely related that it does not sound like something new. To generate just any new stretch from the grammar would probably not constrain the production enough, that is, the new thing would not necessarily be close enough to the earlier thing. Isobel realizes that the particular derivation sequence that produced her piece-tilnow would constrain new production more than the grammar as a whole, but in a way that need not produce something too close. She can theorize the production sequence, as shown in the “meta-grammar” in Example 14, as itself a kind of finite-state machine. This meta-grammar is the bottom diagram of Example 14. Isobel can operate this machine to get a next slice of her piece that is closely, but not too closely related, to what has come before. We have come to the end of this for now, leaving further development and applications of all these ideas to later research. I will feel successful if this has focused attention on the formal modelling of creative musical acts, and has illustrated and encouraged thinking in terms of models that are not simplistic, but are complex enough to be credible.

References Kilp, M., Knauer, U., Mikhalev, A.: Monoids, Acts, and Categories, with Applications to Wreath Products and Graphs. de Gruyter, Berlin (2000) Lewin, D.: Music Theory, Phenomenology, and Modes of Perception. Music Perception 3(4), 327–392 (Summer 1986) Lewin, D.: Generalized Musical Intervals and Transformations. Yale University Press, New Haven (1987) Lewin, D.: Klumpenhouwer Networks and Some Isographies That Involve Them. Music Theory Spectrum 12(1), 83–120 (Spring 1990) Morris, R.: Composition with Pitch Classes. Yale University Press, New Haven (1987)

302

J. Rahn

Morris, R.: Class Notes for Advanced Atonal Theory. Frog Peak Music, Lebanon HN (2001) Rahn, J.: Repetition. Contemporary Music Review 7, 49–58 (1993) Rahn, J.: Some Remarks on Network Models for Music. In: Atlas, R., Cherlin, M. (eds.) Musical Transformation and Intuition: Essays in Honor of David Lewin. Raphael Atlas and Michael Cherlin, pp. 225–235. Ovenbird Press, Roxbury (1994) Rahn, J.: The Swerve and the Flow: Music’s Relationship to Mathematics. Perspectives of New Music 42(1), 130–150 (Winter 2004) Rahn, J.: Cool Tools: Polysemic and noncommutative Nets, subchain decompositions and cross-projecting pre-orders, object-graphs, chain-hom-sets and chain-label-hom-sets, forgetful functors, free categories of a Net, and ghosts. Journal of Mathematics and Music 1(1), 7–22 (2007) Starr, D.: Sets, Invariance, and Partitions. Journal of Music Theory 22(1), 1–42 (1978)

A Transformational Space for Elliott Carter's Recent Complement-Union Music* John Roeder University of British Columbia School of Music [email protected]

Abstract. Elliott Carter's recent music exploits a special combinatorial property of the all-trichord hexachord. I show how this property can be reconceived in terms of interesting and analytically significant musical transformations: three involutions on the pitch-class aggregate which constitute a Klein four-group, and which have a natural interpretation as the symmetry group on a particular 12-vertex geometrical structure. Accordingly the opening of Carter's Figment II for solo cello can be analyzed transformationally as a complete traversal of this structure by just a few, striking, characteristic gestures.

Scholars of Elliott Carter's music have noticed refinements in his compositional technique over the last two decades. The composer affirms, "From about 1990, I have reduced my vocabulary of chords more and more to the 6-note chord no. 35 and the 4note chords nos. 18 and 23, which encompass all the intervals" (Carter 2002, ix).1 The hexachord set-class (sc) 012478 he mentions is unique in the sense that it contains an instance of every one of the 12 classes of trichord. Accordingly scholars call it the alltrichord hexachord (or ATH). It also exhibits what Robert Morris (1990, 182) has theorized as the "complement union property" (CUP): there exist two set-classes X and Y such that the all-trichord hexachord always results from the union of any Xclass set and any disjoint Y-class set. Only 21 of the 50 hexachord set-classes exhibit this property (Considered as a pair, the all-interval tetrachord-classes Carter mentions also exhibit it). But the all-trichord hexachord is nearly unique in manifesting the complement union property two ways: X and Y can be set-classes 048 and 016, or 0167 and 04. In other words, the union of any augmented trichord with any three other notes that form a 016 trichord will always be an ATH; and the union of any 0167 tetrachord with two other notes that form a major third or minor sixth will always be an ATH. A particularly clear exploitation of the complement union property of the alltrichord hexachord is the first section of Carter's Figment II–Remembering Mr. Ives *

This research is supported by a Standard Research Grant from the Social Sciences and Humanities Research Council of Canada. 1 In this paper I will refer to sets of pitch classes (pcs) by enclosing pitch-letter names—or their standard integer representations, C=0, C#=Db=1, etc.—in set brackets. Set-classes (scs), that is, sets of pc-sets related by transposition or inversion, are named by their prime form, without brackets or commas. Carter, though, uses his own system for denoting set classes; his tetrachord no. 18 means the set-class with prime form 0146, his tetrachord no. 23 is 0137, and his hexachord no. 35 is 012478. T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 303–310, 2009. © Springer-Verlag Berlin Heidelberg 2009

304

J. Roeder

for solo cello (2001). Example 1 shows an annotated score in which segments of the music are bracketed and labeled with circled integers. During each segment an alltrichord hexachord is formed the same way. Two perfect fifths are presented a tritone apart; so they form a member of sc 0167. To them is added an 04-class dyad (ic4), the notes of which are labeled and boxed on the Example. The result of this combination is, in every case, an all-trichord hexachord. Different types of brackets on the Example indicate how the successive ATHs group into regions characterized by the member of sc 0167 that they hold in common. For instance, every one of the dottedbracketed ATHs, numbered 5 to 8, contains the pc set {E,F,Bb,B}, or {4,5,t,e}, so their succession is labeled as the "45te region." This bracketing clarifies one aspect of the musical form. The piece begins with what I will call an opening fanfare, in which the fifths and major tenth are exposed for the first time. Starting in m. 4 the music then presents four ATHs, each composed of

Example 1. Carter, Figment II–Remembering Mr. Ives, mm. 1-22, and its all-trichord hexachords

A Transformational Space for Elliott Carter's Recent Complement-Union Music

305

the same 0167-type set combined with an ic4 dyad. The ic4 dyads are disjoint with each other and with the 0167-type set, so the series of four ATHs completes the pc aggregate. Next, the music presents four more similarly related ATHs, sharing the same 0167-type set, and combining it in turn with four mutually disjoint ic4 dyads. Then the music does the same thing based on a third 0167-type tetrachord. Starting from m. 4, then, twelve ATHs are presented, based on three disjoint members of sc 0167. These three 0167-type sets combine to complete the aggregate on a larger scale. Many subtle details of the passage support this abstract compositional design. After {D#4, G#3} is repeated in the fanfare, a strong contrast of pitch content articulates the beginning of the 0167 ATH-region in m. 4, specifically, the two new pcs F# and C#, the latter of which is registrated as the highest pitch yet. Also at this moment, the dynamics begin to continuously crescendo and diminuendo, rather than change abruptly as they did in the preceding fanfare. Each ic4 in mm. 4-9 is orchestrated as a major-third pitch-simultaneity, giving the 0167 ATH-region aural consistency. Starting at the breath mark at m. 9, the ic4s change to minor-sixth pitch intervals, helping to articulate the beginning of the next ATH-region, 45te, centered on the {Bb, F, E, B} tetrachord. The registral, rhythmic, and dynamic climax of the passage, in m. 13, also ups the harmonic ante by presenting in quick succession a minor sixth, a major third, then a major seventh. This last simultaneity interval, which has not appeared previously in the piece, triggers the beginning of the 8923 ATH-region, which is also marked by a temporary disruption of the display of the complement union property. That is, rather than forming 0167s, the perfect fifths are incorporated into the two all-interval tetrachords marked on the score. This contrast, however, helps prepare for closure, which is achieved around m. 17 by restoring the complement-union partition of the all-trichord hexachords into 0167s and 04 dyads; the sense of return is supported further by the 04 dyads' presentation as major-third pitch intervals, as at the beginning of this piece. Indeed, Carter cleverly contrives mm. 19-21 to recall both the dyads of the opening fanfare and the cadential-sounding openstring C2 of the 0167 region. This account implies a simple combinatorial conception of this music, identifying how each instance of the all-trichord hexachord is composed of the same subset types. It is also consistent with the composer's conception to the extent that it is apparent in the recently published Harmony Book, in which Carter systematically tabulates these set combinations (Carter 2002, 159). Example 2a shows the relevant excerpt from that book, in which hexachord no. 35 is shown to result from combining tetrachord no. 2 with four different major thirds. Accordingly, Guy Capuzzo (1999; 2004) analyzes CUP-manifesting passages of ATHs in Carter's Gra, Changes, and other recent compositions, as realizations of compositional spaces (graphs) whose nodes contain sets belonging to two different subset types, and whose edges signify how they combine into ATHs. An example of the kind of graph Cappuzo draws for the ATH is shown in Example 2b. The central node of the graph contains the pc set {4,5,t,e}, conforming to Carter's Harmony Book example; surrounding it are nodes containing instances of ic4 that do not intersect with {4,5,t,e} or with each other. Edges connecting two nodes indicate that the contents of the nodes combine into an ATH. Capuzzo describes such spaces for other CUP-manifesting set-classes, including the pair of AITs.

306

J. Roeder

(a)

(b)

Example 2. (a) Carter, Harmony Book, p. 159 (b) Graph showing the CUP of ATH

However, the highly limited hexachordal vocabulary of Figment II also seems ideally suited to be understood in terms of musical transformations (Lewin 1987; 1993). This involves defining a family of objects of the same type, structured by a semigroup (at least) of transformations, each of which changes one object to another in the same family. On that basis one may construct a transformational space — a network whose nodes represent the objects, linked into a structure by a minimal set of transformations (Gollin 2000). Transformational analysis also identifies characteristic ways that objects are transformed during the piece. These characteristic transformations are conceived as gestures that correspond to distinctive motions in the space, which give the gestures location, direction, landmarks, and orientation. Repetitions of gestures, or changes of gestures, can be understood to articulate musical form. Generally speaking, transformational analyses add focus and temporal orientation to more combinatorial accounts. Capuzzo himself provides a basis for a more transformational conception of Carter's recent complement-union music. If one considers the family of ATHs that share a member of 0167, and the twelve-tone operations that hold the 0167-type subset fixed, then those operations form a simply transitive group of characteristic transformations that engage the [0167] structure at another level. Capuzzo does not develop this idea much further, though, perhaps because the space (and the excerpt he analyzes from Gra) involves only four ATHs. However, a recent article by Adrian Childs (2006) develops a transformational system for CUP-manifesting all-interval tetrachords (AITs). Childs conceives of the transformations among AITs as contextual, partial transpositions, which hold one dyad of an AIT fixed while transposing the other to yield another AIT. Although partial transpositions in general do not have a semigroup structure, the constraints of CUP ensure that these particular ones do. Childs's analyses demonstrate how these transformations succinctly express AIT-preserving voiceleading in Carter's music; however, he does not propose a transformationally structured space in which characteristic gestures can be heard to act. Now, following Lewin's typical strategies, one might analogously define transformations that preserve the 0167s in each quadruple of related ATHs (Capuzzo suggests this as a possible extension of his work). Two of these would be contextual inversions and the other would be T6. For the task of analyzing Figment II, however, this solution seems overly general in two respects. For one thing, the passage does not obviously feature hexachords; rather, it focuses on dyads, with the ATHs acting as a


307

less immediately salient harmonic background. Also, only three of the six 0167s, and only 12 of the 24 ATHs, appear, so it does not seem appropriate to invoke the idea of a contextual inversion that would apply to all ATHs. A more appropriate and analytically expressive structure results from considering three involutions on the pitch-class aggregate that are not pc inversions. Written in cyclic notation, they are: S = (01) (67) (23) (89) (45) (te), F = (07) (16) (29) (38) (4e) (5t), and T = (06) (17) (28) (39) (4t) (5e). The label S stands for Semitone, because it preserves specific (but not all) dyads in ic1. F (for Fifth) preserves specific dyads in ic5, and T (for Tritone) preserves all the ic6 dyads. They also preserve every member of tetrachord class 0167 that appears in Figment II; for example S({2,3,8,9}) = {2,3,8,9}, because S transforms 2 to 3, 3 to 2, 8 to 9, and 9 to 8 (They do not have the same effect on ATHs that do not appear in this passage; for example F({1,2,7,8}) = {0,3,6,9} not {1,2,7,8}). To see how these involutions are manifested musically, consider in Example 1 the succession from ATH 1 to ATH 2, which pivots on the two perfect fifths in mm. 6-7 that they hold in common. The succession from the first of these fifths, , to the second, , can be regarded as the transformation of {6,1} by any of these three transformations, S, F, or T, into {0,7} (The T transformation even supports hearing these fifths as having harmonic roots, as it transforms the root of one into the root of the other). Interestingly, the operation T also transforms the ic4 dyad in ATH 1, {4,8}, to the ic4 dyad in ATH 2, {t,2}. Indeed, the series of dyads at the beginning of ATH2, , is the retrograde of the T transformation of the series of dyads that end ATH1, . So we can regard the background change from ATH 1 to ATH 2 as an elision of these two series, that is, what Lewin (1987) would call a retrograde T-chain of the ATH1 dyad succession. Similarly, the change from ATH 3 to ATH 4 can be heard as the elision of the dyad succession , in mm. 8-9, with its retrograde Ttransform, .

Example 3. Transformational analysis of all of Figment II, mm. 1-22

Example 3 shows how every ATH succession within each region can be analyzed similarly as the transformation of an ic4 dyad and two perfect fifths by one of the involutions S, F, and T. Each region alternates two transformations in a distinctive way, creating form. The first region focuses on T, using it twice in alternation with F. The beginning of the second region is articulated by the introduction of a contrasting,

308

J. Roeder

as yet unheard transformation S, but makes some accommodation with established procedure by alternating it with the T transformation that was focused on earlier. The last region focuses on the hitherto subsidiary transformation F. The return of T in alternation with F here gives the passage not only consistency but also closure, via the rounding of an ABA form. As shown by the brackets at the bottom of the example, in each region appears a different number of retrograde chains (involving at some point every one of the three involutions), peaking in the middle region, the only time span when ATHs are transformed by S. The involutions also suggest a transformational space suitable for analyzing Figment II. Since SF = FS = T, FT = TF = S, and TS = ST = F, they, along with the identity, constitute a Klein four-group, which has a natural geometrical interpretation as the group of symmetries of a rectangle (a subgroup of the symmetries of a square). To illustrate, Example 4 shows three corresponding symmetries acting on squares whose vertices are labeled with the pcs of {0,1,6,7}. Under this particular labeling scheme, T acts like flipping the square about a vertical axis, as shown from the first square to the second in the example, F acts like flipping the square about a horizontal axis, and S acts like the rotation of the square by 180 degrees. Comparing the leftmost and rightmost squares, which are identical, shows that by successively applying F then S after T, the square is transformed back to its original orientation.

Example 4. Isomorphism of the group of ATH-preserving involutions to the Klein 4-group

Now consider the toroidal figure in Example 5a. It is composed of three squares, each with its vertices labeled by the pitch classes from one of the three members of sc 0167 in the music's regions (pc t is shown as 10, and e as 11). Vertical sides of the squares represent F transformations, and their horizontal edges represent T transformations; for example, F transforms the pc 2 at the lower right corner to the pc 9 at the vertex connected vertically to it, and T transforms 2 to the pc 8 at the vertex connected horizontally to it within the same square. The vertices of different squares are connected by edges to form triangles whose vertex pcs belong to sc 048. Each ATH in Example 1 is represented on such a figure by the vertices of a sc-0167 square, plus two vertices, one from each of the other two squares, that form a 048-type trichord with one of the vertices of the first square. For instance, ATH 1 is composed of pcs {0,1,6,7}, the vertices of one square, plus the pcs 4 (from the {4,5,t,e} square) and 8 (from the {8,9,2,3} square), which form a 048-type trichord with 0. In Example 5a, the pcs of the {0,1,6,7} square are shown as two arrows indicating two members of ic5s, the dyads {0,7} and {1,6}, and the pcs of the 048 trichord are shown as the tail of one of those arrows (at the pc 0) plus the two gray balls on pcs 8 and 4 forming ic4. Shading highlights the 0167 square and the 048 triangle. Similarly, Example 5b shows ATH 2 to be composed of the ic5 vertex pairs of the same {0,1,6,7} square plus the pcs t and 2, an ic4 that forms a 048-type trichord with pc 6.


309

In this space the pitch-class transformations that manifest CUP in Example 2 can be visualized as distinctive gestures that change the location of the ic5s and the ic4. This is suggested by the juxtaposition of Examples 5a and 5b. The Ttransformation of ATH 1 to ATH 2 takes the solid arrow directed from 6 to 1 in Example 5a and translates it horizontally across the {0,1,6,7} square, so that now connects 0 to 7. T similarly takes the dotted arrow directed from 0 to 7 and translates it horizontally to connect 6 to 1. The vertices of the triangle involving the pcs 0, 4, and 8 are translated horizontally across their respective squares to the vertices 6, 10, and 2, respectively. Thus the T transformation preserves the square- and triangle-relationships among the ATH-forming vertices, and it manifests the complement union property of the ATH by pairing the same {0,1,6,7} square with two different ic4 dyads, {4,8} and its T-transform {10,2}. Likewise the juxtaposition of Examples 5b and 5c clearly displays the invariants that characterize the vertical transformation F from ATH2 to ATH3, and so highlights their complement union property Example 5. Geometrical representation of the under its action. F takes the succession from ATH1 to ATH2 to ATH3 dotted arrow directed from 6 to 1 in Example 5b and translates its ends vertically, so it is now directed from 1 to 6. Similarly F exchanges the pcs 0 and 7 connected vertically by the solid arrow. The vertices of the triangle involving the pcs 6, 10, and 2 are translated vertically across their respective squares to the vertices 1, 5, and 9, respectively. The S transformation, not shown in the Example, effects a different and distinctive geometrical action, moving balls and arrow ends

310

J. Roeder

diagonally across the square, but also maintains the geometrical relationships among them, thus manifesting CUP. The change from one 0167 region to another, as between ATH4 and ATH5 in mm. 9-10, involves another symmetry of this threedimensional figure—a rotation by 120°, along the edges of the triangles, from one square to another. In effect then, the space is a Cayley diagram that manifests the coset structure of the group actions. It also provides a striking medium to visualize the succession of twelve ATHs that structures the passage, which was described in connection with Example 1. As suggested by Example 5, the succession of four CUP-related ATHs corresponds to the systematic union of an 0167 square with each of the four unique ic4 pairs of the remaining pcs, each pair consisting of pcs from the other two squares. Visually, the progression through the three CUP-manifesting quadruples of ATH repeats this combinational process for each of the three 0167 squares in the space, so that eventually every vertex functions as a member of both a square and a triangle. Accordingly the passage can be understood transformationally as a complete traversal of this structure by just a few, striking, characteristic gesture. A three-dimensional computer animation that can be viewed at theory.music.ubc.ca/~trx/analyses.html makes vivid these transformational gestures, and the musical form of Figment II that they manifest within the space. As an analysis, it subsumes the combinatorial and contextual-inversion perspectives into a temporally oriented mode of listening.

References Capuzzo, G.: Variety within Unity: Expressive Means and Their Technical Ends in the Music of Elliott Carter. Ph.D. dissertation, University of Rochester (1983–1994) Capuzzo, G.: The Complement Union Property and the Music of Elliott Carter. Journal of Music Theory 48(1), 1–24 (2004) Carter, E.: Harmony Book. Hopkins, N., Link, J.F. (eds.). Carl Fischer, New York (2002) Childs, A.: Structural and Transformational Properties of All-Interval Tetrachords. Music Theory Online 12(4) (2006) Gollin, E.: Representations of Space and Conceptions of Distance in Transformational Music Theories. Ph.D. diss., Harvard University (2000) Lewin, D.: Generalized Musical Intervals and Transformations. Yale University Press, New Haven (1987) Lewin, D.: Musical Form and Transformation: 4 Analytic Essays. Yale University Press, New Haven (1993) Morris, R.: Pitch Class Complementation and its Generalizations. Journal of Music Theory 34(2), 175–245 (1990)

Networks Tom Johnson 75 rue de la Roquette 75011 Paris Tel.: 33 1 43 48 90 57 [email protected] www.tom.johnson.org

I was pleased to learn that it would be possible to make a little exhibition of drawings here in Berlin, as well as presenting my lecture, because the Networks I’m working on now are as much visual as aural, and I think these structures need to be seen as well as heard. Of course, those who are simply reading this text will only see the drawings that are included here, but hopefully this will be enough to convey the general idea. Many composers make graphs, tables, or charts of some sort to calculate the details of their music, and I have been doing this for a long time, but finding the system visually has become a particularly important part of the Networks I have been working on since 2005, which have mostly to do with harmony and with defining groups of chords. My earlier music also sometimes concerned chord groups and combination theory, the most obvious example being The Chord Catalogue, but this interest took a new turn when a young Dutch Composer, Samuel Vriezen, showed me how he had composed a group of 11 five-note chords such that each chord had two notes in common with each other chord, and a mathematician friend, Jean-Paul Allouche, suggested that I investigate block designs. This is a relatively new kind of combination theory that I don’t think Vriezen had ever studied either, and which, in fact, is not widely know even to mathematicians. The principles are rather simple, however, and after studying the subject a bit, I realized that these networks of subgroups or chords could lead me to rich and yet unknown musical materials. Let me begin with the example of a block design known as (7, 3, 1). That means that 7 notes (elements) are divided into chords (sub-groups) of 3 notes, in such a way that each pair of notes comes together in one chord. One can do it in this way: (1,2,3), (3,4,7), (2,4,6), (2,5,7), (1,6,7), (3,5,6) (1,4,5), but one can also solve the problem in this way: (1,2,4), (2,3,7), (4,6,7), (2,5,6), (1,5,7), (1,3,6), (3,4,5). How can we see the relationships between these three-note chords? Is there a way to combine the two solutions? How do we find the beginning, the end, the continuity of the logic? How can we find the nerve of the system? Well, the answer to all these questions is to begin drawing pictures. In this way I found four different representations, all of which are in the exhibition, but I particularly want you to see the network in this way, where the white triangles represent the first solution, the shaded triangles represent the second, and the 14 chords are all shown twice: T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 311–317, 2009. © Springer-Verlag Berlin Heidelberg 2009

312

T. Johnson

Of course, mathematicians have been studying such structures for a long time, and I was fortunate to make contact with two mathematicians from the University of Vermont who took an interest in my drawings and wrote some comments, which are posted along with the drawings in the exhibition space. About this particular drawing, Jeffrey Dinitz, who specializes in combinatorial designs, and Dan Archdeacon, who studies topological graph theory, sent me the following text: This shows four representations of the universal coverings of K7 (the complete graph on 7 vertices) on the torus. In each case the seven shaded triangles form one Fano plane (the projective plane of order 2) and the seven white triangles form another. One can see here how (7,3,2) combines two (7,3,1) systems, one of white triangles and one of shaded triangles. To hear this network, we can simply assign the numbers to a scale of seven notes, and read the white circle followed by the gray circle, as I have done in the music notation below. Of course the symmetries would be the same using any seven-note

Networks

313

scale, but I rejected dozens of candidates before finding this one, where the notes and chords sound truly equal and the music seems to homogenize. I am going to play the 14 chords on the piano, but of course it is not really piano music, and could be heard equally well on another instrument (or instruments):

Could you hear the order? Could you hear that each note occurred the same number of times, that each pair of notes came together the same number of times, that everything is in perfect balance? Well, I must admit that I don’t really hear this either, but it is remarkable how clearly one hears if there is a mistake, so the ear is somehow sensitive to what is going on. And of course, this is a new way of listening, and any new way of listening does require a bit of training. No doubt in a few years we will have learned how to perceive such patterns more easily. Another question is whether we can call this a piece of music. I didn’t really compose it. I simply found it within a mathematical phenomenon, and for some time I felt that such things were simply models or prototypes, rather than actual pieces of music. At the same time, when I try to expand the progression, to add a melody, to move the music through a series of variations, the result always seems vulgar. The sequence is much more satisfying in its natural state, and since the natural numbers really are a part of nature, this can be regarded as a little gem found in nature. It is quite lovely just as a diamond in the rough and does not need to be cut and polished. Another block design with musical potential is (13,4,1), a collection of 13 four-note chords. In other cases there are many ways of forming a group of symmetrical subgroups, but in the case of (13,4,1) there is only one solution. Of course, you can always exchange the twos with the eights, for example, but this will just be a morphism of the basic block design. Again each note and each pair of notes occur the same number of times, and again it seems necessary to draw some pictures in order to uncover the nerve of the system. In this case the essence of the structure seems to emerge best in the following diagram. Let us listen to the sequence of two times 13 chords by alternating between the inner and the outer circles. You probably won’t be able to count fast enough to be sure that all 13 notes occur the same number of times, and that the chords appear twice each, but I think you can hear that the music is somehow turning in a circle, and if I play a wrong note, you will hear that there was a mistake.

314

T. Johnson

Networks

315

The comment offered by Dinitz and Archdeacon concerning (13,4,1) is probably too technical for most musician readers, but let me quote it for those who will understand: This is the projective plane of order 3. It can be obtained from the (9,3,1)-design by adding 4 new points {1 , 2 ,3 ,4} and a new block containing them. Then to each block in the ith parallel class of the (9,3,1) design add the point i . Again the resulting chord sequence is terribly short to be considered a musical composition, but it is a complete system nonetheless, and as I continue to study all these networks of chords, I have gradually concluded that such systems must be considered finished objects. If I try to “develop” the material the way composers are taught to do, I don’t really improve anything. Now, in August 2007, as I revise this text for the printed edition, I am also preparing an edition of about a dozen of these little Networks.1 Of course, some combinatorial designs have more blocks and take more time. A good example is (9,3,1). The basic solution here consists of only 12 three-note chords, in which each note is used four times. But this group of 12 chords can be expanded to what the mathematicians call a large (9,3,1), combining seven different solutions to the problem. Now we have a system of 7 * 12 = 84 chords, in which all 84 combinations of the nine notes, taken three at a time, are included exactly once. That is difficult to draw, so I’ll just give you the music.

1

Available from Editions 75. Visit www.tom.johnson.org for more info.

316

T. Johnson

Networks

317

I will not take the space here to include the three unique solutions of (10,4,2), but since we are so attached to our 12-tone chromatic tradition, I want to show you my realization of (12,4,3), so that you can see some 12-tone music that comes directly from a block design. Incidentally, Jeffrey Dinitz told me that mathematicians can prove the existence of 14 million non-isomorphic solutions for the (12,4,3) problem. Many of these are known as “resolvable,” which means that the 33 chords of this group can be divided into 11 groups of three chords, each of which includes all 12 notes. Here is the drawing and the music notation for my (12,4,3). It is resolvable, so each measure, each group of three chords, contains the complete scale: It is also possible to form a large (15,3,1), which makes the 13 * 7 * 5 = 455 chords of Kirkman’s Ladies, a 13 – minute piece that I wrote in 2005. Another block design, known as 4-(12,6,10) produced the 330 chords of Block Design for Piano (2006), an 18-minute composition. Now I am completing a Septet, which transforms the 11 chords of (11,5,2) into 10 different solutions. And enough new possibilities arise that 2 I expect to be exploring this area for some years to come. This is of course only a miniscule introduction to block designs and to the chord groups one can find within this branch of combination theory. The definitive book on the subject, the Handbook of Combinatorial Designs (Chapman and Hall/CRC, second edition 2007), edited by Charles J. Colbourn and Jeffrey H. Dinitz, provides about a thousand pages of supplementary reading for those who wish to go further.

2

After the Berlin conference and finishing the "Networks" edition, I wrote a 12-minute Septet taken from (11,5,2) that was premiered in The Hague in September. The following months I spent most of my composing time with the 55 chords of a particular solution of (11,4,6), which became a series of drawings and also a 20-minute piece for two electric keyboards. Currently I am concentrating on a number of different solutions to (12,4,3), again with both drawings and compositions. There are over 17 million ways to form the 33 blocks of this design, and the resulting music is totally devoted to the equality of the 12 notes and at the same time has almost no resemblance to serial traditions. TJ, March 2008.

From Mathematica to Live Performance: Mapping Simple Programs to Music Katarina Miljkovic New England Conservatory of Music [email protected]

Abstract. This paper focuses on selected simple programs used to model generative processes for basic elements of music material such as rhythm, pitch and texture, as well as large-scale works of music. After presenting decisions on sound mapping procedures, I’ll introduce the system NKM, A New Kind of Music, designed by Peter Overmann, director of software technology for the Mathematica programming environment. NKM is a system controlled by cellular automata (CA), modeling a number of processes in nature. The CA presented in the paper belong to a group of elementary rules that encapsulate four classes of complexity, from simple to universally complex, conceived by Stephen Wolfram and presented in his book A New Kind of Science. All of the examples were generated in Mathematica, the software by Wolfram Research Inc. Mathematical basis for the examples can be found in the book A New Kind of Science.1

1 Background …just as a researcher into nature strives to discover the rules of order that are the basis of nature, we must strive to discover the laws according to which nature, in its particular form “man”, is productive. And this leads us to the view that the things treated by art in general, with which art has to do, are not “aesthetic”, but that it is a matter of natural laws … all discussion of music can only take place along these lines. (Anton Webern, The Path to New Music) My interest in generative music came from a deep belief in close connections between processes in nature and the organization of sound in music. Graphics generated by simple programs in A New Kind of Science that can “capture the essence of the complexity--and beauty--of many systems in nature” (Wolfram Research), attracted my attention soon after the book was published. The remarkable gestural content resembled graphic scores (Ex.1-5). It seemed as if the pattern propagation, ranging from simple, repetitive, self-similar, random, to universally complex, captured the essence of musical processes across styles and cultural traditions. 1

The book can be accessed online at http://www.wolframscience.com/nksonline/toc.html


From Mathematica to Live Performance: Mapping Simple Programs to Music

319

The following examples show two kinds of graphic representation of elementary rules. While the graphs in the left column model the process of evolution generated by two-dimensional, two-neighbor dependent CA, examples in the right column represent the oscillation of a total number of active cells in each step of evolution. The first column suggests the nature of pattern propagation while the second can easily be perceived as dynamic potential of a statistical sound complex. Class 1 Ex.1a-b

Rule 4

Rule 172

Class 2 Ex.2a-b

Rule 9

Ex.3a-b

Rule 90

Rule 82

Class 2-3 Rule 210

Class 3 Ex.4a-b

Rule 45

Rule 45

320

K. Miljkovic

Class 4 Ex.5a-b

Rule 110

Rule 110

Examples 1-5 demonstrate distinctly different behavior in each class of complexity that can be associated with the following musical equivalents: Class 1, with stable, pedal-like, prolonged sonorities (Ex.1); Class 2, with a range of simple repetitive modules (Ex.2); Class 2-3, (on the edge of randomness) with self-similar, repetitive musical gestures on multiple scales (Ex.3); Class 3, with random, irregular textures, statistical complexes, outbursts of sound (Ex.4); Class 4, with multilayered, well balanced, complex musical texture (Ex.5). Musical potential of the elementary rules was evident; the main question was how to proceed with mapping. As frequently noted by the authors on generative music, careful programming can produce good results (Miranda 2001, 1), potentially mimicking any style or composer. Here, the intent was to investigate intricate structures of elementary rules and reveal them through new configurations of sound. In terms of mapping, the premise was that sound mapping corresponding to pattern propagation (McAlpine, Miranda, and Hoggar 1999, 25) would most likely reach this goal.

2 Data Gathering The first trials were directed toward modeling basic elements of music. They were limited to a small number of steps in the evolution of different rules. Each cell on a vertical axis, black or white, from bottom up, corresponded to an ascending step of a previously chosen scale (see Ex. 6). A basic rhythmic unit, an eight note, was a step on a horizontal axis. A black cell was note-on, a white cell was note-off, a rest. Using elementary rules, it was quite easy to model stepwise motion, interval leaps and various textures. Ex. 6a-d Monophonic texture Rule 16

Rule 2

Homophonic texture Polyphonic texture Rule 18 Rule 110


321

The next phase was data gathering through aural observation of thousands of 30second long sound samples generated by simple programs. In addition to elementary rules, substitution systems, random walk and Turing machines were tested. The process was like mining and unearthing patterns that were sometimes musically associative (Cope 2003, 11) and often pleasing. With significant help in programming on part of the Wolfram Research team, different mapping procedures were developed. Soon, Wolfram Research grouped the rules according to their potential to generate specific musical genres: world music, pop, ambient, jazz, to mention only a few. In 2005, Peter Overmann created the NKM system, based on CA. Features of the system made cycling through rules, changing initial conditions and generating midi files fast and easy. Multiple filters articulated sound structures making them more transparent. Possibilities of partitioning CA regions offered multiple choices for orchestration. A variety of scales, metric articulation and tempi, together with a wider register range and longer duration of samples, were provided.2 In spite of significant work on mimicking musical styles, the advancement was, from my perspective, too fast, lacking necessary time for reflection and detailed musical analysis of data. A limited duration of sound samples associated with a frequent cycling through the rules prevented observation of long-range processes and development of musical material, a finding often associated with generative music in particular, and algorithmic composition in general (Klouche 2005, 331 and 353). The next important step was to try to create longer musical pieces. After intense collaboration with Peter Overmann, I decided to generate a piece for violin and piano by a single rule, a predetermined set of initial conditions and musical parameters thus testing the potential of rules to preserve an interesting musical activity for an extended time period.

3 Large Scale Piece 3.1 Choice of a Rule The texture, with a melodic line and piano accompaniment, was an important factor in considering the choice of a rule and initial conditions. Dynamic unfolding of a melodic line, rhythmic differentiation between piano and violin and constant variation of musical material seemed to be necessary components for building the piece. After testing a number of simple programs, Rule 30 proved to be the best choice because of its high level of rhythmic activity and constant, unpredictable interval shifts. 3.2 Partitioning Mapping included multiple steps. The sound fabric of Rule 30 was generated by the procedure outlined in example 6. The algorithm for a melodic line traversing the fabric of the rule was developed by analyzing and then modeling Cantus Firmi and a number of themes from classical repertoire. Three melodic lines were generated, one for the violin part (Ex.7a) and two in the lower region. For the piano accompaniment (Ex.7b), blocks of black cells longer than five units were articulated by pitches of extended duration as an additional layer in the piano part (Ex. 7c). 2

A reduced version of the application can still be accessed at http://tones.wolfram.com

322

K. Miljkovic

Ex. 7a (excerpt)

Ex.7b (excerpt

Ex.7c (excerpt)

4 Initial Conditions The width of the rule was limited to 16 cells with initial conditions {1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1}3. It is worth mentioning that a search for initial conditions that created a texture revealing counterpoint of three well-shaped melodic lines took some time.

5 Choice of Musical Parameters The chosen scale was characterized by an equal number of semitones and whole tones, 2,1,2,2,2,1,1,1,4 and interesting internal symmetries and asymmetries.5 It was, to some extent, consistent with random data of the rule: The probability of occurrence of white cells, disrupting steady pulsation and open chromatic motion thus creating intervals larger than a semitone, is roughly 50%. 6 Tempo was 180 basic units per minute. Because of the non-periodic nature of the rule, it was difficult to identify metric grouping. The score (derived from the sound file) was written in 4/4, with measures being orientation points rather than metric units.

6 The Outcome Example 7a shows development of self-similar gestures in the violin part while example 7b demonstrates counterpoint between three lines. The occurrence of a large 3 4

1 represents a black, active cell, 0 represents a white cell. 1 represents a semitone, 2 represents a whole tone (two semitones).

5

The pitch collection is known as Raga Bahar and Miyan Ki Malhar according to their underlying scales in Indian music.

6

Statistical analysis of data that proved my assumptions was performed by Todd Rowland, Wolfram Research Inc. and communicated to me by e-mail on 1/25/2008.


323

field of white cells in the middle of the example proved to be an important feature, pushing all three lines to the lower register, narrowing the range and introducing element of contrast into the texture. Another contrasting layer was pointillistic texture across the entire register range, obtained by sustained pitches of six or more durational units. The result was a coherent structure with elements of contrast propelling the musical flow. The composition (titled Awakening) had multiple concert performances. It was received very well. Comments by trained musicians focused on the beauty of the synthesis of a well-known language and “something new” that they couldn’t quite define.

7 Generative Pitch Collections and Rhythmic Grouping The previously described composition proved Rule 30 to be a powerful music generator. Similar procedures applied to other elementary rules pointed to a wide range of possibilities for generating longer compositions. Still, there were unresolved problems: the mapping relied on pitch collections outside of the system of a chosen rule, and the form of the piece was to some extent arbitrary. Characteristics of the rule were preserved through texture and rhythmic activity while the identity of frequency collections was, partially, left out. I felt a need to go back a few steps and make a deeper musical exploration of the elementary rules. A possible solution was to use equal temperament for basic frequency units (12tone chromatic scale) apply it to all of the elementary rules and then explore differentiation between pitch collections and rhythmic grouping generated by: a) different classes, b) different rules of the same class, and c) different initial conditions of the same rule. The outcome was as expected. Different classes generated distinctly different pitch and rhythm, the rules of the same class generated a variety of similar materials, and change of initial conditions created a number of variations of the same material. The following examples demonstrate a proposal for pre-compositional procedures for CA music, based on a detailed examination of features embedded in elementary rules.

8 Mapping Mapping was slightly modified. This time, each cell on the horizontal axis, from left to right, corresponded to an ascending semitone and the duration of an eight note. I alternated between two ways of music mapping: mapping A articulated each black cell while mapping B articulated only the first in a block of consecutive black cells and extended its duration for the length of the entire block. Ex. 8a

Rule 127

324

K. Miljkovic

For example, in mapping A, each step of evolution of Rule 127 (Ex. 8a), Class 1, generates a chromatic scale while in mapping B, each step generates an extended duration of a single pitch, depending on the width of the CA. In mapping B, CA from Class 1, with a careful choice of initial conditions and a filter controlling clusters, could produce interesting vertical sonorities. Sometimes they are a mixture of prolonged and repetitive sounds, as in Rule 5 (Ex.8b). Ex. 8b

Rule 5

8.1 Rule 90 Rule 90 from Class 2-3 generated a revolving whole tone scale and a regular pulsation. Using twelve-tone equal temperament, the mapping of each step in evolution resulted in a combination of major seconds (interval 2) and major thirds (interval 4) outlining a whole tone scale. Steps of evolution with the highest density of active, black cells generated uniform pulsation without rhythmic grouping. Steps of evolution with a lower density, because of the presence of fields of white cells, revealed groups divided by periods of silence (Ex.9a). Simultaneous mapping of multiple rows resulted in a texture characterized by stratification of pitch and rhythm. Because of its transparency and periodicity, the entire Class 2-3 had the potential to quickly generate pleasing musical results. Ex. 9a-d

Rule 90

Steps 15-16, graphic representation

Step 15, notation


325

Step 16, notation

8.2 Rule 30 Time and frequency intervals in musical texture generated by Rule 30 were impossible to reduce to any pattern. Still, a surprising level of coherence emerged through aural observation of single rows (randomly chosen from different regions of the rule) that could be explained by statistical analysis of a large quantity of data from multiple rows. Example 10c represents step 32 of Rule 30 in mapping A. We can identify open chromatic motion disrupted by insertion of larger intervals caused by frequent occurrences of blocks of white cells. In order to reduce open chromatic motion, I applied mapping B. In spite of a lower level of activity, the main characteristics of the rule remained untouched (example 10d). The absence of periodic behavior produced an effect of constant shift, forward motion and musical expectation. Clearly, the rule’s most important feature was its potential to constantly generate fresh aural information. Ex. 10a-d

Rule 30


Step 32, mapping A

326

K. Miljkovic

Step 32, mapping B

Further, I mapped each step of evolution as a vertical sonority. The first 13 steps, beginning with one cell and then gradually widening (Ex.10a), resulted in a slow emergence of vertical sonorities. Regardless of randomness of the rule, the chords in a sequence were differentiated enough to be perceived as some kind of a progression. The reason was the property of the rule. Statistical analysis of Rule 30 revealed that 47% of the total number of intervals larger than a semitone were major seconds (interval 2); 26%, minor thirds (interval 3); 12%, major thirds (interval 4); 6%, perfect fourths (interval 5); 3%, tritones (interval 6). The interval content of Rule 30 seemed to be best described by exponential decay. This balance created a musical texture with unified chromatic and diatonic segments and potential to generate interesting and unpredictable tension-resolution vertical relationships. 8.3 Rule 110 Mapping a randomly chosen step in the evolution of Rule 110 (Ex. 11c-d) at first did not show any obvious periodicity (Ex. 11c). The cells exhibited a slight tendency to group by seven, but only in the first 14 basic units. The reason was the choice of region of the rule. Ex. 11a-d

Rule 110


Step 32, Mapping A


327

Step 32, Mapping B

Example 11a clearly demonstrates wide regions of a periodic background disrupted by occasional propagation of irregular descending structures. The steps, presented in the examples 11c-d, belonged to a region where transition from periodic background to irregular structures occurred. Examples 12a-d present four steps of evolution of Rule 110, each 36 cells wide (three octaves). The first two (12a-b) are from a region with uniform background while the second two (12c-d) are from regions with irregular structures. Ex. 12a-b

Steps from a region of periodic background, mapping A

In both of the preceding examples, a revolving cycle of 14 basic units can be identified. Each subsequent cycle will be transposed by a major second up (interval 2). After six cycles (84 basic units), the first one will repeat, thus creating a new cycle on a large scale. On a small scale, it is possible to subdivide the 14-unit cycle into two sub-cycles of seven basic units (sometimes clearly articulated) where the transposition interval will then be a perfect 5th up (interval 7). Three possible cycles, each on a different scale, introduce structural hierarchy and interesting new possibilities for composing. The following steps of evolution demonstrate two degrees of deviation from periodic background. While there are small irregularities at the beginning of examples 12a-b, examples 12c-d illustrate the emergence of randomness. Ex. 12c-d

Steps from a region of irregular behavior, mapping A

328

K. Miljkovic

In example 12c, the 14-unit cycle is preserved in spite of modifications. Example 12d demonstrates dissolution of the cycle preserving only a few segments. It is difficult to identify it even if we are to take deviations into account. In case of a much longer example, we would observe the settling of random behavior and transition back to 14-unit cycles and periodic structure. In conclusion, Rule 110, incorporating cycles, a fine balance of simple periodicity, small deviations and randomness, has the potential to generate multilayered musical structures. It is interesting that generating longer compositions with this rule proved to be more difficult than in the cases of rules from Classes 1-3. Working with the cycles, capturing transitions from periodic to random behavior with the pre-determined set of initial conditions is not an easy task and requires sophisticated programming.

9 New Ground The Previously described steps are an attempt to lay out a ground for composing with the aid of simple programs from A New Kind of Science. Mappings were introduced that transparently and intuitively reveal characteristics of pattern propagation, allow analysis and classification of sound structures and provide tools for navigation through the music by means of 256 elementary rules leading to an endless number of simple programs. These steps have become a point of departure for a broadly expanded exploration of the vast sound world of cellular automata.

Acknowledgements I am indebted to Stephen Wolfram, Peter Overmann, Jason Cawley, Todd Rowland and Ed Pegg for their support, work on programming and generosity to introduce me to the world of cellular automata and A New Kind of Science.

References Cope, D.: Computer Analysis of Musical Allusions. Computer Music Journal 27(1), 11–28 (2003) DuBois, R.L.: Applications of Generative String-Substitution Systems in Computer Music (2003), http://www.music.columbia.edu/~luke/dissertation/ dissertation.pdf Klouche, T.: Aspekte einer Systematik computerbasierter Musikwissenschaft. In: Kulturbesitz, P. (ed.) Jahrbuch des Staatlichen Instituts für Musikforschung, Günther Wagner, pp. 316– 353. Mainz (2005) Weisstein, E.W.: MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com McAlpine, K., Miranda, E., Hoggar, S.: Making Music with Algorithms: A Case-Study System. Computer Music Journal 23(2), 19–30 (1999)


329

Miranda, E.R.: Granular Synthesis of Sounds by Means of Cellular Automata. Leonardo 28(4) (1995) Miranda, E.R.: Evolving Cellular Automata Music: From Sound Synthesis to Composition. In: Proceedings of the Workshop on Artificial Life Models for Musical Applications - ECAL 2001, Prague University of Economics (September 2001) Miranda, E.R.: Composing Music with Computers. Focal Press, Oxford (2002) Webern, A.: The Path to New Music. In: Reich, W. (ed.) Translated by Leo Black, Bryn Mawr, Pa. T. Presser Co. in assoc. with Universal, London (1963) Wolfram, S.: A New Kind of Science. Wolfram Media, Champaign (2002) Xenakis, I.: Formalized Music. Pendragon Press, New York (1992)

Nonlinear Dynamics of Networks: Applications to Mathematical Music Theory Jonathan Owen Clark Brunel University, London, UK [email protected]

1

Introduction and Musical Motivation

Algebraic approaches to modelling and the theory of dynamical systems are important aspects of theories of mathematics and music. Group-theoretic approaches have been used for some time in models of pitch-class, tuning and interval etc. More recent approaches by Mazzola (2002) and others strikingly extend this algebraic formulation into the realm of modules and categories. And the theory of dynamical systems has found musical applications in both algorithmic music creation (for example in the compositions of Agostino Di Scipio), and the physical modelling of musical instruments (in the work of Xavier Rodet and others at IRCAM). This paper presents a variation on how these dual approaches may be brought together, with particular applications to Mathematical Music Theory. In particular we consider the nonlinear dynamics of a generalised form of networks. In the work of mathematicians Ian Stewart and Martin Golubitsky these networks are considered as directed graphs of cells and edges, where the cells have internal dynamics given a system of ordinary differential equations (ODEs) and the edges represent coupling effects between the cells. A simplified summary of some of the main results of the theory is given in Section Two. Algebraic factors are crucial- group symmetries of the network play an important role in understanding the time series outputs of particular cells in the network and the resulting global pattern formation. They imply a ‘catalogue’ of different forms of behaviour from which the actual behaviour is ‘selected’. In addition, in cases of approximate symmetry or networks with repeated sub-units, both groupoids and local internal symmetries can be shown to provide partial generalisations. From the point of view of Mathematical Music Theory, what is striking about the network dynamics of such systems is their idiomatically ‘musical’ nature. Equilibria, periodic, multirhythmic and rotating wave states are supported, and the bifurcation theory (the study of abrupt changes in the existence and distribution of singularities in the associated vector field) reveals even more exotic behavior, such as synchronised chaos and ‘bubbling’. Traditional work on utilising nonlinear methods in both composition and synthesis has concentrated on setting up a particular dynamical system whose parameters are varied and mapped musically in real time- a ‘bottom-up’ approach. The paper stresses a potential route for a different methodology. We consider a ‘top-down’ model that takes as its starting point the catalogue of solutions T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 330–339, 2009. c Springer-Verlag Berlin Heidelberg 2009

Nonlinear Dynamics of Networks

331

of a network itself. The group-theoretic properties of the network force these behavioural motifs irrespective of the local parametric conditions. A discussion of the musical and philosophical implications of this proposed methodology is given in Section Three, and aims at providing a connection between the mathematical ontology of Gilles Deleuze and the morphogenetics of musical events. In this section we also consider some applications to algorithmic composition and show a method where generative music derived from the catalogue of different oscillation patterns may be ‘switched’ using a simple Markov Chain architecture.

2


The material in this section follows the material in a number of papers and publications by Golubitsky and Stewart (1988, 2000, 2006). A coupled cell network is considered as a directed, finite graph whose nodes (or ‘cells’) represent state variables and whose edges represent couplings between the variables. The cells have intrinsic systems of ODE’s which govern the internal dynamics of each, so that the network is at a basic level an interacting high-dimensional dynamical system. To make this precise, suppose x˙ = f (x) where x = (x1 , x2 , ..., xn ) ∈ Rn and f : Rn → Rn is smooth, and γ : Rn → Rn is a linear map. We say the system has symmetry γ if γx(t) is a solution to the above equation whenever x(t) is a solution. It follows easily that γ is a symmetry if and only if f satisfies the condition f (γx) = γf (x). In contrast the symmetry group of the network is defined as the automorphism group of its graph. Denote this by Γ . This group permutes the nodes and edges of the graph and each such network symmetry induces a symmetry of the corresponding ODE’s as defined above, so that these two definitions of symmetry are in fact equivalent. If Σ ⊂ Γ is a subgroup, then Fix(Σ) = {x ∈ Rn : σx = x, ∀σ ∈ Γ } is a flow-invariant subspace, and the set of all such σ is a subgroup of Γ known as the isotropy subgroup of x. Now suppose that x(t) is a periodic solution to the system, so that x(t) = x(t + T ) for some period of time T . Define H = {γ ∈ Γ : γ{x(t)} = {x(t)}},

K = {γ ∈ Γ : γx(t) = x(t), ∀t}

The first are spatiotemporal symmetries, the second spatial symmetries. The first correspond to phase-relations in the behavior of the cells, that is one or more of the cells have identical periodic dynamics except for fixed phase shifts, and the second to synchrony of cells, that is cells with identical waveforms that are in phase. Phase-shifts correspond to actions of the circle group, S1 , i.e. for each h ∈ H, there is a phase shift θ(h) ∈ S1 such that hx(t) = x(t + θ(h)). Moreover, θ : H → S1 is a homomorphism with kernel K. It follows that H/K is isomorphic to a finite subgroup of S1 and hence is cyclic. Moreover, since fixedpoint subspaces are flow-invariant, K is an isotropy subgroup for the action of

332

J.O. Clark

Γ on Rn . In fact, one can show that, with certain caveats, the converse holds too, namely that cyclicity of H/K and isotropy of K implies the existence of periodic solutions - we state the full result below. H/K-Theorem (Golubitsky and Stewart 2006, p.316). Let Γ be the symmetry group of a coupled cell network in which all the cells are coupled and the internal dynamics of each cell is at least two-dimensional. Let K ≤ H ≤ Γ be a pair of subgroups. Then there exist periodic solutions to some coupled cell system with spatiotemporal symmetries H and spatial symmetries K if and only if H/K is cyclic and K is an isotropy subgroup. Moreover the system can be chosen so that the periodic solution is asymptotically stable. But the possible states of the system are only half the story. Typically in dynamical systems theory one is concerned with the states that the system supports, and then with transitions between them as parameters are varied; the latter is known as bifurcation theory. Having classified the possible spatiotemporal and spatial symmetries of a network with the H/K Theorem we now ask how these may arise from symmetry-breaking bifurcations. This is a question, mathematically, about the existence of solutions. The usual context for bifurcations is the loss of stability, as a single parameter λ is varied, of an Γ -invariant equilibrium x0 in a Γ -invariant system of ODE’s x˙ = f (x, λ). We may assume that the bifurcations are located at λ=0 and occur when the Jacobian J = (df )x0 ,0 has eigenvalues on the imaginary axis. If these are at ±i then the bifurcation is defined as a Hopf bifurcation. This represents a conversion in the cell dynamics from a steady-state equilibrium to a periodic state. The major result in this area that classifies the existence of periodic states through Hopf bifurcation is the following. Equivariant Hopf Theorem (Golubitsky and Stewart 2000, p.91). With the above conditions, define a C-axial subgroup, Σ ⊂ Γ × S1 to be an isotropy subgroup of the action of Γ ×S1 on the center subspace E c of J with dimFix(Σ) = 2. Then for each such Σ, there exists a unique branch of periodic solutions to the system with spatiotemporal symmetries Σ. We now illustrate these results with a number of examples of coupled cell networks (each with a small number of cells). Example 1. A three-cell network with identical cells and identical bidirectional coupling (indicated by the bidirectional arrow). O 1 aC CC CC CC C! = 3 {{ { { {{ }{{ 2


333

The ODE’s for each cell may be written x˙ 1 = g(x1 , x2 , x3 ),

x˙ 2 = g(x2 , x3 , x1 ),

x˙ 3 = g(x3 , x1 , x2 )

where x1 , x2 , x3 ∈ Rk , k ≥ 2, and some function g : (Rk )3 → Rk . The overline indicates that g is invariant under permutation of the second and third coordinates. This property reflects the identical coupling whose effect comes from the precise form of g. Here, the network has symmetry group S3 ∼ = D3 generated by σ = (12) which swaps cells 1 and 2, and τ = (123) which cycles all three cells. By the H/K Theorem, periodic solutions are characterized by pairs (H, K) equal to (Z3 (τ ), 1), (Z2 (σ), Z2 (σ)), and (Z2 (σ), 1). The first corresponds to a rotating wave where each cell has identical dynamics but with successive T /3 phase shifts. The second forces two cells to be in phase, and the third forces two of the cells to be out of phase (by exactly half a period) with third cell frequency twice that of the other two. This latter solution is of course of special interest in Mathematical Music Theory, indicating as it does a ‘symmetry-forced’ frequency-doubling oscillation, something that has obvious parallels with Fourier Spectra. All three periodic solutions occur as Hopf bifurcations. To see this, write the action of D3 × S1 on R4 (identified in this case with with C2 ) as φ(z1 , z2 ) = (e−iφ z1 , eiφ z2 ),

κ(z1 , z2 ) = (z2 , z1 ),

θ(z1 , z2 ) = (eiθ z1 , eiθ z2 )

where φ ∈ Z3 acts by rotation through 2π/3 and θ ∈ S1 . Then there are three C-axial subgroups, Z2 ((κ, 0)), Z3 = , and Z2 (κ, π) each with twodimensional fixed-point subspaces, and corresponding to the above H/K pairs (Z2 (σ), Z2 (σ)), (Z3 (τ ), 1), and (Z2 (σ), 1) respectively. Example 2. A three-cell network with identical cells and identical unidirectional coupling. 1 aC CC CC CC C 3 {= { {{ {{ {{ 2 In this case the ODE’s take the form x˙ 1 = g(x1 , x3 ),

x˙ 2 = g(x2 , x1 ),

x˙ 3 = g(x3 , x2 )

This time, the unidirectionality of the coupling reduces the symmetry and hence the number of periodic solutions. The symmetry group this time is Z3 , and since this group has no non-trivial isotropy subgroups, the only supported periodic state inherited from the bidirectional ring is the rotating wave state, resulting from a Hopf bifurcation corresponding to the C-axial subgroup Z3 = .

334

J.O. Clark

Example 3. The above scenario (Example 1) for a three-cell symmetric network can be generalised. In the case of rings of N identical cells with bidirectional coupling, the symmetry group is DN and it can be shown (Golubitsky et al. 1988, p.368) that the required C-axial subgroups form triples (here ζ = 2π/N , and Zc2 = {(0, 0), (π, π)}). (Z2 (κ), ZN , Z2 (κ, π),

(Z2 (κ) ⊕ Zc2 , ZN , Z2 (κ, π) ⊕ Zc2),

(Z2 (κ) ⊕ Zc2 , ZN , Z2 (κζ) ⊕ Zc2 )

depending on whether N is odd, or, congruent to 2 or 0 modulo 4 respectively. For convenience we show below a complete table of waveforms for N = 12. 0 1 2 3 4 5 6 7 8 9 10 11 A A A A A A A A A A A A A A+1 A+2 A+3 A+4 A+5 A+6 A+7 A+8 A=9 A+10 A+11 A B C 2D C1/2 B1/2 A1/2 B1/2 C1/2 2D C B A B C C1/2 B1/2 A1/2 A1/2 B1/2 C1/2 C B A A A+2 A+4 A+6 A+8 A+10 A A+2 A+4 A+6 A+8 A+10 A B B1/2 A1/2 B1/2 B A B B1/2 A1/2 B1/2 B 2A B B 2A B1/2 B1/2 2A B B 2A B1/2 B1/2 A A+3 A+6 A+9 A A+3 A+6 A+9 A A+3 A+6 A+9 A 2B A1/2 2B A 2B A1/2 2B A 2B A1/2 2B 2A B 2A B1/2 2A B 2A B1/2 2A B 2A B1/2 A A+4 A+8 A A+4 A+8 A A+4 A+8 A A+4 A+8 A B B A B B A B B A B B 2A B B1/2 2A B B1/2 2A B B1/2 2A B B1/2 A A+5 A+10 A+3 A+8 A+1 A+6 A+11 A+4 A=9 A+2 A+7 A B C 2D C1/2 B1/2 A1/2 B1/2 C1/2 2D C B A A1/2 B C C B A A1/2 B1/2 C1/2 C1/2 B1/2 A A1/2 A A1/2 A A1/2 A A1/2 A A1/2 A A1/2 Here A, B, C, D are waveforms subject to ‘internal’ (κ, θ) symmetry. The notation A + n indicates the waveform A with a phase-shift of n/12ths of the total period T of the network, 2A means an oscillation at twice the frequency, and A1/2 indicates a waveform with a T /2 phase-shift. The table also contains periodic solutions arising from the nonstandard as well as standard actions. Essentially the solutions split into blocks of q sets of size h where q = gcd(h, j), 0 ≤ j ≤ N − 1. We omit the details. One of the surprising aspects of the theory of Stewart and Golubitsky is that non-trivial global symmetry is not, in general, necessary to prove the existence of periodic solutions in a coupled cell network, as the next example shows. Example 4. A coupled cell network that has trivial symmetry group but that supports a travelling wave solution. = 2 CC CC {{ { CC { CC {{ { { ! 3 1 l

/ 4

/ 5

/ 6

/ 7


335

Here the ODE’s may be given the form x˙ 1 = g(x1 , x3 ),

x˙ j = g(xj , xj−1 ),

j = 2, ..., 7

If we restrict our attention to the states in which x1 = x4 = x7 ,

x2 = x5 ,

x3 = x6

then the equations of this network are identical to those in Example 2 (the unidirectional ring on three cells). Hence this network is a ‘quotient’, and it can then be shown (Golubitsky and Stewart 2006, p. 324) that the rotating wave of the 3-cell network ‘lifts’ to the 7-cell network, producing a periodic solution in which cells 1,4 and 7 are synchronous, as are 6 and 3 (with a 2T /3 phase-shift), and as are 5 and 2 (with a T /3 phase-shift). This notion can be made precise by defining an equivalence relation, on the cells, such that cells that are in the same equivalence class have related ‘input sets’ (defined as the set of cells from which a particular cell receives coupling and whose associated variables appear in the ODE component). The dynamics are then determined by a network, the quotient of the whole G by the equivalence relation, G/ (the graph-theoretic quotient) whose cells correspond to clusters of synchronous cells (equivalence classes) and whose edges are defined to preserve the input type defined by the equivalence relation . In the above case the equivalence classes are {1, 4, 7}, {2, 5}, {3, 6}. See Golubitsky and Stewart 2006 (Section 9) for more details, where this leads to the conclusion that the structure is governed not by the symmetry group of the network (which in this case is trivial), but the symmetry groupoid of the network, which measures, in some sense, the existence of ‘local’ symmetries. Other Examples. In Golubitsky and Stewart 2006, many examples of lowdimensional coupled cell networks are given, including a 3-cell network with asymmetric coupling, with an associated synchrony subspace which equations defining the standard R¨ ossler attractor. This is an intriguing example in which the synchronous dynamic of cells can be chaotic. Such a network, when perturbated, exhibits transverse dynamics that are approximately synchronous- a state in which cells repeatedly obtain and then lose synchrony intermittently (a phenomenon called ‘bubbling’). The details are complex and are omitted.

3 3.1

Discussion and Applications Nonlinear Dynamics and Musical Ontology

We now discuss some connections between the theory of nonlinear dynamics and the topic of the morphogenesis of musical or sonic events, and we will refer here to Manuel DeLanda’s reading of the mathematical ontology of Gilles Deleuze (DeLanda 2002). Central to this reading is DeLanda’s interpretation of the Deleuzian concept of multiplicities. He defines these in relation to wellknown concepts in nonlinear dynamics and complexity theory as ‘nested sets

336

J.O. Clark

of vector fields related to each other by symmetry-breaking bifurcations, together with the distributions of attractors which define each of its embedded levels’, (DeLanda 2002, p. 30). An ontological distinction is then drawn between the parts of this model that carry information about the actual world (possible ‘states’ of the model are actualised as the trajectories of a particular dynamical system) from that part that which, in principle, is never actualised, something that Deleuze calls the virtual. This latter concept should not be confused with the common usage of the word ‘virtual’, that of the ‘virtual reality’ of digital simulations, but as the embedded levels of singularity structure on manifolds which can change through bifurcation. As Deleuze states in Difference and Repetition, p.208, ‘the reality of the virtual consists of the differential elements and relations along with the singular points that correspond to them’, and is, in some sense is part of the object itself, coiled up inside it, ‘as though the object had one part of itself in the virtual into which it is plunged as though into an objective dimension’. That the singularities themselves are never actualised is clear from the mathematics; trajectories in dynamical phase space approach attractors asymptotically, getting ever closer but never reaching them. But they are nevertheless real entities that structure the possible phase space paths- as Deleuze puts it ‘the singularities preside over the genesis of the trajectories’. The above discussion of the nonlinear dynamics of networks can now be interpreted similarly in this context and related to an ‘immanent’ musical ontology. Events in sonic space have complex spectrotemporal and spatiotemporal unfoldings which can be interpreted as one of so many possible actualisations of that event- each structured by singularities and differential relations. We saw in Section Two how a coupled cell system can be ‘dormant’ in a steady-state equilibria, but as a result of a codimension-one bifurcation, spontaneously oscillate with a characteristic period. And further bifurcations are possible, from period-doubling cascades to chaos. It has been known for some time that, in particular, Hopf bifurcations, like those described earlier, play a role in the emerging spectral dynamics of musical tones, and many authors have utilised this phenomenon in the physical modelling of musical instruments (for example see Rodet and Vergez 1999). Particularly important here is how such bifurcations can be seen to structure a (different) sense of musical time; spontaneous bifurcations could result in the emergence, or not, of spectral components in a musical tone. And the timings of these events suggest a conception of musical time as being more like a nested set of levels of periodicities which are fundamentally asymmetric. As complexity theorist Ilya Prigogine puts it, a process ‘in the regime of uniform steady state...ignores time. But once in the periodic regime, it suddenly ‘discovers’ time in the phase of the periodic motion....We refer to this as the breaking of temporal symmetry’, (Nicolis and Prigogine 1989, p.21). In addition, other parts of the morphogenetic evolution of sonic events can be interpreted in this framework, such as attack transients, a phenomenon that certainly relates to the concept, in nonlinear dynamics, of ’relaxation time’the time taken for a trajectory to settle into a behavior within its basin of attraction. Equally important is what this ontological perspective entails for the


337

relationship between chance and determinism in sonic morphogenetics. In Example Three we saw that a ring of 12 identically coupled cells can undergo Hopf bifurcation so that an equilibrium is replaced by one, and only one, of 17 different periodic solutions. The question is- which one? Whilst the bifurcation itself is in some sense deterministic (when the critical value is reached, we now the system will bifurcate), only chance, in the form of a particular perturbation of the environmental circumstances at the time of bifurcation will decide which periodic state is reached. Many solutions are possible for the same parameter value. For the most part however, the study of dynamical systems within musical contexts has often involved consideration of particular systems, which are modelled in real-time, and bifurcation points simulated deterministically etc. And one must perhaps draw a distinction here between theoretical models for musical morphogenesis and ’real-world’ applications. These latter applications propose a model which is then ‘matched’ to a ‘real’ musical situation, so that for example, a dynamical system modelling the tone production of a particular instrument is deemed successful if the output matches the spectrogram of the instrument being modelled via a quantitative analysis. But in Deleuzian ontology, there is a deeper isomorphism here- ‘that a given phase space and the physical system it models may be both actualizations of the same virtual multiplicity. That is turn, implies the crucial relation between model and reality is not one of resemblance (phase space trajectories resembling plotted series of laboratory measured values) but one of co-actualization’, (DeLanda 2002, p. 180). It could be argued however that such qualitative descriptions of the spatial and temporal unfolding of sonic events are often subsumed to more quantitative or informatical approaches. What is perhaps missing from Mathematical Music Theory at present are theoretical models of the inherent uncertainty and contingency that underlie the temporal evolution of even the most simple sonic events. Indeed, this is really a question of scale. Large scale properties of musical events (rhythm, harmony, even pitch) are relatively well-studied, and such concepts represent musical entities that have, in some sense, already ‘coalesced’ from small-scale microevents. But a qualitative theory of these ‘intensive’ and immanent events is perhaps still needed. And it could be that the generality of the theory of coupled cell networks outlined above could form part of such a theory. One of the advantages of this approach is that it is, to a certain degree, model-independent. We have assumed nothing in the examples about the nature of the coupling function for instance, or for that matter, the internal dynamics of each individual cell. To a large extent, these considerations are secondary to the local or global symmetry conditions imposed on the system that ‘force’ the cell dynamics. But some significant problems remain; many assumptions need to be made, such as identical coupling and lack of plasticity of edge-formation (new coupling arising in the network spontaneously etc.). Factors that probably do arise ‘in nature’. And in the groupoid case, which may be the most flexible, due to less stringent requirements on the global symmetries, still has a sketchy bifurcation theory, which is known to be related to the (intractable) representation

338

J.O. Clark

theory of nonsemisimple associative algebras (see Golubitsky and Stewart 2006, p. 360). But it nevertheless may represent a step in the right direction, and more research is required here. 3.2

Applications to Algorithmic Composition

We end with a description of a sound installation (Clark 2005) that takes the catalogue of the possible oscillations in a ring of N cells (described in Example Three) as a starting point. The installation incorporates an eight-channel speaker system linked to a MaxMSP patch that dictates the sonic output of each cell, or speaker. Larger numbers of speakers may also be used; the diffusion throughout the speaker grid handled by third-party MaxMSP spatialisation tools. The periodic solutions act as modulators of the amplitude envelope of particular audio buffers, to create a polyphonic texture in which the phase-locking and phase-shifts of the individual channels can clearly be heard. This reflects one of the artistic motivations behind the installation, namely an approach to sound spatialisation that specifically seeks to explore both phase and synchrony as compositional material. Phase particularly is perhaps often overlooked as a compositional determinant, the tape loop experiments of Steve Reich in the 1960’s notwithstanding, and it is perhaps surprising that the opportunities for spatial separation afforded by multi-channel installation set-ups have not explored this parameter more fully. In the MaxMSP patch that generatively creates phase and synchrony patterns in the loudspeakers, the choice of a particular (bifurcatory) oscillation type is dictated by chance and uses the Markov model of Visell (2004). Visell uses the standard implementation, considering a set of K states S = {s1 , s2 , ..., sK } modelling a time progression of events, and a set of transition probabilities Tij = P (sj |si ), together with an M -dimensional parameter space from which a generated sequence of vectors of parameters v1 , v2 , ... is selected. A set of probability distributions P (vj |sk ) are defined and indicate the probability for a vector vj to have been generated by state sk . Visell models the P (vj |sk ) as weighted Gaussian mixtures. One can also perform a simple training algorithm which updates the matrix Tij so that the model ‘learns’ from its own output. We can consider now this model in our setting. Each Markov state is identified with a single branch of periodic bifurcating solutions. We assume these are preceded in each case by a neutral ‘off’ state, and that for musical reasons, the output decays back to this state after a certain period of time. Similarly, we may treat the period T as a variable in our model. Small values of T can be used, for example, to modify microsound parameters. The M -dimensional parameter vector sampled from the Gaussian mixtures is used to model musical parameters in each individual channel, including spectral content of the waveforms. One can also, as Visell points out, link multiple networks simultaneously into a ‘network of networks’, the sonic result of which is a kind of ‘network polyphony’.


339

References Clark, J.O.: Immanence. A sound installation first shown as part of the ’HTMLes’ CyberArt Bienalle at the Monument National, Montreal (2005) DeLanda, M.: Intensive Science and Virtual Philosophy. Continuum, London (2002) Golubitsky, M., Stewart, I.: Nonlinear Dynamics of Networks: The Groupoid Formalism. Bulletin of the AMS 43(3), 305–364 (2006) Golubitsky, M., Stewart, I.: The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space. Birkh¨ auser, Basel (2000) Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. II. Springer, Berlin (1988) Mazzola, G.: The Topos of Music. Birkh¨ auser, Basel (2002) Nicolis, G., Prigogine, I.: Exploring Complexity. W. H. Freeman, New York (1989) Rodet, X., Vergez, C.: Nonlinear Dynamics in Physical Models: From Basic Models to True Musical-Instrument Models. Computer Music Journal 23(3), 35–49 (1999) Visell, Y.: Spontaneous Organisation, Pattern Models, and Music. Organised Sound 9(2), 151–165 (2004)

Form, Transformation and Climax in Ruth Crawford Seeger’s String Quartet, Mvmt. 3 Edward Gollin Williams College [email protected] Abstract. The paper reckons the progression of permuted registral states in Ruth Crawford Seeger’s String Quartet as a walk through a Cayley graph of the symmetric group, S4. The graph privileges the 2-cycles that exchange adjacent voices in the four-voice quartet texture, imposing a metric upon the group elements that allows one to distinguish permutations that are contextually close or ‘smooth’ from those are distant or ‘agitated.’

The paper explores the transformational structure of voice permutations in the third movement of Ruth Crawford Seeger’s String Quartet (1931), extending and offering an alternative to an analysis presented in an article by Ellie Hisama (Hisama 1995). Hisama argues against a ‘traditional’ view of the movement’s structure in which the music reaches a singular climax (of dynamics, ambitus, density) three-quarters of the way through the piece. She instead suggests that the intertwining lines of the ensemble project multiple climaxes in the structural domain of ‘twistedness’, i.e. the degree to which the relative registral positions of the ensemble depart from a ‘standard’ quartet ordering. Example 1. Ruth Crawford Seeger, String Quartet, iii, measures 25–32.

Example 1 presents mm. 25–32 of the work, illustrating a texture typical of the movement in its approach to the traditional climax: sustained tones in each instrument lie within a narrow ambitus; individual voices leap over or under those of its neighbors, creating a variety of ‘registral states’, particular registral dispositions of the instruments T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 340–346, 2009. © Springer-Verlag Berlin Heidelberg 2009

Form, Transformation and Climax in Ruth Crawford Seeger’s String Quartet, Mvmt. 3

341

in the ensemble. The distinct registral states in Example 1 are identified beneath the score. The first state shown in m. 25 is the standard quartet ordering (high to low: vn1, vn2, vla, vc). On the third beat of m. 25, violin 1 crosses below violin 2, creating a new registral state (high to low: vn2, vn1, vla, vc). On beat 2 of m. 26, the cello leaps up and crosses above the viola, creating a third distinct registral state (high to low: vn2, vn1, vc, vla). Additional voice crossings in the subsequent measures lead to the final registral state of the passage at m. 32, a complete registral inversion of the standard quartet ordering (high to low: vc, vla, vn2, vn1). At the bottom of Example 1, I indicate what Hisama calls the “degree of twist” of each registral state. For a given sonority, the degree of twist tallies the number of voice pairs whose registral ordering represents an inversion of the standard quartet ordering.1 For instance, the ‘standard,’ referential ordering at the opening of Example 1 has no inverted pairs, and hence has a degree of twist 0. The second state has one inverted voice pair (violin 2 over violin 1) and thus has degree of twist 1. The third state has two inverted voice pairs (violin 2 over violin 1, cello over viola) and thus has degree of twist 2. The sonority on beat three of bar 29 has four inverted voice pairs (violin 2 over violin 1, cello over violin 1, viola over violin 1, cello over viola) and thus has degree of twist 4. The final state, in bar 32, has degree of twist 6—all voice pairs are inverted relative to those of the standard ordering—and thus represents a state of maximal twistedness in a four-voice sonority. Example 2. A summary of registral states in measures 1–63.

1

Hisama (1995, 298).

342

E. Gollin

Example 2 lists the registral states of instruments in the first 63 measures of the quartet movement. Each registral state is identified by the measure in which it first appears.2 Hisama’s degree of twist is indicated above each state. The two maximally-twisted four-voice states (asterisked at mm. 32 and 56), in Hisama’s view, represent climaxes in the domain of “twistedness.” The gradual motion toward and away from these two maxima, progressing through more and less twisted states, supports what Hisama has termed a ‘feminist,’ as opposed to the traditional ‘masculine,’ narrative of form in the work.3 Rather than considering the notion of ‘climax’ to reside simply in the twisted registral states of the ensemble, the present paper adopts a transformational interpretation of Hisama’s analysis, considering the ways in which climax can be understood to reside also in the structure of the permutations between them. In one sense, the ‘structure’ of the permutations is manifest as the symmetric group, S4, comprising the transformations between the 24 possible permuted states of the quartet. Yet a transformational view of the quartet movement as simply a reification of S4 through projection of the largely undifferentiated members of the group is not particularly informative. Indeed, it simply exchanges a view of the work as a succession of states with a view of the work as a succession of permutations between those states, themselves objects of a different sort. One way to assert distinctions and impose a hierarchy among group elements is through a combinatorial representation of the group, that is, through a group presentation.4 A group presentation represents a group as a set of generators, {a, b, c, ...}, together with a set of defining relators, {W, U, V, ...}, written < a, b, c, ...; W, U, V, ... >.

(1)

The generators are group elements that (together with their inversions) can be combined to express any group element. Group elements are expressed symbolically as words in the symbols a, b, c, … that represent the ordered combination (reading from left to right) of the generating elements signified by those symbols. The defining relators are words equivalent to the identity element that express how the group structure constrains the combination of generators. Most groups admit multiple distinct presentations. For instance, the symmetric group S3 (equivalently the dihedral group D3) can be represented by a presentation either on two distinct order-2 generators: < b, c; b2, c2, (bc)3 >

(2)

or by a presentation on an order-3 generator and an order-2 generator: < b, c; b3, c2, cbcb-1 >.

(3)

Examples 3(a) and (b) depict the presentations given by [2] and [3], respectively, as Cayley graphs. The graphs reveal how the two presentations differently organize the six elements of the group, imposing distinct hierarchies upon those elements based 2

Brackets embracing violins 1 and 2 at mm. 19 and 19a indicate that the two instruments play in unison; the default position of violin 1 above violin 2 in those two states is assumed. 3 Hisama (1995, 305). 4 Magnus et al. (1976) is the classic work on combinatorial group theory. Gollin (2000) explores various music-transformational applications of a combinatorial group perspective.


343

upon the distances between elements in the graph. In Example 3a, for instance, any group element has only two immediate neighbors, two elements are distant by two steps, and one element is antipodal, lying three steps distant on the graph. The graph of Example 3b (and its underlying presentation) by contrast imposes a less articulated hierarchy among elements of the group: any element has three immediate neighbors and the remaining two elements are only two steps distant. Example 3. Two presentations of S3, represented as Cayley graphs.

Example 4. A concrete instantiation of the presentation from Example 3a.

The graphs of Example 3 are graphs of abstract presentations of S3: nodes are unlabeled, and can be instantiated by a variety of elements of the abstract group. Example 4 instead presents a concrete instantiation of the abstract graph of Example 3(a), associating the abstract generators b and c with particular elements of a concrete group. In this case, the graph represents the group of registral permutations of three instruments (vn2, vla, vc) in a three-voice texture. Generators have been instantiated by the particular order 2 elements that exchange instruments in neighboring voices: b = (AT), the permutation that exchanges instruments in the Alto and Tenor voices of a given three-voice sonority; c = (TB), the permutation that exchanges instruments in the Tenor and Bass voices of a given three-voice sonority. Each node has been labeled to represent a particular registral ordering of the instruments. In the case of the Crawford Seeger quartet, given its gradual twisting and untwisting of adjacent lines, (AT) and (TB) form a natural set of generators against which to compare and measure permutations in the three-voice texture that precedes entry of the fourth part (mm. 14–19).

344

E. Gollin

On Example 2, arrows beneath the three-voice registral states, are labeled using words in the symbols b and c that describe the permutations between successive states. The words, moreover, describe a path through the graph of Example 4, a path that represents the progression of states. Two aspects of the path/progression are notable. First, word lengths increase over the course of the three-voice section: two words, c and b (describing a motion on the graph from 12:00 to 10:00, and 10:00 to 8:00 respectively) are followed by two statements of the word cb (describing motions from 8:00 to 4:00 and 4:00 to 12:00 respectively). Because the presentation, by virtue of its choice of generator elements, values ‘smooth’ transitions (those that exchange single pairs of neighboring elements), the progression from simple to compound transformations reflects an increase in transformational or permutational complexity or ‘agitation.’ Second, the progression of states completes a full circuit on the graph; the return to the opening state formally ends the section, and the introduction of a fourth voice ensues. Example 5. A Cayley graph of S4 on the order-2 generators that exchange adjacent voices in a quartet texture.


345

One can similarly present S4 in generating symbols {a, b, c}, instantiated by the three permutations that exchange adjacent voice parts in a four-voice texture (SATB), a = (SA), b = (AT), c = (TB). The complete presentation < a, b, c; a2, b2, c2, (ac)2, (ab)3, (bc)3>

(4)

is represented as a Cayley graph on Example 5.5 The graph is analogous to the familiar Tonnetz in pitch space; like the graph of Example 4, it imposes a metric upon the quartet’s registral states, allowing one to measure the transformational ‘distance’ between any two states as the word length of that transformation in the group presentation, a distance measured in terms of the space’s ‘privileged’ generating permutations. Beneath the four-voice registral states of Example 2, the permutations between successive states of the quartet are expressed as words in the presentation [4], or equivalently, as pathways through the voice-permutation space of Example 5. Inspection of the voice permutations on Example 2, reckoned as words in the symbols {a, b, c}, reveals that while the progression of registral states becomes twisted and untwisted during the approach to the traditional climax (Hisama’s multiclimax view), the word lengths between successive states gradually increase as one approaches the traditional dynamic/registral climax. That is, whereas the first 45 measures exclusively feature permutations of word length 1 and 2, two words of length 3 (both statements of the word cba) lead to the registral states at mm. 49 and 56. The words of length 3, which signify a greater degree of transformational ‘agitation,’ fortify, from a transformational perspective, a traditional understanding of the movement’s form as a gradual progression toward a singular climax. At the same time, the sequence of word lengths in voice-permutation space can be understood to support Hisama’s view that states of maximal twistedness constitute points of local climax: the first state with maximal degree of twist (m. 32) arrives amid a run of four length-2 permutations; the second state with maximal degree of twist (m. 56) is attained by a length-3 permutation. The transformational structure of the permutations thus articulates and underscores the structure of the registral-state maxima.6

References Hisama, E.: The Question of Climax in Ruth Crawford’s String Quartet, Mvt. 3. In: Marvin, E.W., Hermann, R. (eds.) Concert Music, Rock, and Jazz since 1945: Essays and Analytic Studies, pp. 285–312. University of Rochester Press, Rochester (1995) Gollin, E.: Representations of Space and Conceptions of Distance in Transformational Music Theories. Ph.D diss., Harvard University (2000)

5

The graph is a Schlegel diagram (a distorted planar projection) of a truncated octahedron. Schlegel diagrams are discussed in Loeb (1991). 6 A different transformational view of the permutations in the passage can be gleaned by observing permutations among instruments rather than among voices. Such a view might reflect the intuition that incremental changes of state are those that exchange instruments pairs that are timbrally similar. Lewin (2007, xvii–xx) explores distinctions between voice permutations and instrument permutations in the analysis of works in triple counterpoint.

346

E. Gollin

Lewin, D.: Generalized Musical Intervals and Transformations (Corrected edn. with New Author’s Preface). Oxford University Press, New York (2007) Loeb, A.L.: Space Structures: Their Harmony and Counterpoint. Birkhäuser, Boston (1991) Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. Dover Publications, New York (1976)

A Local Maximum Phrase Detection Method for Analyzing Phrasing Strategies in Expressive Performances Eric Cheng1,* and Elaine Chew2 1 Hsieh Department of Electrical Engineering, University of Southern California Viterbi School of Engineering [email protected] 2 Epstein Department of Industrial and Systems Engineering, Hsieh Department of Electrical Engineering, University of Southern California Viterbi School of Engineering [email protected]

Abstract. This paper proposes a Local Maximum Phrase Detection (LMPD) method for the analysis of phrasing strategies in expressive performances. The LMPD method systematically extracts a quantitative representation of phrasing strategy by equating the occurrence of a local maximum in the loudness curve with the occurrence of a phrase or sub-phrase. We further define mathematical descriptors for phrase strength and volatility, and phrase typicality, for comparing phrasing strategies among performances. Phrase strength measures the prominence or clarity of a phrase, and the volatility is defined as the standard deviation of the phrase strengths within a performance. Phrase typicality quantifies the degree to which a phrase loudness peak location is characteristic among the performances polled. The ideas behind these descriptors extend to phrase information derived from tempo variation. We illustrate the LMPD method using preliminary results from its application to eleven commercially available audio recordings of a solo violin Bach Sonata.

1 Introduction Previous research into expressive phrasing strategies has generally been local in nature. That is, the majority of studies have discussed how performers vary musical parameters within a single phrase or near a single phrase boundary (see, for example, Cambouropoulos (2001), Gabrielsson (1987), Langner and Goebl (2003), and Repp (1992)). While certain local phrasing strategies - such as the clarifying of boundaries with declines in tempo and dynamics - are well documented, relatively little is known about higher level phrasing strategies, i.e., how performers choose to segment a piece into phrases. In this paper, we aim to develop tools for understanding phrasing strategies from this more global perspective. To that end, we propose the Local Maximum Phrase *

Eric Cheng is now a Research Analyst with Areté Associates.


348

E. Cheng and E. Chew

Detection (LMPD) method, which derives a quantitative representation of phrasing strategies by identifying the number and locations of phrases throughout a performance, and defines mathematical descriptors to quantify the characteristics of each phrase. These methods provide the means to quantitatively compare and contrast global phrasing strategies. We extract beat-level1 tempo and loudness data from music audio using manual onset detection and a psycho-acoustic model of loudness. By superimposing author-annotated phrase boundaries over the tempo/loudness data, we find that local maxima in the loudness curve are reliable indicators for phrase or sub-phrase occurrences in the piece examined. This observation leads to the LMPD method, which first equates the occurrence of a local maximum in the loudness curve with that of a phrase or subphrase. Then, the phrase strength, which measures how strongly the phrase stands out against the background, is calculated as the average loudness increase from the adjoining local minima; and, the phrase volatility is computed as the standard deviation of all phrase strengths. Finally, each phrase is assigned a value, the phrase typicality, that quantifies the popularity of its location among the performances polled. To demonstrate the efficacy of the LMPD method, we apply the method to performance data obtained from commercially available audio recordings, and present the preliminary results.

2 The Method This section presents our method for performance analysis, and consists of: (1) a description of our method for tempo and loudness extraction; (2) arguments for focusing on loudness data; and, (3) the proposal of the local maximum phrase detection method, including the extraction of phrases, and of the mathematical descriptors. 2.1 Data Extraction We first extract beat onset times manually using a marking tool in Final Cut Pro. Then, we use the onset times to calculate beat-level tempo. To compute loudness data, we first calculate a loudness waveform, in Sones, using a MATLAB implementation of the PEAQ2 standard (Kabal 2002). The waveform is then smoothed using a Gaussian window, and sampled at each onset time, to obtain beat-level data. To validate the accuracy of the extracted data, we compare loudness values for a single recording to a manually plotted reference curve representing our own perception. The smoothing window width is optimized so that the loudness data best matches the reference curve. 2.2 The Case for Loudness We extracted performance data (tempo and loudness) from eleven commercially available audio recordings of the Andante movement of J. S. Bach's Sonata No. 2 for 1

We use the term “beat-level” to refer to the estimation of tempo and loudness through a sampling of onset times and loudness values at the natural frequency with which one would tap along to the beat of the piece (cf. Sethares, in this volume). 2 Perceptual Evaluation of Audio Quality – a standard of the International Telecommunications Union.

A Local Maximum Phrase Detection Method for Analyzing Phrasing Strategies

349

solo violin, BWV 1003. We chose this piece for its regular pulse and unambiguous phrase structure - qualities that simplified both data extraction and analysis. The performances were by Ehnes, Enescu, Grumiaux, Heifetz, Kremer, Menuhin, Milstein (1956 and 1975), Mintz, Szerying, and Szigeti. To devise a systematic method of phrase detection, we first annotated phrase boundaries using only the score as our guide. Then, as shown in Figures 1 and 2, we superimposed these boundaries over plots showing tempo and loudness data for all performances to see whether any identifiable patterns arose.

Fig. 1. Phrase boundaries superimposed over tempo data

Fig. 2. Phrase boundaries superimposed over loudness data

350


In the two figures, vertical lines denote starts of phrases, and each trajectory represents a single performer's performance data. Observe that the loudness trajectories appear to be more consistently related to the annotated phrase boundaries than their tempo counterparts. In particular, phrases are well characterized by a crescendo/decrescendo arch similar to that mentioned in several past studies, such as Gabrielsson (1987), Sundberg, Friberg and Bresin (2003), and Todd (1992). In some cases, phrases are characterized by two or more sub-arches, suggesting that those performers chose to divide the annotated phrases into subphrases. In contrast, the tempo strategies are less systematically related to the phrase boundaries, with a greater diversity of trajectories. This is not entirely unexpected. Beran and Mazzola (1999) and Repp (1992) found pronouncedly different tempo strategies in their investigations of piano performances of Schumann’s “Träumerei.” The greater diversity of tempo strategies, which was confirmed by analyzing the average interperformer correlations for the tempo and loudness data ( r tempo = 0.4862 and

r loudness = 0.7627 ), led us to conclude that, for this piece, loudness is a more reliable parameter for phrase detection, and in particular, that a crescendo/decrescendo arch is a reliable indicator for the occurrence of a phrase. 2.3 Local Maximum Phrase Detection If we assume that each phrase is characterized by a crescendo/decrescendo arch, then each phrase should also be associated with a local maximum in loudness. The LMPD method uses this local maximum as a mathematical indicator for the existence of a phrase. The method consists of two steps: (1) record number and locations of local loudness maxima for each performance; and, (2) interpret each local maximum as a phrase or sub-phrase. The total number of local maxima in a performance provides a global measure of the degree to which a performer highlights local vs. global phrase structure, while the locations of the local maxima allow us to compare different phrase subdivision strategies. This method also allows us to define additional mathematical descriptors to further quantify the characteristics of phrasing strategy. These descriptors are discussed in the next two sections. 2.3.1 Phrase Strength and Volatility We define the phrase strength (P.S.) of a phrase to be equal to the average loudness difference between its local maximum and the two adjoining local minima:

1 P.S. = [(M i − m j ) + (M i − mk )] , 2

(1)

where Mi is the loudness value of the local maximum, and mj, and mk are the loudness values of the two adjoining local minima as shown in Figure 3. P.S. values allow us to measure the prominence or clarity of a particular phrase.


351

Fig. 3. Phrase strength parameters

Fig. 4. Top: Average performance trajectory. Bottom: Number of performers placing a local maxima at a particular location.

The phrase volatility (P.V.) of a particular performance is defined to be the standard deviation of all P.S. values in the performance. Thus, the greater the variability in phrase strengths, the greater the phrase volatility.

352


2.3.2 Phrase Typicality The phrase typicality (P.T) of a phrase quantifies the popularity of its location. It is defined to be the proportion of other performers who also place a local maximum at the location of the phrase in question. Mathematically, the phrase typicality is given by:

P.T. =

1 [M(i) −1], N −1

(2)

where M(i) is the total number of performers placing a local maximum at location i, and N is the total number of performers. Thus, the greater the number of performers placing a local maximum at a particular location, the greater the phrase typicality of a phrase at that location. Figure 4 shows how M(i) varies from location to location. In the sample performance trajectory, on the top half of Figure 4, the vertical dotted lines indicate local maxima. The histogram on the lower half of Figure 4 shows the number of performers placing a local maximum at a particular location, equivalent to M(i) in Equation 2. The histogram shows, for example, that the peak at bar five is highly typical, and shows up in eleven of the twelve recordings, while a peak at the beat just before it was observed in only one recording.

3 Conclusion and Discussion In conclusion, we have presented a novel method for the analysis of phrasing strategies in expressive performances by equating the occurrence of a local maximum in loudness with the occurrence of a phrase. Preliminary results suggest that a local maximum in loudness is a meaningful expressive event that reliably indicates the presence of a phrase. We defined mathematical descriptors to quantify the characteristics of each phrase. Until we conduct listening tests, we can only hypothesize about the perceptual significance of these descriptors. The fact that we found the loudness data to vary more systematically with the phrase structure, more so than the tempo information, in our case study using Bach’s Sonata for solo violin, BWV 1003, may be an artefact of this particular sonata, Bach’s compositions, the Baroque genre, or of solo violin performances of this kind of music. In a study by Chuan and Chew (2007), the authors found tempo data to reflect well the phrases projected in performances of Chopin’s Preludes (Nos. 1 and 7) for piano, and used tempo variations in Kissin’s and Rubinstein’s recorded performances to analyze the performers’ phrasing strategies. While tempo variation in expressive performance is common in performances of Chopin’s music, or of other compositions of the Romantic genre, performers typically refrain from significant tempo deviations in the performances of music by Bach and his contemporaries. Thus, the expressive strategies employed in the projection of perceptual groupings in such music may be more confined to the management of dynamics in the performance. We posit that the LMPD method and its accompanying mathematical descriptors, while defined here based on loudness information, would extend well to the quantifying of properties of phrases extracted from tempo data. More comprehensive analyses of the twelve Bach recordings will follow.


353

Acknowledgements This material is based upon work supported by a Frank H. Buck Scholarship, and by the National Science Foundation under grant No. 0347988. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors, and do not necessarily reflect the views of the Frank H. & Eva B. Buck Foundation, or of the National Science Foundation.

References Beran, J., Mazzola, G.: Analyzing musical structure and performance – a statistical approach. Statistical Science 14(1), 47–79 (1999) Cambouropoulos, E.: The local boundary detection model (lbdm) and its application in the study of expressive timing. In: Proc. of the Intl. Computer Music Conf. (2001) Chuan, C.-H., Chew, E.: A Dynamic Programming Approach to the Extraction of Phrase Boundaries from Tempo Variations in Expressive Performances. In: Proc. of the Intl. Conf. on Music Information Retrieval (2007) Gabrielsson, A.: Once again: the theme from Mozart’s piano sonata in A major (K. 331). In: Action and Perception in Rhythm and Music, pp. 81–103. Royal Swedish Academy of Music, Stockholm (1987) Kabal, P.: An Examination and Interpretation of ITU-R BS. 1387. McGill University, Perceptual Evaluation of Audio Quality (2002) Langner, J., Goebl, W.: Visualizing expressive performance in tempo-loudness space. Computer Music Journal 27(4), 69–83 (2003) Repp, B.H.: Diversity and commonality in music performance: An analysis of timing microstructure in Schumann’s Träumerei. Journal of the Acoustical Society of America 92(5) (1992) Sethares, W.A.: Rhythm and Transforms, Perception and Mathematics. In: Klouche, T., Noll, T. (eds.) MCM 2007. CCIS, vol. 37, pp. 347–353. Springer, Heidelberg (2009) Sundberg, J., Friberg, A., Bresin, R.: Attempts to reproduce a pianist’s expressive timing with director musices performance rules. Journal of New Music Research 32(3), 317–325 (2003) Todd, N.P.M.: The dynamics of dynamics: a model of musical expression. Journal of the Acoustical Society of America 91(6), 3540–3550 (1992)

Subgroup Relations among Pitch-Class Sets within Tetrachordal K-Families Jerry G. Ianni1 and Lawrence B. Shuster2 1

Department of Mathematics, Engineering, and Computer Science LaGuardia Community College/CUNY Long Island City, New York, United States of America [email protected] 2 Department of Music and Dance University of Massachusetts at Amherst Amherst, Massachusetts, United States of America [email protected]

In 1990 and 1991, Henry Klumpenhouwer and David Lewin introduced Klumpenhouwer networks (K-nets) as theoretical tools that display transformational interpretations of dyads contained within pitch-class multisets (Lewin 1990; Klumpenhouwer 1991). Informally, K-nets are directed graphs that employ pitch classes as nodes and elements of the T/I group as edges. In order for a K-net to be well defined, its edges must commute throughout the directed graph and its nodes must map to adjacent nodes according to the corresponding edge transformations. Several types of K-nets emerged by varying the cardinalities of the underlying pitch-class multisets, the number of constituent dyads subject to transformational interpretation, the number of transpositional and inversional operators employed, and the relative positions of these operators. We will work exclusively with two common types of K-nets: trichordal K-nets and box-style tetrachordal K-nets. See Examples 1a and 1b, respectively, for representatives of these two types. Example 1. Two Klumpenhouwer Networks

We will assume that the reader is familiar with various forms of isography, Kclasses, and K-families (O’Donnell 1998; Lambert 2002). However, it is important to note that Lambert's conception of K-classes and K-families as harmonic spaces places emphasis on the sonorities defined by the pitch-class multisets. Accordingly, these objects are defined as collections of pitch-class multisets, not as collections of K-nets. T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 354–364, 2009. © Springer-Verlag Berlin Heidelberg 2009

Subgroup Relations among Pitch-Class Sets within Tetrachordal K-Families

355

The sole criterion for a pitch-class multiset to be a member of a K-class or a K-family is that there exists a transformational interpretation of its dyads that satisfies the relations of (one of) the generating K-net diagram(s). Another way to understand this criterion is to restrict formally the notion of strong isography to pitch-class multisets as follows: the pitch-class multisets S and T are called strongly isographic if there are K-net interpretations of S and T, respectively, that are strongly isographic as K-nets. Under this perspective, the pitch-class set [1, 2, 4] is observed to be a member of the I3/I5 trichordal K-class from the T2 K-family via the K-net of Example 1a. On the other hand, if the transformational interpretation of the dyad [2, 4] is changed to a T10arrow emanating from 4, then [1, 2, 4] is observed to be a member of the I5/I3 trichordal K-class from the T10 K-family. One also finds that the T2 trichordal K-family is identical to the T10 trichordal K-family using this point of view. Our main goals are to probe the algebraic structure of trichordal and box-style tetrachordal K-classes, to develop some voice-leading applications among the pitch-class multisets that appear in a given K-class, and to show how these relations inform the harmonic design of arrays in the music of Igor Stravinsky. In 2002, Philip Lambert observed the following characteristic in many K-classes: whenever a pitch-class multiset from a given K-class is transposed by six semitones, the resulting pitch-class multiset belongs to the same K-class. He called it the T6 – partner (Lambert 2002). It can be proven easily that this characteristic holds in general for arbitrary K-classes. Thus, if C is a K-class, we are motivated to define a symmetry of C to be an element of the T/I group that acts on C by permuting its members. The group action under consideration is function evaluation. It then follows that both T0 and T6 are symmetries of every K-class. Other elements of the T/I group can be symmetries of one or more particular K-classes. For example, I4 and I10 are symmetries of the I3/I5 trichordal K-class, and I5 and I11 are symmetries of the I0/I10 Kclass from the box-style tetrachordal T3/T7 K-family. See Examples 2 and 3, respectively, for details on the actions of these symmetries. The three results below demonstrate that symmetries of K-classes can be sorted algebraically in very meaningful ways. We will only prove the first result in this paper. The other two are more technical and require supporting lemmas. However, we will provide a heuristic and a brief outline of the arguments for Theorems 2 and 3 after presenting their statements. Example 2. Four Symmetries – I3/I5 trichordal K-class

356

J.G. Ianni and L.B. Shuster

Proposition 1. Let C be a K-class. Then the set G consisting of all symmetries of C is a subgroup of the T/I group. Proof. The action of the T/I group on the set consisting of all pitch-class multisets induces an action on its power set. C is an element of this power set, and G is the stabilizer subgroup of C under the induced action. ❚ We call the stabilizer subgroup G of Proposition 1 the (full) group of symmetries for C. As an exercise, the reader can verify that the groups of symmetries for the K-classes given in Examples 2 and 3 are {T0, T6, I4, I10} and {T0, T6, I5, I11}, respectively. Example 3. Four Symmetries – I0/I10 K-class from the Box-style tetrachordal T3/T7 K-family

Theorem 2. Let C be a trichordal K-class, and let G be its group of symmetries. Suppose that the transformational operators that specify C are Tn, Ij, and Ij+n, where n 6 and j 11 are non-negative integers. a. b. c. d. e.

If n = 1, 3, or 5, then G = {T0, T6}. If n = 0, then G = {T0, T6, Ij, Ij+6}. If n = 2, then G = {T0, T6, Ij+1, Ij+7}. If n = 4, then G = {T0, T6, Ij+2, Ij+8}. If n = 6, then G = {T0, T3, T6, T9, Ij, Ij+3, Ij+6, Ij+9}.

Informal Sketch of Proof. To facilitate discussion of the proof of Theorem 2, we introduce some terminology. A transpositional dyad is a dyad whose constituent pitch classes have been declared to be related transpositionally. Similarly, an inversional dyad is a dyad whose constituent pitch classes have been declared to be related inversionally. Using this terminology, observe that a pitch-class multiset of cardinality three contains one transpositional dyad and two inversional dyads relative to each of its K-net interpretations. Let C be a trichordal K-class, and let σ be a symmetry of C. Since σ is an element of the T/I group, its action preserves the interval classes of the constituent dyads of each member of C. It can be proven that C contains a member S whose transpositional dyad’s interval class is distinct from the interval classes of the two inversional dyads. Thus, σ maps the transpositional dyad of S onto the transpositional dyad of


357

σ(S) and the two inversional dyads of S onto the two inversional dyads of σ(S). The identification of all the symmetries proceeds by examining the general forms of S, each transpositional symmetry Tk, and each inversional symmetry Il, respectively: S = [a, b, a + n], Tk(S) = [a + k, b + k, a + n + k], and Il(S) = [l – a, l – b, l – a – n]. Use the facts that the transpositional (inversional) dyads map to each other to obtain algebraic restrictions on the value(s) of k and l. By analyzing these restrictions, we can derive Theorem 2. ❚ In preparation for the statement of Theorem 3, we introduce a standard form for each box-style tetrachordal K-class C. It can be proven that C is generated by a unique Knet diagram whose transpositional arrows Tm and Tn satisfy the following conditions: m n are both non-negative integers, both transpositional arrows are oriented horizontally from left to right as in Example 1b, the Tm arrow is on the bottom, and either. a. b.

m v 0 and m + n 12; or m = 0 and n 6.

We state that C is a member of the box-style tetrachordal K-family indexed by (m, n). From the restrictions on possible values of m and n, we can deduce that there are forty-three distinct box-style tetrachordal K-families. We now state Theorem 3. Theorem 3. Let C be a box-style tetrachordal K-class, and let G be its group of symmetries. Suppose that the transformational operators that specify C are Tm, Tn, Ij, and Ij+m+n, where j 11 is a non-negative integer and the ordered pair (m, n) indexes the standard form of C. a. If m + n is odd and n v 6, then G = {T0, T6}. b. If m + n is odd and n = 6, then G = {T0, T3, T6, T9}. c. Suppose m + n is even, and choose r such that m + n = 2r. If n v 6, then G = {T0, T6, Ij+r, Ij+r+6}. d. Suppose m + n is even, and choose r such that m + n = 2r. If n = 6, then G = {T0, T3, T6, T9, Ij+r, Ij+r+3, Ij+r+6, Ij+r+9}. Informal Sketch of Proof. Unfortunately, it does not seem possible to use the same heuristics to discover a proof for Theorem 3 as we did for Theorem 2. To illustrate one difficulty, observe that the I11/I5 K-class from the box-style tetrachordal T1/T5 K-family consists of pitch-class multisets that each contain inversional and transpositional dyads with the same interval class. Thus, we cannot easily rule out the possibility of a symmetry for a box-style tetrachordal K-class mapping a transpositional dyad onto an inversional dyad. However, we can make progress by considering the general form of a member of C: [a, a + m, j – a, j – a + n]. Observe that the sum of the pitch classes in each member of C is the same, namely 2j + m + n. The only transpositional operators that preserve this sum are T0, T3, T6, and T9. So, none of the other transpositional operators can possibly be a symmetry of C. By exploring the invariant sum property in conjunction with general forms for transpositional symmetries Tk and in❚ versional symmetries Il, we can derive Theorem 3 through algebraic analysis. If S is a member of a trichordal K-class C, then S can be written in the form [a, b, a + n] for suitably chosen non-negative integers a, b, and n. In this representation, the

358


transpositional dyad of S is [a, a + n], and the inversional dyads are [a, b] and [b, a + n]. Klumpenhouwer specifically noted that strong isography is preserved throughout the progression of trichordal pitch-class multisets obtained by setting the transpositional dyad [a, a + n] in contrary motion to the singleton [b] (Klumpenhouwer 1991). This overall progression can be represented by [a + j, b – j, a + n + j] for j = 0, 1, 2, …, 11. This voice-leading context provides a basis to perceive aurally that T6 is a symmetry of every trichordal K-class: contrary motion by 6 semitones leads to the same pitch-class multiset as parallel motion by 6 semitones. This fact corresponds to the mathematical observation that [a + 6, b – 6, a + n + 6] is the same pitch-class multiset as [a + 6, b + 6, a + n + 6]. Is there a basis to perceive aurally the inversional symmetries that sometimes appear? In the trichordal case, suppose that the two voices of the transpositional dyad [a, a + n] are set in contrary motion towards each other. At the same time, suppose that the singleton voice b is split into two voices that move in contrary motion away from each other. This overall progression is represented by tetrachordal pitch-class sets [a + j, b – j, b + j, a + n – j] for j = 0, 1, 2, …, 11. The inversional dyads [a, b] and [b, a + n] of the original trichord correspond to the inversional dyads [a + j, b – j] and [b + j, a + n – j], respectively, in the progression of tetrachordal pitch-class sets. Observe that a + b = (a + j) + (b – j) and b + (a + n) = (b + j) + (a + n – j). Thus, even though the number of semitones separating the voices of the transpositional dyad [a + j, a + n – j] varies, we find that double axial isography is preserved throughout the progression. Moreover, if the pair of voices from the original transpositional dyad meet, then the number of semitones of motion j satisfies a + j = a + n – j or 2j = n. At the precise moment of meeting, the original singleton voice b will have expanded into a dyad [b – j, b + j] whose voices are (b + j) – (b – j) = 2j = n semitones apart. We obtain a trichordal pitch-class set [b – j, a + j = a + n – j, b + j] that belongs to the same Kclass as [a, b, a + n]. However, the roles of the voices have switched, and the overall correspondence of the pitch-class sets manifests the inversional symmetry Ia+b+j. Since n = 2j is even and a + b + j is the average of a + b and a + b + n, this disclosure of Ia+b+j as an inversional symmetry is consistent with Theorem 2. A similar basis for the aural perception of inversional symmetries of box-style tetrachordal K-classes is also available. The key idea for this voice leading is to set the two voices of one transpositional dyad in contrary motion towards each other and to set the two voices of the other transpositional dyad in contrary motion away from each other. If these pairs of dyads exchange their transpositional distances, then the resulting tetrachordal pitch-class multiset will be an inversional image of the original tetrachord that belongs to the same K-class. We emphasize that our voice-leading model does not preserve strong isography among the tetrachordal pitch-class multisets obtained in the progression because the corresponding transpositional dyads do not have identical interpretations. However, double axial isography is preserved through the unfolding of the two wedge formations between the individual voices. This fact provides the basis for analytical application of various linear relationships in the arrays of Igor Stravinsky, and it also extends the theoretical framework developed by Philip Stoecker in his work on (single) axial isography (Stoecker 2002). In this context, recall that two trichordal K-nets are axially isographic if they have an inversional operator in common.

Example 4. Tetrachordal sets in Stravinsky’s Threni (mm. 405-417)


359

360


Example 4 presents the concluding chorale from Stravinsky’s Threni, measures (405-417) as analyzed by Joseph Straus (2001) in his text Stravinsky’s Late Music. Example 5 demonstrates the four-part pre-compositional array Stravinsky employed to generate the harmonies of the chorale. The four-part array consists of a series of twelve vertical columns and four horizontal rows. Each horizontal row contains a complete statement of the four primary row forms, designated P, I, R and IR respectively. Each vertical column of the array consists of a collection of four pitch-class multisets that Stravinsky employs as the harmonies of the chorale, evident by the segmentations indicated by Straus. The twelve sonorities of the array are evenly distributed between two phrases that combine to form a melodic period (see Example 5). Phrase 1, performed by the chorus (mm. 405-408), contains sonorities 1-7. Phrase 2, performed by the soloists (mm. 408-411), contains sonorities 7-12. The cadence in measure 408 establishes a formal articulation that defines the binary phrase structure of the passage. Continuity is established by use of harmonic dove-tailing with the final sonority of phrase 1, set 7, also serving as the initial sonority of phrase 2. In the second period of the chorale (mm. 412-419), the same binary phrase structure recurs with the twelve sonorities of the array now appearing in retrograde order. Example 5. Four-part array from Stravinsky’s Threni (mm. 405-417)

A diverse collection of sonorities appear within the array as shown in Example 5. These include seven distinct set-class types representing six trichords, five tetrachords and a single dyad {(03), (0235), (015), (026), (0127), (0347), (0369)}. Isolated “pockets” of harmonic correspondence result from the repetition of the trichordal set class (026) in order positions 4 and 5 and again in order positions 8 and 9 of the array. In both instances, the transformation that maps the two subclasses onto one another is I9. Similar instances of set-class equivalence include the (015) trichords that appear in order positions 3 and 6, and the (0127) tetrachords present in order positions 7 and 10. In both cases, the transformation that maps each respective subclass onto the other is also I9. The distribution of these various set-class types establishes a simple symmetrical pattern that provides an important source of harmonic shaping and reinforces the binary phrase structure of the passage. While successful in characterizing the individual sonorities that comprise the array and revealing instances of harmonic correspondence between them, set-class theory remains incapable of disclosing the structural forces responsible for promoting unity and coherence between all the sonorities within the array.


361

Example 6. Double Axial Isography in Stravinsky’s Threni, (mm. 405-417)

Additional forms of correspondence demonstrated by Straus prove more rewarding. Straus illustrates how Stravinsky’s four-part array can be divided along a horizontal axis in order to produce pairs of inversionally related dyads as demonstrated in Example 6. For each respective order position within the array, pitch classes contained in the P and I row forms sum to the common index sum 6, whereas all pitch classes in the R and IR row forms sum to the index sum 0. The succession of inversionally related pairs of dyads establishes an important source of abstract unity and consistency amongst the sonorities of the array in pitch-class space. Different organizational features are revealed when we shift focus from consideration of dyads to sets of larger cardinalities. Example 6 interprets the twelve sonorities of the array as a series of box-style tetrachordal K-nets. In order to maintain the important inversional relationships reflected within the array, the I-related dyads remain paired in the subsequent K-net interpretation. Accordingly, the first dyad consisting of elements of the P and I row forms appears as the left-hand inversional dyad within the tetrachordal box-style graph configuration and the second dyad that includes pitch classes from the R and IR row forms appears as the right-hand inversional dyad in each box-style tetrachordal graph. Limited instances of isographic correspondence are apparent within the succession of tetrachordal K-nets. Graphs 11 and 12 are related by strong isography at . The strong correspondence exhibited between these sonorities provides a stark contrast within the otherwise disparate succession of unrelated K-classes. As a result of the retrograde distribution of array sonorities, these same K-classes recur as the initial sonorities of the first phrase of the second period (mm. 412-417). Thus, strong isography characterizes both the termination of the first melodic period (graphs 11 and 12) and the onset of the second period (graphs 12 and 11). These correspondences emerge as important aural markers within the harmonic design of the passage. Moreover, the harmonic overlap observed between melodic periods reflects, on a larger structural

362


level, Stravinsky’s previous use of harmonic dovetailing apparent in the repetition of graph 7 located at the juncture between phrases 1 and 2 in first melodic period. Graphs 4 and 5 are also related by strong isography, as well as by subclass affiliation as members of the same (026) trichordal set-class. Apart from these localized instances, however, traditional forms of isographic correspondence are unable to establish a broader context in which to characterize all the sonorities of the array as a unified, as opposed to diffuse, succession of harmonies. As a consequence of inversional symmetry, we observe that each of the twelve tetrachordal K-nets share an identical collection of I-operators as demonstrated in Example 6. Each graph contains an I6 operator located on the left-side of the graph, and an I0 operator on the right-side. The common I-operators typify the large-scale double wedge voice-leading motion that connects each sonority within the array to its neighbors. In each case, the successions of dyads that sum to 6 establish one wedge motion, and those dyads that sum to 0 establish the remaining wedge motion. Whereas Stoecker’s original formulation of single axial isography preserves one inversional operator between two trichordal K-nets, double axial isography preserves both of the inversional operators between two box-style tetrachordal K-nets. While conventional isographies require an angle of hearing that emphasizes the presence of invariant, or in the case of negative isography, complementary, transpositional operators, double axial isography requires us to shift orientation in order to focus on the invariant inversional operators present in each tetrachordal K-net. The invariant inversional operators function as dual inversional axes with each successive tetrachordal pitch-class set providing a different rotation of pitch classes about the inversional axes. Our notation for double axial isography, as it appears in Example 6, is an operator designated by where the terms, x and y, correspond to the difference between respective top and bottom T-operators between contiguous graphs. These inversional relationships are manifest in pitch-class space in the form of double wedge voice-leading patterns that incorporate all sonorities of the array as members of a single large-scale voice-leading configuration. While successful in revealing a linear perspective of how the sonorities within the array are connected, axial isography provides little information regarding the vertical organization of the sonorities and the factors that establish harmonic affiliations between them. Consideration of the groups of algebraic symmetries that generate the pitch-class sets associated with individual K-class representatives provides additional perspective concerning the harmonic organization of the array sonorities. Whereas axial isography is construed on the basis of identical I-labels reflected between network graphs, correspondences involving algebraic symmetries are contingent upon common inversional sums apparent when the corresponding I-operators of one graph sum to the same amount as the I-labels of another graph. Because each graph in the array from Threni consists of identical pairs of inversional operators (I0 and I6), their corresponding sums will consequently be the same: 6. However, in other cases, pairs of I-operators belonging to discrepant K-classes will differ while their respective sums remain the same. For instance, despite their different labels, we know from Theorem 3 that the group of symmetries for each of the following Kclasses contains at least four common operators: I1/I5, I2/I4, I3/I3, I4/I2, I5/I1, I6/I0. In each case, the sums of respective I-operators sum to 6. Specifically, the group of symmetries for each representative K-class contains the operators T0, T6, I3, and I9.


363

The single exception is presented in graph 9 (see Example 6). Due to the high degree of internal symmetry inherent in this pitch-class set [T, 2, 4] and its corresponding K-net interpretation (K-class I0/I6), additional symmetrical correspondences are also manifest. In addition to the operators {T0, T6, I3, I9} that are shared between all K-classes in the example, this particular set also includes the additional operators {T3, T9, I0, I6} and thus manifests the fourth case stated in Theorem 3, part d. Despite the presence of these additional operators, all sonorities in the array are unified by the fact that every K-class consists of the subgroup of common symmetries {T0, T6, I3, I9}. This unity extends beyond the fact that only T0 and T6 are guaranteed as common operators. The result is an elegant algebraic unity that transcends traditional K-class distinctions by establishing a new theory of harmonic correspondence based on the presence of equivalent transformational operators, as opposed to equivalent objects. The specific pitch-class collections that appear in the array from Threni represent one of many possible transformational pathways within the larger harmonic network characterized by the common group of symmetries. Any given transformational pathway within the network can potentially employ all of the transformational operators included in the algebraic subgroup, fewer, or even none in cases where only a single set-class representative is present in the particular pathway realization. All potential pathways, however, will involve a voice-leading design consisting of the double wedge motions characteristic of axial isography. In the array from the Threni excerpt, Stravinsky elected to navigate a transformational pathway that emphasizes the I9 operator within the greater harmonic space informed by the subgroup of common symmetries {T0, T6, I3, I9}. Of the twelve sonorities present in the array, there are four instances of set-class duplication. Order positions 4, 5, 8 and 9 present (026) trichordal subclasses related by I9. The (0127) tetrachords that appear in order positions 7 and 10 are also related by I9 as are the (015) trichordal subclasses that appear in order positions 3 and 6. The four remaining sets in order positions 1, 2, 11 and 12 employ only the identity transformation T0 due to the fact they are not affiliated by subclass relation with any other member sonority of the array. Axial isography and correspondences involving subgroups of common symmetries go hand in hand. Axial isography provides a model that characterizes how successive sonorities within the array are connected through wedge-voice leading pathways in pitch-class space. Together, axial isography and subgroup correspondences combine to establish a means in which to confirm our musical intuitions of Stravinsky’s chorales as distinct, unified wholes. Similar organizational features are reflected in the arrays that appear in numerous other works by Stravinsky including Movements, Agon, Epitaphium, and The Flood amongst others. Acknowledgements. Jerry G. Ianni wishes to acknowledge two sources of grant support at LaGuardia Community College for the research presented in this paper: the Educational Development Initiative Team and the Division of Academic Affairs. Both authors are very grateful to Philip Lambert for many helpful comments and suggestions. In addition, we also gratefully acknowledge the valuable comments, suggestions, and insights offered by Norman Carey, David Headlam, Patty Howland, Robert Morris, Thomas Noll, and Joe Straus.

364


References Armstrong, M.A.: Groups and Symmetry. Springer, NewYork (1988) Klumpenhouwer, H.: A Generalized Model of Voice-Leading for Atonal Music. Ph.D. dissertation, Harvard University (1991) Lambert, P.: Isographies and Some Klumpenhouwer Networks They Involve. Music Theory Spectrum 24(2), 165–195 (2002) Lewin, D.: Klumpenhouwer Networks and Some Isographies that Involve Them. Music Theory Spectrum 12(1), 83–120 (1990) Lewin, D.: A Tutorial on Klumpenhouwer Networks Using the Chorale in Schoenberg’s Op. Journal of Music Theory 11(2) 38(1), 79–101 (1994) O’Donnell, S.: Klumpenhouwer Networks, Isography, and the Molecular Metaphor. Integral 12, 53–80 (1998) Stoecker, P.: Klumpenhouwer Networks, Trichords and Axial Isography. Music Theory Spectrum 24(2), 231–245 (2002) Straus, J.N.: Stravinsky’s Late Music. Cambridge University Press, Cambridge (2001)

K-Net Recursion in Perlean Hierarchical Structure Gretchen C. Foley University of Nebraska-Lincoln [email protected]

1 Introduction Klumpenhouwer networks, or K-nets, are graphic representations of the intervallic relationships among elements of a set.1 Theorist David Lewin has suggested that Knets may be applicable to Perle cycles, entities referred to by George Perle in his theory of twelve-tone tonality as cyclic sets. These entities are created through the alternation of inversionally related interval cycles. The present study seeks to broaden the applicability of K-nets in Perle’s theory by exploring their recursive nature at varying levels of structure.

2 K-Nets and Perle Cycles The interval cycle provides the foundation for Perle’s system of twelve-tone tonality. Perle uses two inversionally related interval cycles to form a cyclic set, hereinafter referred to by the more recently coined term Perle cycle. Ex. 1 shows one such Perle cycle formed by inversionally related interval 7 cycles. In this formation any given pitch class (pc) forms a pair of sums with its neighbor notes. These sums repeat throughout the Perle cycle and are called tonic sums. The tonic sums provide the Perle cycle with its name; the Perle cycle in Ex. 1 is 2,9. Further, the cyclic interval itself may be determined by subtracting the first tonic sum from the second (tonic sum 9 – tonic sum 2 = 7, the cyclic interval). Example 1. Perle cycle 2,9

Although the two bracketed trichordal segments in the example belong to two different set classes (scs) – sc 3-9 (027) and sc 3-4 (015), respectively – they share similarities in their internal structure since they both derive from the same symmetrical Perle cycle. Fig. 1 displays these segments graphically as K-nets in order to view their internal relationships. 1

Lewin named these networks after his former student Henry Klumpenhouwer, who extended Lewin’s original conception of networks to include inversional as well as transpositional relationships (Lewin 2003).


366

G.C. Foley

Fig. 1. K-nets of trichordal segments from Perle cycle 2,9

The nodes of the K-nets are filled with the pcs of the two segments, and the arrows connecting the nodes show the specific relationships between them. The first two networks of Fig. 1 are strongly isographic because their graphs are isomorphic: they have identical node and arrow configurations, with the same transformations associated with corresponding arrows. The T7 value represents the cyclic interval 7, and the I2 and I9 values represent the tonic sums formed by the adjacent pcs in the Perle cycle. Thus, despite the fact that the two segments are members of different scs, they both comprise identical internal relationships, as uncovered by the K-nets. Moreover, any pair of trichordal segments from the particular Perle cycle of Ex. 1 can be shown to be strongly isographic. The third network in Fig. 1 displays this recursion in the more abstract K-graph; the arrows indicate the transformational operations shared by all trichordal segments of the Perle cycle. Any three adjacent pcs from this Perle cycle can be fitted into the nodes of this K-graph. The alignment of the ascending and descending interval 7 cycles in Ex. 1 is one of twelve possible alignments. The interval cycles may rotate in relation to one another, creating a different set of repeating tonic sums. Rotating the descending interval 7 cycle twice results in a new Perle cycle, identified in Ex. 2 as 0,7. Example 2. Perle cycle 0,7

The K-nets of all trichordal segments from this new Perle cycle will be strongly isographic with one another. However, a comparison of these bracketed segments with those from Ex. 1 reveals that their K-nets are not strongly isographic, due to the new tonic sums resulting from the shifted cycle. Instead, the K-nets are said to be positively isographic. In Fig. 2 K-graphs representing trichordal segments from both Perle cycles show identical T values between corresponding nodes, but with the consistent difference of ten (t) in their corresponding I values. Lewin expressed this relationship with the symbol .2 This value also represents the difference

2

Lewin (2003, 88).

K-Net Recursion in Perlean Hierarchical Structure

367

Fig. 2. relationship between Perle cycles 2,9 and 0,7

between the corresponding tonic sums of the two Perle cycles, as given below the K-graphs.3

3 K-Nets, Arrays, and Axis-Dyad Chords Any two Perle cycles in a vertical alignment form an array, which takes its name from the combined Perle cycles; in Ex. 3 two aligned Perle cycles form the array 2,9/0,7. Furthermore, just as the interval cycles within a Perle cycle are not fixed in relation to one another, the Perle cycles themselves may rotate within the larger array. Such rotation creates two types of vertical alignments between the Perle cycles. Ex. 3 illustrates the first type, in which the corresponding ascending and descending forms are aligned. Perle refers to this type as a difference alignment. In a sum alignment (not shown), the ascending cycle of one Perle cycle is aligned with the descending cycle of the other. Example 3. Array 2,9/0,7

The array functions as a resource from which to reap melodic and harmonic material. Perle segments arrays into units of varying size; the main unit is the axisdyad chord, a hexachordal collection formed by combining trichordal segments from each of the Perle cycles. In the array above, one such axis-dyad chord may be

3

Lewin identified one other relationship exhibited by K-nets, negatively isographic networks. Sets in such relationships have networks of identical node and arrow configurations, but their T values are complementary rather than identical, and their corresponding I values show the same sum rather than the same difference. Lewin symbolized negatively isographic networks as ; here n equals the sum of the analogous I arrows.

368

G.C. Foley

constructed by two vertically aligned trichords 7 7 2 and 7 5 2.4 If the 0,7 Perle cycle were rotated two places to the left, the resulting axis-dyad chord would comprise trichords 7 7 2 and 2 t 9. Due to the symmetrical nature of the array, all axis-dyad chords from the same array in the same type of alignment (either difference or sum) will be isomorphic. The first K-graph in Fig. 3 represents the recursive fit of all axis-dyad chords drawn from that same array’s difference alignment, while the second K-graph represents the recursive fit of all other axis-dyad chords drawn from that array’s same sum alignment. Chords with strongly isographic K-nets may also be generated by symmetrical rotation, that is, moving the upper and lower trichordal segments by the same amount in the opposite direction. Hence, the dotted lines connecting the nodes in the upper and lower trichords indicate the variable status of the two Perle cycles of the array in relation to one another. Conversely, axis-dyad chords resulting from an asymmetrical shifting of the Perle cycles instead will have positively or negatively isographic K-nets. The latter (not shown) comprise complementary I values between corresponding nodes and identical sums formed by corresponding T values.5

Fig. 3. K-graphs of axis-dyad chords in difference and sum alignments

4 K-Nets and Array Relationships The closest relationships among arrays are those of transposition and inversion. Transposing an array entails adding a constant even integer to each of the four tonic sums. For example, the arrays 0,7/4,e and 4,e/8,3 are related by T4, since 4 is added to each of the tonic sums. At the same time, transposed arrays also retain the same cyclic intervals, which together form the array’s interval system. The interval system for the two transpositionally related arrays mentioned above is 7,7. Consequently, all arrays related by transposition will share strongly isographic relationships among their tonic sums, as shown in Fig. 4. 4

Perle recognizes repetitions of a pc within a segment as independent entities rather than multiple instances of a single member of a collection. Since any collection may include pc duplication, a hexachordal collection in Perle’s terms may contain less than six distinct elements. The same obtains for dyads, trichords, and tetrachords. 5 For more information on these relationships, see Foley (2002).


369

Arrays related by inversion share the same sum between corresponding tonic sums. In addition, their interval systems are complementary rather than identical. In Fig. 5 the array 0,7/4,e comprises interval system 7,7, while the inversionally related array 2,7/t,3 is built on the interval system 5,5. Fig. 5 also demonstrates that inversionally related arrays will have negatively isographic K-nets.

Fig. 4. Strong isography between tonic sums of transpositionally related arrays

Fig. 5. Negative isography between tonic sums of inversionally related arrays

5 K-Nets, Interval Systems, Modes, and Keys Every array contains three pairs of tonic sums, the primary, secondary, and tertiary tonic sum pairs, formed by the horizontal, vertical, and diagonal pairs, respectively. Fig. 6 represents these three pairs as K-nets. In the first K-net, Fig. 6a, the interval system of array 0,7/4,e is realized as the intervallic distance between the tonic sums in each horizontal pair, namely 7,7. All arrays with the same interval system of 7,7, such as array 2,9/1,8 or array 6,1/5,0 can be fitted into the corresponding K-graph below the first K-net. Arrays of differing interval systems may also be related, via modes and keys. These relationships establish connections across the Perle cycles within the array. The

370

G.C. Foley

Fig. 6 a-c. Three pairs of tonic sums in array 0,7/4,e

mode reflects the difference between the corresponding tonic sums between the two Perle cycles. In the second K-net, Fig. 6b, array 0,7/4,e has a modal designation of 4,4 (as 4-0 and e-7). Arrays of any combination of interval systems may belong to the same mode if they preserve the same differences between their corresponding tonic sums. For example, the array e,2/3,6 (interval system 3,3) also belongs to mode 4,4, as does array 0,1/4,5 (interval system 1,1). All such arrays whose corresponding tonic sums differ by four may recur in Fig. 6b’s corresponding K-graph of mode 4,4. In the same way, mode 0,0 contains all arrays whose tonic sums differ by zero, and so on. A mode with two different elements in its name, such as mode t,8, results from arrays whose interval systems comprise different cyclic intervals, such as arrays 1,6/e,2 (interval system 5,3) and 1,7/e,3 (interval system 6,4). Each array may belong to only one mode, of which there are 144 in the universe of twelve-tone tonality. The key reflects the sums of symmetrically aligned tonic sums between the two Perle cycles in an array. The third K-net, Fig. 6c, shows how array 0,7/4,e has a key designation of e,e (0+e and 7+4). As with modes, keys may relate arrays of the same or different interval systems. Therefore, key e,e comprises all arrays whose oppositely aligned tonic sums sum to eleven, such as arrays 3,6/5,8 (interval system 3,3) or 4,5/6,7 (interval system 1,1). The corresponding K-graph of Fig. 6c represents all such arrays in key e,e. Likewise, key 7,3 contains all arrays whose symmetrically aligned tonic sums add up to 7 and 3, such as array 1,t/5,6 (interval system 9,1), all of which can be fitted into a K-graph whose diagonal arrows indicate I7 and I3 transformations. Each array may belong to only one key, of which there are 144 in total.

6 K-Nets and Synoptic Arrays As described above, relationships among arrays include transposition, inversion, and membership in the same mode or key. Yet collections of arrays lacking these associations may be related at more fundamental levels. Perle establishes such


371

connections in his concepts of synoptic arrays, in which reside arrays with related interval systems. A synoptic array divides into two categories, synoptic modes and synoptic keys.6 Just as how a mode includes all those arrays whose individual tonic sums in the component Perle cycles differ by the same value, a synoptic mode includes all those arrays whose cyclic intervals between the tonic sums differ by the same value. There are seven different synoptic modes, numbered from 0 to 6, determined by the interval class difference between the two cyclic intervals in the interval system. Fig. 7 illustrates the array 0,7/4,e and its interval system of 7,7 in the first K-net. The difference between the cyclic intervals in this interval system is zero, and so the array belongs to synoptic mode 0. Likewise, other arrays in synoptic mode 0 display a difference of zero between their cyclic intervals. For example, the interval system of array 5,6/3,4 is e,e and the interval system for array 1,4/6,9 is 3,3. The more abstract K-graph to the right of the K-net in Fig. 7 represents the recursion of all arrays in synoptic mode 0. Consequently, all arrays in the same synoptic mode have interval systems related by transposition. Each of the seven synoptic modes represents a large grouping of arrays. There are twelve combinations of interval systems that differ by 1; within each of these interval systems there are 144 arrays. Thus, each synoptic mode contains 1,728 (as 12x144) different arrays whose interval system differs by the same interval class.

Fig. 7. K-nets of arrays in synoptic mode 0

In contrast to the synoptic mode, a synoptic key includes all those arrays whose component cyclic intervals sum to the same value. Just as how a key comprises all arrays whose symmetrically-aligned tonic sums in the component Perle cycles add up to the same sum, a synoptic key comprises all those arrays whose cyclic intervals between the tonic sums add up to the same sum. There are seven synoptic keys, numbered from 0 to 6. While the arrays 0,7/4,e, 1,6/3,0 and 2,5/2,1 all have differing interval systems, their component cyclic intervals sum to the same amount, namely 6

Perle’s nomenclature is more precise, first dividing “master arrays” into master modes and master keys, to distinguish between arrays given in difference and sum alignments (Perle 1996, 103), then dividing “synoptic arrays” into synoptic modes, comprising master arrays with corresponding cyclic interval differences, and synoptic keys, comprising master arrays with corresponding cyclic interval sums (Perle 1996, 195).

372

G.C. Foley

Fig. 8. K-nets of arrays in synoptic key 2

sum 2, thereby relating the arrays via membership in synoptic key 2. These relationships are evident in the K-nets and K-graph of Fig. 8. As with the synoptic mode, there are 1,728 different arrays in each synoptic key.

7 K-Nets and Tonality Tonality represents the most abstract relation among arrays in Perle’s theory, and is based on the concept of the axis of symmetry. Any symmetrical collection of pcs theoretically contains an axis around which the various pcs are symmetrically positioned. The smallest such collection is a dyad. For every even sum n the two axial points are calculated as n/2 and n/2 + 6. For every odd sum n the two axial points are calculated as (n±1)/2 and ((n±1)/2) + 6. Arrays represent concatenations of dyads; therefore arrays with the same aggregate sum will be symmetrically related and have the same axis of symmetry. Arrays so related fall into one of three tonalities as categorized by Perle. Tonality 0 includes all arrays whose tonic sum aggregates are 0, 4, or 8. The axial points consist of two repeated even integers, and so are simply transpositions of one another by an even value of Tn, as shown in Fig. 9. Tonality 1 constitutes all arrays whose tonic sum aggregates are represented by an odd integer. Their axes are also transpositionally related, and include one even and one odd integer that differ by 1.7 Third, all arrays whose tonic sum aggregates are 2, 6, or t belong to Tonality 2. Their axes consist of a pair of repeated odd integers, also transpositionally related by an even value of Tn. Although Tonality 0 and Tonality 2 both contain arrays of even aggregate sums, the arrays of the former are not symmetrically equivalent to those of the latter, because their axes are from even/even and odd/odd integer pairs, respectively. Perle contends that all arrays from the same category of aggregate sum belong to the same tonality and share the same axis of symmetry. Thus, the symmetrical relations extend from pcs within a single dyad to massive groupings of arrays. 7

Perle further subdivides Tonality 1 based on transpositional equivalence: the first encompasses arrays with aggregate sums 1, 5, and 9; the second, arrays of aggregate sums 3, 7, and e (Perle 1996, 141).


(a)

(b)

(c) Fig. 9. (a) Tonality 0. (b) Tonality 1. (c) Tonality 2.

373

374

G.C. Foley

8 Summary This study focused on the relational aspects of Perle’s theory, from the most basic level of Perle cycles through increasingly abstract entities, culminating in Perle’s conception of tonality. The recursion of K-nets effectively demonstrated the structural integrity at the core of Perle’s twelve-tone tonality and infused in all its hierarchical layers.

References Foley, G.C.: Arrays and K-Nets: Transformational Relationships Within George Perle’s Theory of Twelve-Tone Tonality. Indiana Theory Review 23, 69–97 (2002) Lewin, D.: Thoughts on Klumpenhouwer Networks and Perle-Lansky Cycles. Music Theory Spectrum 24(2), 196–230 (2003) Perle, G.: Twelve-Tone Tonality, 2nd edn. University of California Press, Berkeley (1996)

Webern’s Twelve-Tone Rows through the Medium of Klumpenhouwer Networks Catherine Nolan University of Western Ontario [email protected]

The theory of Klumpenhouwer networks (K-nets) in contemporary music theory continues to build on the foundational work of David Lewin (1990) and Henry Klumpenhouwer (1991), and has tended to focus its attention on two principal issues: recursion between pitch-class and operator networks and modeling of transformational voice-leading patterns between pitch classes in pairs of sets belonging to the same or different Tn/TnI classes.1 At the core of K-net theory lies the duality of objects (pitch classes) and transformations (Tn and TnI operators and their hyper-Tn and hyper-TnI counterparts). Understood in this general way, K-net theory suggests other avenues of investigation into aspects of precompositional design, such as connections between Knets and Perle cycles, K-nets and Stravinskian rotational or four-part arrays, and between K-nets and row structure in the “classical” twelve-tone repertoire. K-net theory and analytical application are contingent on consistently textured musical spaces. That is to say, isographic relations depend in the first instance on graphs of the same size, which in turn reflect some basic musical consistency, such as the same number of voices in a musical passage, the same number of transformations in multi-level network recursions, or the same size of pitch-class segments in a cycle or row. The twelve-tone rows of Webern, known for their high degree of intervallic control, invite closer investigation of their even partitions from a transformational perspective. K-nets offer an effective technology for such an investigation because of their predisposition toward trichords and tetrachords, and because of their capacity for revealing dynamic relations within and between equivalent and non-equivalent pitchclass sets of the same size. The transformational as opposed to taxonomic perspective on relations between successive row segments offers new insights into the integrity of Webern’s rows, and poses beneficial questions about K-net interpretations of aggregate partitions for future investigation. This essay focuses on twelve-tone rows of Webern whose disjunct trichords and tetrachords belong to one or two set classes, and represents the rows as successions of three- or four-node K-nets in order to examine properties and constraints of network isographies. For the most part, I refrain from discussing Webern’s instantiations of the rows in his twelve-tone works, and focus on the implications of their K-net representations. That is, the K-net representations of Webern’s rows are, in general, considered for their abstract properties rather than as models of specific passages of music. 1

Tn/TnI classes refer to the equivalence classes of pitch-class sets related by transposition (Tn) and/or inversion (TnI).


376

C. Nolan

Rows whose disjunct segmental trichords or tetrachords belong to a single set class Three of Webern’s rows are composed of segmental trichords or tetrachords belonging to a single set class: the row for the Concerto for Nine Instruments, Op. 24; the row for the String Quartet, Op. 28; and the row for the First Cantata, Op. 29. The row for Webern’s Concerto for Nine Instruments, Op. 24, is derived from a generating 3-3 [014] trichord, and the trichords are internally ordered so as to relate to one another by the four serial transformations. The row can be interpreted as a series of isographic three-node K-nets in three different ways, each reflecting one of the trichord’s three interval classes as a transposition operator, as shown in Figure 1.

(a)

(b)

(c) Fig. 1. Op. 24 row: K-net interpretations of disjunct trichords

The succession of graphs in Figure 1a shows the shared T4-arrow in each network, and its networks are all strictly identical, or strongly isographic. The successions in Figure 1b and 1c preserve a T1- and T3-arrow in each network respectively, and their constituent networks are positively, not strongly, isographic. In Figure 1b and 1c,

Webern’s Twelve-Tone Rows through the Medium of Klumpenhouwer Networks

377

pairs of complementary and operations separate alternating strongly isographic networks.2 The first and third graphs in Figure 1b belong to the same Kclass, as do the second and fourth graphs, while successive graphs do not, and likewise for the graphs in Figure 1c. Philip Lambert employs the K-net representation shown in Figure 1a to model the opening statement of the first movement of Op. 24, and points to the registral separation of pitches of the T4 dyad from the remaining pitch in each trichord to support this segmentation. (See Lambert 2002, 174-76.) While the correspondence between registral separation and K-net construction is convincing, I would argue that the K-net representations shown in Figures 1b and 1c show equally convincing correspondences with musical parameters other than registral separation. For example, in the same passage, the T1 dyads of Figure 1b articulate the total registral space occupied by the trichord, and the T3 dyads of Figure 1c articulate each trichord’s total durational space. We must consequently question whether a succession of strong isographies such as that shown in Figure 1a is somehow phenomenologically or otherwise superior to a succession of positive isographies as shown in Figures 1b and 1c. The row for the String Quartet, Op. 28, is similarly derived from serial transformations of a generating 4-1 [0123] trichord, and can be represented in three different ways as a series of four-node K-nets, as shown in Figure 2.3 Each network includes a disjunct pair of the six intervals of the 4-1 tetrachord indicated by the two transposition operators. While the individual networks are configured differently in each representation, each succession displays a series of positively isographic relations. The singular positive isographic type in the three K-net representations of the row for Op. 28 seems appropriate given the limited intervallic resources of the 4-1 [0123] row segments, but the absence of strong isography between adjacent or even alternate networks may seem at odds with the K-net representations of the derived row for Op. 24, given the similar generation of the two rows. Like the row for Op. 24, the disjunct segmental trichords of the row for the First Cantata, Op. 29, all belong to set class 3-3 [014], but in the Op. 29 row the transformations do not extend to order relations within the trichords.4 (This, of course, has no effect on K-net formation.)Yet the disjunct trichords of the Op. 29 row can again be represented in three different ways as a succession of isographic three-node K-nets, as shown in Figure 3. 2

The abbreviations “t” and “e” are used for 10 and 11 respectively. Wherever possible, I show only strong or positive isographies, and invoke negative isography only where strong or positive isography does not obtain. The disjunct tetrachords in Figure 2a, for example, because of the inversional as well as transpositional symmetry of the 4-1 set type, can be reconfigured to show a succession of followed by network isographies. Adjacent disjunct trichords in Figure 1a are inversionally related; hence alternate networks can similarly be reconfigured to show a succession of network isographies. 4 Despite the superficial similarity of the rows for Op. 24 and Op. 29 as aggregate partitions of the same type (34) and their identical set-class membership, they belong to different mosaics, because the mosaics are unrelated by Tn or TnI. Mosaics that are generated by the same set class(es), but are distinct, are described by Morris and Alegant as Z-related. (See Morris and Alegant 1988, 79 - 81.) The Op. 24 and Op. 29 rows also have different degrees of symmetry; the Op. 24 row has a degree of symmetry 4 (the unordered pitch-class content of its disjunct trichords is preserved under four Tn or TnI operations), while the Op. 29 row has a degree of symmetry 2. 3

378

C. Nolan

(a)

(b)

(c) Fig. 2. Op. 28 row: K-net interpretations of disjunct tetrachords

Unlike the network isographies of the disjunct trichords of the Op. 24 row shown in Figure 1, however, no representation results in a succession of strongly isographic networks. The representation shown in Figure 3a, however, repeats the pattern of alternating and isographies seen in Figures 1b and 1c, reflecting the membership of the first and third trichords in the same K-class, and likewise the second and fourth trichords. In Figures 3b and 3c a new pattern of network isographies emerges in the recurrence of the operation between the first and second network and between the third and fourth network, a consequence of the T2 relation between corresponding pitch classes. While all the K-nets in Figure 3b and 3c belong to the same family, no pairs of K-nets belong to the same K-class. (See Lambert 2002.)


379

(a)

(b)

(c) Fig. 3. Op. 29 row: K-net interpretations of disjunct trichords

Since any pair of pitch-class sets belonging to the same set class can be modeled by isographic K-nets, the exclusivity of strong and positive isography in the K-net representations of the rows just discussed is to be expected. The perception of a more direct phenomenological viability when networks are related by strong rather than by positive isography comes into question, however, when one considers multiple K-net representations of the same objects in which contrasting types of isography may be equally convincingly supported by musical intuition. Rows whose disjunct segmental tetrachords belong to two set classes The rows for the Symphony, Op. 21, the Variations for Orchestra, Op. 30, and the First Cantata, Op. 29, share not only a pattern of set-class membership in their disjunct segmental tetrachords, but also, as mosaics, the same degree of symmetry. Each of these rows shares a simple a b a pattern in the set-class membership of its constituent tetrachords: that is, the first and last tetrachords belong to one set class, while the middle tetrachord belongs to a different set class.

380

C. Nolan

The row for the Symphony, Op. 21, is unique in that any K-net representation of its disjunct tetrachords (belonging to set classes 4-1 [0123], 4-9 [0167], and 4-1 [0123]) exhibits only strong isography. Figure 4 shows the succession of strongly isographic four-node K-nets whose complementary T1 and Te arrows reflect the only interval class shared by the two set classes. Each network can be reconfigured to show two T1arrows, which results in contrasting I-arrows with no change to the strong isography between successive networks.

Fig. 4. Op. 21 row: K-net interpretation of disjunct tetrachords

(a)

(b) Fig. 5. Op. 30 row: K-net interpretations of disjunct tetrachords

The row for the Variations for Orchestra, Op. 30, is constructed similarly to the row for Op. 21 in terms of the pattern of set-class membership of its disjunct tetrachords (4-3 [0134], 4-17 [0347], 4-3 [0134]), but two different successions of isographic K-nets are possible, one built upon the sharing of two instances of interval class 3, the other built upon the common 3-3 [014] subset. The succession of


381

networks for the Op. 30 row shown in Figure 5a appears similar to that of the Op. 21 row shown in Figure 4, with the complementary T-arrows and identical I-arrows in each network, but here the adjacent networks are positively, not strongly, isographic. The succession of networks for the Op. 30 row in Figure 5b highlights the trichordal set class (3-3 [014]) that is a shared subset of set classes 4-3 [0134] and 4-17 [0347]. While the networks in Figure 5a and 5b bear different structural features and the network operators between successive network pairs differ, the composite operation relating the first to the third network in both cases is the same, : that is, + = + = . This, of course, correlates to the equivalence of set-class membership between the first and third tetrachords. The disjunct segmental tetrachords of the row for the First Cantata, Op. 29, display the same a b a pattern of set-class membership: 4-2 [0124], 4-1 [0123], and 42 [0124]. Two representations of four-node K-nets are possible, each highlighting one of the two shared subsets (3-1 [012] and 3-2 [013]). These are shown in Figure 6. Figure 6a reveals a succession of positively isographic networks, while Figure 6b reveals alternating positively and negatively isographic networks.

(a)

(b) Fig. 6. Op. 29 row: K-net interpretations of disjunct tetrachords

The presence of two set classes among the disjunct tetrachords in Webern’s rows results in K-net successions whose isographies range from exclusively strong isography to a repeating positive isography to contrasting positive isographies to contrasting positive and negative isographies in the same succession. The wide range of possibilities in such a small sample reflects the relatively large number of tetrachordal set classes and interrelations. (See Lambert 2002, 178.) Three rows that have been discussed because of the set class membership of their disjunct tetrachords, the rows for Opp. 21, 28, and 30, also possess disjunct trichords

382

C. Nolan

that similarly belong to only two set classes, and also display a symmetrical pattern, a b b a, in their succession. The rows for Op. 21 and Op. 30 consist of disjunct trichords belonging to the same two set classes in complementary ordering: 3-2, 3-3, 3-3, 3-2 in the row for Op. 21, and 3-3, 3-2, 3-2, 3-3 in the row for Op. 30. The disjunct trichords of the row for Op. 28 belong respectively to set classes 3-2, 3-4, 3-4, 3-2. Figures 7 and 8 show the two three-node K-net successions for the rows for Opp. 21 and 30, reflecting the two shared interval classes between set classes 3-2 and 3-3, and Figure 9 shows the single three-node K-net succession for the row for Op. 28, reflecting the single shared interval class between set classes 3-2 and 3-4. The K-net representations of the rows just mentioned reveal the recurrence of three isographic patterns that we have already observed: the association of strong isography with complementary and in the same succession, creating strong isography between adjacent and non-adjacent networks (Figure 7a and 7b); the alternation of two positive isographies (Figure 8a and 8b); and a succession of identical positive isographies (Figure 9).

(a)

(b) Fig. 7. Op. 21 row: K-net interpretations of disjunct trichords

Although I have referred only in passing to Webern’s actual musical settings of twelve-tone rows (particularly since Webern set single rows in linear fashion infrequently), I will briefly discuss in greater detail one setting from the first movement of the First Canata, Op. 29. Because of Webern’s proclivity for creating polyphonic spaces out of quartets of row transformations consisting of pairs of Tnand TnI-related rows, K-nets can provide an effective resource for interpreting the coherence of such passages, particularly in homophonic moments of extreme textural consistency.


383

(a)

(b) Fig. 8. Op. 30 row: K-net interpretations of disjunct trichords

Fig. 9. Op. 28 row: K-net interpretations of disjunct trichords

Figure 10a shows a score reduction of Op. 29, first movement, mm. 1-6. Figure 10b sketches the pitch-class content of m. 1, and 10c sketches the pitch-class content of m. 6. Each horizontal strand in 10b and 10c represents the opening 3-3 [014] trichord of two pairs of inversionally related rows, and the four 3-3 trichords in m. 6 complete the rows begun in m. 1. In both m. 1 and m. 6, the lower two strands are intoned by the cello and viola, while the upper two strands are intoned by the trombone and trumpet. The T- and I-arrows in the K-net representations reflect the transformational relation between the pairs of rows, linking the strong isography of the four-node K-nets representing three different set classes directly to the work’s compositional design. In m.6, the return to the homophonic texture, the instrumentation, and the pitch classes of the first sonority (with the exchange of dyads between the string and the brass pairs) announce a clear reference to the opening. The identical graphs of the six networks in the two passages reflect coherence of register and texture in a manner that is more concrete through its association with the textural setting than the relationships among the row transformations alone can express.

384

C. Nolan

(a)

(b)

(c) Fig. 10. Op. 29 (First Cantata): (a) mm. 1-6, score reduction; (b) K-net representations, m. 1; (c) K-net representations, m. 6


385

K-net theory has established itself as an important area of music theory, but it seems that it is still in its youth. Further investigation of properties of K-classes and families of four-node and larger networks is needed, as well as further consideration of the explanatory power of network isography. Also needed perhaps is the extension of the range of isographic types beyond the canonical strong, positive, and negative isographies. Michael Buchler’s recent critical study of K-net theory (Buchler 2007) raises important theoretical and analytical questions about K-nets and dual transformations, recursive hierarchies, and other issues, and the invited responses to Buchler’s study affirm the continuing currency and vibrancy of K-net theory. (See Foley 2007, Klumpenhouwer 2007, Losada 2007, Nolan 2007, O’Donnell 2007, Stoecker 2007, and Tymoczko 2007.) This brief study of Webern’s rows through the medium of Klumpenhouwer networks identifies the need for further investigation of the interaction of K-net formation and even partitions of the aggregate. The representations of Webern’s rows as successions of three- and four-node K-nets confirm the fluidity of set-class identification in connection with network isography, yet they also reinforce the significance of pitch-class subset content in assessing alternative K-net representations of the same objects, and raise questions about the perceived hierarchy of strong, positive, and negative isography. It is clear that the range of questions that arise through the continuing study and application of K-net theory will not recede any time soon.

References Buchler, M.: Reconsidering Klumpenhouwer Networks. Music Theory Online 13(2) (2007) Foley, G.: The Efficacy of K-Nets in Perlean Theory. Music Theory Online 13(3) (2007) Klumpenhouwer, H.: Reconsidering Klumpenhouwer Networks: A Response. Music Theory Online 13(3) (2007) Klumpenhouwer, H.: A Generalized Model of Voice-Leading for Atonal Music. Ph.D. dissertation. Harvard University (1991) Lambert, P.: Isographies and Some Klumpenhouwer Networks They Involve. Music Theory Spectrum 24(2), 165–195 (2002) Lewin, D.: Klumpenhouwer Networks and Some Isographies That Involve Them. Music Theory Spectrum 12(1), 83–120 (1990) Losada, C.: K-nets and Hierarchical Structural Recursion: Further Considerations. Music Theory Online 13(3) (2007) Morris, R., Alegant, B.: The Even Partitions in Twelve-Tone Music. Music Theory Spectrum 10, 74–101 (1988) Nolan, C.: Thoughts on Klumpenhouwer Networks and Mathematical Models: The Synergy of Sets and Graphs. Music Theory Online 13(3) (2007) O’Donnell, S.: Embracing Relational Abundance. Music Theory Online 13(3) (2007) Stoecker, P.: Without a Safety (k)-Net. Music Theory Online 13(3) (2007) Tymoczko, D.: Recasting K-nets. Music Theory Online 13(3) (2007)

Isographies of Pitch-Class Sets and Set Classes Tuukka Ilomäki Sibelius Academy, Finland [email protected]

1 Introduction One of the major differences between tonal and atonal music is the significantly larger number of available pitch-class sets in the latter. The categorization of pitchclass sets and the analysis and classification of their relations stand as a major strand in the 20th century music theory. The equivalence relation induced by the group of transpositions and inversions is the de facto standard classification of pitch-class sets. Music theorists, however, have recently suggested alternative approaches, for instance, relations based on similarity or voice leading. Klumpenhouwer networks or K-nets present yet another approach, this one based on transformation theory. K-nets provide a way to represent relations beyond the limits of set classes. Several authors have contributed to this research: Henry Klumpenhouwer, David Lewin, Philip Lambert, and many others. Furthermore, the idiomatic theory of George Perle provides means to tackle the K-nets (Perle 1977; Lewin 2002). A K-net contains some specified contents as vertices and transformations as arrows between the vertices. Typically, the vertices are pitch classes and the arrows are transpositions and inversions. The diagrams commute and for all arrows the pitch classes at the arrowheads are images of the pitches classes at the arrowtails under the transformations which are attatched to the arrows. The motivation for this paper is to examine what the isography of K-nets (with distinct pitch classes as vertices) reveals about the pitch-class sets involved. T5

I1

T5

I6

I8

T5

T5 1 I1

6 I6

I1

0

K-graph

omit nodes

0

5 8

K-net

1

6

I1

I8

omit transformations

0

0

5 8

Isographic pcsets

Fig. 1. From K-nets to K-graphs and from K-nets to the isography of pitch-class sets

The middle part of Figure 1 depicts a typical pair of positively isographic K-nets. Two K-nets with pitch classes {0, 1, 6} and {0, 5, 8} are isographic via . Taking this isography of K-nets as the starting point, the focus can be set either on the transformations or on the contents of the vertices. In other words, some information is omitted or ignored. First, to the left the contents of the vertices are omitted, leaving a K-graph. David Lewin (1990) has pursued this direction in his studies of the relations between transformation networks. Second, to the right the transformations are T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 386–391, 2009. © Springer-Verlag Berlin Heidelberg 2009

Isographies of Pitch-Class Sets and Set Classes

387

omitted, leaving the underlying pitch-class sets. This direction, and the distinction between moving left or right in this way, has been less explored. Philip Lambert (2002) has studied trichordal K-class families with fixed configurations of transpositions, inversions and pitch-class locations. In this paper, however, I focus on the underlying pitch-class sets, with fixed configurations of transpositions and inversions, but allowing any configuration of pitch-class locations. My aim here is to explore what kind of pitch-class sets can be arranged into strongly or positively isographic K-nets using maximally even configurations 2+1 for trichords, 2+2 for tetrachords, etc.1 I will move from isographic pitch-class sets to isographic set classes and group set classes by shared isographic potential.

2 Isography of Pitch-Class Sets and Set Classes The term ‘isography’ has been used to denote an automorphism of transformation networks (Lewin 1990) or, more specifically, a certain type of automorphism, since the automorphisms involving the M-operation are usually excluded. The change of context from transformation networks to pitch-class sets makes it necessary to redefine isography. I retain the term even if the two are algebraically rather different domains. Definition 1. Two pitch-class sets are (strongly) isographic if and only if the pitch classes can be arranged into K-nets that are (strongly) isographic. By definition, isography of K-nets implies isography of the pertinent pitch-class sets. Two K-nets with pitch classes of two isographic pitch-class sets are not necessarily isographic, however, since the isography of K-nets depends both on the pitch classes and on their configuration. For instance, K-nets with pitch classes {0, 1, 6} and {0, 5, 8} are isographic only if the transpositional relation in the K-nets is between pitch classes 1 and 6 and pitch classes 0 and 5, respectively (as in Figure 1). Note that the isography relation between pitch-class sets is not transitive as the concrete legitimating K-nets may change from one related pair to the next. In Definition 2 the isography of set classes is derived from the isography of pitchclass sets. Proposition 1 and Corollary 1 guarantee that the isography of set classes is well defined. Hence, the isography of set classes does not depend on the selection of pitch-class sets representing the set classes. Definition 2. Two set classes are (strongly) isographic if and only if there are two pitch-class sets belonging to the set classes that are (strongly) isographic. Note that Definition 2 does not imply that any two pitch-class sets belonging to (strongly) isographic set classes are necessarily (strongly) isographic.2 1

Without loss of generality it suffices to deal only with positive isography since, in the context of set classes, negative isography of pitch-class sets A and B implies positive isography of pitch-class sets A and TnIB. 2 Definition 2 is thus analogous to the definition of the inclusion relation between set classes. For example, we may state that the set class 3-1[012] is included in the set class 4-1[0123]. This does not imply, however, that an abritrary member of 3-1[012] would necessarily be a subset of an abritrary member of 4-1[0123].

388

T. Ilomäki

Proposition 1. If A and B are (strongly) isographic pitch-class sets, then TnA and TnB are also (strongly) isographic pitch-class sets and so are TnIA and TnIB. Corollary 1. Let A and B be (strongly) isographic pitch-class sets that belong to set classes SC(A) and SC(B), respectively. If A' is an arbitrary pitch-class set belonging to SC(A) then there exists a pitch-class set B' belonging to SC(B) such that A' and B' are (strongly) isographic. Pitch-class sets related by transposition are isographic but not necessarily strongly isographic. Pitch-class sets related by inversion are negatively isographic but not necessarily positively isographic. For instance, transpositionally related pitch-class sets {0, 1, 2} and {1, 2, 3} are positively isographic but not strongly isographic; inversionally related pitch-class sets {0, 1, 2, 3, 4, 6} and {0, 2, 3, 4, 5, 6} are negatively isographic but not positively isographic.

3 Tonality and Whole-Tone Scale Proportion Pitch-class sets can be classified using two concepts: tonality and whole-tone scale proportion. Originally, George Perle defined the tonality of tetrachords as the sum of the four pitch classes modulo 4 (Perle 1977, 138–141). In addition, sums 1 and 3 are considered equivalent.3 The following definitions generalize tonality in all cardinalities 1 n 12. For the sake of coherency, the definition must be stated in terms of the greatest common divisor of the cardinality n and 12. Hence, tonality is a useful concept only in cardinalities n such that gcd(n, 12) > 1. Definition 3. The tonalities of cardinality n are the equivalence classes that the automorphisms of the group Zgcd(n, 12) induce. Definition 4. The tonality of a pitch-class set of cardinality n is the equivalence class of the sum of its pitch classes mod n. For example, the tonalities of cardinality 4 are {0}, {1, 3} and {2}, and the tonalities of cardinality 6 are {0}, {1, 5}, {2, 4} and {3}. The tonality of pitch-class set {0, 1, 2, 3, 4, 5} is {3}. Proposition 2. Pitch-class sets A, TnA and TnIA have the same tonality. Since – according to Proposition 2 – all pitch-class sets that belong to the same set class have the same tonality, the concept of tonality can be inherited from sets to set classes. This motivates the following Definition: Definition 5. The tonality of a set class is the tonality of any of its constituent pitchclass sets. An interval between two pitch classes is even if and only if the two pitches belong to the same whole-tone collection. Therefore, the proportion of odd and even intervals in a pitch-class set depends only upon the distibution of its pitch classes between the two 3

I use sums of pitch classes as notational shorthand; I do not posit a group structure on the set of pitch classes.


389

whole-tone scales. As isography of set classes involves analogous behaviour with respect to odd and even intervals, the two whole-tone scales play a decisive role in the determination of isography. The proportion between even and odd intervals is invariant under transpositions and inversions (as well as under the M-operation). Hence, the number of odd and even intervals in pitch-class sets is constant for all sets belonging to the same set class. This is also evinced by the fact that pitch-class sets in the same set class have identical interval-class vectors.

4 Relations of Set Classes The relations that isography defines on the set of pitch-class sets and the set of set classes are reflexive and symmetric but not transitive. Hence, the resulting relations are not equivalence relations. Set classes are grouped into categories with two layers, however, since strong isography exists only within each category and positive isography (without strong isography) exists only between certain categories. In all cardinalities, the whole-tone scale proportion defines a necessary condition of isography. In the even cardinalities tonality imposes an additional necessary condition. In cardinalities 2 and 10, however, tonality and whole-tone scale proportion coincide. Furthermore, in cardinality 3 set classes with the same wholetone scale proportion are either strongly isographic or not isographic. Hence, positive isography without strong isography appears only between set classes with different whole-tone scale proportions. Correspondingly, in cardinalities 4 and 6 set classes of the same tonality are either strongly isographic or not isographic. Hence, positive isography without strong isography appears only between set classes of different tonalities. These conditions do not apply to the complement cardinalities 8 and 9.

015 012

014 027

037

036

013

025

016

024

026

048

Fig. 2. Strong isographies of trichords

Figure 2 depicts the strong isographies of the trichords. The set classes are divided into those that contain pitch classes from only one whole-tone scale and those that contain pitch classes from both whole-tone scales. Figure 3 depicts the strong isographies of the tetrachords. Whole-tone scale proportion and tonality divide the set classes into five distinct categories. Hence, the relations defined by strong isography are subsets of the equivalence relations that the whole-tone scale proportion induces

390

T. Ilomäki Tonality 0

0+4

0246

Tonality 1

Tonality 2

0268

0248 0236

0258

0124 0135 0148 0247

1+3

0126 0137 0134

0136

0358 0156 0125

2+2 0237

0146 0157

0123 0145

0127 0158 0257

0147 0167

0347

0369

0235

Fig. 3. Strong isographies of tetrachords

on the set of trichords and the whole-tone scale proportion and tonality induce on the set of tetrachords. Propositions 3 and 4 involve complementary pitch-class sets and set classes of complementary cardinality. Their proofs follow immediately from the observation that gcd(n, 12) = gcd(12 - n, 12) for all 1 n 12 and that there exists an automorphism of group Zgcd(n, 12) that maps k mod gcd(n, 12) into 66 - k mod gcd(n, 12).4 Proposition 3. The tonalities in cardinalities n and 12 - n are identical. Proposition 4. Complementary pitch-class sets have the same tonality. Propositions 3 and 4 show that complementary set classes group together correspondingly. Octachords, for instance, divide into five categories and three tonalities like tetrachords. The complementary set classes have the same tonality. The larger cardinalities are more flexible, however: there are more ways to arrange eight pitch classes than four pitch classes into the vertices of a K-net. Consequently, the larger cardinalities permit more strong isographies than the smaller ones. Even if the corresponding set classes in complementary cardinalities divide into corresponding tonalities, we cannot deduce the isography of two set classes from the (strong) isography of the corresponding set classes of the complement cardinality. For instance, the all-interval tetrachord set classes 4-Z15[0146] and 4-Z29[0137] are strongly isographic, but the corresponding complementary set classes 8-Z15[01234689] and 8Z29[02346789] are not even positively isographic. Similar examples can be found between pentachords and septachords (5-1[01234] and 5-Z12[01356] versus 7-1 [0123456] and 7-Z12[0123479]) and trichords and nonachords (3-1[012] and 3-11[037] versus 9-1[012345678] and 9-11[01235679A]). 4

The sum of all pitch classes 0, 1, 2, ..., 11 is 66. Consequently, if the tonality of a pitch-class set of cardinality n is k mod gcd(n, 12) then the tonality of the complementary pitch-class set is 66 - k mod gcd(12 - n, 12) = 66 - k mod gcd(n, 12).


391

The Z-related set classes provide a convenient test on whether the necessary conditions of the isography of set classes depend only on the interval-class content of the set classes. The answer is negative. The specific pattern of pitch classes in the pitch-class sets forming the set classes must be taken into account. For example, set class 6-Z4[012456] is strongly isographic with seven hexachords, whereas its Z-related set class 6-Z37[012348] is strongly isographic with four hexachords. Indeed, these two Z-related set classes are not even isographic.

References Lambert, P.: Isographies and Some Klumpenhouwer Networks They Involve. Music Theory Spectrum 24(2), 165–195 (2002) Lewin, D.: Klumpenhouwer Networks and Some Isographies That Involve Them. Music Theory Spectrum 12(1), 83–120 (1990) Lewin, D.: Thoughts on Klumpenhouwer Networks and Perle-Lansky Cycles. Music Theory Spectrum 24(2), 196–230 (2002) Perle, G.: Twelve-Tone Tonality. University of California Press, Berkeley (1977)

The Transmission of Pythagorean Arithmetic in the Context of the Ancient Musical Tradition from the Greek to the Latin Orbits During the Renaissance: A Computational Approach of Identifying and Analyzing the Formation of Scales in the De Harmonia Musicorum Instrumentorum Opus (Milan, 1518) of Franchino Gaffurio (1451–1522)* Herbert Kreyszig1 and Walter Kreyszig2 1

New York, New York, U.S.A. [email protected] 2 Saskatoon, Saskatchewan, Canada, and Vienna, Austria [email protected]

Abstract. On the occasion of the three hundredth anniversary of Leonhard Euler (1707-1783) and the focus of his contribution to the mathematical explanation of scales, the quest arises to examine similar contributions of music theorists that lived prior to Euler. In this paper we selected from the trilogy (Theorica musicae, 1492; Practica musicae, 1496; De harmonia musicorum instrumentorum opus, 1518) of the eminent Renaissance scholar, musician, theorist and composer Franchino Gaffurio (1451-1522) his third work of 1518. In particular, using a modern elementary number theoretic approach, we discovered a number generating function in closed form, which we called a Gaffurio number generating function, that generates the tones of the Pythagorean scale. The number generating function is of the form f (n, m) = pn . m q , where p and q are primes, and n and m are elements of /N0. Here, Gaffurio chooses the smallest primes p = 2 and q = 3, thereby with this ratio p / q denoting the interval of the diapente and he varies the integral exponents n and m from 0, 1, 2, 3, … and in the process constructing a series of numbers signifying the precise placement of the tones within the gamut. His approach is in line with medieval mathematics, which focuses on elementary number theory and can be seen as a precursor to Euler’s Potenzennetz (grid of proportions). This paper embeds our results in Gaffurio’s oeuvre and his times and includes a *

The authors gratefully acknowledge most valuable comments from the editor Timour Klouche, Staatliches Institut fr Musikforschung (National Institute for Music Research), Berlin, Germany; the latter and Thomas Noll, Department of Theory and Composition, Escola Superior de Musica de Catalunya, Barcelona, Spain for organizing and leading the First International Conference of the Society for Mathematics and Computation in Music, Berlin, May 18-20, 2007; technical help from Ana Miranda Kreyszig, LLM, JD, Attorney, New York, NY, David G. Olson, Ph.D., Political Scientist, New York, NY and some valuable discussions with the late Paul V. Reichelderfer (1913-1996), first as Professor of Mathematics at Ohio State University, Columbus, Ohio, USA and since 1968 at Ohio University, Athens, Ohio, U.S.A. See www.math.ohio-state.edu/history/biographies/PaulReichelderfer for biographical information.


The Transmission of Pythagorean Arithmetic

393

concise of overview of his biography with an emphasis on his pedagogical and literary contributions. Our approach fills a gap in research, because although Latin music theorists from the era of musical humanism frequently include mathematical approaches to the Greek systema teleion and Guidonian system, as projected onto the monochord, they do not disclose the nature and specific details of the mathematical calculations which inform the natural numbers included in both the manuscript and early printed editions.

In 2007, on the occasion of the three hundredth anniversary of the birth of the mathematician and philosopher Leonhard Euler (1707-1783), our attention in mathematical and music theoretical circles is squarely focused on Euler’s mathematical explanation of musical intervals,1 with recourse to arithmetic proportions in the calculation of the intervals of the scale, beginning with the unison and continuing systematically through the octave and beyond. Though in his systematization of the calculations in the so called Potenzennetz (grid of proportions),2 Euler’s important contribution, was widely reviewed and partly adopted, beginning in the eighteenth century, with Denis Diderot (1713-1784) and continuing in the nineteenth century with François-Joseph Fétis (1784-1871),3 Euler’s arithmetic approach to the explanation of the scale had its precedent in the era of the Renaissance, especially in the musictheoretical thought of the late fifteenth and early sixteenth centuries. Latin music theorists from the era of musical humanism frequently include mathematical approaches to the Greek systema teleion4 and the Guidonian system,5 as projected onto the monochord,6 without disclosing the nature and specific details of the mathematical calculations which inform the natural numbers included in both the manuscripts and early printed editions. The music-theoretical discourse of Franchino Gaffurio provides a case in point.7 One of the key musical humanists to emerge during the last decades of the fifteenth century, Franchino Gaffurio, well rounded as a musicus in his embracing of both the practica musicae and the theorica musicae interests first kindled in his youth and continued in his adult years, concentrated on researching the history of the disciplina musicae, thereby following a mode of inquiry common among the musical humanists of his era.8 Gaffurio’s numerous appointments at court, in church and in the gymnasium as well as his manifold acquaintances with scholars, artists and poets in Italy9 offered an ideal platform as those contacts together with his quest for the retrieval of the ancient tradition, including the commissioning of Latin translations of

1

Euler. For an overview of Euler’s calculations, see Hesse. 3 Ibid. 4 On the systema teleion, see Dreyer; see also Neubecker; Michaelides. 5 On the Guidonian system, see Oesch; Allaire. 6 For an overview of the monochord tradition, see Adkins; Smith. 7 For an overview of Gaffurio’s writings, see Young (1954). 8 See, for example, Palisca (1985). 9 For a summary of activities, see Figure 1. 2

394

H. Kreyszig and W. Kreyszig

Greek documents,10 and the collecting of the books on the artes liberales with a focus on the origin of the disciplina musicae11 paved the way for an exceptionally broad inquiry situated within the artes liberales (comprising grammar, dialectic, rhetoric, geometry, arithmetic, astronomy, and harmonics). While his pre-Milan years showed rather modest literary activity, with a focus largely on the study of earlier and contemporary humanist writers, some attention to original writing, and above all, the careful planning of his three principal treatises, namely, the Theorica musice (Milan, 1492),12 the Practica musicae (Milan, 1496),13 and the De harmonia musicorum instrumentorum opus (Milan, 1518),14 as gleaned from a number of specific references to these volumes surfacing as early as the Theoricum opus musice discipline (Naples, 1480),15 the immediate precursor of the Theorica musice, the prestigious appointment as maestro di cappella at the Cathedral of Milan in 1484, as well as, the subsequent intimate association with the Court of the Sforzas offered Gaffurio the necessary infrastructure, that is, a balance between practical experience in his working as organist, choir director and composer of predominantly sacred repertoires destined by-and-large for the performance at the Cathedral of Milan16 and the exposure to music-theoretical thought, in his sharing of the ancient tradition of music as a theorist, with important access to the holdings of manuscripts and early printed materials at the Court of the Sforzas17 which proved to be an indispensable resource in terms of his own scholarship. Central to Gaffurio’s trilogy is the detailed discussion of Pythagorean arithmetic,18 necessary for a full comprehension of the scales, including both the systema teleion with its tonoi and octave species of the Greeks19 and the modes of the Latin West.20 Undoubtedly inspired by the lengthy and venerable tradition of Greek arithmetic, beginning with Pythagoras whose calculations sparked interest in a number of important writers, including Euclid (flourished ca. 300 BCE)21 and Nicomachus of Gerasa (flourished late 1st – early 2nd century CE),22 Gaffurio, in his wholehearted 10

Barbaro; Burana; Ficino. For an overview of these documents, see Kreyszig (1993); see also Gallo; Kreyszig (1998). 11 For an inventory of Gaffurio’s library, see Caretta. 12 For the early and facsimile editions, see Gaffurio (1492). For an English translation, see Kreyszig (1993). For an Italian translation, see Illuminati. 13 For the early and facsimile editions, see Gaffurio (1496). For an English translation, see Miller (1968a); Young (1969). For an overview of the Practica musicae, see Miller (1968b). 14 For the early and facsimile editions, see Gaffurio (1518). For an English translation, see Miller (1977). 15 For the early and facsimile editions, see Gaffurio (1480). 16 For an overview of his repertory, see Blackburn; Finscher and Kreyszig; Kreyszig (2002); Merkley. 17 Pellegrin. 18 For a survey of Pythagorean arithmetic, see Barbera (1980); Münxelhaus; see also Zaminer. On the application of Pythagorean arithmetic within the realm of musica practica, see, for example, Busse-Berger (1990), Busse-Berger (1993). 19 For an overview of the Greek tonoi within the Greek double octave systema teleion as an integral part of the Greek harmoniai, see Palisca (1984). 20 For an overview of the Latin modes, see Markovits. 21 Mathiesen; see also Barbera (1991); Bowen (1991); Busch. 22 Levin (1975); Levin (1994).


395

emulation and embracing of the Greek tradition, much of which received ample vetting through the eminent auctoritas of Boethius (born ca. 480; died ca. 524),23 focuses on the significance, definition, classification, principle and progression of numbers, discrete and continuous quantity, genera (simple and composite), and species of proportions, as well as the concepts of equality and inequality, geometric and arithmetic proportions, proportionality and mean topics which lead to a consideration of the intervals on two plains, namely, as logos (proportionate relationship) and diastemata (distance between two pitches).24 At the forefront of the studia humanitatis lay the inquiry into the origin of the disciplina musicae. For Gaffurio, the agenda of his trilogy is one of exceptional breadth, as he engages in a self-imposed juxtaposition of the Greek tonoi as part of the systema teleion and of the Western modes as part of the Guidonian system of hexachords and solmization. Perhaps inspired by the opinion of Vitruv, who in Chapter 3 of Book 5 of his De architectura indicated that “Harmonics is an obscure and difficult subject to read and write about particularly for those who do not know Greek letters.”25 Gaffurio regards the similar terminology accorded to both Greek and Latin scalar systems as an ideal point of departure for his examination, obviously unaware of the vastly different meaning assigned to the respective systems of scales in the Greek and Latin orbits. Gaffurio’s self-tailored manner of inquiry culminates in the confusion of the terms octave species,26 modes,27 and tonoi,28 as the result of his misreading of the De institutione musica of Boethius with the ultimate resolution of Gaffurio’s misreading occurring only in his De Harmonia. Here, Gaffurio's study of two important treatises in Latin translation, namely the Introduction to Harmonics of Cleonides in a 1497 Latin translation by Giorgo Valla (1447-1500)29 and the Harmonics of Ptolemy in a 1499 Latin translation by Nicolo Leoniceno (1428-1524)30 helped clarify the scalar systems of the Latin and Greek traditions with the Pythagorean tuning at the core of the examination in the De harmonia. Notwithstanding the authors of the primary sources completed during the era of musical humanism, modern scholars, including editors and translators of Latin music theoretical texts from the era of musical humanism as well as twentieth-century commentators on the scientia musicae31 generally proceed little beyond the original documents in clarifying the arithmetic squarely placed at the root of the construction of the scales and our understanding thereof. 23

For an edition of Boethius’s two principal treatises, namely, the De institutione arithmetica and De institutione musica, see Friedlein. For English translations of these treatises, see Masi; Bower (1989). The pre-eminent status of Boethius has received copious acknowledgment in the secondary literature; see, for example, Heller; Bower (1981); Caldwell; Palisca (1990); also Palisca (1993, 168-188). 24 Riethmüller; see also Stenzel. 25 Vitruvius; as cited in Kreyszig (1993, 172). 26 For an overview of the octave species, see Barbera (1984). 27 For an overview of the modes, see Bower (1983). 28 For an overview of the tonoi, see Solomon (1984). 29 Palisca (1985, 67-87). For an English translation of Cleonides’s Harmonics, see Solomon (1980). 30 Palisca (1985, 117-122). For an English translation of Ptolemy’s Harmonics, see Solomon (2000). 31 See, for example, Haase; Bowen, William; Sachs.

396


In his calculations of the tones and intervals of the Pythagorean scale, Gaffurio uses a number generating function of the form f (n, m) = pn .qm , where p and q are primes, and n and m are elements of /N0. Here, Gaffurio chooses the smallest primes p = 2 and q = 3, with this ratio p / q denoting the interval of the diapente and he varies the integral exponents n and m from 0, 1, 2, 3, …, thereby constructing a series of numbers signifying the precise placement of the tones within the gamut. His approach is in line with medieval mathematics, which focuses on elementary number theory, details which follow below. Within Gaffurio’s trilogy, we shall now focus our attention on his calculations of tones and intervals as most comprehensively and succinctly shown in his De harmonia musicorum instrumentorum opus which reveals the following mathematical constructs. Elementary number theory gives us the unique factorization theorem,32 that is, Theorem. The Unique Factorization Theorem Every positive integer can be written as a product of primes in a unique way. Expressed as an equation this means that any positive integer n can be written as

n = a product of r primes

p1

p2

p3

pr ,

pi , where r is any positive integer. This equation can be

written more compactly by grouping the same primes together and using exponentiation. If we denote the integral exponents by ei and the unique primes by

qi we have

n = q1 1 e

q2

e2

q3

e3

qs s . e

This equation allows us to represent n in a standard form by grouping the primes by size, that is,

q1 < q 2 < q3 < < q s , where s is a positive integer. Since these equations are for any positive integer n, where n > 1, they lend themselves to generating infinitely many such numbers by varying either the exponents or the prime numbers or both. This is in tune with Medieval mathematics in which generating numbers and more importantly discovering number generating functions are of great importance to the mathematicians of the times who want to discover prime number generating functions.33 In a most general form such a number

32 33

See Dudley (1969, 14-15). To further look into this line of thought see Dickson. This seminal work is an attempt to give a short description of every work published in number theory from Antiquity to 1918. Further valuable references include Ore; Shanks; Crandall and Pomerance.


397

generating function 34 based on the ideas derived from the unique factorization theorem would look like

f (q1 , q 2 , q 3 ,

, q s , e1 , e2 , e3 ,

, e s ) = q1 1 e

q2

e2

q3

e3

qs s . e

Examining the tones and intervals of the Pythagorean scale that Gaffurio calculated, we discover that a number generating function Gaffurio uses is of the form

=

f (e1 , e2 )

2 e1 ∗ 3e2 .

To simplify the notation, and since henceforth we concentrate only on this what we shall call Gaffurio Number Generating Function we rewrite the last equation in a much simpler form, where we denote the exponents by n and m, thereby avoiding the more complicated e1 , e 2 , and thus obtain in a simple form the following function consisting of a product of exponentiated primes

f (n, m)

=

2 n ∗ 3 m.

Here Gaffurio chooses the smallest prime numbers p = 2 and q = 3, with the ratio p / q denoting the interval of the diapente.35 He varies the integral exponents n and m from 0, 1, 2, 3, …, thereby constructing a series of numbers signifying the precise placement of the tones within the gamut.36 This is demonstrated most significantly in a table created by Gaffurio himself, and explained in terms of the number generating function we discovered, as shown in Figure 2, directly taken from Gaffurio37 and then interpreted. The center piece of this figure is a Gaffurio Number Generating Function in the third column. It is linked to the first column listing the frequencies of the tones. Here frequency denotes a number of proportionality which allocates each tone to a particular position within the Pythagorean scale. The system in itself is closed but movable. It extends over two octaves and represents the complete systema teleion. As is the case with medieval musicology, the scales and frequencies are in inverse 34

This number generating function is strictly in line with elementary number theory based on factorization, orginating with Euclid; see Shanks, 4-7. One of the first theorems on primes is by Euclid in his Elements, Book IX, Proposition 20 and deals with the existence of infinitely many primes. The next important result is a method of finding primes by Eratosthenes (276194 BCE), called the Sieve of Eratosthenes. But the systematic studies of finding primes is only invigorated with the studies by Pierre-Simon de Fermat (1601-1665) and Marin Mersenne (1588-1648), more than two generations after the age of Gaffurio. For details on this historical development, see Ore (1948, 50-85). 35 For simplicity we denote the two primes present by p and q, independent of the notation p i and q i , used prior in the formula for the most general number generating function. Indeed, our Gaffurio number generating function is a specific instance of such a general function. 36 Note that we allow the exponents n and m to take on the value of zero in this Gaffurio number generating function. This is less restrictive than in our more general theory but sacrifices in compactness of presentation. 37 Franchino Gaffurio, De harmonia musicorum instrumentorum opus, Book 1, Chapter 10.

398


movements, that is, increasing scale corresponds to decreasing frequency, and conversely. Thus it differs from our modern (Hertzian) understanding of (fixed) frequency. However, the Gaffurian system with its proportions can be seen as an early precursor to the Hertzian system. Date Beginning of ? until September 1473 16 September 1473

Place Monastery of San Pietro in Lodi Lodi

late 1473 or 1474 1474

Lodi

14 May 1474 May 1474 – 1475 ? ca. 1474

Lodi Mantua

1475-1476

?

1477

Genoa

1477 November 1478

Genoa Naples

1479

Naples

8 October 1480 1480

Naples ?

?

Lodi

1480-1483

Monticelli d’Ongino (territory of Cremona) Monticelli

ca. 1482

Lodi

Verona ?

Activity Theological study

Completion of the copying of the Lucidarium of Marchetto of Padua (flourished 1305-1319); this manuscript also comprising Pomerium of Marchetto of Padua and the Ars cantus mensurabilis of Franco of Cologne (flourished mid to late 13th century) Ordination to priesthood Studies of composition with Johannes Bonadies also known as Godendach (flourished in the 15th century) Participation as singer in Cathedral of Lodi Research in music Employment as public teacher Completion of Extractus parvus musicae [Manuscript Parma, Biblioteca Palatina, sezione Musicale 1158], consisting of extracts from Marchetto of Padua and Ugolino of Orvieto, with references to Johannes de Muris (born ca. 12901295; died after 1344), the Ars nova of Philippe de Vitry (1291-1361) , the Tractatus figurarum of Philippus de Caserta (flourished ca. 1370); several anonymous treatises, and unknown treatise by Guillaume Dufay (1397-1474); Compilation of Tractatus brevis cantus plani [Manuscript Parma, Biblioteca Palatina, sezione Musicale 1158] Completion of Flos musice [Manuscript lost], with reference in Theorica musice, Book 5, Chapter 8 and of Musice institutionis collocutiones [Manuscript lost] Appointment by Doge Prospero Adorno; teaching of music theory and composition (sacred and secular repertories) Performance of some of his motets in the Church of San Lorenzo Move to exile with Adorno to Court of King Ferdinand of Aragon; studies of musica speculativa, upon the recommendation of Phylippinus Bononius (Secretary of King Ferdinand) discussions with Johannes Tinctoris (ca. 14351511), Guglielmo Guarneri, Bernhard Ycart (flourished ca. 1470-1480); also directing of music at Church of San S. Annunziata Completion of Theorice musice tractatus [Manuscript London, British Library, Hirsch IV.1441] as early version of Theoricum opus musice discipline Publication of Theoricum opus musice discipline Musices practicabilis libellum [Manuscript Cambridge, Massachusetts, Harvard University, Houghton Library Mus. 142], with materials incorporated into Book 2 of Practica musicae. Return at invitation of Bishop Carlo Pallavicino of Lodi (1456-1497), owing to plague and Turkish invasion of Puglia Appointment as teacher of singers; preparation of Practica musicae

Completion of Micrologus vulgaris cantus plani [Manuscript Bologna, Biblioteca Civico Museo Bibliografico Musicale A 90] Completion of Tractatus practicabilium proportionum [Manuscript Bologna, Biblioteca, Civico Museo Bibliografico musicale A 69] with the material incorporated into Book 4 of Practica musicae

Fig. 1. The Biography of Franchino Gaffurio (born Lodi, 14 January 1451; died Milan, 25 June 1522) with Emphasis on His Pedagogical and Literary Contributions: An Overview


399

Appointment as choirmaster of Church of San Maria Maggiore; supervision of installation of organ pedal of Battista da Martinengo Return from Bergamo because of war in Ferrara Appointment as choirmaster at Cathedral of Milan and as teacher and composer (after appointment of the theologian, lawyer and vicar of Lodi, Romanus Barnus, as Archbishop of Milan); publication of revision of Theorica musice (1492) and Practica musicae (1496) Completion of Liber primus musices practicabilis [Manuscript Bergamo, Biblioteca Civica Angelo Mai S 4.37] with material incorporated into Book 1 of Practica musicae Contact with Bartholomeo Ramos de Pareia (ca. 1440-after 1490) Persuasion of architect Luca Paperio to work on Cathedral of Milan Appointment to Gymnasium founded by Ludovico d’Este with thirty-eight-year tenure, first music theorist to call himself “professor musices” Publication of Theorica musice

19 May 1483

Bergamo

27 October 1783 22 January 1484

Lodi Milan

1487

Milan

since 1489 1490 1492

Milan Mantua Milan

15. December 1492 1493

Milan

July 1494 22 April 1495 1495 30 September 1496 14 December 1496 1497

Milan Milan Bergamo Milan

Series of written exchanges over dispute concerning music theory with Giovanni Spataro (1458-1541) Appointment as Rector of the Church of San Marcellino Letter to Ludovico Sforza (1452-1508), requesting a benefice Appointment as Cleric of the Church of Pontirolo Publication of Practica musicae

Milan

Letter to Marco Sanudo, accompanying copy of Practica musicae

Milan

1499

Milan

1500

Milan

1504 1506 1508

? Varese Milan

1509

Milan

24 March 1517

Milan

27 November 1518 1518 22 August 1520 4 October 1520

Milan

Appointment as Professor of Music at University founded by Ludovico Sforza in Milan Completion of Glossemata quaedam super nonnullas partes theoricae Johannis de Muris [Manuscript Milan, Biblioteca Ambrosiana H 165 inf.] Capture of Milan by French; Gaffurio’s stay as Regius musicus; completion of De harmonia musicorum instrumentorum opus Visit of Spas [places unknown] Organizing of Chapel of Santa Maria al Monte Publication of Angelicum ac divinum opus musice, largely based on Practica musicae Gaffurio’s publication of oration by Jacopo Antiquario of Perugia (1474? – 1512?), extending welcome to Louis XII after victory over Venice Criticism of the Libri tres de institutione harmonica of Pietro Aaron (ca. 1480after 1545) Publication of De harmonia musicorum instrumentorum opus

1520

Turin

1521 1521

Milan Milan

Milan

Milan Milan Milan

Donation of his materials to the Church of the Incoronato in Lodi Letter to Deputies of the Church of the Incoronato in Lodi, recommending a cleric Letter to Deputies of the Church of the Incoronato in Lodi, acknowledging the hiring of cleric Apologia Franchini Gafurii Musici adversus Joannem Spatarium et complices musicos bononienses Epistola primo in solutions obiectorum Io. Vaginarii Bononiensis Epistola secunda apologetica, addressed to Antonio Alberti in Florence

The biographical information has been compiled from: Pantaleone Melegulo of Lodi, [Biographical Sketch of Franchino Gaffurio], included at the end of Franchino Gaffurio, De harmonia; see English translation of biographical sketch in Miller (1977, 212-214); Blackburn; Finscher and Kreyszig; Kreyszig (2002). Fig. 1. (continued)

400


A Gaffurio Number

Frequency Intervals

2304

Greek Terminology

=

Generating Function f (n, m) 28 * 32

Helmholtz Nomenclature

Nete hyperboleon

a

=

25

* 34

Paranete hyperboleon g

=

22

* 36

Trite hyperboleon

f

=

210

* 31

Nete diezeugmenon

e

=

2 7 * 33

Paranete diezeugmenon

d

=

24

* 35

Trite diezeugmenon or Paranete sinemenon

c

212

* 3

Paramese

b

21 * 3 7

Trite sinemenon

bb

=

29

Mese

a

=

2 6 * 34

Lychanos meson

g

=

23

Parhypate meson

f

tonus

2592 tonus

2916

semi[tonium] [minus]

3072 tonus

3456 tonus

3888


=

4096

0

semi[tonium] maius

=

4374


4608

* 32

tonus

5184 tonus

5832

* 36


6144

=

211 * 31

Hypate meson

e

=

28

* 33

Lychanos hypaton

d

=

25

* 35

Parhypate hypaton

c

213

* 30

Hypate hypaton

B

22

* 37

210

* 32

tonus

6912 tonus

7776

semi[tonium] minus

8192

= semi[tonium] maius

8748

=

?

Bb

semi[tonium] minus

9216

=

Proslambanomenos

A

(For explanations and sample calculation see text.) Fig. 2. Integrum Systema Harmonicum Diatonici Generis

For example, in the third shaded row one computes the frequency of a particular tone

frequency of tone = 2916 = f (n, m) = f (2, 6) = 2 2 * 36.


401

Furthermore, Column 4 shows the correspondence of this generated number38 to Trite hyperboleon. Column 5 converts the Pythagorean tone to the modern tone of f ’. The latter correspondence between the Pythagorean tone and the modern tone lends itself to modern computational tools in that a matching algorithm can be developed which determines the modern tone that is the best match for the frequency of an ancient tone.39 This tone by tone matching is naturally imperfect requiring that one subdivides a range of frequencies for the modern tone and examines into which of these ranges the ancient tone falls. We have shown a novel approach to the problem of examining a Latin music theoretical text by using modern elementary number theory. We have discovered a Gaffurio Number Generating Function that was able to generate a complete Pythagorean scale as shown in Figure 2. This modern approach to medieval problems would have undoubtedly satisfied the ancient and medieval music theorists. Indeed, according to Calvin M. Bower, “the means of coming to know music is uncompromisingly Pythagorean: number [emphasis by Kreyszig]. Boethius states and restates the Pythagorean creed, insisting on the quantitative nature of sound.”40 His view has elegantly paved the future of the analytical and computational investigation concerning the systema teleion, which is at the forefront of Gaffurio’s own deliberations and at the center of the research presented in this paper.

Bibliography Adkins, C.D.: The Theory and Practice of the Monochord. Ph.D. Dissertation, State University of Iowa (unpublished, 1963) Allaire, G.G.: The Theory of Hexachords, Solmization and the Modal System: A Practical Application. In: Carapetyan, A. (ed.) Musicological Studies and Documents, vol. 24. American Institute of Musicology (1972) Barbaro, E.: Translated Paraphrasis in Aristotelem. Treviso: Bartolomeo Confalomeri and Morello Gerardino (February 15, 1481); in modern edn. Barbaro, E.: translated, Paraphraseos de anima libri tres, Heinze, R. (ed.) Berlin (1899) Barbera, C.A.: The Persistence of Pythagorean Mathematics in Ancient Musical Thought. Ph.D. Dissertation, University of North Carolina at Chapel Hill (unpublished, 1980) Barbera, C.A.: Octave Species. The Journal of Musicology: A Quarterly Review of Music History, Criticism, Analysis, and Performance Practice 3, 229–241 (1984) Barbera, C.A.: The Euclidean Division of the Canon: Greek and Latin Sources. In: Mathiesen, T.J., Solomon, J. (eds.) Greek and Latin Music Theory. University of Nebraska Press, Lincoln, Nebraska and London (1991)

38

An actual factorization of 2916 is obtained by successive divisions by prime numbers as follows: 2916 : 2 = 1458; 1458 : 2 = 729; 729 : 3 = 243; 243 : 3 = 81; 81 : 3 = 27; 27 : 3 = 9; 9 : 3 = 3. Here the prime numbers (factors) are 2 and 3 and counting the number of divisions gives two divisions by 2 and five divisions by 3 with a remainder of 3 (together six 3s), hence 2916 = 2

39 40

6

2 ∗ 3 . Details for developing such types of algorithms are given in the seminal work by Knuth. Bower (1989, xx-xxi). Further on Boethius’s preoccupation with the concept of the number, see Illmer; Snider, 5-33 (Chapter 2).

402


Blackburn, B.J.: Gaffurius, Franchinus. In: Sadie, S. (ed.) The New Grove Dictionary of Music and Musicians, 29 vols., vol. 9, pp. 410–414. Macmillan, London (2001) Bowen, A.C.: Euclid’s Sectio canonis and the History of Pythagoreanism. In: Bowen, A.C. (ed.) Science and Philosophy in Classical Greece. Sources and Studies in the History and Philosophy of Science, vol. 2, pp. 164–187. Garland, New York and London (1991) Bowen, W.R.: Music and Number: An Introduction to Renaissance Harmonic Science. Ph.D. Dissertation, University of Toronto (unpublished, 1984) Bower, C.M.: The Role of Boethius’s De institutione musica in the Speculative Tradition of Western Musical Thought. In: Masi, M. (ed.) Boethius and the Liberal Arts: A Collection of Essays. Utah Studies in Literature and Linguistics, vol. 18, pp. 157–174. Peter Lang, Berne and Frankfurt am Main (1981) Bower, C.M.: The Modes of Boethius. The Journal of Musicology: A Quarterly Review of Music History, Criticism, Analysis, and Performance Practice 3, 252–263 (1984) Bower, C.M.: Translated. Anicius Manlius Severinus Boethius: Fundamentals of Music. In: Palisca, C.V. (ed.). Music Theory Translation Series. Yale University Press, New Haven, Connecticut and London (1989) Burana, G.F.: Translated. Bacchius the Elder, Introductio artis musicae, preserved in Manuscript Verona, Bibliotheca Capitolare CCXL (201) (April 15, 1494); also in modern edn. In: Bellermann, F. (ed.) Anonymi scriptio de musica Bacchi senioris introduction artis musicae, Berlin (1841); also in Greek-Latin edn. In: Meibom, M. (ed. & translator) Bacchii senioris introduction artis musicae, Marcus Meibom, Antiquae musices auctores septem graece et latine, 2 vols. Monuments of Music and Music Literature in Facsimile: Second Series – Music Literature, vol. 51. Broude Brothers, New York. facsimile of Amsterdam (1977); In English translation. Steinmeyer, O.: Bacchius Geron’s Introduction to the Art of Music (1652). Journal of Music Theory 29, 271–298 (1985) Busch, O.: Logos syntheseos: Die euklidische Sectio canonis, Aristoxenos, und die Rolle der Mathematik in the antiken Musiktheorie, im Anhang (in the appendix): KATATOMH KANONOȈ / Die Teilung des Kann: Die euklidische Sectio canonis in deutscher Übersetzung. In: Ertelt, T., von Loesch, H. (eds.) Studien zur Geschichte der Musiktheorie, vol. 3; and Veröffentlichungen des Staatlichen Instituts für Musikforschung, vol. 10; Olms, G.: Hildesheim und New York (1998). 2nd unchanged edn. Staatliches Institut für Musikforschung Preuischer Kulturbesitz, [Berlin] (2004) Busse-Berger, A.M.: Musical Proportions and Arithmetic in the Late Middle Ages and Renaissance. Musica Disciplina: A Yearbook of the History of Music 44, 89–118 (1990) Busse-Berger, A.M.: Mensuration and Proportion Signs: Origins and Evolution. Clarendon Press, Oxford (1993) Caldwell, J.: The De institutione arithmetica and the De institutione musica. In: Gibson, M. (ed.) Boethius: His Life, Thought and Influence, pp. 135–154. Basil Blackwell, Oxford (1981) Caretta, Alessandro, et al.: Franchino Gaffurio. Edizioni Dell’Archivio Storico Lodigiano, Lodi (1951) Crandall, R., Pomerance, C.: Prime Numbers: A Computational Perspective, 2nd edn. of 2001. Springer, New York (2005) Dickson, L.E.: History of the Theory of Numbers, vol. 3. AMS [American Mathematical Society]. Chelsea Publishing, Providence, Rhode Island (1919); Carnegie Institution of Washington, Washington, DC (reprint, 1999) Dreyer, E.-J.: Das Tonsystem der Griechen. Musiktheorie 3, 3–25 (1988) Dudley, U.: Elementary Number Theory. W.H. Freeman, San Francisco, California (1969) Euler, L.: Tentamen novae theoriae musicae. In: Leonhard Euler, Opera omnia, vol. 3/1, St. Petersburg (1739)


403

Ficino, M.: Translated. Plato: Opera omnia. Officina of Heinrich Petri, Basel (1576) Finscher, L., Kreyszig, W.: Gaffurio, Franchino. In: Die Musik in Geschichte und Gegenwart: Allgemeine Enzyklopädie der Musik, begründet von (founded by) Blume, F.: zweite neubearbeitete Ausgabe von (second revised edition by) Finscher, L.: 29 vols., Bärenreiter, Kassel; Metzler, Stuttgart (1994–2008), vol. 7 (Personenteil, 2002) cols. 393–403 (2002) Friedlein, G. (ed.): Anicii Manlii Torquati Severinii Boetii: Institutione arithmetica libri duo e Institutione musica libri quinque accedit Geometria quae fertur Boetii. Frankfurt am Main, Minerva (1867); Teubner, B.G.: Leipzig (reprint, 1966) Gaffurio, F.: Theoricum opus musice discipline. Francesco di Dino, Naples (1480); also In: Ruini C. (ed.) vol. 15, Musurgiana: Collana di trattati di teoria musicale, storiografia e organologia in facsimile, ed. by Istituto di Bibliografia Musicale di Roma, Libreria Musicale Italiana, Lucca (1996) Gaffurio, F.: Theorica musice. Filippo Mantegazza, Milan (1492); also In: Cesari, G. (ed.) facsimile reprint, Presso l’Accademia, Rome (1934); also as facsimile reprint as part of Monuments of Music and Music Literature in Facsimile – 2nd series: Music Literature, vol. 21. Broude Brothers, New York (1967); also In: Vecchi, G. (ed.) facsimile reprint. Bibliotheca musica Bononiensis, vol. II/5. Forni, Bologna (1969) Gaffurio, F.: Practica musicae. Guglielmo Signerre, Milan (1496); also In: facsimile reprint. Gregg Press, Farnborough (1967); also In: Vecchi, G. (ed.) facsimile reprint, Bibliotheca musica Bononiensis, vol. II/6. Forni, Bologna (1972) Gaffurio, F.: De harmonia musicorum instrumentorum opus. Gotardus Pontanus Calographus, Milan (1518); also In: Vecchi, G. (ed.) facsimile reprint. Bibliotheca musica Bononiensis, vol. II/7. Forni, Bologna (1972) Gallo, F.A.: Le traduzioni dal Greco per Franchino Gaffurio. Acta Musicologica 35, 172–174 (1963) Haase, R.: Geschichte des harmonikalen Pythagoreismus. Publikationen der Wiener Musikakademie, vol. 3. Elisabeth Lafite, Vienna (1969) Heller, B.: Boethius im Lichte der frühmittelalterlichen Musiktheorie. Ph.D. Dissertation, Universität Wien (unpublished, 1939) Hesse, H.-P.: Intervall. In: Die Musik in Geschichte und Gegenwart: Allgemeine Enzyklopädie der Musik, begründet von (founded by) Blume, F.; zweite neubearbeitete Ausgabe von (second revised edition by) Finscher, L., 29 vols., Bärenreiter, Kassel; Metzler, Stuttgart (1994–2008), vol. 4 (Sachteil, 1996), cols. 1069–1097, especially cols. 1083–1084 (1996) Illmer, D.: Die Zahlenlehre des Boethius. In: Zaminer, F. (ed.) Rezeption des antiken Fachs im Mittelalter. Geschichte der Musiktheorie, vol. 3, pp. 220–252. Wissenschaftliche Buchgesellschaft, Darmstadt (1990) Illuminati, I.: Translated with commentary, and introduction by Illuminati, I., Ruini, C. Franchino Gaffurio: Theorica musicae. In: La tradizione musicale, vol. 11 and Le regole della musica, vol. 2. SISMEL Edizioni del Galluzzo, Florence (2005) Knuth, D.E.: The Art of Computer Programming. Sorting and Searching, vol. 3. AddisonWesley, Reading, Massachusetts (1973); 2nd edn. of 1973 (1998) Kreyszig, W.K.: Translated with introduction and notes. Franchino Gaffurio: The Theory of Music. In: Palisca, C.V. (ed.) Music Theory Translation Series. Yale University Press, New Haven, Connecticut and London (1993) Kreyszig, W.K.: Franchino Gaffurio und seine Übersetzer der griechischen Musiktheorie in der Theorica musice (1492); Ermolao Barbaro, Giovanni Francesco Burana und Marsilio Ficino. Musik als Text: Bericht über den Internationalen Kongreß der Gesellschaft für Musikforschung, Freiburg im Breisgau (1993); Danuser, H., Plebuch, T. (eds.) vol. 1, pp. 164–171. Bärenreiter, Kassel (1998)

404


Kreyszig, W.K.: Research and Teaching During the Era of Musical Humanism: Defending the Scholar-Teacher in Response to the Principles of Creation and Dissemination of Knowledge in the Italian University Curriculum and Cultural Milieu of the Court of the Sforzas, with Special Reference to Franchino Gaffurio (1451–1522). In: Marken, R. (ed.), What is a Teacher-Scholar?: Symposium Proceedings, November 9-10, 2001, pp. 97–132. University of Saskatchewan, Saskatoon, Saskatchewan (2002) Levin, F.R.: The Harmonics of Nicomachus and the Pythagorean Tradition. American Classical Studies, vol. 1. The American Philological Association, University Park, Pennsylvania (1975) Levin, F.R. (ed.): The Manual of Harmonics of Nicomachus the Pythagorean: Translation and Commentary. Phanes Press, Grand Rapids, Michigan (1994) Markovits, M.: Das Tonsystem der abendländischen Musik im frühen Mittelalter. Series II, Publikationen der Schweizerischen Musikforschenden Gesellschaft, vol. 30. Verlag Paul Haupt, Berne and Stuttgart (1977) Masi, M.: Translated, Boethian Number Theory: A Translation of the “De institutione arithmetica”. Studies in Classical Antiquity, vol. 6. Rodopi, Amsterdam (1983) Mathiesen, T.J.: An Annotated Translation of Euclid’s Division of a Monochord. Journal of Music Theory 19, 236–258 (1975) Merkley, P.A., Merkley, L.L.M.: Music and Patronage in the Sforza Court. In: Dunning, A. (ed.) Studi sulla storia della musica in Lombardia: Collana di testi musicologici, vol. 3. Turnhout, Brepols (1999) Michaelides, S.: The Music of Ancient Greece: An Encyclopaedia. Faber and Faber, London (1978) Miller, C.A.: Translated, Franchinus Gaffurius: Practica musicae. In: Carapetyan, A. (ed.) Musicological Studies and Documents, vol. 20. American Institute of Musicology (1968a) Miller, C.A.: Gaffurius’s Practica musicae: Origin and Contents. Musica Disciplina: A Yearbook of the History of Music 22, 105–128 (1968b) Miller, C.A.: Translated, Franchinus Gaffurius: De harmonia musicorum instrumentorum opus. In: Carapetyan, A. (ed.) Musicological Studies and Documents, vol. 33. American Institute of Musicology (1977) Münxelhaus, B.: Pythagoras musicus: Zur Rezeption der pythagorischen Musiktheorie als quadrivialer Wissenschaft im lateinischen Mittelalter. In: Vogel, M. (ed.) OrpheusSchriftenreihe zu Grundfragen der Musik, vol. 19. Verlag für Systematische Musikwissenschaft, Bonn-Bad Godesberg (1976) Neubecker, A.J.: Altgriechische Musik: Eine Einführung, part of Die Altertumswissenschaft; Einführungen in Gegenstand, Methoden und Ergebnisse ihrer Teildisziplinen und Hilfswissenschaften. Wissenschaftliche Buchgesellschaft, Darmstadt (1977) Oesch, H.: Guido von Arezzo: Biographisches und Theoretisches unter besonderer Berücksichtigung der sogenannten odonischen Traktate. Verlag Paul Haupt, Berne (1954) Ore, Ø.: Number Theory and Its History. Dover Publications, New York (1948); reprint of McGraw-Hill, New York (1988) Palisca, C.V.: Round Table The Ancient Greek Harmoniai, Tonoi and Octave Species, organized and chaired by Claude V. Palisca at the National Meeting of the American Musicological Society in Louisville, Kentucky (October 27, 1983). The Journal of Musicology: A Quarterly Review of Music History, Criticism, Analysis, and Performance Practice 3, 221 (1984) Palisca, C.V.: Humanism in Italian Renaissance Musical Thought. Yale University Press, New Haven Connecticut and London (1985)


405

Palisca, C.V.: Boethius in the Renaissance. In: Barbera, A. (ed.) Musical Theory and Its Sources: Antiquity and the Middle Ages. Notre Dame Conferences in Medieval Studies, vol. 1, pp. 259–280. University of Notre Dame Press, Notre Dame (1990) Palisca, C.V.: Studies in the History of Italian Music and Music Theory, pp. 168–188. Clarendon Press, Oxford (1993) Pellegrin, E.: La bibliothèque des Visconti et des Sforzas ducs de Milan, auc XVe siècle. Publications de l’institut de recherche et d’histoire des textes, vol. 5. C.N.R.S., Paris (1955) Riethmüller, A.: Logos und diastemata in der griechischen Musiktheorie. Archiv für Musikwissenschaft 42, 18–36 (1985) Sachs, K.-J.: Musikalische Elementarlehre im Mittelalter. In: Zaminer, F. (ed.) Rezeption des antiken Fachs im Mittelalter. Geschichte der Musiktheorie, vol. 3, pp. 106–161. Wissenschaftliche Buchgesellschaft, Darmstadt (1990) Shanks, D.: Solved and Unsolved Problems in Number Theory, vol. 1. Spartan Books, Washington (1962) Smith, F.J.: The Medieval Monochord. Journal of Musicological Research 5, 1–33 (1984) Snider, G.C.F.: In Defense of Music’s Eternal Nature: On the Pre-eminence of musica theorica Over musica practica. Master of Arts Thesis in Philosophy, University of Saskatchewan January 2005 (unpublished, 2005), http://library2.usask.ca/theses/available/etd-01282005-145722 Solomon, J.: Cleonides ISAYOYI ARMONIKI: Critical Edition, Translation and Commentary. Ph.D. Dissertation, University of North Carolina at Chapel Hill (unpublished, 1980) Solomon, J.: Towards a History of Tonoi. The Journal of Musicology: A Quarterly Review of Music History, Criticism, Analysis, and Performance Practice 3, 242–251 (1984) Solomon, J.: Ptolemy Harmonics: Translation and Commentary. Mnemosyne: Bibliotheca Classica Batava – Supplementum, vol. 203. Brill, Leiden and Boston, Massachusetts (2000) Stenzel, J.: Zur Theorie des Logos bei Aristoteles. Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik 1, 34–66 (1931) Vitruvius Pollio, M.: De architectura; in modern edn. and translation, Vitruvius, On architecture, 2 vols., Granger, F. (ed. and translator). The Loeb Classical Library. W. Heinemann, London; Putnam, New York (1931–1934) Young, I.: Franchinus Gaffurius: Renaissance Theorist and Composer (1451–1522). Ph.D. Dissertation, University of Southern California (unpublished, 1954) Young, I.: Translated, The Practica musicae of Franchinus Gaffurius. Madison. University of Wisconsin Press, Madison, Wisconsin (1969) Zaminer, F.: Pythagoras und die Anfänge des musiktheoretischen Denken bei den Griechen. Jahrbuch des Staatlichen Instituts für Musikforschung Preußischer Kulturbesitz, 203–211 (1979–1980)

Combinatorial and Transformational Aspects of Euler's Speculum Musicum Edward Gollin Williams College [email protected]

Abstract. The paper examines the structural and conceptual differences between the Speculum musicum from Euler's 1774 De Harmoniae, and the nineteenth-century Tonnetz, but also examines how Euler's conception of intervals as paths within the Speculum anticipates a combinatorial conception of interval that underlies contemporary transformational theories.

Example 1 presents the diagram from Euler’s 1774 De Harmoniae Veris Principiis per Speculum Musicum Repraesentatis, known as the Speculum Musicum (Speculum), which has gained prominence in recent years for certain obvious structural similarities to the now well-known Tonnetz or Tonverwandtschaftstabelle, popularized in the nineteenth-century writings of Hugo Riemann. The similarity, first noted by Martin Vogel, has since been observed in a number of writings that also suggest that the Speculum was a forerunner or precursor to the nineteenth-century Tonnetz.1 Example 1. Euler’s Speculum Musicum.

The Speculum represents twelve tones in a 3 by 4 network, a graph of tones-classes in pure intonation, organized by pure major thirds vertically, and by perfect fifths horizontally. This organization of tones in Cartesian orthography with a major-third axis and a perfect-fifth axis is what Vogel and others have seized upon in claiming precedent for the nineteenth-century Tonnetz. The present paper examines why I believe this claim is not quite true, but also explores a less discussed aspect of Euler’s thought, reflected in his writing about the Speculum, that bears not only upon Riemann’s conception of harmonic relations, but also upon aspects of contemporary transformational theory. 1

Vogel (1960) observes the relation of the Speculum to the Tonnetz of Oettingen and Riemann. The idea that the Tonnetz evolved from the Speculum has been suggested by, among others, Mooney (1996) and Cohn (1997, 7).


Combinatorial and Transformational Aspects of Euler's Speculum Musicum

407

Euler uses the Speculum in De Harmoniae to demonstrate and encapsulate the total pitch-class and interval content of what he calls the diatonic-chromatic genus in music. We shall discuss Euler’s genera shortly, but for the moment, it is enough to know that Euler conceives of the Speculum as a mathematical graph—the nodes represent tones, the edges represent the fundamental intervals of the perfect fifth and major third. Euler represents motion along each horizontal edge with the Roman numeral V for fifth; ascending motion by fifth—that is, motion to the right—is labeled +V; descending motion by fifth, motion to the left, is labeled -V. Similarly, downward motion along a third-edge, equivalent to ascent by a major third, is labeled +III; upward motion on the diagram, equivalent to descent by major third, is labeled -III. This symbolic language allows Euler to represent the interval between any two tones by the shortest pathway of connected edges between those tones. In some cases, several equally short paths can reify the interval between two tones, and the same symbol can sometimes reify the interval between several different tone pairs. Example 2. The interval of the augmented second, represented by specific pathways on the Speculum.

For instance, the top of Example 2 shows how Euler symbolically represents the interval from F to G# using three different pathways: one can ascend by two major thirds then ascend by one perfect fifth (+III +III +V); one can ascend by a major third, then perfect fifth, then major third (+III +V + III); or, one can ascend by perfect fifth, then by two major thirds (+V +III +III). Each different reckoning of the interval corresponds to a different path through the Speculum. Euler’s term for each such pathway is transitus. I call these classes of shortest paths between two tones a transitus class—in this case, a class that defines the interval of the augmented second. On Example 2, Euler also shows how the transitus class of the augmented second defines not only the interval from F to G#, but also the intervals from C to D# and from G to B-flat.2 Table 1 lists the fifteen distinct composite transitus classes Euler discusses in De harmoniae. 2

Euler, out of deference to the conventions of German tablature notation still prevalent in the eighteenth century, labels the tone in the lower right corner B (= B-flat). He nonetheless conceives the tone as A# (i.e. as the pure fifth above D# and the pure major third above F#). In Tentamen novae, Euler goes so far as to present his system as an improvement over an otherwise similar tuning system presented in Johann Mattheson’s Grosse General-Bass Schule (Hamburg, 1731), in which a true B-flat is derived as the perfect fifth below F (Euler 1739, 133). Euler provides tuning instructions in Tenatmen novae, which make explicit the A# derivation of the nominal B (Euler 1739, 146–47).

408

E. Gollin Table 1. Composite transitus classes on the Speculum

Transitus

Ratio

Name

+V+V

9:8

major whole tone

+V+III

16:15

major semitone

+V-III

5:6 or 5:3

minor third/major sixth

+III+III

16:25

augmented fifth

+V+V+V

32:27 or 16:27 Pythagorean minor third

+V+V+III

32:45 or 45:64 augmented fourth/diminished fifth

+V+V-III

5:9 or 10:9

greater minor seventh/minor whole tone

+III+III+V

64:75

augmented second

+III+III-V

24:25

minor chroma

+V+V+V+III

135:128 (*)

major chroma

+V+V+V-III

27:20 (*)

greater fourth

+V+V+III+III

225:128 (*)

augmented sixth

+V+V-III-III

36:25 (*)

greater diminished fifth

+V+V+V+III+III 675:512 (*)

augmented third

+V+V+V-III-III

—

27:25 (*)

One reason I am reluctant to believe that Euler’s Speculum was a precursor to the nineteenth-century Tonnetz is that there is no evidence that the nineteenth-century theorists involved in issues of tuning or tone relations—Helmholtz, Drobish, Naumann, Oettingen, or Riemann—appear to have known the particular article, published in the Proceedings of the St. Petersburg Royal Academy of Sciences in 1774, in which the Speculum appears. Euler was certainly well known to these theorists, but primarily through his 1739 treatise, Tentamen novae. Euler’s legacy in the nineteenth century, which derives from Tenatmen novae, was his observation that the frequency ratios of all intervals in just intonation can be exclusively represented as combined powers of 2, 3, and 5. It was this observation, I would argue, that underlies the structural convergence between the Speculum and the later nineteenth-century Tonnetze.


409

But there are a number of structural and conceptual differences that underscore the difference between Euler’s diagram and those of the next century. The nineteenthcentury Tonnetz is a finite representation of an infinite array of tones in just intonation. Euler’s Speculum, in contrast, is a bounded, finite graph—there are no further tones beyond the twelve he shows. Moreover, unlike a nineteenth-century Tonnetz, in which one can conceive, at least potentially, any available interval of the system extending from any tone in the system, intervals of the Speculum are not extensible beyond the borders of the graph; the transitus underlying the augmented second, for example, can only be ‘walked’ from the three tones F, C and G—from any other tone, there exists no destination tone in potentia or in actu. Euler’s finite conception of the Speculum, and, consequently, its differences from the Tonnetz, proceed from Euler’s conception of a genus itself. For Euler, a genus is a collection of musically-distinct, octave-equivalent tones. Every genus can be represented by a composite number that is exclusively comprised by the prime factors 3 and 5. In Euler’s view, a genus is a manifestation, in a Platonic sense, of that composite number, sharing its fundamental properties with that number; just as every composite number has a unique prime factorization, every genus has a unique factorization into distinct tone classes. Example 3 illustrates how the diatonic-chromatic genus manifests the number 675: reckoning F as one, each tone of the Speculum represents a unique factor of 675. Euler’s numerical conception of the Speculum represents a fundamental inversion of relations from those of the Tonnetz. Whereas in the Tonnetz, a set of fundamental intervals generates a collection of tones, in the Speculum, a set of tones defines a collection of intervallic relationships. The tone F in Euler’s Speculum, notably is not a generator, but rather is a reference point; it, like the other tones, is generated by the genus of which it is a factor. Example 3. Tones on the Speculum represented by the factors of 675.

But it is Euler’s combinatorial conception of intervals on the Speculum—his recognition of intervals, and interpretations of intervals, as pathways therein—I would argue, that constitutes a more subtle connection between Euler, the nineteenthcentury tradition, and contemporary transformational theories. Hugo Riemann, for example, came to understand composite intervals as particular pathway classes on the Tonnetz, and even adopted a symbolic language using combinations of the symbols Q and T (for Quint and Terz, respectively) to identify those different pathways. For instance, Riemann (1914/15) symbolized the interval of the ascending whole tone as

410

E. Gollin

2Q (= QQ), a pathway comprising two ascending fifths; he symbolized the minor third as Q/T (= QT-1), a pathway comprising an ascending fifth and descending major third, and so on.3 I call Riemann’s intervals “pathway classes” because, unlike Euler, he does not make distinctions between differently-ordered steps or symbols (i.e. Riemann’s symbols commute). Riemann’s symbolic language and his path conception allowed him to make functional distinctions among tones, even when he later abandoned the premise of just intonation on the Tonnetz: to derive a functionally subdominant tone D in C-major Riemann goes down two fifths and up a major third from the tonic (Q-2T); a functionally dominant tone D, by contrast, lies two fifths above the tonic (Q2). I would argue that this path-combinatorial conception (consciously or unconsciously) underlies our contemporary interpretations of transformations as gestures within music-transformational spaces.4 For example, when we use the label PL to describe a harmonic gesture from E major to C major in neo-Riemannian theory, the label reflects our understanding of that gesture as a particular composite pathway within a space defined by fundamental relations such as L, P and R.5 Although Euler lacks the group-theoretical foundation of these later theories, his Speculum and his symbology for paths therein capture their essential idea.

References Cohn, R.: Neo-Riemannian Operations, Parsimonious Trichords, and Their Tonnetz Representations. Journal of Music Theory 41(1), 1–66 (1997) Euler, L.: Tentamen novae theoriae musicae ex certissimis harmoniae principiis dilucide expositae. St. Petersburg (1739) Euler, L.: De harmoniae veris principiis per speculum musicum repraesentatis Novi commentarii academiae scientarium Petropolitanae. 18, 330–353 (1774) Gollin, E.: Representations of Space and Conceptions of Distance in Transformational Music Theories. Ph.D. diss., Harvard University (2000) Mooney, M.K.: The Table of Relations and Music Psychology in Hugo Riemann’s Harmonic Theory. Ph.D. diss., Columbia University (1996)

3

Strings of letters such as QQ or QT-1 are known mathematically as “words” in the symbols Q and T. I explore words and their application to transformational music theory in “Form, Transformation and Climax in Ruth Crawford Seeger’s String Quartet, Mvmt. 3,” in the present volume, as well as in Gollin (2000). 4 I examine this idea at length in Gollin (2000). 5 L, P and R stand respectively for Leittonwechsel, Parallel and Relative, transformations that correspond to the German Leittonwechsel, Variante and Parallel relations respectively (relations that underlie the Scheinkonsonanzen in Hugo Riemann’s Funktionstheorie). In neoRiemannian theory, L, P and R are privileged because they represent ways of progressing between triads using minimal (or parsimonious) voice leading. For example, L maps a C-major triad to an e-minor triad (E and G are common tones and C moves by semitone to B) or viceversa; P maps an e-minor triad to an E-major triad (E and B are common tones and G moves by semitone to G#) or vice-versa. PL implies a composite motion in a space defined by parsimonious relations (i.e. the progression from E major to C major is implicitly mediated by e minor). Voice-leading parsimony and the centrality of the transformations L, P, and R in neoRiemannian theory is discussed in Cohn (1997).


411

Riemann, H.: Ideen zu einer Lehre von den Tonvorstellungen. Jahrbuch der Musikbibliothek Peters, 21–22, 1–26, (1914/1915) Vogel, M.: Die Musikschriften Leonhard Eulers. In: Speiser, A. (ed.) Leonhardi Euleri Opera omnia sub auspiciis Societas scientiarum naturalium Helveticae. Ser. 3, pp. 11–12. Orell Füssli, Zurich (1960)

Structures Ia Pour Deux Pianos by Boulez: Towards Creative Analysis Using OpenMusic and Rubato Yun-Kang Ahn, Carlos Agon, and Moreno Andreatta Institut de Recherche et Coordination Acoustique/Musique, 1 place Igor Stravinsky, 75004 Paris, France {yun-kang.ahn,agonc,moreno.andreatta}@ircam.fr

Abstract. Pierre Boulez introduced the concept of creative analysis (Boissière 2002) in the late 1980s suggesting that the aim of analysis should be the production of new pieces. Marcel Mesnage and André Riotte followed this path in their work on computer-aided analysis and composition (Mesnage and Riotte 1993). Our current study focuses on Ligeti’s analysis of Structures pour deux pianos Ia (first book in 19511952) by Boulez, where the compositional process is described in detail and further set a model. In Structures, Boulez uses series consisting of 12 pitches, 12 durations, 12 attacks and 12 dynamics borrowed from Messiaen’s Mode de valeurs et d’intensité (1949). Following Boulez’s analytic concept, our task is to start from his compositional model and consider how it might be used to produce another piece. Our study uses the software OpenMusic, which was developed at the IRCAM and is built on a functional paradigm. This graphical environment can be exploited to imitate the model used in Structures, stressing a functional point of view. Parallel to this, a second approach will be implemented using Rubato, an universal music software environment that has been developed at the University of Zurich. The keystone of this application is a categorical point of view, theorized by Guerino Mazzola. Since category theory and functional languages are strongly linked, the two software applications are complementary. However, Rubato brings a different level of abstraction and therefore offers new possibilities that have yet to be developed, for instance creating metapieces that could give different Structures.

1

Introduction

Musicology traditionnally distinguishes between analysis and composition. Indeed these represent two different disciplines although they cannot be completely separated. Boulez aims at reuniting them into what he calls creative analysis (Boissière 2002; Boulez 1989). Thus, a musical work may be considered as containing potentiality, i.e. the possibility of opening perspectives to new compositions. T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 412–418, 2009. c Springer-Verlag Berlin Heidelberg 2009

Structures Ia Pour Deux Pianos by Boulez

413

Evolution of the link between mathematics and music (Andreatta 2003) combined with the development of music through computer science, has been stimulating both composers and analysts to change their habits and has fostered interest in the formalisation of music. We begin by introducing the compositional process used by Pierre Boulez, and then present its implementation in OpenMusic. Afterwards we discuss the proposed implementation in Rubato, and how that would guide the future development of this work.

2 2.1

Compositional Process in Structures Ia Analysis of Constructional and Serial Principles: Decision and Automatism

As a basis for our study, we mainly use Ligeti’s famous analysis of this piece (Ligeti 1975; see also Eimert 1959; Feldman et al. 1952). He describes the respective roles played by compositional decision-making and a kind of automatism resulting from the serial techniques. Boulez was aiming to extend serial methods of composition (for details on his musical technique, see (Boulez 1963 and 2002)). He chose a twelve-tone series, which is arranged to follow the note-succession of “Division” from Messsiaen’s Mode de valeurs et d’intensités as an homage to his teacher (Fig. 1).

Fig. 1. The original series

From this series, Boulez generated two 12 ∗ 12 matrices : one with the 12 transpositions of the original series and a second one with the 12 transpositions of the inverted (by the first note) original series (Fig. 2). Paths through these matrices are chosen, organising serial threads defined by 12 pitches, durations, attacks, and dynamics, and thus extending usual serialist practice. 2.2

How to Create from an Analysis

This analysis of the compositional procedure leads us to ponder on the possibilities in order to compose. Ligeti copes with this problem by examining the balance between automatism and decision, i.e. what concerns the composer’s choice and what does not. From this model we can undertake a kind of metacomposition with two softwares packages proposing different approaches: OpenMusic and Rubato. Guerino Mazzola realised a piano sonata from an analysis he made of the Hammerklavier (op. 106) sonata written by Beethoven.

414

Y.-K. Ahn, C. Agon, and M. Andreatta

Fig. 2. One matrix and some paths used for the composition

3 3.1

An Implementation in OpenMusic: A Visual and Functional Environment Patches and Circularity

OpenMusic offers a human-machine interface which was originally explicitly dedicated to composition. Yet, as we will see, it may also be used in the context of compositionnally-oriented musical analysis (Agon et al. 2004). Furthermore, since at this time theorists are not so commonly using computers in their work there are few accessible softwares packages specifically designed for musical analysis. The basic unit of programming in OpenMusic is a patch. A patch may consist of functions linked up to create a flowchart representing a task. Based on a functional paradigm (LISP/CLOS), OpenMusic offers the advantage of recursivity a patch can contain other patches, inclunding copies of itself. Figure 3 shows the patch allowing the creation, given the original series as input, of the two sections of the piece. 3.2

Composing Following the Model with the Benefit of a Graphical Composition Environment

Given a twelve-note series, the patch applies the same process that Boulez defined. The patch produces a score where bundles defined by Boulez are displayed in tracks (Fig. 4). Each track can then be modified directly. According to Ligeti’s analysis, register (i.e. which octave do we select for the related pitch-set) is an important parameter which may be modified in the process of composition. We choose not to control dynamics as they are difficult to set up in OpenMusic. This emphasizes the distinction between automatism (the procedures contained in the patch) and decision (the allowed modifications on the result). Thus the user is able to create “errors” like Boulez did in his piece (according to Ligeti (1975)).


415

Fig. 3. The patch in OpenMusic reproducing the compositional process of Structures Ia

Fig. 4. The produced score in OpenMusic

4 4.1

Rubato: A Higher Level of Abstraction with a Categorical View Different Perspectives Delivered by Rubato

Rubato is based on a music theoretical paradigm introduced by Mazzola (2003) and on mathematical category theory (especially the theory of topos), pointing out transformations rather than objects (this concept exists in musical analyses using the methods of transformational analysis developed by David Lewin 1982). Figure 5 shows the Rubato environment.

416


Fig. 5. The Rubato environment and an example of network

4.2

Possibilities Brought by Rubato

This software involves higher levels of abstraction. For instance, it deals with generalized musical objects. Indeed, objects like melody are defined, allowing new perspectives. This frame fits the combinatoric point of view of Boulez’compositional process, allowing new perspectives in developing a kind of metacomposition which will be able to foster different pieces. 4.3

Scheme of the Construction

The aforementioned theory allows us to speak of generalized points in space (i.e. addressed points instead of traditional points known from standard mathematical approaches). This means that we are rather interested in morphisms (here module morphisms) than in objects. The key concept is to rely on the module Z11 which is strongly related to the structure of the piece, with a twelve-tone qualities series and the 12 ∗ 12 matrices. Then the idea is to associate the 12 basis vectors of Z11 to each note (i.e. create a module morphism) (Fig. 6), and generalize this process associating them to each row of the matrices. Then permutations of basis vectors will allow to generate the piece. The result is displayed in figure 7, which is a pianoroll -like display of the score produced by Rubato.

Fig. 6. Association of the 12 basis vectors of Z11 with 12 notes


417

Fig. 7. The Rubato “score” of Structure Ia

5

Conclusion

OpenMusic and Rubato offers tools to plan compositional models and consequently allows composers to modify their vision and enlarge their possibilities. The respective general models brought about by these analyses may be subjected to further modifications which subsequently generate new compositions. This approach may be initially generalized to other parameters (like instruments, etc.). Furthermore, the process used by Boulez may be thought of in terms of permutations (and permutation-groups) which could lead to new directions, i.e. metacomposition realised by another group of permutations which may create a set of new Structures.

References Agon, C.: OpenMusic: Un langage visuel pour la composition musicale assistée par ordinateur. PhD diss., Paris VIII University (1998) Agon, C., Andreatta, M., Assayag, G., Schaub, S.: Formal Aspects of Iannis Xenakis’ Symbolic Music: A Computer-Aided Exploration of Compositional Processes. Journal of New Music Research 33(2) (2004) Andreatta, M.: Méthodes algébriques dans la musique et musicologie du XXe siècle: aspects théoriques, analytiques et compositionnels. PhD diss., EHESS/Ircam (2003) Babbitt, M.: The function of Set Structure in the Twelve-Tone System. PhD diss., Princeton University (1946/1992) Boissière, A.: Geste, interprétation, invention selon Pierre Boulez. Revue DEMéter. Lille-3 University (2002) Boulez, P.: Jalons (dix ans d’enseignement au Collège de France). Bourgeois (1989) Boulez, P.: L’écriture du geste. Bourgeois (2002) Boulez, P.: Penser la musique aujourd’hui. Ed. Gonthier (1963) Eimert, H.: The composer Freedom and Choice. Die Reihe III. Universal Edition London and Theodore Presser Company (1959)

418


Feldman, M., et al.: 4 Musicians at Work. Transformation 1(3) (1952) Forte, A.: The Structure of Atonal Music. Yale University Press. (1973) Lewin, D.: Generalized Musical Intervals and Transformations. Yale University Press (1982) Ligeti, G.: Pierre Boulez - Decision and Automatism in Structure IA. Die Reihe 4, 36–62 (1975) Mesnage, M., Riotte, A.: Modélisation informatique de partitions, analyse et composition assistées. Cahiers de l’Ircam 3 (1993) Mazzola, G.: The Topos of Music. Birkh¨ auser, Basel (2003)

The Sieves of Iannis Xenakis Dimitris Exarchos Goldsmiths, University of London, United Kingdom [email protected]

Abstract. Xenakis’s article ‘Sieves’ was published in 1990, but the first extended reference to Sieve Theory is found in the final section of ‘Towards a Metamusic’, published 1967. These two writings mark two periods, where there is a progression from the decomposed formula to the simplified one. This progression reflects Xenakis’s aesthetic of sieves as timbres and is here explored under the light of the idea of inner periodicities.

1 Introduction A sieve refers to a selection of points on a straight line and involves the combination of two or more modules. A module is notated by an ordered pair (M, I) that indicates a modulus (period) and a residue (an integer between zero and M – 1). These modules can be combined by the logical operations of union (+), intersection (·), and complementation (–). In this paper I will demonstrate that, in his later period, Xenakis used only the operation of union and that he abandoned the other two. The diatonic major scale (with period 12 semitones in twelve tone equal temperament) can be expressed as follows (where C = 0 and the unit distance is the semitone): (4, 0)·[(3, 0) + (3, 1)] + (4, 1)·[(3, 0) + (3, 2)] + (4, 3)·[(3, 1) + (3, 2)] + (4, 2)·(3, 2). In this expression, M = 12 is decomposed into elementary moduli M1 = 4 and M2 = 3. The Lowest Common Multiple (LCM) of the elementary moduli is equal to the period (12). There might be several equivalent formulae for a given sieve as we can choose among several alternative decompositions of the modulus. This redundancy of formulae is overcome by prime factorisation, as this is aimed at rendering a decomposed form of a composite number. According to the Unique Factorisation Theorem any integer α > 1 , can be uniquely written as

α = p1l ⋅ p 2 m ⋅ ... ⋅ p k n , where p1, p2, …, pk are prime numbers, p1 < p2 < … < pk and l, m, n ∈ *; this is called the canonical form of Į. It is easily inferred from the theorem, that since two

numbers Į, ȕ, are coprime then α , β (where Į, ȕ, m, n ∈ *) are coprime as well. The canonical form of 12 is 22·3. When these factors correspond to moduli, the literal intersection would have to be (2, I1)·(2, I2)·(3, I3). It is obvious that (2, I1)·(2, I2) is not a valid option. In the case that I1 = I2 the intersection implies that (2, I1) = (2, I2). m

n


420

D. Exarchos

In the case they are different the intersection is empty. We should therefore resolve any exponentials before treating prime factors as the elementary moduli of a period:

12 = 2 2 ⋅ 3 = 4 ⋅ 3 . Consequently, there is one practical limitation in this process: the resolution of all powers prior to the decomposition of the moduli is not possible for all composite moduli. A modulus cannot be decomposed if it is a prime power. Thus, both primes and prime powers represent moduli that are non-decomposable.

2 Types of Formulae When the period of a sieve is taken into account its formula can have two forms: decomposed or simplified. The decomposed formula is the one that employs only moduli that are primes or prime powers. These are the prime factors (or the prime powers that derive from them) of the sieve’s period. Of this type is the formula that uses moduli 4 and 3 to express a sieve whose period is 12. The decomposition of a module is expressed by intersection; thus, a decomposed formula frequently includes intersections. Less frequently, a decomposed formula might involve only union of modules, such as sieve (4, 0) + (3, 0), with period 12.1 A simplified formula that expresses the period of the sieve consists only of unions of single modules and includes composite moduli.2 In the example of the diatonic major scale it is the formula that is based on the octave: (12, 0) + (12, 2) + (12, 4) + (12, 5) + (12, 7) + (12, 9) + (12, 11).

3 Symmetries/Periodicities A formula might not necessarily represent a sieve according to a single modulus, i.e. its period. There is another distinction between two types of formulae: the one that is based on the external period and the one that ignores it. Xenakis constructed an algorithm (and a computer program based on it) that produces a formula by assigning the smallest possible modulus to each point of the sieve. I will examine the construction of such a formula in Section 3. This distinction is parallel to the decomposed or simplified one: we can start with a simplified formula that ignores the external period and decompose each modulus, if it is a composite number. The result is a decomposed formula that is not based on the external period of the sieve. The information that a formula reveals when it is not based on the external period is a very important issue which is pertinent both to the aesthetics of sieve-construction and to sieveanalysis. 1

Note that one can always construct a formula with elementary moduli that are not primes or prime powers; e.g. 60 can be decomposed to 5·12 instead of 4·3·5 (60 = 22·3·5). This decision depends on whether one wishes to express the period as the LCM of two factors instead of three. This particular decomposition (i.e. 60 = 5·12) was used by Xenakis for the rhythmic sieves of Persephassa (1969, for six percussionists) (see Gibson 2003, 58ff). 2 Ariza refers to ‘sieve models’ instead. What I refer to as ‘decomposed formula’ and ‘simplified formula’, he refers to as ‘complex sieve’ and ‘simple sieve’. He argues that the latter model ‘fails to incorporate aspects of the original [the former]’ (2005, 44).

The Sieves of Iannis Xenakis

421

In Xenakis’s sieves the intervallic structure is highly irregular and asymmetric, while the period of the sieve was rarely intended to be audible. Sieve Theory was developed in order to study hidden symmetries. Xenakis demonstrated that an elementary modulus is a kind of tempered chromatic scale, with unit set to an interval other than the semitone. He pointed at an observation Bertrand Russell made in relation to the axiomatics of numbers and he showed that the tempered chromatic ‘has no unitary displacement that is either predetermined or related to an absolute size’ (Xenakis 1992, 195). We can have chromatic (or equidistant) scales of quarter-tones or semitones, but also of whole tones, perfect 4ths, etc (see Varga 1996, 94-5). The chromatic scale of the perfect 4th is a module with M = 5. The sieves following Jonchaies (1977, for orchestra) share a certain aesthetic: they are characterised by an irregular distribution of a set of intervals, between the semitone and the major 3rd. The construction of these sieves is inspired by the Javanese pelog with its interlocking 4ths (see Varga 1996, 144-5). In the following quotation Xenakis talks about the rhythmic structures of Jonchaies, but the same principle was applied to the construction of his sieves: We can illustrate regular events by points an equal distance apart. On a second, lower parallel line, more points represent other regular patterns with a different time unit, so they are shifted with respect to the first line’s points even if they start together. This procedure can be repeated with regular points on other lines. When we hear all these lines together, we obtain a flow of events which consists of a regular intervallic series, but which as a whole is impossible to grasp (Xenakis 1996, 148). The modules in a simplified formula represent ‘chromatic’ scales (with different units and starting points), periodicities, or symmetries (cf. Xenakis 1992, 270). Here symmetry has a broad meaning: it stands for regularity in general. In this sense, every irregular scale can be broken down into a multiplicity of regularities.

4 Inner-Periodic Formula 4.1 Inner Periodicities and Formulae Redundancy The inner periodicities of the sieve are shown by a simplified formula in the form of modules. The multiples of each modulus produce some of the points of the sieve. When the period is not known (or not taken into account), a simplified formula is based on the inner periodicities and is inner-periodic. The redundancy of inner-periodic formulae can be overcome by checking every single point of the sieve and assigning to it the smallest possible modulus. We can find for each point, the module with the smallest M, that either starts on this point or produces it later. These are in fact two different approaches. The former method was for Xenakis an earlier stage before arriving at the latter, implemented in his final analytical algorithm. 4.2 Construction of the Inner-Periodic Simplified Formula In order to demonstrate the progression to this inner-periodic conception of sieves, let us take for example the sieve of Akea (1986, for piano and string quartet) as found in

422

D. Exarchos

Xenakis’s pre-compositional sketches (see Figure 1).3 It consists of 37 points, in a range of 80 semitones, and its intervallic structure is asymmetric (non-palindromic). In Akea the sieve is not used in any way that might reveal the existence of a periodicity; this is confirmed by the fact that in the sketches Xenakis used only an innerperiodic simplified formula (where the bottom pitch C1 = 0 and C4 = middle C): (5, 40, 8) + (14, 9, 5) + (14, 52, 2) + (15, 10, 4) + (15, 36, 2) + (17, 28, 3) + (18, 5, 4) + (19, 18, 3) + (19, 22, 3) + (19, 27, 2) + (19, 32, 2) + (20, 31, 2) + (23, 14, 2) + (24, 7, 3) + (25, 0, 3) + (25, 16, 2).

Fig. 1. Sieve of Akea

There are 16 modules in the sieve and they share 10 moduli. Here the modules are notated as (M, I, R), which stands for: Modulus, Initial point, Reprises of the Modulus. We see that Xenakis did not reduce the residues according to the moduli. Thus, with (5, 40) Xenakis indicates that a periodicity of 5 semitones starts at point 40 (here E4). The third entry in the brackets (R) shows the number of repetitions of each module.4 4.3 Analytical Algorithm: Early Stage The method Xenakis used to arrive at the above formula is not identical to the algorithm he presented in his article ‘Sieves’ (1990, 64-5; 1992, 274-5). It is a precursor of this algorithm. This earlier algorithm goes through the following steps: (a) Each point is considered as a point of departure (I) of a modulus (M). We start testing the first point (In) with M = 2 and check if: (i) its multiples produce only points that belong to the given sieve, and (ii) it produces at least one of the not-yet-covered points of the given sieve. (b) If (i) is not satisfied we pass on to M + 1. If it is satisfied we keep the module and check if (ii) is satisfied: if yes, we keep the module and pass onto the next point (In+1); if not, we ignore the module and pass onto In+1. (c) We stop when every point of the sieve has been covered by a module.

3 4

This sieve was also used in Ata (1987, for orchestra). This is slightly different from Xenakis’s program, which uses R to denote the number of points covered. Following Xenakis’s own practice in the sketches, I will use R to denote the occurrences of a modulus, instead of the number of points covered. This indicates more effectively the contribution of each modulus to the inner-periodic structure of a sieve.


423

4.4 The Condition of Inner Periodicity The modules in a simplified formula sieve can be considered as inner periodicities only if they repeat at least twice. Thus, a module must cover at least three points. Three equally distant elements produce two equal intervals (modulus) that make it possible to compare them and identify them as (two occurrences of) a periodicity. This is precisely Xenakis’s view of sieves as outside-time structures. In his article ‘Symbolic Music’ he referred to the stages of temporal perception: three successive events a, b, c, ‘divide time into two sections [that] may be compared and then expressed in multiples of a unit’ (Xenakis 1992, 160). Although he talks about time here, he stressed that the temporal algebra time-intervals require is identical to that of outside-time structures (such as sieves). In order for R 2, the size of the modulus must be less than half the distance between the starting point of the module and the highest point of the sieve. If n is the highest point of the sieve, this condition of inner periodicity is formulated as follows: for each module in a simplified formula it must be true that

M≤

n−I 2

in order for R 2, and for the sieve to be inner-periodic. I will refer to modules with R 2 as periodic modules. The formula in the sketches of Akea includes only periodic modules. This corresponds to a an ‘inner-periodic’ conception of sieves that Xenakis maintained throughout his application of Sieve Theory in his later music. 4.5 Analytical Algorithm: Final Stage The method Xenakis used to produce the formula of the sieve of Akea differs from the final algorithm of 1990 in one significant aspect: the starting point of a module is not necessarily smaller than the size of the modulus. In other words, if it is enough for a modulus to end at a distance from n smaller than its own size, it is not necessary to depart as close to the lowest point. Such is the case of module (5, 40): modulus 5 is present only for the second half of the sieve. This means simply that the perfect 4th is characteristic for only a part of the sieve and it is a consequence of not limiting the size of I to be smaller than M. On the contrary, module (25, 0), whose range is 75 semitones, is present from the bottom almost to the top. More specifically, both the initial and the final points of (25, 0) lie at a distance from the edges of the sieve smaller than the size of M. As the algorithmic process goes through the higher points, the possibility of finding smaller periodicities naturally increases. A sieve is conceived as a multiplicity of periodicities and analysis should not depend on an unbalanced favouring of smaller moduli when progressing towards higher points. This is the reason why Xenakis added one more step in the final version of his analytical algorithm. The additional condition in the final algorithm reintroduces the idea of the residue: in all modules it must be true that M > I. This is shown in the final step of the algorithm as published in the 1990 article: ‘we ignore all the [modules] which, while producing some of the not-yet-encountered points of the given series, also produce, upstream of the index I, some parasitical points other than those of the given series’ (Xenakis 1990, 65; 1992, 275; italics added). Therefore, the algorithm

424

D. Exarchos

does not look merely for the smallest modulus that departs from the point under consideration, but for the smallest modulus that starts at a point smaller than its own size and that produces the point under consideration (unless this point is located early enough in the sieve so that it is itself the starting point). This is the algorithm that Xenakis’s analytical computer program is based on (see Xenakis 1990, 76ff; 1992, 285-8).5 The program checks for the residue class with the smallest modulus, whose members belong to the sieve and include the point under consideration. The formula that the program suggests for the sieve of Akea has 17 modules that share 13 moduli: (14, 9, 5) + (15, 10, 4) + (18, 5, 4) + (19, 18, 3) + (20, 5, 3) + (23, 14, 2) + (24, 7, 3) + (24, 22, 2) + (25, 0, 3) + (25, 16, 2) + (26, 10, 2) + (27, 5, 2) + (27, 25, 2) + (28, 0, 2) + (28, 27, 1) + (29, 22, 2) + (31, 9, 2). We see that the moduli are now in average greater than the ones in the formula Xenakis used. As I increases, the final version (contrary to the earlier) finds gradually greater moduli. 4.6 The Condition of Inner Symmetry If it is true that M > I for all modules in a simplified formula, then this formula can be used to produce the inversion of the sieve. To construct the formula of the inversion, we keep the moduli of the original and replace every I by n − ( R ⋅ M + I ) . When all the inner periodicities extend as far as possible to both directions in a sieve, and therefore could also produce its inversion, then the sieve bears a certain kind of symmetry. But this symmetry is still a hidden one: the intervallic structure of such a sieve might very well be asymmetric, i.e. non-palindromic. Thus another level of symmetry is revealed through the distribution of the inner periodicities. In analogy with the inner-periodic nature of Xenakis’s sieves of the later music, I will refer to this kind of symmetry as inner symmetry and to sieves that exhibit inner symmetry as inner-symmetric.6 An inner-symmetric sieve is necessarily inner-periodic. This is because the algorithm always finds a modulus that meets at least two points when I < n 2 .7 But with the condition of inner periodicity a module must meet at least three points, i.e. R 2, which affects the maximum size of M. With the additional limiting of the size of M to values greater than I, M depends on the varying values of I for both its minimum and maximum values. Since we operate in discreet space, I < M is equivalently expressed as I + 1 M. Therefore, the condition of inner symmetry, which incorporates that of inner periodicity, is formulated as follows:

I +1 ≤ M ≤ 5

n−I . 2

(1)

This program is provided in Squibbs (1996, 291-303) with no typographical errors. In (1996, 149) Xenakis refers to ‘inner symmetries’; this is what I here refer to as ‘inner periodicities’. 7 Since the absolute value of I affects the size of M, we should calculate the formula with the sieve’s lowest point set to 0. 6


425

Inner periodicity was defined only as a condition for inner symmetry. It is important to stress here that both of them are integrated into one. I used the former to demonstrate the earlier stage of Xenakis’s method, which was extended by his final algorithm. Xenakis’s prompt to study the hidden symmetry of a sieve referred to both symmetries and periodicities, in the form of moduli. These two notions are then themselves integrated; inner symmetry is achieved by the analysis and synthesis of inner periodicities. Thus, an outside-time characteristic (symmetry) is achieved through the treatment of an inside-time one (periodicity). In this sense, an inner-symmetric sieve is necessarily inner-periodic. 4.7 Inner-Symmetric Analysis In fact, most of Xenakis’s sieves of the later period include one or two non-periodic modules. In the sieve of Akea this is the case with module (28, 27). The points that module (28, 27) covers are 27 and 55 (D#3 and G5); from these points, 27 is covered only by this module, but 55 is also covered by two additional periodic modules. Therefore, D# is the only element of the sieve of Akea that does not belong to an inner periodicity. This slight deviation from complete inner symmetry is perhaps one of Xenakis’s typical aesthetic criteria; I will examine this further after I formulate in more detail the consequences of the condition of inner symmetry. The condition of inner symmetry also implies that the minimal and maximal permissible values of M for a given I, increase and decrease respectively as I increases. This allows for the determination of the absolute maximum permissible value for I, for inner-symmetric sieves, given a constant n. The smallest permissible I is naturally 0. The maximum permissible value of I in inner symmetric sieves is found at the point of convergence of the two tendencies of the value of M as I increases. Let us indicate the value of M at this point as Mc. Then from (1) follows that

I +1 = Mc =

n−I . 2

Consequently,

I +1 =

n−I n−2 2 I + 2 = n − I 3I = n − 2 I = . 2 3

Therefore, the maximum value of I for any inner-symmetric sieve is

I=

n−2 . 3

For the sieve of Akea, or for any inner-symmetric sieve with n = 80, the greatest possible I is (80 − 2) 3 = 26 . Since the value of I when M = Mc is the maximum and since it depends only on n, we can locate the value of M at this point (Mc), according to (1), as follows:

I +1 = M c I = M c −1

426

D. Exarchos

and

n − ( M c − 1) n−I n +1 Mc = 2 M c = n − M c + 1 3M c = n + 1 M c = 2 2 3 That is, in inner-symmetric sieves, Mc (i.e. the M-value when I is maximal) is

Mc =

Mc =

n +1 8 . 3

For any inner-symmetric sieve with n = 80, the inner-periodicity starting at the maximum I is M c = (80 + 1) 3 = 27 . In the sieve of Akea, module (28, 27) has the smallest M that could be assigned to I = 27, but this value of I exceeds by one semitone the maximum limit: I = ( n − 2) 3 = 26 . The values M can take a range 9

between 1 and n 2 . The critical point Mc is the value that determines, not merely the highest permissible I, but the behaviour of the values of I as M increases. In fact, Mc is the value of M that determines which part of the inequality in (1) defines the maximum values for I: when

M ≤ Mc ,

I +1 ≤ M I ≤ M −1;

when

Mc ≤ M ,

M≤

n−I 2M ≤ n − I I ≤ n − 2M . 2

As the value of M increases towards Mc, the maximum value of I also increases; as the value of M increases beyond Mc, the maximum value of I decreases. The upper curve in the chart of Figure 2 shows the maximal values of I for every possible M in any inner-symmetric sieve of n = 80. The x-axis represents all the consecutive values of M between 1 and 80 2 = 40 ; the y-axis represents the values of I. We see that the values of I increase linearly up to Mc: for the left part of the chart I M – 1. As the value of M increases beyond Mc, the values of I decrease linearly: for the right part of the chart I n – 2M. The chart demonstrates a general property of inner-symmetric sieves; for a given n, the extreme values of M and I are related as follows: when M is minimal (M = 1), I is minimal (I = 0); when M is maximal ( M = n 2 ), I is minimal; when I is minimal, M can take any permissible value; when I is maximal ( I = ( n − 2) 3 ), M = Mc. The graph of Figure 2 is a chart for the values of M and I, that accounts for the inner symmetry of a sieve. Since these values depend only on the size of n, the above relations of the values and their limits enable one to construct sieves that satisfy the condition of inner symmetry. If a module appears below the upper curve it repeats at least twice. Therefore, when all modules appear under this limit, the sieve is inner-symmetric. The modules in the formula of the sieve of Akea are shown in this chart by dots followed by the pair (M, I). The chart offers a synoptic view of the inner symmetry of the sieve: as I have shown, module (28, 27) repeats only once and therefore lies outside the upper curve. 8 9

When the result is not a whole number, Mc is equal to the lower integer. Although Xenakis (1992, 274-5) defines the first step of his algorithm to start at M = 2, his program starts testing at M = 1. Module (1,0) is then by extension the total chromatic.


427

30

28, 27 25

27, 25 24, 22

29, 22

20

19, 18 25, 16

I 15

23, 14 15, 10 14, 9

10

26, 10

31, 9

24, 7 5

18, 5

20, 5

27, 5 25, 0

28, 0

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 29 30 31 32 33 34 35 36 37 38 39 40 M

Fig. 2. Inner Symmetry Chart for the Sieve of Akea

In inner-symmetric sieves the number of reprises of the modulus (R) depends also on the value of I. Whereas the condition of inner symmetry determines the minimal R (R 2), the value of I determines whether R will take its maximal or its sub-maximal value: when

I ≤ n(mod M ) ,

when

n(mod M ) < I < M , R =

R=

n − n(mod M ) ; M n − n(mod M ) − 1. M

The value of R for a given module can be seen easily in the inner symmetry chart. The values of n(modM) for every M are shown by the dotted zigzag curve. When a module lies on or below the n(modM) curve [which implies that I ≤ n(mod M ) ] R is maximal; when a module lies above the n(modM) curve [which implies that n(mod M ) < I < M ] R is sub-maximal. When a module is located towards the right part of the chart, its M is higher and its R-value is lower. The additional n(modM) curve helps one to locate, in a glance, the most periodic modules in the sieve: these would be located towards the left and under the dotted curve. When the modules of a simplified formula tend to concentrate to this area they are more periodic (their R-value is higher). The area of the chart where 21

M 26 between the two curves and M 26 below the upper curve would accommodate modules with R = 2. The area bellow the dotted curve for 21 M 26 and above the dotted curve for 17 M 20 is for R = 3. The R-values increase as we move to the previous peaks of the n(modM) curve. I will refer to this type of chart as inner symmetry chart. Seven modules of the sieve of Akea lie beneath (or on) the n(modM) curve and therefore their R is maximal.

428

D. Exarchos

4.8 Modules and Degree of Symmetry

As I have shown, the greater the size of M, the smaller the value of R. The value of R also depends on the value of I. Specifically, the value of I determines whether R will have its maximal or its sub-maximal value for a given M. Finally, the value of R depends on the total number of modules in a formula. In the sketches of Akea Xenakis noted that there are many modules in the formula and this means that only few of them repeat continuously.10 In other words, only few will have a high R-value. The characteristic set of intervals of Xenakis’s sieves of the 1980s affect their density, which remains in a constant average value for most of the sieves of that period. This density can be denoted by the ratio of the number of elements to the range of the sieve. The sieve of Akea has density D = 37 80 ≈ 0.46 . Note that the number of modules in the simplified formula does not reflect this density, since a point might be covered by more than one module. For this reason, the relationship between density and the number of modules cannot be formulated rigorously. In general though, the larger the number of the modules in a formula, the lower the average value of R; and the lower the average value of R, the larger the average size of M. Consequently, the average size of M depends on the number of modules: for a given density, the large number of modules is compensated by large moduli. Sieves with similar density to the one of Akea are expected to appear either similarly in the chart of Figure 2, or with less modules but concentrated to the left. The large number of modules in a formula accounts for the ‘hidden’ symmetry. The leftmost part of the chart indicates a more superficial symmetry and the rightmost part a deeper one. When the inner-periodic simplified formula includes only periodic modules, a large number of modules indicates a low degree of symmetry, in the sense that absolutely symmetric sieves have only one module (such as a chromatic scale). But the sieves of Xenakis’s later music usually include one or two non-periodic modules: the irregularity of their intervallic structure reaches the limit of inner-symmetry. Analysing the inner symmetry of a sieve, is not aimed merely at classifying sieves in one of the two categories: (inner-)symmetric and asymmetric. Both notions of symmetry and asymmetry are crucial to the aesthetics of Xenakis’s music: they are ‘the two poles between which music goes back and forth, and the first suggestion of a solution comes from distributing points on a line’ (Xenakis 1996, 147). With the help of the inner symmetry chart, we can see not only whether a module is periodic or not (for this the R-value would suffice), but also how distant a module is from the limits of inner symmetry; and with the inclusion of the n(modM) curve, the inner symmetry chart shows both a general and a more detailed picture of the play between symmetry and asymmetry. 10

‘Beaucoup de (M, I, R) car #S = 17 vue[?] 37 points du crible. Donc peu de périodes d’une seule traite’. In my translation: ‘Many (M, I, R), since #S = 17, given the 37 points of the sieve. Therefore, few periods without stopping once’ (Xenakis, Iannis. Pre-compositional sketches of Akea. Bibliothèque Nationale de France, Musique). Here Xenakis uses ‘S’ to denote the number of modules. When he calculated the formula by hand, he apparently made a mistake and kept module (19, 46) which is included in (19, 27); this is not shown in the formula as reproduced in this paper (therefore the number of modules is shown to be 16). The question mark denotes illegible handwriting.


429

References Ariza, C.: The Xenakis Sieve as Object: A New Model and a Complete Implementation. Computer Music Journal 29(2), 40–60 (2005) Gibson, B.: Théorie et pratique dans la musique de Iannis Xenakis: À propos du montage. PhD diss., École des hautes études en sciences sociales (2003) Squibbs, R.: An Analytical Approach to the Music of Iannis Xenakis: Studies of Recent Works. PhD diss., Yale University (1996) Varga, B.A.: Conversations with Iannis Xenakis. Faber & Faber, London (1996) Xenakis, I.: Sieves. Perspectives of New Music 28(1), 58–78 (1990) Xenakis, I.: Formalized Music: Thought and Mathematics in Composition, edited by Sharon Kanach. Pendragon Press, Stuyvesant (1992) Xenakis, I.: Determinacy and Indeterminacy. Organised Sound 1(3), 143–155 (1996)

Tonal, Atonal and Microtonal Pitch-Class Categories Fernando Gualda Sonic Arts Research Centre Queen's University Belfast, UK [email protected]

Abstract. This paper reviews and generalizes Pitch-Class Set Theory using Group Theory (groups acting on pc-sets) and Category Theory, which provides methods for mapping the structure of a n-tone system onto another m-tone system. This paper also suggests a new implementation approach that represents pitch-class sets as bit-sequences, which are equivalent to integer values. Forte’s best normal order is shown to be equivalent to the smallest integer, among the cyclic permuted pc-sets. Further, transposition of pc-sets is shown to be equivalent to bit-shifts; and their inversion, to bit-reversal. The tonal (diatonic) pc-category is presented as a subset of the atonal (12-tone) pc-category, which, similarly, can also be contained in a microtonal pc-category. Functors between those categories present properties that preserve relationships while still using the same operations: tonal relationships are preserved even though atonal music operations, such as transposition and inversion are applied, allowing motives to be mapped into different modes, scales, or even microtonal scales. The appendix offers an implementation of this new approach to calculate and represent pc-sets with an arbitrary number of pitch-classes.

1 Introduction Pitch-classes are generally defined as musical notes assigned to fixed integer values given by the reminder of octave division by 12, as depicted in the top row of figure 1:

Fig. 1. Two representations of pitch-classes: 12-tone (top) and a diatonic major subset (bottom) T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 430–440, 2009. © Springer-Verlag Berlin Heidelberg 2009

Tonal, Atonal and Microtonal Pitch-Class Categories

431

This paper suggests different representations of pitch-classes depending on their context, such as the subset of white piano keys (bottom row of figure 1), which could be seen as a diatonic major scale starting at the pitch C. Pitch-classes are generally represented as mathematical sets. Nevertheless, agreements on a standard notation for sets of pitch-classes have not yet been reached. This paper uses the following notation conventions: [2, 0, (n-1), …]n representing unordered sets, and [0123…(n-1)]n meaning partially ordered sets. In both cases n represents the total number of pitch-classes in context, therefore not the cardinality of proper subsets. Whenever n is not given, it follows the traditional context of 12 pitch-classes. For the sake of clarity, nomenclature related to pitch-class set theory is henceforth italicized. Representation of notes as pitch-classes assumes octave and enharmonic equivalences. Within tonal music, however, enharmonic equivalence is not assumed: besides rules of voice-leading, pitch spelling depends on musical key or tonal center, according to which pitches are contextualized. Further, octave equivalence is not assumed in case of extended chords such as ninth or eleventh chords that are common in jazz and other musical styles.

2 Applying Pitch-Class Set Theory on Sets with Cardinality (Pitch-Classes) Other Than 12 The term Pitch-Class Set (pc-set) as used in musical analysis and composition was introduced by Babbitt (Forte 1973). Pitch-Class Set theory and terminology was developed by Forte (1973), Rahn (1980), and Straus (1990), among others. Equivalence of pc-sets had been extensively discussed. Morris (1982) compared four distinct approaches to partitioning the universe1 (power set) of pc-sets. The main operations on pc-sets are transposition and inversion. Forte defined the terminology for classification of pitch-class sets: pc-sets can be reduced to a normal order, in which pc-sets are ordered according to their pitch-classes. Normal orders are cyclic permutations (transpositions) of the best normal order, which is the permutation that presents the smallest, ordered intervals, starting at pitch class zero. Finally, the prime form is the ‘best’ (smallest) between the best normal order of a given pc-set and the best normal order of its inversion. Transposition equivalence of a pc-set within the universe (power set) of a set of twelve pitch-classes is isomorphic to chromatic transposition, in which all (chromatic) intervals between pitch-classes are maintained. Considering, however, a power set of a set of seven pitch-classes that form a diatonic major scale, diatonic triads can be seen as circular permutations of the same pc-set7, preserving their tonal relationships, while still being a result of the same operations used on pc-set12: 1

“U*, the universal set of 4096 pcsets, has been partitioned into collections of "equivalent" sets in a number of different ways. In the four systems we shall examine, U* is initially divided into collections of sets of equal size. Pcset equivalence is further defined by intervalclass content or by identity under certain twelve-tone operators, which may be any combination of Tn, I, and/or M.” (Morris 1982, 102) The ‘universal set’ described by Morris is a power set. Classification of pc-sets in this paper is based on group action and is in accordance with Forte (1973). The discussion on classification regarding interval classes is beyond the scope of this paper.

432

F. Gualda

Fig. 2. Permutations of diatonic triad chords: as pc-set7 (top) and pc-set12 (bottom)

Inversion equivalence of 12-tone pc-sets does not necessarily preserve tonal relationships. The prime form of major triads [047]12 are [037]12, which are minor triads. Mappings that preserve tonal relationships under transposition and inversion are presented in section 3 of this paper.

3 Pitch-Class Set Theory within a Bit-Sequence A power set is defined as the set of all subsets of a given set. Its cardinality is thus given by:

ȁܲሺܵሻȁ ൌ ʹ௡ ,

(2.1)

in which n is the number of elements (cardinality) of the underlying set. Conveniently, power sets are isomorphic to binary representation, in which each element (of the underlying set) corresponds to a binary digit. So, if n elements of a set are assigned to values ranging from 0 to n-1, the kth element of the power set is given by:

ܲ௞ǡ௡ ൌ σ ʹ௣ ,

(2.2)

where k ranges from 0 to 2n-1, and p are integers assigned to the elements of the underlying set, ranging from 0 to n-1. Thus, the universe of pc-sets with n pitchclasses is isomorphic to the set of all bit-sequences with n bits. Table 1. Some examples of equivalences between transposed pc-sets and their binary and integer representations Set Class2 0-1 1-1 3-11 3-11B 4-27 4-27B 6-35 7-35

2

T0 ∅ [0] [037] [047] [0258] [0368] [02468A] [013568A]

Binary (T0) 000000000000 000000000001 000010001001 000010010001 000100100101 000101001001 010101010101 101010110101

Forte’s set name (set-class).

Integer 0 1 137 143 293 329 1365 2741

T5 ∅ [5] [058] [059] [157A] [158B] [13579B] [13568AB]

Binary (T5) 000000000000 000000100000 000100100001 001000100001 010010100010 100100100010 101010101010 110101101010

Integer 0 32 289 545 1186 2338 2730 3434


433

Table 2. Some examples of equivalences between inverted pc-sets and their binary representations Set Class3 1-1 3-11 3-11B 4-27 4-27B 6-35 7-35

T0 [0] [037] [047] [0258] [0368] [02468A] [013568A]

Binary (T0) 000000000001 000010001001 000010010001 000100100101 000101001001 010101010101 010110101011

Integer 1 137 143 293 329 1365 1451

I [0] [059] [058] [047A] [0469] [02468A] [13568AB]

Binary (I) 000000000001 001000100001 000100100001 010010010001 001001010001 010101010101 010110101011

Integer 1 545 289 1169 593 1365 1451

Table 3. Nomenclature correlations between pc-set theory, group theory and the new, suggested implementation PC-Set Theory PC-Set Total pitch-classes Transposition Inversion Best Normal Order Prime Form

Group Theory (group action) Element Group order Circular permutation (cyclic group) Reflection (dihedral group) Equivalence-class representative (cyclic) Equivalence-class representative (dihedral)

Binary representation Bit-sequence Number of bits Circular bit-shift Bit-reversal Smallest integer (Tn) Smallest integer (Tn / I)

Transpositionally equivalent pc-sets share the same best normal order: there is a group action that partitions the power set of pc-sets into equivalence classes (or setclasses) under an equivalence relation. The equivalence class created by the transpositions of a pc-set is isomorphic to a cyclic group. The best normal order is defined as the representative of an equivalence class under the binary relation4 smallest intervals5 among all circular permutations of a pcset. Similarly, this binary relation is isomorphic to the smallest powers of two within a bit-sequence. Hence, considering two natural numbers represented as bit-sequences, this binary relation is isomorphic to if and only if both natural numbers are transpositions of the same pc-set. Although this relation does not linearly order6 the transpositions of a pc-set, there is a unique element within the equivalence class that is minimal under this binary relation ( ). Inversion equivalence is isomorphic to reflection (flip) of a 2D dihedral group. In pc-set theory, inversion is given by subtracting7 the value of each pitch-class in the pc-set from the total number of pitch-classes.

ܲ௞ǡ௡ ൌ ෍ ʹ൫ሺ௡ି௣ሻ୫୭ୢ௡൯

3 4

5 6

7

(2.3)

Forte’s set name (set-class). Although a binary relation that well-orders all elements within an equivalence class does not exist for all equivalence classes, any element of a given equivalence class could be chosen to be its representative under a binary relation. Forte (1973, 3-5). With exception of pc-set 1-1 (or, more generally [0]n) which is linearly ordered under the binary relation ( ). a’ = 12-a (mod 12) (Forte 1973, 8).

434

F. Gualda

Inversion of pc-sets is thus equivalent to bit-reversal. In the example below, [059] is I([037]), which is also the T5([047]); similarly [058] is I([047]), which is T5([037]), as expected. The total number of best normal orders can be seen as the quotient of a group action; and prime forms, the quotient of a dihedral group action.

4 Pitch-Class Categories Let Pitch-Class Category (PCCn) be a category consisting of the power set of pc-sets as objects, and transposition and inversion of pc-sets as morphisms, with n being the total number of pitch-classes. In every PCCn exists a unique initial object (the empty set) and a unique terminal object (a set of all n pitch-classes), and they complement each other. Besides transposition and inversion, PCCn presents two other operations: sum, given by the union of objects (composition of inclusion functions); and product, which is the intersection of objects (pitch-classes shared by two pc-sets). The sum and the product obey commutative and associative laws. For any non-terminal object, there are several identity-preserving products. In addition, the most important features of the theory of pc-category are the functors between categories and their natural transformations. The internal structure (morphisms) of PCCn is completely preserved in PCCm provided8 that m n. Functors map every object9 and morphisms of PCCn onto objects and morphisms of PCCm according to PCCn. Table 4. Objects and morphisms of a pitch-class category

Objects and Morphisms

Symbol

Binary implementation

Initial

0

0000...000000 (all zeros)

Terminal

1

1111...111111 (all ones)

Identity

id

(pc-set bit-sequence itself)

Sum

∪

A | B (bitwise or)

Product

∩

A & B (bitwise and)

Id preserving product

⊆

(A & B) == A (Boolean equal)

Transposition

Tn

circular bit-shift

Inversion

8

I

bit-reversal

In the case of functors from PCCn to PCCm with n > m, only a subset of the domain is mapped onto the codomain, thus the functor becomes a ‘partial function’. Musically, notes that do not belong to the diatonic scale cannot be mapped as diatonic, even though they might have some relationship with the scale. 9 Objects being the sets in the power set of a set with cardinality (total number pitch-classes) m and n respectively.


435

In figure 2, diatonic triads were represented as all seven transpositions (T0 to T6) of [024]7, whereas the same triads mapped onto 12 pitch-classes are some of the transpositions of [047], [037], and even [036]. Therefore, the cyclic permutations of [024]7 are isomorphic to diatonic triads as represented in pc-set12. Figure 3 exemplifies functors between two pitch-class categories:

Fig. 3. Tonal mapping from PCC-12 to PCC-7, shift and subsequent mapping from PCC-7 to PCC-12

In the first row, a major triad is represented as pc-set12; a functor maps from twelve (first row) onto seven pitch-classes (second row); the third row is a transposition of the second row within PCC7; finally, another functor maps from seven onto twelve pitch-classes. As binary implementation, both functors are represented as the bitsequence: 101010110101, which is isomorphic to [024579B]12, [0123456]7, and the integer 2741. Natural transformations () guarantee structural equivalence. In the next example, the scale on which the triads are transposed is changed from diatonic major to harmonic minor.

Fig. 4. Change of scale as natural transformation

In figure 4, the functor that maps form PCC12 to PCC7 is based10 on [024579B]12 (row one to row two). The functor that maps from PCC7 to PCC12, however, is based on [023578B]12 (row three to row four). Now, permutations of the [024]7 will generate triads of the harmonic minor scale. Both [024579B]12 and [023578B]12 are isomorphic to [0123456]7, which serves as a ‘basis’ for this natural transformation. In fact, any seven-note scale is isomorphic to PCC7. 10

A category is built on the subset of pc-set12 given by the seven pitch-classes 0,2,4,5,7,9,B: a power set evolves from [024579B]; to it all morphisms (operations) in pc-set theory are added. The functor is then a mapping from PCC7 based on the original pitch-classes 0,2,4,5,7,9,B to another PCC7 (based on[0123456]7) therefore a bijective function.

436

F. Gualda

Let F and G be functors from category C to category D; A and B are objects; and f, morphisms from A to B; and , natural transformations. Then the following diagram commutes:

Fig. 5. Natural transformation

A musical analogy: the objects of a category can be seen as all subsets11 of a scale; morphisms, as transpositions and inversions of those subsets; a scale can be seen as functor (from a category with the seven pitch-classes to another category, with twelve pitch-classes). Then, the change of scale would be a natural transformation.

5 Discussion and Future Work “What happens when we change the “12” in [the] twelve-tone system? How does this affect the Z-relation or the possible properties of entities?” (Morris, this volume). This paper addressed the first question since it provided methods for mapping the structure of any n-tone system onto another m-tone system. It did not, however, explore Z-relations or even the implications of changing the number of interval classes. A pc-set with seven pitch-classes presents only three interval classes: seconds/sevenths, thirds/sixths, and fourths/fifths. Another important direction for further work is the tonal implications of pitch-class categories. It would most certainly correlate the underlying structure (PCCn and functor to PCCm) with tonal weights for each Tn-set. Parncutt (this volume) discusses tonal implications of Tn-sets (with cardinality three) within the context of harmonic profiles. Assuming octave similarity, pitch-classes can be seen as quantization of the division of one octave. In this case, microtonal relationships can be described and analyzed by selecting a higher number of pitch-classes within one octave. Conversely, higher number of pitch-classes could also be used to represent chords and motives that comprise more than one octave. Since this work focused on general theory for pitch-class categories, it did not offer musical examples. Tonal and serial-music examples along with microtonal scales and ragas could be explored in future works. 11

Combination of pitch-classes of a scale.


437

6 Conclusion This paper presented an alternative approach to pitch-class set theory. It was demonstrated that pitch-class categories do not only contain all features from pc-set theory, but also generalize its operations to any number of pitch-classes. The theory offers new possibilities for musical analysis and composition. Additionally, a binary representation of the theory has been implemented, in which objects (pc-sets) and functors are both represented as bit-sequences; and morphisms, as bitwise operations. Computer software for studying aspects of the theory described in this paper has been developed and offered here for academic research. For new versions of the software and source code, please, feel free to contact the author.

References Babbitt, M.: Twelve-Tone Invariants as Compositional Determinants. Musical Quarterly 46(2), 246–259 (1960) Babbitt, M.: Set Structure and a Compositional Determinant. Journal of Music Theory 5(1), 72– 94 (1961) Barr, M., Wells, C.: Category Theory for Computer Science, 2nd edn. Prentice Hall, London (1995) Forte, A.: The Structure of Atonal Music. Yale University Press, Haven (1973) Huron, D.: Sweet Anticipation: Music and the Psychology of Expectation. MIT Press, Cambridge (2006) Kopp, D.: Chromatic Transformations in Nineteenth-Century Music. Cambridge University Press, Cambridge (2002) Lawvere, F.W., Schanuel, S.H.: Conceptual Mathematics. Cambridge University Press, Cambridge (1997) Mazzola, G.: The Topos of Music. Birkhäuser, Basel (2006) Guerino, M., Weissmann, J., Milmeister, G.: Comprehensive Mathematics for Computer Scientists 1, 2nd edn. Springer, Berlin (2006) Morris, R.: Set Groups, Complementation, and Mappings among Pitch-Class Sets. Journal of Music Theory 26(1), 101–144 (1982) Morris, R.: Mathematics and the twelve-tone system: past, present, and future. In: Klouche, T., Noll, T. (eds.) MCM 2007. CCIS, vol. 37, pp. 427–437. Springer, Heidelberg (2009) Parncutt, R.: Tonal implications of harmonic and melodic Tn-types. In: Klouche, T., Noll, T. (eds.) MCM 2007. CCIS, vol. 37, pp. 427–437. Springer, Heidelberg (2009) Rahn, J.: Basic Atonal Theory, pp. 31–39. Longman, New York (1980) Schoenberg, A.: Theory of Harmony, 3rd edn. (1978/1922). University of California Press, Los Angeles (1922); Translated by Roy E. Carter Berkeley Straus, J.N.: Introduction to Post-Tonal Theory. Prentice-Hall, Cliffs (1990) Walters, R.F.C.: Categories and Computer Science. Cambridge University Press, Cambridge (1991)

438

F. Gualda

Appendix C++ Source code for calculating pc-sets with arbitrary number of pitch-classes (up to 31 or 63) //---------------------------------------------------------------------//-- TPitchClassSet -------------------------------------PitchClass.h -//---------------------------------------------------------------------//-- Copyright (C) 2006 --- Fernando Gualda ---------------------------//---------------------------------------------------------------------class TPitchClassSet { public: TPitchClassSet(int = 12, bool = true); ~TPitchClassSet(){}; const inline

int SetSize; void Reset(void) {shiftindex = primeset = normalset = pitchset = 0;};

// Accessor Members inline long PrimeForm(void) inline long BestNormalOrder(void) inline long InvertedForm(void) inline int Transposition(void)

{return {return {return {return

primeset;}; normalset;}; invertset;}; LastShiftIndex;};

// Independent Members static long BitReverse(long, int = 0, int = 0); static long BitShift(long, int, int); // Operations long operator * (long pcs); long operator * (TPitchClassSet *); // 'Multiplication' stands for intersection bool operator == (TPitchClassSet *); bool operator != (TPitchClassSet *); inline bool Isomorphism(long pcs) {return Best(pcs) == normalset;}; inline long operator = (long pcs){Best(pcs); return pitchset;}; inline long operator() (long pcs){Best(pcs); return pitchset;}; inline long operator() (void) {return pitchset;}; inline inline inline inline inline inline

long long long long long long

operator++(void) operator--(void) operator(int s) operator=(int s)

private: long pitchset; long normalset; long invertset; long primeset; bool invertible; int LastShiftIndex; long Best(long); };

// // // // // // //

{return {return {return {return {return {return

BitShift(pitchset,1,true);}; BitShift(pitchset,-1,true);}; BitShift(pitchset,s);}; BitShift(pitchset,-s);}; BitShift(pitchset,s,true);}; BitShift(pitchset,-s,true);};

a pcs (pitch class set) Best Normal Order bit-reversal of pitchset prime form Inversion Equivalence of pitchset Calculates the Best Normal Order


439

//---------------------------------------------------------------------//-- TPitchClassSet ------------------------------------- PitchClass.cpp //---------------------------------------------------------------------//-- Copyright (C) 2006 --- Fernando Gualda ---------------------------//---------------------------------------------------------------------#include “PitchClass.h” //--------------------------------------------------------------------// TPitchClassSet – Constructor //--------------------------------------------------------------------TPitchClassSet::TPitchClassSet(int s, bool inv) :pitchset(0), primeset(0), normalset(0), LastShiftIndex(0), invertible(inv), SetSize(s) {}; //--------------------------------------------------------------------// PrimeForm – Calculates the prime form of a pc-set //--------------------------------------------------------------------long TPitchClassSet::PrimeForm(long pcs) { if (pcs == 0) return pitchset = normalset = invertset = primeset = 0; pitchset = pcs; /////////////////////// // Best Normal Order // /////////////////////// normalset = Best(pcs); /////////////////////////////////////////////////// // Inversion (bit-reversal) of a Pitch Class Set // /////////////////////////////////////////////////// if (invertible) { // Inversion Equivalence (dihedral group) invertset = BitReverse(pcs, SetSize); /////////////// // Prime Set // /////////////// pcs = Best(invertset); primeset = (pcs<normalset)? invertset: normalset; return primeset; } invertset = primeset = 0; return normalset; } //--------------------------------------------------------------------// BitReverse - Reverses and shifts bits of a pc-set //--------------------------------------------------------------------long TPitchClassSet::BitReverse(long p, int size, int shift) { if (size = sizeof(long)) return 0; while (shift < 0) shift += size; long q = 0; for (int i = shift, k = size+shift; i < k; i++) { if ((1 = 1; // (shift only) if (min > pcs) { min = pcs; key = i; } } LastShiftIndex = (key+1)%SetSize; return min; } //--------------------------------------------------------------------// BitShift - Shifts a pc-set //--------------------------------------------------------------------long TPitchClassSet::BitShift(long p, int size, int shift) { if (size = sizeof(long)) return 0; while (shift < 0) shift += size; if (shift == 0) return p; /////////////////////////////////////////////////// // Fast Circular Bit-Shift (up to 31 or 63 bits) // /////////////////////////////////////////////////// long one = ((11, let S = {nθ–¬nθ¼⏐0 2 minimizing the difference of cardinality #A − #Ac (condition of minimality). If there is no such scale (N = 15 is the only known example so far), the trivial 2-generated scale might be chosen as the generalized diatonic scale. So for each N , there is always one generalized diatonic scale. In order to compare the pseudo-diatonic scales of model A with the generalized diatonic scales of model B, we implement an algorithm for searching the scales. For each chromatic number m/N of the shuffled Stern-Brocot tree, we look at its predecessor m1 /N1 and construct the generic cycle of fifths as part of the (generalized) Balzano subsystem U = {m1 k mod N1 , k = 0, 1, ..., N1 − 1} The rational number m2 /N2 = (m − m1 )/(N − N1 ) is the Farey complement of m1 /N1 satisfying m m1 m2 = ⊕ N N1 N2

Pseudo-diatonic Scales

495

With m2 /N2 , we complete the (generalized) Balzano subsystem in terms of the cycle: V = {m2 k mod N2 , k = 0, 1, ..., N2 − 1} The permutation σ(x) = x mod N induces an embedding of U and V in the chromatic universe ZN . This algorithm produces the two types of scales A and B. The N1 white keys are the set σ(U ) and the N2 black keys are a transposition of σ(V ). In model A, the generator m1 = 2r is even, and the number of white keys N1 = 2n+ 1 is always odd. Moreover, the rational number m1 /2N1 = r/N1 must be a chromatic number. In model B, the generalized diatonic scale is determined by the condition of minimality. In both models, the chromatic number m/N and the number (N − m)/N leads to the same scales.

3

Comparison of the Two Models

Doubles of chromatic numbers are not chromatic. In model A, since the chromatic number m1 /2N1 ends by R, the word of the generic pseudo-fifth m1 /N1 ends by L. If ω is the word of m2 /N2 (the black keys), the chromatic number m/N of the N -chromatic universe ends by LR. In model B, for 12 ≤ N ≤ 42, the generalized diatonic scales have the same associated words ωLR if N = 13, 14, 15, 18, 21. For these values (N = 13, 14, 15, 18, 21), there are no pseudodiatonic scales in model A. This suggests to compare the two models only in this case (N = 13, 14, 15, 18, 21). A computer program shows the following results (see also the following table). • The pseudo-diatonic scales of model A are not uniquely determined by the chromatic cardinality N since — for example — there are two pseudo-diatonic scales for N = 42. In model B, for each N , there is by definition only one generalized diatonic scale. • The pseudo-diatonic scale of model A do not always exist for each N ( in contrast to model B), even though m1 is even. In the quarter tone universe (N = 24), there is no pseudo-diatonic scale, because the number m1 /2N1 = 3/13 is not a chromatic number. In model B, the pseudo-fifth has 13 quarter tones. It is composed of two different types of pseudo-thirds. In model A, the two pseudothirds are always of the same size, determined by the number r/N1 = m1 /2N1 . • In model A, the number N1 of white keys is always odd and the generator m1 is always even. In model B, the number of notes N1 of a generalized diatonic scale is sometimes odd and sometimes even. Moreover, contrary to model A, the number N1 can be even and the generator m1 odd (N = 17, 25, 27, 29, 37, 39, 41) or the number N1 is odd and the generator is also odd (N = 36). • If m1 is even, the scales of the two models are different in two cases; (1) if the number m1 /2N1 is not a chromatic number (N = 24, 28, 32, 35, 40) and (2) if the pseudo-diatonic scale does not satisfy the condition of minimality. In this case, m1 is even in model A and odd in model B (N = 29, 37, 39, 41). • If, for both models, the pseudo-fifths or their complements (v) are evaluated in cents 1200 log2 (v) it turns out, that the range of the pseudo-fifth in model

496

F. Jedrzejewski

A varies between 660 and 1057 cents and in model B it varies between 630 and 978 cents. This shows that the pseudo-fifth must be understand as a new interval of the N -chromatic universe, and should not literally be compared with the tempered fifth (700 cents). N Model m/N = m1 /N1 ⊕ m2 /N2 12 A, B 7/12 = 4/7 ⊕ 3/5 16 A, B 7/16 = 4/9 ⊕ 3/7 17 B 5/17 = 3/10 ⊕ 2/7 19 A, B 7/19 = 4/11 ⊕ 3/8 20 A, B 11/20 = 6/11 ⊕ 5/9 22 A, B 17/22 = 10/13 ⊕ 7/9 23 A, B 7/23 = 4/13 ⊕ 3/10 24 B 11/24 = 6/13 ⊕ 5/11 25 B 9/25 = 5/14 ⊕ 4/11 26 A, B 7/26 = 4/15 ⊕ 3/11 27 B 5/27 = 3/16 ⊕ 2/11 28 B 15/28 = 8/15 ⊕ 7/13 29 A 17/29 = 10/17 ⊕ 7/12 29 B 9/29 = 5/16 ⊕ 4/13 30 A, B 7/30 = 4/17 ⊕ 3/13 31 A, B 11/31 = 6/17 ⊕ 5/14 32 A 27/32 = 16/19 ⊕ 11/13 32 B 15/32 = 8/17 ⊕ 7/15 33 A, B 7/33 = 4/19 ⊕ 3/14 34 A, B 25/34 = 14/19 ⊕ 11/15 35 B 11/35 = 6/19 ⊕ 5/16 36 B 17/36 = 9/19 ⊕ 8/17 37 A 7/37 = 4/21 ⊕ 3/16 37 B 13/37 = 7/20 ⊕ 6/17 38 A, B 29/38 = 16/21 ⊕ 13/17 39 A 17/39 = 10/23 ⊕ 7/16 39 B 16/39 = 9/22 ⊕ 7/17 40 A 7/40 = 4/23 ⊕ 3/17 40 B 19/40 = 10/21 ⊕ 9/19 41 A 25/41 = 14/23 ⊕ 11/18 41 B 13/41 = 7/22 ⊕ 6/19 42 A, B 11/42 = 6/23 ⊕ 5/19 42 A 37/42 = 22/25 ⊕ 15/17

ω Pseudo-Fifths Pseudo-Third R3 700 2/7 = RL2 R RLRL 675 2/9 = RL3 R RL2 R 847 − RLR2 758 2/11 = RL4 R R 3 L2 660 3/11 = RL2 R2 2 2 R L R 927 5/13 = RLR3 2 RL RL 835 2/13 = RL5 R 3 RLRL 650 (3/13 = RL3 RL) 2 RLR L 768 − RL2 R2 877 2/15 = RL6 R RL4 R 978 − R 3 L4 643 (4/15 = RL2 R2 L) R3 LR 703 5/17 = RL2 RLR 2 2 RL RL 828 − RL3 RL 920 2/17 = RL7 R RLR2 L2 774 3/17 = RL4 R2 2 4 R L R 1013 8/19 = RLRLR2 5 RLRL 638 (4/17 = RL3 RL2 ) 3 2 RL R 945 2/19 = RL8 R 2 2 R LR L 882 7/19 = RLR2 LR 2 3 RL RL 823 (3/19 = RL5 RL) 6 RLRL 633 − RL4 RL 973 2/21 = RL9 R RLR2 L3 778 − R2 L2 RL2 916 8/21 = RLR4 RLRL2 R 677 5/23 = RL3 R3 RLRLRL 708 − RL4 R2 990 2/23 = RL10 R RLRL7 630 (5/21 = RL3 RL3 ) 5 R L 732 7/23 = RL2 RL2 R 2 4 RL RL 820 − RL2 R2 L2 886 3/23 = RL6 R2 R 2 L6 R 1057 11/25 = RLRL2 R2

The table gives for each N -chromatic universe (first column) the Farey relation (third column) needed to construct the pseudo-diatonic scale of model A or the generalized diatonic scale of model B (given in the second column). The last column provides the chromatic number (m1 /2N1 ) for the pseudo-third of model A. Some numbers (m1 /2N1 ) in parenthesis are not chromatic numbers. The

Pseudo-diatonic Scales

497

word ω associated with the generic pseudo-fifth is given in the fourth column. The value in cents of the pseudo-fifth is in column 5. Some scales, available in both models, have a chromatic number 2/(2n + 3) associated with a word of the form RLn R, n ≥ 2. In this case, it is easy to show that the pseudo-diatonic and the diatonic scale of models A and B are the same. If n = 2k, (k = 1, 2, ...) is even, in the (7k + 5)-chromatic universe, this scale has (4k + 3) white keys and (3k + 2) black keys, 7 4 3 = ⊕ 7k + 5 4k + 3 3k + 2 and if n = 2k +1 is odd, the scale has, in the (7k +9)-chromatic universe, (4k +5) white keys and (3k + 4) black keys: 4 3 7 = ⊕ 7k + 9 4k + 5 3k + 4 Acknowledgements. I would like to thank Thomas Noll and the organizers of the Conference for helpful information and comments.

References Agmon, E.: A Mathematical Model of the Diatonic System. Journal of Music Theory 33(1), 1–25 (1989) Agmon, E.: Linear transformations between cyclically generated chords. Musikometrika 3, 15–40 (1991) Brocot, A.: Calcul des rouages par approximation, nouvelle méthode. Revue Chronométrique 3, 186–194 (1862) Carey, N.: Coherence and Sameness in Well-formed and Pairwise Well-Formed Scales. Journal of Mathematics and Music 1(2), 79–98 (2007) Carey, N., Clampitt, D.: Aspects of Well-Formed Scales. Music Theory Spectrum 11(2), 187–206 (1989) Clough, J.: Aspects of Diatonic Sets. Journal of Music Theory 23, 45–61 (1979) Clough, J., Myerson, G.: Variety and Multiplicity in Diatonic Systems. Journal of Music Theory 29(2), 249–270 (1985) Farey, J.: On a Curious Property of Vulgar Fractions. Philosophical Magazine 47, 385– 386 (1816) Gould, M.: Balzano and Zweifel: Another Look at Generalized Diatonic Scales. Perspectives of New Music 38(2), 88–105 (2000) Jedrzejewski, F. (ed.): La loi de la Pansonorité. Ivan Wyschnegradsky, Contrechamps, Genève (1996) Jedrzejewski, F.: Mathématiques des systèmes acoustiques, Tempéraments et modèles contemporains. L’Harmattan, Paris (2002) Jedrzejewski, F.: Dictionnaire des musiques microtonales. L’Harmattan, Paris (2003) Jedrzejewski, F.: Mathematical Theory of Music. Editions IRCAM/Delatour, Sampzon (2006) Jedrzejewski, F.: Generalized Diatonic Scales. Journal of Mathematics and Music 2(1), 21–36 (2008)

498

F. Jedrzejewski

Lewin, D.: Generalized Musical Intervals and Transformations. Yale University Press, New Haven (1987) Noll, T.: Facts and Counterfacts: Mathematical Contributions to Music-Theoretical Knowledge. In: Bab, S., et al. (eds.) Models and Human Reasoning - Bernd Mahr zum 60. Geburtstag. W&T Verlag, Berlin (2006) Noll, T.: Musical Intervals And Special Linear Transformations. Journal of Mathematics and Music 1(2), 121–137 (2007) ¨ Stern, M.: Uber eine zahlentheoretische Funktion. Journal f¨ ur die reine und angewandte Mathematik 55, 193–220 (1858) Vicinanza, D.: Paths on the Stern-Brocot Tree and Winding Numbers of Modes. In: Proceedings of the ICMC, Barcelona (2005) Wyschnegradsky, I.: 24 Préludes in Vierteltonsystem. M.P. Belaieff, Frankfurt (1979)

Affinity Spaces and Their Host Set Classes José Oliveira Martins Eastman School of Music, University of Rochester [email protected]

Abstract. This paper proposes the organization of pitch-class space according to the notion of affinities discussed in medieval scale theory and shows that the resultant arrangement of intervallic affinities establishes a privileged correspondence with certain symmetrical set classes. The paper is divided in three sections. The first section proposes a pitch-class cycle, the Dasian space, which generalizes the periodic pattern of the dasian scale discussed in the ninthcentury Enchiriadis treatises (Palisca 1995). The structure of this cycle is primarily derived from pitch relations that correspond to the medieval concepts of transpositio and transformatio.1 Further examination of the space’s properties shows that the diatonic collection holds a privileged status (host set class) among the embedded segments in the cycle. The second section proposes a generalized construct (affinity spaces) by lifting some of the intervallic constraints to the structure of the Dasian space, while retaining the relations of transpositio and transformatio, and the privileged status of host set classes.2 The final section examines some of the properties of host set classes, and in turn proposes “rules” for constructing affinity spaces from their host sets. The study of affinity spaces will give us insights regarding scalar patterning, inter-scale continuity, the combination of interval cycles, voice leading, and harmonic distance.3

1 Affinities in the Medieval Dasian Scale The idiosyncratic dasian scale discussed in the late ninth-century Enchiriadis treatises is constituted by a series of tone-semitone-tone tetrachords consistently separated by a tone of disjunction (see Example 1a). This arrangement replicates the four modal qualities (protus, deuterus, tritus, and tetrardus) of the finales tetrachord throughout 1

For a discussion of the interpretative attitudes involving these concepts see Pesce (1986 and 1987). 2 In earlier work, I propose the Dasian and other affinity spaces as suitable analytical frameworks to address harmonic and melodic aspects that result from the combination of different scales in twentieth-century music. For analytical accounts of Béla Bartók’s polymodality and Witold Lutoslawski’s “12-note” music see Martins (2006a); for the analysis of Igor Stravinsky’s harmonic syntax in a neoclassical work, see Martins (2006b); and for the interpretation of Darius Milhaud’s polymodality see Martins (2007). 3 This study intersects in interesting ways with work developed by several theorists: Philip Lambert and Edward Gollin on the combination of interval cycles and multi-sets, see Lambert (1990) and Gollin (2007); Carey and Clampitt (1996); and Tymoczko (2004) on scalar theory. T. Klouche and T. Noll (Eds.): MCM 2007, CCIS 37, pp. 499–511, 2009. © Springer-Verlag Berlin Heidelberg 2009

500

J.O. Martins

the scale, since the semitone placed in the center of each tetrachord between deuterus and tritus creates equidistant intervallic markers and allows for establishment of intervallic affinities at the perfect fifth throughout the scale. This scale structure allowed the transfer of melodic segments at both the upper and lower fifth to retain their modal identities. Such transfer was referred to as transpositio and preserved the segments’ local intervallic structure, as illustrated in Example 1b, where the arrows point to the preservation of the modal quality deuterus. Example 1. The dasian scale (Enchiriadis treatises)

The intervallic arrangement of the dasian scale, however, overrides modal affinity at the octave since the same pitch class is assigned to different modal qualities in different tetrachords. In Example 2, the circled pitch class (pc) A appears as deuterus, protus and tetrardus in the scale.4 The use of B-flat in the Graves tetrachord

4

Guido d’Arezzo considered affinity (likeness) to be completed only at the octave, which is the reason why Guido and subsequent theorists rejected the dasian scale. This is not to say that the Enchiriadis treatises do not acknowledge the privileged status of the octave relation in their musical system, but rather that affinities at the fifth are of a different consonant “quality” (consonae) than octave “hyper-consonant” relations (aequisonae). See Pesce (1986, 337), and Mengozzi (2006).

Affinity Spaces and Their Host Set Classes

501

inflecting B-natural in the Superiores changes the modal quality of the adjacent pc A from protus in the Superiores to deuterus in the Graves tetrachord; similarly, the use of F# in the Excellentes tetrachord instead of F-natural makes pc A sound as tetrardus. In other words, pc A has different modal qualities in each tetrachord given the different distribution of tones and semitones surrounding each appearance in the scale. The same could be said for other pcs in the scale. In the medieval notation of chant, Guido d’Arezzo warned against the change of modal quality created by note alterations to a pure diatonic framework. He referred to such modal change as transformatio. But even though disapproved by Guido, transformatio became a powerful practice extending the flexibility of the tonal system. Transpositio and transformatio thus capture significant pitch relationships: transpositio refers to the transfer of modal quality (local interval pattern) to various positions in the tonal system; whereas transformatio refers to the change of modal quality (local interval pattern) on a given pitch class effected by a note alteration, as well as to the note alteration itself. Example 2. Different modal qualities for the same pitch-class in the dasian scale

2 The Dasian Space Example 3 proposes a pitch-class generalization of the dasian scale by extending the species of fifth beyond the medieval scale’s boundaries (the disjunct tetrachords of the scale are marked by brackets and Roman Numerals). This generalization closes the space in two ways: by exhausting the 12 transpositions of the tetrachordal interval pattern, and by creating a pitch-class cycle.5 This framework renumbers the modal quality of each note in the dasian space, such that tritus is 0, tetrardus is 1, protus is 2, and deuterus is 3. In this arrangement, semitones always fall between adjacent positions 3 and 0, and each of the 12 semitones occurs only once in the space.

5

Edward Gollin (2007) refers to pitch-class cycles that exhaust the aggregate multiple times as multi-aggregate cycles. Although Gollin does not address specifically the structure of the Dasian space, the space exhausts the aggregate four times.

502

J.O. Martins

The affinity of the space is expressed by the interval of transpositio 7, which encompasses as a modular unit of three recurrent whole tones and a semitone, i.e., 7 = 3 • 2 + 1 (mod 12). Example 3. The Dasian space and the transpositio relation.

Example 4 shows that the transformatio relation has two outcomes in the Dasian space: it either produces a note substitution, by ascending a pitch class by a semitone; or it retains the same pitch class. In both cases, transformatio corresponds to moving seven steps in clockwise direction.


503

Example 4. The transformatio relation in the Dasian space.

3 Four Properties of the Dasian Space The examination of continuous segments embedded in the Dasian space allows us to establish the following four properties: (1) Set-class consistency: every continuous 7-note segment is a diatonic collection. Segments of different cardinalities yield different set-classes, but 7-note segments yield a single set class, which I call the host set class of the space. (2) Set-class completion: The space contains all 12 transpositions of this scale, which are related by perfect fifth, the interval of transpositio (t). (3) Set locality: Each diatonic scale appears in a single region of the space, extending through a major-10th segment. Regions are defined by Carey and Clampitt (1996, esp. 124-125) and constitute the largest space in which both parallel octaves and fifths can be maintained.

504

J.O. Martins

(4) Every single “step” or “shift” of a 7-note segment in clock- or counterclockwise direction either substitutes a pitch class for its replication or for a pitch class a semitone away. Both of these relations are structured by transformatio. In Example 5a, successive clock- or counterclockwise steps of 7-note segments yield the same collection four times before a new pitch is introduced. Example 5b shows that the transformatio relation that produces note change corresponds to what David Lewin refers to as a Cohn flip, i.e., when inverted around a given axis, a diatonic set produces a near symmetrical mapping, where one of the notes is substituted by another adjacent note in the space (Lewin 1996, 181-216). In the example, the whitenote diatonic collection can Cohn-flip in two familiar ways, each retaining six of the pitches and exchanging B for Bb, and F for F#. The transformatio relation thus has an important function in defining the host set of the Dasian space as the number of steps spanned by the relation coincide with the cardinality of the host set class. Example 5. “Step” motion and transformatio in Dasian space

The role of the diatonic collection as a host set class, however, is not unique to the Dasian space. Example 6 shows three additional cycles: the Guidonian space, the Hauptmann’s line-of-thirds, and the circle-of-fourths.6 In all of these spaces, diatonic collections are maintained as 7-note segments and the transformatio relation spans seven contiguous steps. As with the dasian space, transformatio either retains the same pitch-class seven steps away, or as shown by the dotted arrows, connects pitchclasses related by semitone belonging to adjacent host sets in the space. In addition, the four properties that characterize the pitch structure of the Dasian space also apply to these spaces. 6

The Guidonian space derives its name from maximally overlapping the Guidonian hexachordal arrangement.


505

Example 6. The diatonic collection as the host set-class of several cycles

4 Affinity Spaces Example 7 proposes a formula (t = n • r + s (mod 12)) that models the interval relations in the modular unit of a generalized construct I call affinity spaces. Let us call affinity spaces to any concrete realization of such construct. The formula lifts the constraints on interval size and interval recurrence specific to the Dasian space, but retains the roles performed by the relations of transpositio and transformatio, and continues to grant privileged status of host set classes.7 In this formula, t corresponds to the interval of transpositio, measuring the affinity of the space; r is the interval of recurrence, and generalizes the “whole tone” of the dasian space; n counts the recurrence of r; and s generalizes the “semitone,” the unique interval of the modular unit. There are two outcomes for transformatio in any affinity space (Example 7b): it either retains the pitch class, in which case the order position of that pitch class descends one order position within the modular unit. When transformatio is applied to a pitch class in the zero order position within a modular unit, it substitutes it for another one in a position n, such that the interval between the two pitch classes is given by r – s. When the interval of transpositio t is co-prime with 12 (i.e., when it is 1, 5, 7, or 11), we can then calculate the number y of steps a transformatio spans (Example 7c). y is also the cardinality of the host set class. In the case of the Dasian space as addressed above, a transformatio corresponds to moving seven steps. 7

Although conceived independently, this generalization intersects in a number of ways with the work of Gollin (2007).

506

J.O. Martins

Example 7. Affinity spaces

Let us now examine the embedding of host set classes in some affinity spaces. Example 8 shows twelve different affinity spaces, which share the same interval of transpositio (t = 5), the recurrence n = 2, and consequently also the cardinality of space (mod 36). A circle-of-fourths in each space is highlighted in order to point out the affinity at-the-fourth in all spaces. In each space, arrows indicate the number of STEPs spanned by transformatio.8 Solid arrows indicate pitch-class retention and dotted arrows indicate note substitution. In each space, the cardinality of the host set-class equals the number of STEPs involved in a transformatio. Notice that six of the spaces have positive STEP values, and the other six have negative values. I have represented the spaces this way in order to keep the shortest number of STEPs spanned by the spaces’ transformatio operation.

8

Let us define STEP as the operation that moves clockwise in a cycle, such that STEP1 moves clockwise one station, between adjacent elements; STEP2 moves clockwise 2 stations, etc. STEP-1 moves counterclockwise one station, etc.


507

Example 8. Twelve affinity spaces structured by 5 = 2 • r + s

The table of Example 9 organizes the twelve affinity spaces presented in Example 8. Going down the list of spaces in the left column shows that the interval r (the recurrence within the modular unit) increases by five semitones (or a perfect fourth) and the interval s (the unique single interval) increases by two semitones. The second column shows that are only three values for the interval of transformatio that produces note substitution, i.e., 1, 4, 7, and 10 (mod 12). The third column shows that the increasing number of STEPs involved in a transformatio increases by three for positive values and then decreases by three for negative values. The fourth column is subdivided, by the dotted line, pairing complementary host set classes for each space.

508

J.O. Martins

Example 9. The table for the twelve affinity spaces under 5 = 2 • r + s

Example 10 examines this complementary relation in two spaces: 5 = 2 • 2 + 1 (mod 12) (the Guidonian space), and 5 = 2 • 3 + 11 (mod 12). Example 10a shows that starting with pc C in position 0, the transformatio relation spans STEP -7 from C to C#. This relation signals the boundaries of two adjacent host sets, which in the case of the Guidonian space is the diatonic collection. Another way of inflecting C to C# in the same positions of the space is to go clockwise by STEP 29. This path produces another set class that is also consistent throughout the space. The host set corresponding to STEP 29 is the pentatonic collection plus two-times-the-aggregate. In other words, the space in fact has two host set classes, which are complements of each other plus twice the exhaustion of the aggregate. Example 10b presents another space, whose transformatio spans STEP 8, and the host set class has cardinality 8. The alternative counterclockwise path of STEP -28 produces a complementary set class (plus twice the aggregate) to the host set produced by STEP 8. In short, the transformatio relations produced by clock- and counterclockwise paths signal complementary host sets (plus a number of aggregates) for a given affinity space.9

9

It is also interesting to notice that those spaces that share the same host set class (plus or minus the aggregate) also share the interval mod 12 of the transformatio.


509

Example 10. Complementary host set classes for 5 = 2 • 2 + 1 and 5 = 2 • 3 + 11

5 Three Properties of Host Set Classes We are now in a position to ask the question: What are the properties of host set classes? Can any set class be a host in an affinity space? Generalizing from the host sets addressed so far, (1) all are inversionally symmetrical (i.e., they map into themselves under some axis); and (2) all produce a near self-mapping “flip,” that is, they map into themselves except for one note. The interval between unmatched notes is the interval between substituted and substituting notes in a transformatio. However, unlike what we observed for Cohn-flips in the diatonic collection, the flip that corresponds to a transformatio does not have to substitute adjacent notes in other host sets (i.e., it produces a kind of unrestricted Cohn-flip). Finally, (3) host sets are either interval-cycle segments or a combination of i-cycle segments.10

6 Generating Affinity Spaces Let us now address the reverse case: Given a set class with the three host-set properties, what are the “rules” for generating an affinity space?11 We first arrange the host set into an ascending i-cycle, and partition it into ascending i-cycle segments of cardinality c, and one segment of cardinality c - 1. Then, we arrange all segments to reflect a constant transpositional relation between their first elements (there might be a few ways of doing this). These transpositional relations correspond to possible 10

Host set classes can be arranged into i-cycle segments when t = 1, 5, 7, or 11; or arranged into a combination of i-cycle segments when t = 2, 3, 4, 6, 8, or 10. 11 I’ll address here only the cases for t = 1, 5, 7, or 11.

510

J.O. Martins

values for t (transpositio). Finally, we determine the values for f (transformatio) by producing the two near self-mapping flips.12 For instance, consider the i-cycle segment {1, 2, 3, 4, 5, 6, 7}, which has the three host set properties. Example 11a arranges the set as an i-cycle segment and partitions or de-cycles it in eight different ways. In Example 11b, the two near self-mapping flips give us the values of the two possible transformatio relations. Example 11c shows three spaces that result from the eight possible arrangements of i-cycle segments.13 Several of the arrangements give rise to what amounts to equivalent affinity spaces: the if r or s = 0 then t = s or t = r in (1) and (2); retrograde cyclic orderings produce the same space when “read” clock or counterclockwise—for instance (3) and (4), or (7) and (8); and the permutation of r and s in a 2-element modular unit results in the same ordering of elements in the space—for instance (3) and (5), or (4) and (6). Example 11. Generating affinity spaces from the set {1, 2, 3, 4, 5, 6, 7}

12

As we’ve seen for the case of the diatonic collection, there’s no one-to-one correspondence between a host set class and an affinity space. 13 The 7-note set is not the only contiguous segment that can serve as host set in the complete icycle 1 or “chromatic scale” of Example 11c; rather, any contiguous segment can function as a host set for the i-cycle. In general, complete i-cycles are trivial realizations of affinity spaces and do not exclusively embed a single host set class. A similar condition applies to the circle-of-fourths in Example 6.


511

7 Conclusion The transformatio relation enables a gradual harmonic “modulation” across members of the host set class by retaining all but one note in the near self-mapping flip between adjacent host sets of an affinity space. The note change involved in the harmonic modulation can take the form of parsimonious voice leading when the value of (the note change in) transformatio is 1 (or 11), as is the case for diatonic or pentatonic host sets.14 Example 11 demonstrates, however, that harmonic modulation can rely on less parsimonious transformatio values, which nevertheless entail a harmonic gradation across set-class members. The several possibilities for de-cycling a given host set create a relation (one-to-several: between a given host set class and the corresponding affinity spaces) that is theoretically rich and analytically useful.

References Carey, N., Clampitt, D.: Aspects of Well-Formed Scales. Music Theory Spectrum 11(2), 187– 206 (1989) Carey, N., Clampitt, D.: Regions: A Theory of Tonal Spaces in Early Medieval Treatises. Journal of Music Theory 40(1), 113–147 (1996) Gollin, E.: Multi-Aggregate Cycles and Multi-Aggregate Serial Techniques in the Music of Béla Bartók. Music Theory Spectrum 29(2), 143–176 (2007) Lambert, J.P.: Interval Cycles as Compositional Resources in the Music of Charles Ives. Music Theory Spectrum 12(1), 43–82 (1990) Lewin, D.: Cohn Functions. Journal of Music Theory 40(2), 181–216 (1996) Martins, J.O.: The Dasian, Guidonian, and Affinity Spaces in Twentieth-century Music. Ph. D diss., University of Chicago (2006a) Martins, J.O.: Stravinsky’s Discontinuities, Harmonic Practice and the Guidonian space in the ‘Hymne’ for the Serenade in A. Theory and Practice 31, 39–64 (2006b) Martins, J.O.: Diatonic reorientation in dual-organization spaces: interpreting polymodality in works of Milhaud. In: Presented at the annual meeting of the Music Theory Society of New York State (2007) Mengozzi, S.: Virtual Segments: The Hexachordal System in the Late Middle Ages. Journal of Musicology 23(3), 426–467 (2006) Palisca, C.V. (ed.): Musica enchiriadis and Scolica enchiriadis. Trans. Raymond Erickson. Music Theory Translation Series. Yale University Press, New Haven (1995) Pesce, D.: B-Flat: Transposition or Transformation? The Journal of Musicology 4(3), 330–349 (1986) Pesce, D.: The Affinities and Medieval Transposition. Indiana University Press, Bloomington (1987) Tymoczko, D.: Scale Networks and Debussy. Journal of Music Theory 48(2), 219–294 (2004)

14

The value transformatio f = 1 can take the form of an augmented unison (or chromatic inflection) as in the case of the diatonic collection or of a diatonic semitone as in the case of a pentatonic collection. In both of these cases the transformatio relation corresponds to what Carey and Clampitt (1989, 192–93) refer to as “primary intervals” of well-formed scales.

The Step-Class Automorphism Group in Tonal Analysis Jason Yust University of Alabama, Tuscaloosa

Until recently, researchers who have dealt formally with tonal hierarchy (prolongation) have considered only models in which the objects of the hierarchy are musical events (where a musical event might be a chord or a note in a particular voice).1 In contrast, Yust (2006) proposes a concept called “dynamic prolongation” in which the objects of tonal hierarchy are motions between events rather than events themselves. The events in the model of Yust (2006) are chords made up of harmonically related pitches from several voices. In the present study I develop a different approach to dynamic prolongation. Rather than expressing harmonic relations between notes by grouping them into chords, we can treat harmonic relations as intervals and mix them in a hierarchy with melodic motions. This creates a model that can posit long-range harmonic relationships and blur the boundaries between intervals of harmonic and melodic significance. A particularly beautiful possibility opened up by this approach is to consider intervals in terms of step-class values and to view prolongational relationships between intervals in terms of step-class symmetries. A step-class is the residue modulo seven of the number of diatonic steps that separate two notes (referred to as “unison,” second,” third,” etc.). The group of step-class intervals is therefore isomorphic to the integers under addition modulo seven (Z7). The automorphisms (i.e. symmetries) of Z7 (isomorphic mappings of Z7 to itself) can be thought of as multiplications modulo 7, and form a group (Aut(Z7)) isomorphic to Z6, which is in turn isomorphic to the direct product Z2 × Z3. Step-class inversion, which I’ll denote I, represents the order-two component of step-class interval automorphism group. The “halving” operation, H, which takes fifths to thirds, thirds to seconds, seconds to fifths, and so forth, is a representative order-three operation. Together these two operations generate the group, {1, H, H-1, I, HI, H-1I}. The order-three operations of this group are of particular interest as prolongational operations on intervals. This is illustrated in Example 1a, which shows intervals at various voice-leading levels with, slurs. Moving from one level to the next, the operation H relates a prolonged interval to two motions that divide it in half. Three types of basic prolongational units result from the operation: the prolongation of a second by the framing fifths of a dominant–tonic progression, the triadic filling-in of a fifth, and the filling-in of a third with a passing motion. The relationships between step-class motions in Example 1a take the form of a binary tree. We can also view this structure as a maximal outerplanar graph, or “MOP,” as in Example 1b, with notes as vertices and intervals of the hierarchy as edges. (Yust 2006 deals extensively with mathematical properties of MOPs and their musical consequences.) The staff above shows the MOP shows intervals in a modified Schenkerian notation that shows third-motions with slurs and fifth-motions with beams. Using step-class automorphisms as prolongational transformations suggests the following interpretation of the resulting MOP: step-class intervals define a space that 1

Rahn 1979, Smoliar 1980, Lerdahl and Jackendoff 1983, Cohn and Dempster 1992.


The Step-Class Automorphism Group in Tonal Analysis

513

Example 1. H as a prolongational operation can occupy a number of orientations. The different orientations define familiar musical perspectives on step-class space: a melodic space where notes a second apart are close together, a space of thirds that defines harmonic closeness, and a fifths-space that relates harmonies and keys. The presence of a particular step-class motion in a piece of music can be seen as an orientation of musical space in terms of that motion. That is, while the listener experiences that motion, thinking at the prolongational level where it occurs means thinking of a corresponding orientation to step-class space. The listener then moves conceptually between different prolongational levels by means of automorphic transformations of the space. The simplest relationship between levels of musical motion is the automorphism H, which defines a symmetrical binary filling-in of an interval. For example, a diatonic sequence by fifths once initiated implies continuation through further motion by fifth. But after hearing a couple repetitions of the sequence the listener can mentally move to a higher level of musical organization, hearing a larger sequence by second that suggests continuation by second. By hearing at various such levels at once the sequence by fifths sounds not like a homogenous motion that could start and end at any time, but at certain points gives a sense of completion of larger-scale motions from earlier tonal events in the piece and initiation of motions that may receive later completion. The interval hierarchy includes not only such melodic motions, however, but also includes intervals that are found in the music as harmonic relationships at various levels. We are motivated to mix these interval types because the familiar structures that result from prolongational step-class automorphisms include passing figures, which are best expressed melodically, triads, which can be realized vertically, and fifth-progressions, which can come about through a combination of vertical and melodically articulated intervals. The total ordering of pitches implied by MOPs combined with the interpretation of intervals in the MOP both as melodic and harmonic intervals evokes Heinrich Schenker’s important idea of horizontalization or unfolding—that intervals of harmonic significance can be transformed into melodic intervals and thereby subjected to melodic elaboration. In fact, it takes the idea one step further by projecting a direction onto undirected foreground harmonic intervals so that intervals that remain vertical on the musical surface are horizontalized in the analysis.

514

J. Yust

The interaction between unordered step-class sets and ordered, prolongationallyinterpreted sets elevates the musical meaning that we can extract from these prolongational structures. Example 2 classifies the small number of transpositional types of step-class sets of order three and four. (Sets are shown on a generic staff to abstract them from particular pitch-class instantiations). The trichords symmetrical with respect to inversion function as the basic units of prolongational structures and provide an immediate musical interpretation of the structures in terms of the familiar musical-theoretic entities of passing motions, triads, and fifth-progressions. Figure 3 shows prolongational interpretations of these trichords and how they are transformed by the step-class automorphisms.

Example 2. A classification of step-class sets

Example 3. Automorphic transformations of step-class trichords


515

Example 4. Automorphic transformations of step-class tetrachords On the other hand, the step-class tetrachords that play a significant direct role in prolongational structures are not the more familiar inversionally-symmetrical sets but the H–symmetrical sets. These sets function as minimal tonal regions because they include exactly one instance of each type of inversionally-symmetrical trichord, and because a step-class tetrachord of this sort can prolong itself indefinitely due to its symmetry with respect to H. Example 4 shows prolongational instantiations of these tetrachords and their transformations under the step-class automorphisms. Note that H and H–1 don’t change the tetrachord but reorder it to change the background interval and the types of trichords that make up the prolongational structure. The possibility of such prolongational reorientations allows these tetrachord types to prolong themselves indefinitely. Example 5 shows a few possible prolongational relationships between tonal regions. Example 5a shows how a region can right-prolong itself through prolongational reorientation, and Examples 5b–d show possible relations between a region and its dominant, supertonic, and mediant regions. Note that there are two possible pitch-class manifestations of the step-class set for the basic tonal region: one major and one minor. This is because the fifths occurring in the set are always perfect fifths, and the remaining step-class not related to the others by fifth can be a minor or major third with respect to the set’s triad. The shading of triads in the MOP facilitates reading of the structures and will be used as a convention for the remaining examples. The method of step-class symmetries also generates an interesting approach to the prolongational interpretation of seventh chords (an inversionally-symmetrical type of step-class set). Example 6a shows the typical configuration of a dominant-tonic progression, and Example 6b shows how the addition of a seventh to the dominant may be prolongationally interpreted in terms of a passing motion. The nature of the dominant seventh chord by itself implies this sort of interpretation, in fact. Consider such a chord presented in isolation, as in Example 6c.

516

J. Yust

Example 5. Relationships between basic tonal regions

Example 6. Prolongational interpretations of dominants


517

Though the chord contains two triads, only one is bounded by a perfect fifth, so this major triad stands out as a prolongational unit. Yet, there’s no way for the remaining note, C, to make a complete prolongational structure with this triad, so it implies a continuation that will make a completion of the structure possible. A continuation of the stepwise motion D–C to B incorporates the note C into a prolongational unit, but the entire structure is incomplete and requires further continuation to the note G. In this sense, the dominant seventh chord by itself implies a continuation to its tonic. The (pitch-class) intervallic makeup of the chord is essential to this interpretation; Example 6d shows the simpler prolongational implications of a half-diminished seventh chord. The theory of step-class symmetries also leads to new outlooks on harmonic elaborations of the basic dominant-tonic structure. As Example 7a shows, the supertonic triad is a natural expansion of the stepwise motion from the third scale degree in the basic structure. The subdominant triad, on the other hand, occurs more naturally in an expansion of stepwise descent from the fifth of the tonic mirroring the progression of dominant to tonic on the right, as shown in Example 7b.

Example 7. Prolongational interpretations of pre-dominant harmonies There is one step-class interval that is not related to others through automorphism: the unison step-class interval. Therefore a theory in which harmonic intervals and melodic motions relate entirely through symmetries of the step-class group excludes the idea of repetition as a kind of motion or unison as a kind of interval. An analysis cannot express a melodic neighboring motion as a prolongation of a melodic repetition as in Example 8a, because the repetition doesn’t relate to the step intervals through any automorphism. This idea can be replaced, however, with the more compelling notion of embedded repetition of a prolongational structure. When we observe a departure and return, we first put the departure into a larger context of continuation. For instance, the motion B–A, a departure from B, implies a continuation to G (which is provided in Example 8a). We then interpret the return as a delay of the continuation by repetition.

518

J. Yust

Example 8. Embedded repetitions and extensions of prolongational structures In harmonic context, the best interpretation of the melodic motions shown in Example 8a would be in the form shown in Example 8b, a common type of background structure (as in a sonata first movement, for example). This analysis consists of the (right) embedded repetition of a tonal region, D–B–A–G, which can be accomplished entirely through symmetrical elaborations (i.e. H prolongations). Here, the motion B–A is part of a larger structure whose resolution to G is interrupted with a return to D–B. However, in certain cases the analyst may want to directly embed a repetition of a simple trichord structure, which one cannot do solely with symmetrical elaborations (Hprolongations). Example 8c accomplishes this with an asymmetrical prolongational unit where one motion is related to the prolonged motion by I (shown with a loop-arrow) and one is related to the prolonged motion by H–1 (shown with a double line). A different application of this sort of asymmetrical prolongation can replace the type of neighboring motion shown in Example 8d with a prolongational extension of a trichord (here a passing motion), as in Example 8e. In this case the motion B–C is not an initiation of a prolongational unit with an eventual continuation, but the reversal of a passing motion to A which then finds continuation. We can understand prolongational extension better by considering the asymmetrical prolongation as an involution or reflection of the normal symmetrical type. Example 9a shows how an asymmetrical prolongation results from reflecting a normal passing figure along one of its more foreground motions (B–A), pulling the other foreground motion (A–G) in behind what would ordinarily be the background interval (B–G). The loop-arrow indicates the flipped interval while the double line indicates the exposed background interval. Examples 9b–c show how the idea of involution can be extended to larger prolongational structures, drawing any foreground interval into the background. The advantage of deriving asymmetrical prolongations in this way is that one can base an interpretation of the resulting structure on the symmetrical one that it involutes. For instance, the structure of Example 9c gives a prolongation of a descending minor second by making it the resolving seventh of a V7–I progression.


519

Example 9. I/H-1 prolongations as a reflections or involutions of H-prolongations Another way to think of the I/H–1 prolongations is as an elision from a normal structure, which is illustrated in Example 10. Excising the root of the dominant, D, collapses two levels of structure into one (indicated by the double line from A to F# in the resulting structure), and turns the indirect leading-tone motion F#–G into an element of the prolongational hierarchy.

Example 10. The I/H–1 prolongation as an elision Example 11 uses the tools developed so far to give a representation of Schenker’s middleground analysis of the Largo of Beethoven’s op. 10 no. 3 (excluding details of the recapitulation) from Der Freie Satz. The analysis begins (at the most background level) from the basic D minor tonal region, elaborated with the standard interruption structure. The resolving tonic of this background structure is pulled back to the left in two places. Bringing the tonic back as the prolongation of A–F provides the foundation for the initial ascent, and as the prolongation of F–E it establishes the initial D-minor tonality of the second theme. The extended A minor region of the exposition occurs in an embedded A minor repetition of the background descent through the dominant (A major). The first part of this descent hosts the C major area of the exposition. (I depart from Schenker here in showing the dominant of C major following the arrival on E, m. 13, as it does in measures 14 and 16.) Schenker shows the startling F major area of the middle section initiating an ascending motion to regain the E of the fundamental line. The step-class symmetry analysis produces this ascending motion as an inversion of the preceding E–A motion. The chord F major is a triadic extension of the overriding A minor.

520

J. Yust

Example 11. Schenker’s analysis of Beethoven’s op. 10 no. 3, 2nd mvt., and a step-class automorphism analysis

Bibliography Cohn, R., Dempster, D.: Hierarchical Unity, Plural Unities: Towards a Reconciliation. In: Disciplining Music: Musicology and its Canons, edited by Katherine Bergeson and Philip V. Bohlman, pp. 156–181. University of Chicago Press, Chicago Lerdahl, F., Jackendoff, R.: A Generative Theory of Tonal Music. MIT Press, Cambridge (1983) Rahn, J.: Logic, Set Theory, Music Theory. In: College Music Symposium, vol. 19(1), pp. 114– 127 (1979) Schenker, H.: Neue Musikalischen Theorien und Phantasien III: Der Freie Satz. In: Jonas, O. (ed.), 2nd edn. Universal Edition, Vienna (1956) Schenker, H.: Free Composition. Phantasies, Trans. Ernst Oster, vol. III. Longman, New York (1979) Smoliar, S.: A Computer Aid for Schenkerian Analysis. Computer Music Journal 4(2), 41–59 (1980) Yust, J.: Formal Models of Prolongation. Ph.D. diss., University of Washington (2006)

A Linear Algebraic Approach to Pitch-Class Set Genera Atte Tenkanen Department of Musicology University of Turku, Finland [email protected]

Abstract. The concept of interval-class vector (ICV) plays an important role in musical set theory. ICV can be seen as a six-dimensional representative of the intervallic content of a set class (SC), which often forms the basic tool in harmonic analysis of twentieth century music. Interrelations between SCs have been evaluated by means of similarity functions and clustering techniques in many contexts. SCs have also been classified into ’families’ called genera. Among others, six-, sevenand twelve-part systems have been outlined. Using the linear algebraic concept of spanning, the six-part genera is unambiguously justified. A group of interval-class vectors, which actually represent points, restrict a finite area in a six-dimensional space. Using the determinant of a matrix, the volume of the area formed by a six-member set class combination1 can be calculated. All possible set groups among TnI-type trichords or hexachords, which produce maximum volumes, are detected. Both the extreme points on the edges of SC space as well as the most neutral set classes in the middle of SC space are recognized using three different methods derived from linear algebra, namely the determinant of a matrix, cosine distance and principal component analysis. A short final demonstration concerns volumes of hexachord combinations used by Finnish composer Magnus Lindberg in his chaconne chains.

1

‘Corner-Stone Set-Classes’

The harmonic possibilities of our equal-tempered system are, in respect of all possible pitch combinations, endless. However, we can reduce this information and find some primary intervallic characteristics which originate from the mathematical properties of the octave divided into 12 equal intervals. When classifying pitch-class sets into families, music set-theorists talk about the concept of pitchclass set genera (Forte 1988) or simply genera (Quinn 2004). Usually six main sonorities have been distinguished and seen as projections of the six interval classes (ibid., p.2). This scheme of the six-part genera can also be justified by the linear algebraic concept of ’spanning’, which we utilize in this paper. When comparing 224 TnI-type set classes (SC) some of them seem to emerge as somewhat divergent from the others, 3-12, 4-28 and 6-35 just to mention a few. 1

That means type M6x6 (R) matrix.


522

A. Tenkanen

These sets can be seen as sort of ’corner-stones’ of a six-dimensional pitch-class set space, because they are literally on the corners of the space they form with the set-classes of the same cardinality. The reasons for their special position have been well documented (ibid., passim). To form an overall picture of their relative positions among others they have been visualized in two-dimensional space (see e.g. Eriksson 1986). At times this has been done computationally using dimensionality reduction techniques such as multidimensional scaling (Samplaski 2005) and hierarchical clustering (Quinn 2001). Our purpose in this presentation is to find the corner-stone SCs and to provide an unambiguous numerical interpretation of the limits of spaces which are generated by combinations of 6 interval-class vectors (ICV) by using the determinant of a matrix.2 If we thereafter want to outline relationships between the extreme point SCs and other SCs, it can be done using some similarity or distance measure. However, it is impossible to reduce six-dimensional information visually to two dimensions, as is done by multidimensional scaling without partly odd results, regardless of the method used. This is not a problem with the numerical methods presented here.

2

Applying Cosine Distance and the Determinant of a Matrix with Musical Set Classes

There is one well-known and much used similarity measure for SCs which has a direct connection to linear algebra, namely the cosine of angle between two vectors (Rogers 1999). The idea behind this measure is: when the angle between two (interval-class) vectors decreases, the similarity between them increases. If two vectors are parallel to each other (maximally similar), the value of the cosine is 1. In case of the minimum (0) the two vectors are orthogonal to one another. The other basic linear algebraic method which we apply here, namely the determinant of a matrix, is also connected to orthogonality. However, the two aforementioned applications belong to different categories according to their applicational scope. The determinant is a single number which give information about certain properties of an nxn-type (square) matrix. There are many useful applications of the determinant. For example, it gives the volume of an n-dimensional ’box’ (called parallelepiped) which has been formed from row or column vectors of a matrix. In this study, these rows are built up by six ICVs. To generate or ’span’ the whole six-dimensional interval-class space, it is sufficient to select six linearly independent vectors. Vectors v1 , v2 , ..., vn are linearly independent if and only if there is not any linear combination of these vectors which gives the zero vector (except the zero combination, where all coefficients ci are zeros): c1 v1 + c2 v2 + ... + cn vn = 0 For an obvious example of the dependency we have chosen two linearly dependent ICVs, [002001] and [004002], of SCs 3-10 and 4-28. Both of them can 2

Formal mathematical definitions of the determinant can be found in most elementary guides of linear algebra (see e.g. Lipschutz and Lipson 2001, ch. 8).

A Linear Algebraic Approach to Pitch-Class Set Genera

523

be constituted from the other one by multiplying it with a scalar, in this case by 2 or 1/2: 2 ∗ v1 − v2 = 0 ⇔ 2 ∗ [002001] = [004002] Because they are parallel to each other, the other one doesn’t bring any new information to (or span a new dimension in) the six-dimensional interval-class space. That all vectors are independent in a more complex system of several vectors is not, however, as easy to see as in the case of the two vectors. The volume of the six-dimensional box formed by six ICVs can be calculated using the determinant of a matrix. If the vectors in a matrix are dependent, the volume of this set is 0 and the value of the determinant is 0. This effect is easier to understand in three-dimensional space: by shrinking just one dimension of a box to 0, we get a plane, the volume of which is 0. If the ICVs are standardized by dividing them by their Euclidean norm, the value of the maximum volume they are able to form is 1. This is exactly the case with the six interval classes, which form a six-dimensional orthogonal ’cube’. It is pertinent to mention herein that the interval classes establish an example of a standard basis consisting of six standard vectors. A basis for a vector space is a sequence of vectors v1 , v2 , ..., vn with two properties: (i) they are independent (det(Mnxn ) = 0) and (ii) they span the whole space. The determinant of a matrix is denoted with two lines around the matrix: 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 = 1 0 0 0 0 1 0 0 0 0 0 0 0 It is easily proved that using linear combinations of the standard vectors, the whole R6 -space can be generated. In that space, 208 ICVs form, however, only a tiny bunch of points.

3

Volume Tests with Interval-Class Vectors

Allen Forte (1988) based his system of pitch-class set genera on 12 TnI-type trichords. According to the concept of spanning, only six vectors are needed to generate all the dimensions of ICV space. In order to find the largest space among all six-member TnI-type trichord combinations, we calculated the volumes of all 924 six-vector combinations. The rows of the determinant below include the ICVs of those trichords which form the largest volume. SCs in question are 3-1, 3-8, 3-10, 3-12, 3-9, and 3-5.3 The determinant value has been calculated using the standardized (by their length) vectors. 3

All these belong to Eriksson’s (1986) ’maxpoint’ sets, a chord species containing the maximum number for its size of at least one interval class. However, the maxpoint set for ic2, which is 3-6 [020100], is missing.

524

A. Tenkanen

0.89 0.45 0 2 1 0 0 0 0 0 0 0 0 0.58 0 0.58 0 0.58 0 1 0 1 0 1 0 0 2 0 0 1 0 0.89 0 0 0.45 ⇒ 0 ≈ 0.48 0 0 0 0 3 0 0 0 0 1 0 0 0 0.45 0 0 1 0 0 2 0 0 0.89 0 0.58 0 1 0 0 0 1 1 0 0 0.58 0.58 This set of trichords is unequivocal: there are no other alternatives of the same volume. In his system of trichordal ’progenitors’, Forte called these six SCs ’chroma’, ’whole-tone’, ’diminished’, ’augmented’, ’dia’ and ’atonal’ (Forte 1988, p. 201). What is missing here are Forte’s ’semigroups’ (’semichroma’, ’chromadia’ etc). By using the concept of spanning we rejected Forte’s proposition of twelve trichordal progenitors4, but on the other hand, we have thus managed to catch his main trichords. The determinant formed by the trichords is < 1 which means that these ICVs do not form an orthogonal six-dimensional cube: some of them are in rectangular position to each other but that is not the case among all the pairs. In fact, there are only 18 TnI-type SCs of cardinality domains 3 − 12 whose ICVs form perpendicular pairs with other ICVs. Among them 7 belong to cardinality 4, the remaining 11 to cardinality 3. By constructing six-vector linear combinations out of these 18 vectors, the largest determinant among 18564 possible combinations5 is ≈ 0.396. There are only four such combinations. If we, moreover, prefer the sets of cardinality 4, the result combination includes 3-6, 3-12, 4-1, 4-9, 4-23, and 4-28. Each of them emphasizes a certain interval class (2, 4, 1, 6, 5, 3 respectively). Some further observations can be made about this group of sets. It consists of two transpositionally symmetric and maximally even SCs, 3-12 and 4-28, which are complementary to each other. One might say that this pair chrystallizes several ’harmonic odours’. Even though there is only one possible union to be attained with these two sets6 , the union set 6-Z28 (0,1,3,5,6,9) contains not only all four triads and the dominanth seventh chord like its better known sister 634A (0,1,3,5,7,9)7, but also both of the important tetrachords used in enharmonic modulations, namely the diminished seventh chord and the ’Tristan chord’. It is, however, good to remember, that in order to span the whole six-dimensional interval-class vector space, six linearly independent vectors are needed.8

4

5 6 7 8

If we have six-dimensional ICV space, we can find the remotest points only in six directions, not more. " # 18 = 18564. 6 This is so called ’complement union property’ defined by Morris (1990). I.e. ’Prometheus chord’. Though the ICV [111111] of 4-Z15 and 4-Z29 can be formed by the union of all interval classes, this vector spans only one dimension: we can’t build a box using one side only!


4

525

‘Strangest’ Hexachords

After the excursion with orthogonal vectors we will now focus on hexachords, which have often been of special interest to the twentieth century composers and music theorists. Before using the determinant with hexachords, some general observations can be made about the extreme SCs indirectly by applying a similarity measure. By calculating all the similarity values between all hexachord pairs, we get an overview of how ’distant’ or ’strange’ each of them are among all others. In order to produce distinctions between Z- and Tn-type SCs, we extend the well-known cosine measure, proposed in musical set theory by Rogers (1999), a little by using the subset-class vectors 2-10CV instead of using mere ICVs (2CVs)9 . Such similarity functions, which exploit subset classes, have been called ’total measures’ (Castrén 1994, p. 4). Therefore, we call this measure costotal. If the subset-class vectors of the two SCs sc1 and sc2 are denoted by subcv1 and subcv2 , we get the formula costotal(sc1 , sc2 ) =

subcv1 · subcv2 subcv1 subcv2

where the dividend is the dot product of the subset-class vectors and the divisor is the product of the Euclidean norms of the subset-class vectors. After calculating all the costotal-values (ranging from 0.167 to 1.0) between all the hexachord pairs and thereafter all the mean values 10 related to each hexachord, the strangest among others with their ’strangeness-values’ appear to be: 6-35, 0.357; 6-20, 0.491; 6-7, 0.525; 6-30A/B, 0.560; 6-1 and 6-32, 0.568; 6-27A/B, 0.591 (see Figure 1). There seems to be a connection between the lower values and the number of symmetrical axes. That ’poor guy 6-35 without friends’ (except Debussy!), or as Forte puts it, ’notoriously antisocial creature’ (Forte 1988, 192), has 6 symmetrical axes, 6-20 three, 6-7 two, 6-30A/B, 6-1 and 6-32 one and 6-27A/B none. On the ’social’ side we meet Z-vectors only: 6Z11A/B and 6-Z40A/B, 0.696; 6-Z10A/B, 6-Z24A/B, 6-Z39A/B and 6-Z46A/B, 0.69711 . The same ’strange’ sets 6-35, 6-20, 6-7, 6-30, 6-32, 6-1 form the corners of the space with the greatest determinant value 0.131 among all hexachord combinations.12 9

10 11

12

nCV stands for subset-class vectors which means that all the subset occurrences of cardinality class n, embedded within a SC, are taken into consideration (Castrén 1994, p. 3). Note that, in the case of the mean values, there are no differences between A- and B-type SCs, because the hexachords themselves form a symmetrical group of sets. Among the trichords the ’middlemost’ sets are 3-3, 3-4 and 3-11 with mean value 0.408. " # 35 There are = 1623160 combinations of six-member hexachord groups. Note 6 that the determinant is calculated using a square matrix (of type 6x6 here). Thus the ICVs are used instead of 2-10CV’s. These 6 SCs belong also to Eriksson’s maxpoint sets. SC 6-27 is missing, which is a maxpoint for ic3. This is substituted by cyclically symmetric 6-30.

526

5

A. Tenkanen

Principal Component Analysis: A Flexible Approach for Mapping ICV-Space

There is one more flexible method which finds the corner SCs and in addition to this, the most neutral or ’odourless’ SCs in the middle of set-class space, namely principal component analysis (PCA). The central idea of PCA is to reduce the dimensionality of a data set consisting of a larger number of interrelated variables, while retaining as much as possible of the variation present in the data set (Jolliffe 2002, p.1). PCA transforms the data to a new coordinate system with orthogonal axes. Those axis vectors are called principal components (PC). The first PC is such a vector, which has the maximum variance with all the data points. The second greatest variance lies on the second PC and so on. These axis vectors are, in fact, real-valued eigenvectors which are a special set of vectors associated with square matrix. However, PCA is a flexible method also in terms of initial requirements: it is not bound to dimensions or to square matrices. We present next an example which is a bit tricky but well-founded. We can calculate principal components for, say all the 2-3CVs of Tn-type hexachords. In this presentation, notation 2-3CV means that not all the subsets but only the 6 interval classes (2CV) and 19 Tn-type trichord subsets (3CV) embedded within a hexachord are worked out as a representative of each hexachord-SC in question. These vectors thus belong to R25 . As an example, the 2-3CVs of set-classes 6-1 and 6-35 are presented in Figure 2. There are only 35 different ICVs (2CVs) but 79 different 2-3CVs among all 80 Tn-type hexachord SCs (6-14A and 6-14B have identical 2-3CVs). Thus, using the 2-3CVs both A- and B-type as well as Z-type SCs can be distinguished. After calculating all the real-valued principal components, 25 pieces in all, we chose the first six of them and calculated the nearest representatives for them among

Fig. 1. Costotal means calculated for 80 Tn-type hexachords. The six ‘strangest’ hexachords have been marked with Forte notation.


527

Fig. 2. The 2-3-class-vectors of set classes 6-1 and 6-35

actual 2-3CVs by using the cosine distance. The best ’discriminator’ vectors among hexachords thus found are again of the symmetrical 6-35, 6-1, 6-7, 620, 6-32, 6-30A/B. Similarly, the middlemost SCs can be found, and they are 6-Z24A/B, 6-Z39A/B, 6-Z40A/B, 6-Z11A/B, 6-Z46A/B and 6-Z10A/B. Principal components can be calculated for any combination of vectors of equal length. Therefore, if we are able to derive, for example, all the ICVs that a musical piece one way or another includes13 , all the six PCs specific to that piece can be computed. This information, for one’s part, can be used, for instance, for classifying musical pieces.

6

Using Corner-Stone Vectors for Producing a System of Genera

The goal in different genera systems is to place all the set classes into a certain pcset family (genus). This is easier said than done. There are many borderline SCs which do not seem to belong clearly to any group (Samplaski 2004). Several approaches have been proposed for solving this problem. For example, multidimensional scaling can be instrumental in mapping set classes into more illustrative two-dimensional space. However, this method is problematic: a lot of data is thrown away by reducing six-dimensional information to two dimensions. For instance, SCs such as 6-1, 6-8 and 6-32 with fairly similar ICVs, [543210], [343230] and [143250], may be projected - among all other hexachords - to the same point in two- or three-dimensional space even though they are very different in their chordal nature.14 13 14

Some kind of an automated segmention is naturally required to produce the intervalclass vectors. This may happen with multidimensional scaling if the two projection vectors are the first two principal components calculated by using the ICVs of all hexachords. In that case, PC-vectors are based on greatest variance between different interval classes. Among Tn-type hexachords the ic-variances are: ic1, 0.82; ic2, 0.91; ic3, 0.87; ic4, 0.94; ic5, 0.82; ic6, 0.50. Ic2 and ic4 have the greatest variance among the ICVs of hexachords and hence these interval classes are emphasized by the projection vectors. But what is more important in this case is that because the sums of the ic1- and ic5-components with distinctive values are the same (6) in all three set classes with similar ’weights’ (0.82), these set classes project to the same point in two-dimensional space.

528

A. Tenkanen

One straightforward solution for generating families from the pitch-class sets of the same cardinality is 1) to select the corner-stone set classes for the heads of the families and 2) to connect all the rest of the SCs to the heads according to their distance from them. If, in the case of hexachords, the corner-stone SCs 6-1, 6-7, 6-20, 6-30, 6-32 and 6-35 are accepted as basis vectors and the Euclidean norm is used to evaluate the distances between the normalized 2-10CVs, the resulting genera is as seen in Figure 3. The figure is analogous to what Eriksson (1986, 106) represents about the genera of hexachords15.

Fig. 3. Corner-stone hexachords and their nearest relatives. The criterion for inclusion is the Euclidean distance.

7

Harmonic Space in Composition

Let us finally make another short experiment with determinants. In discussing the pioneering work of Howard Hanson in the 1960s, Michael Buchler (1998, p.5) states that Hanson’s work was, in part, ’meant to show composers how they could project certain intervallic structures using particular Scs’. An example of a composer who uses SCs systematically this way is the contemporary Finnish composer Magnus Lindberg (b. 1958). Many of his works are ’based upon an extended chaconne principle with chord chains cycling around, undergoing constant transformation and being articulated in a very gestural way’ (Oramo 2004, p. 5). According to Lindberg, these chords are based on hexachords 6-2, 6-5, 6-9, 6-15, 6-16, 6-18, 6-21, 6-22, 6-27, 6-30, 6-31, 6-33 and 6-34 (ibid., p.7). Lindberg seems to avoid the corner SCs such as 6-1, 6-35 and ’break colours’ using 6-2, 634 etc. instead. Our hypothesis was that Lindberg’s SCs could still form a fairly large volume. If five of those special corner SCs (6-35, 6-20, 6-7, 6-32 and 6-1) 15

In fact, the corner-stone sets are comparable to Eriksson’s maxpoint sets, which collect the rest around them.


529

are rejected, using the remaining 30 ICVs, the largest determinant16 (≈ 0.025) is formed by SCs 6-2, 6-Z6, 6-14, 6-22, 6-30 and 6-33. Four of these belong to Lindberg’s ’vocabulary’. We evaluated all the possible determinants formed by the six SCs that are used in the compositions Corrente and GranDuo 17 . The biggest volume in GranDuo (0.00276) is attained by SCs 6-2, 6-16, 6-21, 6-27, 6-30 and 6-34 and in Corrente (0.00072) by 6-15, 6-21/6-34, 6-22, 6-27, 6-30 and 6-31. The mean of the determinant values in GranDuo is appr. 0.0009 and in Corrente 0.0004. It seems that especially the nearly chromatic SC 6-2 widens the harmonic space of GranDuo compared to the corresponding space of Corrente (without a chromatic relative of genus 6-1, cf. Figure 3).

8

Conclusions

Cosine distance and the determinant of a matrix as well as principal component analysis can well be used to evaluate the ’corner’ SCs and the ’middlemost’ SCs in the set-class space. Using the concept of spanning, the number of the basic SCs, which are needed to determine the pitch-class set genera, can be unambiguously defined in the case of six-dimensional ICV space. Regardless of the methods used, the same groups of SCs seem to emerge. They belong to the group of Eriksson’s ’maxpoint’ sets or maximally symmetrical set classes.

References Buchler, M.H.: Relative Saturation of Subsets and Interval Cycles as a Means for Determining Set-Class Similarity. Ph. D. dissertation, University of Rochester (1998) Castrén, M.: Recrel: A Similarity Measure for Set-Classes. Ph.D. dissertation, Sibelius Academy (1994) Castrén, M.: Aspects of Pitch Organization in Magnus Lindberg’s GranDuo for 24 Wind Instruments. In: Humal, M. (ed.) Proceedings of the Fourth International Conference of Music Theory, Tallinn, April 3-5, 2003. A Composition as a Problem, vol. 4(1) (2004) Eriksson, T.: The IC Max Point Structure, MM Vectors and Regions. Journal of Music Theory 30(1), 95–111 (1986) Forte, A.: Pitch-Class Set Genera and the Origin of Modern Harmonic Species. Journal of Music Theory 32(2), 187–270 (1988) Jolliffe, I.T.: Principal Component Analysis, 2nd edn. Springer Series in Statistics (2002) Lipschutz, S., Lipson, M.: Schaum’s Outline of Theory and Problems of Linear Algebra, 3rd edn. McGraw-Hill, New York (1968) (2001) Morris, R.: Pitch Class Complementation and its Generalizations. Journal on Music Theory 34(2), 175–245 (1990) 16 17

The statistical values for minimum, 1st quartile, median, mean, 3rd quartile and maximum of such determinants are 0.0000, 0.0002, 0.0005, 0.0009, 0.0011 and 0.0252. The basic hexachords used in Corrente are 6-15A/B, 6-21A/B 6-22A/B, 6-27A/B, 6-30A/B, 6-31A/B, 6-34A/B (Oramo 2004, p. 9) and in GranDuo 6-2A/B, 6-16A/B, 6-21A/B, 6-22A/B, 6-27A/B, 6-30A/B, 6-31A/B and 6-34A/B (Castrén 2004).

530

A. Tenkanen

Oramo, I.: Chaconne-periaate ja muoto Magnus Lindbergin Correntessa (ChaconnePrinciple and Form in Magnus Lindberg’s Corrente). S¨ avellys ja musiikin teoria 11, 5–15 (2004) Quinn, I.: Listening to Similarity Relations. Perspectives of New Music 39(2), 108–158 (2001) Quinn, I.: A Unified Theory of Chord Quality in Equal Temperaments. Ph.D. dissertation, Eastman School of Music (2004) Rogers, D.W.: A Geometric Approach to PCSet Similarity. Perspectives of New Music 37(1), 77–90 (1999) Samplaski, A.: Mapping the Geometries of Pitch-Class Set Similarity Measures via Multidimensional Scaling. Music Theory Online 11(2) (2005)

Author Index

Adilo˘ glu, Kamil 67, 220, 247 Agon, Carlos 412 Ahn, Yun-Kang 412 Amiot, Emmanuel 469 Anagnostopoulou, Christina 247 Anders, Torsten 52 Andreatta, Moreno 412 Buteau, Chantal

59, 247

Carey, Norman 449 Cartwright, Julyan H.E. 168 Cheng, Eric 347 Chew, Elaine 11, 240, 347 Clampitt, David 464, 477 Clark, Jonathan Owen 330 Dom´ınguez, Manuel

477

Ebeling, Martin 140 Erkkil¨ a, Jaakko 156 Exarchos, Dimitris 419

Lartillot, Olivier 156, 230, 247 Lewis, David 107 Luck, Geoff 156 Majchrzak, Miroslaw 257 Martins, José Oliveira 499 Miljkovic, Katarina 318 Milmeister, Gérard 19 Morris, Robert 266 M¨ ullensiefen, Daniel 107 Nolan, Catherine 375 Noll, Thomas 477 Obermayer, Klaus 67 Oehler, Michael 189 Parncutt, Richard 124 Peck, Robert 489 Piro, Oreste 168 Pound, Eleri Angharad 198

Ferkov´ a, Eva 250 Foley, Gretchen C. 365 Fran¸cois, Alexandre 11

Rahn, John 289 Reuter, Christoph 189 Rhodes, Christophe 107 Riikkil¨ a, Kari 156 Roeder, John 303

Garbers, J¨ org 78, 97 Giesl, Peter 117 Gollin, Edward 340, 406 Gonz´ alez, Diego L. 168 Grijp, Louis 78, 97 Gualda, Fernando 430

Schlingmann, Dirk 441 Scotto, Ciro 25 Sethares, William A. 1 Shuster, Lawrence B. 354 Sidl´ık, Peter 250

Honingh, Aline

Tanir, G. Ada 220 Tenkanen, Atte 211, 521 Toiviainen, Petri 156

88

Ianni, Jerry G. 354 Ilom¨ aki, Tuukka 386 Jedrzejewski, Franck Johnson, Tom 311

493

Kranenburg, Peter van 78, 97 Kreyszig, Herbert 392 Kreyszig, Walter 392 Kulp, Christopher W. 441

Veltkamp, Remco C. 78, 97 Vipperman, John 59 Volk, Anja 78, 97, 204 Wiering, Frans Yust, Jason Zd´ımal, Milan

78, 97 512 250

Index

algebraic combinatorics on words 464, 484 all-interval trichord 454 all-trichord hexachord 303 analysis (musical) 156,211,412,441,443 - automated 1,347 - computer-aided 250 - melodic 59, 247 - Schenkerian 512 - tonal 250, 251 Argus algorithm 240, 241 Atonality 124, 125 audio signal processing 10 autocorrelation 142-148 automorphism group 512 axis-dyad chord 367, 368 bayesian model selection 107, 111 beat tracking 4, 5, 7 block design 311, 313, 317, 454 Boethian music theory 404 Burnside's lemma 271 canon, rhythmic 469,473 canonical form 419 category theory 19,20,412,415,430 Cayley graphs 310,342,343 cellular automata 318,328 centralizer subgroup 489-491 chord 367 - axis-dyad 367,368 -labels 107 - mystic 214,217,218 -rootofachord 124,127,128 - stability 78-87,97,98, 105 Christoffel word 479,480,481 Christoffel duality 480, 481 cluster, chromatic 472 clustering - melodic 59 Cohn flip 464, 504, 505 combination theory 311,317 combinatorial group theory 342 combinatorial word theory 477 common tone theorem 273, 275

comparison set analysis 211-212,218 complement union property 303, 305 composition - algorithmic 321, 338,436 - by creative analysis 412 - constraint based 52, 53 - mathematically inspired 311 compositional design 305 connectivity component 70,72-74 consonance 134-135, 140-141 constraint satisfaction problem 24, 52, 53 content-based retrieval 97 convex sets 88 correlation coefficient - Pearson 81, 82 - Spearman 253 cycle - maximally smooth 464, 467 - modulating 464 cyclical spectra 190 data format 19 denotator 19,20 devi's staircase 171, 172 diagram - co1imit of a 22, 24 - in category theory 19, 20 -limit of a 20,21 diatonic modes 477,483,484 Dirichlet distribution 107, 110 dispersion (statistics) 253 dissonance 134, 140, 141 distance - earth movers (EMD) 79, 82, 85, 98 - transportation 105 dynamics - nonlinear 168, 185 - nonlinear (of networks) 331 dynamical systems 117-119,330,332 Euler lattice 88, 89 expectation - violated 14 expressive performances 347, 352

534

Index

Fano plane 454, 455 Fareysum 172-174,185,493 feature extraction 156 flat interval distribution 469,474 folk song (dutch) 97,99 forced oscillator 168, 171, 185 formal concept analysis 235 formant 189,191 Fourier analysis 3--4, 143, 168 Fourier Transform - discrete, in audio signal processing 10 - discrete, in music theory 469 Fourier spectrum 170 frequency ratio 392 fundamental (pitch) - missing 168, 170, 174 Gaffurio number generating function 392, 397,401 Galois connection 236 generalized coincidence function (in pitch perception) 141,148,153 generative processes 318 gestures - characteristic (in transformational theory) 303, 306, 310 golden mean 168,176,179 graph 305 - as compositional space 305 - Cayley 310,342-345 - outerplanar, maximal 512 - theory 489,490 - underlying a K-net 365, 371-372 group - action 289, 292,489, 490 - centralizer subgroup 489,491 - Klein four-group 303, 308 - SchrittlWechsel 489 - stabilizer subgroup 356 - symmetric 340, 342 - theory 266, 283, 285, 342, 430 - theory, combinatorial 342, 430 - TranspositionlInversion 271,275,489 Guidonian system 393, 395 harmonic - analysis (in mathematics) 469 - analysis (in music) 116, 121, 122,407 - labeling 97, 107, 250

harp 198, 199,201 hexachord - all-trichord 303, 305 - theorem 274 Hopf Bifurcation 332, 333 HopfTheorem (equivariant) 332 host set 499,503,504,508 Hungarian minor 464, 467 improvisation 156, 159, 160 inferior nucleus 143 information Theory 441 inheritance property 67, 70, 73 inner metric analysis (IMA) 78 interval - all-interval 271, 281 - preservation 489 - System generalized 489, 490 intervallic content 470 isography 386, 390 - axial 358, 363 - double-axial 358, 362 - positive 389 - strong 355,361,387,389,390 isomorphism 282, 292 key finding algorithms 12, 88, 250, 257 key range 258, 265 K-relation/Kh -relation (Forte) 283 Klumpenhouwer networks (K-nets) 290, 354,365,375,386,492 - tetrachordal K-nets 354, 362 - trichordal K-nets 354, 362 line of fifths - spiral array 11, 12,246 - straight line 257-265 linkage cluster 221 local meter 80,81,204-205,208 Lyapunov function 118, 119 matrix multiplication 130 Mathematica (Software) 318,441 maximally even - *scale, tightly generated 494 - sets 469-471 MaxMSP (Software) 338 mental disorder 156, 159, 165 metric analysis 78, 79, 204

Index metric weight - inner 78,79, 204, 209 -levels 80,81,97, 100 microtones 198 MIDI 156, 158, 165 modulation 88, 92, 94 monoid 290,291 - action 289, 290, 292 morphisms 416,434,436 motif - description 53, 54, 56, 57 - motivic space 59, 65 - motivic structure 59,60 - musical 52, 53 - representation 52, 54, 56 - variation 52, 53, 55 motivic - analysis 59,61,230,247 - pattern 167,230,247,248 multiplicities (Deleuze) 335 musical action 289, 296 musical analysis - computer aided 220 musical form 304, 306, 310 musical humour 11, 17 musical prosody 11,240,347 music therapy 156, 157, 165 music theory - Boethian 404 - medieval 499,501 Musica Enchiriadis 511 Muugle (Software) 97,99 naturalness (of wind instrument sound) 196 nature (and music) 313,318,319 neighboorhood - (melodic) content 68,73,220 - (melodic) presence 68, 72, 75 neo-Riemannian theory 410 network 306,311,330,335,338,365,367, 406 - isographies 375, 378 - isomorphism 293 neuronal processing 141 nonlinear circles 198, 199 nonlinear time series analysis 443 normal form 25,26,31,48 number - chromatic 494, 495 - elementary 392,401 - factorization (Chinese remainder theorem) 396, 397

535

- natural 213 - prime 396,397,401 - theory 392, 396 octave-complex tones 128, 132 Open Music (Software) 412 ordinary differential equations (ODE) 330 pairwise well-formed scales 464, 466, 467 partials 127 pattern extraction 230, 232 perceptual analysis 135 perceptual salience 127 Perle cycles 365,367,370 periodicity (in pitch height) 422,425,495 - quasiperiodicity (of a response) 171, 172 permutation 341,345 phrase detection 347,350 phrasing strategies 347,348 physical modeling 189 pitch-class category 434 pitch-class space 499 pitch perception 140, 143, 168, 175, 186 pitch stability 99, 100, 105 polyphony - implicit 22-29,67-71,247-249 prime numbers 396,397,401 programming paradigm (functional/object oriented) 412,414 prominence profile 221, 224 proportions 392, 395 psychoacoustics 124, 127 pulse forming 189,190,193,196 pulse width function 195 pure intonation 406 Q-relation 467 redundancy reduction 227,247 representation - musical, acoustic 95,347,348, 353 - musical, symbolic 107, 108,211,213 - of a motiv 58 residue behaviour 169, 173, 174 rhythm 1,2,5, 10 rhythm analysis 214,216,217 rhythm-class set 215 roughness 140, 141 Rubato (Software) 19,98,101,412,413, 416

536

Index

scale 88,89,91,198,200 - chromatic (or equidistant) 421 - Dasian 499,501,502,505 - diatonic 449,451,452 - golden 168,176,179,180,182 - Hungarian minor 464, 467 - maximally even*, tightly generated 494 - pseudo-diatonic 493-495 - theory (musical) 477,482,497 - Wyschnegradsky diatonic 493 Schlegel diagram 345 Schumann's principles of Timbre 189 segmentation, automatic 68,213,216, 228,247 self-similarity 318,320,322 set (in mathematics) - convex 88,89,91 set (of pitch classes) - comparison 211,213,215 - complex 268, 283 - maximally even 469,471,482,484 - theory 28,48,125, 135,219,266,273, 430,431,436,449,451,499 - theory, diatonic 449,451,458,512,513 Shannon entropy 444 sieve theory (Xenakis) 419,421,423 similarity - contour 220 - neighborhood, melodic 59,67,69,70, 97,101,220,221,247 - rhythmic 78, 82, 86, 204 similarity measures 213 space - topological 59, 60, 64 specificity relation 230, 236, 237 spiral array model 11, 12 statistical modeling 156, 159 Steiner triple system 454, 455 step-class - automorphism 512,515,517 - interval 512,513,515,517 - tetrachord, automorphic 513,515 - trichord, automorphic 449,452, 454 Stem-Brocot Tree 477,493,494 - shuffled 493, 494 structure - melodic 67,69,70,76 - motivic 58, 59, 230, 232 symmetry 303,310,421,424,428 - in pitch space 11-13, 347

synthesis and analysis framework 193 196 synthesizer 189,190, 193 ' systema teleion 393, 394, 397, 401 temperament 469 - LehmannlBach 469 - equal 198,199,203 - non-equal ("non-linear") 198, 199 tetrachord 354,357,358,361,362, 375,379,499,500,501 theorem of Wiener 143 timbre 189, 195, 196 tonal - analysis 512,517,519 - fusion 140,153 - harmony, irrelative system in 257,263 - implications 124, 126, 135 - region 515,518,519 tonality 11,15,124,133,134 - Perle 386, 388 tonnetz 345,406,408,409,410 topology - melodic 67,69,70,71,73-75 - motivic 60,61 topological model 60, 64, 67 topos 415,418 transfer entropy 441,445,447 transformation - prolongational 512, 515, 517, 518 - triadic, uniform 491,492 transformatio 499,502,504,506,508 510 transformational 375,383,406,409,410 - analysis 290, 365-374 - theory 344, 406 transitus class 407 transpositio 499-503 transpositional type (T-type) 125 Tree 289, 301 - Christoffel 477,479,480,485 - Stem-Brocot 477,493,494 trope system (Hauer) 271 tuning systems 198, 199,200,203 twelve tone -invariance 268,269,270 - system 266,267,271-273,281, 286 - tonality 365, 370, 374 unique factorisation theorem 419

Index variation - of a folk song 78,81,97, 101,106 - of a motiv 52 Variophon 189,193,195-197 virtual pitch 127, 128 visualization - of melodic clusters 59,247 - of tonal content 388 voice leading 449, 454, 456 volley principle 141

537

weight - inner metric 78, 79, 81 - spectral (metric) 205,206,208 weighted Gaussian mixture 338 well-formed scales 464,465,467,477,486 whole-tone scale proposition 388-390 words - Christoffel 464,477-488 - Sturrnian 464 Z-relation (Forte) 469,470,475

Mathematics and Computation in Music: First International Conference, MCM 2007, Berlin, Germany, May 18-20, 2007. Revised Selected Papers (Communications in Computer and Information Science)

Mathematics and Computation in Music - MCM 2011

Constructing Ambient Intelligence: AmI 2007 Workshops Darmstadt, Germany, November 7-10, 2007, Revised Papers (Communications in Computer and Information Science)

Software and Data Technologies: Second International Conference, ICSOFT ENASE 2007, Barcelona, Spain, July 22-25, 2007, Revised Selected Papers (Communications in Computer and Information Science)

E-Voting and Identity: First International Conference, VOTE-ID 2007, Bochum, Germany, October 4-5, 2007, Revised Selected Papers

Bioinformatics Research and Development: First International Conference, BIRD 2007, Berlin, Germany, March 12-14, 2007, Proceedings

Information Theoretic Security: Second International Conference, ICITS 2007, Madrid, Spain, May 25-29, 2007, Revised Selected Papers (Lecture Notes in Computer Science Security and Cryptology)

Information Theoretic Security: Second International Conference, ICITS 2007, Madrid, Spain, May 25-29, 2007, Revised Selected Papers (Lecture Notes in Computer Science Security and Cryptology)

Research and Education in Robotics -- EUROBOT 2008: International Conference, Heidelberg, Germany, May 22-24, 2008. Revised Selected Papers (Communications in Computer and Information Science, 33)

Computer Vision and Computer Graphics. Theory and Applications: International Conference VISIGRAPP 2007, Barcelona, Spain, March 8-11, 2007, Revised Selected ... in Computer and Information Science)

AsiaSim 2007: Asia Simulation Conference 2007, Seoul, Korea, October 10-12, 2007, Proceedings (Communications in Computer and Information Science)

Mathematics and Computation in Music: Second International Conference, MCM 2009, New Haven, CT, USA, June 19-22, 2009. Proceedings (Communications in Computer and Information Science)

Algebraic Informatics: Second International Conference, CAI 2007, Thessalonkik, Greece, May 21-25, 2007, Revised Selected and Invited Papers

Mathematics and Computation in Music: Second International Conference, MCM 2009, New Haven, CT, USA, June 19-22, 2009. Proceedings (Communications in Computer and Information Science)

Algebraic Informatics: Second International Conference, CAI 2007, Thessalonkik, Greece, May 21-25, 2007, Revised Selected and Invited Papers (Lecture Notes in Computer Science)

Graph-Theoretic Concepts in Computer Science: 33rd International Workshop, WG 2007, Dornburg, Germany, June 21-23, 2007, Revised Papers

Parallel Processing and Applied Mathematics: 7th International Conference, PPAM 2007, Gdansk, Poland, September 9-12, 2007, Revised Selected papers

Formal Methods for Industrial Critical Systems: 12th International Workshop, FMICS 2007, Berlin, Germany, July 1-2, 2007, Revised Selected Papers

Runtime Verification: 7th International Workshop, RV 2007, Vancover, Canada, March 13, 2007, Revised Selected Papers (Lecture Notes in Computer Science)

Algebraic Informatics: Second International Conference, CAI 2007, Thessalonkik, Greece, May 21-25, 2007, Revised Selected and Invited Papers (Lecture Notes ... Computer Science and General Issues)

Euro-Par 2007 Workshops: Parallel Processing: HPPC 2007, UNICORE Summit 2007, and VHPC 2007, Rennes, France, August 28-31, 2007, Revised Selected Papers ... Computer Science and General Issues)

Communications and Networking in China: 1st International Business Conference, Chinacombiz 2008, Hangzhou China, August 2008, Revised Selected Papers (Communications ... in Computer and Information Science)

Information and Communications Security: 9th International Conference, ICICS 2007, Zhengzhou, China, December 12-15, 2007, Proceedings

Topics in the Theory of Computation: Selected International Conference Papers

Infinity in Logic and Computation: International Conference, ILC 2007, Cape Town, South Africa, November 3-5, 2007, Revised Selected Papers (Lecture ... Lecture Notes in Artificial Intelligence)

Scale Space and Variational Methods in Computer Vision: First International Conference, SSVM 2007, Ischia, Italy, May 30 - June 2, 2007, Proceedings (Lecture Notes in Computer Science)

Software and Data Technologies: First International Conference, ICSOFT 2006, Setúbal, Portugal, September 11-14, 2006, Revised Selected Papers (Communications in Computer and Information Science)

Business Information Systems Workshops: BIS 2010 International Workshop, Berlin, Germany, May 3-5, 2010, Revised Papers

Web Information Systems and Technologies: Third International Conference, WEBIST 2007, Barcelona, Spain, March 3-6, 2007, Revised Selected Papers (Lecture Notes in Business Information Processing)

Medical Imaging and Informatics: Second International Conference, MIMI 2007, Beijing, China, August 14-16, 2007, Revised Selected papers

Emerging Technologies in Knowledge Discovery and Data Mining: PAKDD 2007 International Workshops, Nanjing, China, May 22-25, 2007, Revised Selected

Mathematics and Computation in Music: First International Conference, MCM 2007, Berlin, Germany, May 18-20, 2007. Revised Selected Papers (Communications in Computer and Information Science)

Mathematics and Computation in Music - MCM 2011

Constructing Ambient Intelligence: AmI 2007 Workshops Darmstadt, Germany, November 7-10, 2007, Revised Papers (Communications in Computer and Information Science)

Software and Data Technologies: Second International Conference, ICSOFT ENASE 2007, Barcelona, Spain, July 22-25, 2007, Revised Selected Papers (Communications in Computer and Information Science)

E-Voting and Identity: First International Conference, VOTE-ID 2007, Bochum, Germany, October 4-5, 2007, Revised Selected Papers

Bioinformatics Research and Development: First International Conference, BIRD 2007, Berlin, Germany, March 12-14, 2007, Proceedings

Information Theoretic Security: Second International Conference, ICITS 2007, Madrid, Spain, May 25-29, 2007, Revised Selected Papers (Lecture Notes in Computer Science Security and Cryptology)

Information Theoretic Security: Second International Conference, ICITS 2007, Madrid, Spain, May 25-29, 2007, Revised Selected Papers (Lecture Notes in Computer Science Security and Cryptology)

Research and Education in Robotics -- EUROBOT 2008: International Conference, Heidelberg, Germany, May 22-24, 2008. Revised Selected Papers (Communications in Computer and Information Science, 33)

Computer Vision and Computer Graphics. Theory and Applications: International Conference VISIGRAPP 2007, Barcelona, Spain, March 8-11, 2007, Revised Selected ... in Computer and Information Science)

AsiaSim 2007: Asia Simulation Conference 2007, Seoul, Korea, October 10-12, 2007, Proceedings (Communications in Computer and Information Science)

Mathematics and Computation in Music: Second International Conference, MCM 2009, New Haven, CT, USA, June 19-22, 2009. Proceedings (Communications in Computer and Information Science)

Algebraic Informatics: Second International Conference, CAI 2007, Thessalonkik, Greece, May 21-25, 2007, Revised Selected and Invited Papers

Mathematics and Computation in Music: Second International Conference, MCM 2009, New Haven, CT, USA, June 19-22, 2009. Proceedings (Communications in Computer and Information Science)

Algebraic Informatics: Second International Conference, CAI 2007, Thessalonkik, Greece, May 21-25, 2007, Revised Selected and Invited Papers (Lecture Notes in Computer Science)

Graph-Theoretic Concepts in Computer Science: 33rd International Workshop, WG 2007, Dornburg, Germany, June 21-23, 2007, Revised Papers

Parallel Processing and Applied Mathematics: 7th International Conference, PPAM 2007, Gdansk, Poland, September 9-12, 2007, Revised Selected papers

Formal Methods for Industrial Critical Systems: 12th International Workshop, FMICS 2007, Berlin, Germany, July 1-2, 2007, Revised Selected Papers

Runtime Verification: 7th International Workshop, RV 2007, Vancover, Canada, March 13, 2007, Revised Selected Papers (Lecture Notes in Computer Science)

Algebraic Informatics: Second International Conference, CAI 2007, Thessalonkik, Greece, May 21-25, 2007, Revised Selected and Invited Papers (Lecture Notes ... Computer Science and General Issues)

Euro-Par 2007 Workshops: Parallel Processing: HPPC 2007, UNICORE Summit 2007, and VHPC 2007, Rennes, France, August 28-31, 2007, Revised Selected Papers ... Computer Science and General Issues)

Communications and Networking in China: 1st International Business Conference, Chinacombiz 2008, Hangzhou China, August 2008, Revised Selected Papers (Communications ... in Computer and Information Science)

Information and Communications Security: 9th International Conference, ICICS 2007, Zhengzhou, China, December 12-15, 2007, Proceedings

Topics in the Theory of Computation: Selected International Conference Papers

Infinity in Logic and Computation: International Conference, ILC 2007, Cape Town, South Africa, November 3-5, 2007, Revised Selected Papers (Lecture ... Lecture Notes in Artificial Intelligence)

Scale Space and Variational Methods in Computer Vision: First International Conference, SSVM 2007, Ischia, Italy, May 30 - June 2, 2007, Proceedings (Lecture Notes in Computer Science)

Software and Data Technologies: First International Conference, ICSOFT 2006, Setúbal, Portugal, September 11-14, 2006, Revised Selected Papers (Communications in Computer and Information Science)

Business Information Systems Workshops: BIS 2010 International Workshop, Berlin, Germany, May 3-5, 2010, Revised Papers

Web Information Systems and Technologies: Third International Conference, WEBIST 2007, Barcelona, Spain, March 3-6, 2007, Revised Selected Papers (Lecture Notes in Business Information Processing)

Medical Imaging and Informatics: Second International Conference, MIMI 2007, Beijing, China, August 14-16, 2007, Revised Selected papers

Emerging Technologies in Knowledge Discovery and Data Mining: PAKDD 2007 International Workshops, Nanjing, China, May 22-25, 2007, Revised Selected

Recommend Documents