Formalized Music: Thought and Mathematics in Composition

Formalized Music THOUGHT

AND MATHEMATICS

IN

COMPOSITION

Revised Edition

Iannis Xenakis Additional material compiled and edited by Sharon Kanach

HARMONOLOGIA SERIES No. 6

PENDRAGON PRESS STUYVESANT NY

Other Titles in the Harm onologia Series No. 1 Heinrich Schenker: Index to analysis by Larry Laskowski (1978) ISBN 0-918 728-06-1 No. 2 Matpurg's Thorou ghbass and Composition Handbook: A narrative translation and critical study by David A. Sheldon ( 198 9) ISBN 0-918728-55-x No. 3 Between Modes and Keys: German Theory 1592-1802 by Joel Lester (1990) ISBN 0-918728-77-0 No. 4 Music Theory from Zarlino to Schenker: A Bibliography and Guide by David Damschroder and David Russell Williams (1991) ISBN 0-918728-99-1

The Sense of Order by Barbara R. Barry (1990) ISBN 0-945193-01-7

No. 5 Musical Time:

Chapters I-VIII of this book were originally published in French. Portions of it appeared in Gravesaner Blatter, nos. 1, 6, 9, 11/12, 1822, and 29 (1955-65).

Chapters I-VI appeared originally as the book Musiques Formelles, copyright 1963, by Editions Richard-Masse, 7, place Saint-Sulpice, Paris. Ch ap te r VII was first published in La Nef, no. 29 (1967); the English translation appeared in Tempo, no. 93 (1970). Chapter VII was originally published in Revue d'Esthitique, Tome XXI (1968). Chapters IX and Appendices I and II were added for the

English-lan� ge edition by Indiana University Press, Bloomington 1971.

Chapters X, XI, XII, XIV, and Appendi.x III were added for this ed i tion, and all lists were updated to 1991. Library of Congress Cataloging-Publication Data

Xenakis, Iannis, 1922a Formalized music : thought and m thematics in composition 1 Iannis Xenakis. "

P·

c.m.

__

(Harmonologia series ; no. 6)

New expanded edition"--Pref.

ences and index. Includes bibliographical refer ISBN 0-945193-24-6 y and aesthetics. 2. 1. Music--20th century-Philosoph

century. 4. (Music) 3. Music--The· o·ry--20th · · Compos1t1on · · . II . I. Title cnucJsm. and story --Hi ury Music--20th cent

Series.

ML3800.X4 781.3--dc20

1990

. h t 1 99 ,.n•p en dragon Press C opyng 1

I

; .1

Contents P r e face Preface to

Preface

to

vii

Musiques formelles the Pend r ago n

ix

Edition

xi

I Free Stochastic M u s ic

II Markovian III Markovian

IV

Stochastic Music

Theory

-

Stochastic Music-Applications

Musical Strategy

V Free S toch a stic Music by Computer

VI Symbolic Conclusions

and Extensions for Chapters I- VI

VII Towards VIII

Music

a

Metamusic

Towards a Philosophy of Music

IX New Prop osals in Microsound Structure X

Concerning Time, Spac e and

XI Sieves XII Sieves: A User's XIII Dynamic

Music

I

& II Two

Guide

Stochastic Synthesis

III The

Bibliography D is cog rap hy

Laws

of Continuous ProbabiliLy

New UPIC

Biography: D egre es

79 llO 131 155 178 180 201 242 255

268

XIV More Thorough Stochastic Music

Appendices

I 43

System

277

2 89

295

323,327 3 29 335

and Honors

Notes

Index

3 65 371

373

3 83

v

Preface

The

attempted in trying to reconstruct part of the used, for want of time or of capacity, the most a d van c ed aspects of philosophical and scientific thought. But the escalade is started and others will certainly enlarge and extend the new thesis. formalization that I

musical e difice

ex

nihilo has not

This book is addressed to

tion

a

hybrid public, but interdisciplinary hybridiza­

frequently produces superb specimens.

c ou l d sum up twenty y e ar s of p er s o n a l efforts by the progressive in of the following Table of C o here nces. My musical, architectural, and visual works arc the chi ps of this mosaic. It is like a net whose variable lattice s capture fugitive virtualities and entwine them in a multitu � e of ways. T his table, in fact, sums up the true coherences of the successive

I

filling

chronological chapters of this book. The chapters stemmed fr o m mono­ graphs, which tried as much as possible But the profound l esso n of

theory or

sol u t i on given

such

to

a

avoid

table

overlapping.

of cohercnces is that any

on one level can be assigned to

problems on another level. Thus the sol u ti o n s Families level

( p rogr a m m e d

an d more powerful usual

new

trigonometric

heavy

solution of

stochastic

perspectives in the shaping ofmicrosounds than the

(periodic) functions

can. a

clouds of points and their distribution over bypass the

the

in macrocomposition on the mechanisms) can engender simpler T h ere fore ,

in

considering

pressure-time plane,

we

can

harmonic analyses and syntheses and create sounds that

have never before existed. Only then will sound s y n th esis by computers digital-to-analogue converters find its

true

position,

free

and

of the roo ted but

ineffectual tradition of electronic, concrete, and instrumental music that

makes use of Fourier this book,

qu esti ons

more diversified and

tion

as

synth es is

despite

the

failure of this theory. H ence , in

having to do mainly with orchestral sounds

more manageable)

find

a

(which

are

rich and immediate applica­

soon as they are transferred to the Microsound level in the pressure­

time space. All music is thus automatically homogenized and unified.

vii

Preface to the Second

viii

l

Edition

I

everywhere" is the word of this book and its Table of Coherences; Herakleitos would say that the ways up and down are one. The French edition, Musiques Formelles, was produced thanks to Albert Richard, director of La Revue Musicale. The English edition, a corrected and completed version, results from the initiative of Mr. Christopher Butchers, who translated the first six chapters. My thanks also go to Mr. G. W. "Everything is

\

Mrs. John Challifour, who translated Chapters VII and VIII, respectively; to Mr. Michael Aronson and Mr. Bernard Perry of Indi an a University Pres s, who decided to publish it; and finally to Mrs. Natalie Wrubel, who edited this difficult book with infinite patience,

Hopkins, and Mr. and

correcting and

rephrasing many obscure passages.

I. X.

1970

(MOSAIC)

TABLE Phi/o,o}J.g

sense)

Thrust towards truth, revelation. (in the etymological

creativity.

Chapters (in the sense

of the

Quest

ARTS ( VISUAL,

SONJC1

MIXI!.D

• • •

)

COHERENCES

in everything, interrogation, harsh criticism, active knowledge through

methods followed)

Partially inferential and experimental

OF

\

Other methods

Entirely inferential and experimental

SCIENCES (OP MAN,

NATURAL

to come

?

)

This is why the arts are freer, and can therefore guide the sciences, which are entirely inferential and experimental.

C·· r;J :���:. ....,.. ,.:.. �c:r=.::= :·

A

-..�....:-:.: ·,;;;, -:-.= ,....

�

Fig.l-1. Score

of

���--­

Metastasis, 1953/54,

Bars

309-17

3i'

I

I

I

I

_,_

j

$ Cl> Cl>

:::1

c. "' :::1

-5

0

-

.. ., .,., c

>

c:

:::1 0 0

:9,=

G> > CD G>

:0

(Ill II]

2RS&

·�

-�

c

0 z

c;,

u;

] Qua l ific a t i o n s

��

....

g u

�

!

ci �

Free Stochastic Music

29

However, we are not s p e a ki ng here of cases where on e merely p l ays heads and tails in order to ch o ose a p ar ti cul ar alternative in some trivial circu mstance. The problem is much more serious than that. It is a m a tte r here of a philosophic and aesthetic concept ruled by the l aws of probability and by the m athematical functi_ons th at formulate that theory, of a coh er ent

concept in a new region of coherence. The analysis that

fol lo w s

is

taken from

Achorripsis.

For convenience in calculation we shall choose a p riori a mean d e n sity of events

>.

Applying Poisson's formula,

=

0.6 events /unit.

we obtain the table of probabilities :

P0 pl p2

P3 P,

P5

=

0. 5488

=

0.3293

=

0.0988

=

0.0 1 98

=

0 . 0030

=

(1)

0. 0004.

P1 is the probability that the event will occur i times in the unit of volume , time, etc. In choosin g a priori ] 96 units or cells, the distribution of the frequencies a mong the cells is obtained by multiplying the values of

pj by 1 96.

Number of cells

0 1

2 3 4

1 96 PI

1 07 65

19

4

(2)

1

The 1 96 cel l s may be arr a n ge d in one or several groups of cells, quali­

fied as to timbre and time, so that th e number of groups of timbres times

of groups of du rations = 1 96 cells. Let there be 1 distinct timbres ; then 1 96/ 7 28 units of time. Thus the 1 96 cells arc distributed over a two-dimensional space as shown in (3) .

th e nu mber

=

30

Formalized Music Timbre Flute Oboe

String

gliss.

Percussion

(3)

Pizzicato

Brass String

arco 0

1

2

3

. . . . . . . . . . . . 28 :rime

If the musical sample is to last 7 minutes ( a subj ective choice) the unit of U1 will equal 15 sec., and each U1 will contain 6.5 measures at

time

MM

=

26.

of ze ro, single, double, triple, space of Matrix (3) ? Consider th e 28 columns as cells and distribute the zero, single, double, triple, and q u adruple events from table (2) in these 28 new cells. Take as an example the si n gl e event ; from table ( 2 ) it must occur 65 times. Everything h appens as if one were to distribute events in the cells with a mean density A 65/28 2.32 single events per cell (here cell column ) . In applying anew Poisson's formula with the mean density A 2 .32 How shall we distribute the frequencies

and q u adruple events per cell

=

in

the two-dim ensional

=

(2.32

«

we obtain table

30)

Poisson Frequency

K 0 1 2 3 4 5 6 7

Totals

=

=

(4) .

Distribution

Arbitrary Distribution

No. of

Pro du c t

3

0

6

6

Columns col

x

Frequency No . of Product K Columns col x K

K

0 1 2 3 4 5

10 3 0 9 0

0 3 0 27 0 5

8 5 3 2

16

I

6

6

5

30

0

7

0

0

28

65

0

28

15

12 10

65

(4 )

Totals

(5)

Free

31

Stochastic Music

One could c hoos e

single events equals But in

space,

we

this

must

any o th er distribution

on

65. Table (5) shows such

a

condition that

the sum of

distribution.

axiomatic research , where chance must bathe all of so ni c

rej e c t

every distribution which

departs from Poisson's

law. And the Poisson distribution must be effective n o t only for the columns but also

for

the rows of

the matrix. The same reasoning holds true for the

diagonals, etc.

Contenting ourselves just with rows and columns, we obtain

geneous distribution which follows Poisson. It was

a h omo­ in th i s way that the

distributions i n rows a nd columns of Matrix (M) (Fig. 1-9) were calculated . the

So a unique law of chance, the law of Po i ss on

me d i u

m of the arbitrary mean

( for r are

events) t h ro u gh

A is ca p able of conditio ning, on the one

hand, a whole sample matrix, and on t h e other,

the partial distribu tions The a p riori, arbitrary choice a d m i tte d at concerns the variables of the " vector-m atrix. "

following t h e rows and columns. the

beginning

therefore

Var i a b l es or e n t r i e s of t h e "vect o r - mat r i x "

I . Poisson's Law 2 . The mean ;\ 3. The number of cells, rows, and columns The distributions entered in this matrix ar e not always rigorousl y d efined . They really depend, for a given A, on the number of r o w s or col­ umns. The greater the n umber of ro ws o r columns, the more rigorous is the definiti on. This is the law of l arge numbers. B u t this ind eterminism allows free will if the artistic i n spiration wishes i t . It is a se con d door that is open

to the subj ectivism of t h e com p oser, t he first being the " state of entry " of

the

" Vector-Matrix "

defined above .

specify the unit-events, whose frequencies were adjusted in the standard matrix ( M ) . We shall take as a single event a cloud of sounds with linear density 8 sounds/sec. Ten sounds/sec is abo ut the limit th a t a normal o rche s t ra can play. We shall ch oos e S 5 sounds/measure at MM 26, so that 8 2 . 2 soundsfsec ( � 1 0/4) . We shall now set out the following correspondence : Now we must

=

=

Formalized

32 Cloud of density S

Ev e n t

Sounds/ measure 26MM

zero

0

5

single d ouble

I

(15

2.2 4.4 6.6 8.8

20

sec) 0

0

15

qu ad ru pl e

Mean number of sounds/cell

sec

10

triple

=

Sounds/

Music

32.5 65

97.5

1 30

The h atchings in matrix (M) show a Po i sson distribution of frequencies, that

homogeneous and verified in terms of rows and columns. We notice the

rows are

columns. This

interchan geable leads

us

(

=

interchan geable

timbres) . So are the

to admit that the determinism of this matrix is weak

basis for though t-for thought wh ich of all kinds. The true work of molding sound consists of distributi ng the clouds in the two-dimensional space of the matrix, and of anticipating a priori all the sonic encounters before th e calculation of details, eliminating prejudicial positions. It is a work of

in part, and that it serves chiefly as a manipulates frequencies of events

patient research which exploits all the creative faculties instantaneously. This matrix is like a game of chess for a single p l a ye r

who must follow certain rules of the game for a prize for which he himself is the judge. This game matrix h as no unique strategy. It is not even possible to disentangle any balanced goal s . It is very general and incalculable by pure reason. Up to this point we have pl aced the c loud densities in the matrix. Now with the aid of calculation we must proceed to the coordination of aleatory sonic elements. HYPOTHESES

OF

the

CALCULATION

a s a n example cell III, t z o f the matrix : third row, continuous variation (string glissandi) , seventeenth unit of time (measures 1 03-l l ) . The density of the sounds is 4.5 sounds/measure at MM 26 ( S 4.5) ; so that 4.5 sounds/measure times 6.5 me as u re s 29 Let us analyze

sounds of

=

=

for this cell. How shall we place the 2 9 glissando sounds in this cell ? Hypothesis 1. The acoustic characteristic of the glissando sound is

sounds

assimilated to the speed

v

=

df{dt of

a

uniformly continuous movement.

(See Fig. I- 1 0.) Hypothesis 2. The quadratic mean a: of all the possible values of v is proportional to the sonic density S. In this case a: 3 . 38 (temperature) . Hypothesis 3. The values of these speeds are distributed according to the most complete asymmetry (ch an c e) . This distribution follows the law of =

33

Free Stochastic Music

Fig .

1-1 0

Gauss. The probability f(v)

function

for

f(v)

the existence of the speed v i s given by

=

Q�'/T

r v�/a� ;

and the probability P (>.) that v will lie between

8(>.) Hypothesis

=

:'IT f'

4. A glissando s o u n d

moment of its departure ; b. its spe e d register.

Hypothesis

is

vm

the

r .,.,�

v1

( normal

d)..

essen tially =

an d v 2 , by the fu nction

distribution) .

characterized by

tifldt, (v1



� ""=

(n' m'1

� (110 Fig. 1 1-24

Yt )

(, m)

E:==== I

Markovian

69

S tochastic Music-Theory LI N K I N G T H E S C R E E N S

o r music co ul d b e de­ lexicographic order of the pages of a book. If we represent e ac h screen by a specific symbol ( o ne to one coding) , the sound or the music can be translated by a succession of symbols Up

t o now w e

have admitted that

any

sound

scribed by a number of screens arranged in the

-

called

each

a

-

protocol :

a b g k a b · ··b g · ··

letter identifying screens

and mo m e n ts

t for

particular

Without seeking the causes of a

isochronous

t:.. t 's.

succession of screens,

i.e . ,

without entering into either the p hysical s tructure o f the sound o r the l o gic al structure of the composition, we can disengage certain mo de s of succession and s p ecie s of protocols [ 1 6] . We shall quickly review the elementar y definitions.

its unique symbol is called a term. Two successive terms materialize. The second term is called the tra nsform and the change effected is represented by term A --+ term B, or A --+ B.

Any matter

cause a

A

transition

or

to

transformation is

a colJection of transitions . The following example

drawn from the above protocol :

l

another tra nsformation

with

a

b

b

g

g

is

k

k

a

musical n otes :

lC

B

D

G

E A

3

rn �

be closed when the col lec t io n of transforms elements b elonging to the collection of terms, for exam p le :

A transformation is said to

contains only

0

the alphabet, b

c

c

d

z

a

Formalized

70

Music

musical notes,

D Ej,

E

G

F

c

G j,

F

B

G

D j,

A

B

Bj,

A

Ej,

Bj,

E

musical sounds,

1.

infinity of terms,

-1

.. . :· � .

ll

6

single

transform,

�= �= :� .;:f .. �. .

2

3

4

7

4

1 00

l

�

c.

'

d i rect i o n

3. N etwork of p a ra l l e l

5

g l issa n d i in two

d i rection s .

6 2

singl e-val ued ( mapping) when each term

for example :

l

a.

b. j

e . g . , pizzicati

2. N etwork o f paral l e l g l issa n d i i n one

�

..

a

h

a

c

c

h

e

d

The following are examples of transformations that are

J

1 . C l o u d of so u n d - p o i nts.

3

. . :' ; .•· �

A transformation is univocal or has a

"'

� ·�tlll1V

� � "'

an

3

2.

i

l h�

b

c

d

c

,.--- 5 : 4 --,

J ))J)

r-

L._ 4 : 3 -.l

of a

change

clarinets

oboes

s trings

bassoon

univocal :

m, n, p

tJ

timbre

not

�

'

group of values

strings

Timbres timpani, timpani, brass

ti mpani

oboes

f

b r ass strings, oboes

�

•

r

Markovian S t oc h as ti c

" �fan ner ,

I

71

Music-Theory

and d. concrete

characteriology [4, 5]

music

nil

vibrated

cyc lical o r

irregular

trembled

trembled

cyclical

nil

trembled

or

nil or

irregular

vibrated

cyclical

A t ran sfo r ma t io n is a one-to-one mapping when

transform and when each transform is derived

example :

MATR ICAL

irregular

c

b

l:

a

a

term has a single single term, for

d c

d

R E P RE SENTA T I O N

e ach

from

or

A transformation :

can be

l:

represented by a table as .j.

a

b

This

c

a

0 0

c

follows :

c

b

0 0

0 0

+

+

+

c

b

c

.j.

a

b

or

c

b

c

l 0 0 0 0 0 0

table is a matrix of th e transitions of the collection of terms to a

collection

P R ODUCT

of transforms.

Let there b e two transformations T and T:

a

a

b

J:

d

In ce rt a in cases we

c

a

d b

and

U:

U:

can apply to a term

n

of

J:

Ta

b

c

d

c

d

b

tra n s form a ti o n

transformations T and U, on condition that the transforms of U. Thus, first

V

T, the n

transformation U. This is written : U[ T(n) ] , and is the product of the two

=

UT: a -+ c.

T:

a ---+

b, then

U:

b ---+

c,

of T are

terms

which is summarized as

72

Formalized

Music

To calculate the product applied to all the terms of T we shall use the matrical representation :

following

T:

c

a

b

a

0

0

b

1

0

0

1

c

0

0

0

0

c

d

0

I

0

0

d

,j,

t

d

1

a

0

b

c

d

0

0

0

0

0

0

1

I

0

0

1

0

0

b

U:

a

0

0

the total transformation V equals the product of the two matrices T and U

in

the

order U, T. 0

0

0

0

0

1

u

0

0 K I N EMATI C

0 0

0

0 I

0

1

X

0

0

T

v

0

l

0

0

0

0

0

0

0

1

0

1

0

0

0

0

0

1

0

0

0

1

1

0

0

0

0

0

DIAG RAM

The kinematic or transition diagram is a graphical expression of draw it each term is connected to its transform by an arrow pointed at the transform. The representatiue point of a kinematic dia­ gram is an imaginary point which moves in jumps from term to term following the arro ws of the diagram ; for an example see Fig. 11-25. transformation. To

T:

lA

D

p� Fig . 11-26

�/

C

D

D I

I

L

N

P

.A

A

N

A

N

N

N

A transformation i s really a mechanism and theoretically all the mechan isms of the physical or biological universes can be re p resented by

Markovian

73

Stochastic Music-Theory

transformations u nder five condi tions of correspondence :

1.

states

Each state of the mechanism (continuity is

as

clos e together as

is

broken

down into di s c re t e

desired) is in a one-to-one correspondence with

a term of the transformation . 2. Each sequence

internal

structure

of states

the mechan ism by reason of i ts uninterrupted seq uence of the terms of

c rossed by

cor res p o n ds to an

the transform ation . mec

h anis m reaches a state and rem ains there (absorbi n g or corresponds to this state h as no tra nsform . If the states of a mech anism reproduce th e m se l v es i n the same man­

3. If the

stationary state) , 4.

the term which

a

ner without end, the transformation has circuit.

kinematic d i agram in closed

5. A halt of the mechanism and i ts start from another state

is

sented in the diagram by a d i s p la cem e n t of the re p resentative point,

is not due to an arrow but to an arbitrary action on the

repre­

which

paper.

the corresponding transformation is univocal and closed . The mechanism is not determined when the corre­ sponding transformation is man y-val ued . In this case the transformation is said to be stochastic. In a stochastic mech a n ism the n u m be rs 0 and I i n the The mechanism is determined when

transformation matrix must be replaced by relative freq uencies. These are

the alternative probabili ties of vari ous

transformations. The determi ned mechanism, in w h i ch the

mechanism is a particular case of the stochastic pro babilities of t ra nsi t i on are 0

and

l.

Example : All the h armonic o r po l y p h o n i c rules o f classical music could be represented by mechanisms. The fugue is one of the most accom pl ished

and determined mechan isms.

One

co u l d

avan t-garde composer is n o t con ten t

age but proposes new ones, for both are

eve n generalize and say that the

with

fol lowing

the mechanisms

detail and general form.

If these prob abili ties are constant o v e r

a

long period

i ndependen t of the s t a tes of origi n , the stochastic

more particularly, Let

there

be

a

Markov chai n .

two

screens

A and B

and

a

of

h is

of time, and i f they sequence is cal led ,

p ro t o co l of 50

transit ions :

ABABBBABA A BA BA BA BBBBABA A B A BBA A BABBABAAABABBA A BBABBA . The real

freq uencies

of the

A -+ B A - A

transitions 1 7 times

6 t i mes

23 times

arc :

B - A

B- R

1 7 t i mes 1 0 times

27

times

76

Formalized M usic

diminution of the entropy. If melodic or harmonic liaisons are effected in the same distribution, unpredictability and entropy are both di minished.

a

and perceived Rate of ataxy

A

/

B Time

D

�

E

c�

v

F i g . 1 1-26 A . The evolution i s n i l . B . The rate of d i sorder a nd the rich ness i n crease. C . Ataxy decreases. D. Ataxy i n c reases and then d e c reases. E . Ataxy decrea ses and then i ncreases. F. The evo l ution o f t h e ataxy is very complex, b u t it m a y be a n a l yzed from the fi rst t h ree d i a g ra m s . Thus after the first u n fo ldi ng

of

a

of twelve sounds of the tem­ is maxi­ mum , the choice is nil , and th e en tropy is zero. Richness a n d hence i nterest are d i s p l ace d to other fields, such as harmonies, timbres, and durations, and many other com positional wiles are aimed at reviving entropy. In fact sonic discourse is no thing but a perpetual fluctu ation of entropy in all its forms series

pered scale, the unpredictability has fallen to zero, t h e constraint

[1 7] .

does not necessarily follow the variation if it is logarithmic to an appropriate base . It is rather a su ccession or a protocol of strains and rel axations of every degree that often exci tes the li s te n e r in a di rection contrary to th at of entropy. Thus Ravel's Bolero, in which the on l y variation is in the dynamics, has a vi rtually zero entropy after the third or fourth repe tition of the fu ndamental idea. How­ However, human sensitivi ty

in en tropy even

ever, the i n terest,

or

rather the psychological agi tation, g rows with time

the very fact of this immobility and banality. All i n ca n t a to ry manifestations aim at an effect of maximum with m i n imum entropy. The inverse is equ ally true, and seen from a through

tension

certain

Markovian Stochastic Music-

Theory

77

n o i s e with its maximum e n t ro py i s soon tiresome . I t wo u l d there is no correspondence aesthetics - entropy. These two

angle, white seem that

.

e n t i t ie s are l i nked in quite an independent m a n n e r at each oc c as i o n This

statement still leaves some respite for the free will of the composer even i f

this free w i l l is buried under t h e rub bish o f c u l t u re and civi l i zation a n d is

only a sh adow, at the l e a s t a tendency, a simple stochasm . The great o bs t a c l e to a too hasty gen eral i zation is chiefly one o f l og i c a l order ; for an object is only an o bj e ct as a fu n c t io n of its d e fi n i t i o n , and there is, especially in a r t , a near- i n finity of definitions and hence a near- i n fi n i ty of entropies, for the notion of en tropy is an epiphenomenon of the d efi n i t i o n . Which of these is valid ? The ear, the eye, and the brai n u n r av e l sometimes inextricable situations with what i s called intu ition, taste, and in telligence. Two definitions w i th two d i fferent e n trop i e s can be perceived as identical, b u t it i s also tru e that t h e set of defini tions of an object has i ts own degree of disord er.

We

are not concerned here

wi

th

investigating such a difficult,

complex, and un explored si tu ation, b u t si m p l y with looking over the

possibilities that co n nec t e d realms of c o n te m p o r a ry thought p ro m i s e , with a view to action .

To conclude

q u en t

than

screens can

brie fl y , si n ce the

explanatory

texts,

be expressed arc

by

we

applications

wh ich follow are

mor

elo­

shall accept that a coll ection or book of

matrices of

transition probabil ities havi n g

entropy which is in order to render the an alysis and t h e n the synthesis of a son i c work within reach of understand i ng and the slide rule, we s hal l establish three criteria fo r a screen :

p arameters. They

calculable

I .

a ffected by a d egree

of

ataxy or

under certain conditions. However,

TOPOGR A PHIC

CRITER I O N

The posi t i o n of the cells 6.F6. G on the audible area is q u al i tat ively

i mportant, and an

creating a

g ro u p

enumeration of th e i r possi ble combinations is capable of d e fi n e d terms to which we can apply the concept

of wel l

of entropy and i ts calc u l atio n .

2.

DENSITY

C R I TERI ON

The superficial density o f the g r a in s o f a cell !J.F!J.G also constitu tes a q ua l i ty wh ich is immedi ately perceptible, and we cou l d equally well d e fi n e t erms to w h ich the conce p t and calculation o f entropy wou l d be appl ica b l e .

3 . CRITERION s c re en )

OF

P U RE

ATAXY

A cell has t h ree variables :

mean

( defined

i n relation to t h e gra i n s o f

freq u ency, mean am p l i t u de,

and

a

m e an

78

Form alized

Music

density of the grains. For a screen we can therefore establish three indep en­ dent or connected protocols, then three m a trices of transition probabilities which may or may not be coupled. Each of the m atri ces wil l have its entropy and the three coupled m atrices will h ave a mean entropy. In the procession of sound we can establish several series of three matrices and h e nce several series of mean entro pies, their variations constitu ting the criterion of ataxy. The fi rst two criteria, which are general and on t h e scale of screens or cells, will not concern us in wh at follows . But the third, more conventional criterion will be taken up in detail in the next ch apter.

C h a pter I l l

Markovian S tochastic Applications In th is ch apter

we

w i l l d iscuss two m u si ca l a p plications :

string orchestra, and A nalogi que B , lor sin usoidal 1 958-59. W e s h a l l confi ne o u rs e l ves

ponents G,

F, D

Music­

to

a

of the screen take

sounds,

s i m ple c a s e in which

o n ly

Analogique

A , for

both composed in

each of t h e co m­

two values, following matrices of

by means of parameters . In matrices will be made in s u ch a way that we shall h av e only the regular c a s e , conforming to th e chain o f even ts th eory a s it h a s been defined i n t h e work of Mau rice Frechet [ 1 4] . transition probability which will be cou p led

addi t io n , the

choice of probabilities

I t i s obvious

that richer

highly in teres tin g to

the

and more com plex stochastic me ch a n isms

c o ns tr u c t

and to

siderable volume of calculations

to undertake them by

in

put

which

hand, but very

in work,

but in view of

the

are

con­

they necessi tate it would be. u sel e ss

desirable

to pro gram

computer.

them for t h e

Nevertheless, despite the stru ctural simplicity of what follows, the

s to c has ti c

jaccnt

mechanism which wi l l emerge will be

a

mod e l ,

a

s t an dard

sub­

that are far more complex, and will serve to catalyze of greater elaboratio n . For although we confine ourselves here to the s tudy of scr e ens as they h ave been defined in this st udy (sets of elemen tary grains) , it g oes without saying that noth i ng prevents the gen­ to any others

fu rther studies

eralization of this m etho d of s tructu ralization (composition) for defi nitions of sonic entities of more

than three d i mensions. Thus, let us no longer suppose

screens, but criteria of definitions of a sonic entity, such that for the

deg ree

of ord er, density, v a ri a t i o n , and even 79

the criteria

timbre,

of more or less

80

Fonn a

"

«-

.

�

e .........

�,_

.,.

� �... .,.. ..,...� .

...

..

��

�

:� :

��

1-q.

�:-��2--�� -

7+ .,.

'>l

� ,.,

.

...�

.,_ + .,.: ....__

...

·.

.

.

- ��

\ .

�

� '

0�

{ , � " Fig . 11 1- 1 . S wrnos for 1 8 s tring

s

tiz cd J\1usi c

Markovian Stochastic Music-Applications

81

complex elementary structu res (e .g. , melodic and temporal structures of groups of sounds, and instrumental, spatial, and ki nem ati c structures)

s a me

stochastic scheme is adaptable.

well and to be

It is

enough to define the

the variations

able to classify them even in a rough manner. thus obtained is not gu aranteed a priori by calculation. experience must always play their part in guiding, deciding,

The sonic result

Intuition and

and

testing.

{ d ef i n i t i o n We shall define process. It

stochastic

A N A LYS I S

o f the scheme o f a mecha n i s m }

t h e scheme o f a m e ch a ni s m as t h e "analogue " of a

will serve for the production

their transformations over time. These sonic entities

will show the followin g characteristics freely

of sonic entities and for will have screens which

chosen :

1 . They will permit two d i sti nc t combinations

fo and j1 (see

of frequency regions

Fig. I I I-2) .

,__----�---;,F Half axis Audible frequencies

,______.__________._._t----: F

Fig . 1 1 1-2

I.

Audible frequencies

of frequencies in sem itones

H a l f axis of frequencies in semitones

Syrmos, written i n 1 959, is b u i l t o n stoc h a stic tra nsformations of eight bas i c text u res : para l lel horizo nta l bowed notes, para l l e l

ascend i n g bowed gl issa ndi, pa ra l l e l desce n d i n g bowed g l i ss a n d i , crossed (asce n d i ng a n d desce n d i n g ) p a ra llel bowed notes, pizzicato c l o u ds, atmos p h e res made up of col leg no struck notes with s h o rt col leg no gl issa n d i , geo metric config u rat i o n s of conve rgent or d i vergent g l i ssa n d i , a n d g l issando confi g u ra t i o n s t reated as u nd evelopa b l e r u l ed surfaces. The mathematica l str u c ­ ture o f this work is t h e s a m e as t h a t o f Analogique A a n d Analogique B .

Formalized Music

82

2. Th ey will permit two distinct combinations of i n te n s ity regions (see Fig. III-3). G

( P hones)

( P h o n e s)

.,

.,

·"' ., c "'

.� -�

.2!

c

E

-�

"'

..

;e

:0

·a �

i5.

o --...1

Fig. 111-3 3. They

will

permit

(see Fig. I II-4) .

�

"'

D..

o..

Q;

two distinct combin ations of density regions

(Terts • or sounds/s ec)

(Terts • or sounds/sec)

0

D

• Ternary logarithms

F i g . 1 1 1-4

4. Each of these three variables will present a protocol which may be two matrices of transition probabilities (MTP) .

summarized by

t

( p)

The letters ( p )

X y

y

0.2

0.8

0.8

(a)

0.2

and (a) constitute

the

t

X

y

X

0.85

0.4

y

0. 1 5

0.6

parameters

of the (MTP) .

MTPF (of frequencies)

t

(a)

X

fa !1

fo

11

0.2

0.8

0.8

0.2

t

([3)

fo

j�

fo

11

0.85

0.4

0. 1 5

0.6

Markovian

Stochastic Music-Applications (of intensities)

MTPG

t

go

(y)

gl

go

gl

0.2

0.8

0.8

0.2

.J,

(e)

MTPD

do

dl

0.2

0.8

0.8

0.2

t

do

(A)

dl

83

go

gl

0.85

0.4

0. 1 5

0.6

do

dl

do

0.85

0.4

dl

0. 1 5

0.6

go

gl (of d e nsi ties ) .J,

(p.)

The transformations of the variables are indeterminate at the (MTP) (digram processes) , but on th e other hand their (MTP) will be conne c ted by means of a determined cou p l i n g of parameters. The coupling is given by the fo l l o w i ng transformations : 5.

interior of e ac h

lh

h � � A p. a. f3

&

A

& p.

&

{3

& a.

h h 4 � y

e

y

e

we have described th e structure of a m e ch a ni sm It is by three p a irs of (MTP) : (MTPF) , (MTPG) , ( M TPD) , and by the group (e0) of the six couplings of these (MTP) . Significance of the coupling . Let f0 be the state of the frequencies of the screen at an instant t of the sonic e vo l u tion of the mechanism d u ri n g a slice of time !J.t. Let g1 and d1 be the valu es of the other vari ables of the screen a t the moment t. At the next moment, t + !J.t, the term fo is bound to change, for it obeys one of the two (MTPF) , (a.) or (�) . The choice of (ex ) or (f3) is conditioned by the values g1 and d1 of the moment t, conforming to the transformation of the cou pling. Thus g1 p roposes the parameter (a.) and d1 the parameter (/3) simultaneously. In other words the term fo must e i t h e r remainf0 or yield i ts place to f1 according to mechanism (a.) or mech anism (�) . Imagine the term f0 standing before two urns (a) and ({1) , each con­ taining two colors of balls, red for fo and blue for j1, in the followin g By these rules

.

thus constitu ted

prop ortions :

Urn (a.) (f0) , 0.2

Urn ({3) red balls (f0) , 0.85 b l u e balls (j1) , 0 . 1 5

red balls

blue balls (j1 ) , 0 . 8

The choice is free and

the term f0

can

take its successor from either urn (a.)

Formalized Music

84 or urn (/3) with

a

t ( to tal

probabil i ty e q u al to

probab i l i ties) .

Once the urn has been chosen , the c h o i c e of

a

b l u e or a red

ball will

have a p robabi lity equal t o t h e proportion of co l o rs in t h e chosen

Applyin g t h e l a w o f c ompou n d probabilities, t h e probability

moment

and

the

t will

remainf0 at the moment t

probabil i ty

th at

+

6.t is

(0. 20

+

thatf0

0.85 ) /2

i t wi l l change to f1 is (0.80 + 0 . 1 5 ) / 2

=

urn.

from

0.525,

0.475.

=

The five characteristics of the composi tion of the screens hav e estab­

lished

a

stochastic m ec h a n i s m

.

the

We

the

the

sl ices

At

of

th ree variables _h, g1 , d1

round of unforeseeable combinations, always

three (MTP) and

the sonic follow a c h an gi n g according to the

Thus in each of

evolution of the created mech anism,

cou p l i n g which connects terms and parameters .

have established this mechanism witho u t taki n g � n lo consideration

any of th e screen criteri a . That is to say,

d i s tri b u ti on

we

h ave implied a to p o g ra p hic

of grain regions at the time of the choice ofj0 , j1 and g0, g1, but without s p e cifyi n g i t. The sa m e is true for the density distribution . We shall give two examples of very different realizations in which these two criteria will b e e ffective . But b e fo re setting them o u t we sh a l l p u rs u e fu rther the study of the c ri t er i o n of ataxy. We shall neglect the e n t ro p i e s

of the

th r ee variables at the grain

level,

for what m atters is the ma c ros c o p i c mechanism at the screen leveL The

fundamental questions p o s ed by these transformation summarized by

an

Let us consider the ( MTP) :

and

m ech an i s ms

t

X

Y

X

0.2

0.8

y

0.8

0.2

all

to set

s u p p o se one hun dred m ech anisms identified by

(MTP) . We shall allo w them

a re,

" Where does the

( MTP) go ? What is its destiny ? "

out from

the l aw

of

this single

X and evolve fr e el y . The

preceding q u e s t i o n then becomes, " Is t h e r e a gen'eral tendency for the states

if so, what is i t ? " (See App endix I I . ) be transformed into 0. 2 ( 1 OOX) � 2 0 X, and 0.8 ( l OOX) ---+ BO Y. At the third stage 0.2 o f th e X's and 0.8 of the Y's will become X's. Conversely 0.8 of the X's will become Y's and 0.2 of the Y's will remain Y's. This general a rg u m e nt is t ru e for all stages and can

of th e hundred mechanisms, a nd

After the first stage

the

I OOX will

be written :

X' Y'

=

=

0.2X + 0.8 Y 0.8X + 0.2 Y.

Markovian Stochastic Music-App lications

85

If this is to be applied to the I 00 mechanisms X as above, we sh a ll ha vc : Mechanisms

Mechanisms

X

y

Stage

0

1 00

0

I

20

80

2

68

32

3

39 57

5

46

43

54

48

52

6

We

61

4

7

49

8

50

51 50

9

50

50

notice oscillations that show

a

general tendency towards

a

station­

th en , that of the 1 00 m ech a­ nisms that leave from X, the 8th stage will in all prob ability send 50 to X and 50 to Y. The same s tationary prob!lbility distribution of the Markov chain, or the fixed probability vector, is calculated in the following manner : At equil i b rium the two probability values X and Y rem ain u n ch an ge d and the preceding system becomes

ary state at t h e 8th s tage. We may conclude,

X

Y

=

=

0.2X + 0.8Y +

0.8 Y

0.2 Y

or

0 0

= =

- O.BX + O.B Y + O.BX - O.BY.

Since the number of mechanisms is constant, in this case 1 00 (or 1 ) , one of the two e q uations may be replaced at the stationary distribution by l = X + Y. The syst e m then becomes 0

=

O.BX - 0.8 Y

I =X+ Y

and the stationary probability values X, Y a re X 0.50 and Y 0.50. The same method can be applied to the ( MTP) (a) , which will give us stationary prob abilities X = 0.73 and Y 0.27. =

=

=

86

Formalized Another method, particularl y i nterest i n g i n the case

to

man y terms, which forces us

Music

of an (MTP)

w i th

resolve a large syst e m of l i near equations

in

order t o fi n d the stationary pro b a b i l i ties, is t h a t w h i c h makes u sc o f m a trix

calculus.

Thus the first

( MTP) with the

s tage

may be considered as the m a trix pro d u c t of the

u n i co l u mn

c

� l �O �

1 0.8 1 I I I 1 I · I 1 1 I I l l l m atrix

X:

0

Y:

0.8

.

2

0. 2

X

00

=

20

80

0

The se ond stage will be

0 2 .

0.8

and the

0

.8 0. 2

X

20 80

=

68

4

64 + 16 + J6

=

32 '

nth stage

Now that we know how to calculate the stationary probabilities of

a

Markov chain we can e asily calcul ate its mean en tropy . The definition of th e

entropy of a system

is

The calculation of the en tropy of an (MTP) is made fi rst by columns C'J. P1 I ) , the p1 bei ng the probability of the transi tion for the ( MTP) ; t h en this result is weighted with the c orre spondi n g stationary probabilities. Thus for the (M T P) (a) : =

t

X

The

e n tr o py

y

X

0.85

0. 1 5

y

0.4 0.6

of the states of X will be - 0 .85 log 0.85 - O. l 5 log 0 . 1 5

0 . 6 1 1 bits ; the e ntropy of the states of Y,

0 . 9 7 0 b its ; the stationary prob ability of X

bility of Y

=

-

=

0.4 log 0 . 4 - 0.6 log

=

0.6 1 1 (0.73) + 0.970(0.27)

=

=

0. 73 ; the station ary p ro b a

0. 2 7 ; the m ean entropy at the stationary

Ha

0.6

=

stage is

0. 707 bits ;

­

87

Markovian Stochastic Music-Applications

and the mean

The for if we

two

look

e n t r op y

e

n tro p i es

of the

(MTP) (p) Hp =

at the s ta tio n ary stage i s

0. 722 bits.

do not differ

by much,

and this is to be expected,

at the respective ( MTP) we observe that the great contrasts

o f probabilities inside the

matrix

(p) are compensated by an external equ ality (MTP) (a) the interior

of st a tion a ry probabilities, and conversely in the

qu asi-equality, 0.4 and 0.6, succeeds i n coun teracting th e interior contrast,

0.85 and in

0. 1 5 , and t he exterior

At this

level

we

may

contrast, 0 . 7 3 and 0.27. t he (MTP) of the three

modify

such a way as to obtain a n ew pair of

repeatab l e

we c a n

(MTP) of pairs

entropies.

As

variables };, g h d1 this

operatio n

is

form a p ro t o c o l of pairs of e n tropies and therefore an

of en tropies. These sp ecul ations and investigations

are no

doubt interesting , but we s h a l l confine ourselves to the first calculation made ab ov e and we shall pursue the investigation on an even more general plane. EXTENDED

MARKOV CHAIN On p.

83

SIMULTANEOUSLY

FOR

}; ,

gf >

dl

we a na l y ze d the me c h a n ism of transformation of fo to fo or f1

when the probabilities of the two v a ri ab le s g1 and d1 arc give n . We the same arguments for each of the

o t h e rs are given.

three

variables };,

can

apply

g1 , d1 when the two

Example for g1• Let there be a screen at the moment t whose variables (f0 , g 1 , d1) . At the mo m e n t t + tlt the value of g1 will be transformed into g1 or g0• From f0 comes the parameter (y) , and from d1

have the values comes

the

parameter (e) .

Wi th ( M T P ) (y ) the probability that g1 will remain g1 is

0.2. With

( MTP) (e) the probability tha t g 1 wil l remain g1 is 0.6. Applyi ng the rules

of compo und probabil ities and /or probabilities of m u tually exclu sive events

as on p.

8 3 , we

that the pro b abi l i ty that g1 wi l l rem ain g1 at the momen t simultaneous effects ofj0 a n d d1 is eq u a l to (0.2 + 0 . 6) /2

find

t +

tlt under the

and

for

0.4. 1l1e same holds fo r the cal cu l a ti on of the We

the transformations

of d1 •

shall now attempt to emerge from this j u ngle of probability

binations, which is i m poss i bl e

to

=

transformation from g1 i n t o g0

com­

manage, and look for a more general

viewpoint, if it exists. In general , each screen is constituted by a tri ad of s pecific valu es o f the variables F, G , D s o tha t we can enumerate the d i ffe rent screens emer­ gin g fro m the mechanism that we are given ( se e Fig. II I-5) . The possible

88

Formal ized Music

Do

/ ""'

fo

combinations

(f1 g 1d0 ) ,

"

/

91

Fig . 1 1 1-5

are :

/

do Uo

d, do

"

dl

'1

/ ""'

/

do

"

d,

/

U1

do

"

dl

Uogodo) , Uol1od! ), Uog1 do) , Uog1d1 ) , (J!godo), (f1 god1) ,

different screens, which, with their protocols, up the sonic evolution . At each moment t of the composi tion we shall encounter one of t h e s e eight screens and no others. What are the rules for the passage from one com b i n ation to another ? Can one construct a matrix of transition probabi l i t i es for these eight screens ? Let there be a screen (f0g1d1 ) at the moment t. Can one calculate the probability that at the m om e n t t + at this screen will be tra n sfo rm e d into ( f1 g 1 d0 ) ? The above op e r a ti o ns have e na b l e d us to calcul ate the probability that/0 will be transformed i n to /1 under the influence of g1 and d1 and that g1 will remain g1 u nder the i nflu ence of fo and d1• These operations are schemati :1.ed in Fig. 111-6, and the probabil ity that screen (f0g1d1) will be transformed into (f1g1 d0) is 0. 1 1 4. (!J g1d1 ) ; i . e . , eigh t

wil l make

Screen at

the moment t :

Para meters derived fro m transform ations : S c reen

at

the moment t

Values of proba bilities correspo nding to the

the c o u pling + 11t :

taken from the ( M T P )

coupling para meters :

Compound proba b i l ities :

Compound probabilities for

independent events :

0.80 0.1 5

0.6 0.2

0.4 0.8

0.475

0.4

0.6

0 . 475

·

0.4

•

0.6

=

0.1 1 4

Fig . 1 1 1-6

We can therefore ex t e n d the calcul ation to the eight screens and construct

the

matrix of transition

probabilities. It

rows and eight colu mns .

will be squ are and will h ave eight

Markovian Stochastic Music-Ap plications

89

MTPZ

l

( foUodo)

(foUodt l

0.021

0.357

B (fogodd

0.084

C(fogtdo ) D ( f0g1 d1 )

8

A

A (fogodo)

c

( fo g t do)

D

(foUtdt)

E

F

(ftgodo) (ftgodt )

G

(ftgtdo)

H

( ftgl dl)

0.084

0 . 1 89

0 . 1 65

0 . 204

0.408

0 . 0 96

0.089

O.o7 6

0. 1 26

0.1 50

0 .1 3 6

0.072

0 . 1 44

0.084

0.323

0.021

0. 1 26

0 .1

50

0. 036

0.272

0 . 1 44

0.336

0.081

0.01 9

0.084

0.1 3 5

0.024

0.048

0.21 6

0.01 9

0.063

0.336

0.1 7 1

0.1 1 0

0.306

0.1 02

0.064

0.076

0.01 6

0.304

0. 1 1 4

0.1 00

0,01 8

0.096

G ( flgtdo)

0.204

0 .07 6

0.057

0.084

0.1 1 4

0.1 00

0.054

0.068

0.096

H(f1g1 d1 )

0 . 304

0,01 4

0.076

O.o76

0.090

0.036

0.01 2

0 . 1 44

E(f1g0d0)

F(ftgodt )

dA

Does the matrix have a region of stability ? Let there be l 00 mecha­ Z whose scheme is summarized by (MTPZ) . At the moment t, m ec h an is m s will have a screen A, dB a screen B, . . . , d1-1 a screen H. At the m o me n t t + llt all 1 00 mechanisms will p rod u c e screens according to the probabilities written in ( MTPZ) . Thus,

nisms

0.02 1

dA will stay in

be 0. 084 de will be

0. 3 5 7 dB will

A,

A, to A,

transformed to transformed

0.096 dH will be transformed to A . The dA screens a t the moment t will become d� screens a t the moment t + llt, a n d this number will be e q ual to the sum of all the screens t h at will be produced by the remaining mechanisms, in accordance with the corre­ s pondi ng probabilities.

{d���

Therefore :

(e1)

d�

� =

.=

dH

At the

=

0.Q2 J dA

0.084dA +

0.304dA +

stationary

rem ai n constant

+

0.084dA +

and

O.Ol4dB

0.357dB

+ Q.Q84dc +

+

0.323d8

0. 089d8

+ 0.02 l dc +

+

+

0. 096dH 0. J 44dH 0. J 44dH

0.076dc

+

0. 1 44du.

+

0.0 76dc

+

+

+

state the frequency of the screens A, B, C, . . . , H will preceding eq u a tions will become :

the eigh t

{

90

(e2)

Formalized

0 0 0

=

0

=

=

=

- 0.9 79dA + 0 . 3 5 7d8 + 0.084dc + 0.084dA - 0.9 1 1 d8 + 0.076dc + 0.084-dA + 0.323d8 - 0.979dc + 0.304-dA +

0 . 0 1 4d8

+

0.076dc

+

Music

+ 0 .096dH + 0. 1 #dH + O. I 44dH •

•

•

-

0 . 856dH

On the o th e r hand

of

If we replace one of the eigh t eq u ati o ns by the last, we ob t a in a system with eight unknowns. Solu tion by the classic

eight linear equations

method of determinants gives the val ues :

(e3)

{dA

dG

=

=

0. 1 7, d8 0. 1 0, dH

=

=

0. 1 3, de 0. 1 0,

=

0. 1 3, dn

=

0. 1 1 , dE

=

0. 1 4, dF

=

0. 1 2,

the sc re e ns at the stationary stage . This the chance of error is very high (unless a

which are the probabilities of

method

is very

laborious, for

is available) . method (see p.

calculating machine

second

85) , which is more approximate but in making all 1 00 mechanisms Z set out from a single screen and letting them evolve by themselves. After several more o r less long oscillations, the stationary state, if it exists, will be a t t a i n e d and the proportions of the screens will r em ai n invariable. We notice that th e system of equations (e1 ) m ay be broken down into :

The

adequate, consists

1 . Two vectors V' and V which matrices :

may

V' =

and V

=

be represented by two u nicolumn

d(; d�

Markovian

91

S tochastic Music-A pp lications

2 . A l i near o p erator, the matrix of transition probabilities Z. Conse­ system (e 1 ) can be summ arized in a matrix equation :

quently

To cause all

m ea ns

allowing

Z =

a

1 00 mechanisms linear

0.02 1

0.357

0.084 0.084 0.336 0.0 1 9 0.076 0.076 0.304

0.089 0.323 0.08 1 0.063

0. 0 1 6

0. 057

0. 0 1 4

operator :

Z

to leave

0 . 084 0. 1 89 0.076 0. 1 26 0.02 1 0. 1 26 0. 0 1 9 0.084 0.336 0. 1 7 1 0.304 0. 1 1 4 0. 084 0. 1 1 4 0 . 0 7 6 0.076

screen X and evolve " freel y "

0. 1 65 0. 1 50 0. 1 50 0. 1 3 5

0.1 1 0 0. 1 00 0. 1 00 0. 0 9 0

to perform on t h e column vector

0.204

0.408

0. 1 36 0.072

0.096 0. 1 44

0.036 0.272 0.024 0.048 0.306 0 . 1 0 2 0.204 O.D I B 0.054 0.068

0. 1 44 0.2 1 6 0.064 0. 096 0.096

0.0 1 2

0. 1 44

0.036

0 0

V=

1 00 0 0

continuous m an n e r at each moment t. Since we have b ro ke n down a discontinuous succession of thickness in tim e !lt, the equa­ tion (e4) will be applied to each stage !lt. Thus at the beginning (moment t 0 ) the population vector of the 0 mechanisms will be V • After the first stage (moment 0 + !lt) i t will be V' = ZV 0 ; after the second stage (moment 0 + 2/lt) , V" ZV' Z 2 V0 ; and at the nth stage (moment n !lt) , V1"' Z" V0• In applying these data to the vector in

a

continuity i n to

=

=

=

0 0

V}l

=

0 0 0 0 0 1 00

=

Formal ized Music

92 after the first moment /}.t :

Vii

after

=

after the second stage moment 2 f}. t :

stage at the

zv�

=

9.6 1 4.4 1 4.4 2 L6 6.4 9.6 9.6 1 4.4

1 8. 94 1 1 0.934

v;;. = ZVfi

1 4 .4 7 2

=

8 .4 1 6

8.966

the third stage at the

and after the fourth moment

46-t :

1 6.860 1 0.867

=

zv;;.

=

1 3 . 1 18 1 3 . 143 1 4.575

v;;

=

zv;;

=

8. 145

1 1 . 046

Thus after the fourth stage, an av er ag e o f 1 7 out o f the

screen B,

14 screen C,

stage

at the

1 7. 1 1 1 1 1 .069 13.792 1 2.942 1 4.558 12.1 1 1

1 2 .257

have screen A, 1 1

1 1 . 1 46 1 !5 . 1 64 1 1 .954

moment 3 /}.t :

v;;

at the

8.238 1 0. 7 1 6

1 00 mechanisms will

. . . , 1 1 screen H.

we compare the components of the vector V'"' with the values (e3) that by the fourth stage we have almost attained the stationary state. Conseq uently the mechanism we have built shows a very rap id abate­ ment of the oscillations, and a very great convergence towards final stability, the goal (stochos) . The perturbation Pu, which was im p osed on the mecha­ n ism (MPTZ) w hen we considered that all the mechanisms ( h e re 1 00) left from a single screen, was one of the strongest we could create. Let us now calculate the state of the 1 00 mechanisms Z aft e r the first stage with the maximal perturbations P applied.

we

If

notice

:tic Music-Applications PA

v�

=

PH

2. 1 8.4 8.4 33.6 1 .9 7.6 7.6

0

V2 =

1 00 0 0 0 0 0

30.4

Pc

8.9

v�

32.3 =

=

vo

1.9

-

n -

33.6 3 0.4 8.4 7.6

1 00

0 0

0 0

=

1 .4

1 8.9

1 2 .6 1 2.6

8.4 Vb = 1 7. 1 1 1 .4 1 1 .4 7.6 PF

16.5 15.0 15.0 13.5

20.4 1 3.6 3.6

0 0

voF

1 1 .0

1 0. 0 1 0.0

9.0

-

-

0 0 0 1 00 0 0

Po

40.8

0

V8 =

6.3

1 .6

0 0 0

8. 1

5.7

Pn

PE

v�

35.7

0

8.4 7.6 2. 1 v�

93

0

7.2

0

27.2

0 0

0 1 00 0

v�

=

4.8

1 0.2 1 .8 6.8 1 .2

v�

=

2.4

30.6 20.4 5.4 3.6

Formalized Music

94 R e ca pitu l ation of t h e A n a l y s i s

Having arrived at t h i s s t a ge of t h e analysis

On the level of the screen cel ls we

now

we

h ave : I .

take our bearings . partial mechanisms of

mu s t

transformation for fre q uency, in tensity, and d e n sity ranges, expressed by the

(MTPF) ,

( MTPG) ,

between the three fundamental the coupl ing ( e0) ) .

mations of

On

A, B, C,

( M TPD) ; an d

2.

which

are

an interaction

variables F, G, D of the screen

(transfor­

the level of the sc re e n s we now have : I . eight different screens, D, E, F, G, H; 2. a general mechanism, the ( MTPZ) , which sum­

marizes all the partial mechan isms and their i n teractions ;

equilibrium ( the goal, stochos )

3.

a

final state of

of the system Z towards which it tends quite quickly, the st ationary distribution ; and 4. a p rocedu re o f l:l isequilibrium in system Z with the help of the p ertu rbations P w h i ch a r e imposed on it.

Mechanism Z which

SYNTH ES I S w e h ave

just

constru cted docs n o t i m pl y

a real

evolution of the screens. It only establishes a dynamic situa tion and

a

evolution . The natural process is that provoked by a pertu rbation P imposed on the s ys te m Z and the advancement of this system towards its goal , i ts s tationary state, once the perturbation h as ceased its action . We can therefore act on this mechanism through the intermediary of a perturbation such as P, which is st ronge r or weaker as the case may be. From this it is only a brief step to imagining a whole series of s u cc e ss iv e perturbations potential

which would force the apparatus Z to be di sp l aced towards exceptional regions at odds wi lh

its behavior

at equilibrium.

created lies in the fact be. The perturbati ons which apparently change many negations of this existence. And if we create

In effect th e intrinsic value of the organism th us

that it must manifest i tse l f,

its structure represent so a su c ce s s i on of perturbations or ne g ations, on the one hand, and station ary states or existences on the other, we are only ajjirming mechanism Z. In other words, a t first we argue positively by proposing and offering as evi d en c e the e x is ten c e itself; and then we confirm it negatively by opposing it with p e r t u rb a tory states. The hi-pole of being a thing and not being thi s thing creates th e whole -the obj ect which we intended to construct a t the b e g i n n i ng of Chapter III. A dual dialectics is t h u s at the bas i s of this compositional attitude, a dia­ lectics that sets th e pace to be followed. The " experimen tal" sciences are an expression of this a rg u m e n t on an an a l ogo u s plane. An experiment estab­ lishes a body of data, a web which it disen tangles from the magma of

95

Markovian Stochastic Music-Applications

obj ective reality with t h e help of negations and transformations i m p os e d

on this

body. The repetition of these dual operations is a fu ndam ental

condition on which

th e

wh ol e un iverse of knowledge rests. To s ta t e some­

thing once is not to define i t ; the cau sal ity is confounded with

of p h e n o me n a consi dered to be identical.

the rep e t iti on

In con cl u sio n , this dual dialectics with which we are armed in order

to compose within

the framework of our mechanism is homothetic with th a t

of the experimental s c ie nc e s ; and we can extend the comparison to the

dialectics of biological beings or to nothing more than the dial e c t i cs of

being. This it.

br i n gs us

back

to the point of departure.

Thus an e n ti ty must be p r op os e d and then a modification i m posed

our

on

that to propose the entity or its mo d i fi c a tion in

It goes without saying

particular case of musical composition is to give

a human observer the Then th e

means to perceive the two propositions and to compare t h e m .

an titheses, to

entity and modification,

be identified. What does

are repeated enough times for the entity

identification mean in the

c a s e of our mechanism

Z?

Parenthesis. W e h ave su pposed i n the course of the analysis that 1 00 mec h an i s ms Z were present simultaneously, and that we were following the rules of the game of these mechanisms at each moment of an evolu tion

created by a displacement beyond the stationary zone. We were therefore

com parin g the s tate s of 1 00 mechanisms in a 6. t with the states of these 1 00 me ch an i s ms in the next t, so that in c omp a ring two successive stages of the group of 1 00 simu l taneous states, we enumerate I 00 states twice. Enumera­ tion, that is, i nsofar as abstract

a ct

ion

i m plies ordered operations, means to

observe the 1 00 mechanisms one by one, classify th e m , and test them ; then

start agai n with

1 00

at th e following stage, and finally com p are the classes

number by n u m ber. And i f tates

a

fraction of tim e

x,

the

observation of each

it would take

200x of

mechanisms.

mechanism n ec es s i ­

ti m e to e n u merate 2 00

This a rg u me n t therefore allows us to transpose abstractly a s i m u l ta n ­

eity i n t o

a

lexicographic (temporal ) succession

wi thou t subtractin g

any­

thing, however little, from the d e fi n i ti o n of transforma tions engendered by

scheme

Z.

Thus to compare two su ccessive stages of the

1 00 m ec h an i sms Z interval of time 1 00x of time l OOx (see Fig. III-7) .

co me s down to comparing I 00 states produced in an

with 1 00 o the rs produced MATER IAL

in

an equal interval

IDE NTI F I CATI O N

OF

MECHANISM

Z

Identification of mechanism Z m e a ns essentially

a

comparison between

Formalized Music

96

Period of 1 stage

"'

2!

�

0 0

1-

I k4

f==� · 1

time

=

1 OOx

Period of 1 stage

�r---time -

=

1

OOx

I

�� �,=-J

stage

1 stage

Fig . 1 1 1-7

all its possibilities of bei ng : per t u rbe d states compared to stationary states, independent of order. Identification will be established over equ al periods of time IOOx following the diagram :

Phenomenon :

PN ---+

Time :

IOOx

in which PN

an

E ---+

l OOx

PM ---+

E

l OOx

l OOx

d PM represent any perturbations and E is the state of Z

equilibrium (stationary state) .

at

An alternation of P and E is a protocol in which 1 OOx is the unit of time ( l OOx period of the stage) , for example : =

PA

PA

E

E

E

PH

Pa

Pa

E

Pc

A new mechanism W may be constructed with an ( MTP) , etc., which would control the identification and evolution of the compos ition over more general time-sets. We shall not pursue the investigation along these lines for it would lead us too far afield . A realization which will follow will use a very simple kinematic diagram of perturbations P and equilibrium E, conditioned on one hand by the degrees of perturbation P, and on t he other by a freely agreed selection.

�

E ---+ � ---+ � - E - � - � - � - � ---+ E ---+ �

Markovian

Stochastic Music-Applications

97

Def i n i t i o n of State E and of the P e rturbat i ons P

From the above, the stationary s t a te E wi l l be expressed by a se q uence of screens such as :

Protocol E(Z)

ADFFECBDBCFEFADGCHCCHBEDFEFFECFEHBFFFBC HDBABADDBADA DAHHBGADGAHDADGFBEBGABEBB

·

·

· .

To carry out this p ro toc ol w e shall u tilize eight urns [A ] , [B] , [C ] , [D] , [E] , [F ] , [G] , [H], each containing balls of eigh t different colors, whose proportions are given by the probabilities of (MTPZ) . For example, urn [G] wil l contain 40.8% red balls A, 7 . 2 % orange balls B , 2 7 . 2 /0 yellow balls C, 4.8/0 m aro on balls D , 1 0 . 2 /0 green balls E, 1 .8% blue balls F, 6.8% wh i t e balls G, and 1 . 2 70 black balls H. The comp osition of the other s e ve n urns can be read from (MTPZ) in similar fashion . We take a yellow ball C at random from urn [G] . We note the result and return the ball to urn [G] . We take a gre e n ball E at random from urn [C] . We note the result and re turn the ball to urn [C] . We take a black ball H at random from urn [E] , note the re s u l t, and return the ball to urn [E] . From urn [H ] we take . . . . The protocol so far is : GCEH Protocol P� ( V�) is obviously •

AAAA

Protocol P� ( V�) . Consider an are in t h e following proportions :

C,

33.6% color D, color H. After each be the following :

·

·

.

..

· •

urn [ Y] in which the eight col ors of balls A, 8.470 color B, 8.470 c ol or 1 .9% color E, 7 . 6 % color F, 7 . 6 '70 color G, and 30.4% draw return

2 . 1 i'o color

the ba ll to urn Y.

A

likely protocol might

GFFGHDDCBHGGHDDHBBHCDDDCGDDDDFDDHHHBF FHDBHDHHCHHECHDBHHDHHFHDDGDAFHHHDFDG ·

PTOtocol Pb

( V� ) . The same

method fu rnishes us

with a p ro to co l

EEGFGEFEEFADFEBECGEEAEFBFBEADEFAAEEFH ABFECHFEBEFEEFHFAEBFFFEFEEAFHFBEFEEB · · · .

Protocol P8 ( V8) :

CCCC · · · .

Protocol PB ( VB) : BBBB · · · .

·

· .

of P' :

98

Formalized Music

Protocol P(: ( V(:) : AAADOCECDAACEBAFGBCAAADGCDDCGCA DGAA GEC CAACAAHAACGCDAACDAA BDCCCGACACAACACB

R EA LI ZATI O N O F ANA L O G/Q UE A

·

·

· .

F O R O R C H ESTRA

The instrumental composi tion fol lows the preceding exposi tion point within the l i m i ts of orchestral instru ments and conve n tional execution and notation. The m e c h a n i s m which will be used is system Z, which has already been treated n u meri cally. The choice of vari ables for the s creens are shown in Figs . III-8, 9, I 0. by

(fo )

point,

Regions

0

J

r----1 I

If

I

r===I=J 2

"

3

m

lJl

s

'

lr===J ,c=J,

•

Freq u e n ci es

( s e m i t o n es)

c.

(f1 )

Regions

I, Jl, JJI

.B

. ----+1--...:11-!1-�...:1+---:::-+--:::--t---4� o-_,...--lI

.:2.

3

�

S

Fre q u e n c i es (semiton es)

c.

(A3 = 440 Hz) Fig. 1 1 1-8. Freq u e ncies

N uances of i ntensity

R egions F i g . 1 1 1-9. I nte nsities

N uances of i n te nsity

R e g i o ns

Markovian Stochastic Music-App lications

99

(d.) F i g . 1 1 1-1 0. D e nsities

This ch o i c e gives us the partial s creens FG (Fig. III- 1 1 ) and FD (Fig. 111-1 2) , the partial screens GD being a consequ ence of FG and FD. Th e Ro man numerals are the l i aison agents be twee n all the cells of the three planes of reference, FG, FD, and GD, so that the different combinations (j;, g1, dk) which are perc eived theoretically are made possible. Fo r exam pl e , let there be a screen (J1, g1 , d0) and the sonic entity C3 corresponding to fre q uency regi o n no. 3 . From the partial screens above, th i s entity will be the arithmetic sum in t h ree dimensions of the grains of cells I, II, and III, lying on frequency regio n no. 3 . C3 I + II + I I I . The dimensions of the cell corres p onding to I are : 6.F regi o n 3, 6..G region 1 , 6.D = region 2. T h e dimensions of the cell corres p ondin g to II are : 6.F region 3, 6.G re g i o n 2, !1D region I . The dimensions =

=

=

=

=

=

of the cell corresp onding to III are : 11.F !1D

=

region

l . Consequently

=

region 3, !1 G

in this sonic en tity the

gr ai ns

=

region 2,

will have

fre­

quencies included in region 3, i ntensities i n c l u d e d in regions 1 and 2, and they

will form densities i n clu d ed set fiorth above.

in regions

1 and 2, with the

corresponden��

-/��:i;nJVNs• wl�'·-...

u {· :,, �· :o' {l t� u � ....· � �: . tj s �:) ; :.; h•.� t � : t: l· �.\.\) , \«1. ,) '!:hOr . .. �r··r-t;./ --.;.:;������� :::�-· · · •.

Formalized Music

1 00 G

c;

II 2

u

j

r

1/

2 .0

H

I

Ill

G

J/J

J:r

:J

I

2 1 ./

2

.0 Ill 3

If

(/1 /o)

s-

,;

2

1I 1/l

f

I

D

F

Fig. 111-1 1 . Partial Screens for FG D 3 2 f

.a I

D 3

ll II

Jll

f)

1!1

3

Fig .

D /0

1 11-1 2. Partial Screens for FD

J/J .N

(

p J

I

I

z

f

I D l/1 n.

S toch as tic Music-Applications

Markovian

101

The eight principal scree ns A , B , C, D , E, F, G , H which derive from the combinations in Fig. 11 1-5 are shown in Fig. 111- 1 3 . The duration l:!.t of each screen is 1 . 1 1 sec. ( 1 half note 5 4 MM). Within this duration the densities of the occupied cells must be realized. The pe riod of time necessary for the exposition of the protocol of each stage (of the p ro toco l at the station­ ary stage, and of the protocols for the perturbations) is 301:!.. t , which becomes 1 5 whole notes ( 1 whole no te 27 MM) . =

=

Screen JJ .a.

J I

f

Screen

C

#

lJl

flo (., d.,}

.J I

f'i

Eo

!!

Screen E

.I I

N

9 .N

ff{ if�II

Screen

I

G

f

f

H

Jj[. E1 02 Opa c. 84 A5

1

B

.I N

ti#

.9 r

{ N

r:J

H

�

J

1ll

JL

{I

(rf J r

rt· �� d�

Screen F ( /1 1..

Screen

1'+11

D

I

/I, (I do)

I

Eo

9

E C4 84 As

(I, fo do} J I

j

.Jil

J

1/l

E1 02 Opa

J

J1

f

1!

-1

.a

.9 I

Screen

,

I

I

;;

1

I!

II

(I� t'• dij

B

+E

H

{ff!1 df)

�:J

+1/1 �

1

:r 1Y

Fig. lll-1 3

N OTE : The numbers written in the cells are the mean densities in grains/sec.

Formal ized Music

1 02

The linkage of the perturbations and the stationary state of (MTPZ) is followin g kinematic diagram, which was chosen for this purpose :

given by the

F i g . 1 1 1-1 4. B a rs 1 05-1 5 of AnaJogique A

1 03

Markovian Stochastic Music-Ap p lications Fig.

III-14, b ars

1 05- 1 5 of the score of Analogi.que

of perturbations P2 and P�. The

A, comprises a sectio n period occu rs at bar 1 09. The Fig. III- 1 5 . Fo r technical reaso n s

ch ange of

disposition of the screens is given in

screens E, F, G, and H h ave been simplified slightly. 1 05

End of t h e period of perturbatio n pg Fig.

115

1 09

-> J - .

In the case of Analogique B we could o btain 206 + 4 + 7> 227 1 34,2 1 7, 728 different screens. Important comment. At the start of this chapter we would h ave accepted the richness of a musical evolution, an evol utio n based o n the method of stoch astic pro tocols of the coupled screen variables, as a function of the transformations of the entropies of these variables. From the precedin g calculation, we now see that without modifying the entropies of the ( MTPF) , (MTPG) , and ( M TPD) we may ob t ai n a su pp lementary subsidiary evo­ =

=

lution by utilizing the different combinations of regions (topographic criterion) .

Thus in Analogique B the (MTPF) , (MTPG) , and ( MTPD) will not On the contrary, in time th e ft , gi > d1 will have new structures, corol­

vary.

laries of the changing combinations of their regions.

Com p l e menta ry Conc l u s i o n s about Screens and Their

Transform ati ons

I . Rule. To form

a

screen one may choose any combination

of regions

on F, G, and D, the };, g1, dk. 2 . Fundammtal Criterio n . Each region of one of the vari ables F, G, D

must be associable with a region corresponding to the other two variables in all the chosen couplings.

(This is accomplished by the Roman numerals. )

1 09

Markovian S tochastic Music-Appl ications

3. The preceding association is arbitrary (free choice) for two pairs, but obli g ator y for the third p air, a consequence of the first tw o . For e x ampl e ,

the

associ ations of the Roman nume rals off.. with those of g ; and with t h ose

of d,. are bo th free ; the association of the Ro man n umerals of gi with those of d,. is obli g atory, because of th e first tw o associ a tions.

4.

The co mp on e nt s f.. , g1, d,. of the screens generally have stochastic

protocols which correspond, stage by stage .

5.

The ( MTP) of these protocols will, in general, be c o u p l ed with the

help of parameters.

6. If F, G, D are the " variations " ( n u m b e r of components f.. , gh di, respectively) the 'maximum n u mb e r of cou p l i ngs between the com p onents and the p a r a m e ter s of (MTPF ) , (MTPG) , (MTPD) is the sum of the p ro duc ts GD + FG + FD . In an example from Analogique A or B : F

=

G

=

D

=

2

2 2

and th ere are

Indeed, FG

7.

(10 a n d j1 ) ( g0 and

(d0 and 12

l� +

the p a r ame t e r s of the

d1 )

couplings :

fl fa 11 ll

FD

If F, G ,

g1 )

.\

GD D a re the +

p. =

"

4 4

(MT P )

'

s are :

a,

y,

.\, p.

gl go gl de dl do dl � a >. p. a fl y ll

go

+ 4 +

=

12.

v ar ia t i o ns " (number o f components f.. ,

respectively) , the number of possi bl e screens T is the p roduct

example,

T

=

2 8.

X

if F = 2 (10 2

X

2

=

8.

and

{3

ll

j1) ,

G

=

2

( g0 and g1) ,

D

=

g1 , dk, FGD. For

2 (d0

and

d1) ,

The p ro t o co l o f the screens is stochastic (in the broad sense) and can

be summarized

(MTPZ) .

when th e

chain is ergodic (tending to regularity) , by an FGD columns.

This matrix will have FGD rows and

S P A T I A L P R OJ E C T I O N

all has been made in this chapter of the spatialization The subj ect was confined to t h e fu ndamental concept of a sonic complex and of its evo l u ti o n in itself. However n o thing would p revent broade n i ng of th e technique set o u t in this chapter and " leapi ng " i n to space . We can, for example, imagine protocols of screens a ttached to a particular p o i n t i n s p ace , with t r an s i ti o n probabil ities, space-sound co u p ­ lings, e t c . T h e method is ready a n d the g e n e r a l application is possible, along with the recip rocal enrichments it can create . N o mention a t

of sound.

C h a pte r I V

Musical Strate gy-Strate gy, Linear Pro g rammin g , and Musical Composition Before

pa ss i n g

to the

problem of the

the use of compu ters, we shall take of games, their

theory, and

a

mechanization

of stochasti c music by

stroll in a more enjoyable realm, th at

application

in

musical

AUTO N O M O US M US I C

composition .

composer establishes a scheme o r pattern which the con­ ductor and the instrumentalists are called u pon to follow more or less rigor­ o us ly . From the final details-attacks, notes, in te nsiti es , timbres, and styles of performance-to th e fo r m of the whole work, virtually everything is written into the score . And even in the case where the com poser leaves a m argin of i mprovisation to the conductor, the instrumentalist, the machine, or to all three t o ge the r , th e u n folding o f the sonic discourse follows an open lin e withou t loops. The score-model which is presen ted to them once and for all does not give rise to any coriflict other than that between a " good " per­ formance in the technical sense, and its " musical expression " as desi red or suggested by the writer of the score. This opposition between the soni c realization and the symbolic schema which plot s its course migh t b e called internal conflict ; and the role of the conductors, instrumentalists, and their machines is to con trol the ou tput by feedback and comparison with the input signals, a role an alogous to that of servo-m echanisms that reproduce profiles by such m eans as gri nding machines. I n general we can s tate that The musical

1 10

Strategy,

Linear Programming, and Musical Composition

lll

techn i cal oppositions (instrumental and condu ctorial) or aesthetic logic of the musical discourse, is internal the wo r k s written until now. The tensions arc shut up in the score even

the nature of the

even those relating to the to

when more or less d e fi n ed stochastic processes

c l a ss of internal conflict

might be qualified

arc

utili zed . This traditional

as autonomous music.

F i g . IV-1 1 . Conductor 2. O r c h estra 3. Score 4. Aud ience

H ETE R O N O M O U S M U S I C

and probably v ery fruitful t o imagine another of external coriflict between, for instance, two opposing orchestras or instrumen talists . One party's move would influence and condi tion that of the o ther. The sonic discourse would then be identi fied as a very s trict, al though often stochastic, I t would b e i nteresting

class of musical discou rse, which would introduce a concept

succession of sets of acts of sonic opposition. These acts wou ld derive from

tw o ( or more) conductors as well aU in a higher dialectical harmony.

both the will of the

composer,

as

from the

will of the

112

Formalized

Mu sic

Let us imagin e a competitive situation between two o rch es tras , each having o n e

against

conductor. Each

of the condu ctors

di r e c ts sonic operations

the operations of the other. Each operation represents a

tactic and the encounter between two qualitative value which benefits

one

move

or

moves has a n u m eri cal a n d fo r

and harms the other.

This value

a

a

is

of t h e row corres p onding to v i of condu ctor A a n d the co l u m n corres p onding to movej of conductor

written in a grid or matrix at the i n tersection mo e

B . This is the p a r ti a l score iJ, re p resentin g

the paym ent one condu ctor gives two-person zero-sum game. The external conflict, or heteronomy , c a n take all sorts of forms, b u t can a l w ays be summarized by a matrix of payments ij, conforming to the m a the­ matical theory of games. The theory demonstra tes t h a t there is a n op t i m um way of playi ng for A , which , in the long run, guarante�s h i m a minimum advantage o r gain over B whatever B might do ; and that c on v e r s e l y there exists fo r B an op t imu m way of pl aying, which guarantees that his disad­ vantage or loss u n de r A whatever A migh t do will n o t exceed a certain maximum. A's minimum gain and B's maximum loss coincide in absolu te value ; this is called the game value. The in troduction of an external conflict or heteronomy into music is not entirely without precedent. In c e r t a i n traditional folk m u s i c in Europe and other c o n ti n e nt s there exist competitive forms of m u s ic in which two i nstru­ the other. This game, a

duel,

is defined as a

mentalists strive to confound one

a no th e r .

O ne takes the i n i ti ative

and

attempts either rhythmically or melodically to uncouple their tandem

arran gement, all the wh i l e remaining within the musical con text tradition which permits this special kind

di c to r y virtuos i ty is

of

of the

improvisation. This contra­

particularly prevalen t among

the Indians, especially

among tabla and sarod (or sitar) players. th e

A musical heteronomy based on modern science

most conformist eye. But the

is thu s legitimate even to

problem is not the

historical j ustification of

a new adventure ; q u ite the contrar y , i t is the en richme nt and

forward that count. Just as stoch astic processes brough t

a

the

beautifu l

l eap

gen­

co m p l e xi ty of linear polyphony and the deterministic of musical discou rse, and at the same time d is cl os e d an unsuspected opening on a tot ally asymmetric aesthetic form hitherto qualified as non­ sense ; in the s a m e way heteronomy introduces i n to stochastic music a c omp l e ­ eralization to the

l ogi c

ment of dialectical s tructure .

We could equ ally well imagine setting up conflicts between two or more

instrumentalists, between one player and what we agree to call natural environ ment, or between an

orchestra or

several orches tras and the public.

But the fundamental characteristic of this situation is that there exists a gain

Strategy, and

a

L i n e ar Programming,

loss,

a

victory

and

material reward such as

for

a

a n d Musical C o m pos ition

113

a d e fe a t , which may be expressed by a m o r al or prize, m ed al , or cup for one side, and by a p e n alty

the other.

A

a more

is one in wh i ch the parties pl ay arbitrarily fol lowing or less improvised rou te, without an y condi tio n in g for conflict, and

degenerate game

compositional argument. This is a false game. A gambling device with s o u n d or lights would h ave a trivial s e n s e if it were made in a g ra t u i t o u s way, l i k e the usu al slot machines and juke boxes, that is, w i tho u t a new co mpe ti tive inner organization in s p i re d by any heteronom y . A sharp manufacturer might cash in on th i s idea and produce new so u n d and l ight d evices b ased on heteronomic principles. A less trivial u s e would be an e d u c at i o n a l apparatus which would re q uire children (or a d u l ts ) to react to sonic or l u m inou s combinations. The aesthetic i n te r es t , and h ence the rules of the game and the payments , would be determined by the players themselves by means of special input signals. In short the fundamental i n t eres t set forth above lies in the mutual con d i tioning of the two parties, a conditioning w h i ch respects the greater d ivers i ty of the m u s ic a l discourse and a certain liberty for the play e rs, but which i n v o lve s a strong i n fl u en c e by a si n gl e composer. This point of v i ew may be generalized with t h e introduction of a spatial factor in music and with the extension of the g a m e s to the art of light. In the fi e l d of calculation th e problem of games is r a pi dly b eco mi ng difficult, and not all games h ave received adequate mathematical cl ar­ ification, for exam p le, games for several players. We shall therefore con­ fine ou rs elve s to a rel atively s i m p l e case, that of the two-person ze ro - s um the re fo re without any new

game.

A N A LY S I S OF D UEL

This

1 958-59.

orchestras was co m p os e d in to relatively simple concepts : sonic constructions put into

work fo r two condu ctors and two

I t appeals

mutual correspondence

conditioned by the

Event I: A cluster

wooden part ally.

of

the

by the will of the

composer. The of

con d u ctors, who are

followi ng

eve n ts can occur :

themselves

sonic grains s u ch as pizzicati, b lo ws with the brief arco sounds distributed stochastic­

bow, a n d very

Event II: Parallel sustained strings with fluctuations. Event Ill: N et wo r ks of intertwined s tri n g gl i ss a n d i . Event IV: Stochastic pe rc u s sion sou nds.

14

Formalized

Music

Event V: Stochastic wind instrument sounds. Event VI: Silence.

Each of these events is written in the score in a very precise manner and

with

sufficient len g th, so that at any moment, following h i s instantaneou s

choice, the conductor is able to cut out a slice without destroying the iden­

tity

of the event.

imply an overa ll homogeneity in fluctuations.

We therefore

of each event, at the same time maintaining local

make up a list of couples of simultaneous events x, y issuing two orchestras X and Y, with our subj ective evaluations. We can

We can

from

the

the writing

also write this list in the form

of a

Table

qu alitative matrix (M1 ) .

o f Evalu ations

Couple

(x, y ) (1, I )

=

(y, x)

{ I , II ) = ( I I, I ) (1, III) = (III, I)

( I , IV) (IV, I ) (1, V) (V, I ) {II, II) (II, I II) = (III, II) (II, IV ) (IV, II) I V ( I, ) = ( V , II ) =

=

=

Evaluation

passable

(Ill, III) passable (III, IV) = (IV, I II) good + (I I I, V) = (V, II I) good (IV, IV ) passable (V good I I ( V, V) , V) passable (V, V ) =

( p)

(g) good + (g + ) passable + (p + ) very go od ( g + + ) passable (p) p as s ab l e (p) good (g) passable + (p + ) good

(p)

(g + ) (g) (p) ( g)

(p)

Strategy,

1 15

Linear Programm i ng, and Musical Composition

Conductor Y I

I

II

p

g

-

II Conductor X

g+ I--

p

p+

p

g+

g

p

p

g

p

p

p

g

-- -- --

p

p

11-

-- -- -- -

p+

IV

p

p

per row

g+ +

p+

g+

Minimum

V

IV

-- -- --

g r---

III

III

g+

g

-- -- -- -- -

g+ + p+

v

g+ +

Maximum per

column

In

(M1)

g

g

and the smallest maximum per no saddle point and no pure strategy. The introduction of the move of silence (VI) the

largest

g

colu m n do not coincide

minimum per row

(g

=f.

p) ,

and conse q u entl y the game has

modifies (M1 ) , and matrix (M2 ) results.

Conductor II

I

I

II III Conductor X

IV

p I--

g r---

g

g+

g+ +

IV g+

p

p

g

g+ p+

-- -- -- --

p

g+

g

-- -- -- --

g

g+

p

g

-- -- -- -- --

v

g+

p+

g

g

VI

V

-- -- -- --

g+ + p

r---

III

Y

p

p 1

-

1

-

p

p

1-

p

p

g+ +

g

p

p

p

p p

p

p

p

p

1-

-- -- -- -- -- -

VI

p

p-

Formalized

1 16 This

t i m e the

Music

saddle poi n ts . All tactics are possible, is still too sl ack : Conductor Y is interested i n playing tactic VI only, wh e re as conductor X can choose freely a m o ng I, I I , I I I , IV, and V. It m u s t not be forgotten that t h e rules of this matrix were established for t h e benefit of co n d u c to r X and that th e game i n

game has

several

but a closer study sh ows that the conflict

th is fo r m is n o t fair. Moreover t h e rules arc too vague. In order to pursue

o u r study we shall attempt to specify the qualita tive v a lu es by ordering them on an axis and m a k i ng them correspond to a rough numerical scale :

p- p I

If, in

p+

I 1

0

I 2

add i t i o n , we modify the value

becomes (M3) .

g

I 3

of

g+ I

g+ + I

4

5

the

c o u p l e (VI, VI) t h e matrix

Conductor Y I I

II

1 r--

3

1-

II

III IV V VI

3

5

4

4

1

I

3

2

1

3

1

3

I

1

1

1

I

3

4

4

3

---1-

III

5

1

1

4

IV

4

3

4

1

v

4

2

3

VI

I

1

1

5

3

5

Conductor X

I

- - - - 1-

3

---

1

1

1 - - 1-

( M3) has no saddle point and no recessive rows or columns. To find the solution we apply an approximat ion method, which lends i t sel f easily to c om p uter treatment but modifies the relative equilibrium of the en tries as little as possibl e . The p u rp o se of t hi s method is to find a mixed strategy ; that is to say, a weighted multiplicity of tactics of which none may be zero. I t is not possible to give all the calculations h ere [2 1 ] , but the m atrix that results from this method is (M4 ) , with the two unique strategies for X and fo r Y written in the margin of the matrix. Conductor X must therefore play

Strategy,

Li ne a r

Programming, and Musical

1 17

Composition

Conductor Y I I

II III

V

VI

2

3

4

2

3

2

18

3

2

2

3

3

2

4

4

2

1

4

3

1

5

IV

2

4

4

2

2

2

5

v

3

2

3

3

2

2

11

2

l

2

2

4

15

6

8

12

9

14

II III Conductor X

VI

---

I--

2

9 tactics

IV

1-

1-

- 1- -- -1-

(M4)

5 8 Total

in proportions 1 8/58, 4/58, 5/58, 5/58, 1 1 /58, conductor Y pl ays these six tactics in the propor­ tions 9/58, 6/58, 8/58, 1 2/58, 9/58, 1 4/58, respectively. The gam e value from this method is abou t 2.5 in favor of conductor X (g a me with zero-sum but still not fair) . We notice i m m e d i at e ly that the matrix is no longer symmetrical about its diagonal, which means that the t ac t i c cou p les are not commutative, e.g., (IV, II 4) =f. (II, IV 3) . There is an ori e n tati on derived from the adj ustment of the c al c u l a tio n wh ich is, in fact, an enrichment of the game. The following stage is t h e exp erimental control of the matrix. Two methods are possible : l . S i m u lat e the game, i . e . , mentally substitute on e sel f for the two conductors, X an d Y, by following the matrix entries stage by s tage , without memory and without bluff, in order to tes t the le a s t interesting case. I, II,

III,

IV, V, VI

1 5 /58, resp ectively ; wh ile

=

Game value :

=

52/20

=

2.6 points in X 's favor.

Formalized Music

1 18 2.

Choose tactics at random , b u t with frequencies proportiona:l to the

marginal

n um

b e rs

Game value :

the

in

57/2 1

(M4) .

=

2.7 points i n X's favor.

We now establish that the experimental game values are very close to

value

calcu l ated

by a p proximation.

The sonic pl"ocesses d erived from

the two experiments are, moreover, satisfactory. a

method for the definition of the optim um the value of the game by using methods of linear p arti cu lar the simplex m e th o d [22]. This method is based

We may now apply

strategies for X and programming,

on two theses :

in

rigorous

Y and

I . The fu ndamental th e o re m

is that the minimum

of game theory (the " minimax theorem ")

score (maximin) corresponding to X's opti m um

strategy is always equal to the maxim u m s c o re (minimax) corresponding to Y's

optimum strategy. 2. The calculation

of the m ax i m i n or minimax

probabili ties of the optimum s t r a te g i es of a

down to the resolution

(dual simplex method) . Here we

shall

of a

pair of dual

v al u e , j ust as the two - p erson zero-sum game, comes p r o b l e m s of li near programming

simply s t a te the s ys t e m of linear equations for the

of the minimum, Y. Let y 1 , y2, y3, y4 , y5, y6

be the

player

probabilities corresp ondin g

I, II, III, IV, V, VI of Y; y7, y 8, y9, y 1 0, y 1 1 1 y12 be the " slack " and v be the ga m e value which must be minimized. We then h ave following liaisons :

to tactics

variables ; the

2yl 3yl

2y1 3yl 2yl

4yl

+ + +

+

+

+

Y 1 + Y 2 + Ys + y,

3y 2

2y2

4y2 2y2 2y2 2y2

+ 4ya +

+ 2ya

+

+ Ys

+

2y5

+

2y4

+

+ 4y3 + 2y4 + + 3y3 + 3y4 + +

+

Y3

+ 2y4

Y3 +

4y4

+

+

3ys

2ys 2ys

3ys

1 2ys + Y7 = v v 2ys + Ya 2y6 + y9 = v 2ys + Y1o v 4ys + Y 1 1 = v

+ Ys

2y4 + 3ys

+ +

+

+

=

=

=

Ya + Y12

= v.

To arrive at a unique strategy, the c al c u la tion leads to the modification

S trategy, Linear Pro gramming, of the

score

(III, IV

=

4)

For

Tactics

(III, I V

into

ing optimum strategies :

and Musical Composition

5) . The solu tion gives For

X Tactics

Probabilities

I

6/ 1 7

II

III

0

III

3/1 7 2/ 1 7 4/ 1 7

v

VI

v

and for the game value,

42/ 1 7

=

must completely abandon tactic must avoid.

Modifying score (II, IV

strategies :

=

III

3)

2/ 1 7

2/ 1 7 1/17

2/1 7 5/ 1 7

2.47. W e have established that X of III 0) , and this we

(prob ability

to ( I I , IV

For X

Y

5/ 1 7

IV v VI

�

=

= 2 ) , we For

obtai n the following

Y

Probabilities

Tactics

Probabilities

I

1 4/56

6/56 6 /5 6 6/56

I II

1 9 /56

II

Tactics

III IV

v VI

8/56

6/ 56

III

IV

v

VI

7/56 6/56 1 /5 6

7/56

1 6/56

� 2.47 points. been modified a little, the game value has, in fact, not moved. But on the other hand the optimum strategies have varied widely. A rigorous calculation is therefore necessary , and the final matrix accompanied by its calculated strategies is (Ms) ·

and for the game value,

v

1

the follow­

Probabilities

I

2/1 7

II IV

optimum

=

1 19

=

1 38/56

Although the scores have

Formalized Music

1 20

Conductor

I

3

i---

II Conductor X

II III IV V VI

2

I

3

i---

4

III

r-----

IV

2

r-----

v

3

i---

VI

Y 3

2

4

14

2

- - - - -

2

2

2

3

1

5

3

2

6

1

6

-

2

- - - -

4

4

2

2

3

2

!

-

2

6

2

8

(M5)

- -

2

3

2

2

1

19

7

6

2

2

r-----

7

16

4

16

56 Total

By applyi ng the elementary matrix operations to the rows and columns

in such a way as to make th e game fair (game value equivalent matrix (M6) with a zero game value.

=

0) ,

we obtain th e

Conductor Y I I

II

- 13

15

III

IV

43

v

- 13

15

- 13

15

VI 5 6

ll

- 13

-- -- -- -- -- -

15

II

- 13

- 13

5 6

- 13

..L

-- -- -- -- - - -

Conductor X

III

43

- 13

- 41

71

15

- 13

- 13

-- -- -- -- --

- 13

IV

43

43

- 13

- -- -- -- --

15

v

- 13

15

15

- 41

..L

- 13

�-

56

1-

1

5 6

J_ 5 6

- 13

-

- -- -- -- -- -

VI

- 13

- 13

- 41

- 13

5 6

As this matrix is

by

+ 13.

It then

d iffi cul t

5 6

1.§

43 5 6

ll

scores

- 13

(Me)

H. to read ,

it

is

simplified by dividing

becomes (M7) with a game value v

=

-

a ll the

0.07, which

Strategy, Linear

Programming, and Musical Comp osition Conductor

I

X

II

III

IV

-1

+1

+3

1-1--

V

VI

-1

+1

-1

TI

-1

1

+1

-1

_§__

-3

+5

+1

-3

_§__

+3

-1

-1

-1

_§__

+1

-1

-1

-1

- 1

-1

- --

-1

IV

-

--

+3

III

Y

-- --

+1

II

Conductor

I

+3

-- -- -- -- --

+1

v

+1

- 1

- --

-1

-1

VI

121

1-

-3

-- --

5 6

1--

56

5 6

_.!!._ 5 6

+3

TI

1 6

5 6

56

-1

1-

_L

.ll!.

14

(M7)

1 6

TI

means that at the end of the game, at the final score, condu ctor Y should give 0.07m poin ts to condu ctor X, where m is the total number of moves. If w e convert the n umerical matrix (M7) into a qualitative matrix

according to the corres p ondence :

-1

-3

T

p+

p

we obtain

+3

+1 I

I

g

T

g+

(M8) , which is not ve ry different from (lvf2) , ex ce pt for th e silence the opposite of the first value. The calculation is

couple, VI, VI, which is now

+5

finished.

p

p+

g+ +

p

g

p

p+

p

p

p

p+

p

g+ +

p

p

g+ +

p+

p

p

g+ +

g+ +

p

p

p

p+

p

p

p

g+ +

---

p+

p

p+

---

p

p

p

p

Formalized Music

1 22

Mathematical manipulation has brough t abou t a refinement of the

d u el and the emergence of total si l e nc e .

a

Silence is to be

paradox : the

co u p l e

to

avoided, b u t

VI, VI, ch arac terizing

do this it is n ecessary

to aug­

ment its potentiality.

impossible to describe in these pages the fu n d a m e n t a l role of the t h i s p roblem, or the subtle arg u m e n ts we are fo rced to m ake on the way. We must be vig i la n t at every moment and over every part of the m a tr i x area. It is an i n s t a n c e of t h e kind of work where detail is dominated by the whole, an d the whole is dominated by de tail. It was to show the value of this i ntel l e ct ua l l abor that we judged it useful to set out the processes of calculation . The conductors d i r e c t with their backs to each other, using finger or light signals that are invisible to the oppos i n g orchcst;a. If the conductors use illuminated signals operated b y buttons, t he successive partial scores can be announced au tomatically on l ighted p a n els in the hall, the way th e score is d ispl ay e d at fo o tb al l games. If t h e conductors j ust usc their fingers, t h e n a referee can count the poin ts and put up the partial scores manually so they are visible in the hall. At t he end of a certain number of exchanges or minutes, as agreed up on by the conductors, one of the two is declared th e winner and is awar d ed a prize . Now that th e principle has been s e t out, we c a n envisage t h e interven­ tion of the public, who would be invited to evaluate the pairs of tactics of It is

mathematical treatment of

conductors X and composer

Y and

vote immediately on the make-u p of the game

music would then be the result of the conditioning of the who esta b lished the musical score, conductors X and Y, and the

matrix. The

public who construct the matrix of points.

R U LES O F TH E WO R K

The two-headed flow c h ar t

of Duel is

valid for Strategie, composed in 1 9 6 2 .

STRA TEGJE

shown in Fig.

The

IV-2. It

is equally

two orches tras are placed on

back-to-hack (Fig. IV-3), or on auditorium. They may choose and play o n e of six sonic constructions, numbered in the score from I to VI. We call them tactics and they are of stochastic structure. They were c alc u l ate d on the IBM- 7090 in Paris. In addition, each conductor can make his orchestra pl ay simultaneous combinations of two or three of these fu n d am e n ta l tactics.

either side

of the

stage, the conductors

platforms on opposite sides of the

The

six fundamental tactics

are :

1 23

S trategy, Linear Programming, and Musical Composition

I.

Winds

II . Percussion

I I I . S tring s o u n d box struck -

with the hand

IV. String pointi llistic effects

V. VI.

S tring glissandi

S u s tai n e d s tring harmonics.

Th e following are 1 3 compatible and simultaneous combinations of these tactics :

I

& II

=

VII

I & III = VIII

& IV = I X I & V = X I & VI = XI

I

II &

II &

III IV

II & V

I I & VI

XII

I

XI II

I & II &

=

=

= XIV

XV

=

I I

II

&

&

III IV II & V &

& II & VI

Thus there e x i st in all 1 9 tactics which each conductor can tra play,

The

36 1 ( 1 9

x

1 9 ) p o s si b le pairs that m ay

Game

1. Choosing tactics.

H o w will the

play ?

=

XVI

=

XVI I

=

=

XVIII XI X

make

his orches­

be played simultaneously.

c o n d u ct o r s

choose which ta ct i c s t o

a. A first solution consists of ar b i tra ry choice. For example, conductor X chooses tactic I. Con d ucto r Y may then c ho o s e a ny one of the 19 tactic s including I. Conductor X, acting on Y's ch o i c e then chooses a new tactic (see Rule 7 below) . X's seco nd choice is a function of both his taste and Y's choice. I n his turn, cond uctor Y, acting on X's choice and his own t aste, either chooses a new tactic or keeps on with the old one, an d pl ays it for a certain optional length of time. And so on. We thus obtain a continuous ,

succession of couplings of the 19 structures .

b. The conductors draw lots, choosi n g a new tactic by taking one card a pack of 1 9 ; or they might make a d ra w i n g from an urn containing balls n u m b e r ed from I to XIX in d i fferent proportions. These operations can be carried out before the performance and the res u lts of the su ccessive draws set down in the form of a sequential plan which each of th": conductors will

from

h ave before him during the performance. c.

The conductors get together i n ad v a n ce and choose a fixed su ccession will direct. d. Both orchestras are directed by a s i n g l e conductor who establishes the succession of tactics according t o one of the above methods and sets

which they

them down on a mas ter plan, which

he will follow

during the performance.

Formalized Music

1 24

Fig . IV-2

1 . G a m e m a t r i x ( d y n a m ostat, d u a l reg ulator) 2 . Conductor A (dev i c e for c o m pa riso n a nd decision) 3 . Condu ctor B ( d e v i c e fo r c o m p a r i s o n a n d d e c is i o n )

4. Score A (symbolic excitati on) 5. Score B ( sym bo l i c excitatio n ) 6. O r c h estra A ( h u ma n o r e l ectro n i c tra ns­ fo rm i n g d evice) 7. O rc h estra 8 ( h u ma n o r e l ectro n i c t ra ns ­

forming

d evi ce)

8 . A u d i e n ce

B

orchestra :

1 1 1 1 1 1 1 1 2 1 2

Percussion

Bp c l a ri n et

�--

contra bassoon F rench horn

trum pet tro mb ones tu ba perc u ssion 1 vibra p h o n e

4 tom -toms 2 bongos 2 congas 5 temple b l ocks 4 wood blocks 5 bells fi rst viol i ns second viol i n s violas cellos double basses

44 instruments, or 88 players i n both o rchestras

::;;;,c [

"'

c

g" (i

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- 52 - ·· - �·

-Jo - u - to -!'J -v+

1'1' 3.Z

" -Jo

-J�

1lC

Condu ctor Y (columns) � � � -:' "::' : .,. r ,.... . , .. =

I�

16 8

-2

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-ft

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41

1,-1, -51

lr&

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- 18

J

- JJ

-+

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-�

_, B

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'1+ f6

I t>

- I�

Jl>

..

.II

-J.I Jt

-lro ¥

f.l n

J6

_ ,, -1.1 8

i3 6 - 8

- ft

H - 66 - 4 H

12

..A Woodwinds

• N ormal percussion

H Strings striking sou nd-boxes •: Strings pizzicato

.:If Strings g l issando

:= Stri ngs sustained

=

of

_,.

-16 .;t

: -.

Jo

-

u

"0

-1'1.

3.J

-+

u "

-" -.rs

,.

- S'&

2.1

""

- ;t f

-"" - ro

0.

two and three

-J 6 -I�

- II.

- 18

�;!

YIJI'M

K•�

-Jo

" "'

JD

11

p

-I"

1!1

to

-?4-

"·

-4- lr

I�

t,o

.u

- 3/1: - JI!I

/tO

""

-

-U �

- .:

"

lro

.,

- �&

If

- ¥• -" -8

-21

-J.t -.18

- II'

-·· If

2�

- 11 J,o

'!8 -19

-s

"'' -Ji' - H

J lr - J o

- .211 -It: --�

H � 0 16

d ifferent tactics

. ., 11-�.d , .., ""

_ ,., -

-B -3& -.v o

• Combi nations

�

- ��

.t'f

Fig. IV-5. Strategy Two - person G ame. Va l u e of the Game

'

•

&I

_,., - .I& -2o -3.t

"

•

- .Je - �lr - 16

- � - .36

IJ

u

• "

- M - ./0 - 10

- r.z - M

-8 - U - vo

l•

4&

-J.t

8

-r;( 31

9

-.J•

U3

-

-$& �6

6

.u

f,z

-fC - .u - u

-

-

-8

.lc]J - I(¥

,.

,,

'

:t¥ -

n

-J,t

- !12

,,

1,1.(

"' ,

.t

1-1

'"

6o

� - !16

�� _ ,.:z

-J,G l, f

- ¥�

H

-llo

-1(,2 - :J2

,

B

'

and Musical Comp osition

Strategy, Linear Programming,

I;

-6�

Ill

:. •

lH

8 �!

4' 0 Ill •

-1

3

-3

3

.... .

:. . ' *' . ' Ill • •

0

3

• y ti H Y

>< 0

'

· · · � · · · · � ... . ..

n

I�

•

Strategy

for

Y

4

A

{-,

... . 4' 0 m• H O

.. . . :. . '

*' " ' Ill e f

-3

Strategy

F i g . IV-8

Two · person Zero-sum Game. Value of the Game = 1 /1 1 . This game is not fair for Y. •

Y Wooawinas

H Strings striking sou nd - boxes • ••

Normal percussion Strings pizzic ato

ill Strings sustained

Jl-

-1

0

-5

-1

4

3

1

-3

.2 • '

�� : a:;

for

Conductor Y Icolumns) J: • : * ;; • :.t · : if!:;; x - �• • • • • • X " # 'il .. .. .. ., . . . • • . . .. .

3

� t.

.. . .

-5

•o

1 29

Y

A •

Zero-sum Game. Value of the Game = 0. This game is fair for both conductors.

Two - person

' . ' "

' "" ' m O H . :.

., :.

of two dis­

Combinations

tinct tactics

' • H

; :�

' •m

Combi nations

of three dis­ tinct tactics

• IU

Strings glissando

. ,..

Simplification of the 1 9

x

1 9 Matrix

To make first performances easier, the conductors might use an equiva­ lent 3 x 3 matrix derived from the 1 9 x 1 9 matrix in the following manner: Let there b e a fragment of the matrix containing row tactics r + I, • • • , r + m and column tactics s + I, . • • , s + n with the respective probabilities 'lr+ h • • • , 'lr + .. and k, .. , • • • , k,.. •• qr + l 'lr- + f

fr + m

*• + 1

4r+ I .I + l

k, .. ,

k1 + n

Dr+ l , I + J

Dr + l . I + R

Dr + t .a + :l

a,. ... . . . + J

Or + e . a + n

a, + lll ,l + l

a, + • . • + l

Dr + m , a + n

��:a;•gy i

Formalized Music

1 30 This fragment

can

be replaced by the single score

A• + m .• + n

Li.i ::'/ � n (ar + ! ,s + J) (qr + !) (ks + J) Lr q. + , 2J ks + t

-

_

and b y the probabilities

m

Q

=

and

:L q, + , 1=1 n

K

Operating in this way matrix (the tactics will be X

X

-

25

X

1 9 m atri x we obtain the matrices i n Fig.

X

25

592 25

X

45

93 1 4

49

X

30

X

25

26

3088

1 76 1 0

-

30

25

or

x

8296

49

49

45

68 1 8

25

J=!

the same as in

1 45 2 2

25

L ks + J·

wi th the 1 9

7 704 25

=

45

X

26

2496

30

49

X

26

26

2465

- 1 354

1 82

25

- 2581

1 59 7

- 52 8

45

1818

- 1 267

640

30

25

49

26

45 30

the following IV-6) :

C h a pter V

Stochastic

Free

After this interlude , we return

Music

by Computer

to the treatment of composition

by machines.

The theory put forward by Achorripsis had to wait fo u r years before

being realized mechanically. This realization occurred thanks to

M.

Fran /'!! #

IJJ -==== # ::-=--=--;9¥ .f

��

iii -===I ---==---1 .f ==---=-�� - -=/'

1 -=== /.:==- i/1 I'===- ill-==-/

ill'-===..:.

/>-=::::::

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j'

-

!/'!

-====::

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i

#-==-1 -)1';-==:::.f

=-

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I -====.#�Ill I ==-=--=-I -==-"#

I j>

-==:::

II

1-

If'-===-I'

=#�/

I'-====/ /

I� 11P-===-/

/�!>-===/

!!! -== #'

./--======-.#

f�Ill J

} -===-) - 11 -==/ I'

j> -=== # -==- /'

If-==- t

;! -=!==- /'/' -=/-' = )>

.f

I

I -===-- I ===- 1>

I-====. # :::==- /

#

#

/

;I

� )# ;f-= #�)�/ -==::-

#:=-=- _,c --==:#' :::-

144

Formalized Music

Conclusions

A large number of compositions of the same kind as STJ 10-1, 080262

is possible for al r eady

been

a large number of orchestral combinations.

Other

by RTF (France III); soloists.

Atrles for ten soloists; and Morisma-Amorisima, for four

Al th ough this program gives a satisfactory sol ution

structure,

it is, however,

by coupling

a

works have

w ritten : STJ48-J, 240162, for large orchestra, commissioned

to

the

minimal

necessary to jump to the stage of pure composition

digital-to-analogue converter to t he computer. The

calculations would then be changed into sound, whose internal

numerical

organization

had been c onceive d beforehand. At this point one ..:ould bring to fruition and generalize the concepts described in the preceding chapters . The following are several of the advant ages of using electronic compu­ ters in musical composition:

l. The lon g laborious calculation made

The speed of a machin e

by hand is reduced

to nothing.

such as the IBM-7090 is tremendous-of the

order

of 500,000 elementary operations/sec. 2. Freed from tedious calculations the comp oser is able to devote him­ self to the gene ral problems that the new musical form poses and to explore the nooks and crannies of this form while modifying the values of the input data. For example, he may test all inst r u mental combinations from soloists to chamber orchestras, to l arge orchestras. With the aid of elec tr onic com­ puters t he composer becomes a sort of pilot: he presses the b utto ns , intro­ duc es coordinates, and s upe rvi s es the controls of a cosmic vessel sailing in the space of sound , across sonic constellations and g alaxies that he could formerly glimpse only as a distant dream. Now he can explore them at his ease, seated in an armchair. 3. The program, i.e., the list of se q uenti al operations that constitute the new musical form, is an objective manifestation of this form. The program may consequently be d isp atc hed to any point on the ear th that possesses com pute rs of the appropriate type, and may be exploited by any composer pilot.

4. Because of certain uncertainties introduced in the p rogra m , the

composer-pilot can instill his own

personality

in the sonic result he obtains.

145

Free Stochastic Music by Computer Fig. V-3. Stochastic Music Rewritten in Fortran IV c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c

C

C C C C C

C

C

C

C C C

C

C C C

C C C

C

C C

C C C

C

C

C

C C

PROGRAM

FREE

GLOSSAQV A

OF

THE

- DURATION

ALEA

ST OCHAST I C MUSIC

OF

PRINCIPAL

- PARAMET ER USEO -

AIO•A20.AI7.A35•A30 SAME

INPUT

ALFAI31

OF

-

DATA

THREE

lVI

XEN

XEN XEN

ABBREVIATIONS

S EQUENC E

EACH

!FORTRAN

IN SECONDS

NUMBERS FOR GLISSANDO CALCULATION TO ALTE R THE RESULT OF A SECONO RUN

EXPRESSIONS ENTERING

INTO

THE

THREE

SP EED

WITH

9

XEN

10

XEN XEN XEN

II

SLIDING TONES I GLISSANDI I XEN - MAXIMUM LIMIT OF SEQUENCE DURAT I ON A XEN fAMA�(Jl•I�J,KTR1 TABLE OF AN EXPRE SSIO N ENTERING INTO THE XEN CALCULATION OF THE NOTE LENGTH IN PART B XEN BF - DYNAMIC FORM NUMBER• THE LIST IS ESTABLISHED INDEPENDENTLY MEN OF THIS PROGRAM AND IS SUBJECT TO MODIFICATION DELTA - THE RECIPROCAL OF THE MEAN DENSITY OF SOUND EVENTS DURING XEN XEN A �EOUENCE OF DURATION A IEII•�I•I=I•KTRtJ=I•KTEl - PROBABILITIES OF THE KTR TIMBRE CLASS ESXEN INTRODUCED AS INPUT DATA• DEPEND IN G ON THE CLASS NUMBER I=KR AND MEN XEN ON THE PO W ER J=ll ORTAINE"D FROM V3*EXPFIUI=DA EPSI - EPSILON FOR ACCURACY JN CALCULATING PN AND £(1•J)•WHJCH �EN XEN IT IS A D V I SA BL E TO RETAIN• CGN[It�)•I=I•KTRtJ=I•�TSI - lABLE OF THE GIVEN LENGTH OF BREATH �EN FOR EACH INSTRUMENT. DEPENDING ON CLASS J ANO INSTRUMENT J XEN XEN GTNA - GREATEST NUMBER OF NOTES IN TH� SEQUENCE OF DURATION A XEN GREATEST NUMBER OF NOTES I N KW LOOPS GTNS THE

ALIM

CHAMINIJ•JJ•HAMAXII•J)•HBMINII•�)•HBMAXCI,Jl•l•l•KT�•Jcl•KTS)

TABLE OF

INSTRUMENT

COMPASS

LIMIT�•

DEPENDING

ON TIMB�F. CLASS

THE HA OR THE HB TABLE IS FOLLOWEO• THE NUMBER 7 IS A�B ITRARY. JW - ORDINAL NUMBER OF THE SEQUENCE COMPUTED• KNL - NUMBER OF LINES PER PAGE OF THE P RI NT E D RESULToKNL=50 KRJ - NUMBER IN THE CLASS KR=l USED FOR PF.RCUSS10N OR INSTRUMENTS AND

INSTRUMENT �.

TEST

INSTRUCTION

480

IN PART

6

DETERMINES

WHETHER

KTF

-

XEN

XEN

XEN XEN

XEN

XEN

XEN

XEN

POWER OF THE EXPONENTIAL COEFFICIENT E SUCH THAT

XEN

NUMBER

XEN

WITHOUT

A

DEFINITE

PITCH.

KW

-

KTR

-

CU�VE

WHICH IS USEFUL

IN CALCULATING

GLISSANDO

SPEED

13 14 15 16 17 IB 19

20

21

22

23 24 25 26 27 2B

29 30 31 32 33 34 36 37 38 35

40

39

XEN

42 43 44

MAXIMUM NUM BE R

DISTRIBUTION

12

XEN

OF TIMBRE CLASSES XEN OF ,JW KTESTJ,TAVl•ETC - EXPRESSIONS USEFUL IN CALCULATING HOW LONG THE XEN XEN VARIOUS PARTS OF THE PROGRAM WILL RUNo KT! - ZERO IF THE PROGRAM IS BEING RUNo NONZERO DURING DEBUGGING XEN KT? - NUMBER OF LOOPS• EQUAL TO 15 BV A RB IT RARV DEFINITION. XEN IMODIIIX81•IXB=7•1l AUXILIARY FUNC TI ON TO I NTERPOLATE VALUES IN XEN XEN THF. TETA I 2561 TABLE I SEE PART Tl XEN NA - NUMBER OF SOUNDS CALCULATED FOR THE SEQUENCE ACNA=OA*A1 KEN CNT(I)•I•l•KTR) NUMBER OF INSTRUMENTS A��OCATEO TO EACH OF THE XEN KTR T I MB RE CLASSES• IPNIJ•J)•I•t•KTR.J=I•KTS)•CKTS•NTII)•J=l•KTR) TABLE OF PROBABILITYXEN KEN OF F.ACH INSTRUMENT OF THE CLASS 1, IO(II•I•1•KTRI PROBABILITIES 0� THE KTR T I M B R E CLASSES• CDNSJOEREOXEN XEN AS LINEAR FUNCTIONS OF THE DENSITY DAo IS(J)•I=l•KTR) SUM OF THE �UCCESSJVE Oltl PROBABILJTJES• USEO TO XEN XEN CHOOSE THE CLASS KR BY COMPARING IT TO A �ANDOM NUMBER XI !SEE PART 3t LO O P 380 AND PART 5• LOOP 430), XEN StNA - SUM OF THE COMPUTED NOTES JN THE �W CLOUDS NA• ALWAYS LESS XEN THAN GTNS I SEE TEST IN PA�T 10 I• XEN X EN SOPI - SQUARE ROOT OF PI C 3.14159• ••) TA - SOUND ATTACK TIM� AACISSA. XEN TETAC256) - TABLE OF THE 2.S6 VALUES OF THE INTEGRAL OF THE NORMAL WEN 0AfMAX1=V34CE**IKTE-11J

8

XEN THEXEN

VALUES

6 7

XEN

41

45 46

47 48 49 50 51 52 53

54

55 56 57 58 59 60 61 62 63 64 6�

146 c c c c c c c c c c c c c c c c

Formalized Music

ANO SOUND EVENT DURATION• VIGL - GLISSANDO SPEED IVITESSE

GLISSANOO)t �HICH CAN VA�V ASt BE INVERSELY AS THE DENSITY OF THE SEQUENCE• THE ACTUAL MODE OF VARIATION EMPLOYED REMAINING THE SAME FOR THE ENTI�E SEQUENCE ISEF. PA�T 71o VITLIM - MAXIMUM LIMITING GL I SSANDO SPEED fiN SEMITONF.S/SE C )t INDEPENDENT OF•

SU��ECT

OR VARV

TO MODIFICATION.

- MINIMUM CLOUD DENSITY DA lZ1 ClleZ2CI)tlet,SJ TABLE COMPLEMENTARY TO THE

V3

CONSTANTS

�EAD

TETA

TABLE•

HC

MEN XEN

IX=lt"7

XE"N

XEN

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READ 20tfTETAliJ•I=1•2561

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268 269 270 271 272

273 274

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c C c

Stochastic Music by Computer

PART

430

CLASS AND INSTRUMENT NUMBER

�•DEFINE

XI =�ANFC-1) 00 430 t:J•KTR IF

149

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x 2 , x3) . Two points or vectors are said to be equal if they are defined by the same sequence :

X; = y, .

The s e t of these sequences constitutes a vector space i n three dimensions, E3• There exist two laws of composition rel ative to E3 : I . An internal law of compositi on, a d d i ti on : If X = (xh x2, x3) and Y (y1, y2, y3) , the n =

X + Y

=

(xl

+ Y1• X2 + Y2•

Xa

+

Ya) ·

The following properties are verified : a. X + Y Y + X (commutative) ; b. X + ( Y + Z) = (X + Y) + Z (associative) ; and c. Given two vectors X and Y, there exists a sin g le ve cto r Z (z1, z2, z3 ) such that X = Y + Z. We have z1 x1 - y1 ; Z is called the difference of X and Y and is nota­ ted Z = X - Y. In particular X + 0 0 + X = X; and each vector X may be associated with the opposi te v e c t o r ( - X ) , with components ( - x1 , =

=

=

=

- x2, - x3 ) , such that X +

( - X)

=

0.

2. An external law of composition, multipl ication by a scalar : If

p E R and X E E,

then

pX

=

(px1, px2 , px3)

The following properties are verified for

(p, q)

E

E

E3 •

R:

a.

1 ·X

=

X; b. p (qX)

Formalized

1 62

(pq)X ( associative) ; and c. (p (dis tri butive) .

BASIS

are

AND

R EFER ENT

+

OF

If it is impossible to find not all zero, such that

a1X1

+

q) X

=

pX + qX andp( X

A

VE CTOR

a

system of p

a2X2

+

· · ·

+

Y)

=

Music

pX

+ pY

SPACE

numbers

+

a'PXP

=

a11

a2,

a3,

•

•

•

, a'P

which

0,

p vectors X1 1 X2, , Xp of the space En are say that these vectors arc linearly independent. Suppose a vector of En, of which the ith co m p o n e n t is 1 , and the others are 0. This vector l1 is the ith unit vector of En. There exist then 3 unit vectors of E3, for example, li, g, ii, corres p ondi n g to t h e sets H, G, U, respectively ; and these three vectors are linearly independent, for the relation and on

not

th e condition

that the

•

•

•

zero, then we s h a l l

a/i + a2g

en t a ils a1 = a2 be written

=

a3

may

=

+ a3fi

=

0

0. Moreover, every vector

X = (x1 , x2, x3) of E

results from this that there may not exist in E3 more t h an indep endent vectors. The set li, g, il, constitutes a basis of E. By anal ogy with elementary geometry, we can say th at Oh, Og, Ou, are axes of coordinates, and that their set constitutes a referent of E3 • In such a space, all t he referents have the same origin 0. Linear vectorial multiplicity. We say that a set V of vectors of En which is non-empty constitutes a linear vectorial multiplicity if it possesses the following It immediately

3 linearly

properties :

I . If X is a vector of

scalar p may be.

V, every vector pX belongs also

to V whatever

the

2 . If X and

Y are two vectors of V, X + Y also belongs to V. From this a. all linear vectorial multiplicity con tains the vector O (O · X 0) ; and h. every linear combination a1X1 + a2 X2 + . . . + a'PXP of p vec tors of V is a vector of V.

we ded u ce that : =

REMARKS

1 . Every so ni c event may b e expressed as a vectorial multiplicity. 2. There exists only on e base, li, g, ii. Every other quality of the sounds and every other mo re complex c o m po n e n t should be analyzed as a l in e ar combination of these three unit vectors . The dimension of V is therefore 3 .

Symbolic Music

then

1 63

p, the

3. The scalars

move out of

not in

q, m ay

pr ac t i ce take

does not invalidate the g enerality o f these

example, let 0 Og, Ou, as referent, and For

all

v al u es

audible area. But this restriction o f

be

a

the

base

origi n of

/i, g, ii,

for /i, 1

=

arguments and a

trihedral of

a

,

we

for

would

practical o rd er

their applications.

reference

with

with the followi ng units :

Oh,

semi ton e ;

g, 1 = 1 0 decibels ; for ii, 1 second. for

=

The

origin

0 will be chosen arbitrarily on the " absolu te " scales the man n e r of zero on the thermometer. Thus :

established

by tradition, in

/i, 0 will be at C3 ; g, 0 will be at 50 db ; for ii, 0 will be at 1 0 sec ;

(A3

for

for

and the vectors

may be

xl

x2

the same

xl

- 3g

7/i + tg

=

written in traditional notation for

1'1'

In

sfi

=

way

+ X2

= (5

+

7)/i

+

�

( 50 -

( 1 - 3)g

+

= 440 Hz )

+ s u­

1

lu

sec �

:5 0

=

J.

20

(5 - r ) u-

d.B)

=

r 2fi - 2l

+ 4u.

=

-m t

....

c s o - 2 0 = :5 0

a B)

of al l th e preced i n g pro p osi tions. We have established, thanks to vectorial algebra, a workin g language which m ay p ermit both an alyses of the works of the past and new construc­ tions by setting up interacting functions of the com p onents (combinations of the sets H, G, U) . Algebraic research in conjunction with ex p erimen tal

We may similarly p u rsue the verification

research by computers coupled to analogue converters migh t give us

Formalized Music

1 64

the linear relations of a vectorial multiplicity so as to obtain of existing instruments or of other kinds of son i c events . The following i s a n anal ysis o f a fragment o f SoTzata, O p . 5 7 (Appas­ sionata) , by Beethoven (sec Fig. VI- I ) . We do not take the timbre into account si n ce th e p i ano is considered to have only one t i m b re, homogeneous over the register of this fragment.

information on the timbres

A

A

A �----,

B,

F i g . Vl-1 I �

Assume as unit vec tors : /i, for which I � semitone ; g, for which

10

db ; and u,

for which

,

A L G E B RA

=

=

The vector X2 =

T h e vector X3

=

The vector X4 =

The vector X5

=

1 8/i

(18 (18

(18

( O P ERATI O N S +

+

+

+

the g axis, and AND

R E L A T I O N S IN S E T

O.g + 5 u corresponds to G. 3) /i + Og + 4u corresponds to B�. 6) /i + Og + 3 17 corresponds to D p . 9)/i + Og + 2 ii corresponds t o E. 12)/i + Og + 1 il corres ponds to G.

(18 + ( 1,8 + O) li

+

a l so admit the free vector v

X1 (for i = 0, I , 2, 3, 4)

origins

.

OUTSIDE-TIME

The vector X1

us

on the li axis,

ff = 60 db (invariable) on 5J on the u axis.

The vector X0

Let

I � ; . Assume for the

Og + =

J u corresponds

3!i + Og

- I u;

A)

to G.

(See Fig. VI-2 . )

then t h e vectors

are of the form x; = X0 + vi. We n o tic e that se t A consists of two vector families, X1 and iv, combined

by means of addi tion .

Symbolic Music

165

D�

5 11

Fig .

Vl-2 A second law of composition

an ari thmetic pro g ression.

Finally, the scalar

i leads to

e x i s ts

TEMPORAL

ALGEBRA

The sonic statement

( IN

SET

This

boils

on

down t o the axis

the set

Xa

T

T)

xl

T

x2

a

=

0, 1 , 2, 3, 4) ;

of the com­

A is successive :

0 of the base

of A

shifting that has noth ing to

� E3

do

ii.

Thus in

the case of a simu l tanei ty (a

chord)

of the

�

of base

attacks of the

vectors d escribed for set A, the displacement wou l d be zero.

V

with the

ch ange of the base, which is in fact an operation within sp ace E3

fi, g,

is

it

T . . .

saying that the o rigin of time,

(i

variation remaining invariant.

of the vectors X1 of set

T being the operator "before."

displaced

in

an antisymmetric

ponents fi and ii of X1, the second g

is

0

six

In Fig. VI- 3 th e segments designated on the axis of t i m e by the origins 0 of X; are equal and obey the fu nction 6.t1 = 6.t1, which is an internal law

Formalized Music

1 66

of composition in se t T; or consider segment unit

equal to

l::!. t ; then /1

=

an

a

origin 0'

+ il::!. t, for i

on =

the axis of time and 1 , 2, 3, 4, 5 .

a

Vl-3

Fig .

A L G E BRA

IN-TIM E

( RELAT I O N S

AND

B E T W E E N S P A C E E3

SET T)

p on en ts H, G , U, which = il::!. t ; the values are lexicographically ordered and defined by th e increasing order i I , 2, 3, 4, 5. This constitutes an association of each of the components with the ordered set T. It is therefore an algebration of sonic events that is indepen­ dent of time (algebra ou tside-time) , as well as an algebration of sonic events We may say that the vectors X1 o f A have

c om

may be expressed as a function o f a parameter t1• Here 11

=

as a function of time (algebra in-time) . time

that a vector X is a fu nction of the parameter of t if its components are also a fu nction of t. This is written

In ge n er al we admit

X(t )

=

H(t)fi + G(t) g + U (t)ii.

continuous they have differentials. What is of X as a function of time t ? Suppose

When these functions are

the meaning of the variations

dX

If we

neglect

_

dH fi

dt - dt

dG +

dt

dU

_

g +

dt

_

u.

the variation of the component G, we

conditions : For

dHfdt

=

0, H

=

ch ,

and

dUfdt

be independent of the variation of t ; and for

ch

will be of invariable pitch and duration . If ch and (silence) . (See Fig. VI--4.)

will

0, U and Cu t=

=

Cu

h ave the

=

=

cu,

following

H and

U will

0, the sonic

0, there is

no

event sound

For dHfdt 0, H ch , and dUf dt Cu, U Cut + k, if '"' and cu =1- 0, we have an infinity of vectors at the unison. If cu 0, th e n we have d d a singl e vector of constant pitch '"' a n uration U = k. (See Fig. Vl-5) . =

=

=

=

=

167

Symbolic Music

For dHfdt

and dUfdt = f(t ) , U = F (t) , we have an infinite family of vectors at the unison. cht + k, and dUfdt For dHfdt ch , H Cu, if Cu < s, 0, U lim s = 0, we have a constant glissando of a single sound. If Cu > 0, then we have a chord composed of an infinity of vectors of duration cu (thick constant glissando) . (See Fig. VJ.:...5 . ) =

=

Fig.

Vl-4

0,

H

= en, =

=

=

- �-

t,

LL

Fig. Vl-5

Fig .

Vl-6

t.. , u.

Formalized Music

1 68

For dHfdt = ch, H = ch t + k, and dUfdt cu, U cut + r, we have a chord of an infinity of vectors of variable durations and p itch es . (See =

=

Fig. VI-7.)

u = c �-t � 1c U . c".t .

k

Fig. Vl-7 For dHfdt ch , H cht + k, and dUfdt f(t ) , U chord of an infinity of vectors. (See Fig. Vl-8.) =

=

=

=

F (t) ,

we have

a

h

H tJ

"

c h' t ;

+

�

� ( t)

F i g . Vl-8

Fig.

=

j(t) , H

i, G

F (t ) , and dUfdt = 0, U Cu, if Cu < e, lim e = 0, we have a thin variable glissando. If cu > 0, then we have a chord of an infinity of vectors of duration cu (thick variable glissando) . (See For dHfdt

=

=

Vl-9.)

u .. Fig. Vl-9

c"

f,u

169

Symbolic Music For dHfdt f(t ) , H = F (t ) , and dUfdt s(t ) , U chord of an infinity of vectors. ( See Fig. VI-1 0.) =

=

=

S (t ) , w e

h ave a

h

U : S lt) Fig.

Vl-1 0

In the example drawn from Beethoven, set A of the vectors X1 is not a continuous function of t. The correspondence may be written

l

X0

X1

X2

�

�

�

X3 X4 Xs �

4

�

Because of this correspondence the vectors are not commutable. Set B is analogous to set A. The fundamental difference lies in the ch ange of base in space E3 relative to the base of A. But we shall not pursue th e analysis. R emark

If our musical space has two dimensions, e. g . , pitch-time, pitch-intensity, etc., it is interesting to introduce complex variables. Let x be the time and y the pitch, plotted on the i axis. Then z = x + yi is a sound of pitch y with the attack at the instant x. Let there be a plane uv with the following equalities : u v (x, y ) , and w = u + vi. They define u (x, y ) , v a mapping which establishes a correspondence between points in the uv and xy planes. In general any w is a transformation of z. The four forms of a melodic line (or of a twelve-tone row) can be represented by the following complex mappings : pressure-time,

=

w

w

w

w

=

=

=

z,

with

u

- z, with gradation. =

=

x

u =

v =

y, which corresponds to identity (original form) and v = - y, wh i ch corresponds to inversion = - x and v = y, which corres p onds to retrogradation - x and v = - y, which corresponds to inverted retro­

and

j zj 2 /z, with u I z j 2/ - z with u

=

=

x

Formalized

1 70

Music

group.2 unknown, even to present-day musicians, could be envisaged. They could be applied to any prod uct of two sets of sound characteristics . For example, w = (A z 2 + Bz + c) f(Dz2 + Ez + F) ,

These tra nsfo r m ations for m the Klein

Other

transform ations, as yet

which c an be considered as a combi n ation of two bili near tran sformations

the type p = a2• Furthermore, for a two d i mensions we can i n tro d u c e hypercomplex

separated by a transformation of musical sp a c e of more than

systems such

as the

sys t e m of quaternions.

EXT E N S I O N O F T H E T H R E E ALG EB RAS

TO

S ETS

S O N I C EVE NTS ( a n application )

OF

We have noted in the above three kinds of algebras : 1 . The algebra of the components

of

a

son ic even t, with its vector

l angu age, independent of the procession of time, therefore an

outside-time.

2. A temporal algebra,

whi ch the sonic events create on the

time, and which is independent

3.

An

algebra in-time,

space.

axis of metric

issuing from the correspondences and fu nctional

relations be tw e e n time, T,

of the vector

algebra

the clements of the set of vectors X and of the set of metric independent of the set of X.

All that has been said abo u t sonic events themselves , their components,

and about time can

be generalized for sets

of sonic events X and for sets T.

is familiar with the with the concept of the class as i t is interpreted in Boolean al gebra . We shall ad op t this specific algebra, which is i so mo rp h i c with the t he o ry of sets. I n this chapter we have assumed that the reader

concep t of the set, a nd in particular

To simplify the exposition,

we

considering the referential or universal

piano.

We shall

c on s i d e r

shall first take a co n c re te example by

set R, consisting of all

th e sounds of

a

only th e pitcl1es ; timbres, attacks, intensities, and

dura ti o ns will be u tilized in order to clarify the exposi tion of the logical

operations and relations which we shall impose on the set of pitches.

Suppose, then , a This will be set

A,

set

A of keys that have

a

characteristic property.

a subset of set R, which consists of all the keys of the

p ia n o . This subset is chosen a p rio ri and the characteri stic property is

particular choice of a certain nu mber of keys.

the

For the amnesic observer th is class may be presented by p l a yi n g the

keys one after the other, with a period of silence in between. He will deduce from this that he

h as heard a collection of so u n d s

,

or

a

listin g

of elements.

171

Symbolic Music

class, B, consisting of a certain n umber of keys , is c h o s en in the It is stated after class A by causing the e l em e n t s of B to s o u n d . The observer hearing the two cl asses, A a n d B, will note the temporal fact : A b e fore B ; A T B, ( T = before ) . Nexl he begins to notice re l ation­ ships be twee n the clements of the two classes . If c e rt a i n elements or keys are common to both c l a ss es the classes intersect. If no n e are common, they are disjoint. H a ll the elements of B are common to one part of A he deduces that B is a cl ass included i n A. I f all the elements of B are fo und in A, a n d all the elements of A arc fou n d in B, he deduces that the two c la ss es are indistin­ guishable, that they are equal. Let us ch oose A and B in such a way that they have some clements in com mon . Let the observer hear firs t A, then B, then the common part. He will deduce that : 1 . th ere was a choice of keys , A ; 2. there was a second choice of keys , B; and 3. the part common to A and B was considered. The opera­ tion of intersection (conj unction) has there fore been used :

Another

same way.

This

operation has

therefore

A ·B

or

B · A.

engendered a new class, which was symbolized

B. If the observer, having heard A and B, hears a m i xtu re of al l the ele­ ments of A and B, he will deduce that a new class is b e i ng considered, and that a logical summation has b e e n performed on the firs t two classes. This

by the sonic enumeration of th e part common to A and

operation

is

If class

the union

(disjunction) and A + B

or

is written

B

+ A.

A has been s y mbolized or played to him and he is made to hear of R except those of A , he will deduce that the complement

all the sounds

of A wi th respect to R has been chosen . This is a new operation, negation, which is written A. Hitherto we have s h o w n by an i maginary experiment that we can define and

state

classes

of sonic

events (while taking precautions for clarity

in the symbolization) ; and e ffect

three operations

tance : in tersection, union, and negatio n .

of fu ndamental

im p or­

On the other h a n d , an observer m us t undertake a n intellectual task

in o r d e r to deduce from this both cl asses and o p erations. On our plane of

immediate com p rehension, we rep laced gra p hic signs by son i c events. We co n s ide r these sonic events as sy m bo l s of abstract e n tities furnished with abstract l ogical relations on which we may e ffect at least the fundamental operations of the logic of cl asses. We have not allowed special symbols fo r the statement of the classes ; only the sonic enumeration of the gen eric

Formalized Music

1 72 elements was allowed

(though

i n certai n cases, if the cl asses are already

known and if there is no ambigu ity, ment to admit

a sort of

stenosym bolization) .

s p ecial sonic symbols

We have not allowed

which are exp ressed graph ically by these operations arc

shortcuts may be taken

i n the s tate­

mnem otechn ical or even psychophysiological

ex p ressed,

· ,

for the three operations

o n l y the classes resu l ting from

+, - ;

and the operations a re conseq uen tly dedu ced

mentally by the o bserver. In the

same

must deduce the

way the observer

relation of equality of the two classes, and the relation of implic ation based

The e m p ty cl ass,

be symbol i zed by a duly presented silence . In sum, then, we can only state classes, not the operations. The following is a list of corres p ondepces between the sonic symbolization an d the graphical symbolization as we have j ust defi n ed it :

on the concept of inclusion.

Gr a p h ic symbols

Classes A, B, C, . . . Intersection

( ·)

Negation ( -

)

Union ( + )

however, may

Sonic symbols Sonic

enumeration

of the

generic

elements having the properties A, B,

C,

.

•

•

(with

possible shortcuts)

Implication (---+) Membership (e)

A

Sonic

enumeration

R not i n c l u d e d

A ·B

in

of the

A

Sonic enumeration of

A + B

A·B

elements

the elem ents

Sonic enumeration of the elements

A + B

of of

of

A => B A = B This table shows that

we can reason by pinning down our thoughts by

means of sou n d . This is true even in the p r es e nt case

where, because of a remain c l o se to that immedi­ ate intuition from which all sciences are bui l t , we do not yet wish to propose concern for ec o no m y of means, and in o rder to

conventions symbolizing the operations · , + , - , and the relations ---+. Thus propositions of the fo rm A, E, I, 0 may not be symbolized by

sonic = ,

sounds, nor may theorems. Syllogisms and demonstrations of theorems may only

be inferred .

I 73

Sy m bo l ic Music seen

Besides these l og ic a i

that

we

relations and operations o u tside-time, may obtain t e m pora l classes ( T c l ass e s ) issuing fro m

symbolization that defi nes distances or in tervals on the axis of time.

of time is aga i n defined in a new way . It serves primarily

we

have

the sonic

The role

as a crucible, mold,

are inscribed the c l a ss e s whose relations o n e must decipher. is in some ways e q uivalent to the area of a sh e e t of paper o r a black­ b o a rd . It is only in a s e co nd ary sense t h a t it may b e considered as carry in g generic el e m e nt s (temporal distances) and relations or operations between these elements ( temporal algebra) . Relations and corresp ondences may be established between these temporal cl a ss e s and the outside-time classes, and we may recognize in-time operations an d relations on t h e class level . After th ese ge n e ral consid erations , we shall give an ex am p l e of m u si c a l composi tion constructed wi th the aid of the a lgebra of classes. For this we must search out a necessity, a knot of interest.

or s p ac e in which

Time

C o n struct i o n

Every Boolean e x p re ssi o n or fu nc ti o n

three c l a ss e s

A, B, C can

F (A, B, C) ,

be expressed in the form

for example, of the

called

disjunctive canonic :

where a1 0 ; I and k1 A · B · C, A · B · C, A . J1 . C, A · E · C, Jf - B · C, Jf - B - C, A- E · C, .if. J1 . C. A Boolean fun c t i on with n v ar i a b l e s can always be wri tten in such a way as to bring in a maximum of o perati ons + , . , - , equal to 3n 2n - 2 - 1 . For n 3 this number is 1 7, and is fo u n d in the function =

=

·

=

F =

A ·B·C

+

A ·B·C +

.if. B . C + Jf . J1 . c.

(I)

For three classes, e a ch of which intersects w i th

the o t h e r two, fu nction ( I ) can be represented by the Venn d ia gr a m in Fig. VI- 1 1 . The flow chart of the op era t io n s is shown in Fig. VI- 1 2. This s am e function F c a n be obtained with only ten operations : F

=

(A · B

+

A · E) · C + (A · B

+

Jf- .11) · C.

( 2)

flo w chart is gi v e n in Fig. VI-1 3. If we com p are the two expressions ofF, each of which d efi n es a different proced u re in the com posi tion of c l a ss e s A , B, C, we noti ce a more elegant Its

For m alized Music

1 74

F i g . Vl-1 1

F i g . Vl-1 2

1 75

Symbolic Music

F i g . Vl-1 3

(2 ) is more economical ( ten It is this comparison that was chosen for work for piano. Fig. VI - 1 4 shows the flow chart

symmetry in ( 1 ) than in (2) . On

the other h and

op erations as against seventeen ) .

the realization of Herma, a that directs the operations of ( 1 ) and (2) on two parallel planes, and Fig. VI- 1 5 shows the precise plan of the constru ction of Herma. The three classes A , B, C result in an appropriate set of keys of the piano. There exists a stochasti c corres p ondence between the pitch com p onen ts a n d the m om e n ts of occurrence i n set T, which themselves follow a stoch astic l aw. The intensities and densities (number of vectors/sec.) , as wei ! as the silences, hel p clarify the levels of the com position. This work was composed in 1 9 60-6 1 , and was first performed by the extraordinary Japanese pianist

Yuji Takahashi in Tokyo in

Febru ary 1 962 .

1 76

Formalized Music

In conclusion we can say that our arguments are based on relatively simple generic elements. With much more complex generic elements we could still have described the same logical relations and operations. We would simply have changed the level. An algebra on several parallel levels is therefore possible with transverse operations and relations between the various levels.

Fig . Vl-1 4

Symbolic Music

177

u:

I

I

"' ..

I

"

�i

I

..

I I I I 11

I

0

Ill

I I

li

I

1. .i ' I

�I

;: I :- I "' I I

....

I

i

... r; .

Lo

a.

I ."r

1 :-

I

I�

I

I I I I

I

I �I

:I

�I

1

,I

a! � �

.. ""

·� l

� \J 1 "" I I .. I I I

.�

v X

�

I H�

1 ""

�

�

HI

!: 1

I I I I I I I

I� .. X

1 ;.-;

, _4"1

11

\

Conclusions and Extensions for Chapters 1-VI

t h e ge n e ral framework o f an ar tis tic attitude which , for the uses mathematics i n t h ree fundamental aspects : 1 . as a philo­ sophical summary of the enti ty and i ts evol u tion , e.g. , Poisson 's law ; 2 . as a quali tative foundation and m e ch a n is m of the Logos, e.g., sym bolic logic, set theory, th eo r y of chain events , gam e th eory ; and 3 . as an instr u m en t of mensuration wh ich sh arp ens i nvestigation , possible realizations, a n d p er­ ception, e.g. , entropy calculus, m a trix calculus, vector calcul us. To make music means to express human intelligence by so ni c means. This is intel l i gence in its broadest sense, which inclu des not only the pere­ grin a tio n s of pure logic but also the " log ic " of e m o tions and of i n tuition. The tech nics set forth h ere, alth o u gh often rigorou s in their i n te rnal s truc­ tu re, leave many openings through which the most compl ex and mysterious factors of the in tel l i ge n c e may pe n e t r a te These t e ch ni c s carry on steadily between two age-old poles, which are unified by m od e r n science and ph i los o p h y : determinism and fatality on the one hand, and fre e will and uncon d i tioned choice on the oth er. Between the two poles actual everyday life goes on, p a r t ly fatalistic, p a r tl y modifiable, with the w h o l e gamut of I

have sk e t c h ed

first time,

.

i n terpenetrations and interpretations.

In r ea l i ty formalization

guide,

and axiomatization

c onstit u t e

a procedu ral

at the o u ts e t the p laci n g of sonic art on a more universal plane. Once more it can be con­ si d ered on the same level as the stars, the numbers, and the riches of the human brain, as it was in the gre at periods of the ancient civilizations. The better suited to modern thought. They permit,

1 78

,

Conclusions

and

Extensions for Chapters

movements of sounds that

1 79

I-VI

cause movements in

us in agreement with

the m

" procure a common pleasure for those who do not know how to reason ; and

for t hos e

who

do know, a reasoned joy

through the imitation of

harmony which they realize in perishable movem e n ts

The theses advocated

"

(Plato,

the divine

Timaeu.s) .

in this exposition arc an initial sketch, but t h ey

h a v e alr e ady been applied an d extended. Im agine th at all the hypotheses of

in Chapter II were to be of vision. Then, i nstead of acoustic grains, sup­ pose q u a n t a of light, i.e . , p h o t o n s . The c om p o nen t s in the atomic, quan­ tic hypothesis of sound-intensity, frequency, d e nsity , and lexicographic time-are then ad ap te d to the quanta of light. A single s o urce of photons, a p h oton gun, cou l d theoretically reproduce the acoustic sc re e ns described above through the e missi o n of p ho tons of a

generalized stochastic composition as described applied to the ph e n o m e n a

particular choice of frequencies, energies, and densities. In this way we could

create a luminous flow analogous to that of m u s i c issu ing from

then join to this the coordinates of space,

a

s on i c source.

we could obtain a spa ti a l music of light, a sort of space-light. I t would o n ly be necessary to activate photon guns in combination at all c o rne rs i n a gloriously illuminated area of sp a c e . It is technically possible, but p a i n t e rs would h ave to emerge from the le th a rgy of their c ra ft and forsake their brushes an d their hands, u nless a new type of visual artist were to lay hold of these new ideas, technics, and needs. A new and rich work of visual art could arise, whose evol utio n would be ruled by huge com p u ters ( to ol s vital not onl y for the calculation of bombs or p rice indexes, but a l so for the artistic life of the future) , a total a u d i ov i s u a l manifestation ruled in i ts compositional intelligence by machines serving other machines, which are, thanks to the scientific arts, directed by man .

If we

Chapter V I I

Towards

a Metamusic

Today's technocrats and their followers treat mu s ic co mp os er

( source )

sends

to a

listener (receiver) .

as a message which the In this way they believe

that the solution to the problem of the nature of music and of the arts in general lies in formulae taken from i n form ation theory. Drawing up an ac­ count of bits or quanta of information transmitted and received would thus seem to provide them with "obj ective" and sc i en t i fi c criteri a of aesthetic value. Yet apart from e lementary statistical recipes th is theory-which is valuable for tech nological communications-has proved incapable of giving the characteristics of aesthetic v a l u e even for a simple melody of J. S. Bach. Identifications of music with message, wi th communication, and with language are schematizations whose tendency is towards absurdities

in this exception. Hazy music c a nnot be forced into too mold. Perhaps, it will be p ossible l ater when prese nt

and desiccations . Certain Afric an tom-toms cannot be included criticism , but they are an

prec i se a theoretical

theories have been refined and new ones invented .

The followers of information theory or of cybernetics

end two groups :

extreme . At th e other divided into

re presen t one

there are th e intuitionists, who may be broadly

who exalt the grap hic symbol above the sound of make a kind of fetish of it. In this group it is the fashionable thing not to write notes, but to create any sort of design. The " music " is j u d ge d according to the beauty of the drawing. Related to this is the so-called aleatory music, which is an abuse of language, for the true term should be l . The " graphists,"

the music and

English translation or Chapter VII by G. W. Hopkins. 1 80

Towards

a Me tamu si c

th e "improvised"

181

mus i c

grandfathers

our

the fact that gr a p h ica l wri ting , whether

notation, geome tric,

is

as

faithful

k n ew . This group i s i gnor an t of

it

be symbol i c, as in traditional

or nu m e ri c a l , should be

no more than an i mage that

as possible to all the i ns tructions the c o m p o ser gives t o the

orchestra or to the machine . 1 This group is taking music outside itself. 2. Those

who add

a spectacle in the

form of extra-musical scenic acti o n

to accompany the m u sic al performance. Influenced by the " happenings"

which express the confusion

of certain artists, these composers take refuge and thus betray their very limited In fact the y concede certain defeat for their

in mimetics a n d disparate occu rrences

pure

music.

groups

share

confidence in

music in

particular.

The two

a

romantic attitude. They believe

in

immediate

abou t i ts control by the mind. But since musical act io n , unless i t is to risk falling into trivial improvisation, imprecision, and irresponsi bility, im p eri o u s l y demands reflection , these action and are not much concerned

gro u ps are in fact denying music and take it outside i tself. linear

Thought

Aristotle, that t h e mean path is the best, for in middle m e a n s co mpromise. Rather lu cidi ty and harshness of critical thought-in other words, action, reflection, an d self­ trans formation by the sounds themselves-is the path to follow. Thus when I shall not say, like

m us ic-as in politics-th e

scientific and mathematical thought serve music, or any human creative

activity,

it

sh o u l d amalgamate dialectically with intuition. Man is one,

i ndivisible, a nd total. He thinks with his

would like to p ropose what, to my 1 . It makes it.

2. I t

3. It

is

a

belly

is a .

fixing in sou nd . . , arg uments) .

4. It is normative, th at

doing by symp athetic drive.

his mind .

I

is,

a

of

real ization.

i m a gi n ed

unconsciously

it

virtualities

is

a

( cosmological ,

m ode l for being or

for

is catalytic : its mere p resence perm i ts i n ternal psychic or men tal

transformations

6. I t is 7. I t is

in

the sam e way as the crystal ball of the hypno tist.

the gratu itous play of a child . a

Consequently expressions arc only very limited par t icu l ar

my sti ca l (but atheistic) asceticism.

of sadness, joy, love, and dramatic situations i nstances.

and feels wi th

covers the term " music " :

sort of comportment necessary for whoever thinks it and

is an individual pleroma,

philosophical,

5. It

mi n d ,

Formalized

1 82

Musical syntax has u nd e rg o ne considerable u ph e aval

seems that i n n u merable possibili ties c oexi s t in

a

and

Music

today

state of chaos. We h ave

abu ndance of t heories , of (sometim es) i ndividual styles, of ancie n t "school s . " But how docs one make music ? What

cated b y oral teaching ? (A burn i n g q u estion, i f one is

it

an

more or less can be communi­ to re form musical

entire world . ) It can no t be s a i d t h a t the inform ationists or the c y berneticians-much less the i n t u itionists-have pose d the q ucstion of an ideological purge of the dross accumulated over the centuries as well as by presen t-day develop­ ments. In general they all remain ignorant of the su bstratu m on which they found this theory or that action. Yet this substratum exists, and it will allow us to establish for the fi rst time an axiomatic system, an d to bri ng forth a formalization which will u n i fy the ancien t past, the presen t , and the future ; m o reover it will do so on a planetary scale, co m pri sing the still separate

edu cation-a reform that is necessary in the

universes of sound in Asi a, Afri ca, etc. In

19542 I d enoun c ed linear thought (polyphony) , and demonstrated the

contradictions

of

serial music. In its p l ace I proposed

masses, vast groups characteristics such

required d efinitions tic m usic

was

was

a

world of

of sound-events,

clouds, and galaxies go vern ed

and realiz ations

using

as

sound­ by new

density, degree of o rder, and rate of change, which

probability theory.

Thus s tochas­

numbers for it could embrace it as a particu­

born. In fact this new, mass-conception with large

more general than linear polyphony,

lar instance (by No, not yet.

Today these

re d u cing

the

density of the clouds) . General harmony ?

id eas and the realizations which accompany th em

be e n around the world, and the ex pl or a tion seems to

be

have

closed for all

intents and -purposes . However the tempered d i a tonic system-our musical terra

firma

on which all our

music is

fo unded-seems

not to h ave been

breached ei ther by reflection or by music itself.3 This is where the next stage

will come. The exploration and transform ations of this system will herald a new and immensely promising era. In order to understand its determi na­ tive imp o rtance we must look at i ts pre-Christian origins quent development. Thus

I

shall point ou t

the

and

at

stru ctu re of the

its subse­ music of

anci en t Greece ; and then that of Byzantine m u sic, which has best preserved

it while developing

it, and has done so with greater fidelity tha n its sister,

the occidental plainchant.

struction in

a

After demonstrating their shall try to express in a

modern way, I

math ematical and log i c a l language what time (transverse

musicology)

and in s pace

was

abstract

logical

con­

simple but u niversal

and what might be valid

(comparative

musicology) .

in

Towards a

Metamusic

1 83 a

In or de r to do this I propose to make

distinction in musical archi­

t e c t u re s or categories between o u tside-time, 4 in-time, and

sc a l e , for e x am pl e , is a n

temporal. A given pitch ou tside-time architecture, for no horizontal or

vertical combination of its elements can alter it. The event in itself, that is,

its actual o cc u rr e n c e , belongs to the temporal cate gory. Finally, a me l ody or a chord on a gi ven scale is produced by relating the ou tside-time category to the temporal catego ry. Both are realizations i n - tim e of outside-time con­ structions. I have dealt with th i s distinction a l re a d y , but here I s ha l l show how a n ci e n t and Byzan tine music can be ana lyz ed with the aid of th e se cate­ gories . This approach is very general s i n c e it p e rm its b o t h a u niversal axiomatization and a formalization of many of the a s pe ct s of the various k i n ds of music of our pl a n et. St ructu re of A n c i ent M us i c

chant w a s fo u nd e d on the structure of ancient the o thers who accu s e d Hucbald of being behind th e ti m es . The rapid evolution of the mu si c of Western Europe after the ninth century simplified and smoothed out th e plainchant, and theory was Originally t h e Gregorian

music, pace

C o m b ari eu

and

left b e h i n d by practice. But shreds of the ancient theory can stil l be found

in

the sec u l ar

music of

the

fi fteenth and sixteen th centuries, w it n e s s the

Terminorum Musicae dijfinitorium

ofJohannis

Tinctoris . 5 To look at antiq u i ty

of the Gregorian ch a n t and its h ave lon g ceased to be understood . We arc only beginning to glimpse o th e r d i rections in which the modes of the p laincha n t can be ex­ plained . Nowadays the specialists are saying that the modes arc not in fact proto-scales, b u t that they are rather ch aracterized by melodic formulae. To the best of my kno w l ed ge only J acques C h ailley6 has introduced other concepts com plementary to that of the scale, and he would seem to be co rrect . I b e lieve we can go further and affirm that ancient music, at least up to the first centuries of C h ristianity, was not based at all on scales and m o d e s related to the o c t av e , but on tetrachords and systems. Experts on ancient music (with the above exception ) have ignored this fundamental reality, c lo u de d as their minds h ave been b y the ton al con­ stru ction of post-medieval music. However, this is w h a t the Greeks used in their music : a hierarchic structure whose complexity p roceeded by suc ces ­ sive "nesting," and by inclusions and i ntersections from the particular scholars h av e been looking th rough the lens modes, which

to the general ; we can trace its main o u tline if we follow the writings of

Aristoxenos : 7

A. The primary order consists of the

tone and its subdivisions .

The whole

1 84

Formalized

Music

a m o u nt by which the i n terval of a fifth ( the penta­ dia pcnte) exceeds the i n terval of a fo u rth (the tetrachord, or di a tessaron) . The t o n e is divided i n t o halves, called semitones ; thi rds, calle d chromatic dieseis ; and quarters, the extremel y small enharmonic dieseis. No interval sm a l le r than the qu arter-tone was used . B. The secondary order consists of the tetrach ord. It is bounded by the interval of the dia lessaron, which is e q u a l to two an d a half tones, or thirty twelfth-tones, w h i c h we s h a l l call Aristoxenean segments. T h e two outer n o te s always maintain the same i n terval, the fourth , while the two inner notes are mobile. The pos i t i o n s of the inner notes determine the three genera of th e tetrachord (the in tervals o f the fi fth and the octave play no part in it) . The position of the notes in the tetrachord are always c o u n t ed from the lo wes t note up :

tone is defined as the

chord , or

1 . The

3 + 3

+

genus co n t a i n s two enharmonic d iese is , or If X equals t h e value of a t o n e , we can express xs12. X l i4 . X1'4 - X 2

enharmonic

24

=

30 segm ents .

th e enharmonic as

=

2. The chromatic genus co nsists of three types : a . soft, containing two chromatic dieseis, 4 + 4 + 22 30, or X113 . X1'3 - X X 5'2 ; b. h e m iolo n (ses q uialterus) , containin g two hemioloi d i es e is , 4.5 + 4 .5 + 2 1 2 = 30 segments, or X014> X . X7' 4 X5'2 ; and c. "toniaion , " con­ sisting of two semitones and a trihemitone, 6 + 6 + 1 8 = 3 0 segments, xs12, or Xli2 . Xtt2 . xat2 =

•

=

=

=

3 . The

di atonic

consists of: a . soft, containing a five enharmonic dieseis, 6

enharmonic dieseis, then ments,

or X1'2 - X 3 14 . X5'4 = X5'2 ; b. syntonon, tone, and a n o t h e r whole tone, 6 + 1 2

whole Xli2 · X · X

=

xst2 ,

semitone,

+ 9 + 15

then three 30 seg­ =

containing a semitone, +

12

= 30

segments,

a

or

or the system, is essentially a c o mbination of the elements of th e first two-tones and tetrachords eit h e r conjuncted or separated by a tone. Thus we get the p e n ta chord (ou ter interval the perfect fifth) and the o c to c h ord (ou ter interval the octave, some tim es perfect) . The subdivisions of the system follow exactly those of the tetrachord. They are also a function of connexity and of consonance. D. T h e quaternary order consists of the tropes, th e k eys , or the modes, which were probably j u st particularizations of the sys tem s, derived b y means of cadential, melodic, d o m i n a n t , registral, and other formulae, as in Byzantine music, ragas, etc. These orders account for t h e outside-time structure of Hellenic music. After Aristoxenos all the ancient texts one c a n consult on this matter give

C . The

tertiary o rder,

1 85

Towards a Metamusic this same h ierarchical

procedure . Seemi n gly

was

Aristoxenos

u s e d as

a

model. But later, traditions parallel to Aristoxenos, defective i n terpretati o ns,

and sediments d is to r te d this h i e rarchy, even in anc i e n t times. Moreover, it

seems that theoreticians like Aristides Qu inti lianos and Claudios P to le m a eos had b u t little acquaintance with m usic.

This

h ierarchical " tree " · was completed

by transition algorithms ­

the metabolae-from one genus to another, from one system to another, or from one mode to another.

This is

a far cry from the simple modulations or

trans p ositions of post-medieval to n al music.

Pentachords are subdivided i n to the same genera as the tetrachord they contai n . They are derived from tetrachords, but nonetheless arc used as primary concepts, on the same footing as the tetrachord, in order to define the interval of a tone. This vicious circle is accoun ted fo r by Aristoxenos'

determi n ation to rem ai n

faithful to mu sical experience (on which he insists) ,

which alone defines the s t r u c t u re of tetrachords and of

the

en tire harmonic

edifice which results combinatorially from them. His whole axiom ati cs

proceeds from there and his text i s an example of a

me th o d

to b e fo l l o wed

.

Yet the absolute (physical) value of the interval dia tcssaron is le ft undefined,

whereas the Pythagoreans defiQed i t by the

strings.

I

bel ieve this to be

a

ra ti o

3/4 of

the len gths of the

sign of Aristoxenos' wisdom ; the

c ou l d in fact be a mean value. Two Lan g u ages

Atten tion must be d r aw n to the fact that he makes use

rat io

of th e

3/4

addi tive

operation for the i ntervals, t h us foresh adowing logarithms before their time ;

this

co n t r as ts with the

practice

of the Pyth agorcans, who u sed the

geometrical (exponential) language, which is multiplicative. Here, the m e th od

of Aristoxenos is fu ndamental s i nce :

l.

it constitutes one of the two

ways in which musical theory has been expressed over the millennia ;

2.

by

u s ing addition i t i n s titute s a means of " calculation" that is m o r e economi­

cal, s im pler , and better s u i ted to music ;

and 3.

te m pered sc a l e n e a rly twenty cen turies be fo r e Europe.

l ays the foundat ion o f the it was appl ied in Western

it

Over the centu ries the two languages-arithmetic

addition) and geometric

( deri ved

(operating by

from the ratios of s tring l engths, and operating by mu ltiplication) -have a l ways i ntermingled and i n terpene­ trated so as to create much useless confusion in the reckoning of i n tervals and co nsonances, and conseq u ently in theories. In fact they

a rc

both

ex­

pressions of group structure, having two non-identical operations ; th u s they have a formal equivalence.8

Formalized Music

1 86

repeated Greeks," the y say, " had descending scales i n s t e a d of the a sc e n d i n g ones we have today." Yet there is no trac e of this in either Aristoxenos or h i s s uccessors, i ncluding Quintili anos9 and Aly pi os , who give a new and ful l e r version of the steps of many of the trop es. On the con trary, the a n ci e n t writers always begin their theoretical explana� tions and nomencl a tu re of the steps fro m the b ottom . Another bit of foolish ­ ness is the s u pp ose d Aristoxenean scale, of wh ich no trace is to be found in his text. 10 by

There

is a h are-bra i n ed notion t h a t h as been sanctimoniously

musicologists

Structu re of

to

in

rece n t times. " Th e

Byzantine

M us i c

Now we shall look a t the structure of Byzanti n e mu s i c . I t can contribute

an i n fi n i tely better understan d i ng of ancient music, occidental plain­

traditions, and the d i alectics of re ce n t Euro­ its wrong turns and d ead- ends . It can also serve to foresee

chant, non -Euro p ean musical

pean music, with

and constr u ct the fu ture from a view com manding the remote landscapes

of the

past as well as the

electronic

fu ture. T h u s

new

d irections of research

value. By contrast the deficiencies of serial music in certai n d o m a i n s and the d a m a ge it h as done to musical evolution by its ignora n t dogmatism w i l l be i ndirectly expose d .

w o u l d acq uire t he i r full

m e a n s of calculation, the and the Aristoxenean, the mul tiplicative and the additive . l l The fo urth is expressed by the ratio 3/4 of the monochord, or by the 30 te m p e re d segments ( 7 2 to the octave) .12 It contains three kinds of tones :

Byzantine music am algam a tes the two

Pythagorean

1 2 segments) , minor ( 1 0/9 or 1 0 se gm e n ts ) , and minim al ( 1 6( 1 5 or 8 segments) . But smaller and larger intervals are constructed and the elementary u n i ts of the primary order are m o re complex than in

major (9/B or

Aris toxenos. Byzantine music gives

diatonic scale

a

preponderant role

to

the

na tural

( th e s u pp o s ed Aristoxenean scale) whose ste ps are in the follow­

note : 1 , 9/8, 5/4, 4/3, 27/ 1 6, 1 5/8, 2 (in segments 0, 54, 64, 72 ; o r 0, 1 2, 23, 30, 42, 54, 65, 72) . The degrees of this scale bear th e al p h abe ti cal names A , B, r, !1, E, Z, and H. fl. is the lowest n ote and corresponds rou ghly to G2• This scale was propounded at least as far back as the first century by Didymos, and i n the second century by P to lemy , who perm u te d one term and recorded the shift of the tetra­ chord (tone-tone-semitone) , which has re m a i ned unchanged ever since.1a B u t apart from this dia p as o n (octave) attraction, the musical archi tectu re is hierarchical a n d " nested " as in Aristoxenos, as follows :

ing ratios to the fi rs t

1 2, 22,

3 0, 42 ,

A. The

primary order

is b as e d on the three tones 9/8, 1 0/9, 1 6 / 1 5,

a

To w a r d s a

supermajor

187

Metamusic tone 7{6, the trihemitone

6/5 , another

major tone 1 5 / 1 4, the

s em i to n e or leima 2 56/243 , the ap ot om e of the minor tone 1 35 / 1 28, a n d fi n a l l y the comma 8 1 / 8 0 . This complexity results from the mixture of the two means of calculation .

B. The secon dary order con sis t s of the tetrachords, as defined in Aristox­ os, and similarly the pent achords and the octochords. The tetrachords are divided i n to th r e e genera : 3 0 seg­ 1 . Di atonic, subd ivided into : first scheme, 1 2 + 1 1 + 7 ments, or (9/8) ( 1 0/9) ( 1 6/ 1 5) 4/3, st art i n g on 6., H, etc ; second s ch e m e , 1 1 + 7 + 12 30 segm e n ts, or ( 1 0/9) ( 1 6/ 1 5) (9/8) 4/3, starti ng on E, A , etc ; third scheme, 7 + 1 2 + 1 1 30 segm e n ts, or ( 1 6/ 1 5) (9/8) ( 1 0/9) 4/3, starting on Z, etc. Here we notice a developed combinatorial m e t h od that is n o t evident in Aristoxenos ; only t hr e e of the six possible permu tations of the three notes a r c u s ed . 2 . Chromatic, subdivided into : 1 4 a . soft chromatic, derived from the d iat o n i c te trach ords of the first s ch e m e , 7 + 16 + 7 3 0 segments, o r ( 1 6f 1 5) (7f6) ( 1 5/ 14) 4/3 , starting on 6., H, etc. ; b . s yn tonon , or h a rd chromatic, derived from the d i a t o nic tctrachords of the seco n d scheme, 5 + 19 + 6 30 segments, or ( 2 5 6/243 ) (6/5) ( 1 35 / 1 28) = 4/3 , starting on E, A , e tc. 3. Enharmonic, d erived fro m the diatonic by alteration of the m ob i l e notes and s u bdivided into : fi rst scheme, 1 2 + 1 2 + 6 30 segments, or (9/8) (9/8) (256/243) 4/3, starting on Z, H, r, etc. ; s e con d scheme, 12 + 6 + 1 2 30 segmen ts, or (9/8) (256/243) (9/8) 4/3, starting on 6., H, A , etc . ; third scheme, 6 + 1 2 + 1 2 30 se g ments, or (2 56/243) (9/8) (9/8 ) 4/3, starting on E, A, B, etc. en

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

PARENTHESIS

sec a phe no m e no n o f abs o rp t i o n o f the a n c i e n t enharmonic h ave taken pl ace during t h e first c en t ur i es of Ch risti anity, as part of the Church fathers' struggle a gai n s t paga n ism and We c an

by t h e d i a ton i c . This must

certai n of its manifestations i n the sidered sober, severe,

and noble,

arts.

The d i ato n i c h a d always been con­

u n l i ke the o t h e r types. I n fact the chro m atic

especially the enharmonic, d e m a nde d a more advanced m u sical as Aristoxcnos and t h e other th eoretic ians had a l ready po i n t e d o u t, and such a c u l ture was even scarcer amon g the m asses of the Rom a n period . Conseq u ently combinatorial specul ations on the o n e h a nd a n d pra c t i cal usage on the other must have caused t h e spec ifi c ch a racteristics of t h e en­ harmonic to disappe ar i n favor of th e chrom atic, a subd ivision of wh ich f"c ll genus,

and

c u l t u re ,

Formalized Music

1 88 away in Byzantine music, and of the syn tonon diatonic.

This

phenomeno n

of a bsorpti o n is com parable to that of the scales (or mod es ) of the Renais­ sance by the m aj o r diato n i c scale, which pe rpetu a t es th e a n c i e n t syntonon diatonic.

However, this sim p l i fi cation is curious and i t wo u l d be i nteresting to study the exact circums tances a n d c a uses. Apart fro m d i ffe re nces, or rathe r

is b u i l t strictly o n the definitions which s of the Aris toxenean ys te m s ; this was

vari a n ts of an cient interval s , Byza n t i n e typology

anci ent. 1t b u i l ds

up the

next stage w i th tetrachord s, usi ng

sin g u la rl y shed l i g ht on the

expo u n ded in s om e detail

theory

Ptolcmy. 1 5

TH E SCALES

help a:f and asso­ n a nce (paraphonia) . In Byzantine music the principle of iteration and j uxtaposi tion of the system leads very clearly to scales, a development which is still fairly obscure in Aristoxenos and his su ccessors, except for Ptolemy. C.

The tertiary order

by

systems havi ng the same

c o nsi st s

a n c ie n t

of t h e scales constructed with t h e

rules of co n son a n c e , dissonance,

Aristoxenos seems to have seen the system as a category and end in itself, and the co n c e p t

of the

scale did not emerge independently from

the method

which gave rise to it. In Byzantine m usic, o n the o ther hand, the system

was

It i s a s o rt of iterative operator, which starts from the lower category of tetracho rds and their derivatives, the pentachord and th e octochord, and builds up a c h ai n of more complex o rganisms , in the same manner as chromoso mes based on genes. From this point of view, system-scale co u p li n g rea ch e d a s t a ge of fulfi l lm e n t that had been unknown in ancient times. The Byzantines defined the system as the simple or multiple rep et iti o n of two, several, or all the notes of a scale. " Scale" here means a succession of notes that is already organized, such called

as

a method of constru cting scales .

the tetrachord or its derivatives. Three systems are used in Byzantine

m usic :

the octachord or dia pason

the pe n ta chord or wheel ( t rochos)

the

tetrachord

or

triphony.

(synimenon) or disjunct juxtaposition of two tetra­

The system can unite elements by conjunct

(diazeugmenon)

juxtaposition. The d isj u n c t

dia paso n scale spanning a perfec t octave. pason leads to the sca l es and modes with which we arc fam iliar. T h e conj unct juxta­ p osi t i o n of seve r al te trachords ( tri p h o n y) prod u c es a scale in which the

chords one tone apart form the The conjunct juxtaposition

of several of these perfect octave dia

Towards a Metamusic octave is

no

1 89

longer a fixed sound in the tetrachord bu t one of

its mobile

sounds. The same applies to the conj u n c t j uxtaposition ofs eve ral pentachords ( trochos) .

The system

can

be

applied to

the three genera of tetrachords and to

each of their subdivisions, thus creating a very rich collection of scal es. Finally one may even

mix the genera of tetrachords in the same scale ( as i n i n a vast v a r i e ty. Thus the scale

t h e selidia o f Ptolemy) , which will result

order is the product of a combinatori al method-indeed, of a gigantic

are and their derivatives. The

montage (harmony)-by i te rative j uxtapositions of organisms that a l re

a dy

scale as

s trongly d i fferentiated , the tetrachords

it is

c o n ce p t i on

defined here is a richer and m o re universal

t h e impoverished

conceptions

modern

of medieval and

point of view, it is not the tempe red scale so much

as

than all

times. From t h i s

the a b so r p t i o n by the

diatonic tetrachord (and its corresponding scale) of all the o ther com binations or montages

(harmonies) of the

other tetrachords that represen ts a vast loss

of potential. (The d i a to n i c scale is derived

from

a d i sj u n ct system of two

diatonic te trachords separated by a whole tone, and

is

represe n ted by t h e

white keys on the piano.) It is t h i s potential, as much s e nso ri a l as a bstract, that we are seeking here

seen.

to

reinstate, a l b e i t

in

a

modern way, as will be

The following are examples of scales i n segmen ts of

Diatonic

scales.

1 2, 1 1 , 7 ; 1 2 ;

Diatonic t e tra cho rd s :

1 1 , 7, 1 2 , starting

starting on the lower

H or A ;

I I ; 7, 1 2, I 2, I l , starting

on

tem­

on the lower fl, 1 2 ,

1 1, 7; 12;

1 2,

1 1,

7,

system by tetrachord and pen tachord , 7 , 1 2,

the lower Z; wh eel system ( trochos ) ,

1 2 ; 1 1 , 7, 1 2, 1 2 ; 1 1 , 7, 1 2, 1 2 ; etc. Chromatic scales. Soft

Byzantine

is eq ual to 30 segments) : system by disjunct terrachords,

pering (or Aristoxenean, since the perfect fourth

chromatic tctrachords :

I I , 7,

1 2,

wheel system start i n g on

If, 7 , 1 6, 7, 1 2 ; 7 , 1 6, 7, I 2 ; 7, 1 6, 7, I 2 ; e tc. Enharmonic scales. E n ha rmon i c tetrachords, second scheme : sys tem by disj u nct te trachords, s tarti ng o n fl, I 2, 6, 1 2 ; I 2 ; 1 2 , 6 , 1 2 , corres ponding to the mode produced by all the white keys starting with D. The e n h a rmonic

scales p r od u ced by the d i sj u n c t system form a l l the ecclesiastical scal es or

m od es of the Wes t, and others, for exam p le : ch romatic tctrachord , first scheme, by the triphonic s ys te m , starting on low H: 1 2 , 1 2, 6; 1 2, 1 2 , 6 ; 1 2,

I 2 , 6 ; 1 2 , 1 2 , 6.

Mixed scales. Diatonic tctrachords, fi rs t scheme disjunct sys te m , s tart i n g on l o w H, 1 2 , l I , 7 ; 1 2 ; 7, 1 6 ,

+

soft c h rom ati c ;

7 . Hard chro m a tic

te trac hord + soft chromatic ; di�junct system , s tarti ng on low If, 5 ,

12; 7,

1 6,

7;

etc. A11 the montages arc not used, a n d

I 9, 6 ;

one can observe

the

1 90

Formalized Music

p h e n o me n o n of the abso rption

by

of i m perfect oc taves by the perfect octave of the basic rules of consonance. Th is is a l i miting condition. D . The quaternary order consists of the tropes o r echoi (ichi) . The echos

virtue

is defined by :

the genera of tetrachords (or deriva tives) the system of j uxtaposition the attractions the bases or fundamental notes the domin ant notes

the termini or cade nce s

We

constitu ting i t

(katalixis)

the api c h i m a or melodies i n troducing the m o de the e t h o s, which fol l o w s ancient defini tions. shall not concern ou r s e l ve s with the d e tai l s

"of

order.

this

qua ternary

Th us we have succinctl y expou nded our analysis of the outside-time

structure of Byzantine music.

But

this

outside-time

TH E M ETA B O LA E

with a compart­ free circulation between the kinds of tetrachords, between the

structure could not b e satisfied

mentalized hierarchy. It was necessary

to

have

notes and their subdivisions, between the genera, between the systems, and between the echoi-hence t he n ee d for a sketch of the i n-time s tructure, which we will now look at briefly. There exist operative signs which allow alterations, transpositions, mod ulations, and other transformations (metabolae) . These signs are the phthorai and the chroai of notes, tetrachords, systems (or scales) , and echoi. Note metabolae

The

metathesis :

transition

from

a

tetrachord

fo u rth ) to an other te trachord of 30 segm ents.

of 30

segments (perfect

of the i n terval corresponding to the 30 into a larger in terval and vice versa ; or again, distorted tetrachord to another d i s to rte d tetrachord.

The parachordi : distortion

segments of

a

tetrachord

tra nsi ti on from one

Genus

Metaholae

Ph thora characteristic of the genus,

Changing note names

Using

not

changing note n am e s

the parach o rdi chroai.

Using the

System metaholae

Transition from one system to another

us

i ng

the

above

metabolae.

191

Towards a Metamusic tions

Echos metabolae using speci al signs, the martyri kai of the mode i n i tializatio n .

Because o f t h e

com p lex i ty

phthorai o r al tera­

of the metabolae, pedal notes (isokratima)

cannot be " trusted to the ignorant." Isokratima const i tutes an art i n itsel f,

for its functio n is to emphasize and pick o u t all the i n-time fluctu ations of the outside-time structu re that m arks the music. F i rst Com ments

I t can easily be seen that the consum m a tion of t h is outside-time struc­

ture is the most complex and most refined thing that could be invented by monody. What cou l d not be d eveloped in po l yph on y has been brought to such luxu riant fruition tha t to become familiar with it req u i r e s many years of practical studies, such as those followed by the vocalists a n d instru menta­

lists of the high cultures of Asia. I t seems, h owever, that none of the special­

ists in Byzantine music recogn ize the importance

of this structure.

attention to such an extent that they h ave ignored

It wo u l d

has claimed t he i r the livin g tradition of the

appear that in terpret i n g ancient systems of notation

Byzantine Church and h ave p u t th eir names to incorrect assertions. Thus it was o n ly a few years ago th a t one of them16 took the line of the Gregorian s p ecialists in attributing to the echoi characteristics other than those of the oriental scales which h�d been taught them in the conformist sch ools. T h ey have finally discovered that the echoi con t a i ne d certain characteristic m e l ­ odic formulae, t h o u gh of a sedimentary nature. But they have n o t been able or wil l i n g to go fu rther and abandon their soft refuge among the m anuscripts. Lack of understanding

of ancient m u s ic , 1 7 of both Byzantine and Greg­

orian origin , i s doubtless caused b y the blindness resul ting fro m the growth

a high l y origin al invention of t h e barbarous and uncultivated Occid ent fol l owi n g the schism of the chu rches. The passing of centuries and

of polyph on y,

the disappearance of the Byzantine s t a te have san ctioned this n egl e c t and this severance. Thus the effort to feel a " h armonic" language that is m u ch more refined and complex t h a n that of t h e syntonon d i a to n i c and its scales in octaves i s perhaps beyond the us u al ability of a Western music speci alist, even though the music of our own day may h ave been able to liberate him partly from the overwhelmi ng d om i n a nc e of di atonic th i nk i n g . The only exceptions a re the specialists in the m us i c of the Far East, 1 8 wh o h ave always rem a in ed in close co n tac t with m usical practice and, d e a l in g as they were with living music, have been a b l e to look for a harmony other than the t o n al h armony with t we lv e semitones. The heigh t of error is to be found in the transcri p tions of Byzan tin e melodics 1 9 into Western notation using the tempered system . Thus, thou sands of transcribed melodies arc completely

1 92

Formalized Music

wrong !

But

the real criticism one must level at the Byzanti nists is that

remaining aloof from the great musical tradition

have i gnore d the existence of this abs tract and sensual arch itecture,

complex and remarkably interlocking and gen u ine achievement

re t ard ed the

p rogre ss

of th e

in

of the eastern ch urch , they

both

(harmonious) , this d eveloped remnan t

Hellenic tradition. In t h is way they have

of m usicological

re s e a r

ch

in the a re as of:

antiquity

plain chant

folk music of European lands, no t a bly in

the East20

musical cultures of the civilizations of other continents better understanding of the mu sical ev ol u ti on of Western Europe from the middle ages up to the modern period the syntactical prospects for tomorrow's m u si c , i ts enrich ment, and its survival. Secon d Co mme nts

I am mo t i v a t e d to p re s e n t

this architecture, which is linked to antiq uity

a nd dou btless to other cultures, because it is an elegan t and l i vely witness

to what

I have

tried to defi ne

as an ou ts i d e - t i m e category, al geb ra, or struc­

ture of music, as opposed to i ts o t h e r two c a tego ri es , in-time and tem p o ral .

It has often been said (by Stravinsky, Messiaen, and others) that in music

time is everything. Those who express this view fo rge t the basic structures

on

which personal languages, such as "pre- or post-Webernian" s e ri a l music,

rest, however simplified they may be.

past and pres ent ,

In

o rd er

to

u n d er st a n d the u n ivers a l

as well as prepare the fu ture, it is

structures, architectures,

and

n e c es s a ry to

distinguish

sound organisms from their te m p ora l manifes­

tations. It is the re fo r e necessary to t ake "snapshots, " to make a series of veritable tomogr ap h i es over time, to compare the m and bri n g to light their re la tion s and architectures, a n d vice versa. In a d d i t i o n , thanks to the metrical nature of time, one can fu rnis h it too with an outside-time structure, leaving i ts tr u e , unadorned nature, that of immediate reality, of instan­ taneous b e co m i ng, in the fina l a n a l ysi s , to the te m po ra l category alone. I n this way, ti me could be c o n s i d e red as a blank blackboard, on which sym bol s and relationships, arch i tectures and abstract orga n i sms arc in­ sc ri b ed . The clash between organisms and architectures a n d instantan­ eous immediate real i ty gives rise to the pri mo rd ia l quality of the living

consciOusness.

The architectures of Greece and Byza n t i u m are concerned with the

pitches (th e d o m i n a n t character

of the simple sound) of sou nd

entities.

193

Towards a Metamusic Here rhythms

are also

subj ected t o

an organ ization, but

a

much si mpler

o n e . Therefore we shall not refer to it. Certainly these ancient and Byzantine

models cannot serve as examples to be imitated or copied, but rather to

exh ibit a fu ndame n t a l outsid e-time architecture which has been thwarted by the temporal arch itectures of modern ( post-medieval)

polyphonic music.

These systems, including those o f serial m usic, are still a somewhat confused magma of temporal and outside- time structures, for

no one h as yet though t

of unravelling th e m . However we cannot do this here.

P r o g ress i ve Degra d ati o n of O utside-Time St ructures

The tonal organ ization that has resulted from ven turin g into polyphony and neglecting the ancients has leaned strongly, by v i r t ure of its very nature,

on the te mporal category, and defi ned the hierarchies

of

its harmonic

fu nctions as the in-time category . O u tside-time is appreci ably poorer, its "harmonics" being reduced to a single octave sc al e

bases

C and A ) ,

(C m aj o r

corres p onding to the syntonon diatonic

on the two

of the Pyth agore a n

tradition or to the B yz a n tin e enh armonic scales based on two disjunct tetrachords of the fi rst scheme (for C) and on two disj unct te trachords of the

second and t h i rd

scheme (for A ) . Two metabolae have been preserved : that

of transposi tion (sh i fting of the scale) and that of modulation, which consists of

transferring the b ase onto s teps of the same scale. Another loss occurred

with the adoption of the crude temperin g of the semi tone, the twelfth root of two. The consonances have been enriched by the interval of the third,

which, until Debussy, had nearly ousted the traditional perfect fourths and fifths.

The

final s tage of the evolu tion, at on a l i sm,

prep ared by

the theory

and music of the romantics at the end of the nineteen th and the beginning

twentieth centuries, practically abandoned all outside-ti me structure. This was end orsed by the dogm atic suppression of the Viennese school, who accepted only the ultimate total time ordering of the tempered chromatic scale. Of the fou r forms of the series, only the i nversion of the in tervals is rel at ed to a n o u tside-time structure . Naturally the loss was felt, consciously o r not, and symmetric rel ations between i n tervals were grafted onto the

of the

chromatic total i n the choice of the notes of the series, but these always

remained in the in-time category. Since then the situation

has barely

c h a nged in the music of the post-Webcrnians. This degrad ation of the

ou tside-time structures of m u si c since

late m edieval times is p e rh a ps the and

most characteristic fact about the evolutio n of Western Eu ropean music,

it has led to an unparalleled excrescence of temporal and in-time structures. In this lies its originality and its contribution to the universal culture. But herein also lies its impoverishment, its loss of vitali ty, and al s o an apparent

Formalized Music

1 94

h as thus far developed, European music with a field of express i on on a planetary scale, as a universality, and risks i so l a t in g and severing i tse l f from historical necessities. We must o p e n o u r eyes a n d try to build bridges t o wa rd s other cu ltu re s , as well as towards the i mmediate fu ture of m u sical th o ugh t, before we perish suffocating from electronic tech nology, either at the instrumental risk of reaching an impasse. For as it

is

ill-suited

or

level

to providin g

the

world

at the level of composition by compu ters.

R e i n t r o d u ct i o n of t h e O u ts i de-Time Structu re by S t o c h a st i c&

the

By

the

introd uction of th e calculation of probability (stochastic

music)

presen t small h o rizo n of outside-time structures and asymmetries

was

completely explored and enclosed. B u t by the very faQ: of its introduction, stochast i cs gave an i mpe tus to m u s i c al thou g h t that c a r ri ed it over this

e n c l osure towards the clouds of sound events and towards the plasticity of

l arge

numbers articul ated statistically. There was no lon ger any disti nction

between the ve r tic a l and the

horizontal, and the indeterminism

of in-time

structures made a dignified entry i n to the musical edifice . And, to crown t h e Herakleitean dialectic, indeterm i n ism,

fu n ctio ns ,

took on

of organ ization.

by

means of particular stoch astic

color and s t r u c t u re , giving rise to generous possibilities

to indude in i ts scope determinism and , still structures of the past. The c a tego ries t empor a l , unequally amalgamated in the history

It was able

som ew h a t vaguely, the o u ts id e - t i m e

outside-time,

in-time,

and

of music, have suddenly taken on all th eir fundamental significance and for the first time can build a

p resen t ,

and

c oh eren t

and u n ivers al synthesis in the past,

fu ture . This is, I insist, no t only a poss ibility, b u t

even

a direc­

tion having priority. But as yet we have not m an aged to p roceed beyond this stage. To do so we m us t add to our arsenal sharper tools, trenchant axio ma tic s and formalization.

SI EVE TH EO RY It is n ecess ary to give an axiomatization

ture (additive group structure

=

for

the totally

o rd ered struc­ the

addit ive Aristoxenean s tru c t u re ) of

tempered chromatic scale . 21 The axiomatics of the te m p e r ed chromatic

s c ale is b as e d on Peano's ax i om a tics of numbers : Preliminary terms. 0

=

the

stop at the ori gin ;

resul ting from elementary displacement of n ; D

=

n

= a stop ;

n

'

=

a s top

the set of values of the

particular s o u n d characteristic (pitch, d e ns ity , i n te nsi ty , instan t, speed, disorder . . . ). The val u es are iden t i c al with the stops of th e

displacements.

Towards

195

a Metamusic

First propositions (axioms) .

element of D. of D th en the new stop n' is an element of D. 3. If s tops n and m are clemen ts of D t h e n the n ew stops n' and m' are ide n ti c a l if, and only if, st ops n and m are iden tica l . 4. If s top n i s an element of D, it will be different from stop 0 at the origin. 5 . If elements belongi ng to D have a sp ecia l property P, such that stop 0 also has it, and if, for every clement n of D having this property the element n' has it also, all the elements of D w i l l h ave the p ro per ty P. We h ave j u s t defined axiomatically a tempered c h r o m at i c scale not only of pitch , but also of all the sou n d properti es or ch aracteristics referred to above in D (density, intensity . . . ) . Moreover, this abstract scale, as Bertrand R usse l l has rightly observed, a propos th e axiomatics of numbers ofPeano, has no unitary displacement that is e it h e r predetermined or related to an a b so l u te size . Thus it may be constructed w i th tempered semitones, with Aristoxenean segments (twelfth-tones) , with the comm as of Didymos (8 1 /80) , with q u a rter-tones, wi th whole tones, th i rds , fourths, fifths, octaves, e tc . o r with an y other u n i t that is not a fac tor of a perfect octave. Now let us define a n o th er eq uivalent scale based o n this one but having a unitary d isplacement which is a m u lti p le of the fi rs t. It ca n be expressed by t h e concept of congruence modulo m. Definition. Two i n tegers x and n are said to be congruent modulo m when m is a factor of x - n . I t may be expressed as follows : x = n (mod m) . Thus, two integers are congruent m od u lo m wh e n and only when they differ by an exact (posi tive or ne g at ive) multiple of m ; e.g., 4 = 19 ( m o d 5) , 3 = 1 3

1.

S top

0

2 . If stop

is an

n

is an element

(mod 8) , 1 4 = 0 (mod 7) . Consequently, every i n te ge r only one value of n : n

Of each

m ; they n (mod

to

are,

m} is

For

a

(0, l , 2 , . . . , m - 2 ,

modulo m with m

one

and with

- 1).

o f these num h e rs i t is said that i t forms a residual class modulo in fact, the smallest non-negative residues modulo m. x = thus equivalent to x = n + km, wh ere k is an integer. kEZ

given n and

the residual class In

=

is congru e nt

n

=

{0,

for any k e

modul o

m.

± l, ±

Z, the

2,

± 3,

. . . }.

numbers x will belong by defini tion

This class can be

order to grasp these ideas in terms of music,

denoted mn. us take the te m pe red

let

Formalized Music

1 96 semitone of our present-day scale as

the

unit

of displ a cement.

shall again apply the above axiomatics, with say a value of

To this we 4 semitones

(major third) as t h e elementary d i splacement.22 We shall define a new

chromatic s c al e . If the stop at

the origin of the first s c ale is a D:jl:, the second multi p les of 4 s e m i tones, i n other words a " s c a l e " of m aj or thirds : D:jl:, G, B, D':jl:, G', B' ; these a re the notes of the first scale whose order numbers are congruent with 0 mod u l o 4. They all belong to th e

sc a l e will give us all the

residual cl ass

modulo 4. The residual classes I , 2, and 3 modulo 4 will use

0

up all the notes of this

the following manner :

c

h ro m ati c total. These classes

residual class 0 modulo residual class

I

m ay be re presented

4 : 40

modulo 4 : 41

residual class 2 modulo 4 : 42 residual class 3 modulo 4 : 43 re s id u a l class 4 m o d u lo 4 : 40, Since

we

placement by

certain

semitone) ,

one

sieving o f the basic scal e (elementary dis­

continuity to pass through. By extension

will be represented as sieve

be given by sieve 5n, in which

etc.

each resid ual class forms a sieve allowing

elements of the chromatic

the chromatic t o t a l

will

a

arc dealing with

in

n

=

0, I , 2, 3, 4.

1 0 • The scale of fourths will

Every change of the index

n

entail a transposi tion of this gamut. Thus the D e b u ss i a n whole-tone

scale, 2n with

n

=

0, 1 , 20

has

-;.-

two

transpositions :

C, D, E, F#, G#, A#, C

21 -;.- C:jl:, .D#, F, G, A, B, C:j!: S tarting from these elementary

sieves

we

· · ·· · · · ·

can build more complex

scales-all the scales we can imagine-with the help of the three operations of t h e Logic of Classes :

u n i on

(disjunction) expressed as v , intersection

(conj unction ) e x p re ssed as A , and complementation ( negation) expressed

as a bar inscribed over the modulo of the sieve. Thus

20

20

20

v 21

A

21

1 0)

=

chromatic total (also expressible as

=

no notes, or em pty sieve, expressed as 0

= 2 1 an d

21

=

The major scale can be

20•

written as follows :

a

Towards

1 97

Metamusic

By definition,

this notation do es not distinguish between all the modes

on the white keys o f

the

piano, fo r what we are d efining here is the scale ;

modes are the architectures founded on these scales. T h u s the wh ite-key

D, will have the same n otation as t h e C mode. But in distinguish the modes i t would be possible to introduce no n ­ commutativity in the logical expressions. On the other hand each of the 12 transpositions of this scale will be a combination of the cyclic permuta­ tions of the indices of sieves modulo 3 and 4. Thus the major scale transposed a semitone higher (shift to the right) will be written

mode D, starting on

order t o

and in

general

(3n + 2 n

where

II

4n)

V

(§n + l

1\

4n + 1 )

V

(3n + 2

1\

4n + 2)

V

(Sn

can assume any v al ue from 0 to 1 1 , but reduced after the addition

of the sieves ( modul i ) , modulo the of D transposed onto C is written

of the constant index of each

ing

4n + 3) ,

II

sieve. The scale

(3n

1\

4n)

V

(3n + l

4n + l)

II

M usico l ogy Now let

us change

(3 n

V

II

4n + 2)

V

(3n + 2

II

correspond­

4n + 3) .

the basic unit (elementary displacement ELD )

the si e v e s and use the quarter-tone . The

maj or

scale will be written

3n + l) V (8 n + 2 1\ gn + 2) V (8n + 4 1\ 3n + l) V ( 8 n + 6 II Sn) , with n = 0, I , 2, . . . , 23 (modulo 3 or 8) . The same scale with still sievi n g (one octave 72 Aristoxenean se g ments) will be written (B n

1\

of

finer

=

( 8n

II

(9 n

V

9n + s) )

V

V

( 8n + 2 (8 n + 6

II

II

(9n + 3

(9 n

V

V 9n + s) ) 9n + a) ) ,

V

( 8n + 4

II

9n + 3)

0 , 1 , 2 , . . . , 7 1 (modulo 8 o r 9) . of the mixed Byzantine scales, a d isj u nc t system consisting of a ch ro m atic tetrachord and a diaton i c t etr a c h ord , second scheme, separated by a m aj o r tone, is no t a te d in Aristoxenean segments as 5, I 9 , 6 ; 1 2 ; 1 1 , 7, 1 2 , and will be transcribed logically as

with

n

=

One

( 8n

with

n

=

(9n + 6 ( 8n + 5

V

9n + s))

V

0, l , 2 ,

. . ., 7 l

(modulo

II

(9n

V

II

II

8

(8n + 2 ( 9n + 5

or

9) .

V

V

8 n + 4) ) 9n + s))

V

( 8n + 6

V

9n + 3) ,

198

Formalized Music

The Raga Bhairavi of the Andara-Sampurna type (pentatonic as­ heptatonic descending) , 2 3 e x p re sse d in terms of a n Aristoxenean basic sieve ( comp r i s i n g an octave, periodicity 72 ) , will be written a s : Pentatonic scale : cending,

( 8,.

(9,.

A

9n + 3 ) )

V

Heptatonic scale :

(Bn

A

(9n

V

9n + a))

V

V

V

(8n + 2

A

(9n

V

9n + S) )

(8n + 6

V

V 9n + s) ) V ( 8 n + 4 (9,. + 3 V 9 n + 6) ) (modulo 8 or 9) .

(8n + 2

( 8n + S

(9n

A

1\

A

9 n + 3)

A

( 9,. + 4

V

9n + s) )

n = 0, 1 , 2, . . . , 7 1 These two scales expressed in te rm s of a sieve having as its elementary displacement, ELD, the comma of Didymos, ELD 8 1 /80 (8 1 /80 to the power 55.8 2), thus having an oc t ave period ici ty of 56, will be written a s : Pentatonic scale : with

=

=

(7n

A

(7,.

1\

(8n

V

Bn + a) )

V

8,. + 6) )

H e ptato n ic scale :

(8,.

V

V

V

(7n + 2

(7,. + 2 (7n + 4

A

A

A

(Bn + 5

(Bn + S (8 n + 4

V

V

V

8n + 7 ) )

8n + 7) )

8 n + a) )

V V

V

( 7n + 5

A

Bn + l )

(7n + 3

A

8,. u) 8n + l )

(7n + 5

A

= 0, 1 , 2, , 5 5 (modulo 7 or 8) . We have j ust seen how the sieve theory allows us to express- any scale in terms oflogical ( he n ce mech anizable ) functions, and thus unify our study of the structures of su p erior ra n ge with that of the total order. It can be useful in entirely new constructions. To this end let us i m agi ne complex, non-octave-forming sieves.24 Let us take as our sieve unit a tempered quarter-tone. An octave co n ta i ns 24 quarter-tones. Thus we have to con­ st ru ct a compound sieve wi th a periodicity other than 24 or a multiple of 24, thus a periodicity non-congruent with k · 24 modulo 24 (for k 0, l , 2, . . . ) . An example would be any logical function of the sieve of moduli 1 1 and 7 (periodicity I I x 7 'J7 =1 k · 24) , ( 1 1 ,. v 1 1 ,. + 1 ) 1\ 7n + G· This establishes an asymmetric distribution of thc steps of the chrom atic quarter­ tone scale. One can even use a compound sieve which throws periodicity outsi d e the limits of the audible area ; for example, any logical function of

for

n

.

•

.

=

=

modules

17

and 18 (1( 1 7, 1 8] ) ,

S u prastructure&

the

for

17

x

18

=

306

>

(ll

x

24) .

One can apply a stricter structure to a c om p ou n d sieve or simply l e av e choice of elements to a stochastic function. W e shall obtain a statistical

1 99

Towards a Metamusic

coloration of the chromatic t o t a l which

Using metabolae.

has

a

higher

level of complexity.

We know that at every cyc l i c combination of the

sieve

m od u l i of the As examples of m et a b o li c trans­

and

indices ( transposi tions)

at every ch ange in the module or

sieve (modulation) we o btain a m e t a bola. formations l e t us take the smallest res i d u es that are prime to a positive nu mber r. They will form an Abelian (commutative) group when the composition law for th ese resid ues is defined as mul tiplication with re duc ­ tion to the least positive 1·esidue with regard to r. For a numerical ex a m ple let r = 1 8 ; the residues 1 , 5, 7 , 1 1 , 1 3 , 1 7 arc primes to it, and their products after re d u c t io n mod ulo 1 8 will remain within this group ( c losu re ) . The finite comm utative group they form can be exemplified by the following fragment :

5

II

x

x

11

=

Modules I , 7, 1 3 form

logical expression

L (5 , 1 3)

17; 35 - 1 B 1 2 1 ; 1 2 1 - (6 x 1 8) =

7 = 35 ;

a

cyclic sub-group

=

1 3 ; etc.

of o r d er 3. Th e following is a 5 and 1 3 :

of the two sieves h avi n g modules =

( 1 3n + 4 1\

5n + l

V

l 3n + S

(5 n + 2

V

One can i mag ine a transformation

V

V

l 3n + 7

5n + 4)

V

1\

1 3n + 9) 1 3n + 9

of m od u les in

V

1 3n + 6•

p airs, starting from the

Abelian group defined above. Thus the cinematic d i a g r a m (in-time)

L( 5, 1 3) -* L( 1 1 , 1 7) -* L(7, 1 1 ) -* L ( 5, I) -* L(5, 5) -*

so as to return to the initial

This

sieve theory

·

·

·

wi l l be

__.,. L( 5 , 1 3 )

term (closure) .25

can be put into many kinds of archi tecture,

so as

to

create included or successively in tersecting classes, thus stages of increasing complexity ; in other words, orientations t ow a rd s increased in s e l ect i o n , and in topological

textures of neigh borhood.

Subsequently we can put into

of outside-time

music by means

determinisms

i n-time practice this veri table histology

of t em p o ra l fu ncti ons, for

instance by giving

functions of change-of indices, moduli, or unitary displacemen t-in other words, encased log i ca l functions

parametric with

time.

Sieve theory is very general a nd consequently is applicable to any o the r sound characteristics that may be provided with a to t al ly ordered structure, such as intensity, instants, densi ty, degrees of order, sp e ed , e t c . I have al­ ready said this elsewhere, as in the axiom atics of sieves. But this method can be applied equally to visual s ca l es and to the o p t i cal art s of the fu ture. Moreover, in the immediate fu ture we shall wi tness the-1���'7?P,·?(.

':oCl

' (' 1:1>

U n�·f- !; �; i�iA.il:ot t ! � �·

e:,...{

�:t?!?:��-�!!...� :."!1.��'-�::'.::'�·:':

Formalized Music

200 this theory and its widespread

use

with the

help of computers,

for

it is

be a study of partially o rd e re d structures, such as arc to be fou n d in the classification of timbres, fo r exam ple, by m e a n s of l attice or graph techniques. entirely mechanizablc.

Then ,

in a su bseq u ent stage, there will

C o n c l usion

I

believe that music today co u l d s u rpass i tself by research i n to has been atro p h i ed and dominated

side-time category, which

temporal ca tegory. Moreover this

method African,

the out­ by

the

can un i fy the expression of

and Eu ropean music. It has a considerable advantage : its mech anization-hence tests and models of all sorts can be fed into com p u ters, which will effect great progress in the

fundamental stru ctures of all Asian,

musical sciences.

In fact, what we are witnessing is

an

industri alization of music

which in

h a s already st ar t ed , wheth er we l i ke i t or n o t . It already floods our ears

many publ i c places,

shops,

rad i o , T V ,

and

a i r l i nes, t h e world over. I t

permits a consumption of music on a fantastic scale, never b e fo re approached.

But this m u sic is of the l owest k i n d , m ade from a collection of o u t d ated cliches from the dregs of the musical mind. Now it is not a matter of stopping this invasion, which, after all, increases participation in mu sic, even if only passively. It is rather a q uestion of e ffecting a qualitative conversion of this music by exercising a radical but constructive critique of our ways of think­ ing an d of making music. O nly in this way, as I have tried to show in the present study, will the musician su cceed in dom i n ating and transforming this poison that is disch arged into our ears, and only if he sets a b o u t it without further ado. But one must also en visage , and in the same way, a radical conversion of m usical educatio n , from primar y s tudies onwards,

t hro u ghout

the

entire world

(all national

Non-decimal systems and the logic

coun tri es, so why not their application to

sketched out here ?

councils

o f classes a

are

for

music

take note).

already taught i n certain

new musical theory, such

as is

C h a pter V I I I

Towards a Philosophy of Music

P R E LI M I N A R I E S " u n v e i li ng of t h e h i storical of mus ic,l and 2. to construct a m usic . " Reasoning " abo u t p h e n o m e n a and their e x p l a n a tio n was the greatest

We a re g o i n g to attempt briefly : l . an

tradition "

s tep accom plished by man in the course of his l i beration and growth . This

i s why the I on i a n pioneers-Thales, Anaximander, Anaximenes-must be

. considered as the s tarting p o i n t of o u r truest c u l ture, that of " reaso n . " " W h e n I say " reaso n , i t is n o t i n the s e n se of a l ogic a l sequence o f arguments,

syllogisms, or lo g ico-tech nical mechanisms, but that very extraord inary q u ality of feeling

an

uneasi ness,

a

curiosity, then of applying the question,

li.EyXo'>. I t is, in fact, i m po s s ibl e to im agi n e th i s advance , which, in Ionia,

created cosm ology from nothing, which

were

in

spite of religions and pow e rfu l mystiques,

early forms o f " reason ing. " F o r example, Orp h is m , wh ich so

is a fallen god, that its true nature, and that s a c r a men ts (opyw) it can reg a i n

i n fluenced Pyth agorism , taught that the h uman soul only

ek-stasis,

with t he

the depa r t u re from self, can reveal

aid of purifications

(Ka8apf-Loi) and Wheel oj'Birth ( T poxo >

i ts lost position and e s c ape th e

that is to say, the fa te of reincarnations

as an

yEvlaEw Moussorg sky ; and the Far East. "

This rebirth conti n u es ma g nificently

throu g h

Messiae n ,

with his

mod es of l i m i te d transpositions " and " non-retrogradable rhythms , " but

i t never imposes itsel f as a general n ecessity and

never

goes beyond

the

fram ework of the scales. However Messiaen himself abandoned this vein,

yielding

to

th e pressure of serial music.

In order to put things in their proper historical perspective, it is necessary to prevail upon more powerful tools such as mathematics and

go to the b ot to m of thi ngs, to the structure of musical t h o u gh t and I h ave tried to do i n C h apters VI and VII and what I a m going t o d eve lop in the analysis of Nomos alpha. Here, however, I wish to em p h asi z e the fact that i t was D e b u s sy and Messiaen11 in France who reintroduced the category outside-time in th e face of the general evolu tion that resul ted in i ts own atrophy, to the advan­ tage of structures in-time.1 2 In effect, atonality do e s away with scales and accepts the outside-tim e neu trali ty of the half-tone scale. W (This sit u at i on, fu rthermore, has s car ce l y changed for fifty years. ) The i n trodu ction of in-time order by Schonberg made up for this i m poverishment. Later, with the stochastic processes that I i n t ro d u c e d into musical composition, the hypertrophy of the category in-time became overwhelmin g and arrived at a dead e n d It is in this cul-de-sac that music, abusively called aleatory, improvise d , or graphic, is still stirring today. Questions of choice in the category ou tside-ti me are disregarded by m usicians as though they were unable to hear, and especially u na b le to think. In fact, they drift along unconscious, carried away by the agitations of superficial musical fashions which they u n dergo heedlessly. In depth,

logic and

composition. This is what

.

209

Towards a Philosophy of M usic however, the o u tside-time structures

do ex i s t and it is t h e privilege of man

not only to s u s ta i n them, but to construct them and to go b eyo nd them.

th e re are

Sustain them ? Certainly ;

will per mit

us

basic evidences of this order which

to inscribe ou r names i n the Pythagorean- Parmenidean field

and to l ay the platform from which our

ideas will build bridges of under­ aft e r all products of m i l l ions of

s tanding an d insigh t into the past (we are

years of the past) , i n t o the future (we are equally products of the future) ,

and into other sonic civilizations, so badly e xp l a i ned by the present-day

musicolog ies, for want of the o ri gi n a l

them .

tools that we

so graciously set up for

Two axiomaties will open new doors, as we shall see in the analysis

of

Nomos alpha. We sh a ll start from a naive position concernin g the perce ption of s oun d s , naive in Europe as well as in Africa, Asi a , or America. The inhabitants of all these countries learned tens or

hu ndreds of thousands of

years ago to d istinguish (if the s o u n d s were neither too long nor too short)

such c haracteristics as pitch , instants, loudness, roughness, rate of change,

color,

tim bre.

They are even a b l e to speak of the first

in terms of i n tervals.

The first axiomatics leads

us

th ree ch aracteristics

to the cons truction of all possible scales.

We will speak of pitch since it is m o re familiar, but the following ar g uments

to all characteristics which are of the same nature (instants, rou gh n ess , density, degree of disorder, ra t e of ch ange ) . We will start from the obvious as s um pti o n that wi thin certain limits men a r e able to reco g nize whether two modifications or displ acements of pit ch are identical. For example, going from C to D is the same as goi n g from F to G. We will cal l this modification elementary displacement, ELD. (It can be a comma, a halftone, an octave, etc. ) It permits us to define any Equally Tempered Chromatic Gamut as an ETCHG sieve. 1 4 By modifying the will relate

loud ness,

displacement step

ELD,

we engender a new ETCHG sieve with the same

axiomatics. With this m ate ri al we can go no farther. Here we introduce the three logical op e ra t i o ns (Aristotelean logic as seen by Boole) of conjunction

( " an d , " intersecti on, nota ted 1\ ) , di:dunction (" or," u n i o n , notated v ) , and negation (" no," complement, notated - ) , and use them to create cl asses of pitch (various ETCHG sieves) . The following

is the logi ca l expressio n with t h e conventions as indic ated

VII : The major scale (ELD

in Chapter

(Bn

where

n

1\

=

=

:! tone) :

3n+l) V ( 8n + 2 1\ 3 n + 2 ) V ( 8 n + 4 0, I , 2 , , 23, m od u lo 3 or 8 . •

.

.

1\

3 n + l) V (8n + 6

1\

Sn)

210

Formalized

(It i s po ss i b le to modify t h e step ELD by

logical function of the maj or scale with an be based on an ELD

=

1 /3

two s i e ve s , in turn, could

provi d e

be

tone or on

more c o m plex scales.

a " rational metabola."

ELD

a ny

Music

Th u s

the

equal to a quarter- tone can

o

o ther portion of a t ne . These

combined wi th the three logical o p erations to

Finally, " irrational metabolae " of ELD may

a pplied in non-instrumental music. ELD can be taken from the field of real numbers) .

be introduced, which can only be

Accordingly,

the

The scale of limited transposition no 4 1/2 tone) :

3n

A

of Olivier Messiaen15 (ELD

=

(4n + l V 4n + 3) V Sn + l A (4n V 4n + 2) V 4n + 3 V Sn + l A (4n V 4n + 2)

4n + l

n 0, 1 , . . . , modulo 3 or 4. The second axiomatics l e ads us to ve ctor spaces and graphic and n um eri c al rep resentations. 1 6 Two conjunct i n tervals a and b can be combined by a m us i c al operation to produce a new interval c. This o pe ratio n is ca lle d addition. To either an ascending or a descending interval we may add a s eco n d c o nj u n ct interval such that the result will be a u nison ; this second in terval is the symmetric interval of the first. Unison is a neutral interval ; that is, when it is added to any other interval, it does not modify it. We may a lso create intervals by association without changing the result. Finally, in composing intervals we can invert the orders of the intervals without ch anging the result. We have just shown that the naive experience of musici ans since antiqu i ty (cf. Aristoxenos) all over the earth attributes the s tr u c tu re of a commutative

where

=

group to intervals.

Now we are able to combine this group with

a

field

structure. At least

r

two fields are possible : the set of real numbers, R, and the i so m o phi c set of

to combine th e . Abelian group of intervals with the field C of com p l ex numbers or with a field of characteristic P. By definition the combination of the group of intervals with a field forms a vector space in the fo l l ow i ng manner : As we have just said, interval gro u p G possesses an internal law of composition, addition. Let a and b be two elements of the group. Thus we have :

points on

I.

a

2. a

3. a

4.

5.

a

a

a

+

straight line. I t is

b

+ b +

+

o

a

+ b

=

+ '

= =

=

c, c

cE

=

G

(a

o +

a,

b

a

o,

+

+

m o reo v er

b) + c

=

with

a

+

possible

(b

a'

=

+

c)

assoCiattvtty

G the neutral element ( u n i son ) - a t h e symmetric interval of a

with o E

=

commutativity

211

Towards a Philosophy of Music

field

in G with those in the the field of re a l numbers) then we

We notate the external composition of elements

have

C by a dot · . If >.. , p.

E

C

(where C

the foll o wi n g properti es : 6. A · a, f.L · a E G

7. l · a

=

a· I

=

a

multiplication }

(I

=

is the neutral element

m

p.·a}

8. A· (p.a) = (i\. · f.L) · a 9 . (i\ + P.) · a = i\ · a + A · (a + b) A · a + >.. · b =

C with res p ect to

associativity

or A,

p.

distributivity

M U S I C A L N OTATI O N S A N D E N C O D I N G S

The vector space structure

o f intervals o f certain

sound characteristics by

permits us to treat their elements mathematically and to express th em

the set of numbers, which is indispensable for dialogue by the set of points on a straight

convenient. The

line, graphic

two preceding axiomatics may

with

computers, or

expression often b ei ng

be applied to

all

very

sound charac­ m o m e n t it

teristics that possess the same structure. For example, at the

of a scale of t i m b re which might be univer­ the scales of pitch, instants, and intensity are. On the other hand, time, intensity, density (number of events per unit of time) , the quantity of order or diso rd e r (measured by e ntropy ) , etc . , could be put into

would not make sense to speak sally accepted

as

one - to - o ne corres p ondence with the set

points on a straight line . (See Fig.

Fig.

Vl l l-1

Pitches

I nsta nts

of real VIII-1 .)

num bers R and the

De nsities

Intensities

Mo reo ve r, the ph enome n on of soun d is

a

set of

D isord e r

correspondence of sound these axes. The simplest

characteristics and therefore a correspondence of

Formalized Music

212

by Cartesian coord i n ates ; for ex a m p le , the in Fig. VIII-2. The unique p o in t (H, T) corresponds to the sound

c o rres po n d e nce may be sh o w n two axes

that has a pitch H at the

t

r

instant

T.

· - - - - - - - - - - - -- - - - - - - - -

-