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
of QuesliiNU

PHYSICS, MATHBMATJCS, LOGIC

(fragmentation of the directions leading to creative knowledge, to philosophy)

C:A.U5A.LlTYj

AS A CONSRJ;tUENCB OP NEW MENTAL STRUCTURES j

REALITY

Families

(EXISTENTlALITY);

of Solulions or

PrO<ed�ts (of

INFERENCE: CONNBXtTY; COMPACTNESS;

the above

Piocts (examples of particular realization)

�

/�

INDETERMINISM • • •

categories)

STOCHASTIC

materialized by

FREE

a computer pro-

ST/10-1, 080262 ST/48-J, 240162 ATRiES

of Sonie Eltmen�s

with computers

• • •

UBIQ.UITY

DETERMINISM j

• •

�

�

MARKOVIAN

SPATIAL

GROUPS

GAMES

MORSIMA-AMORSIMA

A DUEL

ANALOGIQUE B ANALOCIQ.UE

SYR.MOS

STRAT f: GIE

AKilATA

NOMOS GAMMA NOMOS ALPHA

{sounds that are he ard and recognized as a whole, and classified with respect to their 50urces)

ORCHESTRAL, ELECTRONIC

MicrosouTids

bi-po}e-+

gram

ACHORRIPSIS

Classes

TEMPORAL AND

+-

and

(produced

by analogue devices),

digital-to-analogue converters) ,

,

• ,

CONCRETE

(microphone

collected),

DICITAL

(realized

und s belong or which Forms and structures in the pressure.. time space, recognition of the classes to which mi r oso microstructures produce .

at the Microsound types result from questions and solutions I hat were adopted

levels.

c

cATEGORIES,

PAMILJES, and PIECES

I

Preface

This book is

a

to

collection

Musiques Formelles

m us ical composition pu rsu ed in

of explorations in

several directions. The effort to reduce certain sound sens at i o n s , to

under­

their logical causes, to dominate them, and then to use them in wanted constructions; the effor t to materialize movements of thought through sounds, then to test them in compositions; the effor t to understand better the pieces of the past, by searchin g for an underlying unit which would be identical stand

with that of the scientific thought

while" geometrizing," that than

the impulse

of our

time; the effort to make "art"

is, by gi v ing it a

reasoned support less perishable

of the moment, and hence more serious, more worthy of

intelligence wages in a l l the other domains of the musical compositional act. This abstraction and formalization has found, as have so many other sciences, an unexpected and, I think, fertile support in certain areas of mathematics. It is no t so much the inevitable use of mathematics that characterizes the attitude of these experiments, as the overridin g n e e d to cons i der sound and music as a vast potential rese r voir in

the fierce fight which the human

-all these efforts have led to a sort of abstraction and formalization

which a knowledge of the laws of thought and the structured creations of thought may

find a

completely new medium of materialization,

communication.

For this purpose the qualification "beautiful"

s en se for sound,

or

nor for the music that derives from

i.e.,

of

"ugly" makes no

it; the

quantity of

intelligence carried by the sounds must be the true criterion of the validity

of a particular music.

This does not prevent the utilization of sounds defined

beautiful

according

to the fashion

of the

mo m e

nt ,

nor

as

pl ea s ant or

even their

s

their own right, which may enrich symbolization and al gebr atio n .

is

in itself a sign of intelligence. We are so

conv in ce d

ix

in

Efficacy

historical arts take an

of the

necessity of this step, that we should like to see the visual

t udy

X

Preface to

analogous path-unless, that is, "artists" of

done it in

a new

type have

laboratories, sheltered from noisy publicity.

These studies have

always

been matched

by

Musiques Formelles

actual works

not

already

which mark

out the various stages. My compositions constitute the experimental dossier of this undertaking. In the beginning my compositions and research were

and moral and material chapters in the present work reflect the results of the teaching of certain masters, su ch as H. Scherchen and Olivier Messiaen in music, and Prof. G. Th. Guilbaud in mathematics, who, through the virtuosity and liberality of his thought, has given me a clearer view of the algebras which constitute th e fabric of the chapter devoted to Symbolic Music. I. X.

recognized and

published, thanks

to the friendship

support of Prof: Hermann Scherchen. Certain

1962

Preface

to

the Pendragon Edition

Here is a new expanded edition of Formalized Music. It invites two fundamen­

tal questions:

Have the theoretical propositions

thirty-five years

which I have made over the

past

a) survived in my music?

b) been aesthetically efficient?

To the first question, I will answer a ge ne ra l "yes." The theories which I

in the various chapters preceding this new edition have p resen t in my music, even if some theories have b een mingled with others in a same work. The exploration of the co nc eptu al and sound world in which I have been involved necessitated an harmonious or even have presented always been

conflicting

synthesis

architectural view

procedures.

of earlier

tha n

a mere

theses.

It

necessitated

a

more global

comparative confrontation of the various

But the supreme criterion always

aesthetic efficiency of the music which resulted.

remained the validation, the

was up to me and to me alone to determine the aesthetic not, in virtue of the first principle which one can not get around. The artist (man) has the duty and the privilege to decide, radically alone, his choices and the value of the results. By no mea n s should he choose Naturally, it

criteria, consciously or

any other means; those of power, glory, money, ...

Each time, he mus t throw himself and his ch os e n

all while striving to stan fr om scra tch yet not "monkey" ourselves by virtue of the own

"echolalic"

properties. But

to

be

criteria into question forget. We should not

habits we so easily acquire due to our b orn at each and every instant, like a

re

ch i ld with a new an d "independent" view of things. All of this is

oneself,

as

part of a second p rin ciple :

It is

absolutely necessary

much as possible, from a ny and all contingencies. xi

to free

Xll

Preface This may be con s ide red man's

in ge neral . Indeed, the Being's not, deterministic or chaotic

destiny

c onstant

(or

in particular,

and

the universe's

dislocations, be they continuous or

both simultaneously) arc manifestations of

the vital and incessant drive towards change, towards

return.

An artist can not remain isolated in the universal

freedom witllout

ocean of forms

and

their chang es . His interest lies in embracing the most vast horizon of knowledge and problematics, all p re sented above.

From

in

a ccordance witl1 the two principles

hence comes the new chapter in

this

edition entitled

"Concerning Time, Space and Music."

Finally, to finish with the first question, I have all al o ng continued to

develop certain theses and to open up some new ones. The new cl1apter on

"Sieves" is

an

Appen dix III

example of this alon g with tile comp uter program presented in

which represents a lon g aesthetic and theoretical search. This

resea rch was developed

UPIC.*

approach

Another

as

well as its application in sound synthesis on

to the mystery of sounds is the use of cellular

automata whi ch I have employed in several instrumental compositions these

by an observation which I made: scales of pitch (sieves) automatically establish a kind of global mus ical style, a sort of

past few years. This can be explained macroscopic

"synthesis"

of musica l

works,

their dissymmetries are the

d is ce rn i n g logico-aesthetic

m uch

physics of particles. reason behind this.

frequencies, or iterations," of the

cho ic e

like a

"spectrum

of

Internal symmetries or

Therefore, tluo u gh a

of "non-octave" scales, we can obtain very

rich simultaneities (chords) or linear successions which revive and generalize tonal, mod al or se rial aspects. It is on

this

basis of sieves that cellular

automata can be useful in harmonic progressions which create new and r ic h

timbt·ic

fusions with orchestral instruments. Exa mpl es of this can be

works of mine

such as Ata, Horos,

Today, there is

a

found

in

etc.

who le new fi eld of investigation called "Experimental

Mathematics," that gi ves fascinating insights especially in automatic dynamic

systems, by the

use

of math and computer

graphics.

Thus,

many s truc tu re s possess self-

such as tl1e already- mentioned cel lu lar automata or those which

*UPIC-Unite Polygogique Informatique du CEMAMu. A sort of musical drawi ng board which, through the digitalization of a drawing, enables one to compose music, teach acoustics, engage in musical pedagogy at any age. This machine was developed at the Centre d'Etudes de Mathematiqucs et Automatiques Musicales de Paris.

Preface similarities such

as

Juli a or

Mandelbrot sets,

xiii

arc studi ed and visualized. These

studies lead one right into the frontiers of deter minism and indeterminism.

C h aos to s ym m e try

and

the reverse o rientation a re once ag ain being studied

They

a nd are even quite fashionable!

are novel aspects started d eali n g with about thirty

open up new horizons,

al tho ugh for

of tJ1e equivalent compositional prob l e ms I

me, the results

-

five years ago. The theses

presented in the

earlier editi on s of this book bea r witness to tl1is fact altlwugh the dyn amic of

musical works depe nds on several levels simultaneously

and

not only on tl1e

c al culus level.

An important task of tJ1e research program at CEMAMu is to deve lop

synthesis through quantified sounds but with up-to-date tools capable

of

involving autosimilitudes, symmetries or deterministic ch aos , or stochastics

within a dynamic evolution of amplitude

corresponds to Einstein in the attributed to

a

sound

quantum or

1910s. This research,

Gabor,

frequency

frames where each

"phonon," as which I started

pixel by

already im a gi ne d

in 1958 and wrongly

can now be pursued witl1 much more p ower ful

and

modern means. Some surprises can

In App en d i x

IV

of tl1is

be expected! ed iti on a new, more ,

stochastic sound synthesis can be found

as a

precise formulation of

of tl1e last chapter of

follow-up

the preced ing edition of Formalized Music (presented here as Chapter the inter i m , this approach has been tested and used in my

d'Eer

IX). In

work La ligende

for seven-track tape. This approach was developed at the CEMAMu in

Paris and

worked out

at

WDR, the West-German N a tio n al

tJ1e

Radio

studio

in Cologne. This work was part of the Dia tope which was installed for the inauguration of

the

entirely automated

Pompidou/Beaubourg Center

with

in

Pari s. The event was

a c om p l ete laser installation

and

1600 electr o n ic

flashes. This syntl1esis is part of CEMAMu's permanent research program.

In this same spirit, ran d o m walks or Brownian movements have been

works, especially instrumental pieces such as "breath" or spir i t in Hebrew; for 2 female voices, 2 trombones and 1 'cello. This piece was written at the request

the basis for several of my N'Shi'11UJ,, which means French Horns, 2

of

"

"

Recha Freier, founder of the "Aiiya movement" and premiered at the

Testimonium Festival in Jerusalem.

The answer to Preface

is

the second question posed at the be gi nning of this

not up to me.

In

absolute te rm s , the artisan musician (not to say

creator) must remain doubt ful of the decisions he has made, doub tful ,

however subtly, of the result. The percentage of doubt shoul d not exist in

virtue ofthe pr in ci pl e s elaborated

connoisseurs

(either

above. But in relative terms, the public, or diachronic}, alone decide upon a work's

synchronic or

xiv

efficiency.

Preface However, any culture's validation follows "seasonal" rules, or

varying

m ille nnia We must never forget the nearly-total lack of consideration Egyptian art suffered for over 2000 years, or Meso-American art.

between periods of a fe w years to centuries

even

.

of art, or, let us say, just a work, to the on a document, seal in a bottle which we will throw of the ocean. Will it eve r be found? When and by whom and

One can assimilate a work information we can put into the middle

how will it be read,

interpreted?

My gratitude and thanks

go

to Sharon Kanach, who translated and

supervised the new material in this updated edition Robert Kessler, the courageous

publisher.

of Formalized Music and

to

Formalized

Music

...... ....

....:.. Ol "' Ol

-

I rl.

_,.

-lit_.,,

(:. ..... . �

Preliminary sketch 103-09.

._:-d/.4.�'

A11iLlogique B (1959).

cMtS

-- ---· · "'"' ·

-

See Chapter III,

�:r�Mt'_:

-. -._;c

pp.

-'��;_...... _,-Ad'&

•.

·-'

,U

Chapter I

Free

Stochastic Music

Art, and a bove all, music has a fundamental function, which is to catalyze the sublimation that it can bring about through al l means of expression. It must aim thro ugh fixations which arc landmarks to draw towards a total e xalta ti on in which the individual mingles, losing his consciousness in a truth immediate, rare, enormous, and perfect. If a work of art succeeds in this undertaking even for

dous truth is not ma de

a

single moment, it attains its goal. This tremen­

of objects, emotions, or sensations; it is beyond these, as Beethoven's Seventh Symphony is beyond music. This is why art can lead to realms that religion still occupies for some people.

But this transmutation of every-day artistic material which transforms

trivial products into meta-art is

knowing its" mechanisms."

The

a

secret. The;" possessed" reach it without

others strugg le in

the ideological and tech ­ "climate"

nical mainstream of their epoch which constitutes the perishable and

the stylistic fashion.

Keeping our eyes fixed

on

this supreme meta-artistic

goal, we sha l l attempt to define in a more mod est manner the p aths which

can

lead to

it

There

exists

from our point of

dic tions in present music. a

historical

departure, which is the

magma of contra­

music and the reason. The music of antiquity, cau s al and deterministic, was already strongly influenced by the schools of Pythagoras and Plato. Plato insisted on the principle of c au sali ty , "for it is impossible for anything, to come into b e ing without cause" (Timaeus). Strict causality lasted until the nineteenth century when it underwent a parallel between European

su c cessi ve attempts to explain the world by

The English translation of Chaps. I-VI is by Christopher A. Butchers. 1

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of

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Metastasis, 1953/54,

Bars

309-17

3i'

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I

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12 second violins V II

double basses CB

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4

Formalized Music

brutal and fertile transformation as a result of statistical

Since antiquity the concepts of chance

(tyche),

the ori es in

disorder

(ataxia),

physics. and dis­

organization were conside red as the opposite and negation of reason

(logos),

orde r (taxis), and organization (systasis). It is only r ecen tly that knowledge has been able to penetrate chance and has discovered how to s epar a te its d egrees -in other words to ra t ion al ize it progressively, without, however, su cceed i n g in a definitive and total explanation of the problem of " pure chance."

After

a

time

lag

of several

decades, atonal music

broke

function and opened up a new path parallel to that of the

up the tonal

physical

sciences,

but at the same time constricted by the virtually absolute determinism of

serial music.

It is therefore n ot surprising that the presence or absence of the principle of causality, first in philosophy and then in the sciences, might influence musical co mp os ition . It c au sed it to follow paths that appeared to be diver­ gent, but which, in fact, coalesced in probability theory and finally in polyvalent logic , which arc kinds of generalization and enrich ment s of the p ri n c ip le of causality. The e xplanation of the world, and consequently of the sonic phenomena which surround

us or

and profited from the enl argemen t

which may be created, necessitated

of the principle of causality, the basis of wh i ch enlargement is fo r med by the law of large numbers. This law i mpl ies an asymptotic evolution towards a stable sta te , towards

a

kind of goal, of

stochos, whence comes the adjective "stochastic." But everything in pure determinism or in less pure indeterminism is subjected to the fundamental opera tio n al laws of logic, which were disen­ tangled by mathematical th ou ght under the ti tle of general algebra. These laws

operate on isolated states

or on sets of elements with

the

aid of

o perati ons , the most primitive of which arc the union, notated u,

the

intersection, notated n, and the n egation . Equivalence, implication, and

quantifications are elem en t ary relations from which all current science can be

constructed. Music, then, may be defined

as

an organization of these elementary

operations and rel at i ons between sonic entities

or

between functions

of occupied by also for an alysis

sonic entities. We understand the first-rate position which is set theory, not only for the construction of new works, but

and better comprehension stochastic c on stru c t io n or stochastlcs cannot be

of the w orks of the past. In the same way a an investigation of history wi t h the help of carried thro ug h without the help of logi c-t he queen

of the sciences, and I would even venture to suggest, of the arts--or its mathematical form algebra. For e verything that is said here on the subject

Free Stochastic

Music

5

is also valid for all forms of art (painting, sculpture, architecture, films, etc.). From t his very general, fundamental point of view, from which we wish to examine and make music, primary time appears as a wax or clay on which opera tio ns and relati ons can be inscribed and engraved, first for the purposes of work, and then for communication with a third person. On this level, the asymmetric, noncommutative character of time is use (B after A =I A after B, i.e., lexicographic order). Commutative, metric time (symmetrical) is subjected to the same logical laws and can therefore also aid organizational speculations. What is remarkable is that these fundamental notions, which are necessary for construction, are found in m an from his tenderest age, and it is fa s ci n ating to follow their evolution as Jean Piaget1 has done. After this short preamble on generalities we shall enter into the details of an approach to musical composition which I have de v elo ped over several years. I call it "stochastic," in honor ofprobabi1ity theory, which has served as a logi cal framework and as a method of resolving the conflicts and knots encountered.

The first task is to c ons t ruct an abstraction from all inherited conven­ tions and to exercise a fundamental critique of acts of thought and their materialization. What, in fact, d oes a musical composition offer strictly on the cons t ru ction level ? It o ffers a collection of sequences which it wishes to be causal. When, for simplification, the major scale implied its hierarchy of tonal functions-tonics, dominants, and subdominants-around which the other n ot es gravitated, it constructed, in a highly deterministic manner, linear processes, or melodies on the one hand, and simultaneous events, or chords, on the other. Then the serialists of the Vienna school, not having known how to master logically the indeterminism of atonality, returned to an o rganization which was extremely causal in the strictest sense, more ab­ stract than that of tonality; however, this abs trac tion was th eir great con­ tribution. Messiaen generalized this process and took a great step in sys­ tematizing the abstraction of all the variables of instrumental music. What is paradoxical is that he did this in the modal field. He created a multimodal music which immediately found imitators in serial music. At the outset Messiaen's abstract systematization found its most justifiable embodiment in a multiserial music. I t is from here that the postwar nco-serialists have drawn their inspiration. They could now, following the Vienna school and Messiaen, with some occasional borrowing from Stravinsky and Debussy, walk on with ears shut and proclaim a truth greater than the others. Other movements were growing stronger; chief among them was the systematic exploration of sonic entities, new instruments, and "noises." Vares e was the

1 st P e a k

A.

G r o un d prof i l e of the left half of

t he "sto m a c h .' " The i n t en t i o n was t o

b u i l d a s h e l l , com posed of as few

g r o u nd pl a n . A c o n o id ( e ) is c o n ­

ruled su rfaces as possible, over the

structed

curve ; t h i s w a l l is

s tr a i g ht

t h r o u g h t h e g r o u n d pro f i l e

bounded

by

two

l i n e s : the st raight d irectrix

( r i s i n g from the l eft extre m it y of t h e

g ro u n d prof i l e ) ,

and

t h e o u te r m ost

g e neratrix ( passing t h r o u g h the right produces t h e first " p e a k" o f the

extremity of

t h e gro u n d profi le ) . This

pavi l i o n .

B.

A r u l e d surface consisting of two

conoids, a and d, is laid t h r o u g h the the right half of the

"st o m a c h . " The st r a i g h t di r ec t ri x of d

c u rve

bounding

passes t h r o u g h t h e first peak, a n d the

o ut e rmost g e n e ratrix a t t h i s s i d e forms a t r ia n g u l a r e x it with the g e n eratrix of e. The s tra i g h t d irect ri x of a passes through a se c o n d peak and

is j o i n e d

by an a rc t o t h e directrix of d.

T h i s basic form is t h e

t h e f i rst desig n and was r e tai n e d , w i t h

one

used i n

s o m e modificatio ns, i n the f i n a l

a n aesthetic

s t r u cture. T h e m a i n pro b l e m o f t h e b a l a n ce between t h e two peaks.

design

was t o establ ish

1 st Peak "-"' "b

�

�

C.

At te mp t t o close t h e space between

t h e two ruled s u rfaces

of

the f i r s t

d e s i g n b y f l a t s u r faces (w h i c h m i g h t serve a s project i o n wa l l s ) .

Fig. 1-3 . Stages in the Development of the First Design of the Philips Pavilion

1 st Pea k

D.

Anot h e r a t t e mpt. Above t h e

e n trance c h a n n e l a sm a l l t r i a ng u l a r

2nd

i s formed. flan ked by two

hyperbolic p a r a b o l o i d s ( g a n d

open ing

Peak

k). and

the w h o le is co ve r e d with a h o r i z o n t a l

top

surface.

1 st

E.

Ela boration of D . T h e

third

Peak

peak

begins to take shape (shyly } .

3rd Peak

F.

The

first

2nd P e a k

de sig n completed (s e e

a l so t h e first model, F i g . 1 -4 } . There

flat

t h i rd p ea k i s fully developed and are no longer a n y

surfaces. The

c reates, with its o p p osing sweep, a

counterbala nce for the first two peaks.

The h eigh ts of

the three peaks

h ave

t h e s m a l l a r c c o n n ecti n g the straig h t

been established. The third p e a k a n d

d i rectrixes of c o n 0 1 d s a a n d d ( s e e B . ) fo r m , respect i ve l y, tho a p e x a n d t h e

b a s e of a p a r t of a c o n e /.

2nd P e a k

8

Formalized Music

pioneer in this fi eld , and electrom agnetic music has been the beneficiary (electronic music

bei ng

a

branch of instrumental music) . However, in

electromagnetic music, problems of construction and of morphology were not

faced conscientiously. M u ltise ria l music, a fusion of the multimodality of Messiaen and the Viennese school, re ma i ned , nevertheless, at the heart of

the fundamental problem of music.

But by 1 954 it was already in the process of d e fla t io n , for the com plete ly

deterministic complexity of the operations of compositi on and of the works themselves produced an a u di tory and i deolo g i cal nonsense. I described the

inevitable conc l usi o n in " The Crisis of Seri al Music " :

Linear polyphony d e st roys itself b y i ts very comp l exi ty ; w hat one

hears is in reality nothing but

a

mass of notes in various registers. The

enormous co mpl e xi ty preven ts t h e audience from following twi n in g

of the

l i n es and has as

the inter­

its

macroscopic effect an i rrational

a

con tradiction between the poly­

and fortui taus dispersion of sounds over the w hole extent of the so n i c

spectrum . There is consequently

phonic linear system an d the heard res u l t , which is surface or mass.

This contradiction in heren t in polyphony w i l l disappear independence of sounds

is

total . In fac t , w hen

l i n ear

when

the

combinations

an d their p ol y ph oni c superpositions no longer o p e ra t e , what will count will be the statistical

mean

tions of sonic components at

a

of isolated states and of transforma­

given mom e n t. The m ac rosco pi c effect

can then be control led by the mean of the movements of elemen ts wh ich we select.

The

result is the i n troduction of the notion of p roba­

bility, w h ic h implies, in this particular case, combi natory calculus.

Here, i n a few words, is the p ossi bl e escape route fro m the "linear

category"

in musical thou gh t . 2

This a rticl e served as a bridge

to my introduction of mathematics in

music. F or if, thanks to complexity, the strict, deterministic causality whi c h

the nco-serialists postulated was lost, then it was necessary to replace it by a more general causality, by a probabilistic logi c which would contain strict

serial causality as a pa r ticu l ar case. This is the fu nction of stochastic science. " Stochastics " stud ies

and

formulates the l aw o f l a rg e numbers, wh ich has

already been men ti one d , th e l aws of procedures, etc. As

causes,

a

rare

e vents , the different aleatory

resu lt of the impasse in serial music, as wel l as other

I originated in 1 954 a m usic

con s tructe d from the principle of

indeterminism ; two years later I n am ed it " S to c h a st ic Musi c . " The l aw s of the calculus of probabili ties entered

com position

th ro u gh

musical

necessity.

But other paths also l ed to the same stochastic crossroads-first of a ll ,

9

Free Stochastic M usic

n at u r al events s u c h as the collision of hail or rain with hard surfaces, or the

song of cicadas in a summer field. These s on ic

events

are m a d e out of thou­

sands of isolated sounds ; this multitude of s ou n d s , seen as

a

totality, is a n e w

sonic event. This mass ev e n t is articulated and forms a plastic mold of time,

which itself follows aleatory and stochastic laws. If one then wishes to fo r m

a

s tri n g pizz ica ti, one must know these ma thematical laws, which, in any c as e , are no more than a tight an d concise expression of chain of logical reasonin g . E v e ryon e has observed the soni c phenomena of a political crowd o f d oz e n s or hundreds of tho u s a n d s o f people. The human riv e r shouts a sloga n i n a uniform rhythm . Then a no t he r slogan springs from the head of the demonstration ; it spreads tow a rd s the tail, re p l ac i n g the first. A wave of transition thus passes from the head to the tail. The c l amo r fills t h e city, and the i nh ibiti n g force of voice and rhy t h m reaches a cl imax. It is an event of great power and beauty in its fe ro c i ty . Then the impact between th e demonstrators and th e enemy occurs. The perfect rhy t h m of t h e last s logan breaks up in a huge clu s t e r of ch aotic shouts, which also spr e a d s to the tail. Imagine, in add i t i o n , the re p orts o f dozens of mach ine guns and the whistle of bullets a d d i r g their punctuations to this total disorder. T he crowd is t h e n rapidly dispe rsed, and after son i c and vi su al hell follows a detonating cal m , full of despair, dust, and death . The statistical laws of t h e se events, separated from their p ol it i c al or moral context, are the same as t h os e of the cicadas or the rain. They are the laws of the passage from com pl e te o rd e r to t otal disord er i n a continuous or exp lo ­ sive manner. They are stochastic laws. Here we touch on one of the great p rob le ms that have haunted human intelligence since antiquity : continuous or d iscontinuous transformation. The sophisms of movement (e.g., Achilles a n d the tortoise) or of definition ( e.g., baldness) , especially the latter, are solved by statis ti cal definition ; that is t o say, by stoch astics. One may p rod u c e continu ity with either continuous or discontinuous elements. A multitude of short glissandi on strings can gi ve the impression of continuity, and so can a multitude of p izzicat i . Passage s l arge mass

of point-notes,

such as

from a discontinuous state to

a

c ontinuous state are controllable with the now

I

have been conducting these

fascinating experiments in instrumental works ;

the mathematical char­

aid of probability theory. For some ti m e

acter

of this

but

music has frig h tened musicians and has mad e the approach

especially difficult.

Her� is an oth e r direction that converges on indeterminism . The s t udy

of the

v ari a ti o n of rhythm poses the problem of knowing what the limit of

total asymmetry is, and

among

durations.

of the cons eque n t

complete d isr uption o f c aus al i ty

The sounds of a Geiger counter in the proximity of a

10

Formalized

radioactive

source

necessary l a ws .

give

an

impressive idea of this.

Stoch astics

provides the

Before ending this short insp ection tour of even ts rich i n the

which

were

closed

to the

un derstand ing u n t i l recen tly, I

Music

new

l ogic,

would like to in­

a short parenthesis. I f glissandi are long and sufficiently i n terl a c e d , obtain son i c s p aces of con t i n u ou s evolution . It is poss i b l e t o pr o d u c e r uled surfaces by drawing the gl issandi as straig h t l ines. I performed t h i s experiment with Metastasis ( this w o r k had i ts premiere i n 1 9 55 at D o n au ­ eschingen) . Several years later, whe n the archi t e c t Le Cm· b usicr, whose collaborator I was, asked me to suggest a design for the arch i tectu re of the Philips Pavilion in Brussels, my inspi ration was p i n - p o i n ted by the experi­ ment wi t h Metastasis. Thus I beli eve that on this occasion music and arch i­ tecture found an i n tim ate connect i o n . 3 Fi gs. I- 1 -5 indicate the causal chain of ideas which led me t o formulate t h e arch i tect1.1re of the Philips Pavilion from the score of Metastasis.

clude

we

F i g . 1-4. F i rst M od e l

of

P h i l i ps Pavi l ion

Free Stochastic Music

11

.'

·�

·.

.. ,, \ ' \ l \

·}

' �

\ '

. \�

F i g . 1-5 . P h i l i ps Pavilion, B ru ssels

World's Fair,

1 9 58

12

Formalized Music

STO C H ASTI C LAWS A N D I N CA R N AT I O N S

I sha l l give quickly som e of th e stochastic laws which I introduced i n to

composition several years ago . We s hal l examine one by one the in d e p e n den t components of an instrumental sound. D U RA T I O N S

Time (metrical ) i s considered a s a straight line o n which the p o i nts correspond ing to the variations of the other components are marked. The

interval

possib le

between two point s is i dentical with the d u rati on . Amon g a l l the sequences of poi n ts , which shall we choose ? Put thus, the q ue s t i o n

makes no sense.

If a mean n u m b e r of points is designated on a given

length the question

b ec om e s : Given this mean, what is the number of segments equal to fixed

in

advance ?

The followin g formula, which derives

from th e

a length

principles of continuous

probability, giv es the probabilities for all possi ble lengths when one kn ows

the mean nu m ber of p o ints placed a t random on

a str ai gh t line. (See

in

w hich

8 is the linear d ens i ty of points, and

x

the length of any segment.

If we now choose some po i n ts and compare them to

distribu tion obeying the above law

or

Appendix I.) a

theoretical

any o th e r distribution, we can deduce

choice, or the more or less rigorous distribution, which can even be a bsolu t ely functional . The comparison can be m ade with the aid of te sts , of which the most widely used is the x2 cri terion of Pearson. In our case, where all the co mp one n ts of sound can be measured to a first approxima­ ti on , we shall use in addition the correlation coefficient. It is known that if two populations are in a linear functional relationship, the correlation the amount of chance included in our

adapt at io n of our choice to the

of

law

coefficient is one. If the two populations are ind ependent, the coefficient is

zero. All

intermedi ate

degrees

of relationship are possible.

Cl ouds of So u n d s

Assume a given duration a n d a s e t o f sound - p oin t s defined

in the

intensity-pitch space real i ze d duri n g this d ur atio n . Given th e mean super­

ficial d e n sity of this tone cl uste r , what d e nsity occurring in a given region Law answers this question : Pu

is the probability of a particu l ar of the intensity-pitch space ? Po i sson ' s u

=

P.o e - u . ' P. ·I

13

Free Stochastic Music

where p. 0 is the mean density and p. i s any particular density. As with

durations, comparisons with other distributions of sound-points can fashion the law which we wish our cluster to obey.

I N TE R V A L S

OF I N T E N S I T Y , P I T C H ,

ETC.

For these vari ables the simplest law is

B(y) dy

=

� (I �) dy,

(See Appendix I . )

-

which gives t h e probability that a segment (interval o f intensity, pitch, etc . ) within a s eg me nt of length

y + dy, for S P E ED S

O � y � a.

a,

will have a length included within y and

are particular case of sounds of continuous variation. Among these let us consider glissandi. Of all the possible forms that a glissando sound can take, we shall choose the simplest-the uniformly continuous glissando. This We have been speakin g o f sou nd-points, or granular sounds, which

i n reality a

glissando

can

be assi m ila ted sensorially and physically into the mathematical

concept of speed. In a one-dimensional vectorial represen tation, the scalar size of

the vector can be gi ve n by the hypotenuse of the right triangle

in

which the du ration and the melodic interval covered form the other two sides . Certain mathematical operations on the continuously variable sounds

thus defined are then p e r m itt ed The traditional sounds of wind instruments .

particular cases fre q uencies can be

are, for example, towards higher

q u en c i es as negative.

where the speed is

zero.

A glissando

defined as positive, towards lower fre­

We shall demonstrate the simplest logical hypotheses wh i ch lead us

to a mathematical formula for the distribution of speed s . The arguments

which follow arc in reality one of those " logical po e m s

intelligence creates

"

which the human

in order to trap the superficial incoherencies of physi cal

phenomena, and which can serve, on the rebound, as a poi n t of departure for b u i lding abstract entities, and then incarnations of these entities in sound or light. It is for these reason s that I o ffer t h em as examples :

Homogeneity hypotheses [ I I ] * 1 . The d ensity of speed-animated

sou nds is constant ; i.e. , two regions

of equal extent on the pitch range contain the same average num ber of mobile sounds (glissan d i ) .

* The nu mbers in brackets correspond to the numbers in the Bibl iogra p hy end of the book .

at

the

14

Formalized Music

2. The absol u te value of sp ee d s (ascending o r descending glissandi) is uniformly ; i.e., the m e a n quadratic sp e ed of mobile so un ds is the same in different registers. 3 . There is isotropy ; that is, th e re is no privileged d ir e c t io n for the movements of mobile sounds in an y register. There is an equal number of sounds ascending and descending. From these th ree hypotheses of symmetry, we can define the fu n c t i on f(v) of the probability of the absolute s pee d v. (f(v) is the rel ative frequency of occurrence of v.) L e t n be t h e number of glissandi per u n i t of the pitch range (density of mobile sounds ) , and r a ny portion taken fro m the range . T h en the number of speed-animated sounds between v and v + dv and positive, is, from hypotheses 1 and 3 : spread

n

(the probability that the sign is + is }) .

r !f(v) dv

2 the number of animated sounds with speed o f value l v l is a function which depends o n v2 onl y . Le t cl1 is functio n be g(v2). We then have the equation

From hypothesis

absolute

tf(v) dv

n r

=

n r

g(v2) dv.

Moreover i f x v, the probabi lity function g ( v 2) will be equal to the law of probability H of x, when_cc g(v2) H(x) , or log g(v2) = h(x) . In order that h(x) may d ep e n d only on x2 v2, it is n ecessary and sufficie n t that the differentials d log g(v2) h' (x) dx and v dv x dx have a constant ratio : =

=

=

=

d log g(v2) --"'-'f.!......!. whence

h' (x)

v dv

- 2jx, h(x)

a

from - CIJ to l fa2, it follows =

h' (x) dx --x dx

x· ,

=

- 2.J, .

=

=

=

g(v2)

j( v) =

s ta n t

=

=

and

v

c on

that

tf(v)

fo r

=

-jx2 + c, a n d H(x) ke -ix". function of elementary probabilities ; th e refo re its i ntegral + oo m us t be equal to I . j is p osi tiv e and k y'jjy'TT. If

=

But H(x) is

j

=

=

which is

a

=

Gaussian

H(x)

=

ay'TT

1-

_2_ e - ••ta• ay'TT

distribution.

e - ••ta•

15

Free S tochastic Music This

chain o f

reasoning borrowed fro m

Maxwel l , wh o , wi th gases. The

Boltzmann, was

function f(v)

Paul L evy was established after

res p onsible fo r the kinetic theory of

gives the probability of

the

s p eed v ;

t h e constant

a

defines the " temperature " of thi s soni c atmosphere. The a rithmetic mean

equal to af v'", and the s t a nd a rd deviation is a { \/2. offer as an example several bars from th e work Pithoprakta for s tring orchestra (Fig. 1-6) , wri tten in 1 95 5-56, and performed by Prof. Herman n S c h erchen in Munich in March 1 957.4 The graph ( Fig. 1-7) represents a of v is

We

set of speeds of temperat u re proportional to a 35. The abscissa represen ts time in units of 5 em 26 MM ( M iilzel Metronome ) . This unit is sub­ divided i n to three, four , a n d five equal p arts, which allow very slight =

=

arc d rawn as the ordinates, with the unit 1 sem i tone 0 . 2 5 em. I e m on th e verti cal s c al e corresponds to a m ajor third. There arc 46 stringed instru m e n ts, each represented by a j agged line. Each of the lines represents a speed taken from the table of probabilities

differences of duration. The pi tches =

calculated w i th the formu l a

J(v)

=

_ 2_ e - v 9 t a 2 . a v'TT

in 58 disti nct values according to Gauss's traced for this passage (measures 52-60, with a duration of 1 8. 5 sec.) . The distribution being G a u ssian, the macroscopic configura tion is a pl astic mod u l a tion of the sonic m aterial . The same passage was transcribed into trad itional no tation . To sum up we h av e a so n i c

A total of 1 1 48 speeds, distribu ted

law, h ave

been calcu l ated

and

compound in which :

1 . The durations do not vary. 2. The mass of pitches is freely modulated . 3. The density of sounds at each moment is 4. The d y n a m ic isjf without variation .

constant.

5 . The timbre i s constant.

6.

The speeds determ ine a " temperat u re " which is

fluctuations.

Their distrib u tion is Gaussian .

As we h ave already had occasion

to rem ark,

subject

to local

we can establish more or

less strict relationsh ips between t h e com p onent parts of sounds.5 The most useful coefficient which measures the degree of correlation between two variables

x

and y is

2: (x - x) (y - y) r=v�1,.=(;=x-'---x=-)"' 2v -';.L �(y� --=y= )2 '

16

Formalized

Music

where x and fi are the arithmetic means of the two variables. Here then, is the technic al aspect of the starting point for a utilization of the theory and calculus of p rob abilities in musical composition. With the above, we already know that :

1 . We can control continuous transformations of large sets of gra nular and/or continuous sounds. In fact, densities, durations, registers, speeds, etc., can all be subjected to the law of large numbers with the necessary approximations. We can therefore with the aid of means and deviatio ns shape th ese sets a nd make them evolve in different directions. The best known is that which goes from order to disord er, or vice versa, and which introduces the concept of en tropy. We can conceive of Qther continuous transformations : for example, a set of plucked sounds transforming con­ tinuously into a set of arco sounds, or in electromagnetic music, the passage from one sonic substance to another, assuring thus an organic connection between the two substances. To illustrate this idea, I recall the Greek sophism about baldness : " How ma ny hairs must one remove from a hairy skul l in order to make it bald ? " It is a p rob lem resolved by the theory of probability with the standard deviation, and known by the term statistical definition.

2. A transformation may be explosi ve when devi ations from the mean suddenly become exceptional. 3. We can likewise confront. highly improbable events with average events.

Very rarified sonic atmospheres may be fashioned and controlled aid of formulae such as Poisson's. Thus, even music for a solo instrument can be composed with stochastic methods. 4.

with the

laws, which we have met before in a multitude of fields, are di amonds of contemporary thought. They govern the laws of the advent of b e ing and becoming. However, it must be well understood that they are not an end in themselves, but marvelous tools of construction and logical li fel ines. Here a backfire is to be found. This time it is these stochastic tools that pose a fu n d ament al question : " What is the minimum of l ogical constraints necessary for the construction of a musical pro ces s ? " But before answering this we shall sketch briefly the basic phases in the construction of These

veritable

a musical work.

W. ll

=-- -----1 ---- - ----1 [!!] ---.,---o-l---- -l-. ----f----=t pi••· "Jii: � • � � 5.::-----, ,.......- r -------, .. r ----, r--- • � - " -..:!:

V. l

···--

A.

v. . ..

&, LN. ltJ.S6S

F i g . 1-6. B a rs 52-57 of Pithoprakta

18

F i g . 1-7 . G ra p h of B a rs 52-57 of Pith oprakta

Form alized

Music

Free S tochastic

l'vi usic

19

20

F i g . 1-7

Formalized Music

(continued)

Free Stochastic Music

21

22

Formalized Music F U N D A M E N TA L P H AS E S O F A M U S I C A L W O R K

I.

Initial conceptions (in t u i ti o n s ,

pro vi s i onal o r d efinitive data ) ;

2. Definition of the sonic entities and of their s y m b ol i sm commu nicable with the limits of p ossible means (sounds of m usical instruments, electronic sounds, noises, s e t s of ordered sonic elements, granu l a r or continuous formations, etc.) ;

3. Definition of the t ra nsfo rma tions which t h e s e sonic entities must undergo course of the composi tion (macrocomposition : general choice of l ogical fi·amework, i.e., of the ele me nt a ry alg eb r aic operations and the set­ ting up of relations between enti ties, sets, and their symbols as d efined in 2 . ) ; and the arrangement of these operations in lexicographic time with the aid of succession and simultanei ty) ; 4. Microcomposition (choice and detailed fixing of the fu nctional or s to cha s ti c rel ations of the clements of 2 . ) , i.e., algebra outside-time, and in the

•

algebra i n-time ;

5. Sequential program ming of 3. in its e n tire ty) ;

work

6.

Implementation of calculations,

and

4.

(the schema

and pattern

of the

verifications, fe e d b a cks , a n d definitive

modificati o n s of the se q uential program ;

7.

paper

Final symbolic result of the programming (setting out the music on in traditional n otation, n um e ri c al expressions, graphs, or o ther means

of sol fcggio) ; 8. Sonic

realization

of the p ro gr a m

(direct

manipulations of the type of electromagnetic

orchestral

p erfo r mance ,

music, computerized

construc­

t io n of the sonic entities and t h e i r transform ations) .

The order

really rigid. Permutations of a composition . Most of

of this list is not

the course of the working out

are possible in the

ti m e

these

ph ases are unconscious and defective. However, this list does establish

ideas

that

only

and

a l l ows specul ation about the fut u r e .

ph ases 6 .

and

ph ases 6. and

7.,

7.

an d

even

8. But

In fac t ,

compu ters can t a k e

as a first ap p ro ach,

are i m m ediately accessible. That is

to

it s e e m s

in hand

the final orchestra or

say, that

result, at l east in France, may be realized only by an by manipulations of el ectroacoustic music on tape recorders , emitted by the existi ng electroacoustic chan nels ; and not, as would be desirable in the very symbol i c

ncar

fu tu re, by

an

el aborate

m ec h a n i z ati o n

which would o mi t orchestral or

Lapc i nterpreters, and which wo u l d assume the computerized fabrication of the sonic enti ties and of their transformations. Here now i s a n answer to the q u estion put above, an answer that i s t r u e for i nstrumental m usic, b u t which can be a pp l i e d a s we l l t o all kinds of

23

Free Stochastic Music sound production. For this we shall again t ake u p the ph ases described :

the

2. Defi nition of sonic entities. The sonic enti ties of

tra can be represented in

in depen dent variables, Ca

h1

gi

uk

by

base

(c, h,

Cvlol . arco>

h 39 (

or instrum ental fa mi l y

duration of the sound.

=

defines a

u) .

at

point M in the

c3

played arco an d forte on a viol i n , one eighth

(

=

forte) ,

u5

(

=

=

M are plotted on an axis which we shall

we draw another axis t, at right angles to

axi s , called the

multidimensional space provided coordinates the n u m b e rs

This poi n t M will have as

one eighth no te

c 3 ) , g4

=

classical orches­

b y vectors of fo u r u s u a l l y

i n tensity o f the sound, or dyn amic form

=

'"' h" g,, uk. For example :

note in length ,

approximation

p i t c h of the sound

=

g,

fi rst

E, (c, h, g, u) :

timbre

=

The vector E,

a

a

be represen ted as that these poi n ts call E" a n d that through its origin axis E,. We sh all represent on this 240 MM, can

-!- se c . ) . Su ppose

axis of lexicographic time, the lexicographic-temporal succession

o f the points M. Thus we h ave d efi n e d and conveniently represented

two-dimensional space (E, t) . This will allow us to pass to phase

tion of transform ation ,

and

3.,

a

defin i­

4., microcomposi tion, which must contain the

a nswer to the problem posed concern ing the minimum of constraints. To this end, su pp ose that

the

po i n t s M

defined above c an appear w i th of obeying an aleatory law with o u t m emory. This hypothesis is e q uivalent t o sayi n g that we adm i t a stochastic d istribution of the events E, in the space (E, t) . Admitting a sufficiently weak su perficial distri b u tion n, we enter a region where the l aw of P o i sson no

necessary co n d i tion other than that

is applicabl e :

I ncidentally we can consider this problem as

conveniently chosen linear stochastic processes active bodies) .

a

syn thesis o f several

( law o f radiation from radio­

(The second method is perhaps more favorable for a mecha­

nization of the transformations.)

A su fficien tly long fragment of this distribution constitu tes the m usical

work. The basic

law defined above generates a whole

family of

com positions

as a fu nction of the superficial density. So we have a formal archetype of composition in which the basic

metry

and

aim is

to attain the greatest

(in the etymological sense ) and the

rules. We t h i n k

that from this

poss i bl e

asym­

minimum of constraints, causalities,

archetype, which

is

perhaps the most

Formalized

24

general one, we can red es ce nd the ladder of forms by introducing

Music

progres­

sively more num erous constraints, i . e . , choices, restrictions, and n e ga t ions

In the an alysis in several linear processes we

th os e

cesses :

of Wiener-Levy, P .

ch ains, etc., or

m ixt u res of

Levy's

can

infinitely

several. It is this

method the more fertile.

The exploration of the limits

a

and b of

implies,

in effect, a gradation

which

this

divisibles,

.

pro­

Markov

makes this second

archetype

a

:;;;

n

:;;;

b is

of the mutual com pa r i son of of the i nc rem en ts of n in order

equally interesting, but on another level-th at samples. This

also introduce other

that the differences b e tw e en the families n1 may

be recognizable. Analogous

case of other linear processes . If we opt for a Poisson p rocess, t h e re are two necessary hy po th eses which answer the q uestion of the m in i m u m of constraints : l . there exists in a given space musical instruments and men ; and 2 . there exist means of contact between thes e men and these ins truments which p e rm i t the emission of rare so n i c events. This is the only hypothesis (cf. the ekklisis of Epicurus) . From these two constraints and with the aid of stochastics, I b u i l t an entire com posi tion without admitting any o the r restrictions. Achorripsis for 2 1 instruments was c om p o sed in 1 956-5 7, and h ad its first p erformance in Buenos Aires in 1 958 under Prof. H e rm an n Scherchen. (See Fig. I-8.) At that time I wrote :* remarks arc valid in the

T6 yit.p rxth6 voetv Td yap ONTOLOGY

In

at}r6

�aTlv

e fva' �arlv

Tt

n

Kai ovK

efvatt

o f nothingness. A brief train of waves , s o brief that its end coi nci de ( negative time) disengaging itself endlessly.

a universe

and beginning

Nothingness

resorbs, creates.

It engenders bein g . These rare son i c events can They

Kai elva£

can

be

Time, Causality.

so m e th i n g more than isolated sounds.

be melodic fi g ures, cell structures, or agglomerations whose

"' The following excerpt (through p. 3 7) is from " I n Search of a Stochastic Music," Gravesaner Bliilter, no. 1 1 / 1 2. t "For it is the same to t h i nk as to be " (Poem by Parmeni dcs) ; and my paraph rase, " For it is the same to be as not to be."

25

Free Stochastic Music c h aracteristics are

also ru l ed by the l aws

sou nd-points or speed- temperatures . 6

succession of r a r e sonic events . This s a mp l e may

be

of ch ance, for examp l e, clo uds of

In each case they

represe n ted by e i th er

a

form a sample of

a

simple table of prob abili ties

or a double-entry table, a matrix, in which the cells are filled by the fre­ quencies of events. The rows re present the particu l ar qualifications of th e

events, and the

columns the d ates (see Matrix M, Fig. I-9) .

The freq uencies

in this matrix a re distributed a c cord in g to Poisson's fo rm u la , wh ich is the

law for the appearances of rare random events.

We s h o u l d further define the sense of such a distribution and the manner

in which we realize it. There is an advantage in

defining

chance as

an

aesthetic law, as a n or m a l philosophy. Chance is the limit of the notion of

evolving symmetry . Symmetry tends to asymmetry, which in this s e ns e is

eq uivalent to the n egat i o n of traditionally inheri ted behavioral frameworks. This n eg a t io n not only operates on d etai l s , but most im portan tly on the c o mposit i on of structures, hence te nde n c i es in painti ng, sculpture, architec­ ture, and other realms of thought. For example, in ar ch i te ct u re , plans wo rk e d out with the aid of regulating di agrams are rendered more complex and dynamic by exceptional events. Ever y thin g happens as if there were one-to­ one

oscillations between symmetry , ord er, rationality , and asymmetry,

disorder, i rrationality in the At the

beginning of a

reac tio ns between the e poch s of civilizations.

transformation towards asymmetry, excep tional

events are introduced into symm etry a n d act as aesthetic stimuli. W h e n

th ese e x c ep t i o n a l events mul tiply and become the general case, a j ump to a

higher

level occ u rs .

The level is

one

of disorder, which, at l e as t in the arts

an d i n th e expressions of artists, p roclaims i tself as e n ge n d e re d

plex, vast, and rich

s u c h as abstract

vi

s io n

of

the

bru tal

and decorative ar t

and action

fact. Consequently chance, by w h os e side

com­

l i fe. Forms

painting bear witness to this walk in all our daily occu p a­ co n tro lled disorder ( that which

we

tions, is n o th i ng b u t an extreme c ase of this

s i gn i fies the rich ness

by the

encounters of modern

or poverty of the connections between events and wh ich by

engenders the dependence or independence o f transfo rmations) ; and

virtue of the negation, it conversely enjoys all the benevolent characteristics of an artistic regulator. I t is a

reg

ul ator also of sonic events, their appearance,

and their life . But it is here that the i ron logic of the laws of chance in ter­

ve n es ; this chance

laws.

On

cannot be

created withou t total subm ission to its

own

t h is condition, chance checked by its own force becomes a hyd ro­

electric torren t .

:r-�

/ II oJ

r- 1--,

'

A

·�

'l':e

)(yf.

lt.· BI

•

T

..

3

.--

. � !:l · - - - -

II '-'---'-5= A

[7

7

r--- 1 ;::ro

-

s.

.!(

II

!- - -�11. ':;J: '�� r l -;' ...

I

r-1--,

r

I r3 -, r- li--. . ..

.If

F i g . l-8 . B a rs 1 04-1 1 1 of A chorripsis

�

..

I

•

r!�

• " I 16 - - - -:: t:\

;;. �

•

Js__, L-- 5"--' I

..,

Jl ;:.

.,

II

"'! J '--J....;...o . )(.

.

.

1"-- S�

-I

r;- 5'�

r- 5'� r-=5"--,

��

-

��i !'_- - - - - - -.i

� �· -t

1......- !i --'

·

• .. -

" I

16 - : r.� ·

I

--,

r:-� "

§=J

I

..

�S�'i:t!: 1

r- !i___,

6

"'

...

I "

!'

��

l 'T-T

.:t: �

�

�

t

.....

�

..

r- s--,

�

8· · · :

......

.....

llb.Z

T T1 r;- �'�

'-- 5...::::1

ror

1

r-6"--, r- 5"--. ·� ., . ,·,

r-:il �

II

3

��

:!:

:!:

��1 '1< jol

�5'--'

"' ·" " " - - - - -

1

Vc/.2

�

.P I1'

�

Yl. 2

•,y

7

•

1

r- r--, r-3--.

---;2 r - li

• II

-�

� 6--. r-- 3 --,

-=s=-

B

�-""

r-. 5"--r r

r-•.:..-:t:

,..... fl"r .P. �

i t� *t'"'t�:-:

��-

.ll r.t

r-J-, � 5--, .....

•

r-- J -,

I I ......

Free Stochastic Music "" "

Pice .

27

.P- .fll.

!,.,.�

-....

¥-

�.(I.

l"Ji" S"---,

I "'

Ob .

II£

II

fl

1'1

1
..

/, 7ip t. 7 Xyl.

II

.:a:....t:t

r- 5 ---,

r-:: 3 --,

..Q.-

�: r;-;7 r;-�-, ���� ·"

"

R l'

A I'

. . L.-1 1"

II r- !; --, r-- 5"---, II

II

1

I'

l

v I

.. . - l ���f:

Yl. 7.

· II B · · · II .... ·

, ..

:;_·

-{,

.J

""

3

A

..

'

.__ il ---' M

��-

.--- s--. "

.

( P'•� S--, ..

� I

r- .r--, '.1 " •

r3---. . t • • I r I I

" " I

r;-..5---, .

"I • •

u...J

.....

.

r-- r;-vr-- 5"� ":'

�x

(pizz.) �

I

;;;:!.� � 1!" -=

�

.-s--,

... .... .

� .. , ..., .

t;'l � 6 --.

tf!.- -::Jil l v

y�

s · ;; : A ..

r-

I

.,

,.

•

" I

.- -

11 ..

,

.�

f:.

A "

I I'

..

r---ys--,

�

�

"

,.-- s---,

--

v

,.,

.-- s-,

It

€-!:it:· :

..

r- 5--,

"

,__ 3 __,

-

r- 3 ---,

•

� 3 --, .

r-5"--"")-:-�� I

•

L::J-J

'-- 3"--l 1..- S"--1

�

£_-.

�,:��

..

r- 3 ---.

-

I

I'

r-- 3----!

'-J

�f.

..

........_ 5"---'

....,

i

•

r,1

t-=r-r-•.! l i\f;�� _ _ _ _ _

�·

1-J

.-;�

-·�

Yc/, 2

., •

I

II

s···:

5---, ll r- " , v

1

'-- 5'---1

,--;- 5 ---,

•

"

gr.Tr:

.-� '�-=

lbl.. � r:.- :,;::-- 3 -,

· "' ---

H.·BI

J

��

M

Cl> 0 .0 0

.,.,

"' cii

C> c 0 ·.: u

U) Co

.. c "' > Cl> Cl>

:::1

c. "' :::1

-5

0

-

.. ., .,., c

>

c:

:::1 0 0

:9,=

G> > CD G>

:0

(Ill II]

2RS&

·�

-� <:)

t::

'5

-� ""(

0

co

:E

� ·� co

�

., .,

., .,

c >

Ill

E "'

>

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

<

Vm

<

a.

v2) ; and

the

c. i t s

Assimilate time to a l in e and make each moment of that line. It is as if one were to distri bute a number of points on a line with a linear density S 4.5 poin t s at MM 26. This, then, is a problem of continuous probabilities. These p oin ts define segments and the probability th at the i-th segment will have a length x1 between 5;

departu re a point

on

=

x

and

x

+

=

dx is

Hypothesis 6. The moment of d epar tu re corresponds to a so u n d . We shall attempt to define its pitch. The strings have a range of a b o u t 80 semi­ tones, which may be rep resented by a line of length a 80 semi tones. Since between two successive or simultaneous g l iss a n d i there e x i sts an interval between the pitches at the moments of departure, we can define not only the note of attack for the first glissando, but also the m el od i c interval which separates the two origins . =

Formalized Music

34 s

Put

problem consists of finding the probability that a segment segment of l ength a will have a length between j and j + dj This probability is given by the formula

thus, the

within a l i n e

(0

:::;:

j

:::;:

a) .

O (j)

Hyp othesis

dj

=

� (I - �)

(See Appendix

dj .

7 . The three essential ch aracteristics o f

defined in Hypothesis 4

are

the glissando

independent.

we

1.)

sound

draw up the three tables of probability : and a table of intervals . All these t ables fu r n is h us with the e l e m e n t s which materialize in cell I I I , t z . The reader is encouraged to examine the scor� to see how the results of the calculations have been used. Here also, may we emphasize, a great liberty of choice i s g i v en the com poser. The res t ri c t io n s are more of a general can alizing kind, rather than peremptory . The theory and the calculation define the tendencies of the sonic entity , but they do not con­ stitute a slavery . Mathematical formulae arc thus tamed and subjugated by musical tho u gh t . We have given this example of glissando sounds because it contains all the problems of stoch asti c music, con trolled, up to a certain point, by calculation.

a

From these hypotheses

table

can

of du rations, a table of s p e e d s ,

Table of Durations Unit

4.5 X

·

8

=

x =

6.5

=

Sx

4.5 sounds/measure at MM 2 6

0. 1 0

of t h e

e - 6x

at 2 6 M M i.e., 28 durations

measure

2 9 sounds/cell,

se - 6,.

se - 6X dx

28P..,

0.00

0 . 00

1 .000

4 . 5 00

0 . 36 2

10

0. 1 0

0.45

0.638

2.870

0.23 1

7

0 .9 0

0 . 40 7

1 .830

0. 148

1 .35

0.259

1 . 1 65

0.094

0.40

1 .80

0 . 1 65

0 . 743

0.060

0.50

2 . 25

0 . 1 05

0.473

0.60

2.70

0.067

0.302

0. 7 0

3.15

0.043

0.20 0.30

Totals

0 . 03 8

4

3

2

0.024

1

0. 1 94

0.0 1 6

0

1 2 .41 5

0.973

28

Free S tochastic

35

Musi c

An approximation is made b y considering

dx a s a constant factor.

co

L 8e - 6x dx 0

Therefore

dx I n this case dx

1 / 1 2.4 1 5

=

=

=

= I.

00

I /2. 8e - 6x .

0.805.

0

Table of Speeds

4.5 glissando sounds/measure at 26 MM 3.88, quadratic m e a n of the speeds v is expressed in scmitonesfmeasure at 2 6 MM Vm is the mean speed (v1 + v2) /2 29 glissando sounds/cell. 4.5 6.5 8

=

a =

·

=

v

A = vfa

O(A)

0

0.000

0.0000

0.258 2

0.5 1 6

P(A)

=

0(A2)

-

0(A1)

29

P(A)

0.2869

9

0.5

0.25 1 0

7

1 .5

0. 1 859

5

2.5

0. 1 3 1 0

4

3 .5

0.077 1

2

4. 5

0.0397

1

5.5

0. 0 1 7 9

1

6.5

0.007 1

0

7.5

0.2869 0.5379

3

0. 773

0.7238

4

1 .032

0.8548

5

1 .228

0 .93 1 9

6

1 .545

0.97 1 6

7

1 . 805

0 .9895

Vm

Formalized Music

36 Table S

a

=

=

of Intervals

4.5 glissandi/measure at 26 MM. 80 semitones, or 1 8 times the arbitrary unit

of 4.5 semi tones.

j is ex p ressed in multiples of 4.5 semitones. dj is considered to be constant. Therefore dj 1 /l.O(j) or dj and we obtain a step function. Forj 0, B(j ) dj 2/(m + l ) j 18, O(j) dj 0 . =

=

=

=

=

af (m

=

+ 1 ),

0. 1 0 5 ; for

=

4 . 5 6.5 ·

=

29 glissando sounds p e r cell.

We can construct the

table of probabilities by means of a straight O(j) dj

j

=

P(j )

29

P(j )

2 3 4

3 3 3 3 2

6

2

8

2

5

2

7

2 2

9

10

1

11 12

1 1

13

1

14

0

15

16

0

17

0 0

18 We shall not speak

line.

of the means of verification of liaisons and correla­ used . It would be too long , complex, and

tions between the various values

tedious. For the moment the two formulae :

let us affirm that the basic matrix was verified by

Free

37

Stoch astic Music

an d z

=

r

t log � 1 +

us now imagine music c om posed with the aid of m atrix ( M ) . A n who p e rce i v e d the frequen cies of events of t h e m u s i c a l sample would ded u ce a distribution due to ch ance and following the laws of probability. Now the qu e s ti on is, w h en heard a n u mber of times, w i l l this music keep its su rp ri s e effect ? Will i t not c h a ng e i n t o a set of foreseeable p h en omen a throu gh the existence of mem ory, despite the fac t that the law Let

observer

of ch ance ? o n ly at the first

of freq uencies has been derived from the l aws

In fact, the data will appear al e a to ry

d u ring successive reh e a r in gs th e

relations

hearing. T h e n ,

o f the sample o rd ai n ed by " chan ce " will form a network, which will take on a definite m e a n i n g in the mind of th e listener, and will initiate a special " lo g ic," a new cohesion capable of satisfying his in t el le c t as well as his a e s t h et i c sense ; that is, if the ar ti st has a certain flair. If, on the other hand, we wish the s a m p l e to be u n foreseeable at all times, it is possible to conceive that at each repetition c e r t a i n data m i gh t be tran sfo rm e d in such a w ay that their deviations from theoretical fre­ quencies would not be sign ificant. P e r h a ps a programming useful for a first, second, third , etc. , performance will give aleatory s am ple s tha t are not i den ti ­ cal in an absolute sense, whose deviations will also be distri b u ted by chance. Or agai n a system with el e c tro ni c computers might p e r mi t variations of the parameters of e n tr anc e to the matrix and of the c l ou ds , under certain co nditi o n s . There would thus a rise a music which can be di s to r t e d in t h e co urse of t i m e , giving th e same observer n r es u l ts apparently due to chance for n performances. In the l on g run the m u s i c will fol l o w the laws of pro b a­ bility and the performances will be statistically iden tical with each other, the i d e n t i ty being defined o n c e for all by the " vector-matrix." T h e sonic scheme d e fi n e d under this form of vecto r-matrix is co nse­ quently capable of establishing a m or e or less self-determined regulation of the rare son i c events contained in a musical comp o si t i o n sample. It repre­ sents a comp ositional a t ti t u d e, a fu ndamentally stochastic b eh a vi o r, a unity [ 1 956-5 7] . of s u pe ri o r order.

be tw e e n t h e

events

If the first steps may be summarized by the process vision -+ rules -+ the question concerning the minimum has produced an inverse

works of art,

Formalized Music

38

path : rules ---+ vision.

example of Achorripsis

In

fa c t stochastics perm i ts a phi losoph i c v isi on , as th e

bears witness.

C H A N C E-I M P R OVI S ATI O N

the essence o f m u si ca l composition, we speak of the principle of improvisation which caused a furore among the nco-serialists, and which gives them the r i gh t , or so t h e y think, to s p eak

Before

generalizi ng further o n

must

the ale atory, which they thus i n t ro du ce i n to music. They which certain combinatio ns of sounds m ay be freely chosen in terpreter. It is evident that these composers consider the various

of chance, of

write

scores i n

by the

p o ss i b l e circuits as equivalent. Two l og i c a l infirmities

are

deny them

and " com p osition "

on

to

the righ t

to

speak of chance on

the other ( co m p osi tion in

the

one h and

apparent which

the broad se n s e , that is) ; 1 . The interp ret e r is a highly con di tio n e d being, so that it is not possible

of unconditioned choice, of an i n terp reter a c t ing like a roulette game. The martingale betting at Mon te C a r l o and t h e p ro c ess i on of suicides should c o n v in c e anyone o f th i s . We s h all return to this. accept

the

thesis

2 . The composer commits an act of resignation when he admits several possible and equivalent circu i ts . In the name of a " scheme " the problem of choice is betrayed, and it is the interpreter who is promoted to the rank of co m p o s e r by the co mpo ser himself. There is thus a substitu tion of authors. The extremist extensi o n of th is atti tude is one which uses g raphical si g n s o n a piece of paper wh ich the interpreter r e a ds while improvising the whole. The two infirmities m e n t i o ned above are terribly aggravated here. I would like to pose a question : If this sheet of paper is put before an inter­ preter who is an incomparable expert on Chopin, will the result not be modul ated by the style and writing of C h o p in in the same way that a per­ former who is immersed in t h is style might improvise a Chopin-like cadenza to a no th er composer's concerto ? From the point of view of the composer there is no interest. On the contrary, two conclusions may be draw n : first, that seri al composition h as b e co m e so banal that it can be im provised l i k e Ch opin's, wh ich confirms the general impressio n ; and se c o n d , that the composer resigns his fu nction altogether, that he has nothing to say, and that his function can be taken over by pai ntings or by cuneiform glyphs. Chance needs to b e calcu l ated

To

finish with the thesis of the roulette-musician, I shall add this : a rare th i n g and a sn are. It can be construc ted up to a certain

Ch ance is

39

Free Stochastic Music point with great difficulty, by m ea n s of complex reasoning which is

sum­

marized in mathematical formulae ; it can be constructed a little, but never

or intellectually imitated. I refer to the demonstration of the mathemati­ cian Emile Borel, who was one of the specialists in the calculus of probabil­ ities. In any case-to play with sounds like dice-what a truly simplistic activity ! But on c e one h as emerged from this primary field of chance worth­ less to a musician, the calculation of the aleatory, that is to say stochastics, guarantees fi rst that in a region of precise definition slips will not be made, and then furnishes a powerful m e th o d of reasoning and enrichment of s o n i c improvised

impossibility of i m ita t i n g chance which was made by the great

processes.

STO C H AST I C PAI N TI N G ?

In line with these ideas, Michel Philippot introduced the calculus of probabilities into his p ain t i n g several years ago, thus opening new direc­

tions for investigation in this artistic realm. I n music he recen tly endeavored

to an alyze the act of composition in the form of a .flow

chart for an imaginary to a chain of a l ea to ry or deterministic events, and is based on th e work Comp osi­ tion p our double orcheslre ( 1 960) . The term imaginary machine means that the composer may rigorously define the entities and operating methods, just as on an electronic computer . In 1 960 Philippot commen ted on h i s Composition

machine.

It is a fundamental analysis of volu n tary choice, which leads

pour double orchestre : If,

in

co n n e c t i o n

with this work , I happ ene d to usc t h e

term in

"experimental music , " I s h ou l d specify in w hat sense it was mea n t

this p articular case. It has noth i n g to do w i t h

c o ncr e te

or electronic

music, but with a very banal sc o re w ri tten on the usual ru le d p aper

an d requiring

none

but the most trad i tional orchestral instrume n t s .

the experimen t of w h i ch this com p osi t ion was in s om e sense a by-product does exist ( and on e can think of man y industries t h a t survive onl y through the exploitation of their b y - pro d uct s ) . The end sought was m e rel y to effect, in the con text of a work wh i c h I would have writ ten independent of all expe ri m e n t a l ambi­ tio ns , an exploration of the processes fol lowe d by my own cerebral mech anism as it arranged the sonic elements. I therefore devised the However,

fol lowing steps :

1 . Make t h e mos t complete inventor y possible of the set of my gestu res , ideas, mannerisms, decisions, and choi ces , e t c . , which were mine when I wrote the music.

Formalized Music

40

2. Reduce this set to a succession of simple decisions, binary,

if

possible ; i . e . , accept or refuse a particular note, duration , or silence in a situation determined an d defined by the context on one hand, and

by the conditioning to which I h ad been subjected an d my personal tastes on the other.

3. Establish, i f possi ble, from this sequence of simple decisions, a

scheme ordered according to t h e fol lowing two considerations ( which

were som etimes con trad ictory ) : the manner i n which these decisions

emerged from my imagination in the cou rse of the work , and the

manner in w h ich they would have to emerge in order to be most

usefu l .

4. Present t h i s scheme in t h e form of a flow chart containing the

of these

logical chain

be

con trolled.

decisions, the operation of whi�h could easily

5 . Set in motion a mechanism of simulation respecting the rules

of the game in the flow chart and note the resul t .

6. Compare this resul t with m y musical in tentions.

7. Check the diffe rences be tween result and intentions, detect their

causes, and co r re c t the operating rules.

8. Refer these corrections back to the sequence of exp erimental

phases, i . e . ,

start

again a t

1.

tained.

If

u n t i l a satisfactor y resu l t h as been ob­

we confine ourselves to the most general considerations, it

woul d simply be

a

matter of proceeding to an

ity, considered as an accumu l ation, in

a

a n al ysi s

certai n o rd e r,

of the complex­

of single events,

and then of reconstructing this complexity, at the same time verifying the nature of the elements and their rules of combination . A cursory

look at the flow chart of the first mere glance the

me

th o d I

movement

specifies quite well by a

use d . But to confine oneself to this first

movemen t wou ld be to misunderstand the essentials of musical composition .

In fact the "preludial" character which emerges from this

combination of notes

(elementary constituents of the orchestra)

should remind us of the fact that composition i n its ultimate stage is

also an assembly of groups of notes, motifs, or themes and their transformations. Con sequen tly the task revealed by the flow charts of

the fol lowing movemen ts ough t to make conspicuous a gro u pi ng of

a higher order, in which the da ta of the first movement were used as

a

a

sor t of " p refabricated" material. Thus appeared the phenomenon ,

rather banal one, of au togen eration of complexity by jux taposition

an d combination of a large number of single events an d operations.

A t the end of this experiment I possessed at mos t some insigh t in to

my ow n musical tastes, but to me, the obviously interesting aspect of

Store in me mory

I nitial note +

d uration +

)t i

I I I

H orizontal or

I I I I

vertical

v

and i n te nsity

.----- � ::)_ n? ,_;;---

------ · -

---

I I yes --- - - ---

l

in c

�1

Note

- --

I

I I

Store .m no

note

_

yes

Start

no

�----'-� total 1 2 7

( Is the memory

I

)':::;:----'

next cell

Fig . 1-1 1 . Composition for Double Orchestra, by M i c hel Philip pot, 1 959 Flow Chart of the First Movement

-l

�

Choose i n itial

_j-E-----+--' ---------, -I

1�--D ra_ w_nex _ t_ not�

f Verify if \.._f_ alse relatio n

1r

yes

'� as previously and verify

�

no-

Formalized Music

42

i t (as lon g as there is no error of omission ! ) was the analysis of the

composer,

his men tal processes, and

imagination .

The b iggest difficulty encountered

a

certai n

was

liberation

of the

that of a conscious and

vol u ntary split i n personality. On one han d , was the composer w ho already had

a

cle ar idea and a precise audi tion of the work he wished

to obtain ; and on the other was the experimenter who had to maintain a l u ci d ity which rapidly

became

burdensome i n these conditions-a

lucidity with resp ect to his own ges tu res and d ecisions. We must not

ignore the fact that such experiments must be examined with the

greatest prudence, for everyone knows that no observation o f a phenomenon exists w hich does not disturb that phenomenon, and I

fear that the resul ting disturbance migh t be particularly strong

when

it concerns s uch an ill-defined domain and such a delicate activity.

Moreover, in t h is particular case , I fear that observation migh t pro­

voke i t s own disturbance. I f I accepted this risk, I did not under­

estimate its exte n t . At most, my ambi tion confined itself to the attempt

to proj ect on a m a r ve l o u s un know n , that of aesthetic creat ion, the

timid light of a dark lantern. (The dark lan tern had the repu tation

of bein g used especially by housebreakers . On several occasions I have been able to verify how much my thirst for investigation has mad e

me

appear in the eyes of the majority as a dangerous housebreaker of

inspi ration . )

C h a pter I I

Markovian Stochastic Music-Theory Now we can rapidly generalize the study of musical composition with the aid of stochastics. The first thesis is that stochastics is valuable n ot only in instrumental music, but a l so in electrom agnetic music. We h ave demonstrated this with several works : Diamorphoses 1 95 7-58 (B.A.M. Paris) , Concret PH (in lhe Philips P a v i l i o n at the Brussels Exh i bition , 1 958) ; and Orient- Occident, music for the film of the same name by E . Fulch i gnoni, produced b y UNESCO in 1 960. The second t h esi s is that stochastics can lead to the creation of new sonic materials and to new fo r ms . For th i s p u rpose we must as a preamble put forward a temporary hypothesis which concerns the n a t u r e of so u n d of

all

,

sound [ 1 9] .

B A S I C T E M P O R A RY H Y P OTH ESIS

( lemma )

All so u n d i s a n integr ation o f grains,

. A N D D E F I N IT I O N S

o f elementary sonic particles, of these elementary grains has a th ree fol d nature : duration, freque ncy, and i n tensity . 1 A l l soun d , even al l continuous sonic vari atio n , is conceived as an assemblage of a l a r g e number of elemen tary grains adeq u atel y d i s p o s e d i n time. So every sonic complex can be analyzed as a s e ri es of pure sin usoidal so u n d s even i f the vari ations of these sinusoidal sounds are i n fin i tely close, short, and complex. In the attack, body, and decl ine of a complex sound, t h o u s a n d s of pure s o u n d s appear in a m o re or less short interval of time, 6-t. Hecatombs of pure sounds arc necessary for the cre at i on of a complex s o u n d . A complex s o u n d m ay be i m agined as a m u l ti-colored fi rewo r k in which each poi n t of l igh t appears and i nstansonic quanta. Each of

43

44

Formal ized Music

t an eo u sly d isapp ea rs

ag a in s t

a

black

sky. But in

th is firework there would

be such a quantity of points of l i gh t org anized i n such

a

way that their rapid

teeming s u cce s si on would create forms a nd sp i r als, slowly unfolding, or conversely, br i e f explosions s e tti n g the whole sky aflame. A line oflight would and

b e created by

a

sufficiently large multitude of p o i nts ap p ea ri n g and dis­

appearing instantaneously.

d u ratio n t!. t of th e grain as quite sm all but invariable, follows and consider frequency and in te nsity only. The two physica l substances of a soun d are fre q u e n cy and i n te n sity in a ss o c ia ti o n . They constitute two sets, F and G, inde p endent by their nature. They have a set p roduct F x G, which is the elementary grain of s o un d . Set F can be put in an y kind of correspondence with G : man y-valued, s in g l e ­ valued, o n e- t o - on e mapping, . . . . The corres p ondence can be given by an If we consider the

we can i gnore

it

in what

extensive representation, a matrix representation,

or

tation.

EXAMPLES

OF

Extensive

R E P R E SENTATI O N S

(term by term ) :

fl f2 fa f4 gn g3 g,.

l g3

Frequencies

I nte nsi t i e s

Matrix (in

the form of a

.j.

gl

Canonical

a canonical represen­

(in

table ) :

fl f2 fa f4 f5 fa f7 +

0

+

0

0

0

g2

+

0

+

0

0

0

+

0

g3

0

0

0

+

+

0

0

the form

of a function) :

vf = Kg f frequency =

g

K

=

intensity

=

coefficient.

The corres p ondence may also be indeterminate (stoch astic) , and here

the most

co n ven i e n t representation is the matrical one, which gives the

t ra nsi ti o n probabilities.

45

Markovian Stoch astic Music-Theory Exam ple :

t fl g2 0 ga

The t abl e

0.3

0.5

0

0.3

0.7

be interpreted as

should

0.2

0

0.5

gl

!4

fa

!2

0.5

0

fo l l o w s : for

one or several corresponding i n t en s ity values

For example, the two in tensi ties

g2

each valu e }; off there

g1,

defined by

a

are

probability.

and g 3 corres p ond to the frequency /2,

with 30% and 70% chance of occurrence, respectively. On the other hand,

each of the two sets F and

G

can be fu rnished with a structure-that is to

say, inte rnal relations and laws of composition. Time

t is considered

lexicographic form.

as a to t a l ly ordered set mapped onto F or G in a

Examp les :

a . !1 !2 fa t

b.

t

1 , 2,

=

c.

t

=

fo.5 fa ].;u fx =

0.5, 3 , y l l ,

ft fl 12 fl 12 f2 J,. fa · " · · · " · A B c D E �� ��

� � �� ��

x,

At At

�� = At Example

into sli c e s

makes it

much

GRAP HICAL We

c.

is the most general since continuous e vo l u tio n is sectioned

of a single thickness At,

can

which

t ra nsforms it in d i scon tinuity ; this

easier to isolate and examine

R E P R E SENTATIONS

plot

u nder

the magnifying glass .

the valu es o f pure freq uencies i n units o f octaves o r semi­ axis, and the intensity values in decibels on the ord i n ate

tones on t h e abscissa

axis, using log ari th m i c

scales (see

Fig. II-1 ) . This cloud of points is the

cylindrical projection o n the plane (FG) of the grains contained in a t h i n

slice

The graphical representations Figs . I I-2 and I I-3 the ab s tr a c t p ossibilities ra i s e d up to this poi n t .

�� (see Fig. I l-2) .

m ake m o re tangible

Psychophys i o l ogy

We are confronted with

a

cloud of evolvi ng points. T h i s cloud

is

the

product of the two sets F and G in the slice of time �t. What are the possi blc

46

Formalized Music G

(dB)

Elemen tary grain considered as a n i nsta nta neous associ ­ ation o f an i n te nsity g and a fre q u e ncy f . . .

. .

Fig.

1 1-1

Freq ue ncies i n log arith mic u n its (e.g., sem iton es) L-------------------:-

F

G

G

'

\

�----�--------------------��- F

Fig. I�

F

Fig.

1 1-3

'1.--- F

t

-<- Point i Each point i must be attributed to an i ntensity g1

·.: ...... ,

t

47

Markovi an Stochastic Mu sic-Theory

restrictive limits of h u m a n psychophysiology ? What are the most gen e r a l manipulations which may be i m p o sed on the tions within psychop hysiological limits ?

The

cl o u d s

and their tra n sfo rm a ­

basi c abstract hypo thesis, which is the granular construction of all

p o ss ible sou n ds, gives a very profound meaning to these two q u estions. I n

fact within h u m a n limits, u s i n g

ail

sorts

of m anip u l ations

with these grain

cl us te rs , we can hope to produce not only the s o u n d s of classical instruments and elastic bodies, and those so u nds generally p referred in concrete music, but also sonic perturbations with evolutions, unparalleled and u n i m aginable

u n til now. The basis of the timbre structures and transformations will have noth ing in co m mo n w i th what has b een known u n t i l now. We

can

even express

a m or e

general

p o i n t of these clu sters represents not only

a

intensity, but an already present s t ru c t u r e

su p p osition .

pu r e

S uppose that

frequency

of elemen tary

each

and its satellite

grains, orde re d a

priori. We believe that in this way a sonor i ty of a second, third, o r h i gher order can be p rod uced .

Recent work on hearing h a s given satisfactory answers to certain

problems o f pe rcep tion . The basic p roblems which concern us and which

we shall suppose to be resolved , eve n if so m e of the solutions l a c k i n g,

are

[2, 3] :

fort) o f a sinusoidal sou nd, as a fu nction o f i ts frequency and

2. What arc the

are

in part

com­ its intensity ?

1 . What is the minimum pe rce p tible duration (in

m i n imum values

of inte nsities

in deci bels compatible with

minimum frequ encies and du rations of sinusoidal so u n ds ?

3.

What

are

the

minimum melod ic i n te rval thresholds, as a fu nction of register, intensity, and d u ra ti o n ? A good

e

q u al l o u d n ess

car

approximation is the

conto urs (see Fig. I I-4) .

Fletch er-Munson d i agram

The total n u mber of elemen tary audible grains is a b o u t

is more sensi tive at the center o f the aud i b l e

are a .

340,000.

of

The

At th e extre m i ties i t

pe rceives less a mplitu d e and fewer melodic i n tervals, s o that i f one wished to

represent the

audible area i n

a

homogeneous manner using the coordin ates

F and G, i . e . , with each s u rface element

of g r a in s of perce p tible so u n ds, o ne (Fig. II-5 ) .

6.Ft1G contai n i n g the same density would o b t a i n a sort o f mappa mundi

In order t o simplify the reasoning which will fol lo w without al tering it,

we shall base o u r arg u m e n t o n Fletcher's diagram a n d s u ppose that an a pprop r i a t e

one-to-one trans formation applied to this group of c o ordinates

will change t h i s curved space All the

above

into

an ord i n ary rectangle

(Fig. II-6) .

ex p erimen tal resu l ts were established in id e a l co n d i tions

and without re ference to

the actual

com plexity o f the n atural s o u n ds of the

orchestra and of clastic bodies i n gen eral, not to mention the more com plex

Formalized Music

48 G

+ Limit of the

90

�

�

80 7a Ga

-- - . -

sD

------.....__

r-----

�

�

��I'-"""

·-- -

-MoO O

·-·

r--r- -

"' I'-

r--.-

!'--

��

--

,_

..

-

�....____ ----

�

� '-,

'"-�

---

""'-

·-·-

3D

:----

�

40

Go

-'"'

�� � """

Contours

.._ Th resh old of sou nd perception

F i g . 1 1-5

........___

r=::-

Fig . 1 1-4. Fletcher- M u ns o n D iagram

F

"•'------

(Semitones)

0

/

1)1

v( VI

----....

r----,......__....

J

/_

/ /

I

--

/

---

-- ./

-

8o fDD

Equal Lo ud ness

'·"---

'-----

Audible area

G (d 8)

-

-....___

"'-�

:So 2<1

r--

------- ......___ I /v-

I-

-

�

audible area

· ·- · ·· --

{/0

/

/"'"

/

lloe>O

F

49

Markovian Stochastic Musi c-Theor y

-

--....

� I

F'

sounds of industry or of chaotic n ature [4] .

Theoretically [5]

I

a complex

sound can only be exhaustively represented on a three-di mensional diagram F, G, t, giving the instantaneous frequency and in t e nsi ty as a function of time . But in practice this boils down to saying that in

m o m ent a ry sound,

such as a simple noise made

tions and graphs are necessary. This

order

to represen t a

months of calcula­ impasse is strikingly re m i n is c e n t of by a car,

classi c al mech anics, which claimed that, given sufficient ti m e, it could

accoun t for all physical and even biological phenomena us i n g only a few formulae. But just to describe the state of a gaseous m ass of greatly re duced volume at one instant t, even if simplifications are allowed at the beginning of the calculation, would require several centuries of human work ! This was a fal se problem b e c a u se it is useless ; and as far as gaseous masse s are concerned, the Maxwell-Boltzmann k in et i c theory of gas es, with its statistical method, has been very fru i tfu l [6] . This method re­ established the value of scales of observation . For a macrosco p ic phenomenon it is the massed total result wh i ch counts, and each time

a

p h e n o m e no n is

to be observed the scale rel a tionship between observer and phenomenon

must first be established. Thus if we ob s e r ve galactic masses, we must decide

whether it is the movement of the whole mass, the movement of a s i n gle sta r,

the mo l e cu l ar constitution of a minute region on a star that interests us. The same thing holds true for c o mpl ex as well as quite simple sou nds. It would be a waste of effort to a tte mpt to accou n t analytically or graph i­ cally for the characteristics of complex sounds when they arc to be used in

or

an electromagnetic c o m p osi tio n

.

For t h e manipulation of these sou n d s

m acroscopic m e th o ds are necessary.

I n vers e l y , and this is what p a rt i cul arl y interests us here, to work l i ke architects on the so n i c material in order to construct complex so u nds and

50

Formalized Music

evolutions of these entities means that we must use macroscopic methods o f analysis and construction. Microsounds and eleme n tary g r ai ns h av e

importance

on

no

the s ca l e which we have chosen . Only gro u p s of grains and

the characteristics of these groups have any meaning. N aturally in very particular cases, the single g ra i n will

be reestab lished

in all its glory.

In

a

Wilson chamber it is the elementary particle which carries theoretical and

experimental physics on its shoulders, while in the sun it is the mass of

p articles and their compact interactions which co nst i tu t e the

but

solar obj ect.

Our field of evolution is therefore the curved space described above,

simplified to a rectilinear space by means of complete one-to-one

transformation, which safegu ards the validity of the

re aso n ing which we

shall pursue.

S C R EE N S

The graph i cal representation o f a cloud o f grains i n a sl i c e o f time ll.t

a new concept, that of the density of grains per unit t1Ft1Gt1t (Fig. I I-7) . Every po ssib le sound may therefore be cut up into a precise quantity of el eme n ts !:iFt1Gt1tt1D i n fou r dimensions, distributed in this space and following certain ru les defining this sound,

examined earlier brings of volume,

which are summarized by a function with four variables : s(F, G, D, t) .

G

AF

Pla n e of

reference

{FG)

at moment t

/

F

tlD wi l l be the dimension of the density

t.

F i g . 1 1-7

51

Markovian Stochastic Music-Theory The scale of

the

density will also

be logarithmic with its base between

2 and 3 . 2 To simplify the expl a n ation we will make an abstraction of this

It will always be present in our mind but as an with the three-dimensional element 11FI1GI1t. If time is considered as a procedure of lexico g raphic or d eri n g we can,

n ew coordinate of density. entity associ ated

,

withou t loss, assume that the 11t are equal constants and quite small. We can

thus reason on a two-dimensional space defined by the axes F

condition that

we

and G, on

do not lose sight of the fact that the cloud of grains of

sound exists in the thickness of time

artificially flattened on the plane

Difinition of the

11t and that the grains of sound are only

(FG) .

by a as defined a b o v e the cells of which may or may not be occu pied by grains. In this way any sound and its history may be described by means of a sufficiently large number of sheets of paper carrying a g iven screen S. These sheets are placed in a fixed lexi­ screen. The screen is the audible area (FG) fixed

sufficien tly cl os e and homogeneous grid

,

cographic order (see Fig. I I-8) .

- S c reen

Cell full of grains

A book of scre e n s e q u a ls of a c o m p lex so u nd

Fig . 1 1-8

to

the l ife

The clouds of grains drawn on the screens will differ from one screen another by their geographical or topographical position and by their

surface density

(see Fig.

I I-9) . Screen A contains

a

small elemental rectan­

gle with a small cluster of density d of mean frequencyf and

mean

amplitude

g. It is almost a pure sound. Screen B re p resents a more complex sound with

strong high

and low areas but

with a weak cen ter. Screen

C

represents

a

Formalized Music

52 " white "

sound of we a k density which may

therefore be p er ce i v ed as a

sheen occupying the whole audible area.

has

What is important in all the statem e n ts made up to now is that

sonic

nothing

b ee n said about th e topographic fixity of the grains on the screens . All

of small surface ele m en ts filled intensity. The is fundamen tal, and it is very

natural or instrumental sou nds arc com p osed

with g r a in s which fluctuate arou nd a mean frequency and

same holds for the density. This statement

of e l ec tronic music to c re a te n ew timbres, aside from inadeq uacy of the serial method, is largely due to the fixity of the grains, which form structures like packets of sp agh e tti (Fig. I I- 1 0) . Topographic fixity of the grains is a v ery p ar t i cu l ar case, the most ge n eral case being mobility and the statistical distributim• of gr ai n s around positions of equilibrium. Conse q uently in the majority of cases real sounds can be an alyzed as quite small rectangles, !:l.F!:l.G, in which the topographic positions and the d e n si ties vary from one screen to another following more

likely that the fail u re the

or

less well-defined laws .

Thus the sound of example

collection of rectangl es

D

at this precise instant is formed by

the

(j2g4 ) , (f2g5) , (f4g2) , (f4ga), (fsgt ) , (f5g2) , Uagl) ,

(fag2) , Usgs) , (f1g2) , (f7g3 ), (f7g4), (f7gs) , Uag3), (feg4), (fags) , an d i n e ac h

of the rectangles

manner

the grains are disposed

( se e Fig. 11-1 1 ) .

CONSTRUCTION

OF

TH E

E LEMENTS

in an asymmetric and homogeneous

fj.p[j. G

OF

THE

S C REENS

I . By calculation. W e shall examine the means o fcalculating the elements

!:l.F!:l.Gat/j.D.

How shou ld the g ra i n s

fix the

be

d istributed in an elemental volume ?

mean density of the grains

(

=

number of grains

If we

per unit of volume)

we h ave to resol v e a problem of p robab i l i ty in a four-dimensional space . A simpl er method would be independently. lS :

For

For the

to co n sid er and then calculate the four coordinates

coordinate t the law of distribu tion

P"

=

c e - c" dx

or

Px,

=

of gr ain s on

e- •1v ctu1•

the coordinates G , F, D t h e s to ch a s ti c l a w w i l l

or

2

pi

= 2 . t on

( l

i

2 . wn

(r)

the axis of time

(See Appendix i . )

be :

- 1) .

(Sec Appendix I . )

53

Markovian Stochastic Music-Theory G

G

· :·. ·

.. .. . .

( - - ---- -·

I

G

F

A

Fig. ll-9

.. . .: · . ·-:·:

F

8

::

·.

:

c

G

Fig. 1 1-1 0

�------�� t

G

(dB}

(s

ft

t3

t'L (,

. .. :� : .. ..

/, h h

•

I�

D

F i g . 1 1-1 1

h-

h

�

I',

-

·•

=.: :

-

••

r1

F

(Semitones)

F

Form alized

54

Music

F rom th ese formu l ae we can draw u p tables of the frequencies of the G, F, D (see the a n al ogo u s pr o bl e m i n C h apter 1) . These formulae are in our opinion privileged, for they arise from very simple reasoning, prob ably the very simplest ; and it is essential to start o u t with a m i n imum of t er ms and constraints if we wish to keep to the principle of the tabula rasa ( 1 st and 3rd ru l es of Descartes's Discou rse o n Method) . Let there be o ne of these elemental v ol u m e s /!:,ti:!.DAF!':!.G o f th e screen at the m o men t t. This vo l u m e has a density D taken from the t abl e derived from form u l a (r' ) . Points on !':!.t are d efined with a linear density D c according to the tab le defined by fo r m u l a (r) . To each po i n t is attributed a sonic grain of frequency f and i n te ns i t y g, taken from within the rectangle !':!.F!:,G by means of the table of frequencies derived from fS> rmul a (r') . The corres p ondences are made graphically o r by random successive drawings from urns composed according to t he above ta ble s. 2. Mechanically. a. On the tape recorder : The grains are realized from sinusoidal so u n d s whose durations ar e constant, about 0.04 sec. These grains must co ver the selected elemental area !':!. F !':!.G. U nfo ld i ng i n time is accomplished by using the table of durations for a minimum density c D.

values t,

=

B y mi xi n g sections o f this tape with itsel f, w e geometrically with

we use.

b. On

programmed

ratio

I : 2 : 3 . . . according

can

to

=

obtain densities varying

the number of tracks

computers : The gr a i n s are realized from wave

a cco r d i n g

to Ga b o r ' s signals, for

a

that

forms duly

co mp u t er to which

analogue converter has been coupled. A second p rog r am would provide

an

for

the construction of the elemental volumes !':!.tAD!':!.FAG from formulae (r) and (r') . F i rst G e n e ra l Com ment Take the cell

!':!.F!':!.GAt. Although

occupied in a homogeneous

manner

by g ra i n s of sound, it v ar i es in time by fl uctuating around a mean density dm. We can

these

apply

another

arg um en t which is more synthetic, and admit that most general case anyhow (if tbe sound is

fluctuations will exist in the

long enough ) ,

and will therefore obey the laws of chance. In this case, the put in the following manner : Given a prismatic cloud o f grains o f density dm, o f cross s ec ti o n tl.Fll. G and length 2 !':!.t, what is t h e probability that d grai n s will be found in a n elemental vol ume !':!.F!':!.Gt3. t ? If the number d'" i s small enou g h, t h e prob abil­ i ty is given by Poisson's formula :

pro b le m is

P

�<

=

{dm)" e - d., . K!

Markovian S t oc h a sti c Music-Theory

For the definition of e ach

grain

scribed above .

55 we

s h all

again use the me tho d s de­

Second Genera l C o mment (Vector S pace ) [8] We can construct e l e m e n tal cells tl.FD.. G

el em en tal vectors

po i nts , but w i th

of the screens

no t

only

with

associated with the grains ( v e c to r space ) .

The mean density of 0 . 04 sec/grain really im pl i e s a small vec tor . The p ar tic­

ular

case

of the grain occu rs when the vector is p arallel to the axis of time,

whe n its projection on

the grain is constan t. can be variable

and

�

the

p l ane (FG) is a point,

and when the

frequency of

In general, the fre q uencies a nd intensities of the gr a i n

the

grain

a

very sho r t

glissando (see

s

Fig. 11- 1 2) .

A P

/

/

I

F

AF

F

\ t

Fig. 11-1 2

I n a v ec t o r space (FG) thus d e fi n e d , the construction of screens would e r haps be cumbersome, for it wou l d be necessary to introduce the idea of p speed and the statistical distribution of its values, but the interest in the undertaking wo u l d be enormous. We cou l d imagine screens as the basis of granular fields which are magnetized or completely ne u t r a l (disordered) . I n t h e case of total disorder, we can

the existence of a vector

app l ied to

two

v

c al c u l a t e

the probability f(v) of

o n the plane (FG) using Maxwell's fo rm ul a

d ime n si ons [ I I ] :

f( v)

as

Form alized

56

For the mean v al u e

in

which A1

=

v1

vm

:s;;

[1 3].

v2,

v1fa and

for A.1 :s;; A. :s;; A.2 (normal

matter of a

:s;;

Music

vector space

Gaussian

or

a

scalar

law) [ 1 2] . In any case, wheth er it

is

a

space does not modify the arguments

Su m mary of t h e Scree ns

I . A screen is described by a set of clouds th a t are themselves a set of rectan gles M!l.G, a nd which may or may not contain grains of sound. These conditions exist at the moment t in a slice of t i m e M, as small elemental

as desired.

2. The grains of sound create a density peculiar to each elemental !l.F!l.G and are generally d istribu ted in the rectangles in an er­ god i c manner. (The ergodic p ri nci p l e states that the capricious effect of an operation that depends on chance is regularized more and more as the rec ta n gl e

ope ra tio n is repeated . Here it is understood that screens

is

being considered [ 1 4] . )

3. The

a

very large succession of

volume !l.F!l.G!l. T!l.D is such that Simultaneity occurs when the de n sity is h i gh enough . Its fre q uency is bound up with the size of the density. It is all a question of scale and this paragraph refers above all to realization. The temporal dimension of the grain (vector) being of the order of0.04 sec., no sys tematic overlapping of two grains (vectors) will be accepted when the elementary density is, for ex amp le , D0 1 . 5 grains/sec. And as conception of the elemental

no simu ltaneity of grains is generally

a dmi tte d.

=

the surface distri bu tion of the grains is homogeneous, only chance can

c r e a te th is overlapping. 4.

The limit

for

a

screen

may be only one pure sound (si nusoidal) ,

or even no sound at all (empty scre e n) .

Let

ELE M E NTA R Y O P E RATI O N S O N S C R E E N S

t he re

be

a

vectors on

complex sound. A t

an

i n s t an t

t

of i t s l ife during

a

be re p resented by one or several clouds of grains or the plane (FG) . This is the definition which we gave fo r the

thickness !l.t it can

57

Markovian S toc h as tic Music-Theory

The j u nction of several of these screens in a given order describes of this sonic complex. It would be interesting to envisage in all its generality the m a n n e r of combining and juxtaposing screens to describe, and above all to construct, sonic evolutions, which may be con tinuous or discontinuous, with a -view to playing with them in a composition. To this end we shall borrow the terminology and symbolism of modern algebra, but in an elementary manner and as a form of introduction to a

screen .

or prescribes the life

­

we

fu rther development which

shall not undertake at the moment.

It does not matter whether we place ourselves on the plane

Comment :

of physical phenomena or of perception . In general, on the plane of per­ ception we consider arithmetically that which is ge om e t ri cal on the p hysical plane.

can be ex p ress ed in a m or e rigorous manner.

This

Perception almost isomorphic with a physical a multiplicative grou p . The " almost " is necessary to

constitutes an additive group

excitation

consti tuting

exorcise app roximations. Grains

or vectors on

composed of no grain at

which

is

the plane (FG) co ns t i tu t e

all

or

of several

a cloud. A screen can be or vectors (s e e

clouds of grains

Fig. II-1 3) .

@

B

. .

A

Screen

Screen 1

F i g . 1 1-1 3

To notate that a grain or vector

th e con trary is written

another

cloud

sub-cloud of

Y.

rj: E. I f

belongs to a clo u d

X c

c

c

included in Y or that X is is not ated X c Y (inclusion) . have the following p roper ti e s : for any X.

X Y

Y and Y

and

c

they are indistinguishable

Y

c

Z

imply

X

c

Z.

the re l a t io n i s written :

A clou d m ay contain as little as empty when it contains no grain

a,

a

part

or

X and Y consist of the same grai ns ; X Y (equatio n) . a single grai n . A cloud X is said to b e such that a E X. The empty cl o u d is

X, the clouds

and

3

E, we write a E E X are grains of

all the grains of a cloud

This relation

X

n o ta te d 0 .

a

Screen

2

Y, it is said that X is

Consequently we

When X

a

��

=

Formalized Music

58 E LEMENTARY

These

O P ERAT I O N S

o p e rati o ns app l y equally

well to

clouds and to scree n s . We

can

therefore use the terms " screen " and " cloud " indiscriminately, with cloud and grain as " constitutive elemen ts."

The intersection of two screens

A

an d B is the screen of clouds

b e lo n g to both A and B. This is notated as

A

which B"

II B and read as " A inter

(Fig. 1 1- 1 4) . W h e n A n B 0 , A and B are said to be disjoint (Fig. 11-1 5) . The union of two screens A and B is the set of clouds wh i ch belong to both A an d /o r B ( Fig. II -16) . The complement of a sc re e n A in re l a t i o n to a s cre e n E c o nt ai n ing A is the set of cl o u ds in E which d o not belong to =

A . This is

n o t a te d

(Fig.

1 1- 1 7 ) . The

(A

B)

CEA when t h e re is no possible u n c er t ai n ty about E difference (A - B) of A and B i s the set of clouds of A

B. The immediate co ns e q u e nce is A - B A B) (Fig. 11- 1 8 ) . stop this b o rr o win g here ; however, i t will afford a stro n g e r , conception on the whole, better adapted for the manipu­

which do not belong to n

=

=

CA (A n

W e shall

more precise

l ations and arguments wh ich follow.

D I STI N CT I V E C H A R A CT E R I STI C S O F T H E SC R E E N S

I n our desire t o create sonic complexes from

the te m pora ry accepted

pr i m a ry matter of sound, sine waves (or their replacements o f the Gabor sort) , and to

th an

create

n atu ral s o u n d s

sonic co m pl exes as rich as b u t more

extraordinary very

(using scientifically controlled evolutions on

gen eral abstract p l a n e s ) ,

we h ave implicitly reco g nized the importance of three basic factors which seem to be able to d omin a te both the theoretical construction of a so n ic process and its sensory effectiveness : 1 . the d e nsi ty of the e le m e nt a ry e l e m en ts, 2. the topographic situation of events on the sc re en s, and 3. the order or disorder of events. At first sight then the density of g r ai n s or v e ctors , their topography,

and their degree of order

are

our m acrosco p ic cars. It is

the i ndirect enti ties and aspects perceived by

wonderful that the

ear and the m i n d

follow

objective reality and react directly in spite of g ross inherent or cultural imperfections. Measurement has been the foundation of the experimental

sciences. Man voluntarily treats himself as a s e n so r y invalid, and it is for this reason that he has armed himself,

justifiably,

wi th mach ines that measure

other mach ines. His cars and eyes do measure entities or physical phen­ o m en a , but they

are

transformed as

if a distorting filter c am e

between

im­

mediate perception and consciousness . About a century ago the logarithmic

Markovian Stochastic Music-Theory

F i g . 11-1 5

59

A nB .. :··

� ,. .

·. :.·.\�) ' •"... -:.

U'

A

. : . : . .· . · .

E�:.�t

. . . . ....

.

A U S

Fig. ll-1 6

: )it�(:'' · ·.

" ¢

.

E F i g . ll-1 7

A

A F i g . ll-1 8

B

(A - B )

60

Formalized Music

sensation was discovered ; until now it has not been contradicted. knowledge never stops in its advance, tomorrow's science will with­ out doubt find not only a greater flexibility and exacti tude for th i s l aw, b u t also the begi n n i ngs of an explanation of this distorting filter, which is so astonishing. This sta t isti c a l , but none the less quasi-one-to-one transformation of excitation into perception has up to now allowed us to argue ab ou t physical entities, such as s creens, all the while thinking " perceived events . " A reciprocity of the same kind between perception and its com prehension permits u s to pass from the screens to the consequent distinctive characteris­ tics. Thus the arguments which we shall pursue apply equ ally well to pure concepts and to those re s ulting from perception or sensory events, and we may take the atti tu d e of the craftsman or the listener. We h ave al ready remarked on the density and the topography of grains and cells and we have acknowledged the concepts of orde r and dis­ ord e r in the homogeneous s u p e rfi c i a l distri bution or grains. We shall examine closely the co nc e p t of order, for it is probably hidden behind the o ther two. That is to say, density and topography are rather palpably simplifi ed embodiments of this fleeting and many-sided concept of disorder. When we speak of order or d i sord e r we imply first of all " obj ects " or " element s . " Then, and t hi s is already more complex, we define the very " elements " which we wish to study and from whi ch we wish to construct order or disorder, and their scale in r el a tio n to ours. Finally we qualify and endeavor to measure this order or disorder. We can even d raw up a list of all the orders and disorders of these entities on all scales, from all aspects, for all measu rements, even the characteristics of order or disorder of this very list, and establish anew asp ects and measurements. Take the example of the gases mentioned above . O n the molecular scale (and we could have descended to the atomic level) , the absolute values of th e speeds, directions, and distributions in spac e are of all sorts. We can distinguish the " elements " which carry order or disorder. Thus if we could theoretically isol ate the element " directio ns " and assume that t he re is an obligation to follow certain privileged directions and not all directions, we could impose a certain degree of order which would he i n d epende n t of the other elements constituting the co n cep t " gas." In the same way, given enough time, the values ol" the speeds of a single molecule will be distributed around a m e a n val u e and the size of the deviations will follow Gauss's law. Th ere we will have a certain order since these val u es are vastly more law of

But as

61

Markovian Stochastic Music-Th eory n u merous in the nei g hborhood o f their mean finitely small to infini tely large.

t h an

anywhere else, from i n ­

Let us take another exam p le, more obvious and equally true. A crowd pe rsons is assemb led in a town s q u a re . I f we e x a m i ne the group

of 500,000

d isplacement of this crowd we can prove that

i t does not budge. However,

each individual moves his limbs, his head, his eyes, and displ aces h is center a

of gravity by

terror

If the displacements of the would break u p with ye l l s of

few centimeters in every d i rection.

ce n t e rs of gravity were very large the c ro w d

because of the multiple collisions b et w e en individu als. The statistical

values of these displacements n o rm a l l y

l i e between

they affect immobility, the disorder is

weak.

very n arrow limits which

vary with the density of the c ro w d , From the point of view of these va l u e s as

is the orientation of the fa c es . If an a balcony were to spea k with a calming effect, 499,000 faces would look at the balcony and 998,000 ears wou l d listen to the honeyed words . A Another characteristic of the crowd

orator on

thousand or so faces and 2000 ears would be distracted for various reasons :

annoyance, ima g ination, sexu ality, contempt, theft, etc. We could confirm , along with the mass media, without any possible d i sp u te , that

fatigue,

crowd and speaker were in compl ete accord, that

500,00 1

people,

in

fact,

were unanimous. The d egree of ord e r that the s p e aker was after would attain a maximum

for

a

few minutes at least , and if u n anim i ty

were expressed be per­

equ ally s trongly at the conclusion of the meeting, the orator could suaded

that the

own .

id ea s were as well ord e red

in

the heads

of the crowd

as in his

We c a n establish from these two extreme examples that the concept of order and disorder is basic to a very large n u m b er of phenomen a, and that even the d efin i t i on of a phenomenon or an object is very often attri b u table to

this concept. On the other hand,

fo unde d on

precise

we

can

establish that this concept is

and distinct groups of elements ; that

tant in the choice of elements ;

and

the

sc a l e is impor­

finally, that the concept of order or dis­

order im p li e s the re l a t i o n s h ip between e ffective values over all possible values

that the ele m e n ts of a group can

p ossess . This introduces the concept

of probability i n th e quanti tative estimate of order

or disorder.

We sh all call the number of distinct elements i n

a

grou p its variety. We

or disorder d e fi n a bl e i n a group of elements its entropy. Entropy is lin ked w i th the concept of variety, and for th a t very reason, it is linked to the probability of an eleme nt i n the group. These shall call the degree of order

concepts are those of the theory of communications, which itself borrows from the second law of thermodynamics

(Boltzmann's

th eorem

H) [ 1 5] .

Formalized Music

62

V ari e ty is expressed as a pure n u m b er or as its l og ar i t h m to Thus human sex has two elements, male and or

1 bit : 1

bit

=

L e t there

log2

be a

2.

female , and

the b ase 2 .

its variety i s 2 ,

probabilities (a grou p of r e a l numbers p, I ) . The entropy H of this group is defined as

group o f

positive or zero, whose

sum

is

if we is t, and the entropy of this sequence, i . e . , its uncertainty at e ac h th row, will be 1 bit. If both sides o f the coin were heads, the u ncertainty w o u l d b� removed and the entropy H would be zero . Let us suppose that th e advent of a h e a d or a tail is not controlled by tossi n g the coin, but by a fixed, u n iv oca l l aw, e . g . , heads at each eve n toss and tails at each odd toss . Uncertainty or disorder is always absent and the entropy is zero . If the law becomes very co m pl ex the appearance of heads or t a ils will seem to a h uman o bserver to be ru l e d by the law of chance, and disord er and uncertainty will be reestablished . What the observer could do would be to co u n t the appearan ces of heads and tails, add u p their respecti ve If the logarithmic b a se

is

2, the

entropy is expressed

in b i ts. Thus

have a sequence of heads and tails, the probabi lity of each

frequencies, deduce their probabilities, and then calculate the entropy in

bits. If the frequency of he ad s is equal to that of tails the u n certai nty will be eq u al to 1 bit. This ty p ic al example sh ows roughly the passage from order to disorder and the m e a n s of calibrating th is disorder so that it may be compared with o t he r states of disorder. It also shows the i m p o r t a n ce of scal e . The intelli­

maximum and

gence of the observer would assimil ate a deterministic complexity up to certain limit. Beyond that, in his eyes, the complexi ty wou l d swi n g

over

a

i n to

unforcseeability and would become chance or disorder ; an d the visible (or m acro s c op i c would slide into the invisible ( or microscopic) . Other methods and points of view would be necessary t o observe and con trol the pheno­ mena. At the beginning of this ch apter we admitted th at the mind and es peci ­ ally the ear we re very sensitive to the order or disorder of phenomena. The laws of perception and judgment are probably in a geo me t ric a l or logarith­ mic rel a tion to the laws of excitation. We do not know much about this, and we shall again co n fi ne ourselves to exami ning general e n t i t i e s and to tracing an overalt orientation of the poetic processes of a very ge ner al kind of music, wi thout givi n g figures, moduli, or determinisms. We are still o p timistic enough to think t h a t the interdependent e xp e ri m e n t and action of abstract

Markovian Stochastic

hypotheses

63

M usic-Theory

can cut biologically into the l ivin g conflict between i gn orance

and reality (if there

is any reality) .

S�udy of Ata xy ( o r d e r

or

d i so r d e r ) on t h e P l a n e of a

C l oud of G ra i n s or Vecto rs

Axis

of time : The

degree of ataxy, or the

e n t ro py, is a function of the the

simultaneity of the grai n s and of the distinct intervals of time between

If th e variety of the durations of the emissions is weak, e n tropy is also weak. If, for e x am pl e , in a g ive n l!.. t each gr ai n is emitted at regular intervals of time , the tem p o ra l variety will be 1 and the e ntropy zero. The cloud will have zero ataxy a n d will be c omp le tely ordered. Con­ v ers e ly, if in a fairly long succession of At the gr ai ns are emitted according to the law pX = ae - 6" dx, the degree of ataxy will be much larger. The limit of entropy is infinity, for we can i m agi n e all p oss ible values of time intervals with an equal probability. Th us, if the variety is n -+ oo, the probability for each time interval is p1 1 /n, and the entropy is emission of each grain . the

=

H H for

n

-+

=

ao,

-K

� I L. -

l=o

H -+ ao .

n

I log -

n

=

- K 'L, p; log pi n

=

i=O

I 1 - K n - log n

n

=

I - K log n

=

K l og n

l!.. t wil l never offer a very gre at variety of du r ati o ns and its entropy will be weak. Furthermore a sonic composition w i ll rarely have more than 1 00,000 l!..t 's, so that H � log 1 00,000 and H � I 6 . 6 bits. Axis offrequencies (melodic) : The same arguments a re valid here but with greater re st r i c t io n on the v a rie ty of melodic intervals and on the absol ute frequencies bec au s e of the limits of the audible area. Entropy is zero when the variety of frequencies of grains is I , i.e . , when th e cloud contains only one pure sound. Axis of intensity and density : Th e above observations are valid. There­ fore, if at the l i m it, the entropies following the three axes of an el em en t ll.Fl!.. Gl!.. tl!..D are zero, this element will o nly contain one pure sou nd of co n stan t intensity emitted at regular intervals. In conclusion, a c l ou d m ay contain just one si n g l e pure sound emitted at regu l a r intervals of time (see Fig. 11- 1 9) , in which case its m ea n entropy ( arithmetic mean of the three entropies) would be zero . It may contain chaotically distributed grains, with maximum ataxy and maximum mean This i s less true in p rac ti c e , for a

64

Formalized Music

entropy (theoretically oo) . Between these two l i m i ts the grains may distributed i n an i n fi n i te number of ways with

m e an

be

entropies betwee n 0

a raw,

and the maximum and a b l e to produce both the M arseillaise and

dodecap honic series .

t

Fig.

- - - - - - - - - -

� I I I

I

I

11-1 9

A single gra i n emitted at reg u l a r i n terva ls of time

Parentheses GENERAL

O B S E R V AT I O N S

ON

A T A XY

Taking this l as t possibility as a basis, we shall examine th e very general

formal processes

in all

real ms of thought, in all physical and p syc h i c r�alities.

this end we shall

To

imagine a " Primary

Thing, "

malleable at will ;

capable of deforming instan tan eously, progressively, or step- by-step ;

ible

plex

or

as

It

retractable ; unique or plu ral ; as simple

the

u niverse

will have

(as compared

to

man,

as

that i s ) .

a given mean entropy. At a defined

u n d e rgo a transformation. tion can have one

of three

From

the p o i nt of view

time we w i l l cau se it to

of a taxy

1 . The d egree of complexity de g ree

(variety) d oe s not change ; does not change.

of complexity

increases and so does

the

the transforma­

entropy.

transformation is a simplifying one, and the entropy dimin­

ishes.

T h u s t h e neutral transformation may a c t on a nd

d i so rd e r

this transforma­

effects :

tion is neutral ; and the overall e n t rop y

2. The 3. The

extend­ com­

an electron ( ! ) or as

into perfect d iso rd e r

transform :

perfect

(fluctuations ) , p a r ti a l disorder into p arti al

disorder, and perfect o rde r into perfect order.

Multiplicative

disorder, partial

disorder.

transform ation transforms : perfect dis o rd er into p erfect

disorder into greater disorder, and p e rfe c t order i n to p artial

65

Markovian Stochastic Music-Theory

transforms : perfect disorder into partial p arti al order into greater order, and partial order into perfect ord er. Fig. I I-20 sh ows these transformations in the form of a kinematic d i agr am . And simplifying transformation

disorder,

Perfect

D e g ree

disorde r

of

o rd e r

Max.

e n tropy

Partial

Entropy

Pe rfect

E ntropy 0

«

d isorder

order

Fig .

max.

1 1-20

STUDY

OF

Time ATAXY

AT

THE

LEVEL

S CR E E N S ( SET

OF

OF

CLOUDS)

screen which i s composed o f a set of wi t h densities during a slice of t i m e llt, may be dissociated according to the two characters of the grains, frequency and amplitude, and affected by a mean entropy. Thus we can classify screens according to the criterion of ataxy b y means of two parameters of disorder : the variety of the frequencies and the variety of the intensities. We shall make an abstraction of the temporal distribution of the g r ains in Ill and of the d e n sity , wh ich is implicitly bound up with the varieties of the two fundamental sizes of the grain . In symbolic form : cells

From

the above

discussion, a

l:!.. F!:iG associated

Perfect disorder Partial di s o rd er

Partial order Perfect order From the

point

of view

of ataxy a

=

=

=

=

oo

n

or

m

m or n 0.

s creen

is formul ated by a

pa i r of

entropy values ascribed to a pair of fre q u encies an d intensities of its grains.

Thus the pair (n, oo) means a screen whose frequencies h ave quite a small entropy (partial order or d iso rd e r ) and whose intensities have max i m u m entropy (more or less perfect disorder) .

66

Formalized

Music

CO N ST R U CTI O N O F T H E SC R E E N S

W e shall quickly Fig. 11-2 1 . Partial Perfect disorder disorder F G F G F

survey some o f the screens

Perfect order F G

G

F

G

F G

F

F

G

U n ique scre e n I nfin ite n u mber of screens

oo,

0

U n i que scre e n I nfin ite n u mber

n, co

of screens U n ique screen, pure so unds

I nfin ite n u mbe r of scre ens

n, m

F

G G

co, co

0, co

F

G

Description

Symbol

oo , n

G

F

I nfin ite number

of screens I nfin ite number of screens U n ique scre e n , pure sound

n, O O, n

F G

in

0, 0

the entropy table i n

Diagram

Diagram

.

:�·�=�}��r�---. : :��?��-�

,

.

•••

..: :;

· 1 �•

- .

·;

-.

0•

../

••· , ,

Ifj�fj ,) I I -�}�:] - .. , ... , , . I . . r.J

F i g . 1 1-21 . Screen E ntropy Table S CR E E N

(ro, ro)

Let there be

a

very large number of grains distribu ted at random over

of the audible area and l asting an interval of time equal to /).t, Let there also be a grid fine enough so that the average density will not be more than 30 grains per cell. The distribution law is th en given by Poisson's formula the whole range

p,_ �

=

(dm)k e - dm ' K!

the m e a n density and Pk the probability that there will be k cell . If dm becomes greater than about 30, the distri b u tion l aw will become normal. Fig. II-22 is an ex am p l e of a Poisson distri bu tion for a mean density dm 0.6 grains/cell i n a grid of 1 96 cells for a screen ( oo, oo) . Thus we may construct the (oo, oo) screens by hand, according to the distri butions for the rows and columns, or with suitable c o m p u ter programs.

where dm is grains in

=

a

.

.

. -

. 1-1--+--1--+----j -

-

- -

.

--1- - .

. - -;-· --

. �--l--+--1--1---1-'-l --.

.

.

. . .

--

.

1 - - 1 . -- •

•

.

•

•

I.

+-- � l -4-4- - ��-� -+-l-4-+-r�4-�

-.: · t··

.

.

67

Stochastic Music-Theor y

Markovian

· -

-

.

.

.

.

-

f- :- 1.

I•

.

. . .

.

.

.

.

.

.

.

Fig . 1 1-22

I

.

.

.

.

:

F

a very high m ean density the screens in which disorder is perfect (maximum) will give a very rich sound, almost a white sound, which will never be identical throughout time . If the calculation is d o n e by hand we can construct a l arge number of (oo, ro) s cre en s from the first (ro, ro) sc re e n i n orde r t o avoid work and numerical calculation for each separate screen . To thi s end we permute the cells by c ol u mn and row (see Fig. Il-23) .

For

F i g . 11-23. Exa m p l e of Permutati o n by Col u m n s

Discussion. It is obvious th at for a high mean d ens i ty, t h e greater the number o f cells, the more the distribution of grai n s in one regi on of the screen tends to regularize (ergodism) and the weaker are the fluctuations from one c el l or cloud to another. But the absolute l i m i ts of the densi ty in the cel l s in the audible a re a w i l l be a function of t h e technical means available : sl i d e rules, tables, calculating machines, compu ters, ruled paper, orchestral i n struments, tap e recorders, scissors, programmed impulses of p u re sounds, au t o m a t i c splicing devi ces, programmed record i n gs, a n a l o g u e converters, etc. If each ceiJ is c o n s i d ere d as a symbol defined by the number of grains k, the en tropy of the screen (for a given fineness of grid) will natu r a l l y be a ffected by the mean density of the grains pe r cel l and will gro w at the same

Formalized Mt.. s ic

68

time. It is here that a whole series of statistical experim ents will circumscribe th e perceptible limits of ataxy

for

even express the color nuances of white sound . It is very possible

ear

have to

and that the

these screens (co, co )

classifies in the same file a great n u mb er of sc re e ns whose entropies

tremendously.

There

wou ld

vary

result from this an impoveri shment and

a

simplification of t h e commu nication : physical information --? percepti on, but at least there

ing screens ALL

can

will be

SCREENS

Starting from

a

the work involved in construct­

elementary operations we of th e en tropy table. See Fig. 11-24 for a few I n practice, freq uency and intensity fil ters imitate these elemen­

construct all th e

examples.

the advantage that

will be considerably reduced.

few screens and applying the

screens

tary operations perfectly.

(00 00 )

llllll

.

(n

m>

� ""=

(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 -<-

Beginning of return to c q u i l i brium ( p ertu rbation P�)

11 1-1 5 Analogique A

replaces elementary

sinusoidal sounds by very ordered In any case a

clouds of elementary grains, restoring the string timbres.

realization with classical instruments could not p r od u ce tim b re other than that of s tri ngs because of the limits

screens

of human

h aving a playing.

The hypothesis of a so nority of a second order cannot, therefore, be

firmed or i nval i d at e d under these conditions. O n the o t h e r

con­

hand, a realization using electromagnetic devices as and adequate converters would enable one to prove

mighty as compu ters the existence of

a

s ec on d

order sonority with elementary sinusoidal grains

or grains of the Gabor type as

a base.

While anticipating some such technique, which

has yet to be developed ,

we s h a l l demonstrate how more complex screens are realizable with the

resou rces of an ordinary electroacoustic studio equipped wi th several mag­

netic tapes or synchronous recorders, filters, and sine-wave generators .

E L E CT R O M AG N ETI C M U S I C

TA K E N

(si nuso i d a l sou n d s )-EXA M P LE

F R O M A NAL O G/Q UE 8

We choose : I . Two groups offrcque nc y regions10,/1 , as in Fig. I I I- 1 6.

The p ro t o c ols of these tw o groups will

be s u ch that they

preceding ( MTP) 's :

1o

11

0.2

0.8

0.8

0.2

in which (a) and ({3) arc

the

1o ({3) 11

parameters.

1o

f1

0.85

0.4

0. 1 5

0.6

will obey the

Hz

Reg ions

r If 11 .-------, .-----, r------1 I I I I (12 811 ll ( 3

I Hz 42 Regions

z

a z tr---1-; lr---1-1 f1B MS 7"' 15 9 • s � ,

I

1

r-::1 lfl ¥

1' (!.!!,I 1r=-:-'t

I�

s

'

�

'

8

,,.

,

10

If*' II

ID

,.."

.u.n

II

II

14'

IJ

II

r1-

JS"

�·

,...

I�

(h)

Fig. 111-1 6

2. Two groups of intensity regions g0, g1, as in Fig. III-1 7 . The protocols of this group will again obey the same (MTP) 's with their

parameters

(y)

and

(y)

(e) :

�

go

'";'

0.8

81 G

� � �[ �

go

0.2

� "l

.. "" 0 ·;;,

..

c:

a:

.,

0

..... 0

.., ll.

0 "'

�

c: 0 .c: a..

.. ..

((o) Fig . 1 1 1-1 7

gl

0.8

0.2

(e)

�

gl

0.85

0.4

0.6

0.g,1 5 G

�[ �� �[ �[

�

-..

11(

.. �

c: 0

.,

·c. " a:

0 "' 0 "

11.. 0 ...

� ., "

c 0 .c: a..

(g�)

F

•

1 05

Markovian Stochastic Music-Applications

3. Two grou ps of density regions d0, d1 , as in Fig. III- 1 8. The

protocols

of this group will have the same (MTP) 's with parameters (A) and (p.) :

d1 (A)

Fig.

0.8 0.2

.j.

d0

d1

0.85

0.4

0. 1 5

0.6

1 1 1-1 8

us the principal screens A, B, C, D, E, F, G, H, as in Fig. I I I-1 9. The duration /j.t of each screen is about 0.5 sec. The perio d of exposition of a perturbation or of a stationary state is about 15 sec. We shall choose the same protocol of exchanges between p e rt u rb a tions and stationary states of (MTPZ) , that of Analogique A.

This choice gives

shown

­

�

E-� - � - E - � -� - � - � - E-�

The screens of Analogique B calculated up to now constitute a special ch o ice Later in the c ourse of this composition other screens will be used more particularly, but they will always obey the same ru l es of coupling and the same (MTPZ) . In fact, if we consider the combinations of regions of the variable }; of a screen, we notice that without tampering with the name of the variable }; its structure may be changed. .

Screen A

{;to (odo)

�I

.:;� I Screen 8

/).to a.,]

jl

2.

II c; l �.

Screen C

(f., #., d.)

3

.2 {

'i ,f

Screen D

(1. ;., d,)

3

-1

1

v If Scree n E

(ft (,. d.J

Screen F

{.lt fo dl)

3

.t

I

� " J

.2 f

Screen G

(l., ,., d� )

Screen H

(.ff g, d,J Fig. 1 1 1-1 9

t; 4' 3

:1.

-1

.:; 4'

j

/l 1

Markovian Stochastic Music-Applications Thus

forfo we may have the regions shown

107

in Fig. III-20. The Roman

n u me ral s establish the l i a is on with the reg ions of th e other two variables . Hz ���

Regions

1

2

811

.II

I�

1

Fi g . 1 1 1-20

tJ

7"'

t

Ill

,.,. II

n another combinationj0, as in

But we could have chose H z 42

Reg ions

li!UJ -ttl 1f

2

f

F

Fig. I II-2 1 . erso fL f;1

(/.,)

Fig. 1 1 1-21

This p romp ts the question : " Given n divisio ns

I!.F (regions on

is the total num b er of p ossible combinations of t:J.F regions ?

1st

case.

_,<¥

None of the

n

"'S"

Sf«>

F)

N#Dtl

ftt

what

areas is used . The screen corresponding to this

combination is silent. The number of these combinations will be

n!

2nd will be

case.

(n - 0) !0 ! ( = 1 ) .

One of the n areas is occu p ied. The nu mber of combinations

3rd case. Two of the

will be

mth

will be

case. m

of the

n

n! (n - 1 ) 1 1 !

n areas are

occu pied. The nu mber of combinations

n! (n - 2) !2 !

areas

are

occu p i ed . The

number

of combinations

n!

(n - m) !m ! F I G . 1 1 1 -1 9 : The Arabic n u m bers above the R e m a n n u m erals in the cells indicate the density in logarithmic u n its. Thus cell (1 0,1 ) will have a density of [(log 1 .3/log 3) + 5] rerts, which is 31 5.9

g rai ns/sec e n the overage.

F

1 08

nth

will

be

case.

Form alized Music

n

of the areas

are occupied. The number of the combinations n!

( n - n) !n ! The total number of combinations will be equal to the sum of all the preceding :

n!

..,. ! -, !0,.-: 0),. . ----.,... (n+

+

(n

n! I) ! 1 !

-

n!

n!

+

(n

-

[n - (n - I ) ] ! ( n - I ) !

2) !2 ! +

+ n!

( n - n) !n ! .

=

2n

The same arg um ent operates for the other two variables of the screen . Thus for the intensity, if k is the number of available regions !1G, the total number of variables g1 will be 2k ; and for the density , if r is the number of available regions !1D, the total number of variables d1 will be 2'. Consequently the total number of possi ble screens will be T

=

2(n + k + r> .

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