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 al c u l a tio n wh ich is, in fact, an enrichment of the game. The following stage is t h e exp erimental control of the matrix. Two methods are possible : l . S i m u lat e the game, i . e . , mentally substitute on e sel f for the two conductors, X an d Y, by following the matrix entries stage by s tage , without memory and without bluff, in order to tes t the le a s t interesting case. I, II,

III,

IV, V, VI

1 5 /58, resp ectively ; wh ile

=

Game value :

=

52/20

=

2.6 points in X 's favor.

Formalized Music

1 18 2.

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

marginal

n um

b e rs

Game value :

the

in

57/2 1

(M4) .

=

2.7 points i n X's favor.

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

value

calcu l ated

by a p proximation.

The sonic pl"ocesses d erived from

the two experiments are, moreover, satisfactory. a

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

We may now apply

strategies for X and programming,

on two theses :

in

rigorous

Y and

I . The fu ndamental th e o re m

is that the minimum

of game theory (the " minimax theorem ")

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

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

optimum strategy. 2. The calculation

of the m ax i m i n or minimax

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

down to the resolution

(dual simplex method) . Here we

shall

of a

pair of dual

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

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

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

be the

player

probabilities corresp ondin g

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

to tactics

variables ; the

2yl 3yl

2y1 3yl 2yl

4yl

+ + +

+

+

+

Y 1 + Y 2 + Ys + y,

3y 2

2y2

4y2 2y2 2y2 2y2

+ 4ya +

+ 2ya

+

+ Ys

+

2y5

+

2y4

+

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

+

Y3

+ 2y4

Y3 +

4y4

+

+

3ys

2ys 2ys

3ys

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

+ Ys

2y4 + 3ys

+ +

+

+

=

=

=

Ya + Y12

= v.

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

S trategy, Linear Pro gramming, of the

score

(III, IV

=

4)

For

Tactics

(III, I V

into

ing optimum strategies :

and Musical Composition

5) . The solu tion gives For

X Tactics

Probabilities

I

6/ 1 7

II

III

0

III

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

v

VI

v

and for the game value,

42/ 1 7

=

must completely abandon tactic must avoid.

Modifying score (II, IV

strategies :

=

III

3)

2/ 1 7

2/ 1 7 1/17

2/1 7 5/ 1 7

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

(prob ability

to ( I I , IV

For X

Y

5/ 1 7

IV v VI

�

=

= 2 ) , we For

obtai n the following

Y

Probabilities

Tactics

Probabilities

I

1 4/56

6/56 6 /5 6 6/56

I II

1 9 /56

II

Tactics

III IV

v VI

8/56

6/ 56

III

IV

v

VI

7/56 6/56 1 /5 6

7/56

1 6/56

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

and for the game value,

v

1

the follow­

Probabilities

I

2/1 7

II IV

optimum

=

1 19

=

1 38/56

Although the scores have

Formalized Music

1 20

Conductor

I

3

i---

II Conductor X

II III IV V VI

2

I

3

i---

4

III

r-----

IV

2

r-----

v

3

i---

VI

Y 3

2

4

14

2

- - - - -

2

2

2

3

1

5

3

2

6

1

6

-

2

- - - -

4

4

2

2

3

2

!

-

2

6

2

8

(M5)

- -

2

3

2

2

1

19

7

6

2

2

r-----

7

16

4

16

56 Total

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

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

=

0) ,

we obtain th e

Conductor Y I I

II

- 13

15

III

IV

43

v

- 13

15

- 13

15

VI 5 6

ll

- 13

-- -- -- -- -- -

15

II

- 13

- 13

5 6

- 13

..L

-- -- -- -- - - -

Conductor X

III

43

- 13

- 41

71

15

- 13

- 13

-- -- -- -- --

- 13

IV

43

43

- 13

- -- -- -- --

15

v

- 13

15

15

- 41

..L

- 13

�-

56

1-

1

5 6

J_ 5 6

- 13

-

- -- -- -- -- -

VI

- 13

- 13

- 41

- 13

5 6

As this matrix is

by

+ 13.

It then

d iffi cul t

5 6

1.§

43 5 6

ll

scores

- 13

(Me)

H. to read ,

it

is

simplified by dividing

becomes (M7) with a game value v

=

-

a ll the

0.07, which

Strategy, Linear

Programming, and Musical Comp osition Conductor

I

X

II

III

IV

-1

+1

+3

1-1--

V

VI

-1

+1

-1

TI

-1

1

+1

-1

_§__

-3

+5

+1

-3

_§__

+3

-1

-1

-1

_§__

+1

-1

-1

-1

- 1

-1

- --

-1

IV

-

--

+3

III

Y

-- --

+1

II

Conductor

I

+3

-- -- -- -- --

+1

v

+1

- 1

- --

-1

-1

VI

121

1-

-3

-- --

5 6

1--

56

5 6

_.!!._ 5 6

+3

TI

1 6

5 6

56

-1

1-

_L

.ll!.

14

(M7)

1 6

TI

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

according to the corres p ondence :

-1

-3

T

p+

p

we obtain

+3

+1 I

I

g

T

g+

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

couple, VI, VI, which is now

+5

finished.

p

p+

g+ +

p

g

p

p+

p

p

p

p+

p

g+ +

p

p

g+ +

p+

p

p

g+ +

g+ +

p

p

p

p+

p

p

p

g+ +

---

p+

p

p+

---

p

p

p

p

Formalized Music

1 22

Mathematical manipulation has brough t abou t a refinement of the

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

a

Silence is to be

paradox : the

co u p l e

to

avoided, b u t

VI, VI, ch arac terizing

do this it is n ecessary

to aug­

ment its potentiality.

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

mathematical treatment of

conductors X and composer

Y and

vote immediately on the make-u p of the game

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

matrix. The

public who construct the matrix of points.

R U LES O F TH E WO R K

The two-headed flow c h ar t

of Duel is

valid for Strategie, composed in 1 9 6 2 .

STRA TEGJE

shown in Fig.

The

IV-2. It

is equally

two orches tras are placed on

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

either side

of the

stage, the conductors

platforms on opposite sides of the

The

six fundamental tactics

are :

1 23

S trategy, Linear Programming, and Musical Composition

I.

Winds

II . Percussion

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

with the hand

IV. String pointi llistic effects

V. VI.

S tring glissandi

S u s tai n e d s tring harmonics.

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

I

& II

=

VII

I & III = VIII

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

I

II &

II &

III IV

II & V

I I & VI

XII

I

XI II

I & II &

=

=

= XIV

XV

=

I I

II

&

&

III IV II & V &

& II & VI

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

The

36 1 ( 1 9

x

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

Game

1. Choosing tactics.

H o w will the

play ?

=

XVI

=

XVI I

=

=

XVIII XI X

make

his orches­

be played simultaneously.

c o n d u ct o r s

choose which ta ct i c s t o

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

succession of couplings of the 19 structures .

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

from

h ave before him during the performance. c.

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

which they

them down on a mas ter plan, which

he will follow

during the performance.

Formalized Music

1 24

Fig . IV-2

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

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

forming

d evi ce)

8 . A u d i e n ce

B

orchestra :

1 1 1 1 1 1 1 1 2 1 2

Percussion

Bp c l a ri n et

�--

contra bassoon F rench horn

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

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

44 instruments, or 88 players i n both o rchestras

::;;;,c [

"'

c

g" (i

� !!!. 0 0 :::l "'

:::l

I

(!)

5!:

r::r "'

(")

!e.

�

<

a· iii "'

� t:""'

s·

� "" ....,

...., 0 (lq ...., ""

I

..-------, 0 0 c:

,....

Vibraph one

'i:i

y

X

Cond uctors <

<

§.:

a· :;·

"'

"'

:;·

g: (!) ..

<

()" iii

"'

(")

!e. 0 "'

0 0 c: r::r

;r::r ., 1/) 1/) "' "'

...- Public __,.

+

3 3 s· �

Brasses

! Woodwinds

Woodwinds I

B rasses

� oq

Percussion

Mari mba phone

bassoon

1 bass drum

8 8 4 4 3

P l a cement of the O rc h estras on a S i ngle Stage

piccolo flute E[> c l a r i n et

1 m a ri m ba phone 1 m a racas 1 suspended cym ba l

Ul r+ ...., ""

Fig. I V-3 . Strategy

Composition of the

N OTE: If two stages a re used, each orch estra is arra nged in the cla ss i c m a n n er.

u

s:: -o 0> � �. C") 3 c r::r l::

0> -· -c 0 ::r "' 0 :::l (!)

"" :::1 p.

�

s:: "'

o· e:.. 0 0

3

"0 0 �. ::r. 0 :::

...... }.j U1

Formalized Music

126 e.

Actually a l l these ways const i tu te what one ma y caJI " degenerate "

competi tive situ ations. The only worthwhile setup, which adds something n ew in the c a s e of more than one orchestra, is one that in trod uces dual

conflict between the con d u ctors . In this case the pairs of tactics are per­

formed sim ultaneously without i n terru ption from one choice to the next

(sec

Fig. IV-4) , an d the d ec i sions made by the conductors are conditioned

by the winnings or losses co n t a in ed in the game matrix.

X

C Oh

OR.

COlo

·oR. Y

G • INJ

"TIICTICS

tfA iNJ

TA C TICS

72

7ll

X V///

IX

.5"2

VI/

40

X IX

4IS

X V

XIV 48

XV

J& :28 v

VII

"'"

--- - -

Fig. IV-4

2.

Limiting the game. The game may be limited in several ways : a. The a ce rtain number of po i nts , and the first to reach

conductors agree to play to

it is the winner. b. The conductors agre e in a dva n ce to play n engagements. The one with more poi n ts at th e end of the nth engagement is the winner. c.

The conductors d ec id e on the du ration for the game, m seconds

(or

minutes ) , for instance. The o n e wi th more points at the end of the mth

second (or mi n u te) is the winner.

3. a.

Awarding poinLf.

One me t hod is to h ave one or t wo referees co u n t i ng the points in two columns, one for conductor X and one for conductor Y, b o th in positive numbers. The referees stop the game after the agreed limit and announce

the result to the public.

b. Another m eth od has no referees, but uses an automatic system

that

consists of an i n dividual board for each conductor. The b o ard has the

n x n ce lls of the game matrix used. Each cell has the corres p onding partial score and a p us h b utton . S u ppose that t h e game matrix is the large one of

19 x 19 cells. I f conductor X ch oos es tactiC XV against Y's IV , he presses the b utton at the in tersection of row XV a n d column IV. Corresponding to this in tersection is the cell containing the partial score of 28 points for X and the bu tton th at X m u s t push. Each b u t t o n is con n ected to a

small

adding

machine which totals up the results on an electric p a n el so that t hey can be

seen by the public as the game proceeds, just like the panels in the football

stadium, but on a smaller scale.

4. Assigning of rows or columns is made by the conductors tossing a coin. 5 . Deciding who starts the game is determined by a s e con d toss.

Strategy, Linear Programming, and Musical Composition

127

6. Reading the tactics. The orchestras perform th e tactics cyclically o n a closed loop. Thus the cessation of a tactic is made instantaneously at a bar line, at the discretion of the conductor. The subsequent eventual resumption of this tactic can be made either b y : a . reckoning from the bar line defined above, or b. reckoning from a bar line identified by a particular letter. The conductor will usually indicate the letter he wishes by displaying a large card to the orchestra. If he has a pile of cards bearing the letters A through U, he has available 22 different points of entry for each one of the tactics. In the score the tactics have a duration of at le ast two minutes. When the conductor reaches the end of a tactic he starts again at the beginning, hence the " da capo" written on the score. 7. Duration of the engagements. The du ration of each e ngage ment is optional. It is a good i d ea, however, to fix a lower limit of about l O seconds ; i.e., if a conductor engages in a tactic he must keep it up for at least 10 s econd s This limit may vary from concert t o concert. It constitutes a wish on the part of the composer rather than an obligation, and the conductors have the right to d ecide the lower limit of duration for each engagement before the game. There is no upper limit, for the game itself conditions whether to maintain or to change the tactic. 8. Result of the contest. To demonstrate the dual structure of this compo­ sition and to honor the conductor who more faithfully followed the con­ ditions imposed by the composer in the game m atrix, at the end of the combat one might a. proclaim a victor, or b. award a p rize, bouquet of flowers, cu p , or medal, whatever the concert impresario might care to donate. 9. Choice of matrix. In Strategic there exist three matrices. The large one, 19 rows x 1 9 columns (Fig. IV-5) , contains all the partial scores for pairs of the fundamental tactics I to VI and their combinations. The two smaller matrices, 3 x 3, also contain these but in the following manner : Row I and column I contain the fundamental tactics from I to V I without discrimina­ tion ; row 2 and column 2 contain the two-by-two compatible combinations of the fundamental ta c ti cs ; and row 3 and column 3 co n tai n the three-by­ three compatible combinations of these tactics. The choice between the large 19 x 1 9 m a tri x and one of the 3 x 3 matrices depends on the ease with which th e condu ctors can read a matrix. The cells with positive scores mean a gain for cond u c t or X and autom atically a symmetrical loss for conductor Y. Conversely, the cells with negative scores mean a loss for conductor X and au tom atical ly a symmetrical gain for conductor Y. The two si m ple r, 3 x 3 matri c es with differ e n t strategies are shown in Fig. IV-6. .

1 28

Formalized

Music

M AT R I X O F T H E G A M E

-

-

.r

-lib - �

%

Jt

•

,, - � -U

9'

-� 0

J ll

8 1r

�"

12

- IIi

- flo _,, -!6 Jl(

-.fl, - fJ

11 -J<;

•

liZ

'" - +8

,. -3] t

,

ID

• I - ,,

;;;; JlL

r

__);: "

r 11 6 "'

* I

.

- tl ID

- lfB .fl "

�'

" 166

'"

'9'

to

"

,.

-.r.z - 6

- 1& - . M

J.t

-

lG

"

-·

lolr - (.

3&

I�

.t

-¥ "

f:t

0

-1�

-8

'"'

""

-1t

fl 88

88

Ia"' -.18

,

92.

*' -36 -14

�

0

4

�J

-]tJ

!'2

-28

8

¥

If. -SZ - .>2 -li6

-u -ft

}6 _,, to

,,

"

0

6

-so �

JJ

�"

.t r

#

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-liB -.UJ -H - t,f

_,,

1,! 3�

"'

- 52 - ·· - �·

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

1'1' 3.Z

" -Jo

-J�

1lC

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

I�

16 8

-2

4

-ft

-It

.'lo

-B "

41

1,-1, -51

lr&

_,,

:"

"

-.'I t _ ,,. - n

-fl/ - 1,-8 -

.f'.ll

,,

- "" �2

- 6 - If

.8

""

J

"

&t

-.U -111

B

"

0

0

If+ -I�

-"

-�

B

"'

- tl

J'f

- .J:/

0

u

-If

-B

- f.( ¥8

-+

68

.1&

-f!;

,Zo

-2(} -$0 -.u.

- 18

J

- JJ

-+

-.31/

'f8 - 12

-�

_, B

�

'1+ f6

I t>

- I�

Jl>

..

.II

-J.I Jt

-lro ¥

f.l n

J6

_ ,, -1.1 8

i3 6 - 8

- ft

H - 66 - 4 H

12

..A Woodwinds

• N ormal percussion

H Strings striking sou nd-boxes •: Strings pizzicato

.:If Strings g l issando

:= Stri ngs sustained

=

of

_,.

-16 .;t

: -.

Jo

-

u

"0

-1'1.

3.J

-+

u "

-" -.rs

,.

- S'&

2.1

""

- ;t f

-"" - ro

0.

two and three

-J 6 -I�

- II.

- 18

�;!

YIJI'M

K•�

-Jo

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JD

11

p

-I"

1!1

to

-?4-

"·

-4- lr

I�

t,o

.u

- 3/1: - JI!I

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-

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-21

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2�

- 11 J,o

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-s

"'' -Ji' - H

J lr - J o

- .211 -It: --�

H � 0 16

d ifferent tactics

. ., 11-�.d , .., ""

_ ,., -

-B -3& -.v o

• Combi nations

�

- ��

.t'f

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

'

•

&I

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

"

•

- .Je - �lr - 16

- � - .36

IJ

u

• "

- M - ./0 - 10

- r.z - M

-8 - U - vo

l•

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8

-r;( 31

9

-.J•

U3

-

-$& �6

6

.u

f,z

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-

-

-8

.lc]J - I(¥

,.

,,

'

:t¥ -

n

-J,t

- !12

,,

1,1.(

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.t

1-1

'"

6o

� - !16

�� _ ,.:z

-J,G l, f

- ¥�

H

-llo

-1(,2 - :J2

,

B

'

and Musical Comp osition

Strategy, Linear Programming,

I;

-6�

Ill

:. •

lH

8 �!

4' 0 Ill •

-1

3

-3

3

.... .

:. . ' *' . ' Ill • •

0

3

• y ti H Y

>< 0

'

· · · � · · · · � ... . ..

n

I�

•

Strategy

for

Y

4

A

{-,

... . 4' 0 m• H O

.. . . :. . '

*' " ' Ill e f

-3

Strategy

F i g . IV-8

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

Y Wooawinas

H Strings striking sou nd - boxes • ••

Normal percussion Strings pizzic ato

ill Strings sustained

Jl-

-1

0

-5

-1

4

3

1

-3

.2 • '

�� : a:;

for

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

3

� t.

.. . .

-5

•o

1 29

Y

A •

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

Two - person

' . ' "

' "" ' m O H . :.

., :.

of two dis­

Combinations

tinct tactics

' • H

; :�

' •m

Combi nations

of three dis­ tinct tactics

• IU

Strings glissando

. ,..

Simplification of the 1 9

x

1 9 Matrix

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

fr + m

*• + 1

4r+ I .I + l

k, .. ,

k1 + n

Dr+ l , I + J

Dr + l . I + R

Dr + t .a + :l

a,. ... . . . + J

Or + e . a + n

a, + lll ,l + l

a, + • . • + l

Dr + m , a + n

��:a;•gy i

Formalized Music

1 30 This fragment

can

be replaced by the single score

A• + m .• + n

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

-

_

and b y the probabilities

m

Q

=

and

:L q, + , 1=1 n

K

Operating in this way matrix (the tactics will be X

X

-

25

X

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

X

25

592 25

X

45

93 1 4

49

X

30

X

25

26

3088

1 76 1 0

-

30

25

or

x

8296

49

49

45

68 1 8

25

J=!

the same as in

1 45 2 2

25

L ks + J·

wi th the 1 9

7 704 25

=

45

X

26

2496

30

49

X

26

26

2465

- 1 354

1 82

25

- 2581

1 59 7

- 52 8

45

1818

- 1 267

640

30

25

49

26

45 30

the following IV-6) :

C h a pter V

Stochastic

Free

After this interlude , we return

Music

by Computer

to the treatment of composition

by machines.

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

being realized mechanically. This realization occurred thanks to

M.

Fran<_;ois Genuys of IBM-France and to M. Jacques Barraud of the Regie

Autonome des Transports Parisiens. THE A STO C HASTIC

PARA DOX :

WORK

M U S I C A N D CO M P U T E R S

EXECUTED

The general public has

a

BY

number

THE

IBM-7090

of different

reactions when faced by

the alliance of the machine with artistic creation. They fall into three categories : " It is impossible to obtain a

work cif art,

s i n ce by definition it is a handi­

craft and requires moment-by-moment " creation " an

for

each detail and for

and cannot invent." " Yes, one may play games with a machine or use it for speculative purposes, but the result will not be " finished " : i t will represent o n l y an the entire structure, while a machin e is

inert thing

experiment-interesting, perhaps, but no more . "

The enthusiasts wh o at the o u tset accept witho u t flinch ing the

whole frantic brouhaha of science fiction . " The moon ? Wel l , yes,

within

our reach. Prolonged

life will

also be with

us

tomorrow-why

it's

not a

creative machine ? " These people arc among the credulous , who, in thei r i diosyncratic optimism, have replaced t h e myths of Icarus and the fairies,

which have decayed, by the scientific civil ization

partly agrees with the m In paradox nor all animism, for it progresses foreseeable at too great a distance . a n d science

.

131

of the

twentieth century,

reality, science

is

nei ther all

in limited stages that are n o t

Formalized

1 32

Music

the

arts w h a t we may c a ll ratio nalism in the etymo­ logi cal s e n s e : the search for proport i o n . The artist h a s always c alled upon it o u t of necessity. The r u les of construction have v a r i e d widely over the cen­

There exists in all

tu ries, b u t t h e re h ave alw ays b ee n rules in every epoch because of the neeessi ly of m a k i n g oncscl f u n d e rstoo d _ Those wh o be l i eve the firs t statement

first to re fuse to a p p l y do not understand at alL

above arc the which they

the q u al i fication

Thus the musical scale is a convention

artistic to

which circumscribes

product

a

the

area

of

potentiality and perm i ts construction wi thin those li m i ts in i ts own p articu­ lar

symmetry.

The r u l es of Christian hymnography, of harmony, and of

coun terpoint in the various

ages

h ave al lowed artists to cons t r u c t

and

to

m a k e themselves u n d erstood by those who adopted the same constraints­

through tradi tions, thro ugh col lective taste or imitation, or through sym­

p athetic resonance. The rules of serial ism, for instance, those that

the t r aditiona l

octave doublings of tonal i ty,

partly new but none the less real .

banned

i m p os e d constraints which were

Now everything that is rule or repe ated constraint

is

part of the mental

machi n e . A little " i maginary machine," Philip pot would have said-a

choice, a set of decisions. A musical work can be analyzed of mental machines. A melodic theme in

a

as a

symphony is a mold,

multitude a

mental

machine, in the same way as its structure is. These mental machines

something very restrictive

i n decisive. In the last few years

really a

very general

are

and of mech a n ism is

and determ inistic, and sometimes very vague we

have seen that this idea

one. It flows through every area of h uman knowled ge

and action, from strict logic to artistic manifestations. Just as the wheel was o n c e

one

of the greatest produ cts of human

intelligence, a mechanism which allowed o n e to travel farther and faster with more luggage, so is

the compu ter,

which t o d ay allows the transforma­

tion of man's ideas. Computers resolve logical problems by heu ristic methods .

But computers are not rea l ly responsible for the in trod uction of mathem atics

in t o music ; rather it

is mathematics that m akes use of the computer in if people's minds are in general ready to recognize the u s e ful ne s s of geo met r y in the plastic arts (architectu re, painting, etc.) , th ey have only one more stream to cross to be able to conceive of using more abstract, n o n vi s u al mathematics and machines as aids to musical composi­ tion, which is more abstract than the plastic arts.

co mpo siti o n . Yet

-

To

summarize :

l . The

creativ e thought

of

man gives birth to me n tal mechanisms,

·

1 33

Free Stochastic Music by C ompu ter

which, in the last analysis, are

merely s et s

of constraints

and choices . This

process takes place in all realms of thought, including the arts.

2. Some of these mechanisms can be expressed in mathematical terms. 3 . Some of them are physically realizable : the wheel, motors, bombs,

digital computers, analogue computers, etc.

4. C e r tain mental mechanisms may c orre spond of nature.

to certain me ch a ni sms

5. Cer ta i n mechanizable aspects of artistic creation may be s im u l a t ed by ce r tai n physical mech anisms or machines which exist or may

6. It h appens

Here then

e l ectro n ic

be cre ated .

that computers can be useful in certain ways .

is the theoretical point of d epartu re for

a utilization of

computers in musical composition .

We may further establish that the role of the living composer seems to

have e volve d , on the one hand, to one of i n v e nt in g schemes (previously forms) and exploring the limits of these schemes, an d on the other, to effect­

ing the scientific

synthesis of the new

methods of construction

and of sound

emission. In a short while the se methods must comprise all the ancient and

modern me a ns of m u s i c a l instrument making, whether acoustic or electronic,

with the help, for example, of digital-to-analogue converters; these have already been used in communication studies by N. Guttman, J. R. Pierce, and M.

V.

Mathews of Bell T el e ph on e Laboratories

in New Jersey. Now

these expl ora tions necessitate impressive mathematical, logical, physical,

and psychological i m p ed im e n ta , e s pec i a l ly computers that accelerate the

mental processes necessary for clearing the way for new fields by providing immediate

experimental verifications

at all stages of musical construction .

Music, by its very abstract nature,

is the

first

of

the arts to h ave

at­

tempted the conciliation of artistic creation with scientific thought. Its

industrialization is in evita b le and irreversible. Have we not already seen

attempts to industrialize serial and popular music by the Parisian team of

P.

Barbaud, P. Blanchard, and J e ani n e Ch a rb o n ni er, as well

musicological research of Hiller and Isaacson at the University In the pre c edi n g

ch apte rs we

as

by the

of Illinois ?

of musical mini­ m at hem a ti cs and e spe ci a lly

demonstrated some new areas

cre a tio n : Poisson, Markov processes, musical games, t he thesis of the

mum of constraints, etc. They are all based on on

the

theory of probability. They therefore lend th em selve s to b e i ng

treated and expl ore d by computers. The simplest and most meaningful

s che me is one

of minimum constraints in composition, as exemplified by Achorripsis. Thanks to my friend Georges Boudouris of th e C.N.R.S. I made the

1 34

Formalized

acq uaintance of

Jacques

director of the Ensemble

Barraud , Engineer of the Ecole

Electroniqucs de

des

Gcstion de Ia Societe

Music

Mines,

then

des Petroles

mathematics, and head of the All three arc scien tists, yet they co n s e n t e d to attempt an ex p e r i m e n t which s e e m ed at first far-fe tch ed­ that of a marriage of music w i th one of the m o s t powerfu l machines in the

S hell-Berre, and Fran�ois Gcn u ys, agrcgc i n

E tu d e s Scientifiq u cs Nouvelles at IBM- France .

world .

rel a t i o n s it

rarely pure lo g ic al persuasion which is is material interest. Now in t h i s case it was not l og i c much less self-in terest, that arranged the be­ trothal, but p u rely exp eri m e n t for experiment's sake, or game for game's sake, that indu ced collaboration. Stoch astical ly spe a k i n g my ventu re should have encoun tered failure . Yet the doors were opened, and at the end of a year and a half of contacts and h a rd work " the most u n u s u al event wit­ ness ed by the firm or by this musical season [in Paris] " took place on 24 May 1 962 at the h eadquarters of I BM-France. It was a live concert pre­ senting a work of stoch astic instrumental m u s i c entitled ST/ 1 0-1, 080262, which had been calcu lated on the IBM-7090. It was brilliantly p e rformed by the conductor C. Simonovic and his Ensemble de Musique Contem­ poraine de Paris. By its p assage t h rou g h the m achine, this work made tangible a stochastic method of com position, that of t he minimum of constraints and rules. In most human

is

i mportant ; usually the paramo u n t consideration ,

,

Positi o n of the Problem

w as the drawin g u p of the flow chart, i.e. , down cl e arly and in o rder the stages of the operations of the scheme of Achorripsis, 1 and adapting it to the machine stru cture. In the first chapter we set out the entire sy n th e t i c method of this minimal s tr u ctu re Since the machine is an iterative apparatus and p erfo rm s these iterations with extra­ o rdi n a ry speed, the thesis had to be broken down into a sequential series of op erati ons reiterated in loops. An excerpt from the first flow chart is shown in Fig. V- 1 . The statement o f the thesis o f Achorripsis receives its first machine­ The first working ph ase

writing

.

oriented interpretation in the following manner :

I . The work consists of a succession of sequences or movements each a1 seconds long. Their du ra ti o ns are totally independent (asymmetric) but have a fixed mean du ration, wh i ch is introduced in t h e form of a parameter. These durations and their stochastic succession are given by the formula

(See Appendix I.)

1 35

Free Stochastic Music by Computer

f li s s iHtoli.. YJL

(vic,c)

,..... rr YJI B

TGLb le. : 9(.;1) �t. .. etio .. :Z. S"o va iiA t �

9�-1
- :t- � 5 ' �-1o ot. = B t G� - Bt- i tF �F

j �

=

oJ.. ==

k'll'f = \l r c;; L = To

YUL

F i g . V-1 . E x c erpt from the F i rst Flow C h a rt of A chorripsis

Formalized Music

1 36

2. Dqinition of the mean density of the sounds during a1• During a sequence sou nds are emitted fro m several so n i c s o u rces. If the total n u m be r of these s oun ds or poi n ts during a sequence is Na, • the mean density of this point­ clust e r is Na.fa1 sou nds/sec. In ge ne r a l , for a given instru mental ensemble this d e nsity has limits that depend on the number of instru mentalists, the nature of their i nstruments, and the tech nical difficu l ties of performance. For a large orch estra the upper l i m i t is of the order of 1 50 sounds/sec. The lo we r limit ( V3 ) is arbitrary and positive. We choose ( V3) 0. 1 1 sounds/sec. Previous exp eriments led u s to ad o p t a logarithmic progression for the density sensation wi th a number between 2 and 3 as its base. We adopted e = 2 . 7 1 82 7 . Thus the densi ties a re i ncluded between ( V3) e0 and ( V3) eR sounds/sec. , wh ich we can draw on a line graduated logarithmically ( b ase e) •2 As o u r purpose is total independence, we attribute t o each of the sequences a1 calculated in 1 . a density re pres e n te d by a point drawn at random from the portion of the line men tioned above. However a certain concern for continuity leads us to temper the independence of the densities among seq u e n ce s a1 ; to this end we in troduce a certain " memor y " from sequence to sequence in the fo l l ow i n g manner : Let a1 _ 1 be a seq uence of duration a1 _ 1 , (DA) 1 _ 1 its d ensity, and a1 the next sequence with duration a 1 and d ens i ty (DA ) 1• Density (DA)1 will be given by the formula : =

in

which

length

x

is

a

eq u al to

segment

(R - 0) .

of line PX

and finally,

d rawn at

random

from

The probability of x is given

=

Na,

�

S

=

(1 - �) dx S

a

by

line

segment

s

of

(see Appendix I)

(DA),a,·

3. Composition Q of the orchestra during sequence a1• First the instruments are divided into r classes of timbres, e.g., flutes and clarinets, oboes and bassoons, brasses, bowed strings, pizzicati, col le g n o strokes, glissandi, wood, skin, and metal percussion instruments, etc. (See the table for Atrees.) The

composition of the orchestra is s tochastically conceived, i . e. , the distribution of the classes is not deterministic. Thus d u ri n g a seque n ce of d u rat i on a1 it may happen that we h av e 80/0 pizzicati, 1 0% percussion, 7 /0 keyboard, an d 3 /0 flute class. Unde r actual cond i ti on s the determining factor which would condition the co m p ositio n of the orchestra is density. We therefore

Free Stochastic Music

by

137

Computer

Composition of the Orchestra for Atrties Timbre classes and instruments Class

Timbre

Percussion

as

Instrument

Temple-blocks Tom-toms

Maracas

Susp. cymbal

2 3 4

Horn

5

Glissando

Flute Clarinet

Gong

French horn

Flute Clarinet B� Bass dar. B�

Violin

Trombone Tremolo or

fluttertongue

Flute Clarinet B� Bass clar. B� French horn Trumpet

Trombone

a

Trombone b

(pedal notes )

Violin Plucked

Violin

strings

Cello

Struck

Violin

strings*

Cello

9

Vibraphone

Vibraphone

Trumpet

Trumpet

12

Trombone

Bowed

Violin

strings

Cello

* collegno

I

3

I 2 3 4 5 6 7 8

2

Trombone a Trombone b

( pedal

2

1 2 I

8 10 ]]

l-5

6-9 10 11 12

9

Cello 7

Instrument No.

2

Cello

6

(ST/1 0-3,060962)

on present input data

notes)

1

2 1

2

Formalized Music

138

connect the orchestral composition with d ensi ty by means of a special diagram. An example from ST/10-1, 080262 is sh own in Fig. V-2. Fig. V-2 is expressed by the formula

Q.

=

(n - x) (e,.,. -

inwhichr thenumberofthe class,x such that n :S x :S n + I, and en.r and =

as

=

a function of n.

e,.+l,r

)

+

e,. .•

loge[(DA)1/(V3)],n = 0, 1,2, ,R, en+ 1 •• arc the probabiliti es of cfass T .

•

.

without saying that the composition of this table complexity and delicacy. Once these preliminaries have been completed, we can define, one after the other, the Na., sounds of s equence a1•

is

a

It goes

precise task of great

4. Definition of th e moment of occu"ence of the sound N within the sequence a,.

The mean density o f t he points or sounds to be distributed within

k

=

N4.fa1•

a,

is

The formula which gives the intervals separating the sound

attacks is

(See Appendix I.)

5. Attribution to the above sound of an instrument belonging to orchestra Q, which has already been calculated. First class r is drawn at random with proba­ bility q. from th e orchestra ensemble calculated in 3. (Consider an urn with

balls of r colors in various proportions.) Th en fro m within cl ass r the number

of the instrument is drawn according to the probability Pn given by an arbitrary table (urn with balls of n colors ). Here als o the distribution of i nstruments within a class is delicate and complex. 6. Attribution of a pitch as a function of the instrument. Taking as the zero point the lowest Bj;, of the piano, w e establish a chromatic scale in semitones of about 85 degrees . The range s of each in s trument is thus expressed by a natural n umbe r (distance). But the pitch hu o f a sound is expressed by a decimal number of which the whole number part is related to a note ofthe chromati c s ca le within the instrument's range. Just as for the density in 2., we a ccept a certai n memory of or dependence on the p receding p itch played by the same instrument, so that we have

where

z

is given

by the

probability formula

P" Pz is the pro bability

=

�

s

of the interval

(1 - -=) dz. s

z

(See Appendix I.)

t ak en at random from the ranges, and

139

Free Stochastic Music by Computer

... !',) :::r

1J �

w :I 0

Q

:I .,

r;

c Ill

Ill

i5"

:I

Fig. V-2.

�011. 3

!..

::r "'

-a

ST/10-1, 080262, Density

=

"""

!:!.

::;· �

"'

�

";DJ (J1

'9. ;;;· ... "' :I c. 0

011 01

a: 3

-..1 1J

0

i5'

Composition of the Orchestra

(DA)1

= 0.11 eu, U

=

;;;· N

()' "' .... 0

log. (DA/0.11)

()0 0

2.. co

(Q :I 0

140

Formalized

Music

sis expressed as the difference between the highest and lowest pitches that can be played on the instrument. 7.

Attribution ofa glissando speed if class r is characterized as a glissando. The I led

homogeneity hypotheses in Chap.

us

to the formula

) 2 ., • f(v = -- e-va , a.y''IT vfa =

and by the transformation

T(u)

u

to its homologue:

2 y''IT ru o e-u• du, J

=

for which there are tables.f(v) is the probability of oct:urrence ofthe

(which is expressed

portional to

in semitonesfsec. ) ; it has a parameter

a,

which

speed v is pro­

the standard deviations (a = sy'2). as a function of the logarithm of the density of sequence

a is defined

by : an inversely proportional function

=

a

a,

y'1r(30 � L[(DA)d(V3)] } -

or a directly proportional function

a or a

=

v'IT( 10 � L[(DA)d( V3)] )• +

of density

function independent

a =

where

k is

a

1 7.7 +

0

random number between

35k, and I.

The constants of the preced ing formulae derive from the limits of the

speeds that string glissandi Thus for (DA),

=

may take.

145 sounds/sec. a =

and for (DA)·,

=

2s

75

semitonesfsec. semitonesfsec.,

0.13 sounds/sec. a

2s

a

=

53.2

= =

17.7 semitonesfsec.

25

semitonesfsec.

8. Attribution of a duration x to the sounds emitted.

mean duration for each instrument,

To simplify we establish

which is independent of tessitura and

Free Stochastic Music by Computer

141

nuance. Consequently we reserve the right to modify it when transcribing

in to traditional notation. The following is the list of constraints that we take

into

account for the establishment of duration

G, the

x:

maximum len gth of respiratio n

or desired duration

the density of the sequence the probability of class r Pn, the probability of the instrument (DA)i,

q,,

Then if we define z as a parameter of

the

inversely proportional to

instrument, so

z

that

sound's duration,

probability of.

the

could choose

z, ... =

G.

Instead of letting Z...ax

to freeze the growth of

z.

=

could

law

applies for any

be

of the

and in this case

G, we shall establish a logarithmic law so

This z'

given value of z.

as

G In z/ln Zmax

=

Since we admit a total independence, the distribution of the durations

will be Gaussian:

where m is and

f(x)

=

1

--- e-<x-m)•tZs•'

the arithmetic mean

s

vz:;r

of the durations,

mm

+

4.25s 4.25s

=

=

s

the standard deviation,

0

z'

the linear system which furnishes us with the constants

u

z

occurrence

will be at its maximum when (DA),pnq, is at its minimum,

we

x

a

n

=

(x - m)Jsv'2

we find the

tables.

Finally, the d u ra tion

x x

function T(u),

m

and

s.

By assuming

for which we consult the

of the sound will be given by t h e relation =

±usv'2

+

m.

We do not take into account incom patibilities between instruments, for this would needlessly burden the machine's p ro g ram and calculation.

9. Attribution of dyno.mic forms to the sounds emitted. We define four zones ppp, p, j, if. Taken three at a time they yield 43 = 64

of mean intensities:

Formalized Music

142

permutations, of which 44 are different (an urn wit h 44 colors); for example, PPP--p. I 0. The same operations are begun again for each sound of the cluster Na,· 11. Recalculations of the same sort are madefor the other sequences.

An extract from the sequential s t atem ent was reproduced in Fig. V-1. Now we must p roce ed to the transcription into Fortran IV, a language

"understood" by the machine (see Fig. V-3). It is not our purpose to d e scri b e the transformation of the flow chart into Fortran. However, it would be interesting to sh ow an example of the adaptation of a mathematical expression to machine meth ods . Let us consider the elementary law of probability (density function)

f(x) dx

=

ce-cx dx.

[20]

in ord er for the computer to give us length s x with the probability f(x) dx? The machine can only draw random numbers Yo with equip robability between 0 and 1. We shall "modu l ate " this prob a­ bility: Assume some length x0; then we have How shall we proceed

prob. (0 where F(x0)

S

x

S

x0)

=

Lzo f(x)

dx

= 1

-

e-czo

=

F(x0),

is the distribution function of x. But F(x0)

=

prob.

then

(0

S

y

S

Yo)

=

Yo

and Xo

for all

x0

�

=

ln (1 - y0) c

0.

Once the program is transcribed into language that the machine's internal organization can assimilate, a process that can take several months, we can proceed to punching the cards and setting up certain tests. Short sections are run on t he machine to detect errors of logic and orthography and to determine the values of the entry parame t ers , which are introduced in the form of variables. This is a very imp ortant phase, for it permits us to explore all parts of the program and determine the modalities of its opera­ tion. The final phase is the d ec oding of the re s ul ts into traditional notation, unless an a utoma tic transcriber is available.

Free Stochastic Music by Computer Table

of the

44

143

Intensity Forms

Derived from 4 Mean Intensity Values, ppp, p, f. ff

iN-----;PN

f ========-- p

/1'1

f

I

/'11 --===J ===-)'II' /1!

I

f

!II -= il'!

--==-

/

Ill

1 -===- ))!> /'!! #

IJJ -==== # ::-=--=--;9¥ .f

��

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

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

ill'-===..:.

/>-=::::::

I

j'

-

!/'!

-====::

#:::. I.:::-

i

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

=-

/11-==:::#

I -====.#�Ill I ==-=--=-I -==-"#

I j>

-==:::

II

1-

If'-===-I'

=#�/

I'-====/ /

I� 11P-===-/

/�!>-===/

!!! -== #'

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

f�Ill J

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

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

If-==- t

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

.f

I

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

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

#

#

/

;I

� )# ;f-= #�)�/ -==::-

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

144

Formalized Music

Conclusions

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

is possible for al r eady

been

a large number of orchestral combinations.

Other

by RTF (France III); soloists.

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

Al th ough this program gives a satisfactory sol ution

structure,

it is, however,

by coupling

a

works have

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

to

the

minimal

necessary to jump to the stage of pure composition

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

calculations would then be changed into sound, whose internal

numerical

organization

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

l. The lon g laborious calculation made

The speed of a machin e

by hand is reduced

to nothing.

such as the IBM-7090 is tremendous-of the

order

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

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

composer-pilot can instill his own

personality

in the sonic result he obtains.

145

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

C

C C C C C

C

C

C

C C C

C

C C C

C C C

C

C C

C C C

C

C

C

C C

PROGRAM

FREE

GLOSSAQV A

OF

THE

- DURATION

ALEA

ST OCHAST I C MUSIC

OF

PRINCIPAL

- PARAMET ER USEO -

AIO•A20.AI7.A35•A30 SAME

INPUT

ALFAI31

OF

-

DATA

THREE

lVI

XEN

XEN XEN

ABBREVIATIONS

S EQUENC E

EACH

!FORTRAN

IN SECONDS

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

EXPRESSIONS ENTERING

INTO

THE

THREE

SP EED

WITH

9

XEN

10

XEN XEN XEN

II

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

ALIM

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

TABLE OF

INSTRUMENT

COMPASS

LIMIT�•

DEPENDING

ON TIMB�F. CLASS

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

INSTRUMENT �.

TEST

INSTRUCTION

480

IN PART

6

DETERMINES

WHETHER

KTF

-

XEN

XEN

XEN XEN

XEN

XEN

XEN

XEN

POWER OF THE EXPONENTIAL COEFFICIENT E SUCH THAT

XEN

NUMBER

XEN

WITHOUT

A

DEFINITE

PITCH.

KW

-

KTR

-

CU�VE

WHICH IS USEFUL

IN CALCULATING

GLISSANDO

SPEED

13 14 15 16 17 IB 19

20

21

22

23 24 25 26 27 2B

29 30 31 32 33 34 36 37 38 35

40

39

XEN

42 43 44

MAXIMUM NUM BE R

DISTRIBUTION

12

XEN

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

8

XEN THEXEN

VALUES

6 7

XEN

41

45 46

47 48 49 50 51 52 53

54

55 56 57 58 59 60 61 62 63 64 6�

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

Formalized Music

ANO SOUND EVENT DURATION• VIGL - GLISSANDO SPEED IVITESSE

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

SU��ECT

OR VARV

TO MODIFICATION.

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

V3

CONSTANTS

�EAD

TETA

TABLE•

HC

MEN XEN

IX=lt"7

XE"N

XEN

XEN XEN

READ 20tfTETAliJ•I=1•2561

30e CZJ C I) •221

�ORMATC12F6•6) READ

XEN

XEN XEN

(),Jet t8)

FOR�ATI6CF3e2•F9,SJ/F3e2tF9.8tF.6•2•F9e8J PRINT

THE

40tTETAtZ1e22

�ORMATI*l

40

**

THE ZJ

TETA

TABLE

•

TABLE =

XEN

THE Z2

XEN

*•/•21112Fl0.6•/l•�FI0,6•/////•

*•/e7F6e2tE12,3•///t*

*•IHII

TABLE

�

*•/•8Fl4•B•/XEN

XEN

REAO 50•0ELTAeV3eAIO•A20•A17•A30•A35tBF•SQPI•EPSI•VITLIM•ALEA•

50

60 70

XEN

AXEN

*LIM FORMATCF3t0tF3.3e5F3el•F2e0tFB,7eFS,e,F4t2tFS,B•F5•21 READ

XEN

X EN XEN AXEN J<EN

60tKTI•KT2tKWtKNLtKTRtKTEtKRttGTNAtGTNS•INTCl)ti=I•KTRJ

FORMAT(5J3t21Zt2F6•0tl212)

PRINT

70tOELTAtV3•AIOtA20•A17•A30tA�5tBFtSOPitEPSI•VITLJMeALEA•

itLJM�KT1 tKTZtK\thKNL FORMATt*tOELTA

**

�

A20

*/•*

BF

=

e

tKTReKTE tKRJeGTNA•GTNSt CCI tNTC I)) •1•1 tt
V3

*•F4,0e/•* e

�

4tF6•3•/•*

*•F4el•/••

A10

�

*•F4,1•/•

*•F4el•/•*

A35

•

XEN

*•F4eltXEN

SOP! =*•F1le8t/•* EPSI =*•Fl2•8•/t* ALEA =*•Fl2e8•/•* ALIM = ••F6e2e/•* KTI • *•l3t/t KW = * • l3 t/ • * KN� = *•13•/•* KTR : *t13t/4 *•J2•/•* KRJ = *•12•/•• G TN A • *•F7•0•/•* GTNS = *•F7•0•

+tF4ele/e•

A17

A30

•

*•F3e0•/•*

VJTLIM

�

*•F5•2•/•*

**

KT2

**

KTE

=

C

90

READ

*•13•/•*

=

*/•J2l*

BO

IN

Cl.ASS

••12•*•

THERE

ARE

*•12•"*

tNSTRUMENTS•*•/l)

80•KTEST3eKTEST1eKTEST2

FORMATI5131

KTESTl

PRINT 90tKTEST3tKTESTleKTEST2 FORMATC*

KTEST3

IFIKTEST3oNEo01

R=KTE-t

=

*•13•/•*

PRINT

*tl3t/t* KTEST2

*•13)

830

A10•AID*SOPI A30=A30•SOPJ

C

C

IF ALEA IS NON-ZEROoTHE �ANDOM NUMBER IS GENERATED FROM THE TIME WHEN THE FOLLOWING JNST�UCTION tS EXECUTED• IF ALEA IS NON-ZERO EACH RUN OF THIS PROGRAM WILL PROOUCE DIFFERENT OUTPUT DATA. IFIALEAoNEoOoOI

*XEN

XEN

XEN

XEN XEN XEN XEN XEN

XEN

�N

XEN

XEN XEN

A20•A20*SOPI/� C

:teEN MEN XEN XEN

XEN JO

TO 1•1 .. 1

�0

)(F.N

i 2)(EN

MOCIIIIXF.II=I

20

XEN XEN XEN XEN

•

IXB•B-IX

c

XEN

XEN

TABLES

Q( IZJ tSf 12') •Ef 12•121 tPNt 12•'50) •SPN( 12•'50 I tNTf 1 21 • •HAMINf J 2•501 •HAMAXr 12• SO J •HBMINC 12•50) •HBMA>< C 12•50 J •GNC 12•50) *•SO I tTETAI Z56J tVIGLC 3) tM00117) tZII BJ tZZCB) tALFA( 31 tAMAXC 121 1•1

XEN

XEN AND

DTMF.NSJON

DO

XEN XI!N XEN )(EN

XEN

CALL

�ANFSETITIMEFI1)1

XEN XEN

XEN

XEN XEN XEN

66 67 68 69 70 71 72 73 74 75 76 78 77 79 so 81 82 83 84 85 86 87 BS 89 90

91

92 93 9. 95 96 97 98 99 100 114

IJS 1.26

127 128

129

130 13l

132

133 1 34

135

136

142 143 I145

141

147 148 149 ISO 151 152 153 146

Stochastic Music by Computer

Free

147

830

PRINT

DO 130

*PNC(tJ)t..J�I•KTS)

FORMATC515F2t0tF"3e:1l)

100

1

Pj:;�JNT 110

FORMAT(////•• =

F3,0t*tGN

DO

c

IF

*•F3•0•*•

1:1

•

FORMAT112Fl'o21

HAS

,,JJ

tHBMINC lt..JJ

*•12tC/t•

FOR

*•F3e0t4tHBMJN

PN : *•F6e3l)

=

INSTRUMENT NO, ••F3.0•*•HBMAX

*•12• =

*•

167

XEN

XEN

XEN 4•12•

XEN

••F 6 ,2• / 11

OF

)(EN

16.:\

166

1 68 169 170 171

172 173 1 74

175

176

XEN

177

DO 1 70 I • 1 ti
XEN

178

)(EN

179

IF

fAB5FrY-taOJ,GE.EPSIJ

?00 t=taKTR AMAX ( I l: t • 0./'E t I DO ?00 J•2•KTE

t

XEN XEN

CALL EXIT

I)

IF

PRINT

IKTt.NEeO)

190tAX

TAVl;TJMEF!ll

NLIHE•SO

1 AND 2•

KNA•O KI�O 230 Xl .. RANFI-1 l A�-DELTA *

SEQUENCE

GO

240

GO TO 250

(l
GO TO 230

TO

A�ALJM/�,o

270 UX=�*X2 GO T O 310 IF

XEN XEN XEN )(EN

187

A

SECONDS

AND CLOUD NA DURING

A

IJW.GT,J)

ISS

168 189 190

191

192

XEN

196

XEN

197

XEN

199 200

XEN

XEN

194

193

195

198

201

XEN

202

XEN

203

XE N XEN XE'N

XEN

204

206

205

XEN XEN

20A 20Q

XEN

214

XEN

GO TO 280

183

1 84

XEN XEN

XFN

J
XI=O,O 250 K2=0 260 X2=RANF ( -1 1

DEFINE

LOGFIXIl

1FfA.LE,ALIM1

IF

186

XEN

IKTeSTlsNEt01

PARTS

XEN

)(EN

SINA�O,O

IF

XEN XEN )(EN )(EN

FORMAT(IH t9E12t8l

181

l 80 182

XEN

JW:l

;?4n

165

XEN

LEVEL

162

163

XEN

XEN

DENSITY

160

XEN

XEN

200 IF IAXaGT.AMAXI IJ1 AMAXIJlcAX IF (KTt,NE,O> PntNT 210•AMAX 210 FOQMATI lH e9EI2tB1

220

XEN

XEN

AX•t.O/(E(t.�I*EXPF(AJ))

c C c

XEN

XEN

IN

1!!7 158

161

XEN

XEN

AJ=:.J-1

c

)(EN

155

tHBMA)(( I,..;) tGNC t t..J�XEN

CALL EXIT

C LAS S NUMBER *•12•/•1*

PROBABILITY

DO

190

)(EN

J:I,KTE

DO 180 Y=O,O

lBO

I

160tltfJtECit.J)t..J=t•KTEl

A

156

159

KTR

FORMAT(//////•*

PRINT

154

)(EN

)(EN

140tCECitJ)tJ=t•KTEl

140

170

=

AND

CABSFCV-J,O),GE,EPSil

00 1 50

tHAMAX(

NUMBER

lt.J)=:'f

READ

44

CLASS

J=loKTS

ll'O

SPNC

IN

160

1!150

J ,J)

** HAMIN = *•FJ,O•*•HAMAX •

120 130

t.Jt:rl •KTS)

JO•l•IJ,HAMJNC

••PNI I•.J)

)(EN

XEN

l=l•KTR

v�o.o KTS•NTCI) READ tQO,tHAMTNfltJ)tHAMAXC!tJl,HBMlN(J,J),HSMAXCJ,J)tGNCitJ)•

XEN )(EN XEN

207

211 212

210

213

Formalized Music

148 IF CRANFf-l)eGEeOe�l GO TO 2q0 ux�UP� � � • CJ,O-SORTFCX211 GO TO 300 290 UX:UPR - R * ( t,D-SQRTFCXZJI 300 IF CCUX,GEeOeOJ,ANO•fUX,LEeR)J GO TO 310 IF CK2,GE.KT21 GO TO 270 280

GO 10 310 U=U� DA=V3

K2:el<2+1

1F

260

IF

•

OA +

O,�J

+

IKN4oGEoKT21

GO TO ?.:'30

1

GO

GO TO 320

CGTNAeGTeF�OATFCNAJJ

KNA:KNA+1 320

XEN XEN XEN

TO 330

XEN XEN

)(f':N XF.N l<EN

330 UPR•U

c C c

GO

CKTI,EO.O)

TO

XEN

360

FbRMATC1Hle418•3X•4EtB.B•3Xel8)

350

FORMATCIHDtZI9tFlne21

N4=1CTI

CKTEST3,NE,01

P4RT

350

3•

PRINT

350eJWeNAeA

DEFINE CONSTITUTION

SINA•SINA

OF

ORCHESTRA DuRING SFOUENCE A

+ FL04TFINAJ *XLOGOA

XEN

XEN

)(EN XEN

N;:XINTFCXLOGOAJ IF

fCM+�J.GTeKTE)

�•KTE-2

SR:OeO

M2mM+2 00 380 t=l •KTR ALFX�EC 1eM11 BETA=E'C I ellll21 �M=M OR=f�LOCOA-XM1

*

CRF.TA-ALFXJ

SIIJ=SR

390

FORMAT(lH

TF

CKTt,NE.Ot PRINT •12F9•4J

PART 4t0EFINE

c C

c

+

ALFX

253 25ca

XEN

XEN XEN

INSTANT

XEN X EN

xEN TA

OF

EACH

POINT

IN SEQUENCE A

XEN

T=-LOGFC XJ /DA

TA•TA+T

C�TleNEeOJ

258

259

260

261

262

263 Z64

XEN XEN XEN XEN XEN

FORMATC//•1Bt3E20t8J

256

257

265

XEN

PRINT 420•N•X•TtTA

255

XEN

XEN

l<=RANFI-11

252

XEN XEN

TA=O•O GO TO 410 N=N+l

IF

251

XEN

XEN

T•O.O

410 11\20

250

XEN

N•J

11\00

XEN

XEN

390tC0111•1=1•1
246

248 249

XEN XEN

370tXMtALFXtBETA

230

231 232 233 234 235 236 237 238 23<;1 240 241 242 243 244 245

247

XEN

IF (KTleNE.OJ PRTNT 370 FORMAT(lH •3F20.8) Qll J•OR s��SR+OR

215 216 217 21B 219 220 221 222 223 224 225 226 227 228 229

XEN l<EN

Nt=M+1

380

><EN

XEN

XEN

XLOGDA=U XALOG=A20

><EN ><EN ><EN XEN ><EN ><EN

340e�WtKNAeKltK2eXI•X2tAtOAeNA

340

IF

XEN

XEN

A=OF.L TA

IF

XEN XEN

XEN

CO TO ?50

PRINT

XEN

XEN

* EXPF'CU1

NA•XINTFCA

l<EN

XEN XEN

XEN

XEN

2 66

267

268 269 270 271 272

273 274

275

Free

c C c

Stochastic Music by Computer

PART

430

CLASS AND INSTRUMENT NUMBER

�•DEFINE

XI =�ANFC-1) 00 430 t:J•KTR IF

149

CXleLE•SCJ))

l=KTR

TO

GO

TO EACH

POINT

KRct

460

PtFN•PNCK�•tNST�l

IF CKTt.NEeOl PRINT 470•Xl•SCKR)eKR•XZ•SPIENeiNSTRM 470 FORMAT! II-( •2E20,9ei6•2E20.B•I6 > PART 6t0EFINE C K R. GT,JJ

TO

GO

560

TO

GO TO

CKRtLTe7)

HN FOR

PITCH

C!NSTRMtGEeKR1)

HXsOeO IF

EACH

POINT

480 GO TO 490

OF SEQUENCE A

GO

TO

IF

CHPR.LE,O.Ol

540 �50 560

c C

c

JF

CN,GT,t)

XEN

GO TO

HXcHPR+HM

*

HX=HRR-HM

*

1F

GO

(

520

TO

IRANFr-lJeGE•Ot�)

GO TO 550

GO TO

I KeGE .� T 2l

GO TO 520

510

GO TO 560

HI KR•INSTRM):HX

JF IKTl•NE•O) PRINT 570tKtX.HX �70 FO�MATftH t16•2E20tAJ PART 7t0EFINE SPEED VJGL


GO

TO

560

TO

EACH POINT OF

304 305

30 7 306

308

311

310

XEN

3 13 314 315 316

XEN

318

XEN XEN

320

XEN

322

XEN

324 325

XEN

XEN

XEN

312

317

319

321

323

326

XE"N XEN XEN

327 328 329 330

XEN

333

XEN

XEN

XEN

XLAMBOA•O,O

XE'N

GO TO

XEN

740

303

XEN

XEN

A

297

309

XEN

lteO-SQRTF()())

298 299 300 301 302

XEN

XEN

940

294

XEN XEN

teO-SORTF(X))

292 293

XEN

XEN

JFrCHX,GEtHINF)eANOeCHX.LE•MSUP))

II(:K+l GO TO

XEN

XEN

530 520 HX=HINF•HM*X RANFC-l) GO TO 560

SJO

295 296

XEN XE N )(EN XEN )(EN

XEN

)(:RJ'iNFl-1) IF

XEN

XEN

HPR=HCKR•INSTRM)

290

XEN

XEN

XEN

HINF•HA�INCKR,JNSTRM)

28l

283

291

)(EN

XEN

500

500 HM�HSUP-HINF

510

XEN

XEN

HSuPzHAMAXIKRe!NSTRM}

1<:=0

2B4 285 286 287 288 289

XEN

4qo

HSUP=HBMA�IKR•IN5TRMl HINF=HBMIN(KRtiNSTR�)

490

282

X EN

XEN

GO TO 460

CX2.LEeSPIEN)

tNSTRM-=KTS

GO

XEN

XEN XEN

INSTRMzJ

IF

480

XEN

XEN

SPJE'NeSPNCKRtJ)

276

277

27B 279 280

XEN

450

IF

XEN XEN

XEN

X2sRANFC-I)

00 G50 J=l•KTS

IF

XEN

XEN

XEN 440

440 KTSaNT(J)

c c c

0� A

XFN

332

331

:J34 335 336

150

Formalized Music

sao -:x�t 590 XI ::RANFC 600

l= t 2A

-11

IF

CXt-O,qqQ�)

DO

630

TO

XE'N

6t0e640o6?0

XEN

XEN

630

XEN

1•1-MDDIIIXt

XEN

630

CONTIM..JE

640

XLAMBOA:FLOATFfJ-1)/lnn.n

650

XF.N

XE N

Jat+MOOJCIX, GO

620

XE N

IX=Io7

1FITF.TAIIJ-Xl)

6tn

XEN XEN

�00t6�0•6RO

I"CTETAIII-XII

XEN

XEN

670o640o660

GO TO

C720t76�1•

GO TO

1720o7601oKX

XEN

KX

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XLAMAOA=?.·��

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660

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670

TXI=TETAIII

6!!0

GO TO C 720•760 00 690 I •2• 7

XEN XEN XEN XEN

690

CONTINUE

700

TXJ:t.O

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TX2=ZI C I 1

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GO TO

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XEN

1• �X

XLAMSOA:CFLOATFCt-ll+CXt-TXJJ/CTETACI+tJ-TXIJl/lQO,O

700o71 Oo690

TXI=Z2111

I"IXI-TXII

XEN

XEN XEN

I

XLAMBDA:TX2-IITXJ-XII/ITXI-Z21 1 -I llt•ITX2-ZIII-Ill 7?0o760

710

XLAMBDA=Zll!l

720

GO TOC

720•760

I•

�·

XEN

KX

XEN

KX

ALFAili=AlO+XALOG

XEN XF.N XEN

ALFAC31=A30-XALOG

X EN

X2=RANFI-ll

ALFAC21•Al7+A35*�2 DO 730 t-=lt3

XEN

X EN

I" IVIGLII loLToOoDI V IGLI 11=-VIGLIII tF CVIGLt lltGTeVITLIMJ VIGLC II=VITLIM 730 I" IRANFI-IIoLToOo51 VIGLIIt=-VIGLIII 740 IFIKTitNEtOl PRINT 750eXI•X2eXLAMBDAeVIGL

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VICLCI1=1NTFCAL�ACI1*XLAMBDA�0.SJ

c C c

750

"ORMATIIH

FOR EACH

G-=GNCIO=G/LOGF C ZMAI()

XEN

POINT

OF

A

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XEN

XEN

XEN

XEN

OPNOA-=1•0/fQfi
GE:A8SFIRO*LOGFIOPNDAII

XEN

XMU=GE/2o0

XEN

SJGMA!!!!GE/4.0

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KX=i'

XF.N

GO

XEN

TO �gO

X2=RANF(o-J)

TAU�SJGMA*XLAMBOA*I•4142

XOUR•XMU+TAU GO TO 790 '"

770

XEN XEN

CCKR.E0t71•0R.CKR.�0.8l> GO TO 780

ZMAX=AMAX(�R)/CV3*PIEN)

760

XEN

o6EI9o81

PART BtDEFJNE DURATION IF

XEN

XEN

IX2oGEoOo51

GO

IXDURoGEoOoOI

770

XE N

XEN

XEN

XOUA=X�U-TAU

IF

XE N

XEN

TO

GO

TO 790

XEN XEN

337 338 339 340 341 342 343 344 345 346 347 3&8 349 350 351 352 353 354 355 356 357 3!58 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390

392 393 394 395 396 397 391

Free

c: C c:

by Computer

Stochastic Music

151

7 8C XDUR=O•O 790 IFCKTleNE.OlPRINT 800•ZMAXtXMU,SIGMA,Xl•XLAMBOA•X2tXOUR BOO FORM-TCtH •SEIS•S•Eli•4•E15.8) PART

9•DEFINE

INTENSITY

FORM TO EAC� POINT OF A

CKTJ,EO,Ot GO TO 840 INLINEoLT,KNLI GO TO 610 IF (NLINEtEO,KNL) GO TO 620

IF

IF

TO

900

GO TO QOO 820 PRINT A30 830 FOQMAT r IHI) NLINE=O GO TO 900

NLINEcNLINE+I

640

IF

950 860

GO

GO TO

860

NLINE=NLINE .. 1

870

PRJNT

4),//)

PRINT

650

c: c

PART

c

IF

IOtREPEAT

lN•LT.NAI 11•

SAME

GO TO

REPEAT

(KTF�T2.�0.0J

OFFINITIONS

400

SEQUENCES

GO

TAP2=TIMEFI!I-TAV2

TO

A

910

TAP?.:TAPZ/FLOATFCNA) PRINT

c: 910 c:."ZO <>3C

941'1 c

c c c:

TCH* •6Xt

BAO PRtNT A90,N,TA,KR, J NSTRM,HX.(V(GLrll•l=1•31tXOuR•IFORM SQn FOQMATC IH ,. 17eFI2e21 t9,JB,Fllel ,FJ3•1•2FIO•l•Fl4•2•11 I) IF

407

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411

414

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ALL

POINTS

OF

A

402

AOA 405

406

408

409 41 0 412

liEN

413

415

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416

XEN XEN )(EN

417 416

419 420 022 423 421

XEN

420

XEN

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425

XEN

427

ICEN XEN ICEN X EN

428

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XEN XEN X EN XEN XEN XEN

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750•TAP2

IF (JW,.GEeKW1 GO TO 930 JW:..J\1'+1 IF IGTNSoGT.S!NAI GO TO ?.20 IF fKTFSTteF-OeO) CALL F.MJT

XEN XEN XEN

TAPt:TAPI/FLOATFr�WI TAP1•TIMEFI-11-TAV1

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400

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JW=*•I3t4Xt*Aw*tF8e2t4Xt*NA:*•I6t4Xt*OCIJ=*•12CF4e2t*/4XEN

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c: C c

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R60tJ�tAtNAtl0111•f=1•KTR)

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401

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TO

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398 399

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810

MEN XEN XEN

7'50oTAPI

END

XEN XEN XEN XEN

ornononJtJnnn?26nno339non451onn�fi400067600n7eqn��9010010130�112500I23600

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XEN

426 429

430

431

432 433 434

436

435

438

437

439

440 44 1

442

443 444

445

446

447 448

Formalized Music

! 52

7 6 5 1 0 0 7 7 0 7 0 0 7 7 6 1 0 f' 7 8 1 4 0 0 7 8 6 7 0 0 79 1 8 0 0 79 6 <;1 0 0 8 0 1 9 0 0 8 0 68 0 0 8 1 1 60 0 8 1 6 3 0 0 8 ?. 0 9 0 0 B 2 5 4 0 0 8 ?.9 9 0 0 8 3 4 2 0 0 8 3 R � O O B 4 2 7 0 0 8 4 6 8 0 0 8 5 0 A 0 0 8 5 4 8 � 0 8 5 8 6 0 0 8 6 2 4 0 0 86 6 1 0 0 8 6 9 B O O

B 73 3 0 0 8 76 8 0 0 8 8 0 2 0 08 8 3 5 0 0 8 B 68 0 0 8 9 C 0 0 0 8 9 3 1 0 0 8 9 6 1 0 0 89 9 1 0 0 9 0 2 0 0 0 9 0 4 8 0 0 9 0 7 6 0 0 9 1 0 3 0 0 9 1 3 0 0 0 9 1 5 5 0 0 9 1 8 1 0 0 9 2 0 5 0 0 92 2 9 0 0 92 5 20 0 9 2 7 5 0 0 9 2 9 7 0 0 9 3 1 9 0 0 9 3 4 0 0 0 93 6 1 0 0

9 3 8 1 0 0 9 4 0 0 0 0 9 4 1 9 0 0 9 4 3 8 0 0 9 4 5 7 0 0 9 4 7 3 0 0 9 4 9 0 0 0 9 5 0 7 0 0 9 5 2 3 0 0 9 5 38 0 0 9 5 5 4 0 0 9 5 6 9 0 0 9 5 8 3 0 0 9 5 <;1 7 0 0 9 6 1 1 0 0 9 6 2 4 0 0 9 6 3 7 0 0 9 6 4 9 0 0 966 1 0 0 96 7 3 0 0 9 6 8 4 0 0 9 6 9 5 0 0 97 0 6 0 0 9 7 1 60 0 9 7 2 6 0 � 9 7 3 6 0 0 9 7 4 5 0 0 9 7 5 � 0 0 9 7 6 3 0 0 9 7 7 2 0 0 9 7 B 0 0 0 9 7 8 8 0 0 9 7 9 6 0 0 9 8 0 4 0 0 98 1 1 0 0 9 B I A O O 9 8 2 5 0 0 9 8 3 ?. 0 0 Q 8 3 8 0 0 9 A 4 4 0 0 9 A � C 0 09 8 '56 0 0 9 A 6 1 0 0 98 6 7 < ' CJ 9 8 7 2 0 0 9 A 7 7 0 0 98 8 2 0 0 9 B B 6 0 0 9 8 9 1 0 0 9 8 <;1 � 0 0 9 8 9 9 0 0 9 9 0 3 0 0 9 <;1 07 0 0 9 9 1 1 0 0 99 1 �0 099 1 A 0 0 9 9 ? 2 0 0 9 02 5 0 0 9 9 2 8 0 09 9 3 1 0 0 9 9 3 4 0 0 9 9 3 7 0 0 9 9 3 9 0 0 9 9 4 2 0 0 99 4 4 0 0 99 4 7 0 0 9 9 4 9 0 0 9 9 5 1 0 0 9 9 5 3 0 0 9 95 5 0 0 9 9 5 7 0 0 99� 9 0 0 99 6 1 0 0 996 3 0 0 9 96 4 0 09 9 6 6 0 0 9 9 6 7 0 0 9 9 69 0 09 9 7 0 0 0 99 7 2 0 0 9 9 7 3 0 0 9 97 4 0 0 99 7 5 0 09 9 76 0 0 9 9 7 7 0 0 9 9 7 9 0 0 9 9 7 9 6 0 9 98 0 5 0 9 9 8 1 4 0 9 9 8 2 3 0 9 9 8 3 2 099 8 4 0 0 9 9 8 4 8 0 9 98 5 5 0 9 9 8 6 2 0 99868 0 9 9 8 7 4 0 9 9 8 8 0 0 9 9 8 8 5 0 9 98 9 1 0 9 9 8 9 6 09 9 9 0 1 0 9 9 9 0 6 0999 1 0 0 9 99 1 4 0 9 99 1 8 0 99 9 2 3 0 9 9 9 2 7 0 9 9 9 3 0 0 99 9 3 4 0 9 9 9 3 7 0 9 99 4 0 0 99 9 4 4 09994 7 0 9 9 9 5 0 099 9 5 3 0 <;1 9 9 5 6 0 9 9 95 8 0 9 9 9 6 0 0 9 9 9 6 3 0 9 <;1 0 6 5 0 <;1

Q <;I � 7 0 0 9 Q 6 0 0 9 <;1 9 7 0 0

2 5 5 0 9 9 9 7 0 0 0 0 2 6 3 0 9 99 8 0 0 0 0 ?. 7 5 0 9 99 9 0 00 0 3 1 3 0 9 9 99 9 0 0 0 3 4 6 0 99 9 9 <;19 0 0 3 7 7 099999990 4 06 0 <;1 9 <;19999<;1 1 0 0 E �O I 0 0 0 0 0 0 0 0 0 4 0 0 s O I O n 2 0 0 I 7 7 3 0 o 3 5 5 6 3 1 7 7 2 4 5 3 9 0 I O O o o o 0 7 t o o n n o o o o o n t 20 0 0 0 0 0 0 ! 5 0 5 0 0 5 0 0 1 2 0 72 0 0 0 1 6 0 0 0 2 5 0 0 0 1 2 0 1 0 1 0 2 03 0 9 0 2 02 0 1 0 1 0 2 0 2 0 1 0 1 0 0 0 0 1 0 0 7 0 0 1 0 1 0 0 0 0 1 0 0 9 0 0 1 0 1 0 0 0 0 1 0 1 2 0 0 1 0 1 0 0 00 1 0 1 1 0 0 1 0 1 0 0 0 0 1 0 0 9 0 0 1 0 1 0 0 0 0 1 0 1 2 0 0 1 0 1 0 0 0 0 ! 0 0 A0 0 1 0 1 0 0 0 0 I 0 0 80 0 1 0 1 0 n no 1 0 1 2 0 0 1 0 1 0 0 0 0 1 0 0 8 0 0 1 0 1 0 0 0 0 ! � 0 ? 0 0 1 0 1 0 0 0 0 2 0 0 ?. 0 1 7 5 5 0 0 0 0 1 09<;19 3 9 7 5 0 0 0 0 1 5999 2 9 7 1 0 0 0 0 P 06 0 0 1 7 5 4 0 0 0 0 1 0 4 0 0 3 4 8 5 0 0 0 0 1 54 0 0 1 5 6 3 0 0 0 0 1 54 0 0 1 95 3 0 0 0 0 1 0 2 0 0 3 9 75 0 0 0 0 1 5 1 5 0 2 9 7 1 0 0 0 0 1 0 0 9 0 1 7 5 4 0 0 0 0 0 7 0 9 0 1 7550 0 0 0 ! 0 0 9 0 3 3 6 3 0 0 0 0 1 0 0 9 0 1 95 3 0 0 0 0 1 0 0 70 1 0 1 3 0 0 0 0 1 0 2 0 0 3 4 8 5 0 0 0 0 1 5 2 0 0 1 5 6 30 0 0 0 1 5 0 2 0 0 00 0 3 4 6 7 0 05 0 0 0 0 0 0 1 54800500 0 000346700�000 0 0 0 1 54A00�DO 0 00 0 3 2 6 8 1 0 9 0 0 0 0 0 0 3 36 3 1 09 9<;1 0 0 0 0 1 95 3 1 0 !'1 0 0 0 0 00 1 0 1 � 07 2 0 0 0 00 0 3 4 8 7 1 5 5 0 0 0 0 0 0 1 5 7 � 1 5 5 0 0 2 5 0 8 0 4 0 8 0 1 1 3 09 0807 ! 6020 1 0 1 1 0 0 3 03 0 4 2 0 0 1 0 1 1 0 0 2 0 5 0 32 '1 0 1 0 1 1 ?.

0 3 3 5 0 3 1 � n J 1 0i0 �

0 2 1 0 0 3 D P. I 0 3 9 0 7 0 2 02 0 2 0 3 1 5 0 ? 0 7 0 2 0 2 0 2 0 2 4 1 0 207 0 3 09 0 3 1 7 04 1 60 9 0 3 1 3 2 0 0 3 ?. 0 0 50 9 0 ?. 0 5 2 8 0 1 0 3 0 4 00 4 5 0 1 1 ? 0 ?. 0 ? 0 1 06

Free Stochastic JW=

1

A=

Music

153

by Computer

9.13

N A -=


Q I I I = O . l 2 / 0 o 0 4 / 0 . 0 4 / 0 . 0 5 / 0 . 1 2 / 0 . 2 9 /0 . 0 4 / 0 . 0 4 1 0 . 1 4 / 0 o 0 6 / D . 0 6 / � . 0 3 1

S T AR T CL A S S I N S T R M 1 o . co 1 H' 0 . 10 1 6 Ooll 8 6 0.13 3 l 0.18 4 9 0.25 1 6 0 . 33 1 .. 0 . 34 1 1 (j • 4 0 l 6 0.41 q 6 C . 7 E: 7 Cl . 9 0 8 2 1 1 .00 2 1� 1 . (; 9 1 6 I . (1 9 2 6 1.23 3 6 1 .42 1 10 1.57 1 4 1.65 2 6 1 . 78 B 6 1 .92 3 1 . CJ lt 5 1 4 2.18 1 6 2.18 6 1 :t . l 9 12 9 2 . 20 l 9 2oL3 1 1 2 .32 1 4 2 .33 l 2 .54 1 b 11 2 . 57 2 .. .t. � c; z.n 1 1 3 J 2 . 80 5 5 'H 3 . 2 8 2 1 �5 3.33 7 5 3 E 3 . 38 2 l l' 3 '1 3 . 5 5 1 1 3 1! 3 . 5 6 9 c; 3 9 3 . t l) 1 12 4Q 3 .64 2 (: 4 1 3.65 5 5 lo 2 3 . 7 1 3 6 lo 3 3 . 80 B 6 "" 3.87 2 (: 4 5 3 . 89 7 5 lo t 4 o 1 5 2 5 41 4. 15 1 9 48 4.25 1 12 4 <; lo o 3 1 2 1 5 1) ' � 3 3 1n

J';

1 2 3 4 5 � 7 e c; 10 1] 12 13 14 15 H 17 18 Ic; .0: 0 21 22 23 2 lo 2� u 27 2E 2� �0 �)

P I T C H GL I S S 1 GL I S S 2 G L I S S 3 D UR A T I O N O Y NA M 3 o . r:-o 3 4 .0 o.o o .o o.o 50 �. 4 1 o.o 43 . 2 o .o o.o Oo63 21 o.o 81 . 3 o. o o.o 10 0.18 47 .0 Q.O o.o o.o 29 1 . 9(1 o.o �-" o.o o.o 35 1). 5 1 48.7 o.o 1) . 0 o.o 42 1 1 .4 o. o 0 . 37 o.o o.o 59 n . oo .3 8 . 1 o.o o.o o.o 45 o .o o. o 2 . 20 o.o o.o 0 1 .0 7 o.o 5 5 . 1) o. o (1 . 0 1 0 . 4C llo5 o.o o .o o.o 19 23.2 n . oo o.o � .o o. o 6 o.oo 26.9 o.o o .n o.o 57 lt 6 . 2 n . 32 "·" o. o tl . O 25 1) . 7 1 6 8 .5 o.o o. o (' . 0 32 46.9 1) . 6 4 Q .O o.o o.o 1 44.0 0 . 44 o.o c.n o.o 21 0. 22 36.2 ') . Q o.o (1 . (1 13 32. 5 t . o c; 0 . 1) o.o o.o 6(.\ 12.6 (I . O t o.o o.o o.o 60 o.ss o.n 3 B o CJ o.o l'! o t' 62 74 . 6 0 . 80 7 1 .0 -2 s.o -71.0 50 1 . 50 32 . 6 o.o o.n o.o 26 50 . 9 0 . 60 o. o 0 . (\ o.o 24 4 o 5 1! 0 . () o.o o.o o.o 58 0.02 49 . 3 O . CJ o.o o.o 13 o.o 0. 22 5 1 .0 o.o o.o lt 3 36 . 9 o.o (1 . 0 1) 1) . 0 1:' . 0 56 1 o 3B 31.8 1') . (') o .o o. o 14 n. 2 e I). I) (! . «) () . 0 o.o 40 1 . 6 <; 12.2 0 . (' 0 . 1) o.o 55 48.5 0.37 n.o o.o o.o 58 ll . O n .o 1 . 50 o. o o.o 1'1 . 5 2 1 5 .4 21 49 . 0 5 . (1 - 3 1 . 0 B 1 .38 O . IJ o.o o.o o.o 4 lo 7 . 3 -71.0 1 . ('1 5 46.0 - 1 7.0 24 37 . 6 0 . 14 o.o o.o o.o 0 1 o 30 o .o o.o o.o o.o 1 '3 0. 19 (: 4 . 3 o.o o.o o.o 9 3 .72 52.2 0 . (\ o.o o.o 28 59.0 0.83 o .o c .o f.' e O 38.8 11 25.0 (1 . 0 (1 z .o - 1 5 . 0 7 5 .6 0.43 11 0 . (1 ll . (l o.o 5 1 .5 57 (1 . 7 7 o.o c.o o.o 2 0 . 3 <; 12.1 o.o (! . 0 o. o 1 . 16 4 3 .0 2 -7 1 . 0 24.0 n.o eo . 3 50 0.85 3 6 . 1'1 4.0 22.0 59 .9 10 o. o (' . l(! o.o o.o 2 . 4 Cj 40 . 1 33 o.o o.n o.o o .� n . 4 E: 34 o.o c.o o.o

Fig . V-4. Provis i o nal R esu l ts of O n e Phase o f t h e Ana lysis

1 54

Formalized Music

-

0

10

I

..-

�


>

Lri D)

ii:

C h a pter VI

Symbolic we

Here

shall

atta c k the

Music

thorny p robl e m of the logic underlying musical

composition. Logic, that queen of knowledge, monopolized by m athem atics,

wavers between her own of algebra.

name,

borne

through two millennia, and the name

Let us leave the task of logically co nnecting the preceding chap ters the moment. We shall confine ourselves to following a path which may l e a d us to regions even more harmonious in the not too dis tant fu ture.

for

In

A LOG I CA L AN D ALG E B RAIC S K ETC H M U SI CA L C O M P O SITI O N

OF

we shall begin by imagining that we are suffering from We shall thus be able to reascend to the fo u ntain-head of the mental o p erations used in composi tion and attempt to extricate the general principles that are v al i d for all sorts of music. We shall not make a psycho-physiological study of perception , b u t s h a l l simply try to u n d e rstand more clearl y the phenomenon of h earing and the thought-processes involved when l is te n ing to music. In th i s way we hope to forge a t o ol for the better comprehension of the works of the past and for the constru ction of new music. We shall therefore be obliged to collect, cut up, and sol der scattered as well as organ ized enti ties an d conceptions, while u n ravel i n g the th i n thread of a l ogic, which will certainly p resen t lacunae, b u t which will a t least have the merit o f existing. this chapter

a sudden am nesia.

CASE

OF

A

SIN GLE

GENERIC

E L E M ENT

Let there b e a son i c event wh i ch i s not ! 55

endless .

I t i s see n

as a

whole,

as

Formalized

156

Music

an enti ty, and this overall pe rce p tio n is sufficient for the m o m e nt. Because of our amnesia, we d ecide that it is neuter-neither pl e asant nor unpleasant.

Postulate. We sh al l systematically refuse

every sonic event. What eve n t

or

will

a

q u alitative judgment

cou n t will be the abstract relations

on

within the

and the logical operations wh i ch m ay be of the sonic event is thus a kind of statement, sonic symbol, which may be n ot at ed gra p hically by the

betw ee n several events,

impos ed on them. The emission

in scri p tio n , or

le tte r a.

which and then disappears ; we simply have a. If it is emitted several times in s uccessi o n , the events a re compared and we co n cl ude that th ey are identical, a n d no more. Identity and tautology are t h erefo re implied by a repetition. But simultaneously an other phe­ nomenon, subjacent to the first, is created by reason of this very repeti tion : modulation of time. If the event were a Morse sound, th e temporal abscissa wou ld take a mea n ing external to the sou nd and independent of it. In addi­ tion to the deduction of tautology, then, rep e titio n causes the appearance of a new phe no me non , which is i n scribe d in time and which modulates time. To summarize : If no acco u n t is taken of the temporal element, then a s i ngle sonic event signifies only i ts statement. The sign, the symbol, the generic element a have be e n stated . A sonic event actu ally or mentally repeated signifies only an identity, a tautology : If it is emitted once it means n othing more than a single existence

appears

a v a v a

V

a · • ·

V

a

=

a.

V is an op erator that means " p ut side by side withou t regard to time." The sign means that it is the same thing. This is all that can be done with a single sonic event. =

CASE

OF T W O OR

MORE

G ENERIC

Let there b e two sonic events

and s u ch

a

and

a

ELEMENTS

and

b such that a i s not identical with b ,

that the two are distinct and e asily recognizable, l i k e the letters b, for exam pl e , which are only confused by a near-sighted person or

when they are poorly written .

If no

account is taken of the temporal element, then the two elements

are co nsidered as a pair. Consequently emitting first

then

they

a,

a

then

h,

or

fi rst b

gives us no more information about these disti nct events than when

a re

heard i n isolation after long intervals of

account is

taken of the relation of similitude

silence. And

since no

or of the time factor, we can

1 57

Symbolic Music

write for a

"#

which means

b

th a t a

b

a v

=

b

V

a,

b side by side do not

and

create a new t hi n g , having th e

same meaning as before . Therefore a commutative case

In the

of three distinguish able

of these s on i c sy m b o l s entity

in relation

events, a,

may be considered

to the

third :

(a

v

b)

This is

an

b)

v

v

v c.

c = a

v

(b

of the

the

other hand, when

t he

com b i n a tion of two

another element,

an

nothing more we may write v

c) .

time factor leads therefore to

sition outside-time-the commu tative and the are extensible to the c ase of a single event. ) On

c, a

associative law.

The exclusion

a, b,

law exists.

as forming

But since t his associational operation produces

(a

b,

two rules of compo­ rules

assoc iative . (These two

manifestations of the generic events

c are considered in time , then commutativity may no longer

Thus

be accepted.

a T b "# b T a, the symbol of the law of comp o s i tion which means " anterior to. " asymmetry i s the result o f our tr aditional e xperience, o f our cus­ tomary one-to-one correspondence between eve n ts and time instants. It is raised when we c onsider time by i tsel f without events, and the conse­ quent metric time which admits both the commutative and the associa tive T being

This

properties :

aTh = bTa (a T b) T c a =

CONCEPT

The

OF

D I STANCE

consideration

T

mus t penetrate the

associative law.

(INTERVAL)

of generic clements

permit much of an advance . To exploit

we

commutative law

(b T c)

internal

a,

b,

c,

. . . as en tities doe s not

and clarify w h at has j ust be e n said,

organization of the so nic sy mbols.

is modified dur­ life. On a primary level we perceive pitch, du ration, timbre, attack, rugosity, etc. O n another level we may dis tingu ish compl exities, degrees of

Every sonic

event is perceived as a set of qualities that

i n g its

order, variabilities, den s i ties, homogenei ties, fluctuations, thicknesses, etc.

Our

study will not attem p t

to

elu c i date

these q u estions,

which are not only

1 58

Formalized Music

difficult

but

at this moment secondary. They arc also secondary becau se q u a l i t i e s may be gra duated, even if o n l y broad ly, and may be totally o rdered . We s h a l l therefore c h oo s e one q u a l i ty and what will be said about i t will be extensible to oth ers. Let us, then, consider a series of eve n t s discernible solely by p i tc h , such as is perceived by an observer who has lost his memory. Two clements, a, b, are not enough for him to create the notion of distance or interval . We must look for a third term, c, i n order that the observer may, by successive com­ parisons and through his immediate sensations, form first, the c o nc ept of relative size ( h compared to a and c) , which is a primar y expression of rank­ ing ; and then the notion of distance, of i n terval. This mental toil will end i n the to tally ordered classification not onl y of pi tches, b u t also of melodic in tervals. Given the set of pitch intervals

and the

binary re l a t i o n S

many of the

1 . /zS/z

for al l h

E

2. h4Shb :f:. hbSh4

3.

(greater than o r

equal

)

to , we

have

H, hence reflexivity ;

except for ha

h4Shc,

ha.Shb and hbShc entail

Thus the differen t aspects of

=

hb,

hence antisymme try ;

hence transitivity.

the sensations pro d u c e d by s o n i c events

may eventually totally or partially constitute ordered sets ac c o rd in g to the

adopted. For exam ple, if we ad op te d as the unit inte rv a l of the relationship of the semitone (� 1 .059) but a relationship of 1 .0000 1 , then t h e sets of pitches a n d intervals w o u l d be very vague and would not be totally ordered because the differential sensitiviry of the human ear is inferior to this relationship. Generally for a sufficien tly large u n i t distance, many of t h e qualities of sonic e v e n t s can be to t a l ly ordered. To conform with a first-de g ree acoustic e xp e ri en ce , we shall suppose u n i t i n te rv al

pitch, not

that the ultimate asp ects of sonic

events

are frequency1 (experienced as

pitch) , intensity, and d u ra t i o n , and that every son i c evcn t may be constructed

from these

t h re e

when

duly

interwoven . In

this case the number t hree is events see

i rreducible. For other assumptions on the microstructure of sonic the Pre face and Ch apter I X .

St r u ct u re of t h e Qu a l ities of S o n i c Events*

From a naive

or d i s ta n c e

musical p ractice we

have d efined

the concept of interval

. Now let us examine sets of i nterv al s which are in fact isomorphic

to the equivalence cl asses of the N

x

N pro d u c t set

of natural numbers.

Symbolic

Music

Let

I.

1 59

there be

a

set H of pitch i n tervals (melodic) . The law of internal (h0, hb E H) a third element may be

composition states that to every couple

made to correspond . This is

hb

ha +

=

he,

such that

he

the composi te E

which we shall notate there be three so u n d s

of ha by h b,

H. For example, let

characterized by the pitches I, I I , I I I , and

let h• h(II,III> b e the intervals

in semi tones separating the cou p l es (1, I I ) and ( I I , I I I ) , res p ectively. The interval h(I .III> separating sound I and so u n d I I I will be equal to th e sum of th e semi tones of the other two . We may t h er e for e establish that internal com p osition for conj u ncted intervals is addition.

2.

The law is associative :

3.

There exists a neu tral

e l em

h 0 + ha

en t h 0 such

=

ha +

h0 =

th a t for every

that

4. For every h0

nameless ; and

there exists h� +

ha

a

=

=

h0

h�,

which returns to

the

unison ;

H,

the

zero interval ; for

it is simultaneity .

to

the inverse, such

0.

=

Corresponding to an a scen di n g melodic in te rv al h0,

ing interval

or

for duration

special element h� , called ha + h�

E

law of

h 0•

For pitch the n e u t r al element has a name, unison,

intensity the zero interval is

ha

the

there m ay be

an increasing

a descend­

interval

of

in tensity (expressed in positive decibels) m ay be added a n oth er diminishing

interval

to

(in negative db) , such that it cancels

time i n te r v al there may be the sum of the two is zero, or simultaneity. 5. The law is commutative : ing

a positive

the other's effect ; correspond­

a negative duration, such that

"' Following Peano, we may state an axiomatics of pitch and construct the chro­ matic or whole-tone scale b y m ea n s of three primary terms-origin, note, and the successor of . . . -and five primar y propositions :

the ori g in is a note; the su ccessor of a note is a note ; notes having the same successor are iden tical ; the origin is not the su ccessor of any note ; and if a property applies to t h e origin, and if when it applies to any note it also appl ies to its successor, then it applies to all n o t e s ( principal of induction). 1. 2. 3. 4. 5.

See also Chap. VII, p. 194.

Formalized

1 60

Music

been established for pitch, outside-time. But th e m to the two other fundamental factors of sonic events, and we may state that the sets H (pitch intervals) , G (intensity intervals ) , and U (durations) are fu rnished with an A belian additive group

These

the

five axioms have

examples have extended

structure.

To

sp ecify pro p erly

the

difference and the relati onship that exists

between the temporal set T and the other sets examined outside-time, and in o rd e r not to confuse, for example, set U ( d ura ti o n s characterizing a sonic event) with the time intervals chronologically se parating sonic e ve nts belon gi n g to set T, we shall summarize the su c c e s s iv e stages of our comprehension. S U M M A RY

Let there be t hre e events a, b, c emitted successively. Three e ve n ts are distinguished, and th a t is all. Second stage : A " temporal succession " is distinguished, i.e., s pondence between events and moments. There results fr o m this

First stage :

a before b '# b

before

a

a

corre­

(non-commutativity) .

Third stage : Three sonic events are distinguished which divide time into two sections within the events. These two sections may be compared and then expressed i n multiples of a unit. Time b e c om es metric and the s e c ti o ns constitute generic elements of set T. They thus enjoy commutativity. According to Piaget, the concept of time among children passes through th ese three p h as es (see Bib liography for Chapter VI ) . Fourth stage : Th re e sonic events are distinguished ; the time intervals are distinguished ; and independence be tw e en the s o n i c events and the time intervals is recognized. An algebra outside-time is thus adm i tte d for sonic events, and a secondary temporal algebra exists for temporal intervals ; the two algebras are otherwise identical. (It is u seless to repeat the arguments in order to show that the tem po ral intervals between the events consti tute a set T, which is furnished with an Abelian additive group structure. ) Finally, one-to-one correspondences are admitted between algebraic func­

tions outside-time and temporal an algebra in-time.

alge b r aic

functions. They may

constitute

In con cl u s i o n , most musical anal ys i s and construction may be based on : 1 . the study of an entity, the sonic event, which, ac c ord in g to our temporary assumption groups three characteristics, pitch, intensity, and duration, and which possesses a structure outside-time ; 2. the study of another si mp le r entity,

Symbolic

161

Music

time, which possesses a

temporal structure ; and

3 . the correspondence between

the str u ct u re o u tside-time an d the temporal structure : the

Vector Space

Sets H (melodic intervalsh

G (intensity intervals) ,

U

structu re in-time.

(time intervals) ,

and T ( i n terv a l s of time separating the sonic events, and i ndependent of

them)

are

totally ordered. We also assume that they may be isomorphic

numbers, and that an external law of composition for each of them may be established with set R. For every a E E (E is any one of the above sets) and for every el em ent A E R, there exists an element b = Aa such that b E E. For another approach to vector space, see the discussion of sets of intervals as a product of a group times a field, Chap . VIII, p. 2 1 0. Let X be a sequenc e o f th ree numbers x1 , x2, x 3 , corresp onding t o the under certain conditions with set R of the real

elements of the sets H, G, U, res p ectively, and arranged in a certain order :

X=

(xh x2, x3) . This sequence is

The particular c as e

of

a

the vector in

vector and

x 1 , x2, x3 are

zero vector, 0. It may also be called

th e

components. are zero is a

its

which all the components

origin of the coordinates, and by

analogy with elementary geometry, the vector with th e nu mbers

(x1, x2, x3)

as com ponents will be called point M of coordinates (xl > x 2 , x3) . Two points or vectors are said to be equal if they are defined by the same sequence :

X; = y, .

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

X + Y

=

(xl

+ Y1• X2 + Y2•

Xa

+

Ya) ·

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

=

=

=

- x2, - x3 ) , such that X +

( - X)

=

0.

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

p E R and X E E,

then

pX

=

(px1, px2 , px3)

The following properties are verified for

(p, q)

E

E

E3 •

R:

a.

1 ·X

=

X; b. p (qX)

Formalized

1 62

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

BASIS

are

AND

R EFER ENT

+

OF

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

a1X1

+

q) X

=

pX + qX andp( X

A

VE CTOR

a

system of p

a2X2

+

· · ·

+

Y)

=

Music

pX

+ pY

SPACE

numbers

+

a'PXP

=

a11

a2,

a3,

•

•

•

, a'P

which

0,

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

not

th e condition

that the

•

•

•

zero, then we s h a l l

a/i + a2g

en t a ils a1 = a2 be written

=

a3

may

=

+ a3fi

=

0

0. Moreover, every vector

X = (x1 , x2, x3) of E

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

3 linearly

properties :

I . If X is a vector of

scalar p may be.

V, every vector pX belongs also

to V whatever

the

2 . If X and

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

we ded u ce that : =

REMARKS

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

Symbolic Music

then

1 63

p, the

3. The scalars

move out of

not in

q, m ay

pr ac t i ce take

does not invalidate the g enerality o f these

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

all

v al u es

audible area. But this restriction o f

be

a

the

base

origi n of

/i, g, ii,

for /i, 1

=

arguments and a

trihedral of

a

,

we

for

would

practical o rd er

their applications.

reference

with

with the followi ng units :

Oh,

semi ton e ;

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

=

The

origin

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

established

by tradition, in

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

(A3

for

for

and the vectors

may be

xl

x2

the same

xl

- 3g

7/i + tg

=

written in traditional notation for

1'1'

In

sfi

=

way

+ X2

= (5

+

7)/i

+

�

( 50 -

( 1 - 3)g

+

= 440 Hz )

+ s u­

1

lu

sec �

:5 0

=

J.

20

(5 - r ) u-

d.B)

=

r 2fi - 2l

+ 4u.

=

-m t

....

c s o - 2 0 = :5 0

a B)

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

We may similarly p u rsue the verification

research by computers coupled to analogue converters migh t give us

Formalized Music

1 64

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

information on the timbres

A

A

A �----,

B,

F i g . Vl-1 I �

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

10

db ; and u,

for which

,

A L G E B RA

=

=

The vector X2 =

T h e vector X3

=

The vector X4 =

The vector X5

=

1 8/i

(18 (18

(18

( O P ERATI O N S +

+

+

+

the g axis, and AND

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

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

(18 + ( 1,8 + O) li

+

a l so admit the free vector v

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

origins

.

OUTSIDE-TIME

The vector X1

us

on the li axis,

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

The vector X0

Let

I � ; . Assume for the

Og + =

J u corresponds

3!i + Og

- I u;

A)

to G.

(See Fig. VI-2 . )

then t h e vectors

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

by means of addi tion .

Symbolic Music

165

D�

5 11

Fig .

Vl-2 A second law of composition

an ari thmetic pro g ression.

Finally, the scalar

i leads to

e x i s ts

TEMPORAL

ALGEBRA

The sonic statement

( IN

SET

This

boils

on

down t o the axis

the set

Xa

T

T)

xl

T

x2

a

=

0, 1 , 2, 3, 4) ;

of the com­

A is successive :

0 of the base

of A

shifting that has noth ing to

� E3

do

ii.

Thus in

the case of a simu l tanei ty (a

chord)

of the

�

of base

attacks of the

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

V

with the

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

fi, g,

is

it

T . . .

saying that the o rigin of time,

(i

variation remaining invariant.

of the vectors X1 of set

T being the operator "before."

displaced

in

an antisymmetric

ponents fi and ii of X1, the second g

is

0

six

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

Formalized Music

1 66

of composition in se t T; or consider segment unit

equal to

l::!. t ; then /1

=

an

a

origin 0'

+ il::!. t, for i

on =

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

a

Vl-3

Fig .

A L G E BRA

IN-TIM E

( RELAT I O N S

AND

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

SET T)

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

c om

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

=

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

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

In ge n er al we admit

X(t )

=

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

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

When these functions are

the meaning of the variations

dX

If we

neglect

_

dH fi

dt - dt

dG +

dt

dU

_

g +

dt

_

u.

the variation of the component G, we

conditions : For

dHfdt

=

0, H

=

ch ,

and

dUfdt

be independent of the variation of t ; and for

ch

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

will

0, U and Cu t=

=

Cu

h ave the

=

=

cu,

following

H and

U will

0, the sonic

0, there is

no

event sound

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

=

=

=

=

167

Symbolic Music

For dHfdt

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

=

Fig.

Vl-4

0,

H

= en, =

=

=

- �-

t,

LL

Fig. Vl-5

Fig .

Vl-6

t.. , u.

Formalized Music

1 68

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

=

Fig. VI-7.)

u = c �-t � 1c U . c".t .

k

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

=

=

=

F (t) ,

we have

a

h

H tJ

"

c h' t ;

+

�

� ( t)

F i g . Vl-8

Fig.

=

j(t) , H

i, G

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

=

=

Vl-9.)

u .. Fig. Vl-9

c"

f,u

169

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

=

=

S (t ) , w e

h ave a

h

U : S lt) Fig.

Vl-1 0

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

l

X0

X1

X2

�

�

�

X3 X4 Xs �

4

�

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

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

=

w

w

w

w

=

=

=

z,

with

u

- z, with gradation. =

=

x

u =

v =

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

and

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

=

=

x

Formalized

1 70

Music

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

These tra nsfo r m ations for m the Klein

Other

transform ations, as yet

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

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

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

systems such

as the

sys t e m of quaternions.

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

TO

S ETS

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

OF

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

of

a

son ic even t, with its vector

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

outside-time.

2. A temporal algebra,

whi ch the sonic events create on the

time, and which is independent

3.

An

algebra in-time,

space.

axis of metric

issuing from the correspondences and fu nctional

relations be tw e e n time, T,

of the vector

algebra

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

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

and about time can

be generalized for sets

of sonic events X and for sets T.

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

concep t of the set, a nd in particular

To simplify the exposition,

we

considering the referential or universal

piano.

We shall

c on s i d e r

shall first take a co n c re te example by

set R, consisting of all

th e sounds of

a

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

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

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

Suppose, then , a This will be set

A,

set

A of keys that have

a

characteristic property.

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

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

particular choice of a certain nu mber of keys.

the

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

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

h as heard a collection of so u n d s

,

or

a

listin g

of elements.

171

Symbolic Music

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

Another

same way.

This

operation has

therefore

A ·B

or

B · A.

engendered a new class, which was symbolized

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

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

operation

is

If class

the union

(disjunction) and A + B

or

is written

B

+ A.

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

all the sounds

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

state

classes

of sonic

events (while taking precautions for clarity

in the symbolization) ; and e ffect

three operations

tance : in tersection, union, and negatio n .

of fu ndamental

im p or­

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

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

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

Formalized Music

1 72 elements was allowed

(though

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

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

a sort of

stenosym bolization) .

s p ecial sonic symbols

We have not allowed

which are exp ressed graph ically by these operations arc

shortcuts may be taken

i n the s tate­

mnem otechn ical or even psychophysiological

ex p ressed,

· ,

for the three operations

o n l y the classes resu l ting from

+, - ;

and the operations a re conseq uen tly dedu ced

mentally by the o bserver. In the

same

must deduce the

way the observer

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

The e m p ty cl ass,

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

on the concept of inclusion.

Gr a p h ic symbols

Classes A, B, C, . . . Intersection

( ·)

Negation ( -

)

Union ( + )

however, may

Sonic symbols Sonic

enumeration

of the

generic

elements having the properties A, B,

C,

.

•

•

(with

possible shortcuts)

Implication (---+) Membership (e)

A

Sonic

enumeration

R not i n c l u d e d

A ·B

in

of the

A

Sonic enumeration of

A + B

A·B

elements

the elem ents

Sonic enumeration of the elements

A + B

of of

of

A => B A = B This table shows that

we can reason by pinning down our thoughts by

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

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

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

sonic = ,

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

be inferred .

I 73

Sy m bo l ic Music seen

Besides these l og ic a i

that

we

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

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

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

we

have

the sonic

The role

as a crucible, mold,

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

or s p ac e in which

Time

C o n struct i o n

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

three c l a ss e s

A, B, C can

F (A, B, C) ,

be expressed in the form

for example, of the

called

disjunctive canonic :

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

=

·

=

F =

A ·B·C

+

A ·B·C +

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

(I)

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

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

=

(A · B

+

A · E) · C + (A · B

+

Jf- .11) · C.

( 2)

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

For m alized Music

1 74

F i g . Vl-1 1

F i g . Vl-1 2

1 75

Symbolic Music

F i g . Vl-1 3

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

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

the other h and

op erations as against seventeen ) .

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

Yuji Takahashi in Tokyo in

Febru ary 1 962 .

1 76

Formalized Music

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

Fig . Vl-1 4

Symbolic Music

177

u:

I

I

"' ..

I

"

�i

I

..

I I I I 11

I

0

Ill

I I

li

I

1. .i ' I

�I

;: I :- I "' I I

....

I

i

... r; .

Lo

a.

I ."r

1 :-

I

I�

I

I I I I

I

I �I

:I

�I

1

,I

a! � �

.. ""

·� l

� \J 1 "" I I .. I I I

.�

v X

�

I H�

1 ""

�

�

HI

!: 1

I I I I I I I

I� .. X

1 ;.-;

, _4"1

11

\

Conclusions and Extensions for Chapters 1-VI

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

have sk e t c h ed

first time,

.

i n terpenetrations and interpretations.

In r ea l i ty formalization

guide,

and axiomatization

c onstit u t e

a procedu ral

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

1 78

,

Conclusions

and

Extensions for Chapters

movements of sounds that

1 79

I-VI

cause movements in

us in agreement with

the m

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

for t hos e

who

do know, a reasoned joy

through the imitation of

harmony which they realize in perishable movem e n ts

The theses advocated

"

(Plato,

the divine

Timaeu.s) .

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

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

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

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

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

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

then join to this the coordinates of space,

a

s on i c source.

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

If we

Chapter V I I

Towards

a Metamusic

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

( source )

sends

to a

listener (receiver) .

as a message which the In this way they believe

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

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

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

prec i se a theoretical

theories have been refined and new ones invented .

The followers of information theory or of cybernetics

end two groups :

extreme . At th e other divided into

re presen t one

there are th e intuitionists, who may be broadly

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

the music and

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

Towards

a Me tamu si c

th e "improvised"

181

mus i c

grandfathers

our

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

notation, geome tric,

is

as

faithful

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

it

be symbol i c, as in traditional

or nu m e ri c a l , should be

no more than an i mage that

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

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

who add

a spectacle in the

form of extra-musical scenic acti o n

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

which express the confusion

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

in mimetics a n d disparate occu rrences

pure

music.

groups

share

confidence in

music in

particular.

The two

a

romantic attitude. They believe

in

immediate

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

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

Thought

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

m us ic-as in politics-th e

scientific and mathematical thought serve music, or any human creative

activity,

it

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

i ndivisible, a nd total. He thinks with his

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

2. I t

3. It

is

a

belly

is a .

fixing in sou nd . . , arg uments) .

4. It is normative, th at

doing by symp athetic drive.

his mind .

I

is,

a

of

real ization.

i m a gi n ed

unconsciously

it

virtualities

is

a

( cosmological ,

m ode l for being or

for

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

transformations

6. I t is 7. I t is

in

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

the gratu itous play of a child . a

Consequently expressions arc only very limited par t icu l ar

my sti ca l (but atheistic) asceticism.

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

and feels wi th

covers the term " music " :

sort of comportment necessary for whoever thinks it and

is an individual pleroma,

philosophical,

5. It

mi n d ,

Formalized

1 82

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

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

a

and

Music

today

state of chaos. We h ave

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

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

it

an

more or less can be communi­ to re form musical

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

edu cation-a reform that is necessary in the

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

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

contradictions

of

serial music. In its p l ace I proposed

masses, vast groups characteristics such

required d efinitions tic m usic

was

was

a

world of

of sound-events,

clouds, and galaxies go vern ed

and realiz ations

using

as

sound­ by new

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

probability theory.

Thus s tochas­

numbers for it could embrace it as a particu­

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

more general than linear polyphony,

lar instance (by No, not yet.

Today these

re d u cing

the

density of the clouds) . General harmony ?

id eas and the realizations which accompany th em

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

be

have

closed for all

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

firma

on which all our

music is

fo unded-seems

not to h ave been

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

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

I

shall point ou t

the

and

at

stru ctu re of the

its subse­ music of

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

it while developing

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

the occidental plainchant.

struction in

a

After demonstrating their shall try to express in a

modern way, I

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

musicology)

and in s pace

was

abstract

logical

con­

simple but u niversal

and what might be valid

(comparative

musicology) .

in

Towards a

Metamusic

1 83 a

In or de r to do this I propose to make

distinction in musical archi­

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

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

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

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

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

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

music, pace

C o m b ari eu

and

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

in

the sec u l ar

music of

the

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

Terminorum Musicae dijfinitorium

ofJohannis

Tinctoris . 5 To look at antiq u i ty

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

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

Aristoxenos : 7

A. The primary order consists of the

tone and its subdivisions .

The whole

1 84

Formalized

Music

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

tone is defined as the

chord , or

1 . The

3 + 3

+

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

enharmonic

24

=

30 segm ents .

th e enharmonic as

=

2. The chromatic genus co nsists of three types : a . soft, containing two chromatic dieseis, 4 + 4 + 22 30, or X113 . X1'3 - X X 5'2 ; b. h e m iolo n (ses q uialterus) , containin g two hemioloi d i es e is , 4.5 + 4 .5 + 2 1 2 = 30 segments, or X<3t2>014> X<31 ><114> . X7' 4 X5'2 ; and c. "toniaion , " con­ sisting of two semitones and a trihemitone, 6 + 6 + 1 8 = 3 0 segments, xs12, or Xli2 . Xtt2 . xat2 =

•

=

=

=

3 . The

di atonic

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

enharmonic dieseis, then ments,

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

whole Xli2 · X · X

=

xst2 ,

semitone,

+ 9 + 15

then three 30 seg­ =

containing a semitone, +

12

= 30

segments,

a

or

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

C . The

tertiary o rder,

1 85

Towards a Metamusic this same h ierarchical

procedure . Seemi n gly

was

Aristoxenos

u s e d as

a

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

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

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

This

h ierarchical " tree " · was completed

by transition algorithms ­

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

This is

a far cry from the simple modulations or

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

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

determi n ation to rem ai n

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

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

the

en tire harmonic

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

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

me th o d

to b e fo l l o wed

.

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

whereas the Pythagoreans defiQed i t by the

strings.

I

bel ieve this to be

a

ra ti o

3/4 of

the len gths of the

sign of Aristoxenos' wisdom ; the

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

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

rat io

of th e

3/4

addi tive

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

this

co n t r as ts with the

practice

of the Pyth agorcans, who u sed the

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

of Aristoxenos is fu ndamental s i nce :

l.

it constitutes one of the two

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

2.

by

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

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

and 3.

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

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

it

Over the centu ries the two languages-arithmetic

addition) and geometric

( deri ved

(operating by

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

a rc

both

ex­

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

Formalized Music

1 86

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

There

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

musicologists

Structu re of

to

in

rece n t times. " Th e

Byzantine

M us i c

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

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

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

chant, non -Euro p ean musical

pean music, with

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

of the

past as well as the

electronic

fu ture. T h u s

new

d irections of research

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

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

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

Byzantine music am algam a tes the two

Pythagorean

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

major (9/B or

Aris toxenos. Byzantine music gives

diatonic scale

a

preponderant role

to

the

na tural

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

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

ing ratios to the fi rs t

1 2, 22,

3 0, 42 ,

A. The

primary order

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

a

To w a r d s a

supermajor

187

Metamusic tone 7{6, the trihemitone

6/5 , another

major tone 1 5 / 1 4, the

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

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

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

PARENTHESIS

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

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

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

and noble,

arts.

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

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

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

and

c u l t u re ,

Formalized Music

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

This

phenomeno n

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

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

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

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

anci ent. 1t b u i l ds

up the

next stage w i th tetrachord s, usi ng

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

expo u n ded in s om e detail

theory

Ptolcmy. 1 5

TH E SCALES

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

The tertiary order

by

systems havi ng the same

c o nsi st s

a n c ie n t

of t h e scales constructed with t h e

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

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

of the

scale did not emerge independently from

the method

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

was

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

as

a method of constru cting scales .

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

m usic :

the octachord or dia pason

the pe n ta chord or wheel ( t rochos)

the

tetrachord

or

triphony.

(synimenon) or disjunct juxtaposition of two tetra­

The system can unite elements by conjunct

(diazeugmenon)

juxtaposition. The d isj u n c t

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

chords one tone apart form the The conjunct juxtaposition

of several of these perfect octave dia

Towards a Metamusic octave is

no

1 89

longer a fixed sound in the tetrachord bu t one of

its mobile

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

The system

can

be

applied to

the three genera of tetrachords and to

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

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

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

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

are and their derivatives. The

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

a dy

scale as

s trongly d i fferentiated , the tetrachords

it is

c o n ce p t i on

defined here is a richer and m o re universal

t h e impoverished

conceptions

modern

of medieval and

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

as

than all

times. From t h i s

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

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

(harmonies) of the

other tetrachords that represen ts a vast loss

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

from

a d i sj u n ct system of two

diatonic te trachords separated by a whole tone, and

is

represe n ted by t h e

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

seen.

to

reinstate, a l b e i t

in

a

modern way, as will be

The following are examples of scales i n segmen ts of

Diatonic

scales.

1 2, 1 1 , 7 ; 1 2 ;

Diatonic t e tra cho rd s :

1 1 , 7, 1 2 , starting

starting on the lower

H or A ;

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

on

tem­

on the lower fl, 1 2 ,

1 1, 7; 12;

1 2,

1 1,

7,

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

the lower Z; wh eel system ( trochos ) ,

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

Byzantine

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

pering (or Aristoxenean, since the perfect fourth

chromatic tctrachords :

I I , 7,

1 2,

wheel system start i n g on

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

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

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

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

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

+

soft c h rom ati c ;

7 . Hard chro m a tic

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

12; 7,

1 6,

7;

etc. A11