Formalized Music: Thought and Mathematics in Composition [2 ed.] 1576470792, 9781576470794

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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

c:

:::1 0 0

:9,=

G> > CD G>

:0

(Ill II]

2RS&

·�

-�

c

0 z

c;,

u;

] Qua l ific a t i o n s

��

....

g u

�

!

ci �

Free Stochastic Music

29

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

concept in a new region of coherence. The analysis that

fol lo w s

is

taken from

Achorripsis.

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

>.

Applying Poisson's formula,

=

0.6 events /unit.

we obtain the table of probabilities :

P0 pl p2

P3 P,

P5

=

0. 5488

=

0.3293

=

0.0988

=

0.0 1 98

=

0 . 0030

=

(1)

0. 0004.

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

pj by 1 96.

Number of cells

0 1

2 3 4

1 96 PI

1 07 65

19

4

(2)

1

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

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

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

th e nu mber

=

30

Formalized Music Timbre Flute Oboe

String

gliss.

Percussion

(3)

Pizzicato

Brass String

arco 0

1

2

3

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

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

time

MM

=

26.

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

and q u adruple events per cell

=

in

the two-dim ensional

=

(2.32

«

we obtain table

30)

Poisson Frequency

K 0 1 2 3 4 5 6 7

Totals

=

=

(4) .

Distribution

Arbitrary Distribution

No. of

Pro du c t

3

0

6

6

Columns col

x

Frequency No . of Product K Columns col x K

K

0 1 2 3 4 5

10 3 0 9 0

0 3 0 27 0 5

8 5 3 2

16

I

6

6

5

30

0

7

0

0

28

65

0

28

15

12 10

65

(4 )

Totals

(5)

Free

31

Stochastic Music

One could c hoos e

single events equals But in

space,

we

this

must

any o th er distribution

on

65. Table (5) shows such

a

condition that

the sum of

distribution.

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

rej e c t

every distribution which

departs from Poisson's

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

for

the rows of

the matrix. The same reasoning holds true for the

diagonals, etc.

Contenting ourselves just with rows and columns, we obtain

geneous distribution which follows Poisson. It was

a h omo­ in th i s way that the

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

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

me d i u

m of the arbitrary mean

( for r are

events) t h ro u gh

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

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

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

following t h e rows and columns. the

beginning

therefore

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

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

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

the

" Vector-Matrix "

defined above .

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

=

=

Formalized

32 Cloud of density S

Ev e n t

Sounds/ measure 26MM

zero

0

5

single d ouble

I

(15

2.2 4.4 6.6 8.8

20

sec) 0

0

15

qu ad ru pl e

Mean number of sounds/cell

sec

10

triple

=

Sounds/

Music

32.5 65

97.5

1 30

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

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

rows are

columns. This

interchan geable leads

us

(

=

interchan geable

timbres) . So are the

to admit that the determinism of this matrix is weak

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

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

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

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

OF

the

CALCULATION

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

sounds of

=

=

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

sounds

assimilated to the speed

v

=

df{dt of

a

uniformly continuous movement.

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

33

Free Stochastic Music

Fig .

1-1 0

Gauss. The probability f(v)

function

for

f(v)

the existence of the speed v i s given by

=

Q�'/T

r v�/a� ;

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

8(>.) Hypothesis

=

:'IT f'

4. A glissando s o u n d

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

Hypothesis

is

vm

the

r .,.,�

v1

( normal

d)..

essen tially =

an d v 2 , by the fu nction

distribution) .

characterized by

tifldt, (v1


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

- I�

Jl>

..

.II

-J.I Jt

-lro ¥

f.l n

J6

_ ,, -1.1 8

i3 6 - 8

- ft

H - 66 - 4 H

12

..A Woodwinds

• N ormal percussion

H Strings striking sou nd-boxes •: Strings pizzicato

.:If Strings g l issando

:= Stri ngs sustained

=

of

_,.

-16 .;t

: -.

Jo

-

u

"0

-1'1.

3.J

-+

u "

-" -.rs

,.

- S'&

2.1

""

- ;t f

-"" - ro

0.

two and three

-J 6 -I�

- II.

- 18

�;!

YIJI'M

K•�

-Jo

" "'

JD

11

p

-I"

1!1

to

-?4-

"·

-4- lr

I�

t,o

.u

- 3/1: - JI!I

/tO

""

-

-U �

- .:

"

lro

.,

- �&

If

- ¥• -" -8

-21

-J.t -.18

- II'

-·· If

2�

- 11 J,o

'!8 -19

-s

"'' -Ji' - H

J lr - J o

- .211 -It: --�

H � 0 16

d ifferent tactics

. ., 11-�.d , .., ""

_ ,., -

-B -3& -.v o

• Combi nations

�

- ��

.t'f

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

'

•

&I

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

"

•

- .Je - �lr - 16

- � - .36

IJ

u

• "

- M - ./0 - 10

- r.z - M

-8 - U - vo

l•

4&

-J.t

8

-r;( 31

9

-.J•

U3

-

-$& �6

6

.u

f,z

-fC - .u - u

-

-

-8

.lc]J - I(¥

,.

,,

'

:t¥ -

n

-J,t

- !12

,,

1,1.(

"' ,

.t

1-1

'"

6o

� - !16

�� _ ,.:z

-J,G l, f

- ¥�

H

-llo

-1(,2 - :J2

,

B

'

and Musical Comp osition

Strategy, Linear Programming,

I;

-6�

Ill

:. •

lH

8 �!

4' 0 Ill •

-1

3

-3

3

.... .

:. . ' *' . ' Ill • •

0

3

• y ti H Y

>< 0

'

· · · � · · · · � ... . ..

n

I�

•

Strategy

for

Y

4

A

{-,

... . 4' 0 m• H O

.. . . :. . '

*' " ' Ill e f

-3

Strategy

F i g . IV-8

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

Y Wooawinas

H Strings striking sou nd - boxes • ••

Normal percussion Strings pizzic ato

ill Strings sustained

Jl-

-1

0

-5

-1

4

3

1

-3

.2 • '

�� : a:;

for

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

3

� t.

.. . .

-5

•o

1 29

Y

A •

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

Two - person

' . ' "

' "" ' m O H . :.

., :.

of two dis­

Combinations

tinct tactics

' • H

; :�

' •m

Combi nations

of three dis­ tinct tactics

• IU

Strings glissando

. ,..

Simplification of the 1 9

x

1 9 Matrix

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

fr + m

*• + 1

4r+ I .I + l

k, .. ,

k1 + n

Dr+ l , I + J

Dr + l . I + R

Dr + t .a + :l

a,. ... . . . + J

Or + e . a + n

a, + lll ,l + l

a, + • . • + l

Dr + m , a + n

��:a;•gy i

Formalized Music

1 30 This fragment

can

be replaced by the single score

A• + m .• + n

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

-

_

and b y the probabilities

m

Q

=

and

:L q, + , 1=1 n

K

Operating in this way matrix (the tactics will be X

X

-

25

X

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

X

25

592 25

X

45

93 1 4

49

X

30

X

25

26

3088

1 76 1 0

-

30

25

or

x

8296

49

49

45

68 1 8

25

J=!

the same as in

1 45 2 2

25

L ks + J·

wi th the 1 9

7 704 25

=

45

X

26

2496

30

49

X

26

26

2465

- 1 354

1 82

25

- 2581

1 59 7

- 52 8

45

1818

- 1 267

640

30

25

49

26

45 30

the following IV-6) :

C h a pter V

Stochastic

Free

After this interlude , we return

Music

by Computer

to the treatment of composition

by machines.

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

being realized mechanically. This realization occurred thanks to

M.

Fran /'!! #

IJJ -==== # ::-=--=--;9¥ .f

��

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

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

ill'-===..:.

/>-=::::::

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 B A = B This table shows that

we can reason by pinning down our thoughts by

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

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

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

sonic = ,

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

be inferred .

I 73

Sy m bo l ic Music seen

Besides these l og ic a i

that

we

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

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

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

we

have

the sonic

The role

as a crucible, mold,

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

or s p ac e in which

Time

C o n struct i o n

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

three c l a ss e s

A, B, C can

F (A, B, C) ,

be expressed in the form

for example, of the

called

disjunctive canonic :

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

=

·

=

F =

A ·B·C

+

A ·B·C +

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

(I)

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

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

=

(A · B

+

A · E) · C + (A · B

+

Jf- .11) · C.

( 2)

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

For m alized Music

1 74

F i g . Vl-1 1

F i g . Vl-1 2

1 75

Symbolic Music

F i g . Vl-1 3

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

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

the other h and

op erations as against seventeen ) .

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

Yuji Takahashi in Tokyo in

Febru ary 1 962 .

1 76

Formalized Music

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

Fig . Vl-1 4

Symbolic Music

177

u:

I

I

"' ..

I

"

�i

I

..

I I I I 11

I

0

Ill

I I

li

I

1. .i ' I

�I

;: I :- I "' I I

....

I

i

... r; .

Lo

a.

I ."r

1 :-

I

I�

I

I I I I

I

I �I

:I

�I

1

,I

a! � �

.. ""

·� l

� \J 1 "" I I .. I I I

.�

v X

�

I H�

1 ""

�

�

HI

!: 1

I I I I I I I

I� .. X

1 ;.-;

, _4"1

11

\

Conclusions and Extensions for Chapters 1-VI

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

have sk e t c h ed

first time,

.

i n terpenetrations and interpretations.

In r ea l i ty formalization

guide,

and axiomatization

c onstit u t e

a procedu ral

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

1 78

,

Conclusions

and

Extensions for Chapters

movements of sounds that

1 79

I-VI

cause movements in

us in agreement with

the m

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

for t hos e

who

do know, a reasoned joy

through the imitation of

harmony which they realize in perishable movem e n ts

The theses advocated

"

(Plato,

the divine

Timaeu.s) .

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

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

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

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

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

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

then join to this the coordinates of space,

a

s on i c source.

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

If we

Chapter V I I

Towards

a Metamusic

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

( source )

sends

to a

listener (receiver) .

as a message which the In this way they believe

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

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

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

prec i se a theoretical

theories have been refined and new ones invented .

The followers of information theory or of cybernetics

end two groups :

extreme . At th e other divided into

re presen t one

there are th e intuitionists, who may be broadly

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

the music and

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

Towards

a Me tamu si c

th e "improvised"

181

mus i c

grandfathers

our

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

notation, geome tric,

is

as

faithful

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

it

be symbol i c, as in traditional

or nu m e ri c a l , should be

no more than an i mage that

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

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

who add

a spectacle in the

form of extra-musical scenic acti o n

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

which express the confusion

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

in mimetics a n d disparate occu rrences

pure

music.

groups

share

confidence in

music in

particular.

The two

a

romantic attitude. They believe

in

immediate

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

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

Thought

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

m us ic-as in politics-th e

scientific and mathematical thought serve music, or any human creative

activity,

it

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

i ndivisible, a nd total. He thinks with his

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

2. I t

3. It

is

a

belly

is a .

fixing in sou nd . . , arg uments) .

4. It is normative, th at

doing by symp athetic drive.

his mind .

I

is,

a

of

real ization.

i m a gi n ed

unconsciously

it

virtualities

is

a

( cosmological ,

m ode l for being or

for

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

transformations

6. I t is 7. I t is

in

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

the gratu itous play of a child . a

Consequently expressions arc only very limited par t icu l ar

my sti ca l (but atheistic) asceticism.

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

and feels wi th

covers the term " music " :

sort of comportment necessary for whoever thinks it and

is an individual pleroma,

philosophical,

5. It

mi n d ,

Formalized

1 82

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

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

a

and

Music

today

state of chaos. We h ave

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

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

it

an

more or less can be communi­ to re form musical

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

edu cation-a reform that is necessary in the

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

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

contradictions

of

serial music. In its p l ace I proposed

masses, vast groups characteristics such

required d efinitions tic m usic

was

was

a

world of

of sound-events,

clouds, and galaxies go vern ed

and realiz ations

using

as

sound­ by new

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

probability theory.

Thus s tochas­

numbers for it could embrace it as a particu­

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

more general than linear polyphony,

lar instance (by No, not yet.

Today these

re d u cing

the

density of the clouds) . General harmony ?

id eas and the realizations which accompany th em

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

be

have

closed for all

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

firma

on which all our

music is

fo unded-seems

not to h ave been

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

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

I

shall point ou t

the

and

at

stru ctu re of the

its subse­ music of

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

it while developing

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

the occidental plainchant.

struction in

a

After demonstrating their shall try to express in a

modern way, I

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

musicology)

and in s pace

was

abstract

logical

con­

simple but u niversal

and what might be valid

(comparative

musicology) .

in

Towards a

Metamusic

1 83 a

In or de r to do this I propose to make

distinction in musical archi­

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

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

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

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

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

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

music, pace

C o m b ari eu

and

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

in

the sec u l ar

music of

the

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

Terminorum Musicae dijfinitorium

ofJohannis

Tinctoris . 5 To look at antiq u i ty

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

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

Aristoxenos : 7

A. The primary order consists of the

tone and its subdivisions .

The whole

1 84

Formalized

Music

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

tone is defined as the

chord , or

1 . The

3 + 3

+

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

enharmonic

24

=

30 segm ents .

th e enharmonic as

=

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

•

=

=

=

3 . The

di atonic

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

enharmonic dieseis, then ments,

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

whole Xli2 · X · X

=

xst2 ,

semitone,

+ 9 + 15

then three 30 seg­ =

containing a semitone, +

12

= 30

segments,

a

or

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

C . The

tertiary o rder,

1 85

Towards a Metamusic this same h ierarchical

procedure . Seemi n gly

was

Aristoxenos

u s e d as

a

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

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

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

This

h ierarchical " tree " · was completed

by transition algorithms ­

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

This is

a far cry from the simple modulations or

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

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

determi n ation to rem ai n

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

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

the

en tire harmonic

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

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

me th o d

to b e fo l l o wed

.

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

whereas the Pythagoreans defiQed i t by the

strings.

I

bel ieve this to be

a

ra ti o

3/4 of

the len gths of the

sign of Aristoxenos' wisdom ; the

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

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

rat io

of th e

3/4

addi tive

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

this

co n t r as ts with the

practice

of the Pyth agorcans, who u sed the

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

of Aristoxenos is fu ndamental s i nce :

l.

it constitutes one of the two

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

2.

by

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

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

and 3.

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

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

it

Over the centu ries the two languages-arithmetic

addition) and geometric

( deri ved

(operating by

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

a rc

both

ex­

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

Formalized Music

1 86

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

There

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

musicologists

Structu re of

to

in

rece n t times. " Th e

Byzantine

M us i c

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

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

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

chant, non -Euro p ean musical

pean music, with

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

of the

past as well as the

electronic

fu ture. T h u s

new

d irections of research

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

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

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

Byzantine music am algam a tes the two

Pythagorean

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

major (9/B or

Aris toxenos. Byzantine music gives

diatonic scale

a

preponderant role

to

the

na tural

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

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

ing ratios to the fi rs t

1 2, 22,

3 0, 42 ,

A. The

primary order

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

a

To w a r d s a

supermajor

187

Metamusic tone 7{6, the trihemitone

6/5 , another

major tone 1 5 / 1 4, the

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

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

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

PARENTHESIS

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

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

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

and noble,

arts.

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

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

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

and

c u l t u re ,

Formalized Music

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

This

phenomeno n

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

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

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

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

anci ent. 1t b u i l ds

up the

next stage w i th tetrachord s, usi ng

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

expo u n ded in s om e detail

theory

Ptolcmy. 1 5

TH E SCALES

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

The tertiary order

by

systems havi ng the same

c o nsi st s

a n c ie n t

of t h e scales constructed with t h e

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

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

of the

scale did not emerge independently from

the method

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

was

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

as

a method of constru cting scales .

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

m usic :

the octachord or dia pason

the pe n ta chord or wheel ( t rochos)

the

tetrachord

or

triphony.

(synimenon) or disjunct juxtaposition of two tetra­

The system can unite elements by conjunct

(diazeugmenon)

juxtaposition. The d isj u n c t

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

chords one tone apart form the The conjunct juxtaposition

of several of these perfect octave dia

Towards a Metamusic octave is

no

1 89

longer a fixed sound in the tetrachord bu t one of

its mobile

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

The system

can

be

applied to

the three genera of tetrachords and to

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

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

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

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

are and their derivatives. The

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

a dy

scale as

s trongly d i fferentiated , the tetrachords

it is

c o n ce p t i on

defined here is a richer and m o re universal

t h e impoverished

conceptions

modern

of medieval and

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

as

than all

times. From t h i s

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

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

(harmonies) of the

other tetrachords that represen ts a vast loss

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

from

a d i sj u n ct system of two

diatonic te trachords separated by a whole tone, and

is

represe n ted by t h e

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

seen.

to

reinstate, a l b e i t

in

a

modern way, as will be

The following are examples of scales i n segmen ts of

Diatonic

scales.

1 2, 1 1 , 7 ; 1 2 ;

Diatonic t e tra cho rd s :

1 1 , 7, 1 2 , starting

starting on the lower

H or A ;

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

on

tem­

on the lower fl, 1 2 ,

1 1, 7; 12;

1 2,

1 1,

7,

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

the lower Z; wh eel system ( trochos ) ,

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

Byzantine

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

pering (or Aristoxenean, since the perfect fourth

chromatic tctrachords :

I I , 7,

1 2,

wheel system start i n g on

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

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

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

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

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

+

soft c h rom ati c ;

7 . Hard chro m a tic

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

12; 7,

1 6,

7;

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

I 9, 6 ;

one can observe

the

1 90

Formalized Music

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

by

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

virtue

is defined by :

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

the termini or cade nce s

We

constitu ting i t

(katalixis)

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

"of

order.

this

qua ternary

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

structure of Byzantine music.

But

this

outside-time

TH E M ETA B O LA E

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

structure could not b e satisfied

mentalized hierarchy. It was necessary

to

have

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

The

metathesis :

transition

from

a

tetrachord

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

of 30

segments (perfect

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

The parachordi : distortion

segments of

a

tetrachord

tra nsi ti on from one

Genus

Metaholae

Ph thora characteristic of the genus,

Changing note names

Using

not

changing note n am e s

the parach o rdi chroai.

Using the

System metaholae

Transition from one system to another

us

i ng

the

above

metabolae.

191

Towards a Metamusic tions

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

Because o f t h e

com p lex i ty

phthorai o r al tera­

of the metabolae, pedal notes (isokratima)

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

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

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

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

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

ists in Byzantine music recogn ize the importance

of this structure.

attention to such an extent that they h ave ignored

It wo u l d

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

appear that in terpret i n g ancient systems of notation

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

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

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

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

of polyph on y,

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

1 92

Formalized Music

wrong !

But

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

remaining aloof from the great musical tradition

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

complex and remarkably interlocking and gen u ine achievement

re t ard ed the

p rogre ss

of th e

in

of the eastern ch urch , they

both

(harmonious) , this d eveloped remnan t

Hellenic tradition. In t h is way they have

of m usicological

re s e a r

ch

in the a re as of:

antiquity

plain chant

folk music of European lands, no t a bly in

the East20

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

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

this architecture, which is linked to antiq uity

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

to what

I have

tried to defi ne

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

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

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

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

on

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

rest, however simplified they may be.

past and pres ent ,

In

o rd er

to

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

as well as prepare the fu ture, it is

structures, architectures,

and

n e c es s a ry to

distinguish

sound organisms from their te m p ora l manifes­

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

consciOusness.

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

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

of the simple sound) of sou nd

entities.

193

Towards a Metamusic Here rhythms

are also

subj ected t o

an organ ization, but

a

much si mpler

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

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

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

polyphonic music.

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

no one h as yet though t

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

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

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

on the te mporal category, and defi ned the hierarchies

of

its harmonic

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

bases

C and A ) ,

(C m aj o r

corres p onding to the syntonon diatonic

on the two

of the Pyth agore a n

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

second and t h i rd

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

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

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

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

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

The

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

prep ared by

the theory

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

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

of the

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

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

has barely

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

ou tside-time structures of m u si c since

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

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

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

Formalized Music

1 94

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

is

ill-suited

or

level

to providin g

the

world

at the level of composition by compu ters.

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

the

By

the

introd uction of th e calculation of probability (stochastic

music)

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

was

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

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

l arge

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

between the ve r tic a l and the

horizontal, and the indeterminism

of in-time

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

fu n ctio ns ,

took on

of organ ization.

by

means of particular stoch astic

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

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

It was able

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

outside-time,

in-time,

and

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

p resen t ,

and

c oh eren t

and u n ivers al synthesis in the past,

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

even

a direc­

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

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

ture (additive group structure

=

for

the totally

o rd ered struc­ the

addit ive Aristoxenean s tru c t u re ) of

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

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

=

the

stop at the ori gin ;

resul ting from elementary displacement of n ; D

=

n

= a stop ;

n

'

=

a s top

the set of values of the

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

displacements.

Towards

195

a Metamusic

First propositions (axioms) .

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

1.

S top

0

2 . If stop

is an

n

is an element

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

Of each

m ; they n (mod

to

are,

m} is

For

a

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

modulo m with m

one

and with

- 1).

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

given n and

the residual class In

=

is congru e nt

n

=

{0,

for any k e

modul o

m.

± l, ±

Z, the

2,

± 3,

. . . }.

numbers x will belong by defini tion

This class can be

order to grasp these ideas in terms of music,

denoted mn. us take the te m pe red

let

Formalized Music

1 96 semitone of our present-day scale as

the

unit

of displ a cement.

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

To this we 4 semitones

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

chromatic s c al e . If the stop at

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

sc a l e will give us all the

residual cl ass

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

0

up all the notes of this

the following manner :

c

h ro m ati c total. These classes

residual class 0 modulo residual class

I

m ay be re presented

4 : 40

modulo 4 : 41

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

we

placement by

certain

semitone) ,

one

sieving o f the basic scal e (elementary dis­

continuity to pass through. By extension

will be represented as sieve

be given by sieve 5n, in which

etc.

each resid ual class forms a sieve allowing

elements of the chromatic

the chromatic t o t a l

will

a

arc dealing with

in

n

=

0, I , 2, 3, 4.

1 0 • The scale of fourths will

Every change of the index

n

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

scale, 2n with

n

=

0, 1 , 20

has

-;.-

two

transpositions :

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

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

sieves

we

· · ·· · · · ·

can build more complex

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

u n i on

(disjunction) expressed as v , intersection

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

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

20

20

20

v 21

A

21

1 0)

=

chromatic total (also expressible as

=

no notes, or em pty sieve, expressed as 0

= 2 1 an d

21

=

The major scale can be

20•

written as follows :

a

Towards

1 97

Metamusic

By definition,

this notation do es not distinguish between all the modes

on the white keys o f

the

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

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

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

mode D, starting on

order t o

and in

general

(3n + 2 n

where

II

4n)

V

(§n + l

1\

4n + 1 )

V

(3n + 2

1\

4n + 2)

V

(Sn

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

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

of the constant index of each

ing

4n + 3) ,

II

sieve. The scale

(3n

1\

4n)

V

(3n + l

4n + l)

II

M usico l ogy Now let

us change

(3 n

V

II

4n + 2)

V

(3n + 2

II

correspond­

4n + 3) .

the basic unit (elementary displacement ELD )

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

maj or

scale will be written

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

1\

of

finer

=

( 8n

II

(9 n

V

9n + s) )

V

V

( 8n + 2 (8 n + 6

II

II

(9n + 3

(9 n

V

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

V

( 8n + 4

II

9n + 3)

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

with

n

=

One

( 8n

with

n

=

(9n + 6 ( 8n + 5

V

9n + s))

V

0, l , 2 ,

. . ., 7 l

(modulo

II

(9n

V

II

II

8

(8n + 2 ( 9n + 5

or

9) .

V

V

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

V

( 8n + 6

V

9n + 3) ,

198

Formalized Music

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

( 8,.

(9,.

A

9n + 3 ) )

V

Heptatonic scale :

(Bn

A

(9n

V

9n + a))

V

V

V

(8n + 2

A

(9n

V

9n + S) )

(8n + 6

V

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

(8n + 2

( 8n + S

(9n

A

1\

A

9 n + 3)

A

( 9,. + 4

V

9n + s) )

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

=

=

(7n

A

(7,.

1\

(8n

V

Bn + a) )

V

8,. + 6) )

H e ptato n ic scale :

(8,.

V

V

V

(7n + 2

(7,. + 2 (7n + 4

A

A

A

(Bn + 5

(Bn + S (8 n + 4

V

V

V

8n + 7 ) )

8n + 7) )

8 n + a) )

V V

V

( 7n + 5

A

Bn + l )

(7n + 3

A

8,. u) 8n + l )

(7n + 5

A

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

for

n

.

•

.

=

=

modules

17

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

S u prastructure&

the

for

17

x

18

=

306

>

(ll

x

24) .

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

1 99

Towards a Metamusic

coloration of the chromatic t o t a l which

Using metabolae.

has

a

higher

level of complexity.

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

sieve

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

and

indices ( transposi tions)

at every ch ange in the module or

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

5

II

x

x

11

=

Modules I , 7, 1 3 form

logical expression

L (5 , 1 3)

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

7 = 35 ;

a

cyclic sub-group

=

1 3 ; etc.

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

of the two sieves h avi n g modules =

( 1 3n + 4 1\

5n + l

V

l 3n + S

(5 n + 2

V

One can i mag ine a transformation

V

V

l 3n + 7

5n + 4)

V

1\

1 3n + 9) 1 3n + 9

of m od u les in

V

1 3n + 6•

p airs, starting from the

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

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

so as to return to the initial

This

sieve theory

·

·

·

wi l l be

__.,. L( 5 , 1 3 )

term (closure) .25

can be put into many kinds of archi tecture,

so as

to

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

textures of neigh borhood.

Subsequently we can put into

of outside-time

music by means

determinisms

i n-time practice this veri table histology

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

instance by giving

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

parametric with

time.

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

':oCl

' (' 1:1>

U n�·f- !; �; i�iA.il:ot t ! � �·

e:,...{

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

Formalized Music

200 this theory and its widespread

use

with the

help of computers,

for

it is

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

Then ,

in a su bseq u ent stage, there will

C o n c l usion

I

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

side-time category, which

temporal ca tegory. Moreover this

method African,

the out­ by

the

can un i fy the expression of

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

fundamental stru ctures of all Asian,

musical sciences.

In fact, what we are witnessing is

an

industri alization of music

which in

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

many publ i c places,

shops,

rad i o , T V ,

and

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

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

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

t hro u ghout

the

entire world

(all national

Non-decimal systems and the logic

coun tri es, so why not their application to

sketched out here ?

councils

o f classes a

are

for

music

take note).

already taught i n certain

new musical theory, such

as is

C h a pter V I I I

Towards a Philosophy of Music

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

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

tradition "

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

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

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

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

an

uneasi ness,

a

curiosity, then of applying the question,

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

created cosm ology from nothing, which

were

in

spite of religions and pow e rfu l mystiques,

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

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

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

ek-stasis,

with t he

the depa r t u re from self, can reveal

aid of purifications

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

i ts lost position and e s c ape th e

that is to say, the fa te of reincarnations

as an

yEvlaEw.. · b =

C with res p ect to

associativity

or A,

p.

distributivity

M U S I C A L N OTATI O N S A N D E N C O D I N G S

The vector space structure

o f intervals o f certain

sound characteristics by

permits us to treat their elements mathematically and to express th em

the set of numbers, which is indispensable for dialogue by the set of points on a straight

convenient. The

line, graphic

two preceding axiomatics may

with

computers, or

expression often b ei ng

be applied to

all

very

sound charac­ m o m e n t it

teristics that possess the same structure. For example, at the

of a scale of t i m b re which might be univer­ the scales of pitch, instants, and intensity are. On the other hand, time, intensity, density (number of events per unit of time) , the quantity of order or diso rd e r (measured by e ntropy ) , etc . , could be put into

would not make sense to speak sally accepted

as

one - to - o ne corres p ondence with the set

points on a straight line . (See Fig.

Fig.

Vl l l-1

Pitches

I nsta nts

of real VIII-1 .)

num bers R and the

De nsities

Intensities

Mo reo ve r, the ph enome n on of soun d is

a

set of

D isord e r

correspondence of sound these axes. The simplest

characteristics and therefore a correspondence of

Formalized Music

212

by Cartesian coord i n ates ; for ex a m p le , the in Fig. VIII-2. The unique p o in t (H, T) corresponds to the sound

c o rres po n d e nce may be sh o w n two axes

that has a pitch H at the

t

r

instant

T.

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