Formalized Music THOUGHT
AND MATHEMATICS
IN
COMPOSITION
Revised Edition
Iannis Xenakis Additional material compiled and edited by Sharon Kanach
HARMONOLOGIA SERIES No. 6
PENDRAGON PRESS STUYVESANT NY
Other Titles in the Harm onologia Series No. 1 Heinrich Schenker: Index to analysis by Larry Laskowski (1978) ISBN 0-918 728-06-1 No. 2 Matpurg's Thorou ghbass and Composition Handbook: A narrative translation and critical study by David A. Sheldon ( 198 9) ISBN 0-918728-55-x No. 3 Between Modes and Keys: German Theory 1592-1802 by Joel Lester (1990) ISBN 0-918728-77-0 No. 4 Music Theory from Zarlino to Schenker: A Bibliography and Guide by David Damschroder and David Russell Williams (1991) ISBN 0-918728-99-1
The Sense of Order by Barbara R. Barry (1990) ISBN 0-945193-01-7
No. 5 Musical Time:
Chapters I-VIII of this book were originally published in French. Portions of it appeared in Gravesaner Blatter, nos. 1, 6, 9, 11/12, 1822, and 29 (1955-65).
Chapters I-VI appeared originally as the book Musiques Formelles, copyright 1963, by Editions Richard-Masse, 7, place Saint-Sulpice, Paris. Ch ap te r VII was first published in La Nef, no. 29 (1967); the English translation appeared in Tempo, no. 93 (1970). Chapter VII was originally published in Revue d'Esthitique, Tome XXI (1968). Chapters IX and Appendices I and II were added for the
English-lan� ge edition by Indiana University Press, Bloomington 1971.
Chapters X, XI, XII, XIV, and Appendi.x III were added for this ed i tion, and all lists were updated to 1991. Library of Congress Cataloging-Publication Data
Xenakis, Iannis, 1922a Formalized music : thought and m thematics in composition 1 Iannis Xenakis. "
P·
c.m.
__
(Harmonologia series ; no. 6)
New expanded edition"--Pref.
ences and index. Includes bibliographical refer ISBN 0-945193-24-6 y and aesthetics. 2. 1. Music--20th century-Philosoph
century. 4. (Music) 3. Music--The· o·ry--20th · · Compos1t1on · · . II . I. Title cnucJsm. and story --Hi ury Music--20th cent
Series.
ML3800.X4 781.3--dc20
1990
. h t 1 99 ,.n•p en dragon Press C opyng 1
I
; .1
Contents P r e face Preface to
Preface
to
vii
Musiques formelles the Pend r ago n
ix
Edition
xi
I Free Stochastic M u s ic
II Markovian III Markovian
IV
Stochastic Music
Theory
-
Stochastic Music-Applications
Musical Strategy
V Free S toch a stic Music by Computer
VI Symbolic Conclusions
and Extensions for Chapters I- VI
VII Towards VIII
Music
a
Metamusic
Towards a Philosophy of Music
IX New Prop osals in Microsound Structure X
Concerning Time, Spac e and
XI Sieves XII Sieves: A User's XIII Dynamic
Music
I
& II Two
Guide
Stochastic Synthesis
III The
Bibliography D is cog rap hy
Laws
of Continuous ProbabiliLy
New UPIC
Biography: D egre es
79 llO 131 155 178 180 201 242 255
268
XIV More Thorough Stochastic Music
Appendices
I 43
System
277
2 89
295
323,327 3 29 335
and Honors
Notes
Index
3 65 371
373
3 83
v
Preface
The
attempted in trying to reconstruct part of the used, for want of time or of capacity, the most a d van c ed aspects of philosophical and scientific thought. But the escalade is started and others will certainly enlarge and extend the new thesis. formalization that I
musical e difice
ex
nihilo has not
This book is addressed to
tion
a
hybrid public, but interdisciplinary hybridiza
frequently produces superb specimens.
c ou l d sum up twenty y e ar s of p er s o n a l efforts by the progressive in of the following Table of C o here nces. My musical, architectural, and visual works arc the chi ps of this mosaic. It is like a net whose variable lattice s capture fugitive virtualities and entwine them in a multitu � e of ways. T his table, in fact, sums up the true coherences of the successive
I
filling
chronological chapters of this book. The chapters stemmed fr o m mono graphs, which tried as much as possible But the profound l esso n of
theory or
sol u t i on given
such
to
a
avoid
table
overlapping.
of cohercnces is that any
on one level can be assigned to
problems on another level. Thus the sol u ti o n s Families level
( p rogr a m m e d
an d more powerful usual
new
trigonometric
heavy
solution of
stochastic
perspectives in the shaping ofmicrosounds than the
(periodic) functions
can. a
clouds of points and their distribution over bypass the
the
in macrocomposition on the mechanisms) can engender simpler T h ere fore ,
in
considering
pressure-time plane,
we
can
harmonic analyses and syntheses and create sounds that
have never before existed. Only then will sound s y n th esis by computers digital-to-analogue converters find its
true
position,
free
and
of the roo ted but
ineffectual tradition of electronic, concrete, and instrumental music that
makes use of Fourier this book,
qu esti ons
more diversified and
tion
as
synth es is
despite
the
failure of this theory. H ence , in
having to do mainly with orchestral sounds
more manageable)
find
a
(which
are
rich and immediate applica
soon as they are transferred to the Microsound level in the pressure
time space. All music is thus automatically homogenized and unified.
vii
Preface to the Second
viii
l
Edition
I
everywhere" is the word of this book and its Table of Coherences; Herakleitos would say that the ways up and down are one. The French edition, Musiques Formelles, was produced thanks to Albert Richard, director of La Revue Musicale. The English edition, a corrected and completed version, results from the initiative of Mr. Christopher Butchers, who translated the first six chapters. My thanks also go to Mr. G. W. "Everything is
\
Mrs. John Challifour, who translated Chapters VII and VIII, respectively; to Mr. Michael Aronson and Mr. Bernard Perry of Indi an a University Pres s, who decided to publish it; and finally to Mrs. Natalie Wrubel, who edited this difficult book with infinite patience,
Hopkins, and Mr. and
correcting and
rephrasing many obscure passages.
I. X.
1970
(MOSAIC)
TABLE Phi/o,o}J.g
sense)
Thrust towards truth, revelation. (in the etymological
creativity.
Chapters (in the sense
of the
Quest
ARTS ( VISUAL,
SONJC1
MIXI!.D
• • •
)
COHERENCES
in everything, interrogation, harsh criticism, active knowledge through
methods followed)
Partially inferential and experimental
OF
\
Other methods
Entirely inferential and experimental
SCIENCES (OP MAN,
NATURAL
to come
?
)
This is why the arts are freer, and can therefore guide the sciences, which are entirely inferential and experimental.
C
of QuesliiNU
PHYSICS, MATHBMATJCS, LOGIC
(fragmentation of the directions leading to creative knowledge, to philosophy)
C:A.U5A.LlTYj
AS A CONSRJ;tUENCB OP NEW MENTAL STRUCTURES j
REALITY
Families
(EXISTENTlALITY);
of Solulions or
PrO<ed�ts (of
INFERENCE: CONNBXtTY; COMPACTNESS;
the above
Piocts (examples of particular realization)
�
/�
INDETERMINISM • • •
categories)
STOCHASTIC
materialized by
FREE
a computer pro-
ST/10-1, 080262 ST/48-J, 240162 ATRiES
of Sonie Eltmen�s
with computers
• • •
UBIQ.UITY
DETERMINISM j
• •
�
�
MARKOVIAN
SPATIAL
GROUPS
GAMES
MORSIMA-AMORSIMA
A DUEL
ANALOGIQUE B ANALOCIQ.UE
SYR.MOS
STRAT f: GIE
AKilATA
NOMOS GAMMA NOMOS ALPHA
{sounds that are he ard and recognized as a whole, and classified with respect to their 50urces)
ORCHESTRAL, ELECTRONIC
MicrosouTids
bi-po}e-+
gram
ACHORRIPSIS
Classes
TEMPORAL AND
+-
and
(produced
by analogue devices),
digital-to-analogue converters) ,
,
• ,
CONCRETE
(microphone
collected),
DICITAL
(realized
und s belong or which Forms and structures in the pressure.. time space, recognition of the classes to which mi r oso microstructures produce .
at the Microsound types result from questions and solutions I hat were adopted
levels.
c
cATEGORIES,
PAMILJES, and PIECES
I
Preface
This book is
a
to
collection
Musiques Formelles
m us ical composition pu rsu ed in
of explorations in
several directions. The effort to reduce certain sound sens at i o n s , to
under
their logical causes, to dominate them, and then to use them in wanted constructions; the effor t to materialize movements of thought through sounds, then to test them in compositions; the effor t to understand better the pieces of the past, by searchin g for an underlying unit which would be identical stand
with that of the scientific thought
while" geometrizing," that than
the impulse
of our
time; the effort to make "art"
is, by gi v ing it a
reasoned support less perishable
of the moment, and hence more serious, more worthy of
intelligence wages in a l l the other domains of the musical compositional act. This abstraction and formalization has found, as have so many other sciences, an unexpected and, I think, fertile support in certain areas of mathematics. It is no t so much the inevitable use of mathematics that characterizes the attitude of these experiments, as the overridin g n e e d to cons i der sound and music as a vast potential rese r voir in
the fierce fight which the human
-all these efforts have led to a sort of abstraction and formalization
which a knowledge of the laws of thought and the structured creations of thought may
find a
completely new medium of materialization,
communication.
For this purpose the qualification "beautiful"
s en se for sound,
or
nor for the music that derives from
i.e.,
of
"ugly" makes no
it; the
quantity of
intelligence carried by the sounds must be the true criterion of the validity
of a particular music.
This does not prevent the utilization of sounds defined
beautiful
according
to the fashion
of the
mo m e
nt ,
nor
as
pl ea s ant or
even their
s
their own right, which may enrich symbolization and al gebr atio n .
is
in itself a sign of intelligence. We are so
conv in ce d
ix
in
Efficacy
historical arts take an
of the
necessity of this step, that we should like to see the visual
t udy
X
Preface to
analogous path-unless, that is, "artists" of
done it in
a new
type have
laboratories, sheltered from noisy publicity.
These studies have
always
been matched
by
Musiques Formelles
actual works
not
already
which mark
out the various stages. My compositions constitute the experimental dossier of this undertaking. In the beginning my compositions and research were
and moral and material chapters in the present work reflect the results of the teaching of certain masters, su ch as H. Scherchen and Olivier Messiaen in music, and Prof. G. Th. Guilbaud in mathematics, who, through the virtuosity and liberality of his thought, has given me a clearer view of the algebras which constitute th e fabric of the chapter devoted to Symbolic Music. I. X.
recognized and
published, thanks
to the friendship
support of Prof: Hermann Scherchen. Certain
1962
Preface
to
the Pendragon Edition
Here is a new expanded edition of Formalized Music. It invites two fundamen
tal questions:
Have the theoretical propositions
thirty-five years
which I have made over the
past
a) survived in my music?
b) been aesthetically efficient?
To the first question, I will answer a ge ne ra l "yes." The theories which I
in the various chapters preceding this new edition have p resen t in my music, even if some theories have b een mingled with others in a same work. The exploration of the co nc eptu al and sound world in which I have been involved necessitated an harmonious or even have presented always been
conflicting
synthesis
architectural view
procedures.
of earlier
tha n
a mere
theses.
It
necessitated
a
more global
comparative confrontation of the various
But the supreme criterion always
aesthetic efficiency of the music which resulted.
remained the validation, the
was up to me and to me alone to determine the aesthetic not, in virtue of the first principle which one can not get around. The artist (man) has the duty and the privilege to decide, radically alone, his choices and the value of the results. By no mea n s should he choose Naturally, it
criteria, consciously or
any other means; those of power, glory, money, ...
Each time, he mus t throw himself and his ch os e n
all while striving to stan fr om scra tch yet not "monkey" ourselves by virtue of the own
"echolalic"
properties. But
to
be
criteria into question forget. We should not
habits we so easily acquire due to our b orn at each and every instant, like a
re
ch i ld with a new an d "independent" view of things. All of this is
oneself,
as
part of a second p rin ciple :
It is
absolutely necessary
much as possible, from a ny and all contingencies. xi
to free
Xll
Preface This may be con s ide red man's
in ge neral . Indeed, the Being's not, deterministic or chaotic
destiny
c onstant
(or
in particular,
and
the universe's
dislocations, be they continuous or
both simultaneously) arc manifestations of
the vital and incessant drive towards change, towards
return.
An artist can not remain isolated in the universal
freedom witllout
ocean of forms
and
their chang es . His interest lies in embracing the most vast horizon of knowledge and problematics, all p re sented above.
From
in
a ccordance witl1 the two principles
hence comes the new chapter in
this
edition entitled
"Concerning Time, Space and Music."
Finally, to finish with the first question, I have all al o ng continued to
develop certain theses and to open up some new ones. The new cl1apter on
"Sieves" is
an
Appen dix III
example of this alon g with tile comp uter program presented in
which represents a lon g aesthetic and theoretical search. This
resea rch was developed
UPIC.*
approach
Another
as
well as its application in sound synthesis on
to the mystery of sounds is the use of cellular
automata whi ch I have employed in several instrumental compositions these
by an observation which I made: scales of pitch (sieves) automatically establish a kind of global mus ical style, a sort of
past few years. This can be explained macroscopic
"synthesis"
of musica l
works,
their dissymmetries are the
d is ce rn i n g logico-aesthetic
m uch
physics of particles. reason behind this.
frequencies, or iterations," of the
cho ic e
like a
"spectrum
of
Internal symmetries or
Therefore, tluo u gh a
of "non-octave" scales, we can obtain very
rich simultaneities (chords) or linear successions which revive and generalize tonal, mod al or se rial aspects. It is on
this
basis of sieves that cellular
automata can be useful in harmonic progressions which create new and r ic h
timbt·ic
fusions with orchestral instruments. Exa mpl es of this can be
works of mine
such as Ata, Horos,
Today, there is
a
found
in
etc.
who le new fi eld of investigation called "Experimental
Mathematics," that gi ves fascinating insights especially in automatic dynamic
systems, by the
use
of math and computer
graphics.
Thus,
many s truc tu re s possess self-
such as tl1e already- mentioned cel lu lar automata or those which
*UPIC-Unite Polygogique Informatique du CEMAMu. A sort of musical drawi ng board which, through the digitalization of a drawing, enables one to compose music, teach acoustics, engage in musical pedagogy at any age. This machine was developed at the Centre d'Etudes de Mathematiqucs et Automatiques Musicales de Paris.
Preface similarities such
as
Juli a or
Mandelbrot sets,
xiii
arc studi ed and visualized. These
studies lead one right into the frontiers of deter minism and indeterminism.
C h aos to s ym m e try
and
the reverse o rientation a re once ag ain being studied
They
a nd are even quite fashionable!
are novel aspects started d eali n g with about thirty
open up new horizons,
al tho ugh for
of tJ1e equivalent compositional prob l e ms I
me, the results
-
five years ago. The theses
presented in the
earlier editi on s of this book bea r witness to tl1is fact altlwugh the dyn amic of
musical works depe nds on several levels simultaneously
and
not only on tl1e
c al culus level.
An important task of tJ1e research program at CEMAMu is to deve lop
synthesis through quantified sounds but with up-to-date tools capable
of
involving autosimilitudes, symmetries or deterministic ch aos , or stochastics
within a dynamic evolution of amplitude
corresponds to Einstein in the attributed to
a
sound
quantum or
1910s. This research,
Gabor,
frequency
frames where each
"phonon," as which I started
pixel by
already im a gi ne d
in 1958 and wrongly
can now be pursued witl1 much more p ower ful
and
modern means. Some surprises can
In App en d i x
IV
of tl1is
be expected! ed iti on a new, more ,
stochastic sound synthesis can be found
as a
precise formulation of
of tl1e last chapter of
follow-up
the preced ing edition of Formalized Music (presented here as Chapter the inter i m , this approach has been tested and used in my
d'Eer
IX). In
work La ligende
for seven-track tape. This approach was developed at the CEMAMu in
Paris and
worked out
at
WDR, the West-German N a tio n al
tJ1e
Radio
studio
in Cologne. This work was part of the Dia tope which was installed for the inauguration of
the
entirely automated
Pompidou/Beaubourg Center
with
in
Pari s. The event was
a c om p l ete laser installation
and
1600 electr o n ic
flashes. This syntl1esis is part of CEMAMu's permanent research program.
In this same spirit, ran d o m walks or Brownian movements have been
works, especially instrumental pieces such as "breath" or spir i t in Hebrew; for 2 female voices, 2 trombones and 1 'cello. This piece was written at the request
the basis for several of my N'Shi'11UJ,, which means French Horns, 2
of
"
"
Recha Freier, founder of the "Aiiya movement" and premiered at the
Testimonium Festival in Jerusalem.
The answer to Preface
is
the second question posed at the be gi nning of this
not up to me.
In
absolute te rm s , the artisan musician (not to say
creator) must remain doubt ful of the decisions he has made, doub tful ,
however subtly, of the result. The percentage of doubt shoul d not exist in
virtue ofthe pr in ci pl e s elaborated
connoisseurs
(either
above. But in relative terms, the public, or diachronic}, alone decide upon a work's
synchronic or
xiv
efficiency.
Preface However, any culture's validation follows "seasonal" rules, or
varying
m ille nnia We must never forget the nearly-total lack of consideration Egyptian art suffered for over 2000 years, or Meso-American art.
between periods of a fe w years to centuries
even
.
of art, or, let us say, just a work, to the on a document, seal in a bottle which we will throw of the ocean. Will it eve r be found? When and by whom and
One can assimilate a work information we can put into the middle
how will it be read,
interpreted?
My gratitude and thanks
go
to Sharon Kanach, who translated and
supervised the new material in this updated edition Robert Kessler, the courageous
publisher.
of Formalized Music and
to
Formalized
Music
...... ....
....:.. Ol "' Ol
-
I rl.
_,.
-lit_.,,
(:. ..... . �
Preliminary sketch 103-09.
._:-d/.4.�'
A11iLlogique B (1959).
cMtS
-- ---· · "'"' ·
-
See Chapter III,
�:r�Mt'_:
-. -._;c
pp.
-'��;_...... _,-Ad'&
•.
·-'
,U
Chapter I
Free
Stochastic Music
Art, and a bove all, music has a fundamental function, which is to catalyze the sublimation that it can bring about through al l means of expression. It must aim thro ugh fixations which arc landmarks to draw towards a total e xalta ti on in which the individual mingles, losing his consciousness in a truth immediate, rare, enormous, and perfect. If a work of art succeeds in this undertaking even for
dous truth is not ma de
a
single moment, it attains its goal. This tremen
of objects, emotions, or sensations; it is beyond these, as Beethoven's Seventh Symphony is beyond music. This is why art can lead to realms that religion still occupies for some people.
But this transmutation of every-day artistic material which transforms
trivial products into meta-art is
knowing its" mechanisms."
The
a
secret. The;" possessed" reach it without
others strugg le in
the ideological and tech "climate"
nical mainstream of their epoch which constitutes the perishable and
the stylistic fashion.
Keeping our eyes fixed
on
this supreme meta-artistic
goal, we sha l l attempt to define in a more mod est manner the p aths which
can
lead to
it
There
exists
from our point of
dic tions in present music. a
historical
departure, which is the
magma of contra
music and the reason. The music of antiquity, cau s al and deterministic, was already strongly influenced by the schools of Pythagoras and Plato. Plato insisted on the principle of c au sali ty , "for it is impossible for anything, to come into b e ing without cause" (Timaeus). Strict causality lasted until the nineteenth century when it underwent a parallel between European
su c cessi ve attempts to explain the world by
The English translation of Chaps. I-VI is by Christopher A. Butchers. 1
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Bars
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12 second violins V II
double basses CB
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4
Formalized Music
brutal and fertile transformation as a result of statistical
Since antiquity the concepts of chance
(tyche),
the ori es in
disorder
(ataxia),
physics. and dis
organization were conside red as the opposite and negation of reason
(logos),
orde r (taxis), and organization (systasis). It is only r ecen tly that knowledge has been able to penetrate chance and has discovered how to s epar a te its d egrees -in other words to ra t ion al ize it progressively, without, however, su cceed i n g in a definitive and total explanation of the problem of " pure chance."
After
a
time
lag
of several
decades, atonal music
broke
function and opened up a new path parallel to that of the
up the tonal
physical
sciences,
but at the same time constricted by the virtually absolute determinism of
serial music.
It is therefore n ot surprising that the presence or absence of the principle of causality, first in philosophy and then in the sciences, might influence musical co mp os ition . It c au sed it to follow paths that appeared to be diver gent, but which, in fact, coalesced in probability theory and finally in polyvalent logic , which arc kinds of generalization and enrich ment s of the p ri n c ip le of causality. The e xplanation of the world, and consequently of the sonic phenomena which surround
us or
and profited from the enl argemen t
which may be created, necessitated
of the principle of causality, the basis of wh i ch enlargement is fo r med by the law of large numbers. This law i mpl ies an asymptotic evolution towards a stable sta te , towards
a
kind of goal, of
stochos, whence comes the adjective "stochastic." But everything in pure determinism or in less pure indeterminism is subjected to the fundamental opera tio n al laws of logic, which were disen tangled by mathematical th ou ght under the ti tle of general algebra. These laws
operate on isolated states
or on sets of elements with
the
aid of
o perati ons , the most primitive of which arc the union, notated u,
the
intersection, notated n, and the n egation . Equivalence, implication, and
quantifications are elem en t ary relations from which all current science can be
constructed. Music, then, may be defined
as
an organization of these elementary
operations and rel at i ons between sonic entities
or
between functions
of occupied by also for an alysis
sonic entities. We understand the first-rate position which is set theory, not only for the construction of new works, but
and better comprehension stochastic c on stru c t io n or stochastlcs cannot be
of the w orks of the past. In the same way a an investigation of history wi t h the help of carried thro ug h without the help of logi c-t he queen
of the sciences, and I would even venture to suggest, of the arts--or its mathematical form algebra. For e verything that is said here on the subject
Free Stochastic
Music
5
is also valid for all forms of art (painting, sculpture, architecture, films, etc.). From t his very general, fundamental point of view, from which we wish to examine and make music, primary time appears as a wax or clay on which opera tio ns and relati ons can be inscribed and engraved, first for the purposes of work, and then for communication with a third person. On this level, the asymmetric, noncommutative character of time is use (B after A =I A after B, i.e., lexicographic order). Commutative, metric time (symmetrical) is subjected to the same logical laws and can therefore also aid organizational speculations. What is remarkable is that these fundamental notions, which are necessary for construction, are found in m an from his tenderest age, and it is fa s ci n ating to follow their evolution as Jean Piaget1 has done. After this short preamble on generalities we shall enter into the details of an approach to musical composition which I have de v elo ped over several years. I call it "stochastic," in honor ofprobabi1ity theory, which has served as a logi cal framework and as a method of resolving the conflicts and knots encountered.
The first task is to c ons t ruct an abstraction from all inherited conven tions and to exercise a fundamental critique of acts of thought and their materialization. What, in fact, d oes a musical composition offer strictly on the cons t ru ction level ? It o ffers a collection of sequences which it wishes to be causal. When, for simplification, the major scale implied its hierarchy of tonal functions-tonics, dominants, and subdominants-around which the other n ot es gravitated, it constructed, in a highly deterministic manner, linear processes, or melodies on the one hand, and simultaneous events, or chords, on the other. Then the serialists of the Vienna school, not having known how to master logically the indeterminism of atonality, returned to an o rganization which was extremely causal in the strictest sense, more ab stract than that of tonality; however, this abs trac tion was th eir great con tribution. Messiaen generalized this process and took a great step in sys tematizing the abstraction of all the variables of instrumental music. What is paradoxical is that he did this in the modal field. He created a multimodal music which immediately found imitators in serial music. At the outset Messiaen's abstract systematization found its most justifiable embodiment in a multiserial music. I t is from here that the postwar nco-serialists have drawn their inspiration. They could now, following the Vienna school and Messiaen, with some occasional borrowing from Stravinsky and Debussy, walk on with ears shut and proclaim a truth greater than the others. Other movements were growing stronger; chief among them was the systematic exploration of sonic entities, new instruments, and "noises." Vares e was the
1 st P e a k
A.
G r o un d prof i l e of the left half of
t he "sto m a c h .' " The i n t en t i o n was t o
b u i l d a s h e l l , com posed of as few
g r o u nd pl a n . A c o n o id ( e ) is c o n
ruled su rfaces as possible, over the
structed
curve ; t h i s w a l l is
s tr a i g ht
t h r o u g h t h e g r o u n d pro f i l e
bounded
by
two
l i n e s : the st raight d irectrix
( r i s i n g from the l eft extre m it y of t h e
g ro u n d prof i l e ) ,
and
t h e o u te r m ost
g e neratrix ( passing t h r o u g h the right produces t h e first " p e a k" o f the
extremity of
t h e gro u n d profi le ) . This
pavi l i o n .
B.
A r u l e d surface consisting of two
conoids, a and d, is laid t h r o u g h the the right half of the
"st o m a c h . " The st r a i g h t di r ec t ri x of d
c u rve
bounding
passes t h r o u g h t h e first peak, a n d the
o ut e rmost g e n e ratrix a t t h i s s i d e forms a t r ia n g u l a r e x it with the g e n eratrix of e. The s tra i g h t d irect ri x of a passes through a se c o n d peak and
is j o i n e d
by an a rc t o t h e directrix of d.
T h i s basic form is t h e
t h e f i rst desig n and was r e tai n e d , w i t h
one
used i n
s o m e modificatio ns, i n the f i n a l
a n aesthetic
s t r u cture. T h e m a i n pro b l e m o f t h e b a l a n ce between t h e two peaks.
design
was t o establ ish
1 st Peak "-"' "b
�
�
C.
At te mp t t o close t h e space between
t h e two ruled s u rfaces
of
the f i r s t
d e s i g n b y f l a t s u r faces (w h i c h m i g h t serve a s project i o n wa l l s ) .
Fig. 1-3 . Stages in the Development of the First Design of the Philips Pavilion
1 st Pea k
D.
Anot h e r a t t e mpt. Above t h e
e n trance c h a n n e l a sm a l l t r i a ng u l a r
2nd
i s formed. flan ked by two
hyperbolic p a r a b o l o i d s ( g a n d
open ing
Peak
k). and
the w h o le is co ve r e d with a h o r i z o n t a l
top
surface.
1 st
E.
Ela boration of D . T h e
third
Peak
peak
begins to take shape (shyly } .
3rd Peak
F.
The
first
2nd P e a k
de sig n completed (s e e
a l so t h e first model, F i g . 1 -4 } . There
flat
t h i rd p ea k i s fully developed and are no longer a n y
surfaces. The
c reates, with its o p p osing sweep, a
counterbala nce for the first two peaks.
The h eigh ts of
the three peaks
h ave
t h e s m a l l a r c c o n n ecti n g the straig h t
been established. The third p e a k a n d
d i rectrixes of c o n 0 1 d s a a n d d ( s e e B . ) fo r m , respect i ve l y, tho a p e x a n d t h e
b a s e of a p a r t of a c o n e /.
2nd P e a k
8
Formalized Music
pioneer in this fi eld , and electrom agnetic music has been the beneficiary (electronic music
bei ng
a
branch of instrumental music) . However, in
electromagnetic music, problems of construction and of morphology were not
faced conscientiously. M u ltise ria l music, a fusion of the multimodality of Messiaen and the Viennese school, re ma i ned , nevertheless, at the heart of
the fundamental problem of music.
But by 1 954 it was already in the process of d e fla t io n , for the com plete ly
deterministic complexity of the operations of compositi on and of the works themselves produced an a u di tory and i deolo g i cal nonsense. I described the
inevitable conc l usi o n in " The Crisis of Seri al Music " :
Linear polyphony d e st roys itself b y i ts very comp l exi ty ; w hat one
hears is in reality nothing but
a
mass of notes in various registers. The
enormous co mpl e xi ty preven ts t h e audience from following twi n in g
of the
l i n es and has as
the inter
its
macroscopic effect an i rrational
a
con tradiction between the poly
and fortui taus dispersion of sounds over the w hole extent of the so n i c
spectrum . There is consequently
phonic linear system an d the heard res u l t , which is surface or mass.
This contradiction in heren t in polyphony w i l l disappear independence of sounds
is
total . In fac t , w hen
l i n ear
when
the
combinations
an d their p ol y ph oni c superpositions no longer o p e ra t e , what will count will be the statistical
mean
tions of sonic components at
a
of isolated states and of transforma
given mom e n t. The m ac rosco pi c effect
can then be control led by the mean of the movements of elemen ts wh ich we select.
The
result is the i n troduction of the notion of p roba
bility, w h ic h implies, in this particular case, combi natory calculus.
Here, i n a few words, is the p ossi bl e escape route fro m the "linear
category"
in musical thou gh t . 2
This a rticl e served as a bridge
to my introduction of mathematics in
music. F or if, thanks to complexity, the strict, deterministic causality whi c h
the nco-serialists postulated was lost, then it was necessary to replace it by a more general causality, by a probabilistic logi c which would contain strict
serial causality as a pa r ticu l ar case. This is the fu nction of stochastic science. " Stochastics " stud ies
and
formulates the l aw o f l a rg e numbers, wh ich has
already been men ti one d , th e l aws of procedures, etc. As
causes,
a
rare
e vents , the different aleatory
resu lt of the impasse in serial music, as wel l as other
I originated in 1 954 a m usic
con s tructe d from the principle of
indeterminism ; two years later I n am ed it " S to c h a st ic Musi c . " The l aw s of the calculus of probabili ties entered
com position
th ro u gh
musical
necessity.
But other paths also l ed to the same stochastic crossroads-first of a ll ,
9
Free Stochastic M usic
n at u r al events s u c h as the collision of hail or rain with hard surfaces, or the
song of cicadas in a summer field. These s on ic
events
are m a d e out of thou
sands of isolated sounds ; this multitude of s ou n d s , seen as
a
totality, is a n e w
sonic event. This mass ev e n t is articulated and forms a plastic mold of time,
which itself follows aleatory and stochastic laws. If one then wishes to fo r m
a
s tri n g pizz ica ti, one must know these ma thematical laws, which, in any c as e , are no more than a tight an d concise expression of chain of logical reasonin g . E v e ryon e has observed the soni c phenomena of a political crowd o f d oz e n s or hundreds of tho u s a n d s o f people. The human riv e r shouts a sloga n i n a uniform rhythm . Then a no t he r slogan springs from the head of the demonstration ; it spreads tow a rd s the tail, re p l ac i n g the first. A wave of transition thus passes from the head to the tail. The c l amo r fills t h e city, and the i nh ibiti n g force of voice and rhy t h m reaches a cl imax. It is an event of great power and beauty in its fe ro c i ty . Then the impact between th e demonstrators and th e enemy occurs. The perfect rhy t h m of t h e last s logan breaks up in a huge clu s t e r of ch aotic shouts, which also spr e a d s to the tail. Imagine, in add i t i o n , the re p orts o f dozens of mach ine guns and the whistle of bullets a d d i r g their punctuations to this total disorder. T he crowd is t h e n rapidly dispe rsed, and after son i c and vi su al hell follows a detonating cal m , full of despair, dust, and death . The statistical laws of t h e se events, separated from their p ol it i c al or moral context, are the same as t h os e of the cicadas or the rain. They are the laws of the passage from com pl e te o rd e r to t otal disord er i n a continuous or exp lo sive manner. They are stochastic laws. Here we touch on one of the great p rob le ms that have haunted human intelligence since antiquity : continuous or d iscontinuous transformation. The sophisms of movement (e.g., Achilles a n d the tortoise) or of definition ( e.g., baldness) , especially the latter, are solved by statis ti cal definition ; that is t o say, by stoch astics. One may p rod u c e continu ity with either continuous or discontinuous elements. A multitude of short glissandi on strings can gi ve the impression of continuity, and so can a multitude of p izzicat i . Passage s l arge mass
of point-notes,
such as
from a discontinuous state to
a
c ontinuous state are controllable with the now
I
have been conducting these
fascinating experiments in instrumental works ;
the mathematical char
aid of probability theory. For some ti m e
acter
of this
but
music has frig h tened musicians and has mad e the approach
especially difficult.
Her� is an oth e r direction that converges on indeterminism . The s t udy
of the
v ari a ti o n of rhythm poses the problem of knowing what the limit of
total asymmetry is, and
among
durations.
of the cons eque n t
complete d isr uption o f c aus al i ty
The sounds of a Geiger counter in the proximity of a
10
Formalized
radioactive
source
necessary l a ws .
give
an
impressive idea of this.
Stoch astics
provides the
Before ending this short insp ection tour of even ts rich i n the
which
were
closed
to the
un derstand ing u n t i l recen tly, I
Music
new
l ogic,
would like to in
a short parenthesis. I f glissandi are long and sufficiently i n terl a c e d , obtain son i c s p aces of con t i n u ou s evolution . It is poss i b l e t o pr o d u c e r uled surfaces by drawing the gl issandi as straig h t l ines. I performed t h i s experiment with Metastasis ( this w o r k had i ts premiere i n 1 9 55 at D o n au eschingen) . Several years later, whe n the archi t e c t Le Cm· b usicr, whose collaborator I was, asked me to suggest a design for the arch i tectu re of the Philips Pavilion in Brussels, my inspi ration was p i n - p o i n ted by the experi ment wi t h Metastasis. Thus I beli eve that on this occasion music and arch i tecture found an i n tim ate connect i o n . 3 Fi gs. I- 1 -5 indicate the causal chain of ideas which led me t o formulate t h e arch i tect1.1re of the Philips Pavilion from the score of Metastasis.
clude
we
F i g . 1-4. F i rst M od e l
of
P h i l i ps Pavi l ion
Free Stochastic Music
11
.'
·�
·.
.. ,, \ ' \ l \
·}
' �
\ '
. \�
F i g . 1-5 . P h i l i ps Pavilion, B ru ssels
World's Fair,
1 9 58
12
Formalized Music
STO C H ASTI C LAWS A N D I N CA R N AT I O N S
I sha l l give quickly som e of th e stochastic laws which I introduced i n to
composition several years ago . We s hal l examine one by one the in d e p e n den t components of an instrumental sound. D U RA T I O N S
Time (metrical ) i s considered a s a straight line o n which the p o i nts correspond ing to the variations of the other components are marked. The
interval
possib le
between two point s is i dentical with the d u rati on . Amon g a l l the sequences of poi n ts , which shall we choose ? Put thus, the q ue s t i o n
makes no sense.
If a mean n u m b e r of points is designated on a given
length the question
b ec om e s : Given this mean, what is the number of segments equal to fixed
in
advance ?
The followin g formula, which derives
from th e
a length
principles of continuous
probability, giv es the probabilities for all possi ble lengths when one kn ows
the mean nu m ber of p o ints placed a t random on
a str ai gh t line. (See
in
w hich
8 is the linear d ens i ty of points, and
x
the length of any segment.
If we now choose some po i n ts and compare them to
distribu tion obeying the above law
or
Appendix I.) a
theoretical
any o th e r distribution, we can deduce
choice, or the more or less rigorous distribution, which can even be a bsolu t ely functional . The comparison can be m ade with the aid of te sts , of which the most widely used is the x2 cri terion of Pearson. In our case, where all the co mp one n ts of sound can be measured to a first approxima ti on , we shall use in addition the correlation coefficient. It is known that if two populations are in a linear functional relationship, the correlation the amount of chance included in our
adapt at io n of our choice to the
of
law
coefficient is one. If the two populations are ind ependent, the coefficient is
zero. All
intermedi ate
degrees
of relationship are possible.
Cl ouds of So u n d s
Assume a given duration a n d a s e t o f sound - p oin t s defined
in the
intensity-pitch space real i ze d duri n g this d ur atio n . Given th e mean super
ficial d e n sity of this tone cl uste r , what d e nsity occurring in a given region Law answers this question : Pu
is the probability of a particu l ar of the intensity-pitch space ? Po i sson ' s u
=
P.o e - u . ' P. ·I
13
Free Stochastic Music
where p. 0 is the mean density and p. i s any particular density. As with
durations, comparisons with other distributions of sound-points can fashion the law which we wish our cluster to obey.
I N TE R V A L S
OF I N T E N S I T Y , P I T C H ,
ETC.
For these vari ables the simplest law is
B(y) dy
=
� (I �) dy,
(See Appendix I . )
-
which gives t h e probability that a segment (interval o f intensity, pitch, etc . ) within a s eg me nt of length
y + dy, for S P E ED S
O � y � a.
a,
will have a length included within y and
are particular case of sounds of continuous variation. Among these let us consider glissandi. Of all the possible forms that a glissando sound can take, we shall choose the simplest-the uniformly continuous glissando. This We have been speakin g o f sou nd-points, or granular sounds, which
i n reality a
glissando
can
be assi m ila ted sensorially and physically into the mathematical
concept of speed. In a one-dimensional vectorial represen tation, the scalar size of
the vector can be gi ve n by the hypotenuse of the right triangle
in
which the du ration and the melodic interval covered form the other two sides . Certain mathematical operations on the continuously variable sounds
thus defined are then p e r m itt ed The traditional sounds of wind instruments .
particular cases fre q uencies can be
are, for example, towards higher
q u en c i es as negative.
where the speed is
zero.
A glissando
defined as positive, towards lower fre
We shall demonstrate the simplest logical hypotheses wh i ch lead us
to a mathematical formula for the distribution of speed s . The arguments
which follow arc in reality one of those " logical po e m s
intelligence creates
"
which the human
in order to trap the superficial incoherencies of physi cal
phenomena, and which can serve, on the rebound, as a poi n t of departure for b u i lding abstract entities, and then incarnations of these entities in sound or light. It is for these reason s that I o ffer t h em as examples :
Homogeneity hypotheses [ I I ] * 1 . The d ensity of speed-animated
sou nds is constant ; i.e. , two regions
of equal extent on the pitch range contain the same average num ber of mobile sounds (glissan d i ) .
* The nu mbers in brackets correspond to the numbers in the Bibl iogra p hy end of the book .
at
the
14
Formalized Music
2. The absol u te value of sp ee d s (ascending o r descending glissandi) is uniformly ; i.e., the m e a n quadratic sp e ed of mobile so un ds is the same in different registers. 3 . There is isotropy ; that is, th e re is no privileged d ir e c t io n for the movements of mobile sounds in an y register. There is an equal number of sounds ascending and descending. From these th ree hypotheses of symmetry, we can define the fu n c t i on f(v) of the probability of the absolute s pee d v. (f(v) is the rel ative frequency of occurrence of v.) L e t n be t h e number of glissandi per u n i t of the pitch range (density of mobile sounds ) , and r a ny portion taken fro m the range . T h en the number of speed-animated sounds between v and v + dv and positive, is, from hypotheses 1 and 3 : spread
n
(the probability that the sign is + is }) .
r !f(v) dv
2 the number of animated sounds with speed o f value l v l is a function which depends o n v2 onl y . Le t cl1 is functio n be g(v2). We then have the equation
From hypothesis
absolute
tf(v) dv
n r
=
n r
g(v2) dv.
Moreover i f x v, the probabi lity function g ( v 2) will be equal to the law of probability H of x, when_cc g(v2) H(x) , or log g(v2) = h(x) . In order that h(x) may d ep e n d only on x2 v2, it is n ecessary and sufficie n t that the differentials d log g(v2) h' (x) dx and v dv x dx have a constant ratio : =
=
=
=
d log g(v2) --"'-'f.!......!. whence
h' (x)
v dv
- 2jx, h(x)
a
from - CIJ to l fa2, it follows =
h' (x) dx --x dx
x· ,
=
- 2.J, .
=
=
=
g(v2)
j( v) =
s ta n t
=
=
and
v
c on
that
tf(v)
fo r
=
-jx2 + c, a n d H(x) ke -ix". function of elementary probabilities ; th e refo re its i ntegral + oo m us t be equal to I . j is p osi tiv e and k y'jjy'TT. If
=
But H(x) is
j
=
=
which is
a
=
Gaussian
H(x)
=
ay'TT
1-
_2_ e - ••ta• ay'TT
distribution.
e - ••ta•
15
Free S tochastic Music This
chain o f
reasoning borrowed fro m
Maxwel l , wh o , wi th gases. The
Boltzmann, was
function f(v)
Paul L evy was established after
res p onsible fo r the kinetic theory of
gives the probability of
the
s p eed v ;
t h e constant
a
defines the " temperature " of thi s soni c atmosphere. The a rithmetic mean
equal to af v'", and the s t a nd a rd deviation is a { \/2. offer as an example several bars from th e work Pithoprakta for s tring orchestra (Fig. 1-6) , wri tten in 1 95 5-56, and performed by Prof. Herman n S c h erchen in Munich in March 1 957.4 The graph ( Fig. 1-7) represents a of v is
We
set of speeds of temperat u re proportional to a 35. The abscissa represen ts time in units of 5 em 26 MM ( M iilzel Metronome ) . This unit is sub divided i n to three, four , a n d five equal p arts, which allow very slight =
=
arc d rawn as the ordinates, with the unit 1 sem i tone 0 . 2 5 em. I e m on th e verti cal s c al e corresponds to a m ajor third. There arc 46 stringed instru m e n ts, each represented by a j agged line. Each of the lines represents a speed taken from the table of probabilities
differences of duration. The pi tches =
calculated w i th the formu l a
J(v)
=
_ 2_ e - v 9 t a 2 . a v'TT
in 58 disti nct values according to Gauss's traced for this passage (measures 52-60, with a duration of 1 8. 5 sec.) . The distribution being G a u ssian, the macroscopic configura tion is a pl astic mod u l a tion of the sonic m aterial . The same passage was transcribed into trad itional no tation . To sum up we h av e a so n i c
A total of 1 1 48 speeds, distribu ted
law, h ave
been calcu l ated
and
compound in which :
1 . The durations do not vary. 2. The mass of pitches is freely modulated . 3. The density of sounds at each moment is 4. The d y n a m ic isjf without variation .
constant.
5 . The timbre i s constant.
6.
The speeds determ ine a " temperat u re " which is
fluctuations.
Their distrib u tion is Gaussian .
As we h ave already had occasion
to rem ark,
subject
to local
we can establish more or
less strict relationsh ips between t h e com p onent parts of sounds.5 The most useful coefficient which measures the degree of correlation between two variables
x
and y is
2: (x - x) (y - y) r=v�1,.=(;=x-'---x=-)"' 2v -';.L �(y� --=y= )2 '
16
Formalized
Music
where x and fi are the arithmetic means of the two variables. Here then, is the technic al aspect of the starting point for a utilization of the theory and calculus of p rob abilities in musical composition. With the above, we already know that :
1 . We can control continuous transformations of large sets of gra nular and/or continuous sounds. In fact, densities, durations, registers, speeds, etc., can all be subjected to the law of large numbers with the necessary approximations. We can therefore with the aid of means and deviatio ns shape th ese sets a nd make them evolve in different directions. The best known is that which goes from order to disord er, or vice versa, and which introduces the concept of en tropy. We can conceive of Qther continuous transformations : for example, a set of plucked sounds transforming con tinuously into a set of arco sounds, or in electromagnetic music, the passage from one sonic substance to another, assuring thus an organic connection between the two substances. To illustrate this idea, I recall the Greek sophism about baldness : " How ma ny hairs must one remove from a hairy skul l in order to make it bald ? " It is a p rob lem resolved by the theory of probability with the standard deviation, and known by the term statistical definition.
2. A transformation may be explosi ve when devi ations from the mean suddenly become exceptional. 3. We can likewise confront. highly improbable events with average events.
Very rarified sonic atmospheres may be fashioned and controlled aid of formulae such as Poisson's. Thus, even music for a solo instrument can be composed with stochastic methods. 4.
with the
laws, which we have met before in a multitude of fields, are di amonds of contemporary thought. They govern the laws of the advent of b e ing and becoming. However, it must be well understood that they are not an end in themselves, but marvelous tools of construction and logical li fel ines. Here a backfire is to be found. This time it is these stochastic tools that pose a fu n d ament al question : " What is the minimum of l ogical constraints necessary for the construction of a musical pro ces s ? " But before answering this we shall sketch briefly the basic phases in the construction of These
veritable
a musical work.
W. ll
=-- -----1 ---- - ----1 [!!] ---.,---o-l---- -l-. ----f----=t pi••· "Jii: � • � � 5.::-----, ,.......- r -------, .. r ----, r--- • � - " -..:!:
V. l
···--
A.
v. . ..
&, LN. ltJ.S6S
F i g . 1-6. B a rs 52-57 of Pithoprakta
18
F i g . 1-7 . G ra p h of B a rs 52-57 of Pith oprakta
Form alized
Music
Free S tochastic
l'vi usic
19
20
F i g . 1-7
Formalized Music
(continued)
Free Stochastic Music
21
22
Formalized Music F U N D A M E N TA L P H AS E S O F A M U S I C A L W O R K
I.
Initial conceptions (in t u i ti o n s ,
pro vi s i onal o r d efinitive data ) ;
2. Definition of the sonic entities and of their s y m b ol i sm commu nicable with the limits of p ossible means (sounds of m usical instruments, electronic sounds, noises, s e t s of ordered sonic elements, granu l a r or continuous formations, etc.) ;
3. Definition of the t ra nsfo rma tions which t h e s e sonic entities must undergo course of the composi tion (macrocomposition : general choice of l ogical fi·amework, i.e., of the ele me nt a ry alg eb r aic operations and the set ting up of relations between enti ties, sets, and their symbols as d efined in 2 . ) ; and the arrangement of these operations in lexicographic time with the aid of succession and simultanei ty) ; 4. Microcomposition (choice and detailed fixing of the fu nctional or s to cha s ti c rel ations of the clements of 2 . ) , i.e., algebra outside-time, and in the
•
algebra i n-time ;
5. Sequential program ming of 3. in its e n tire ty) ;
work
6.
Implementation of calculations,
and
4.
(the schema
and pattern
of the
verifications, fe e d b a cks , a n d definitive
modificati o n s of the se q uential program ;
7.
paper
Final symbolic result of the programming (setting out the music on in traditional n otation, n um e ri c al expressions, graphs, or o ther means
of sol fcggio) ; 8. Sonic
realization
of the p ro gr a m
(direct
manipulations of the type of electromagnetic
orchestral
p erfo r mance ,
music, computerized
construc
t io n of the sonic entities and t h e i r transform ations) .
The order
really rigid. Permutations of a composition . Most of
of this list is not
the course of the working out
are possible in the
ti m e
these
ph ases are unconscious and defective. However, this list does establish
ideas
that
only
and
a l l ows specul ation about the fut u r e .
ph ases 6 .
and
ph ases 6. and
7.,
7.
an d
even
8. But
In fac t ,
compu ters can t a k e
as a first ap p ro ach,
are i m m ediately accessible. That is
to
it s e e m s
in hand
the final orchestra or
say, that
result, at l east in France, may be realized only by an by manipulations of el ectroacoustic music on tape recorders , emitted by the existi ng electroacoustic chan nels ; and not, as would be desirable in the very symbol i c
ncar
fu tu re, by
an
el aborate
m ec h a n i z ati o n
which would o mi t orchestral or
Lapc i nterpreters, and which wo u l d assume the computerized fabrication of the sonic enti ties and of their transformations. Here now i s a n answer to the q u estion put above, an answer that i s t r u e for i nstrumental m usic, b u t which can be a pp l i e d a s we l l t o all kinds of
23
Free Stochastic Music sound production. For this we shall again t ake u p the ph ases described :
the
2. Defi nition of sonic entities. The sonic enti ties of
tra can be represented in
in depen dent variables, Ca
h1
gi
uk
by
base
(c, h,
Cvlol . arco>
h 39 (
or instrum ental fa mi l y
duration of the sound.
=
defines a
u) .
at
point M in the
c3
played arco an d forte on a viol i n , one eighth
(
=
forte) ,
u5
(
=
=
M are plotted on an axis which we shall
we draw another axis t, at right angles to
axi s , called the
multidimensional space provided coordinates the n u m b e rs
This poi n t M will have as
one eighth no te
c 3 ) , g4
=
classical orches
b y vectors of fo u r u s u a l l y
i n tensity o f the sound, or dyn amic form
=
'"' h" g,, uk. For example :
note in length ,
approximation
p i t c h of the sound
=
g,
fi rst
E, (c, h, g, u) :
timbre
=
The vector E,
a
a
be represen ted as that these poi n ts call E" a n d that through its origin axis E,. We sh all represent on this 240 MM, can
-!- se c . ) . Su ppose
axis of lexicographic time, the lexicographic-temporal succession
o f the points M. Thus we h ave d efi n e d and conveniently represented
two-dimensional space (E, t) . This will allow us to pass to phase
tion of transform ation ,
and
3.,
a
defin i
4., microcomposi tion, which must contain the
a nswer to the problem posed concern ing the minimum of constraints. To this end, su pp ose that
the
po i n t s M
defined above c an appear w i th of obeying an aleatory law with o u t m emory. This hypothesis is e q uivalent t o sayi n g that we adm i t a stochastic d istribution of the events E, in the space (E, t) . Admitting a sufficiently weak su perficial distri b u tion n, we enter a region where the l aw of P o i sson no
necessary co n d i tion other than that
is applicabl e :
I ncidentally we can consider this problem as
conveniently chosen linear stochastic processes active bodies) .
a
syn thesis o f several
( law o f radiation from radio
(The second method is perhaps more favorable for a mecha
nization of the transformations.)
A su fficien tly long fragment of this distribution constitu tes the m usical
work. The basic
law defined above generates a whole
family of
com positions
as a fu nction of the superficial density. So we have a formal archetype of composition in which the basic
metry
and
aim is
to attain the greatest
(in the etymological sense ) and the
rules. We t h i n k
that from this
poss i bl e
asym
minimum of constraints, causalities,
archetype, which
is
perhaps the most
Formalized
24
general one, we can red es ce nd the ladder of forms by introducing
Music
progres
sively more num erous constraints, i . e . , choices, restrictions, and n e ga t ions
In the an alysis in several linear processes we
th os e
cesses :
of Wiener-Levy, P .
ch ains, etc., or
m ixt u res of
Levy's
can
infinitely
several. It is this
method the more fertile.
The exploration of the limits
a
and b of
implies,
in effect, a gradation
which
this
divisibles,
.
pro
Markov
makes this second
archetype
a
:;;;
n
:;;;
b is
of the mutual com pa r i son of of the i nc rem en ts of n in order
equally interesting, but on another level-th at samples. This
also introduce other
that the differences b e tw e en the families n1 may
be recognizable. Analogous
case of other linear processes . If we opt for a Poisson p rocess, t h e re are two necessary hy po th eses which answer the q uestion of the m in i m u m of constraints : l . there exists in a given space musical instruments and men ; and 2 . there exist means of contact between thes e men and these ins truments which p e rm i t the emission of rare so n i c events. This is the only hypothesis (cf. the ekklisis of Epicurus) . From these two constraints and with the aid of stochastics, I b u i l t an entire com posi tion without admitting any o the r restrictions. Achorripsis for 2 1 instruments was c om p o sed in 1 956-5 7, and h ad its first p erformance in Buenos Aires in 1 958 under Prof. H e rm an n Scherchen. (See Fig. I-8.) At that time I wrote :* remarks arc valid in the
T6 yit.p rxth6 voetv Td yap ONTOLOGY
In
at}r6
�aTlv
e fva' �arlv
Tt
n
Kai ovK
efvatt
o f nothingness. A brief train of waves , s o brief that its end coi nci de ( negative time) disengaging itself endlessly.
a universe
and beginning
Nothingness
resorbs, creates.
It engenders bein g . These rare son i c events can They
Kai elva£
can
be
Time, Causality.
so m e th i n g more than isolated sounds.
be melodic fi g ures, cell structures, or agglomerations whose
"' The following excerpt (through p. 3 7) is from " I n Search of a Stochastic Music," Gravesaner Bliilter, no. 1 1 / 1 2. t "For it is the same to t h i nk as to be " (Poem by Parmeni dcs) ; and my paraph rase, " For it is the same to be as not to be."
25
Free Stochastic Music c h aracteristics are
also ru l ed by the l aws
sou nd-points or speed- temperatures . 6
succession of r a r e sonic events . This s a mp l e may
be
of ch ance, for examp l e, clo uds of
In each case they
represe n ted by e i th er
a
form a sample of
a
simple table of prob abili ties
or a double-entry table, a matrix, in which the cells are filled by the fre quencies of events. The rows re present the particu l ar qualifications of th e
events, and the
columns the d ates (see Matrix M, Fig. I-9) .
The freq uencies
in this matrix a re distributed a c cord in g to Poisson's fo rm u la , wh ich is the
law for the appearances of rare random events.
We s h o u l d further define the sense of such a distribution and the manner
in which we realize it. There is an advantage in
defining
chance as
an
aesthetic law, as a n or m a l philosophy. Chance is the limit of the notion of
evolving symmetry . Symmetry tends to asymmetry, which in this s e ns e is
eq uivalent to the n egat i o n of traditionally inheri ted behavioral frameworks. This n eg a t io n not only operates on d etai l s , but most im portan tly on the c o mposit i on of structures, hence te nde n c i es in painti ng, sculpture, architec ture, and other realms of thought. For example, in ar ch i te ct u re , plans wo rk e d out with the aid of regulating di agrams are rendered more complex and dynamic by exceptional events. Ever y thin g happens as if there were one-to one
oscillations between symmetry , ord er, rationality , and asymmetry,
disorder, i rrationality in the At the
beginning of a
reac tio ns between the e poch s of civilizations.
transformation towards asymmetry, excep tional
events are introduced into symm etry a n d act as aesthetic stimuli. W h e n
th ese e x c ep t i o n a l events mul tiply and become the general case, a j ump to a
higher
level occ u rs .
The level is
one
of disorder, which, at l e as t in the arts
an d i n th e expressions of artists, p roclaims i tself as e n ge n d e re d
plex, vast, and rich
s u c h as abstract
vi
s io n
of
the
bru tal
and decorative ar t
and action
fact. Consequently chance, by w h os e side
com
l i fe. Forms
painting bear witness to this walk in all our daily occu p a co n tro lled disorder ( that which
we
tions, is n o th i ng b u t an extreme c ase of this
s i gn i fies the rich ness
by the
encounters of modern
or poverty of the connections between events and wh ich by
engenders the dependence or independence o f transfo rmations) ; and
virtue of the negation, it conversely enjoys all the benevolent characteristics of an artistic regulator. I t is a
reg
ul ator also of sonic events, their appearance,
and their life . But it is here that the i ron logic of the laws of chance in ter
ve n es ; this chance
laws.
On
cannot be
created withou t total subm ission to its
own
t h is condition, chance checked by its own force becomes a hyd ro
electric torren t .
:r-�
/ II oJ
r- 1--,
'
A
·�
'l':e
)(yf.
lt.· BI
•
T
..
3
.--
. � !:l · - - - -
II '-'---'-5= A
[7
7
r--- 1 ;::ro
-
s.
.!(
II
!- - -�11. ':;J: '�� r l -;' ...
I
r-1--,
r
I r3 -, r- li--. . ..
.If
F i g . l-8 . B a rs 1 04-1 1 1 of A chorripsis
�
..
I
•
r!�
• " I 16 - - - -:: t:\
;;. �
•
Js__, L-- 5"--' I
..,
Jl ;:.
.,
II
"'! J '--J....;...o . )(.
.
.
1"-- S�
-I
r;- 5'�
r- 5'� r-=5"--,
��
-
��i !'_- - - - - - -.i
� �· -t
1......- !i --'
·
• .. -
" I
16 - : r.� ·
I
--,
r:-� "
§=J
I
..
�S�'i:t!: 1
r- !i___,
6
"'
...
I "
!'
��
l 'T-T
.:t: �
�
�
t
.....
�
..
r- s--,
�
8· · · :
......
.....
llb.Z
T T1 r;- �'�
'-- 5...::::1
ror
1
r-6"--, r- 5"--. ·� ., . ,·,
r-:il �
II
3
��
:!:
:!:
��1 '1< jol
�5'--'
"' ·" " " - - - - -
1
Vc/.2
�
.P I1'
�
Yl. 2
•,y
7
•
1
r- r--, r-3--.
---;2 r - li
• II
-�
� 6--. r-- 3 --,
-=s=-
B
�-""
r-. 5"--r r
r-•.:..-:t:
,..... fl"r .P. �
i t� *t'"'t�:-:
��-
.ll r.t
r-J-, � 5--, .....
•
r-- J -,
I I ......
Free Stochastic Music "" "
Pice .
27
.P- .fll.
!,.,.�
-....
¥-
�.(I.
l"Ji" S"---,
I "'
Ob .
II£
II
fl
1'1
1
..
/, 7ip t. 7 Xyl.
II
.:a:....t:t
r- 5 ---,
r-:: 3 --,
..Q.-
�: r;-;7 r;-�-, ���� ·"
"
R l'
A I'
. . L.-1 1"
II r- !; --, r-- 5"---, II
II
1
I'
l
v I
.. . - l ���f:
Yl. 7.
· II B · · · II .... ·
, ..
:;_·
-{,
.J
""
3
A
..
'
.__ il ---' M
��-
.--- s--. "
.
( P'•� S--, ..
� I
r- .r--, '.1 " •
r3---. . t • • I r I I
" " I
r;-..5---, .
"I • •
u...J
.....
.
r-- r;-vr-- 5"� ":'
�x
(pizz.) �
I
;;;:!.� � 1!" -=
�
.-s--,
... .... .
� .. , ..., .
t;'l � 6 --.
tf!.- -::Jil l v
y�
s · ;; : A ..
r-
I
.,
,.
•
" I
.- -
11 ..
,
.�
f:.
A "
I I'
..
r---ys--,
�
�
"
,.-- s---,
--
v
,.,
.-- s-,
It
€-!:it:· :
..
r- 5--,
"
,__ 3 __,
-
r- 3 ---,
•
� 3 --, .
r-5"--"")-:-�� I
•
L::J-J
'-- 3"--l 1..- S"--1
�
£_-.
�,:��
..
r- 3 ---.
-
I
I'
r-- 3----!
'-J
�f.
..
........_ 5"---'
....,
i
•
r,1
t-=r-r-•.! l i\f;�� _ _ _ _ _
�·
1-J
.-;�
-·�
Yc/, 2
., •
I
II
s···:
5---, ll r- " , v
1
'-- 5'---1
,--;- 5 ---,
•
"
gr.Tr:
.-� '�-=
lbl.. � r:.- :,;::-- 3 -,
· "' ---
H.·BI
J
��
M
Cl> 0 .0 0
.,.,
"' cii
C> c 0 ·.: u
U) Co
.. c "' > Cl> Cl>
:::1
c. "' :::1
-5
0
-
.. ., .,., c
>
c:
:::1 0 0
:9,=
G> > CD G>
:0
(Ill II]
2RS&
·�
-� <:)
t::
'5
-� ""(
0
co
:E
� ·� co
�
., .,
., .,
c >
Ill
E "'
>
c
0 z
c;,
u;
] Qua l ific a t i o n s
��
....
g u
�
!
ci �
Free Stochastic Music
29
However, we are not s p e a ki ng here of cases where on e merely p l ays heads and tails in order to ch o ose a p ar ti cul ar alternative in some trivial circu mstance. The problem is much more serious than that. It is a m a tte r here of a philosophic and aesthetic concept ruled by the l aws of probability and by the m athematical functi_ons th at formulate that theory, of a coh er ent
concept in a new region of coherence. The analysis that
fol lo w s
is
taken from
Achorripsis.
For convenience in calculation we shall choose a p riori a mean d e n sity of events
>.
Applying Poisson's formula,
=
0.6 events /unit.
we obtain the table of probabilities :
P0 pl p2
P3 P,
P5
=
0. 5488
=
0.3293
=
0.0988
=
0.0 1 98
=
0 . 0030
=
(1)
0. 0004.
P1 is the probability that the event will occur i times in the unit of volume , time, etc. In choosin g a priori ] 96 units or cells, the distribution of the frequencies a mong the cells is obtained by multiplying the values of
pj by 1 96.
Number of cells
0 1
2 3 4
1 96 PI
1 07 65
19
4
(2)
1
The 1 96 cel l s may be arr a n ge d in one or several groups of cells, quali
fied as to timbre and time, so that th e number of groups of timbres times
of groups of du rations = 1 96 cells. Let there be 1 distinct timbres ; then 1 96/ 7 28 units of time. Thus the 1 96 cells arc distributed over a two-dimensional space as shown in (3) .
th e nu mber
=
30
Formalized Music Timbre Flute Oboe
String
gliss.
Percussion
(3)
Pizzicato
Brass String
arco 0
1
2
3
. . . . . . . . . . . . 28 :rime
If the musical sample is to last 7 minutes ( a subj ective choice) the unit of U1 will equal 15 sec., and each U1 will contain 6.5 measures at
time
MM
=
26.
of ze ro, single, double, triple, space of Matrix (3) ? Consider th e 28 columns as cells and distribute the zero, single, double, triple, and q u adruple events from table (2) in these 28 new cells. Take as an example the si n gl e event ; from table ( 2 ) it must occur 65 times. Everything h appens as if one were to distribute events in the cells with a mean density A 65/28 2.32 single events per cell (here cell column ) . In applying anew Poisson's formula with the mean density A 2 .32 How shall we distribute the frequencies
and q u adruple events per cell
=
in
the two-dim ensional
=
(2.32
«
we obtain table
30)
Poisson Frequency
K 0 1 2 3 4 5 6 7
Totals
=
=
(4) .
Distribution
Arbitrary Distribution
No. of
Pro du c t
3
0
6
6
Columns col
x
Frequency No . of Product K Columns col x K
K
0 1 2 3 4 5
10 3 0 9 0
0 3 0 27 0 5
8 5 3 2
16
I
6
6
5
30
0
7
0
0
28
65
0
28
15
12 10
65
(4 )
Totals
(5)
Free
31
Stochastic Music
One could c hoos e
single events equals But in
space,
we
this
must
any o th er distribution
on
65. Table (5) shows such
a
condition that
the sum of
distribution.
axiomatic research , where chance must bathe all of so ni c
rej e c t
every distribution which
departs from Poisson's
law. And the Poisson distribution must be effective n o t only for the columns but also
for
the rows of
the matrix. The same reasoning holds true for the
diagonals, etc.
Contenting ourselves just with rows and columns, we obtain
geneous distribution which follows Poisson. It was
a h omo in th i s way that the
distributions i n rows a nd columns of Matrix (M) (Fig. 1-9) were calculated . the
So a unique law of chance, the law of Po i ss on
me d i u
m of the arbitrary mean
( for r are
events) t h ro u gh
A is ca p able of conditio ning, on the one
hand, a whole sample matrix, and on t h e other,
the partial distribu tions The a p riori, arbitrary choice a d m i tte d at concerns the variables of the " vector-m atrix. "
following t h e rows and columns. the
beginning
therefore
Var i a b l es or e n t r i e s of t h e "vect o r - mat r i x "
I . Poisson's Law 2 . The mean ;\ 3. The number of cells, rows, and columns The distributions entered in this matrix ar e not always rigorousl y d efined . They really depend, for a given A, on the number of r o w s or col umns. The greater the n umber of ro ws o r columns, the more rigorous is the definiti on. This is the law of l arge numbers. B u t this ind eterminism allows free will if the artistic i n spiration wishes i t . It is a se con d door that is open
to the subj ectivism of t h e com p oser, t he first being the " state of entry " of
the
" Vector-Matrix "
defined above .
specify the unit-events, whose frequencies were adjusted in the standard matrix ( M ) . We shall take as a single event a cloud of sounds with linear density 8 sounds/sec. Ten sounds/sec is abo ut the limit th a t a normal o rche s t ra can play. We shall ch oos e S 5 sounds/measure at MM 26, so that 8 2 . 2 soundsfsec ( � 1 0/4) . We shall now set out the following correspondence : Now we must
=
=
Formalized
32 Cloud of density S
Ev e n t
Sounds/ measure 26MM
zero
0
5
single d ouble
I
(15
2.2 4.4 6.6 8.8
20
sec) 0
0
15
qu ad ru pl e
Mean number of sounds/cell
sec
10
triple
=
Sounds/
Music
32.5 65
97.5
1 30
The h atchings in matrix (M) show a Po i sson distribution of frequencies, that
homogeneous and verified in terms of rows and columns. We notice the
rows are
columns. This
interchan geable leads
us
(
=
interchan geable
timbres) . So are the
to admit that the determinism of this matrix is weak
basis for though t-for thought wh ich of all kinds. The true work of molding sound consists of distributi ng the clouds in the two-dimensional space of the matrix, and of anticipating a priori all the sonic encounters before th e calculation of details, eliminating prejudicial positions. It is a work of
in part, and that it serves chiefly as a manipulates frequencies of events
patient research which exploits all the creative faculties instantaneously. This matrix is like a game of chess for a single p l a ye r
who must follow certain rules of the game for a prize for which he himself is the judge. This game matrix h as no unique strategy. It is not even possible to disentangle any balanced goal s . It is very general and incalculable by pure reason. Up to this point we have pl aced the c loud densities in the matrix. Now with the aid of calculation we must proceed to the coordination of aleatory sonic elements. HYPOTHESES
OF
the
CALCULATION
a s a n example cell III, t z o f the matrix : third row, continuous variation (string glissandi) , seventeenth unit of time (measures 1 03-l l ) . The density of the sounds is 4.5 sounds/measure at MM 26 ( S 4.5) ; so that 4.5 sounds/measure times 6.5 me as u re s 29 Let us analyze
sounds of
=
=
for this cell. How shall we place the 2 9 glissando sounds in this cell ? Hypothesis 1. The acoustic characteristic of the glissando sound is
sounds
assimilated to the speed
v
=
df{dt of
a
uniformly continuous movement.
(See Fig. I- 1 0.) Hypothesis 2. The quadratic mean a: of all the possible values of v is proportional to the sonic density S. In this case a: 3 . 38 (temperature) . Hypothesis 3. The values of these speeds are distributed according to the most complete asymmetry (ch an c e) . This distribution follows the law of =
33
Free Stochastic Music
Fig .
1-1 0
Gauss. The probability f(v)
function
for
f(v)
the existence of the speed v i s given by
=
Q�'/T
r v�/a� ;
and the probability P (>.) that v will lie between
8(>.) Hypothesis
=
:'IT f'
4. A glissando s o u n d
moment of its departure ; b. its spe e d register.
Hypothesis
is
vm
the
r .,.,�
v1
( normal
d)..
essen tially =
an d v 2 , by the fu nction
distribution) .
characterized by
tifldt, (v1
<
Vm
<
a.
v2) ; and
the
c. i t s
Assimilate time to a l in e and make each moment of that line. It is as if one were to distri bute a number of points on a line with a linear density S 4.5 poin t s at MM 26. This, then, is a problem of continuous probabilities. These p oin ts define segments and the probability th at the i-th segment will have a length x1 between 5;
departu re a point
on
=
x
and
x
+
=
dx is
Hypothesis 6. The moment of d epar tu re corresponds to a so u n d . We shall attempt to define its pitch. The strings have a range of a b o u t 80 semi tones, which may be rep resented by a line of length a 80 semi tones. Since between two successive or simultaneous g l iss a n d i there e x i sts an interval between the pitches at the moments of departure, we can define not only the note of attack for the first glissando, but also the m el od i c interval which separates the two origins . =
Formalized Music
34 s
Put
problem consists of finding the probability that a segment segment of l ength a will have a length between j and j + dj This probability is given by the formula
thus, the
within a l i n e
(0
:::;:
j
:::;:
a) .
O (j)
Hyp othesis
dj
=
� (I - �)
(See Appendix
dj .
7 . The three essential ch aracteristics o f
defined in Hypothesis 4
are
the glissando
independent.
we
1.)
sound
draw up the three tables of probability : and a table of intervals . All these t ables fu r n is h us with the e l e m e n t s which materialize in cell I I I , t z . The reader is encouraged to examine the scor� to see how the results of the calculations have been used. Here also, may we emphasize, a great liberty of choice i s g i v en the com poser. The res t ri c t io n s are more of a general can alizing kind, rather than peremptory . The theory and the calculation define the tendencies of the sonic entity , but they do not con stitute a slavery . Mathematical formulae arc thus tamed and subjugated by musical tho u gh t . We have given this example of glissando sounds because it contains all the problems of stoch asti c music, con trolled, up to a certain point, by calculation.
a
From these hypotheses
table
can
of du rations, a table of s p e e d s ,
Table of Durations Unit
4.5 X
·
8
=
x =
6.5
=
Sx
4.5 sounds/measure at MM 2 6
0. 1 0
of t h e
e - 6x
at 2 6 M M i.e., 28 durations
measure
2 9 sounds/cell,
se - 6,.
se - 6X dx
28P..,
0.00
0 . 00
1 .000
4 . 5 00
0 . 36 2
10
0. 1 0
0.45
0.638
2.870
0.23 1
7
0 .9 0
0 . 40 7
1 .830
0. 148
1 .35
0.259
1 . 1 65
0.094
0.40
1 .80
0 . 1 65
0 . 743
0.060
0.50
2 . 25
0 . 1 05
0.473
0.60
2.70
0.067
0.302
0. 7 0
3.15
0.043
0.20 0.30
Totals
0 . 03 8
4
3
2
0.024
1
0. 1 94
0.0 1 6
0
1 2 .41 5
0.973
28
Free S tochastic
35
Musi c
An approximation is made b y considering
dx a s a constant factor.
co
L 8e - 6x dx 0
Therefore
dx I n this case dx
1 / 1 2.4 1 5
=
=
=
= I.
00
I /2. 8e - 6x .
0.805.
0
Table of Speeds
4.5 glissando sounds/measure at 26 MM 3.88, quadratic m e a n of the speeds v is expressed in scmitonesfmeasure at 2 6 MM Vm is the mean speed (v1 + v2) /2 29 glissando sounds/cell. 4.5 6.5 8
=
a =
·
=
v
A = vfa
O(A)
0
0.000
0.0000
0.258 2
0.5 1 6
P(A)
=
0(A2)
-
0(A1)
29
P(A)
0.2869
9
0.5
0.25 1 0
7
1 .5
0. 1 859
5
2.5
0. 1 3 1 0
4
3 .5
0.077 1
2
4. 5
0.0397
1
5.5
0. 0 1 7 9
1
6.5
0.007 1
0
7.5
0.2869 0.5379
3
0. 773
0.7238
4
1 .032
0.8548
5
1 .228
0 .93 1 9
6
1 .545
0.97 1 6
7
1 . 805
0 .9895
Vm
Formalized Music
36 Table S
a
=
=
of Intervals
4.5 glissandi/measure at 26 MM. 80 semitones, or 1 8 times the arbitrary unit
of 4.5 semi tones.
j is ex p ressed in multiples of 4.5 semitones. dj is considered to be constant. Therefore dj 1 /l.O(j) or dj and we obtain a step function. Forj 0, B(j ) dj 2/(m + l ) j 18, O(j) dj 0 . =
=
=
=
=
af (m
=
+ 1 ),
0. 1 0 5 ; for
=
4 . 5 6.5 ·
=
29 glissando sounds p e r cell.
We can construct the
table of probabilities by means of a straight O(j) dj
j
=
P(j )
29
P(j )
2 3 4
3 3 3 3 2
6
2
8
2
5
2
7
2 2
9
10
1
11 12
1 1
13
1
14
0
15
16
0
17
0 0
18 We shall not speak
line.
of the means of verification of liaisons and correla used . It would be too long , complex, and
tions between the various values
tedious. For the moment the two formulae :
let us affirm that the basic matrix was verified by
Free
37
Stoch astic Music
an d z
=
r
t log � 1 +
us now imagine music c om posed with the aid of m atrix ( M ) . A n who p e rce i v e d the frequen cies of events of t h e m u s i c a l sample would ded u ce a distribution due to ch ance and following the laws of probability. Now the qu e s ti on is, w h en heard a n u mber of times, w i l l this music keep its su rp ri s e effect ? Will i t not c h a ng e i n t o a set of foreseeable p h en omen a throu gh the existence of mem ory, despite the fac t that the law Let
observer
of ch ance ? o n ly at the first
of freq uencies has been derived from the l aws
In fact, the data will appear al e a to ry
d u ring successive reh e a r in gs th e
relations
hearing. T h e n ,
o f the sample o rd ai n ed by " chan ce " will form a network, which will take on a definite m e a n i n g in the mind of th e listener, and will initiate a special " lo g ic," a new cohesion capable of satisfying his in t el le c t as well as his a e s t h et i c sense ; that is, if the ar ti st has a certain flair. If, on the other hand, we wish the s a m p l e to be u n foreseeable at all times, it is possible to conceive that at each repetition c e r t a i n data m i gh t be tran sfo rm e d in such a w ay that their deviations from theoretical fre quencies would not be sign ificant. P e r h a ps a programming useful for a first, second, third , etc. , performance will give aleatory s am ple s tha t are not i den ti cal in an absolute sense, whose deviations will also be distri b u ted by chance. Or agai n a system with el e c tro ni c computers might p e r mi t variations of the parameters of e n tr anc e to the matrix and of the c l ou ds , under certain co nditi o n s . There would thus a rise a music which can be di s to r t e d in t h e co urse of t i m e , giving th e same observer n r es u l ts apparently due to chance for n performances. In the l on g run the m u s i c will fol l o w the laws of pro b a bility and the performances will be statistically iden tical with each other, the i d e n t i ty being defined o n c e for all by the " vector-matrix." T h e sonic scheme d e fi n e d under this form of vecto r-matrix is co nse quently capable of establishing a m or e or less self-determined regulation of the rare son i c events contained in a musical comp o si t i o n sample. It repre sents a comp ositional a t ti t u d e, a fu ndamentally stochastic b eh a vi o r, a unity [ 1 956-5 7] . of s u pe ri o r order.
be tw e e n t h e
events
If the first steps may be summarized by the process vision -+ rules -+ the question concerning the minimum has produced an inverse
works of art,
Formalized Music
38
path : rules ---+ vision.
example of Achorripsis
In
fa c t stochastics perm i ts a phi losoph i c v isi on , as th e
bears witness.
C H A N C E-I M P R OVI S ATI O N
the essence o f m u si ca l composition, we speak of the principle of improvisation which caused a furore among the nco-serialists, and which gives them the r i gh t , or so t h e y think, to s p eak
Before
generalizi ng further o n
must
the ale atory, which they thus i n t ro du ce i n to music. They which certain combinatio ns of sounds m ay be freely chosen in terpreter. It is evident that these composers consider the various
of chance, of
write
scores i n
by the
p o ss i b l e circuits as equivalent. Two l og i c a l infirmities
are
deny them
and " com p osition "
on
to
the righ t
to
speak of chance on
the other ( co m p osi tion in
the
one h and
apparent which
the broad se n s e , that is) ; 1 . The interp ret e r is a highly con di tio n e d being, so that it is not possible
of unconditioned choice, of an i n terp reter a c t ing like a roulette game. The martingale betting at Mon te C a r l o and t h e p ro c ess i on of suicides should c o n v in c e anyone o f th i s . We s h all return to this. accept
the
thesis
2 . The composer commits an act of resignation when he admits several possible and equivalent circu i ts . In the name of a " scheme " the problem of choice is betrayed, and it is the interpreter who is promoted to the rank of co m p o s e r by the co mpo ser himself. There is thus a substitu tion of authors. The extremist extensi o n of th is atti tude is one which uses g raphical si g n s o n a piece of paper wh ich the interpreter r e a ds while improvising the whole. The two infirmities m e n t i o ned above are terribly aggravated here. I would like to pose a question : If this sheet of paper is put before an inter preter who is an incomparable expert on Chopin, will the result not be modul ated by the style and writing of C h o p in in the same way that a per former who is immersed in t h is style might improvise a Chopin-like cadenza to a no th er composer's concerto ? From the point of view of the composer there is no interest. On the contrary, two conclusions may be draw n : first, that seri al composition h as b e co m e so banal that it can be im provised l i k e Ch opin's, wh ich confirms the general impressio n ; and se c o n d , that the composer resigns his fu nction altogether, that he has nothing to say, and that his function can be taken over by pai ntings or by cuneiform glyphs. Chance needs to b e calcu l ated
To
finish with the thesis of the roulette-musician, I shall add this : a rare th i n g and a sn are. It can be construc ted up to a certain
Ch ance is
39
Free Stochastic Music point with great difficulty, by m ea n s of complex reasoning which is
sum
marized in mathematical formulae ; it can be constructed a little, but never
or intellectually imitated. I refer to the demonstration of the mathemati cian Emile Borel, who was one of the specialists in the calculus of probabil ities. In any case-to play with sounds like dice-what a truly simplistic activity ! But on c e one h as emerged from this primary field of chance worth less to a musician, the calculation of the aleatory, that is to say stochastics, guarantees fi rst that in a region of precise definition slips will not be made, and then furnishes a powerful m e th o d of reasoning and enrichment of s o n i c improvised
impossibility of i m ita t i n g chance which was made by the great
processes.
STO C H AST I C PAI N TI N G ?
In line with these ideas, Michel Philippot introduced the calculus of probabilities into his p ain t i n g several years ago, thus opening new direc
tions for investigation in this artistic realm. I n music he recen tly endeavored
to an alyze the act of composition in the form of a .flow
chart for an imaginary to a chain of a l ea to ry or deterministic events, and is based on th e work Comp osi tion p our double orcheslre ( 1 960) . The term imaginary machine means that the composer may rigorously define the entities and operating methods, just as on an electronic computer . In 1 960 Philippot commen ted on h i s Composition
machine.
It is a fundamental analysis of volu n tary choice, which leads
pour double orchestre : If,
in
co n n e c t i o n
with this work , I happ ene d to usc t h e
term in
"experimental music , " I s h ou l d specify in w hat sense it was mea n t
this p articular case. It has noth i n g to do w i t h
c o ncr e te
or electronic
music, but with a very banal sc o re w ri tten on the usual ru le d p aper
an d requiring
none
but the most trad i tional orchestral instrume n t s .
the experimen t of w h i ch this com p osi t ion was in s om e sense a by-product does exist ( and on e can think of man y industries t h a t survive onl y through the exploitation of their b y - pro d uct s ) . The end sought was m e rel y to effect, in the con text of a work wh i c h I would have writ ten independent of all expe ri m e n t a l ambi tio ns , an exploration of the processes fol lowe d by my own cerebral mech anism as it arranged the sonic elements. I therefore devised the However,
fol lowing steps :
1 . Make t h e mos t complete inventor y possible of the set of my gestu res , ideas, mannerisms, decisions, and choi ces , e t c . , which were mine when I wrote the music.
Formalized Music
40
2. Reduce this set to a succession of simple decisions, binary,
if
possible ; i . e . , accept or refuse a particular note, duration , or silence in a situation determined an d defined by the context on one hand, and
by the conditioning to which I h ad been subjected an d my personal tastes on the other.
3. Establish, i f possi ble, from this sequence of simple decisions, a
scheme ordered according to t h e fol lowing two considerations ( which
were som etimes con trad ictory ) : the manner i n which these decisions
emerged from my imagination in the cou rse of the work , and the
manner in w h ich they would have to emerge in order to be most
usefu l .
4. Present t h i s scheme in t h e form of a flow chart containing the
of these
logical chain
be
con trolled.
decisions, the operation of whi�h could easily
5 . Set in motion a mechanism of simulation respecting the rules
of the game in the flow chart and note the resul t .
6. Compare this resul t with m y musical in tentions.
7. Check the diffe rences be tween result and intentions, detect their
causes, and co r re c t the operating rules.
8. Refer these corrections back to the sequence of exp erimental
phases, i . e . ,
start
again a t
1.
tained.
If
u n t i l a satisfactor y resu l t h as been ob
we confine ourselves to the most general considerations, it
woul d simply be
a
matter of proceeding to an
ity, considered as an accumu l ation, in
a
a n al ysi s
certai n o rd e r,
of the complex
of single events,
and then of reconstructing this complexity, at the same time verifying the nature of the elements and their rules of combination . A cursory
look at the flow chart of the first mere glance the
me
th o d I
movement
specifies quite well by a
use d . But to confine oneself to this first
movemen t wou ld be to misunderstand the essentials of musical composition .
In fact the "preludial" character which emerges from this
combination of notes
(elementary constituents of the orchestra)
should remind us of the fact that composition i n its ultimate stage is
also an assembly of groups of notes, motifs, or themes and their transformations. Con sequen tly the task revealed by the flow charts of
the fol lowing movemen ts ough t to make conspicuous a gro u pi ng of
a higher order, in which the da ta of the first movement were used as
a
a
sor t of " p refabricated" material. Thus appeared the phenomenon ,
rather banal one, of au togen eration of complexity by jux taposition
an d combination of a large number of single events an d operations.
A t the end of this experiment I possessed at mos t some insigh t in to
my ow n musical tastes, but to me, the obviously interesting aspect of
Store in me mory
I nitial note +
d uration +
)t i
I I I
H orizontal or
I I I I
vertical
v
and i n te nsity
.----- � ::)_ n? ,_;;---
------ · -
---
I I yes --- - - ---
l
in c
�1
Note
- --
I
I I
Store .m no
note
_
yes
Start
no
�----'-� total 1 2 7
( Is the memory
I
)':::;:----'
next cell
Fig . 1-1 1 . Composition for Double Orchestra, by M i c hel Philip pot, 1 959 Flow Chart of the First Movement
-l
�
Choose i n itial
_j-E-----+--' ---------, -I
1�--D ra_ w_nex _ t_ not�
f Verify if \.._f_ alse relatio n
1r
yes
'� as previously and verify
�
no-
Formalized Music
42
i t (as lon g as there is no error of omission ! ) was the analysis of the
composer,
his men tal processes, and
imagination .
The b iggest difficulty encountered
a
certai n
was
liberation
of the
that of a conscious and
vol u ntary split i n personality. On one han d , was the composer w ho already had
a
cle ar idea and a precise audi tion of the work he wished
to obtain ; and on the other was the experimenter who had to maintain a l u ci d ity which rapidly
became
burdensome i n these conditions-a
lucidity with resp ect to his own ges tu res and d ecisions. We must not
ignore the fact that such experiments must be examined with the
greatest prudence, for everyone knows that no observation o f a phenomenon exists w hich does not disturb that phenomenon, and I
fear that the resul ting disturbance migh t be particularly strong
when
it concerns s uch an ill-defined domain and such a delicate activity.
Moreover, in t h is particular case , I fear that observation migh t pro
voke i t s own disturbance. I f I accepted this risk, I did not under
estimate its exte n t . At most, my ambi tion confined itself to the attempt
to proj ect on a m a r ve l o u s un know n , that of aesthetic creat ion, the
timid light of a dark lantern. (The dark lan tern had the repu tation
of bein g used especially by housebreakers . On several occasions I have been able to verify how much my thirst for investigation has mad e
me
appear in the eyes of the majority as a dangerous housebreaker of
inspi ration . )
C h a pter I I
Markovian Stochastic Music-Theory Now we can rapidly generalize the study of musical composition with the aid of stochastics. The first thesis is that stochastics is valuable n ot only in instrumental music, but a l so in electrom agnetic music. We h ave demonstrated this with several works : Diamorphoses 1 95 7-58 (B.A.M. Paris) , Concret PH (in lhe Philips P a v i l i o n at the Brussels Exh i bition , 1 958) ; and Orient- Occident, music for the film of the same name by E . Fulch i gnoni, produced b y UNESCO in 1 960. The second t h esi s is that stochastics can lead to the creation of new sonic materials and to new fo r ms . For th i s p u rpose we must as a preamble put forward a temporary hypothesis which concerns the n a t u r e of so u n d of
all
,
sound [ 1 9] .
B A S I C T E M P O R A RY H Y P OTH ESIS
( lemma )
All so u n d i s a n integr ation o f grains,
. A N D D E F I N IT I O N S
o f elementary sonic particles, of these elementary grains has a th ree fol d nature : duration, freque ncy, and i n tensity . 1 A l l soun d , even al l continuous sonic vari atio n , is conceived as an assemblage of a l a r g e number of elemen tary grains adeq u atel y d i s p o s e d i n time. So every sonic complex can be analyzed as a s e ri es of pure sin usoidal so u n d s even i f the vari ations of these sinusoidal sounds are i n fin i tely close, short, and complex. In the attack, body, and decl ine of a complex sound, t h o u s a n d s of pure s o u n d s appear in a m o re or less short interval of time, 6-t. Hecatombs of pure sounds arc necessary for the cre at i on of a complex s o u n d . A complex s o u n d m ay be i m agined as a m u l ti-colored fi rewo r k in which each poi n t of l igh t appears and i nstansonic quanta. Each of
43
44
Formal ized Music
t an eo u sly d isapp ea rs
ag a in s t
a
black
sky. But in
th is firework there would
be such a quantity of points of l i gh t org anized i n such
a
way that their rapid
teeming s u cce s si on would create forms a nd sp i r als, slowly unfolding, or conversely, br i e f explosions s e tti n g the whole sky aflame. A line oflight would and
b e created by
a
sufficiently large multitude of p o i nts ap p ea ri n g and dis
appearing instantaneously.
d u ratio n t!. t of th e grain as quite sm all but invariable, follows and consider frequency and in te nsity only. The two physica l substances of a soun d are fre q u e n cy and i n te n sity in a ss o c ia ti o n . They constitute two sets, F and G, inde p endent by their nature. They have a set p roduct F x G, which is the elementary grain of s o un d . Set F can be put in an y kind of correspondence with G : man y-valued, s in g l e valued, o n e- t o - on e mapping, . . . . The corres p ondence can be given by an If we consider the
we can i gnore
it
in what
extensive representation, a matrix representation,
or
tation.
EXAMPLES
OF
Extensive
R E P R E SENTATI O N S
(term by term ) :
fl f2 fa f4 gn g3 g,.
l g3
Frequencies
I nte nsi t i e s
Matrix (in
the form of a
.j.
gl
Canonical
a canonical represen
(in
table ) :
fl f2 fa f4 f5 fa f7 +
0
+
0
0
0
g2
+
0
+
0
0
0
+
0
g3
0
0
0
+
+
0
0
the form
of a function) :
vf = Kg f frequency =
g
K
=
intensity
=
coefficient.
The corres p ondence may also be indeterminate (stoch astic) , and here
the most
co n ven i e n t representation is the matrical one, which gives the
t ra nsi ti o n probabilities.
45
Markovian Stoch astic Music-Theory Exam ple :
t fl g2 0 ga
The t abl e
0.3
0.5
0
0.3
0.7
be interpreted as
should
0.2
0
0.5
gl
!4
fa
!2
0.5
0
fo l l o w s : for
one or several corresponding i n t en s ity values
For example, the two in tensi ties
g2
each valu e }; off there
g1,
defined by
a
are
probability.
and g 3 corres p ond to the frequency /2,
with 30% and 70% chance of occurrence, respectively. On the other hand,
each of the two sets F and
G
can be fu rnished with a structure-that is to
say, inte rnal relations and laws of composition. Time
t is considered
lexicographic form.
as a to t a l ly ordered set mapped onto F or G in a
Examp les :
a . !1 !2 fa t
b.
t
1 , 2,
=
c.
t
=
fo.5 fa ].;u fx =
0.5, 3 , y l l ,
ft fl 12 fl 12 f2 J,. fa · " · · · " · A B c D E �� ��
� � �� ��
x,
At At
�� = At Example
into sli c e s
makes it
much
GRAP HICAL We
c.
is the most general since continuous e vo l u tio n is sectioned
of a single thickness At,
can
which
t ra nsforms it in d i scon tinuity ; this
easier to isolate and examine
R E P R E SENTATIONS
plot
u nder
the magnifying glass .
the valu es o f pure freq uencies i n units o f octaves o r semi axis, and the intensity values in decibels on the ord i n ate
tones on t h e abscissa
axis, using log ari th m i c
scales (see
Fig. II-1 ) . This cloud of points is the
cylindrical projection o n the plane (FG) of the grains contained in a t h i n
slice
The graphical representations Figs . I I-2 and I I-3 the ab s tr a c t p ossibilities ra i s e d up to this poi n t .
�� (see Fig. I l-2) .
m ake m o re tangible
Psychophys i o l ogy
We are confronted with
a
cloud of evolvi ng points. T h i s cloud
is
the
product of the two sets F and G in the slice of time �t. What are the possi blc
46
Formalized Music G
(dB)
Elemen tary grain considered as a n i nsta nta neous associ ation o f an i n te nsity g and a fre q u e ncy f . . .
. .
Fig.
1 1-1
Freq ue ncies i n log arith mic u n its (e.g., sem iton es) L-------------------:-
F
G
G
'
\
�----�--------------------��- F
Fig. I�
F
Fig.
1 1-3
'1.--- F
t
-<- Point i Each point i must be attributed to an i ntensity g1
·.: ...... ,
t
47
Markovi an Stochastic Mu sic-Theory
restrictive limits of h u m a n psychophysiology ? What are the most gen e r a l manipulations which may be i m p o sed on the tions within psychop hysiological limits ?
The
cl o u d s
and their tra n sfo rm a
basi c abstract hypo thesis, which is the granular construction of all
p o ss ible sou n ds, gives a very profound meaning to these two q u estions. I n
fact within h u m a n limits, u s i n g
ail
sorts
of m anip u l ations
with these grain
cl us te rs , we can hope to produce not only the s o u n d s of classical instruments and elastic bodies, and those so u nds generally p referred in concrete music, but also sonic perturbations with evolutions, unparalleled and u n i m aginable
u n til now. The basis of the timbre structures and transformations will have noth ing in co m mo n w i th what has b een known u n t i l now. We
can
even express
a m or e
general
p o i n t of these clu sters represents not only
a
intensity, but an already present s t ru c t u r e
su p p osition .
pu r e
S uppose that
frequency
of elemen tary
each
and its satellite
grains, orde re d a
priori. We believe that in this way a sonor i ty of a second, third, o r h i gher order can be p rod uced .
Recent work on hearing h a s given satisfactory answers to certain
problems o f pe rcep tion . The basic p roblems which concern us and which
we shall suppose to be resolved , eve n if so m e of the solutions l a c k i n g,
are
[2, 3] :
fort) o f a sinusoidal sou nd, as a fu nction o f i ts frequency and
2. What arc the
are
in part
com its intensity ?
1 . What is the minimum pe rce p tible duration (in
m i n imum values
of inte nsities
in deci bels compatible with
minimum frequ encies and du rations of sinusoidal so u n ds ?
3.
What
are
the
minimum melod ic i n te rval thresholds, as a fu nction of register, intensity, and d u ra ti o n ? A good
e
q u al l o u d n ess
car
approximation is the
conto urs (see Fig. I I-4) .
Fletch er-Munson d i agram
The total n u mber of elemen tary audible grains is a b o u t
is more sensi tive at the center o f the aud i b l e
are a .
340,000.
of
The
At th e extre m i ties i t
pe rceives less a mplitu d e and fewer melodic i n tervals, s o that i f one wished to
represent the
audible area i n
a
homogeneous manner using the coordin ates
F and G, i . e . , with each s u rface element
of g r a in s of perce p tible so u n ds, o ne (Fig. II-5 ) .
6.Ft1G contai n i n g the same density would o b t a i n a sort o f mappa mundi
In order t o simplify the reasoning which will fol lo w without al tering it,
we shall base o u r arg u m e n t o n Fletcher's diagram a n d s u ppose that an a pprop r i a t e
one-to-one trans formation applied to this group of c o ordinates
will change t h i s curved space All the
above
into
an ord i n ary rectangle
(Fig. II-6) .
ex p erimen tal resu l ts were established in id e a l co n d i tions
and without re ference to
the actual
com plexity o f the n atural s o u n ds of the
orchestra and of clastic bodies i n gen eral, not to mention the more com plex
Formalized Music
48 G
+ Limit of the
90
�
�
80 7a Ga
-- - . -
sD
------.....__
r-----
�
�
��I'-"""
·-- -
-MoO O
·-·
r--r- -
"' I'-
r--.-
!'--
��
--
,_
..
-
�....____ ----
�
� '-,
'"-�
---
""'-
·-·-
3D
:----
�
40
Go
-'"'
�� � """
Contours
.._ Th resh old of sou nd perception
F i g . 1 1-5
........___
r=::-
Fig . 1 1-4. Fletcher- M u ns o n D iagram
F
"•'------
(Semitones)
0
/
1)1
v( VI
----....
r----,......__....
J
/_
/ /
I
--
/
---
-- ./
-
8o fDD
Equal Lo ud ness
'·"---
'-----
Audible area
G (d 8)
-
-....___
"'-�
:So 2<1
r--
------- ......___ I /v-
I-
-
�
audible area
· ·- · ·· --
{/0
/
/"'"
/
lloe>O
F
49
Markovian Stochastic Musi c-Theor y
-
--....
� I
F'
sounds of industry or of chaotic n ature [4] .
Theoretically [5]
I
a complex
sound can only be exhaustively represented on a three-di mensional diagram F, G, t, giving the instantaneous frequency and in t e nsi ty as a function of time . But in practice this boils down to saying that in
m o m ent a ry sound,
such as a simple noise made
tions and graphs are necessary. This
order
to represen t a
months of calcula impasse is strikingly re m i n is c e n t of by a car,
classi c al mech anics, which claimed that, given sufficient ti m e, it could
accoun t for all physical and even biological phenomena us i n g only a few formulae. But just to describe the state of a gaseous m ass of greatly re duced volume at one instant t, even if simplifications are allowed at the beginning of the calculation, would require several centuries of human work ! This was a fal se problem b e c a u se it is useless ; and as far as gaseous masse s are concerned, the Maxwell-Boltzmann k in et i c theory of gas es, with its statistical method, has been very fru i tfu l [6] . This method re established the value of scales of observation . For a macrosco p ic phenomenon it is the massed total result wh i ch counts, and each time
a
p h e n o m e no n is
to be observed the scale rel a tionship between observer and phenomenon
must first be established. Thus if we ob s e r ve galactic masses, we must decide
whether it is the movement of the whole mass, the movement of a s i n gle sta r,
the mo l e cu l ar constitution of a minute region on a star that interests us. The same thing holds true for c o mpl ex as well as quite simple sou nds. It would be a waste of effort to a tte mpt to accou n t analytically or graph i cally for the characteristics of complex sounds when they arc to be used in
or
an electromagnetic c o m p osi tio n
.
For t h e manipulation of these sou n d s
m acroscopic m e th o ds are necessary.
I n vers e l y , and this is what p a rt i cul arl y interests us here, to work l i ke architects on the so n i c material in order to construct complex so u nds and
50
Formalized Music
evolutions of these entities means that we must use macroscopic methods o f analysis and construction. Microsounds and eleme n tary g r ai ns h av e
importance
on
no
the s ca l e which we have chosen . Only gro u p s of grains and
the characteristics of these groups have any meaning. N aturally in very particular cases, the single g ra i n will
be reestab lished
in all its glory.
In
a
Wilson chamber it is the elementary particle which carries theoretical and
experimental physics on its shoulders, while in the sun it is the mass of
p articles and their compact interactions which co nst i tu t e the
but
solar obj ect.
Our field of evolution is therefore the curved space described above,
simplified to a rectilinear space by means of complete one-to-one
transformation, which safegu ards the validity of the
re aso n ing which we
shall pursue.
S C R EE N S
The graph i cal representation o f a cloud o f grains i n a sl i c e o f time ll.t
a new concept, that of the density of grains per unit t1Ft1Gt1t (Fig. I I-7) . Every po ssib le sound may therefore be cut up into a precise quantity of el eme n ts !:iFt1Gt1tt1D i n fou r dimensions, distributed in this space and following certain ru les defining this sound,
examined earlier brings of volume,
which are summarized by a function with four variables : s(F, G, D, t) .
G
AF
Pla n e of
reference
{FG)
at moment t
/
F
tlD wi l l be the dimension of the density
t.
F i g . 1 1-7
51
Markovian Stochastic Music-Theory The scale of
the
density will also
be logarithmic with its base between
2 and 3 . 2 To simplify the expl a n ation we will make an abstraction of this
It will always be present in our mind but as an with the three-dimensional element 11FI1GI1t. If time is considered as a procedure of lexico g raphic or d eri n g we can,
n ew coordinate of density. entity associ ated
,
withou t loss, assume that the 11t are equal constants and quite small. We can
thus reason on a two-dimensional space defined by the axes F
condition that
we
and G, on
do not lose sight of the fact that the cloud of grains of
sound exists in the thickness of time
artificially flattened on the plane
Difinition of the
11t and that the grains of sound are only
(FG) .
by a as defined a b o v e the cells of which may or may not be occu pied by grains. In this way any sound and its history may be described by means of a sufficiently large number of sheets of paper carrying a g iven screen S. These sheets are placed in a fixed lexi screen. The screen is the audible area (FG) fixed
sufficien tly cl os e and homogeneous grid
,
cographic order (see Fig. I I-8) .
- S c reen
Cell full of grains
A book of scre e n s e q u a ls of a c o m p lex so u nd
Fig . 1 1-8
to
the l ife
The clouds of grains drawn on the screens will differ from one screen another by their geographical or topographical position and by their
surface density
(see Fig.
I I-9) . Screen A contains
a
small elemental rectan
gle with a small cluster of density d of mean frequencyf and
mean
amplitude
g. It is almost a pure sound. Screen B re p resents a more complex sound with
strong high
and low areas but
with a weak cen ter. Screen
C
represents
a
Formalized Music
52 " white "
sound of we a k density which may
therefore be p er ce i v ed as a
sheen occupying the whole audible area.
has
What is important in all the statem e n ts made up to now is that
sonic
nothing
b ee n said about th e topographic fixity of the grains on the screens . All
of small surface ele m en ts filled intensity. The is fundamen tal, and it is very
natural or instrumental sou nds arc com p osed
with g r a in s which fluctuate arou nd a mean frequency and
same holds for the density. This statement
of e l ec tronic music to c re a te n ew timbres, aside from inadeq uacy of the serial method, is largely due to the fixity of the grains, which form structures like packets of sp agh e tti (Fig. I I- 1 0) . Topographic fixity of the grains is a v ery p ar t i cu l ar case, the most ge n eral case being mobility and the statistical distributim• of gr ai n s around positions of equilibrium. Conse q uently in the majority of cases real sounds can be an alyzed as quite small rectangles, !:l.F!:l.G, in which the topographic positions and the d e n si ties vary from one screen to another following more
likely that the fail u re the
or
less well-defined laws .
Thus the sound of example
collection of rectangl es
D
at this precise instant is formed by
the
(j2g4 ) , (f2g5) , (f4g2) , (f4ga), (fsgt ) , (f5g2) , Uagl) ,
(fag2) , Usgs) , (f1g2) , (f7g3 ), (f7g4), (f7gs) , Uag3), (feg4), (fags) , an d i n e ac h
of the rectangles
manner
the grains are disposed
( se e Fig. 11-1 1 ) .
CONSTRUCTION
OF
TH E
E LEMENTS
in an asymmetric and homogeneous
fj.p[j. G
OF
THE
S C REENS
I . By calculation. W e shall examine the means o fcalculating the elements
!:l.F!:l.Gat/j.D.
How shou ld the g ra i n s
fix the
be
d istributed in an elemental volume ?
mean density of the grains
(
=
number of grains
If we
per unit of volume)
we h ave to resol v e a problem of p robab i l i ty in a four-dimensional space . A simpl er method would be independently. lS :
For
For the
to co n sid er and then calculate the four coordinates
coordinate t the law of distribu tion
P"
=
c e - c" dx
or
Px,
=
of gr ain s on
e- •1v ctu1•
the coordinates G , F, D t h e s to ch a s ti c l a w w i l l
or
2
pi
= 2 . t on
( l
i
2 . wn
(r)
the axis of time
(See Appendix i . )
be :
- 1) .
(Sec Appendix I . )
53
Markovian Stochastic Music-Theory G
G
· :·. ·
.. .. . .
( - - ---- -·
I
G
F
A
Fig. ll-9
.. . .: · . ·-:·:
F
8
::
·.
:
c
G
Fig. 1 1-1 0
�------�� t
G
(dB}
(s
ft
t3
t'L (,
. .. :� : .. ..
/, h h
•
I�
D
F i g . 1 1-1 1
h-
h
�
I',
-
·•
=.: :
-
••
r1
F
(Semitones)
F
Form alized
54
Music
F rom th ese formu l ae we can draw u p tables of the frequencies of the G, F, D (see the a n al ogo u s pr o bl e m i n C h apter 1) . These formulae are in our opinion privileged, for they arise from very simple reasoning, prob ably the very simplest ; and it is essential to start o u t with a m i n imum of t er ms and constraints if we wish to keep to the principle of the tabula rasa ( 1 st and 3rd ru l es of Descartes's Discou rse o n Method) . Let there be o ne of these elemental v ol u m e s /!:,ti:!.DAF!':!.G o f th e screen at the m o men t t. This vo l u m e has a density D taken from the t abl e derived from form u l a (r' ) . Points on !':!.t are d efined with a linear density D c according to the tab le defined by fo r m u l a (r) . To each po i n t is attributed a sonic grain of frequency f and i n te ns i t y g, taken from within the rectangle !':!.F!:,G by means of the table of frequencies derived from fS> rmul a (r') . The corres p ondences are made graphically o r by random successive drawings from urns composed according to t he above ta ble s. 2. Mechanically. a. On the tape recorder : The grains are realized from sinusoidal so u n d s whose durations ar e constant, about 0.04 sec. These grains must co ver the selected elemental area !':!. F !':!.G. U nfo ld i ng i n time is accomplished by using the table of durations for a minimum density c D.
values t,
=
B y mi xi n g sections o f this tape with itsel f, w e geometrically with
we use.
b. On
programmed
ratio
I : 2 : 3 . . . according
can
to
=
obtain densities varying
the number of tracks
computers : The gr a i n s are realized from wave
a cco r d i n g
to Ga b o r ' s signals, for
a
that
forms duly
co mp u t er to which
analogue converter has been coupled. A second p rog r am would provide
an
for
the construction of the elemental volumes !':!.tAD!':!.FAG from formulae (r) and (r') . F i rst G e n e ra l Com ment Take the cell
!':!.F!':!.GAt. Although
occupied in a homogeneous
manner
by g ra i n s of sound, it v ar i es in time by fl uctuating around a mean density dm. We can
these
apply
another
arg um en t which is more synthetic, and admit that most general case anyhow (if tbe sound is
fluctuations will exist in the
long enough ) ,
and will therefore obey the laws of chance. In this case, the put in the following manner : Given a prismatic cloud o f grains o f density dm, o f cross s ec ti o n tl.Fll. G and length 2 !':!.t, what is t h e probability that d grai n s will be found in a n elemental vol ume !':!.F!':!.Gt3. t ? If the number d'" i s small enou g h, t h e prob abil i ty is given by Poisson's formula :
pro b le m is
P
�<
=
{dm)" e - d., . K!
Markovian S t oc h a sti c Music-Theory
For the definition of e ach
grain
scribed above .
55 we
s h all
again use the me tho d s de
Second Genera l C o mment (Vector S pace ) [8] We can construct e l e m e n tal cells tl.FD.. G
el em en tal vectors
po i nts , but w i th
of the screens
no t
only
with
associated with the grains ( v e c to r space ) .
The mean density of 0 . 04 sec/grain really im pl i e s a small vec tor . The p ar tic
ular
case
of the grain occu rs when the vector is p arallel to the axis of time,
whe n its projection on
the grain is constan t. can be variable
and
�
the
p l ane (FG) is a point,
and when the
frequency of
In general, the fre q uencies a nd intensities of the gr a i n
the
grain
a
very sho r t
glissando (see
s
Fig. 11- 1 2) .
A P
/
/
I
F
AF
F
\ t
Fig. 11-1 2
I n a v ec t o r space (FG) thus d e fi n e d , the construction of screens would e r haps be cumbersome, for it wou l d be necessary to introduce the idea of p speed and the statistical distribution of its values, but the interest in the undertaking wo u l d be enormous. We cou l d imagine screens as the basis of granular fields which are magnetized or completely ne u t r a l (disordered) . I n t h e case of total disorder, we can
the existence of a vector
app l ied to
two
v
c al c u l a t e
the probability f(v) of
o n the plane (FG) using Maxwell's fo rm ul a
d ime n si ons [ I I ] :
f( v)
as
Form alized
56
For the mean v al u e
in
which A1
=
v1
vm
:s;;
[1 3].
v2,
v1fa and
for A.1 :s;; A. :s;; A.2 (normal
matter of a
:s;;
Music
vector space
Gaussian
or
a
scalar
law) [ 1 2] . In any case, wheth er it
is
a
space does not modify the arguments
Su m mary of t h e Scree ns
I . A screen is described by a set of clouds th a t are themselves a set of rectan gles M!l.G, a nd which may or may not contain grains of sound. These conditions exist at the moment t in a slice of t i m e M, as small elemental
as desired.
2. The grains of sound create a density peculiar to each elemental !l.F!l.G and are generally d istribu ted in the rectangles in an er god i c manner. (The ergodic p ri nci p l e states that the capricious effect of an operation that depends on chance is regularized more and more as the rec ta n gl e
ope ra tio n is repeated . Here it is understood that screens
is
being considered [ 1 4] . )
3. The
a
very large succession of
volume !l.F!l.G!l. T!l.D is such that Simultaneity occurs when the de n sity is h i gh enough . Its fre q uency is bound up with the size of the density. It is all a question of scale and this paragraph refers above all to realization. The temporal dimension of the grain (vector) being of the order of0.04 sec., no sys tematic overlapping of two grains (vectors) will be accepted when the elementary density is, for ex amp le , D0 1 . 5 grains/sec. And as conception of the elemental
no simu ltaneity of grains is generally
a dmi tte d.
=
the surface distri bu tion of the grains is homogeneous, only chance can
c r e a te th is overlapping. 4.
The limit
for
a
screen
may be only one pure sound (si nusoidal) ,
or even no sound at all (empty scre e n) .
Let
ELE M E NTA R Y O P E RATI O N S O N S C R E E N S
t he re
be
a
vectors on
complex sound. A t
an
i n s t an t
t
of i t s l ife during
a
be re p resented by one or several clouds of grains or the plane (FG) . This is the definition which we gave fo r the
thickness !l.t it can
57
Markovian S toc h as tic Music-Theory
The j u nction of several of these screens in a given order describes of this sonic complex. It would be interesting to envisage in all its generality the m a n n e r of combining and juxtaposing screens to describe, and above all to construct, sonic evolutions, which may be con tinuous or discontinuous, with a -view to playing with them in a composition. To this end we shall borrow the terminology and symbolism of modern algebra, but in an elementary manner and as a form of introduction to a
screen .
or prescribes the life
we
fu rther development which
shall not undertake at the moment.
It does not matter whether we place ourselves on the plane
Comment :
of physical phenomena or of perception . In general, on the plane of per ception we consider arithmetically that which is ge om e t ri cal on the p hysical plane.
can be ex p ress ed in a m or e rigorous manner.
This
Perception almost isomorphic with a physical a multiplicative grou p . The " almost " is necessary to
constitutes an additive group
excitation
consti tuting
exorcise app roximations. Grains
or vectors on
composed of no grain at
which
is
the plane (FG) co ns t i tu t e
all
or
of several
a cloud. A screen can be or vectors (s e e
clouds of grains
Fig. II-1 3) .
@
B
. .
A
Screen
Screen 1
F i g . 1 1-1 3
To notate that a grain or vector
th e con trary is written
another
cloud
sub-cloud of
Y.
rj: E. I f
belongs to a clo u d
X c
c
c
included in Y or that X is is not ated X c Y (inclusion) . have the following p roper ti e s : for any X.
X Y
Y and Y
and
c
they are indistinguishable
Y
c
Z
imply
X
c
Z.
the re l a t io n i s written :
A clou d m ay contain as little as empty when it contains no grain
a,
a
part
or
X and Y consist of the same grai ns ; X Y (equatio n) . a single grai n . A cloud X is said to b e such that a E X. The empty cl o u d is
X, the clouds
and
3
E, we write a E E X are grains of
all the grains of a cloud
This relation
X
n o ta te d 0 .
a
Screen
2
Y, it is said that X is
Consequently we
When X
a
��
=
Formalized Music
58 E LEMENTARY
These
O P ERAT I O N S
o p e rati o ns app l y equally
well to
clouds and to scree n s . We
can
therefore use the terms " screen " and " cloud " indiscriminately, with cloud and grain as " constitutive elemen ts."
The intersection of two screens
A
an d B is the screen of clouds
b e lo n g to both A and B. This is notated as
A
which B"
II B and read as " A inter
(Fig. 1 1- 1 4) . W h e n A n B 0 , A and B are said to be disjoint (Fig. 11-1 5) . The union of two screens A and B is the set of clouds wh i ch belong to both A an d /o r B ( Fig. II -16) . The complement of a sc re e n A in re l a t i o n to a s cre e n E c o nt ai n ing A is the set of cl o u ds in E which d o not belong to =
A . This is
n o t a te d
(Fig.
1 1- 1 7 ) . The
(A
B)
CEA when t h e re is no possible u n c er t ai n ty about E difference (A - B) of A and B i s the set of clouds of A
B. The immediate co ns e q u e nce is A - B A B) (Fig. 11- 1 8 ) . stop this b o rr o win g here ; however, i t will afford a stro n g e r , conception on the whole, better adapted for the manipu
which do not belong to n
=
=
CA (A n
W e shall
more precise
l ations and arguments wh ich follow.
D I STI N CT I V E C H A R A CT E R I STI C S O F T H E SC R E E N S
I n our desire t o create sonic complexes from
the te m pora ry accepted
pr i m a ry matter of sound, sine waves (or their replacements o f the Gabor sort) , and to
th an
create
n atu ral s o u n d s
sonic co m pl exes as rich as b u t more
extraordinary very
(using scientifically controlled evolutions on
gen eral abstract p l a n e s ) ,
we h ave implicitly reco g nized the importance of three basic factors which seem to be able to d omin a te both the theoretical construction of a so n ic process and its sensory effectiveness : 1 . the d e nsi ty of the e le m e nt a ry e l e m en ts, 2. the topographic situation of events on the sc re en s, and 3. the order or disorder of events. At first sight then the density of g r ai n s or v e ctors , their topography,
and their degree of order
are
our m acrosco p ic cars. It is
the i ndirect enti ties and aspects perceived by
wonderful that the
ear and the m i n d
follow
objective reality and react directly in spite of g ross inherent or cultural imperfections. Measurement has been the foundation of the experimental
sciences. Man voluntarily treats himself as a s e n so r y invalid, and it is for this reason that he has armed himself,
justifiably,
wi th mach ines that measure
other mach ines. His cars and eyes do measure entities or physical phen o m en a , but they
are
transformed as
if a distorting filter c am e
between
im
mediate perception and consciousness . About a century ago the logarithmic
Markovian Stochastic Music-Theory
F i g . 11-1 5
59
A nB .. :··
� ,. .
·. :.·.\�) ' •"... -:.
U'
A
. : . : . .· . · .
E�:.�t
. . . . ....
.
A U S
Fig. ll-1 6
: )it�(:'' · ·.
" ¢
.
E F i g . ll-1 7
A
A F i g . ll-1 8
B
(A - B )
60
Formalized Music
sensation was discovered ; until now it has not been contradicted. knowledge never stops in its advance, tomorrow's science will with out doubt find not only a greater flexibility and exacti tude for th i s l aw, b u t also the begi n n i ngs of an explanation of this distorting filter, which is so astonishing. This sta t isti c a l , but none the less quasi-one-to-one transformation of excitation into perception has up to now allowed us to argue ab ou t physical entities, such as s creens, all the while thinking " perceived events . " A reciprocity of the same kind between perception and its com prehension permits u s to pass from the screens to the consequent distinctive characteris tics. Thus the arguments which we shall pursue apply equ ally well to pure concepts and to those re s ulting from perception or sensory events, and we may take the atti tu d e of the craftsman or the listener. We h ave al ready remarked on the density and the topography of grains and cells and we have acknowledged the concepts of orde r and dis ord e r in the homogeneous s u p e rfi c i a l distri bution or grains. We shall examine closely the co nc e p t of order, for it is probably hidden behind the o ther two. That is to say, density and topography are rather palpably simplifi ed embodiments of this fleeting and many-sided concept of disorder. When we speak of order or d i sord e r we imply first of all " obj ects " or " element s . " Then, and t hi s is already more complex, we define the very " elements " which we wish to study and from whi ch we wish to construct order or disorder, and their scale in r el a tio n to ours. Finally we qualify and endeavor to measure this order or disorder. We can even d raw up a list of all the orders and disorders of these entities on all scales, from all aspects, for all measu rements, even the characteristics of order or disorder of this very list, and establish anew asp ects and measurements. Take the example of the gases mentioned above . O n the molecular scale (and we could have descended to the atomic level) , the absolute values of th e speeds, directions, and distributions in spac e are of all sorts. We can distinguish the " elements " which carry order or disorder. Thus if we could theoretically isol ate the element " directio ns " and assume that t he re is an obligation to follow certain privileged directions and not all directions, we could impose a certain degree of order which would he i n d epende n t of the other elements constituting the co n cep t " gas." In the same way, given enough time, the values ol" the speeds of a single molecule will be distributed around a m e a n val u e and the size of the deviations will follow Gauss's law. Th ere we will have a certain order since these val u es are vastly more law of
But as
61
Markovian Stochastic Music-Th eory n u merous in the nei g hborhood o f their mean finitely small to infini tely large.
t h an
anywhere else, from i n
Let us take another exam p le, more obvious and equally true. A crowd pe rsons is assemb led in a town s q u a re . I f we e x a m i ne the group
of 500,000
d isplacement of this crowd we can prove that
i t does not budge. However,
each individual moves his limbs, his head, his eyes, and displ aces h is center a
of gravity by
terror
If the displacements of the would break u p with ye l l s of
few centimeters in every d i rection.
ce n t e rs of gravity were very large the c ro w d
because of the multiple collisions b et w e en individu als. The statistical
values of these displacements n o rm a l l y
l i e between
they affect immobility, the disorder is
weak.
very n arrow limits which
vary with the density of the c ro w d , From the point of view of these va l u e s as
is the orientation of the fa c es . If an a balcony were to spea k with a calming effect, 499,000 faces would look at the balcony and 998,000 ears wou l d listen to the honeyed words . A Another characteristic of the crowd
orator on
thousand or so faces and 2000 ears would be distracted for various reasons :
annoyance, ima g ination, sexu ality, contempt, theft, etc. We could confirm , along with the mass media, without any possible d i sp u te , that
fatigue,
crowd and speaker were in compl ete accord, that
500,00 1
people,
in
fact,
were unanimous. The d egree of ord e r that the s p e aker was after would attain a maximum
for
a
few minutes at least , and if u n anim i ty
were expressed be per
equ ally s trongly at the conclusion of the meeting, the orator could suaded
that the
own .
id ea s were as well ord e red
in
the heads
of the crowd
as in his
We c a n establish from these two extreme examples that the concept of order and disorder is basic to a very large n u m b er of phenomen a, and that even the d efin i t i on of a phenomenon or an object is very often attri b u table to
this concept. On the other hand,
fo unde d on
precise
we
can
establish that this concept is
and distinct groups of elements ; that
tant in the choice of elements ;
and
the
sc a l e is impor
finally, that the concept of order or dis
order im p li e s the re l a t i o n s h ip between e ffective values over all possible values
that the ele m e n ts of a group can
p ossess . This introduces the concept
of probability i n th e quanti tative estimate of order
or disorder.
We sh all call the number of distinct elements i n
a
grou p its variety. We
or disorder d e fi n a bl e i n a group of elements its entropy. Entropy is lin ked w i th the concept of variety, and for th a t very reason, it is linked to the probability of an eleme nt i n the group. These shall call the degree of order
concepts are those of the theory of communications, which itself borrows from the second law of thermodynamics
(Boltzmann's
th eorem
H) [ 1 5] .
Formalized Music
62
V ari e ty is expressed as a pure n u m b er or as its l og ar i t h m to Thus human sex has two elements, male and or
1 bit : 1
bit
=
L e t there
log2
be a
2.
female , and
the b ase 2 .
its variety i s 2 ,
probabilities (a grou p of r e a l numbers p, I ) . The entropy H of this group is defined as
group o f
positive or zero, whose
sum
is
if we is t, and the entropy of this sequence, i . e . , its uncertainty at e ac h th row, will be 1 bit. If both sides o f the coin were heads, the u ncertainty w o u l d b� removed and the entropy H would be zero . Let us suppose that th e advent of a h e a d or a tail is not controlled by tossi n g the coin, but by a fixed, u n iv oca l l aw, e . g . , heads at each eve n toss and tails at each odd toss . Uncertainty or disorder is always absent and the entropy is zero . If the law becomes very co m pl ex the appearance of heads or t a ils will seem to a h uman o bserver to be ru l e d by the law of chance, and disord er and uncertainty will be reestablished . What the observer could do would be to co u n t the appearan ces of heads and tails, add u p their respecti ve If the logarithmic b a se
is
2, the
entropy is expressed
in b i ts. Thus
have a sequence of heads and tails, the probabi lity of each
frequencies, deduce their probabilities, and then calculate the entropy in
bits. If the frequency of he ad s is equal to that of tails the u n certai nty will be eq u al to 1 bit. This ty p ic al example sh ows roughly the passage from order to disorder and the m e a n s of calibrating th is disorder so that it may be compared with o t he r states of disorder. It also shows the i m p o r t a n ce of scal e . The intelli
maximum and
gence of the observer would assimil ate a deterministic complexity up to certain limit. Beyond that, in his eyes, the complexi ty wou l d swi n g
over
a
i n to
unforcseeability and would become chance or disorder ; an d the visible (or m acro s c op i c would slide into the invisible ( or microscopic) . Other methods and points of view would be necessary t o observe and con trol the pheno mena. At the beginning of this ch apter we admitted th at the mind and es peci ally the ear we re very sensitive to the order or disorder of phenomena. The laws of perception and judgment are probably in a geo me t ric a l or logarith mic rel a tion to the laws of excitation. We do not know much about this, and we shall again co n fi ne ourselves to exami ning general e n t i t i e s and to tracing an overalt orientation of the poetic processes of a very ge ner al kind of music, wi thout givi n g figures, moduli, or determinisms. We are still o p timistic enough to think t h a t the interdependent e xp e ri m e n t and action of abstract
Markovian Stochastic
hypotheses
63
M usic-Theory
can cut biologically into the l ivin g conflict between i gn orance
and reality (if there
is any reality) .
S�udy of Ata xy ( o r d e r
or
d i so r d e r ) on t h e P l a n e of a
C l oud of G ra i n s or Vecto rs
Axis
of time : The
degree of ataxy, or the
e n t ro py, is a function of the the
simultaneity of the grai n s and of the distinct intervals of time between
If th e variety of the durations of the emissions is weak, e n tropy is also weak. If, for e x am pl e , in a g ive n l!.. t each gr ai n is emitted at regular intervals of time , the tem p o ra l variety will be 1 and the e ntropy zero. The cloud will have zero ataxy a n d will be c omp le tely ordered. Con v ers e ly, if in a fairly long succession of At the gr ai ns are emitted according to the law pX = ae - 6" dx, the degree of ataxy will be much larger. The limit of entropy is infinity, for we can i m agi n e all p oss ible values of time intervals with an equal probability. Th us, if the variety is n -+ oo, the probability for each time interval is p1 1 /n, and the entropy is emission of each grain . the
=
H H for
n
-+
=
ao,
-K
� I L. -
l=o
H -+ ao .
n
I log -
n
=
- K 'L, p; log pi n
=
i=O
I 1 - K n - log n
n
=
I - K log n
=
K l og n
l!.. t wil l never offer a very gre at variety of du r ati o ns and its entropy will be weak. Furthermore a sonic composition w i ll rarely have more than 1 00,000 l!..t 's, so that H � log 1 00,000 and H � I 6 . 6 bits. Axis offrequencies (melodic) : The same arguments a re valid here but with greater re st r i c t io n on the v a rie ty of melodic intervals and on the absol ute frequencies bec au s e of the limits of the audible area. Entropy is zero when the variety of frequencies of grains is I , i.e . , when th e cloud contains only one pure sound. Axis of intensity and density : Th e above observations are valid. There fore, if at the l i m it, the entropies following the three axes of an el em en t ll.Fl!.. Gl!.. tl!..D are zero, this element will o nly contain one pure sou nd of co n stan t intensity emitted at regular intervals. In conclusion, a c l ou d m ay contain just one si n g l e pure sound emitted at regu l a r intervals of time (see Fig. 11- 1 9) , in which case its m ea n entropy ( arithmetic mean of the three entropies) would be zero . It may contain chaotically distributed grains, with maximum ataxy and maximum mean This i s less true in p rac ti c e , for a
64
Formalized Music
entropy (theoretically oo) . Between these two l i m i ts the grains may distributed i n an i n fi n i te number of ways with
m e an
be
entropies betwee n 0
a raw,
and the maximum and a b l e to produce both the M arseillaise and
dodecap honic series .
t
Fig.
- - - - - - - - - -
� I I I
I
I
11-1 9
A single gra i n emitted at reg u l a r i n terva ls of time
Parentheses GENERAL
O B S E R V AT I O N S
ON
A T A XY
Taking this l as t possibility as a basis, we shall examine th e very general
formal processes
in all
real ms of thought, in all physical and p syc h i c r�alities.
this end we shall
To
imagine a " Primary
Thing, "
malleable at will ;
capable of deforming instan tan eously, progressively, or step- by-step ;
ible
plex
or
as
It
retractable ; unique or plu ral ; as simple
the
u niverse
will have
(as compared
to
man,
as
that i s ) .
a given mean entropy. At a defined
u n d e rgo a transformation. tion can have one
of three
From
the p o i nt of view
time we w i l l cau se it to
of a taxy
1 . The d egree of complexity de g ree
(variety) d oe s not change ; does not change.
of complexity
increases and so does
the
the transforma
entropy.
transformation is a simplifying one, and the entropy dimin
ishes.
T h u s t h e neutral transformation may a c t on a nd
d i so rd e r
this transforma
effects :
tion is neutral ; and the overall e n t rop y
2. The 3. The
extend com
an electron ( ! ) or as
into perfect d iso rd e r
transform :
perfect
(fluctuations ) , p a r ti a l disorder into p arti al
disorder, and perfect o rde r into perfect order.
Multiplicative
disorder, partial
disorder.
transform ation transforms : perfect dis o rd er into p erfect
disorder into greater disorder, and p e rfe c t order i n to p artial
65
Markovian Stochastic Music-Theory
transforms : perfect disorder into partial p arti al order into greater order, and partial order into perfect ord er. Fig. I I-20 sh ows these transformations in the form of a kinematic d i agr am . And simplifying transformation
disorder,
Perfect
D e g ree
disorde r
of
o rd e r
Max.
e n tropy
Partial
Entropy
Pe rfect
E ntropy 0
«
d isorder
order
Fig .
max.
1 1-20
STUDY
OF
Time ATAXY
AT
THE
LEVEL
S CR E E N S ( SET
OF
OF
CLOUDS)
screen which i s composed o f a set of wi t h densities during a slice of t i m e llt, may be dissociated according to the two characters of the grains, frequency and amplitude, and affected by a mean entropy. Thus we can classify screens according to the criterion of ataxy b y means of two parameters of disorder : the variety of the frequencies and the variety of the intensities. We shall make an abstraction of the temporal distribution of the g r ains in Ill and of the d e n sity , wh ich is implicitly bound up with the varieties of the two fundamental sizes of the grain . In symbolic form : cells
From
the above
discussion, a
l:!.. F!:iG associated
Perfect disorder Partial di s o rd er
Partial order Perfect order From the
point
of view
of ataxy a
=
=
=
=
oo
n
or
m
m or n 0.
s creen
is formul ated by a
pa i r of
entropy values ascribed to a pair of fre q u encies an d intensities of its grains.
Thus the pair (n, oo) means a screen whose frequencies h ave quite a small entropy (partial order or d iso rd e r ) and whose intensities have max i m u m entropy (more or less perfect disorder) .
66
Formalized
Music
CO N ST R U CTI O N O F T H E SC R E E N S
W e shall quickly Fig. 11-2 1 . Partial Perfect disorder disorder F G F G F
survey some o f the screens
Perfect order F G
G
F
G
F G
F
F
G
U n ique scre e n I nfin ite n u mber of screens
oo,
0
U n i que scre e n I nfin ite n u mber
n, co
of screens U n ique screen, pure so unds
I nfin ite n u mbe r of scre ens
n, m
F
G G
co, co
0, co
F
G
Description
Symbol
oo , n
G
F
I nfin ite number
of screens I nfin ite number of screens U n ique scre e n , pure sound
n, O O, n
F G
in
0, 0
the entropy table i n
Diagram
Diagram
.
:�·�=�}��r�---. : :��?��-�
,
.
•••
..: :;
· 1 �•
- .
·;
-.
0•
../
••· , ,
Ifj�fj ,) I I -�}�:] - .. , ... , , . I . . r.J
F i g . 1 1-21 . Screen E ntropy Table S CR E E N
(ro, ro)
Let there be
a
very large number of grains distribu ted at random over
of the audible area and l asting an interval of time equal to /).t, Let there also be a grid fine enough so that the average density will not be more than 30 grains per cell. The distribution law is th en given by Poisson's formula the whole range
p,_ �
=
(dm)k e - dm ' K!
the m e a n density and Pk the probability that there will be k cell . If dm becomes greater than about 30, the distri b u tion l aw will become normal. Fig. II-22 is an ex am p l e of a Poisson distri bu tion for a mean density dm 0.6 grains/cell i n a grid of 1 96 cells for a screen ( oo, oo) . Thus we may construct the (oo, oo) screens by hand, according to the distri butions for the rows and columns, or with suitable c o m p u ter programs.
where dm is grains in
=
a
.
.
. -
. 1-1--+--1--+----j -
-
- -
.
--1- - .
. - -;-· --
. �--l--+--1--1---1-'-l --.
.
.
. . .
--
.
1 - - 1 . -- •
•
.
•
•
I.
+-- � l -4-4- - ��-� -+-l-4-+-r�4-�
-.: · t··
.
.
67
Stochastic Music-Theor y
Markovian
· -
-
.
.
.
.
-
f- :- 1.
I•
.
. . .
.
.
.
.
.
.
.
Fig . 1 1-22
I
.
.
.
.
:
F
a very high m ean density the screens in which disorder is perfect (maximum) will give a very rich sound, almost a white sound, which will never be identical throughout time . If the calculation is d o n e by hand we can construct a l arge number of (oo, ro) s cre en s from the first (ro, ro) sc re e n i n orde r t o avoid work and numerical calculation for each separate screen . To thi s end we permute the cells by c ol u mn and row (see Fig. Il-23) .
For
F i g . 11-23. Exa m p l e of Permutati o n by Col u m n s
Discussion. It is obvious th at for a high mean d ens i ty, t h e greater the number o f cells, the more the distribution of grai n s in one regi on of the screen tends to regularize (ergodism) and the weaker are the fluctuations from one c el l or cloud to another. But the absolute l i m i ts of the densi ty in the cel l s in the audible a re a w i l l be a function of t h e technical means available : sl i d e rules, tables, calculating machines, compu ters, ruled paper, orchestral i n struments, tap e recorders, scissors, programmed impulses of p u re sounds, au t o m a t i c splicing devi ces, programmed record i n gs, a n a l o g u e converters, etc. If each ceiJ is c o n s i d ere d as a symbol defined by the number of grains k, the en tropy of the screen (for a given fineness of grid) will natu r a l l y be a ffected by the mean density of the grains pe r cel l and will gro w at the same
Formalized Mt.. s ic
68
time. It is here that a whole series of statistical experim ents will circumscribe th e perceptible limits of ataxy
for
even express the color nuances of white sound . It is very possible
ear
have to
and that the
these screens (co, co )
classifies in the same file a great n u mb er of sc re e ns whose entropies
tremendously.
There
wou ld
vary
result from this an impoveri shment and
a
simplification of t h e commu nication : physical information --? percepti on, but at least there
ing screens ALL
can
will be
SCREENS
Starting from
a
the work involved in construct
elementary operations we of th e en tropy table. See Fig. 11-24 for a few I n practice, freq uency and intensity fil ters imitate these elemen
construct all th e
examples.
the advantage that
will be considerably reduced.
few screens and applying the
screens
tary operations perfectly.
(00 00 )
llllll
.
(n
m>
� ""=
(n' m'1
� (110 Fig. 1 1-24
Yt )
(, m)
E:==== I
Markovian
69
S tochastic Music-Theory LI N K I N G T H E S C R E E N S
o r music co ul d b e de lexicographic order of the pages of a book. If we represent e ac h screen by a specific symbol ( o ne to one coding) , the sound or the music can be translated by a succession of symbols Up
t o now w e
have admitted that
any
sound
scribed by a number of screens arranged in the
-
called
each
a
-
protocol :
a b g k a b · ··b g · ··
letter identifying screens
and mo m e n ts
t for
particular
Without seeking the causes of a
isochronous
t:.. t 's.
succession of screens,
i.e . ,
without entering into either the p hysical s tructure o f the sound o r the l o gic al structure of the composition, we can disengage certain mo de s of succession and s p ecie s of protocols [ 1 6] . We shall quickly review the elementar y definitions.
its unique symbol is called a term. Two successive terms materialize. The second term is called the tra nsform and the change effected is represented by term A --+ term B, or A --+ B.
Any matter
cause a
A
transition
or
to
transformation is
a colJection of transitions . The following example
drawn from the above protocol :
l
another tra nsformation
with
a
b
b
g
g
is
k
k
a
musical n otes :
lC
B
D
G
E A
3
rn �
be closed when the col lec t io n of transforms elements b elonging to the collection of terms, for exam p le :
A transformation is said to
contains only
0
the alphabet, b
c
c
d
z
a
Formalized
70
Music
musical notes,
D Ej,
E
G
F
c
G j,
F
B
G
D j,
A
B
Bj,
A
Ej,
Bj,
E
musical sounds,
1.
infinity of terms,
-1
.. . :· � .
ll
6
single
transform,
�= �= :� .;:f .. �. .
2
3
4
7
4
1 00
l
�
c.
'
d i rect i o n
3. N etwork of p a ra l l e l
5
g l issa n d i in two
d i rection s .
6 2
singl e-val ued ( mapping) when each term
for example :
l
a.
b. j
e . g . , pizzicati
2. N etwork o f paral l e l g l issa n d i i n one
�
..
a
h
a
c
c
h
e
d
The following are examples of transformations that are
J
1 . C l o u d of so u n d - p o i nts.
3
. . :' ; .•· �
A transformation is univocal or has a
"'
� ·�tlll1V
� � "'
an
3
2.
i
l h�
b
c
d
c
,.--- 5 : 4 --,
J ))J)
r-
L._ 4 : 3 -.l
of a
change
clarinets
oboes
s trings
bassoon
univocal :
m, n, p
tJ
timbre
not
�
'
group of values
strings
Timbres timpani, timpani, brass
ti mpani
oboes
f
b r ass strings, oboes
�
•
r
Markovian S t oc h as ti c
" �fan ner ,
I
71
Music-Theory
and d. concrete
characteriology [4, 5]
music
nil
vibrated
cyc lical o r
irregular
trembled
trembled
cyclical
nil
trembled
or
nil or
irregular
vibrated
cyclical
A t ran sfo r ma t io n is a one-to-one mapping when
transform and when each transform is derived
example :
MATR ICAL
irregular
c
b
l:
a
a
term has a single single term, for
d c
d
R E P RE SENTA T I O N
e ach
from
or
A transformation :
can be
l:
represented by a table as .j.
a
b
This
c
a
0 0
c
follows :
c
b
0 0
0 0
+
+
+
c
b
c
.j.
a
b
or
c
b
c
l 0 0 0 0 0 0
table is a matrix of th e transitions of the collection of terms to a
collection
P R ODUCT
of transforms.
Let there b e two transformations T and T:
a
a
b
J:
d
In ce rt a in cases we
c
a
d b
and
U:
U:
can apply to a term
n
of
J:
Ta
b
c
d
c
d
b
tra n s form a ti o n
transformations T and U, on condition that the transforms of U. Thus, first
V
T, the n
transformation U. This is written : U[ T(n) ] , and is the product of the two
=
UT: a -+ c.
T:
a ---+
b, then
U:
b ---+
c,
of T are
terms
which is summarized as
72
Formalized
Music
To calculate the product applied to all the terms of T we shall use the matrical representation :
following
T:
c
a
b
a
0
0
b
1
0
0
1
c
0
0
0
0
c
d
0
I
0
0
d
,j,
t
d
1
a
0
b
c
d
0
0
0
0
0
0
1
I
0
0
1
0
0
b
U:
a
0
0
the total transformation V equals the product of the two matrices T and U
in
the
order U, T. 0
0
0
0
0
1
u
0
0 K I N EMATI C
0 0
0
0 I
0
1
X
0
0
T
v
0
l
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
0
0
1
1
0
0
0
0
0
DIAG RAM
The kinematic or transition diagram is a graphical expression of draw it each term is connected to its transform by an arrow pointed at the transform. The representatiue point of a kinematic dia gram is an imaginary point which moves in jumps from term to term following the arro ws of the diagram ; for an example see Fig. 11-25. transformation. To
T:
lA
D
p� Fig . 11-26
�/
C
D
D I
I
L
N
P
.A
A
N
A
N
N
N
A transformation i s really a mechanism and theoretically all the mechan isms of the physical or biological universes can be re p resented by
Markovian
73
Stochastic Music-Theory
transformations u nder five condi tions of correspondence :
1.
states
Each state of the mechanism (continuity is
as
clos e together as
is
broken
down into di s c re t e
desired) is in a one-to-one correspondence with
a term of the transformation . 2. Each sequence
internal
structure
of states
the mechan ism by reason of i ts uninterrupted seq uence of the terms of
c rossed by
cor res p o n ds to an
the transform ation . mec
h anis m reaches a state and rem ains there (absorbi n g or corresponds to this state h as no tra nsform . If the states of a mech anism reproduce th e m se l v es i n the same man
3. If the
stationary state) , 4.
the term which
a
ner without end, the transformation has circuit.
kinematic d i agram in closed
5. A halt of the mechanism and i ts start from another state
is
sented in the diagram by a d i s p la cem e n t of the re p resentative point,
is not due to an arrow but to an arbitrary action on the
repre
which
paper.
the corresponding transformation is univocal and closed . The mechanism is not determined when the corre sponding transformation is man y-val ued . In this case the transformation is said to be stochastic. In a stochastic mech a n ism the n u m be rs 0 and I i n the The mechanism is determined when
transformation matrix must be replaced by relative freq uencies. These are
the alternative probabili ties of vari ous
transformations. The determi ned mechanism, in w h i ch the
mechanism is a particular case of the stochastic pro babilities of t ra nsi t i on are 0
and
l.
Example : All the h armonic o r po l y p h o n i c rules o f classical music could be represented by mechanisms. The fugue is one of the most accom pl ished
and determined mechan isms.
One
co u l d
avan t-garde composer is n o t con ten t
age but proposes new ones, for both are
eve n generalize and say that the
with
fol lowing
the mechanisms
detail and general form.
If these prob abili ties are constant o v e r
a
long period
i ndependen t of the s t a tes of origi n , the stochastic
more particularly, Let
there
be
a
Markov chai n .
two
screens
A and B
and
a
of
h is
of time, and i f they sequence is cal led ,
p ro t o co l of 50
transit ions :
ABABBBABA A BA BA BA BBBBABA A B A BBA A BABBABAAABABBA A BBABBA . The real
freq uencies
of the
A -+ B A - A
transitions 1 7 times
6 t i mes
23 times
arc :
B - A
B- R
1 7 t i mes 1 0 times
27
times
76
Formalized M usic
diminution of the entropy. If melodic or harmonic liaisons are effected in the same distribution, unpredictability and entropy are both di minished.
a
and perceived Rate of ataxy
A
/
B Time
D
�
E
c�
v
F i g . 1 1-26 A . The evolution i s n i l . B . The rate of d i sorder a nd the rich ness i n crease. C . Ataxy decreases. D. Ataxy i n c reases and then d e c reases. E . Ataxy decrea ses and then i ncreases. F. The evo l ution o f t h e ataxy is very complex, b u t it m a y be a n a l yzed from the fi rst t h ree d i a g ra m s . Thus after the first u n fo ldi ng
of
a
of twelve sounds of the tem is maxi mum , the choice is nil , and th e en tropy is zero. Richness a n d hence i nterest are d i s p l ace d to other fields, such as harmonies, timbres, and durations, and many other com positional wiles are aimed at reviving entropy. In fact sonic discourse is no thing but a perpetual fluctu ation of entropy in all its forms series
pered scale, the unpredictability has fallen to zero, t h e constraint
[1 7] .
does not necessarily follow the variation if it is logarithmic to an appropriate base . It is rather a su ccession or a protocol of strains and rel axations of every degree that often exci tes the li s te n e r in a di rection contrary to th at of entropy. Thus Ravel's Bolero, in which the on l y variation is in the dynamics, has a vi rtually zero entropy after the third or fourth repe tition of the fu ndamental idea. How However, human sensitivi ty
in en tropy even
ever, the i n terest,
or
rather the psychological agi tation, g rows with time
the very fact of this immobility and banality. All i n ca n t a to ry manifestations aim at an effect of maximum with m i n imum entropy. The inverse is equ ally true, and seen from a through
tension
certain
Markovian Stochastic Music-
Theory
77
n o i s e with its maximum e n t ro py i s soon tiresome . I t wo u l d there is no correspondence aesthetics - entropy. These two
angle, white seem that
.
e n t i t ie s are l i nked in quite an independent m a n n e r at each oc c as i o n This
statement still leaves some respite for the free will of the composer even i f
this free w i l l is buried under t h e rub bish o f c u l t u re and civi l i zation a n d is
only a sh adow, at the l e a s t a tendency, a simple stochasm . The great o bs t a c l e to a too hasty gen eral i zation is chiefly one o f l og i c a l order ; for an object is only an o bj e ct as a fu n c t io n of its d e fi n i t i o n , and there is, especially in a r t , a near- i n finity of definitions and hence a near- i n fi n i ty of entropies, for the notion of en tropy is an epiphenomenon of the d efi n i t i o n . Which of these is valid ? The ear, the eye, and the brai n u n r av e l sometimes inextricable situations with what i s called intu ition, taste, and in telligence. Two definitions w i th two d i fferent e n trop i e s can be perceived as identical, b u t it i s also tru e that t h e set of defini tions of an object has i ts own degree of disord er.
We
are not concerned here
wi
th
investigating such a difficult,
complex, and un explored si tu ation, b u t si m p l y with looking over the
possibilities that co n nec t e d realms of c o n te m p o r a ry thought p ro m i s e , with a view to action .
To conclude
q u en t
than
screens can
brie fl y , si n ce the
explanatory
texts,
be expressed arc
by
we
applications
wh ich follow are
mor<>
elo
shall accept that a coll ection or book of
matrices of
transition probabil ities havi n g
entropy which is in order to render the an alysis and t h e n the synthesis of a son i c work within reach of understand i ng and the slide rule, we s hal l establish three criteria fo r a screen :
p arameters. They
calculable
I .
a ffected by a d egree
of
ataxy or
under certain conditions. However,
TOPOGR A PHIC
CRITER I O N
The posi t i o n of the cells 6.F6. G on the audible area is q u al i tat ively
i mportant, and an
creating a
g ro u p
enumeration of th e i r possi ble combinations is capable of d e fi n e d terms to which we can apply the concept
of wel l
of entropy and i ts calc u l atio n .
2.
DENSITY
C R I TERI ON
The superficial density o f the g r a in s o f a cell !J.F!J.G also constitu tes a q ua l i ty wh ich is immedi ately perceptible, and we cou l d equally well d e fi n e t erms to w h ich the conce p t and calculation o f entropy wou l d be appl ica b l e .
3 . CRITERION s c re en )
OF
P U RE
ATAXY
A cell has t h ree variables :
mean
( defined
i n relation to t h e gra i n s o f
freq u ency, mean am p l i t u de,
and
a
m e an
78
Form alized
Music
density of the grains. For a screen we can therefore establish three indep en dent or connected protocols, then three m a trices of transition probabilities which may or may not be coupled. Each of the m atri ces wil l have its entropy and the three coupled m atrices will h ave a mean entropy. In the procession of sound we can establish several series of three matrices and h e nce several series of mean entro pies, their variations constitu ting the criterion of ataxy. The fi rst two criteria, which are general and on t h e scale of screens or cells, will not concern us in wh at follows . But the third, more conventional criterion will be taken up in detail in the next ch apter.
C h a pter I l l
Markovian S tochastic Applications In th is ch apter
we
w i l l d iscuss two m u si ca l a p plications :
string orchestra, and A nalogi que B , lor sin usoidal 1 958-59. W e s h a l l confi ne o u rs e l ves
ponents G,
F, D
Music
to
a
of the screen take
sounds,
s i m ple c a s e in which
o n ly
Analogique
A , for
both composed in
each of t h e co m
two values, following matrices of
by means of parameters . In matrices will be made in s u ch a way that we shall h av e only the regular c a s e , conforming to th e chain o f even ts th eory a s it h a s been defined i n t h e work of Mau rice Frechet [ 1 4] . transition probability which will be cou p led
addi t io n , the
choice of probabilities
I t i s obvious
that richer
highly in teres tin g to
the
and more com plex stochastic me ch a n isms
c o ns tr u c t
and to
siderable volume of calculations
to undertake them by
in
put
which
hand, but very
in work,
but in view of
the
are
con
they necessi tate it would be. u sel e ss
desirable
to pro gram
computer.
them for t h e
Nevertheless, despite the stru ctural simplicity of what follows, the
s to c has ti c
jaccnt
mechanism which wi l l emerge will be
a
mod e l ,
a
s t an dard
sub
that are far more complex, and will serve to catalyze of greater elaboratio n . For although we confine ourselves here to the s tudy of scr e ens as they h ave been defined in this st udy (sets of elemen tary grains) , it g oes without saying that noth i ng prevents the gen to any others
fu rther studies
eralization of this m etho d of s tructu ralization (composition) for defi nitions of sonic entities of more
than three d i mensions. Thus, let us no longer suppose
screens, but criteria of definitions of a sonic entity, such that for the
deg ree
of ord er, density, v a ri a t i o n , and even 79
the criteria
timbre,
of more or less
80
Fonn a
"
«-
.
�
e .........
�,_
.,.
� �... .,.. ..,...� .
...
..
��
�
:� :
��
1-q.
�:-��2--�� -
7+ .,.
'>l
� ,.,
.
...�
.,_ + .,.: ....__
...
·.
.
.
- ��
\ .
�
� '
0�
{ , � " Fig . 11 1- 1 . S wrnos for 1 8 s tring
s
tiz cd J\1usi c
Markovian Stochastic Music-Applications
81
complex elementary structu res (e .g. , melodic and temporal structures of groups of sounds, and instrumental, spatial, and ki nem ati c structures)
s a me
stochastic scheme is adaptable.
well and to be
It is
enough to define the
the variations
able to classify them even in a rough manner. thus obtained is not gu aranteed a priori by calculation. experience must always play their part in guiding, deciding,
The sonic result
Intuition and
and
testing.
{ d ef i n i t i o n We shall define process. It
stochastic
A N A LYS I S
o f the scheme o f a mecha n i s m }
t h e scheme o f a m e ch a ni s m as t h e "analogue " of a
will serve for the production
their transformations over time. These sonic entities
will show the followin g characteristics freely
of sonic entities and for will have screens which
chosen :
1 . They will permit two d i sti nc t combinations
fo and j1 (see
of frequency regions
Fig. I I I-2) .
,__----�---;,F Half axis Audible frequencies
,______.__________._._t----: F
Fig . 1 1 1-2
I.
Audible frequencies
of frequencies in sem itones
H a l f axis of frequencies in semitones
Syrmos, written i n 1 959, is b u i l t o n stoc h a stic tra nsformations of eight bas i c text u res : para l lel horizo nta l bowed notes, para l l e l
ascend i n g bowed gl issa ndi, pa ra l l e l desce n d i n g bowed g l i ss a n d i , crossed (asce n d i ng a n d desce n d i n g ) p a ra llel bowed notes, pizzicato c l o u ds, atmos p h e res made up of col leg no struck notes with s h o rt col leg no gl issa n d i , geo metric config u rat i o n s of conve rgent or d i vergent g l i ssa n d i , a n d g l issando confi g u ra t i o n s t reated as u nd evelopa b l e r u l ed surfaces. The mathematica l str u c ture o f this work is t h e s a m e as t h a t o f Analogique A a n d Analogique B .
Formalized Music
82
2. Th ey will permit two distinct combinations of i n te n s ity regions (see Fig. III-3). G
( P hones)
( P h o n e s)
.,
.,
·"' ., c "'
.� -�
.2!
c
E
-�
"'
..
;e
:0
·a �
i5.
o --...1
Fig. 111-3 3. They
will
permit
(see Fig. I II-4) .
�
"'
D..
o..
Q;
two distinct combin ations of density regions
(Terts • or sounds/s ec)
(Terts • or sounds/sec)
0
D
• Ternary logarithms
F i g . 1 1 1-4
4. Each of these three variables will present a protocol which may be two matrices of transition probabilities (MTP) .
summarized by
t
( p)
The letters ( p )
X y
y
0.2
0.8
0.8
(a)
0.2
and (a) constitute
the
t
X
y
X
0.85
0.4
y
0. 1 5
0.6
parameters
of the (MTP) .
MTPF (of frequencies)
t
(a)
X
fa !1
fo
11
0.2
0.8
0.8
0.2
t
([3)
fo
j�
fo
11
0.85
0.4
0. 1 5
0.6
Markovian
Stochastic Music-Applications (of intensities)
MTPG
t
go
(y)
gl
go
gl
0.2
0.8
0.8
0.2
.J,
(e)
MTPD
do
dl
0.2
0.8
0.8
0.2
t
do
(A)
dl
83
go
gl
0.85
0.4
0. 1 5
0.6
do
dl
do
0.85
0.4
dl
0. 1 5
0.6
go
gl (of d e nsi ties ) .J,
(p.)
The transformations of the variables are indeterminate at the (MTP) (digram processes) , but on th e other hand their (MTP) will be conne c ted by means of a determined cou p l i n g of parameters. The coupling is given by the fo l l o w i ng transformations : 5.
interior of e ac h
lh
h � � A p. a. f3
&
A
& p.
&
{3
& a.
h h 4 � y
e
y
e
we have described th e structure of a m e ch a ni sm It is by three p a irs of (MTP) : (MTPF) , (MTPG) , ( M TPD) , and by the group (e0) of the six couplings of these (MTP) . Significance of the coupling . Let f0 be the state of the frequencies of the screen at an instant t of the sonic e vo l u tion of the mechanism d u ri n g a slice of time !J.t. Let g1 and d1 be the valu es of the other vari ables of the screen a t the moment t. At the next moment, t + !J.t, the term fo is bound to change, for it obeys one of the two (MTPF) , (a.) or (�) . The choice of (ex ) or (f3) is conditioned by the values g1 and d1 of the moment t, conforming to the transformation of the cou pling. Thus g1 p roposes the parameter (a.) and d1 the parameter (/3) simultaneously. In other words the term fo must e i t h e r remainf0 or yield i ts place to f1 according to mechanism (a.) or mech anism (�) . Imagine the term f0 standing before two urns (a) and ({1) , each con taining two colors of balls, red for fo and blue for j1, in the followin g By these rules
.
thus constitu ted
prop ortions :
Urn (a.) (f0) , 0.2
Urn ({3) red balls (f0) , 0.85 b l u e balls (j1) , 0 . 1 5
red balls
blue balls (j1 ) , 0 . 8
The choice is free and
the term f0
can
take its successor from either urn (a.)
Formalized Music
84 or urn (/3) with
a
t ( to tal
probabil i ty e q u al to
probab i l i ties) .
Once the urn has been chosen , the c h o i c e of
a
b l u e or a red
ball will
have a p robabi lity equal t o t h e proportion of co l o rs in t h e chosen
Applyin g t h e l a w o f c ompou n d probabilities, t h e probability
moment
and
the
t will
remainf0 at the moment t
probabil i ty
th at
+
6.t is
(0. 20
+
thatf0
0.85 ) /2
i t wi l l change to f1 is (0.80 + 0 . 1 5 ) / 2
=
urn.
from
0.525,
0.475.
=
The five characteristics of the composi tion of the screens hav e estab
lished
a
stochastic m ec h a n i s m
.
the
We
the
the
sl ices
At
of
th ree variables _h, g1 , d1
round of unforeseeable combinations, always
three (MTP) and
the sonic follow a c h an gi n g according to the
Thus in each of
evolution of the created mech anism,
cou p l i n g which connects terms and parameters .
have established this mechanism witho u t taki n g � n lo consideration
any of th e screen criteri a . That is to say,
d i s tri b u ti on
we
h ave implied a to p o g ra p hic
of grain regions at the time of the choice ofj0 , j1 and g0, g1, but without s p e cifyi n g i t. The sa m e is true for the density distribution . We shall give two examples of very different realizations in which these two criteria will b e e ffective . But b e fo re setting them o u t we sh a l l p u rs u e fu rther the study of the c ri t er i o n of ataxy. We shall neglect the e n t ro p i e s
of the
th r ee variables at the grain
level,
for what m atters is the ma c ros c o p i c mechanism at the screen leveL The
fundamental questions p o s ed by these transformation summarized by
an
Let us consider the ( MTP) :
and
m ech an i s ms
t
X
Y
X
0.2
0.8
y
0.8
0.2
all
to set
s u p p o se one hun dred m ech anisms identified by
(MTP) . We shall allo w them
a re,
" Where does the
( MTP) go ? What is its destiny ? "
out from
the l aw
of
this single
X and evolve fr e el y . The
preceding q u e s t i o n then becomes, " Is t h e r e a gen'eral tendency for the states
if so, what is i t ? " (See App endix I I . ) be transformed into 0. 2 ( 1 OOX) � 2 0 X, and 0.8 ( l OOX) ---+ BO Y. At the third stage 0.2 o f th e X's and 0.8 of the Y's will become X's. Conversely 0.8 of the X's will become Y's and 0.2 of the Y's will remain Y's. This general a rg u m e nt is t ru e for all stages and can
of th e hundred mechanisms, a nd
After the first stage
the
I OOX will
be written :
X' Y'
=
=
0.2X + 0.8 Y 0.8X + 0.2 Y.
Markovian Stochastic Music-App lications
85
If this is to be applied to the I 00 mechanisms X as above, we sh a ll ha vc : Mechanisms
Mechanisms
X
y
Stage
0
1 00
0
I
20
80
2
68
32
3
39 57
5
46
43
54
48
52
6
We
61
4
7
49
8
50
51 50
9
50
50
notice oscillations that show
a
general tendency towards
a
station
th en , that of the 1 00 m ech a nisms that leave from X, the 8th stage will in all prob ability send 50 to X and 50 to Y. The same s tationary prob!lbility distribution of the Markov chain, or the fixed probability vector, is calculated in the following manner : At equil i b rium the two probability values X and Y rem ain u n ch an ge d and the preceding system becomes
ary state at t h e 8th s tage. We may conclude,
X
Y
=
=
0.2X + 0.8Y +
0.8 Y
0.2 Y
or
0 0
= =
- O.BX + O.B Y + O.BX - O.BY.
Since the number of mechanisms is constant, in this case 1 00 (or 1 ) , one of the two e q uations may be replaced at the stationary distribution by l = X + Y. The syst e m then becomes 0
=
O.BX - 0.8 Y
I =X+ Y
and the stationary probability values X, Y a re X 0.50 and Y 0.50. The same method can be applied to the ( MTP) (a) , which will give us stationary prob abilities X = 0.73 and Y 0.27. =
=
=
86
Formalized Another method, particularl y i nterest i n g i n the case
to
man y terms, which forces us
Music
of an (MTP)
w i th
resolve a large syst e m of l i near equations
in
order t o fi n d the stationary pro b a b i l i ties, is t h a t w h i c h makes u sc o f m a trix
calculus.
Thus the first
( MTP) with the
s tage
may be considered as the m a trix pro d u c t of the
u n i co l u mn
c
� l �O �
1 0.8 1 I I I 1 I · I 1 1 I I l l l m atrix
X:
0
Y:
0.8
.
2
0. 2
X
00
=
20
80
0
The se ond stage will be
0 2 .
0.8
and the
0
.8 0. 2
X
20 80
=
68
4
64 + 16 + J6
=
32 '
nth stage
Now that we know how to calculate the stationary probabilities of
a
Markov chain we can e asily calcul ate its mean en tropy . The definition of th e
entropy of a system
is
The calculation of the en tropy of an (MTP) is made fi rst by columns C'J. P1 I ) , the p1 bei ng the probability of the transi tion for the ( MTP) ; t h en this result is weighted with the c orre spondi n g stationary probabilities. Thus for the (M T P) (a) : =
t
X
The
e n tr o py
y
X
0.85
0. 1 5
y
0.4 0.6
of the states of X will be - 0 .85 log 0.85 - O. l 5 log 0 . 1 5
0 . 6 1 1 bits ; the e ntropy of the states of Y,
0 . 9 7 0 b its ; the stationary prob ability of X
bility of Y
=
-
=
0.4 log 0 . 4 - 0.6 log
=
0.6 1 1 (0.73) + 0.970(0.27)
=
=
0. 73 ; the station ary p ro b a
0. 2 7 ; the m ean entropy at the stationary
Ha
0.6
=
stage is
0. 707 bits ;
87
Markovian Stochastic Music-Applications
and the mean
The for if we
two
look
e n t r op y
e
n tro p i es
of the
(MTP) (p) Hp =
at the s ta tio n ary stage i s
0. 722 bits.
do not differ
by much,
and this is to be expected,
at the respective ( MTP) we observe that the great contrasts
o f probabilities inside the
matrix
(p) are compensated by an external equ ality (MTP) (a) the interior
of st a tion a ry probabilities, and conversely in the
qu asi-equality, 0.4 and 0.6, succeeds i n coun teracting th e interior contrast,
0.85 and in
0. 1 5 , and t he exterior
At this
level
we
may
contrast, 0 . 7 3 and 0.27. t he (MTP) of the three
modify
such a way as to obtain a n ew pair of
repeatab l e
we c a n
(MTP) of pairs
entropies.
As
variables };, g h d1 this
operatio n
is
form a p ro t o c o l of pairs of e n tropies and therefore an
of en tropies. These sp ecul ations and investigations
are no
doubt interesting , but we s h a l l confine ourselves to the first calculation made ab ov e and we shall pursue the investigation on an even more general plane. EXTENDED
MARKOV CHAIN On p.
83
SIMULTANEOUSLY
FOR
}; ,
gf >
dl
we a na l y ze d the me c h a n ism of transformation of fo to fo or f1
when the probabilities of the two v a ri ab le s g1 and d1 arc give n . We the same arguments for each of the
o t h e rs are given.
three
variables };,
can
apply
g1 , d1 when the two
Example for g1• Let there be a screen at the moment t whose variables (f0 , g 1 , d1) . At the mo m e n t t + tlt the value of g1 will be transformed into g1 or g0• From f0 comes the parameter (y) , and from d1
have the values comes
the
parameter (e) .
Wi th ( M T P ) (y ) the probability that g1 will remain g1 is
0.2. With
( MTP) (e) the probability tha t g 1 wil l remain g1 is 0.6. Applyi ng the rules
of compo und probabil ities and /or probabilities of m u tually exclu sive events
as on p.
8 3 , we
that the pro b abi l i ty that g1 wi l l rem ain g1 at the momen t simultaneous effects ofj0 a n d d1 is eq u a l to (0.2 + 0 . 6) /2
find
t +
tlt under the
and
for
0.4. 1l1e same holds fo r the cal cu l a ti on of the We
the transformations
of d1 •
shall now attempt to emerge from this j u ngle of probability
binations, which is i m poss i bl e
to
=
transformation from g1 i n t o g0
com
manage, and look for a more general
viewpoint, if it exists. In general , each screen is constituted by a tri ad of s pecific valu es o f the variables F, G , D s o tha t we can enumerate the d i ffe rent screens emer gin g fro m the mechanism that we are given ( se e Fig. II I-5) . The possible
88
Formal ized Music
Do
/ ""'
fo
combinations
(f1 g 1d0 ) ,
"
/
91
Fig . 1 1 1-5
are :
/
do Uo
d, do
"
dl
'1
/ ""'
/
do
"
d,
/
U1
do
"
dl
Uogodo) , Uol1od! ), Uog1 do) , Uog1d1 ) , (J!godo), (f1 god1) ,
different screens, which, with their protocols, up the sonic evolution . At each moment t of the composi tion we shall encounter one of t h e s e eight screens and no others. What are the rules for the passage from one com b i n ation to another ? Can one construct a matrix of transition probabi l i t i es for these eight screens ? Let there be a screen (f0g1d1 ) at the moment t. Can one calculate the probability that at the m om e n t t + at this screen will be tra n sfo rm e d into ( f1 g 1 d0 ) ? The above op e r a ti o ns have e na b l e d us to calcul ate the probability that/0 will be transformed i n to /1 under the influence of g1 and d1 and that g1 will remain g1 u nder the i nflu ence of fo and d1• These operations are schemati :1.ed in Fig. 111-6, and the probabil ity that screen (f0g1d1) will be transformed into (f1g1 d0) is 0. 1 1 4. (!J g1d1 ) ; i . e . , eigh t
wil l make
Screen at
the moment t :
Para meters derived fro m transform ations : S c reen
at
the moment t
Values of proba bilities correspo nding to the
the c o u pling + 11t :
taken from the ( M T P )
coupling para meters :
Compound proba b i l ities :
Compound probabilities for
independent events :
0.80 0.1 5
0.6 0.2
0.4 0.8
0.475
0.4
0.6
0 . 475
·
0.4
•
0.6
=
0.1 1 4
Fig . 1 1 1-6
We can therefore ex t e n d the calcul ation to the eight screens and construct
the
matrix of transition
probabilities. It
rows and eight colu mns .
will be squ are and will h ave eight
Markovian Stochastic Music-Ap plications
89
MTPZ
l
( foUodo)
(foUodt l
0.021
0.357
B (fogodd
0.084
C(fogtdo ) D ( f0g1 d1 )
8
A
A (fogodo)
c
( fo g t do)
D
(foUtdt)
E
F
(ftgodo) (ftgodt )
G
(ftgtdo)
H
( ftgl dl)
0.084
0 . 1 89
0 . 1 65
0 . 204
0.408
0 . 0 96
0.089
O.o7 6
0. 1 26
0.1 50
0 .1 3 6
0.072
0 . 1 44
0.084
0.323
0.021
0. 1 26
0 .1
50
0. 036
0.272
0 . 1 44
0.336
0.081
0.01 9
0.084
0.1 3 5
0.024
0.048
0.21 6
0.01 9
0.063
0.336
0.1 7 1
0.1 1 0
0.306
0.1 02
0.064
0.076
0.01 6
0.304
0. 1 1 4
0.1 00
0,01 8
0.096
G ( flgtdo)
0.204
0 .07 6
0.057
0.084
0.1 1 4
0.1 00
0.054
0.068
0.096
H(f1g1 d1 )
0 . 304
0,01 4
0.076
O.o76
0.090
0.036
0.01 2
0 . 1 44
E(f1g0d0)
F(ftgodt )
dA
Does the matrix have a region of stability ? Let there be l 00 mecha Z whose scheme is summarized by (MTPZ) . At the moment t, m ec h an is m s will have a screen A, dB a screen B, . . . , d1-1 a screen H. At the m o me n t t + llt all 1 00 mechanisms will p rod u c e screens according to the probabilities written in ( MTPZ) . Thus,
nisms
0.02 1
dA will stay in
be 0. 084 de will be
0. 3 5 7 dB will
A,
A, to A,
transformed to transformed
0.096 dH will be transformed to A . The dA screens a t the moment t will become d� screens a t the moment t + llt, a n d this number will be e q ual to the sum of all the screens t h at will be produced by the remaining mechanisms, in accordance with the corre s pondi ng probabilities.
{d���
Therefore :
(e1)
d�
� =
.=
dH
At the
=
0.Q2 J dA
0.084dA +
0.304dA +
stationary
rem ai n constant
+
0.084dA +
and
O.Ol4dB
0.357dB
+ Q.Q84dc +
+
0.323d8
0. 089d8
+ 0.02 l dc +
+
+
0. 096dH 0. J 44dH 0. J 44dH
0.076dc
+
0. 1 44du.
+
0.0 76dc
+
+
+
state the frequency of the screens A, B, C, . . . , H will preceding eq u a tions will become :
the eigh t
{
90
(e2)
Formalized
0 0 0
=
0
=
=
=
- 0.9 79dA + 0 . 3 5 7d8 + 0.084dc + 0.084dA - 0.9 1 1 d8 + 0.076dc + 0.084-dA + 0.323d8 - 0.979dc + 0.304-dA +
0 . 0 1 4d8
+
0.076dc
+
Music
+ 0 .096dH + 0. 1 #dH + O. I 44dH •
•
•
-
0 . 856dH
On the o th e r hand
of
If we replace one of the eigh t eq u ati o ns by the last, we ob t a in a system with eight unknowns. Solu tion by the classic
eight linear equations
method of determinants gives the val ues :
(e3)
{dA
dG
=
=
0. 1 7, d8 0. 1 0, dH
=
=
0. 1 3, de 0. 1 0,
=
0. 1 3, dn
=
0. 1 1 , dE
=
0. 1 4, dF
=
0. 1 2,
the sc re e ns at the stationary stage . This the chance of error is very high (unless a
which are the probabilities of
method
is very
laborious, for
is available) . method (see p.
calculating machine
second
85) , which is more approximate but in making all 1 00 mechanisms Z set out from a single screen and letting them evolve by themselves. After several more o r less long oscillations, the stationary state, if it exists, will be a t t a i n e d and the proportions of the screens will r em ai n invariable. We notice that th e system of equations (e1 ) m ay be broken down into :
The
adequate, consists
1 . Two vectors V' and V which matrices :
may
V' =
and V
=
be represented by two u nicolumn
d(; d�
Markovian
91
S tochastic Music-A pp lications
2 . A l i near o p erator, the matrix of transition probabilities Z. Conse system (e 1 ) can be summ arized in a matrix equation :
quently
To cause all
m ea ns
allowing
Z =
a
1 00 mechanisms linear
0.02 1
0.357
0.084 0.084 0.336 0.0 1 9 0.076 0.076 0.304
0.089 0.323 0.08 1 0.063
0. 0 1 6
0. 057
0. 0 1 4
operator :
Z
to leave
0 . 084 0. 1 89 0.076 0. 1 26 0.02 1 0. 1 26 0. 0 1 9 0.084 0.336 0. 1 7 1 0.304 0. 1 1 4 0. 084 0. 1 1 4 0 . 0 7 6 0.076
screen X and evolve " freel y "
0. 1 65 0. 1 50 0. 1 50 0. 1 3 5
0.1 1 0 0. 1 00 0. 1 00 0. 0 9 0
to perform on t h e column vector
0.204
0.408
0. 1 36 0.072
0.096 0. 1 44
0.036 0.272 0.024 0.048 0.306 0 . 1 0 2 0.204 O.D I B 0.054 0.068
0. 1 44 0.2 1 6 0.064 0. 096 0.096
0.0 1 2
0. 1 44
0.036
0 0
V=
1 00 0 0
continuous m an n e r at each moment t. Since we have b ro ke n down a discontinuous succession of thickness in tim e !lt, the equa tion (e4) will be applied to each stage !lt. Thus at the beginning (moment t 0 ) the population vector of the 0 mechanisms will be V • After the first stage (moment 0 + !lt) i t will be V' = ZV 0 ; after the second stage (moment 0 + 2/lt) , V" ZV' Z 2 V0 ; and at the nth stage (moment n !lt) , V1"' Z" V0• In applying these data to the vector in
a
continuity i n to
=
=
=
0 0
V}l
=
0 0 0 0 0 1 00
=
Formal ized Music
92 after the first moment /}.t :
Vii
after
=
after the second stage moment 2 f}. t :
stage at the
zv�
=
9.6 1 4.4 1 4.4 2 L6 6.4 9.6 9.6 1 4.4
1 8. 94 1 1 0.934
v;;. = ZVfi
1 4 .4 7 2
=
8 .4 1 6
8.966
the third stage at the
and after the fourth moment
46-t :
1 6.860 1 0.867
=
zv;;.
=
1 3 . 1 18 1 3 . 143 1 4.575
v;;
=
zv;;
=
8. 145
1 1 . 046
Thus after the fourth stage, an av er ag e o f 1 7 out o f the
screen B,
14 screen C,
stage
at the
1 7. 1 1 1 1 1 .069 13.792 1 2.942 1 4.558 12.1 1 1
1 2 .257
have screen A, 1 1
1 1 . 1 46 1 !5 . 1 64 1 1 .954
moment 3 /}.t :
v;;
at the
8.238 1 0. 7 1 6
1 00 mechanisms will
. . . , 1 1 screen H.
we compare the components of the vector V'"' with the values (e3) that by the fourth stage we have almost attained the stationary state. Conseq uently the mechanism we have built shows a very rap id abate ment of the oscillations, and a very great convergence towards final stability, the goal (stochos) . The perturbation Pu, which was im p osed on the mecha n ism (MPTZ) w hen we considered that all the mechanisms ( h e re 1 00) left from a single screen, was one of the strongest we could create. Let us now calculate the state of the 1 00 mechanisms Z aft e r the first stage with the maximal perturbations P applied.
we
If
notice
:tic Music-Applications PA
v�
=
PH
2. 1 8.4 8.4 33.6 1 .9 7.6 7.6
0
V2 =
1 00 0 0 0 0 0
30.4
Pc
8.9
v�
32.3 =
=
vo
1.9
-
n -
33.6 3 0.4 8.4 7.6
1 00
0 0
0 0
=
1 .4
1 8.9
1 2 .6 1 2.6
8.4 Vb = 1 7. 1 1 1 .4 1 1 .4 7.6 PF
16.5 15.0 15.0 13.5
20.4 1 3.6 3.6
0 0
voF
1 1 .0
1 0. 0 1 0.0
9.0
-
-
0 0 0 1 00 0 0
Po
40.8
0
V8 =
6.3
1 .6
0 0 0
8. 1
5.7
Pn
PE
v�
35.7
0
8.4 7.6 2. 1 v�
93
0
7.2
0
27.2
0 0
0 1 00 0
v�
=
4.8
1 0.2 1 .8 6.8 1 .2
v�
=
2.4
30.6 20.4 5.4 3.6
Formalized Music
94 R e ca pitu l ation of t h e A n a l y s i s
Having arrived at t h i s s t a ge of t h e analysis
On the level of the screen cel ls we
now
we
h ave : I .
take our bearings . partial mechanisms of
mu s t
transformation for fre q uency, in tensity, and d e n sity ranges, expressed by the
(MTPF) ,
( MTPG) ,
between the three fundamental the coupl ing ( e0) ) .
mations of
On
A, B, C,
( M TPD) ; an d
2.
which
are
an interaction
variables F, G, D of the screen
(transfor
the level of the sc re e n s we now have : I . eight different screens, D, E, F, G, H; 2. a general mechanism, the ( MTPZ) , which sum
marizes all the partial mechan isms and their i n teractions ;
equilibrium ( the goal, stochos )
3.
a
final state of
of the system Z towards which it tends quite quickly, the st ationary distribution ; and 4. a p rocedu re o f l:l isequilibrium in system Z with the help of the p ertu rbations P w h i ch a r e imposed on it.
Mechanism Z which
SYNTH ES I S w e h ave
just
constru cted docs n o t i m pl y
a real
evolution of the screens. It only establishes a dynamic situa tion and
a
evolution . The natural process is that provoked by a pertu rbation P imposed on the s ys te m Z and the advancement of this system towards its goal , i ts s tationary state, once the perturbation h as ceased its action . We can therefore act on this mechanism through the intermediary of a perturbation such as P, which is st ronge r or weaker as the case may be. From this it is only a brief step to imagining a whole series of s u cc e ss iv e perturbations potential
which would force the apparatus Z to be di sp l aced towards exceptional regions at odds wi lh
its behavior
at equilibrium.
created lies in the fact be. The perturbati ons which apparently change many negations of this existence. And if we create
In effect th e intrinsic value of the organism th us
that it must manifest i tse l f,
its structure represent so a su c ce s s i on of perturbations or ne g ations, on the one hand, and station ary states or existences on the other, we are only ajjirming mechanism Z. In other words, a t first we argue positively by proposing and offering as evi d en c e the e x is ten c e itself; and then we confirm it negatively by opposing it with p e r t u rb a tory states. The hi-pole of being a thing and not being thi s thing creates th e whole -the obj ect which we intended to construct a t the b e g i n n i ng of Chapter III. A dual dialectics is t h u s at the bas i s of this compositional attitude, a dia lectics that sets th e pace to be followed. The " experimen tal" sciences are an expression of this a rg u m e n t on an an a l ogo u s plane. An experiment estab lishes a body of data, a web which it disen tangles from the magma of
95
Markovian Stochastic Music-Applications
obj ective reality with t h e help of negations and transformations i m p os e d
on this
body. The repetition of these dual operations is a fu ndam ental
condition on which
th e
wh ol e un iverse of knowledge rests. To s ta t e some
thing once is not to define i t ; the cau sal ity is confounded with
of p h e n o me n a consi dered to be identical.
the rep e t iti on
In con cl u sio n , this dual dialectics with which we are armed in order
to compose within
the framework of our mechanism is homothetic with th a t
of the experimental s c ie nc e s ; and we can extend the comparison to the
dialectics of biological beings or to nothing more than the dial e c t i cs of
being. This it.
br i n gs us
back
to the point of departure.
Thus an e n ti ty must be p r op os e d and then a modification i m posed
our
on
that to propose the entity or its mo d i fi c a tion in
It goes without saying
particular case of musical composition is to give
a human observer the Then th e
means to perceive the two propositions and to compare t h e m .
an titheses, to
entity and modification,
be identified. What does
are repeated enough times for the entity
identification mean in the
c a s e of our mechanism
Z?
Parenthesis. W e h ave su pposed i n the course of the analysis that 1 00 mec h an i s ms Z were present simultaneously, and that we were following the rules of the game of these mechanisms at each moment of an evolu tion
created by a displacement beyond the stationary zone. We were therefore
com parin g the s tate s of 1 00 mechanisms in a 6. t with the states of these 1 00 me ch an i s ms in the next t, so that in c omp a ring two successive stages of the group of 1 00 simu l taneous states, we enumerate I 00 states twice. Enumera tion, that is, i nsofar as abstract
a ct
ion
i m plies ordered operations, means to
observe the 1 00 mechanisms one by one, classify th e m , and test them ; then
start agai n with
1 00
at th e following stage, and finally com p are the classes
number by n u m ber. And i f tates
a
fraction of tim e
x,
the
observation of each
it would take
200x of
mechanisms.
mechanism n ec es s i
ti m e to e n u merate 2 00
This a rg u me n t therefore allows us to transpose abstractly a s i m u l ta n
eity i n t o
a
lexicographic (temporal ) succession
wi thou t subtractin g
any
thing, however little, from the d e fi n i ti o n of transforma tions engendered by
scheme
Z.
Thus to compare two su ccessive stages of the
1 00 m ec h an i sms Z interval of time 1 00x of time l OOx (see Fig. III-7) .
co me s down to comparing I 00 states produced in an
with 1 00 o the rs produced MATER IAL
in
an equal interval
IDE NTI F I CATI O N
OF
MECHANISM
Z
Identification of mechanism Z m e a ns essentially
a
comparison between
Formalized Music
96
Period of 1 stage
"'
2!
�
0 0
1-
I k4
f==� · 1
time
=
1 OOx
Period of 1 stage
�r---time -
=
1
OOx
I
�� �,=-J
stage
1 stage
Fig . 1 1 1-7
all its possibilities of bei ng : per t u rbe d states compared to stationary states, independent of order. Identification will be established over equ al periods of time IOOx following the diagram :
Phenomenon :
PN ---+
Time :
IOOx
in which PN
an
E ---+
l OOx
PM ---+
E
l OOx
l OOx
d PM represent any perturbations and E is the state of Z
equilibrium (stationary state) .
at
An alternation of P and E is a protocol in which 1 OOx is the unit of time ( l OOx period of the stage) , for example : =
PA
PA
E
E
E
PH
Pa
Pa
E
Pc
A new mechanism W may be constructed with an ( MTP) , etc., which would control the identification and evolution of the compos ition over more general time-sets. We shall not pursue the investigation along these lines for it would lead us too far afield . A realization which will follow will use a very simple kinematic diagram of perturbations P and equilibrium E, conditioned on one hand by the degrees of perturbation P, and on t he other by a freely agreed selection.
�
E ---+ � ---+ � - E - � - � - � - � ---+ E ---+ �
Markovian
Stochastic Music-Applications
97
Def i n i t i o n of State E and of the P e rturbat i ons P
From the above, the stationary s t a te E wi l l be expressed by a se q uence of screens such as :
Protocol E(Z)
ADFFECBDBCFEFADGCHCCHBEDFEFFECFEHBFFFBC HDBABADDBADA DAHHBGADGAHDADGFBEBGABEBB
·
·
· .
To carry out this p ro toc ol w e shall u tilize eight urns [A ] , [B] , [C ] , [D] , [E] , [F ] , [G] , [H], each containing balls of eigh t different colors, whose proportions are given by the probabilities of (MTPZ) . For example, urn [G] wil l contain 40.8% red balls A, 7 . 2 % orange balls B , 2 7 . 2 /0 yellow balls C, 4.8/0 m aro on balls D , 1 0 . 2 /0 green balls E, 1 .8% blue balls F, 6.8% wh i t e balls G, and 1 . 2 70 black balls H. The comp osition of the other s e ve n urns can be read from (MTPZ) in similar fashion . We take a yellow ball C at random from urn [G] . We note the result and return the ball to urn [G] . We take a gre e n ball E at random from urn [C] . We note the result and re turn the ball to urn [C] . We take a black ball H at random from urn [E] , note the re s u l t, and return the ball to urn [E] . From urn [H ] we take . . . . The protocol so far is : GCEH Protocol P� ( V�) is obviously •
AAAA
Protocol P� ( V�) . Consider an are in t h e following proportions :
C,
33.6% color D, color H. After each be the following :
·
·
.
..
· •
urn [ Y] in which the eight col ors of balls A, 8.470 color B, 8.470 c ol or 1 .9% color E, 7 . 6 % color F, 7 . 6 '70 color G, and 30.4% draw return
2 . 1 i'o color
the ba ll to urn Y.
A
likely protocol might
GFFGHDDCBHGGHDDHBBHCDDDCGDDDDFDDHHHBF FHDBHDHHCHHECHDBHHDHHFHDDGDAFHHHDFDG ·
PTOtocol Pb
( V� ) . The same
method fu rnishes us
with a p ro to co l
EEGFGEFEEFADFEBECGEEAEFBFBEADEFAAEEFH ABFECHFEBEFEEFHFAEBFFFEFEEAFHFBEFEEB · · · .
Protocol P8 ( V8) :
CCCC · · · .
Protocol PB ( VB) : BBBB · · · .
·
· .
of P' :
98
Formalized Music
Protocol P(: ( V(:) : AAADOCECDAACEBAFGBCAAADGCDDCGCA DGAA GEC CAACAAHAACGCDAACDAA BDCCCGACACAACACB
R EA LI ZATI O N O F ANA L O G/Q UE A
·
·
· .
F O R O R C H ESTRA
The instrumental composi tion fol lows the preceding exposi tion point within the l i m i ts of orchestral instru ments and conve n tional execution and notation. The m e c h a n i s m which will be used is system Z, which has already been treated n u meri cally. The choice of vari ables for the s creens are shown in Figs . III-8, 9, I 0. by
(fo )
point,
Regions
0
J
r----1 I
If
I
r===I=J 2
"
3
m
lJl
s
'
lr===J ,c=J,
•
Freq u e n ci es
( s e m i t o n es)
c.
(f1 )
Regions
I, Jl, JJI
.B
. ----+1--...:11-!1-�...:1+---:::-+--:::--t---4� o-_,...--lI
.:2.
3
�
S
Fre q u e n c i es (semiton es)
c.
(A3 = 440 Hz) Fig. 1 1 1-8. Freq u e ncies
N uances of i ntensity
R egions F i g . 1 1 1-9. I nte nsities
N uances of i n te nsity
R e g i o ns
Markovian Stochastic Music-App lications
99
(d.) F i g . 1 1 1-1 0. D e nsities
This ch o i c e gives us the partial s creens FG (Fig. III- 1 1 ) and FD (Fig. 111-1 2) , the partial screens GD being a consequ ence of FG and FD. Th e Ro man numerals are the l i aison agents be twee n all the cells of the three planes of reference, FG, FD, and GD, so that the different combinations (j;, g1, dk) which are perc eived theoretically are made possible. Fo r exam pl e , let there be a screen (J1, g1 , d0) and the sonic entity C3 corresponding to fre q uency regi o n no. 3 . From the partial screens above, th i s entity will be the arithmetic sum in t h ree dimensions of the grains of cells I, II, and III, lying on frequency regio n no. 3 . C3 I + II + I I I . The dimensions of the cell corres p onding to I are : 6.F regi o n 3, 6..G region 1 , 6.D = region 2. T h e dimensions of the cell corres p ondin g to II are : 6.F region 3, 6.G re g i o n 2, !1D region I . The dimensions =
=
=
=
=
=
of the cell corresp onding to III are : 11.F !1D
=
region
l . Consequently
=
region 3, !1 G
in this sonic en tity the
gr ai ns
=
region 2,
will have
fre
quencies included in region 3, i ntensities i n c l u d e d in regions 1 and 2, and they
will form densities i n clu d ed set fiorth above.
in regions
1 and 2, with the
corresponden��
-/��:i;nJVNs• wl�'·-...
u {· :,, �· :o<>' {l t� u � ....· � �: . tj s �:) ; :.; h•.� t � : t: l· �.\.\) , \«1. ,) '!:hOr . .. �r··r-t;./ --.;.:;������� :::�-· · · •.
Formalized Music
1 00 G
c;
II 2
u
j
r
1/
2 .0
H
I
Ill
G
J/J
J:r
:J
I
2 1 ./
2
.0 Ill 3
If
(/1 /o)
s-
,;
2
1I 1/l
f
I
D
F
Fig. 111-1 1 . Partial Screens for FG D 3 2 f
.a I
D 3
ll II
Jll
f)
1!1
3
Fig .
D /0
1 11-1 2. Partial Screens for FD
J/J .N
(
p J
I
I
z
f
I D l/1 n.
S toch as tic Music-Applications
Markovian
101
The eight principal scree ns A , B , C, D , E, F, G , H which derive from the combinations in Fig. 11 1-5 are shown in Fig. 111- 1 3 . The duration l:!.t of each screen is 1 . 1 1 sec. ( 1 half note 5 4 MM). Within this duration the densities of the occupied cells must be realized. The pe riod of time necessary for the exposition of the protocol of each stage (of the p ro toco l at the station ary stage, and of the protocols for the perturbations) is 301:!.. t , which becomes 1 5 whole notes ( 1 whole no te 27 MM) . =
=
Screen JJ .a.
J I
f
Screen
C
#
lJl
flo (., d.,}
.J I
f'i
Eo
!!
Screen E
.I I
N
9 .N
ff{ if�II
Screen
I
G
f
f
H
Jj[. E1 02 Opa c. 84 A5
1
B
.I N
ti#
.9 r
{ N
r:J
H
�
J
1ll
JL
{I
(rf J r
rt· �� d�
Screen F ( /1 1..
Screen
1'+11
D
I
/I, (I do)
I
Eo
9
E C4 84 As
(I, fo do} J I
j
.Jil
J
1/l
E1 02 Opa
J
J1
f
1!
-1
.a
.9 I
Screen
,
I
I
;;
1
I!
II
(I� t'• dij
B
+E
H
{ff!1 df)
�:J
+1/1 �
1
:r 1Y
Fig. lll-1 3
N OTE : The numbers written in the cells are the mean densities in grains/sec.
Formal ized Music
1 02
The linkage of the perturbations and the stationary state of (MTPZ) is followin g kinematic diagram, which was chosen for this purpose :
given by the
F i g . 1 1 1-1 4. B a rs 1 05-1 5 of AnaJogique A
1 03
Markovian Stochastic Music-Ap p lications Fig.
III-14, b ars
1 05- 1 5 of the score of Analogi.que
of perturbations P2 and P�. The
A, comprises a sectio n period occu rs at bar 1 09. The Fig. III- 1 5 . Fo r technical reaso n s
ch ange of
disposition of the screens is given in
screens E, F, G, and H h ave been simplified slightly. 1 05
End of t h e period of perturbatio n pg Fig.
115
1 09
-> J -<-
Beginning of return to c q u i l i brium ( p ertu rbation P�)
11 1-1 5 Analogique A
replaces elementary
sinusoidal sounds by very ordered In any case a
clouds of elementary grains, restoring the string timbres.
realization with classical instruments could not p r od u ce tim b re other than that of s tri ngs because of the limits
screens
of human
h aving a playing.
The hypothesis of a so nority of a second order cannot, therefore, be
firmed or i nval i d at e d under these conditions. O n the o t h e r
con
hand, a realization using electromagnetic devices as and adequate converters would enable one to prove
mighty as compu ters the existence of
a
s ec on d
order sonority with elementary sinusoidal grains
or grains of the Gabor type as
a base.
While anticipating some such technique, which
has yet to be developed ,
we s h a l l demonstrate how more complex screens are realizable with the
resou rces of an ordinary electroacoustic studio equipped wi th several mag
netic tapes or synchronous recorders, filters, and sine-wave generators .
E L E CT R O M AG N ETI C M U S I C
TA K E N
(si nuso i d a l sou n d s )-EXA M P LE
F R O M A NAL O G/Q UE 8
We choose : I . Two groups offrcque nc y regions10,/1 , as in Fig. I I I- 1 6.
The p ro t o c ols of these tw o groups will
be s u ch that they
preceding ( MTP) 's :
1o
11
0.2
0.8
0.8
0.2
in which (a) and ({3) arc
the
1o ({3) 11
parameters.
1o
f1
0.85
0.4
0. 1 5
0.6
will obey the
Hz
Reg ions
r If 11 .-------, .-----, r------1 I I I I (12 811 ll ( 3
I Hz 42 Regions
z
a z tr---1-; lr---1-1 f1B MS 7"' 15 9 • s � ,
I
1
r-::1 lfl ¥
1' (!.!!,I 1r=-:-'t
I�
s
'
�
'
8
,,.
,
10
If*' II
ID
,.."
.u.n
II
II
14'
IJ
II
r1-
JS"
�·
,...
I�
(h)
Fig. 111-1 6
2. Two groups of intensity regions g0, g1, as in Fig. III-1 7 . The protocols of this group will again obey the same (MTP) 's with their
parameters
(y)
and
(y)
(e) :
�
go
'";'
0.8
81 G
� � �[ �
go
0.2
� "l
.. "" 0 ·;;,
..
c:
a:
.,
0
..... 0
.., ll.
0 "'
�
c: 0 .c: a..
.. ..
((o) Fig . 1 1 1-1 7
gl
0.8
0.2
(e)
�
gl
0.85
0.4
0.6
0.g,1 5 G
�[ �� �[ �[
�
-..
11(
.. �
c: 0
.,
·c. " a:
0 "' 0 "
11.. 0 ...
� ., "
c 0 .c: a..
(g�)
F
•
1 05
Markovian Stochastic Music-Applications
3. Two grou ps of density regions d0, d1 , as in Fig. III- 1 8. The
protocols
of this group will have the same (MTP) 's with parameters (A) and (p.) :
d1 (A)
Fig.
0.8 0.2
.j.
d0
d1
0.85
0.4
0. 1 5
0.6
1 1 1-1 8
us the principal screens A, B, C, D, E, F, G, H, as in Fig. I I I-1 9. The duration /j.t of each screen is about 0.5 sec. The perio d of exposition of a perturbation or of a stationary state is about 15 sec. We shall choose the same protocol of exchanges between p e rt u rb a tions and stationary states of (MTPZ) , that of Analogique A.
This choice gives
shown
�
E-� - � - E - � -� - � - � - E-�
The screens of Analogique B calculated up to now constitute a special ch o ice Later in the c ourse of this composition other screens will be used more particularly, but they will always obey the same ru l es of coupling and the same (MTPZ) . In fact, if we consider the combinations of regions of the variable }; of a screen, we notice that without tampering with the name of the variable }; its structure may be changed. .
Screen A
{;to (odo)
�I
.:;� I Screen 8
/).to a.,]
jl
2.
II c; l �.
Screen C
(f., #., d.)
3
.2 {
'i ,f
Screen D
(1. ;., d,)
3
-1
1
v If Scree n E
(ft (,. d.J
Screen F
{.lt fo dl)
3
.t
I
� " J
.2 f
Screen G
(l., ,., d� )
Screen H
(.ff g, d,J Fig. 1 1 1-1 9
t; 4' 3
:1.
-1
.:; 4'
j
/l 1
Markovian Stochastic Music-Applications Thus
forfo we may have the regions shown
107
in Fig. III-20. The Roman
n u me ral s establish the l i a is on with the reg ions of th e other two variables . Hz ���
Regions
1
2
811
.II
I�
1
Fi g . 1 1 1-20
tJ
7"'
t
Ill
,.,. II
n another combinationj0, as in
But we could have chose H z 42
Reg ions
li!UJ -ttl 1f
2
f
F
Fig. I II-2 1 . erso fL f;1
(/.,)
Fig. 1 1 1-21
This p romp ts the question : " Given n divisio ns
I!.F (regions on
is the total num b er of p ossible combinations of t:J.F regions ?
1st
case.
_,<¥
None of the
n
"'S"
Sf«>
F)
N#Dtl
ftt
what
areas is used . The screen corresponding to this
combination is silent. The number of these combinations will be
n!
2nd will be
case.
(n - 0) !0 ! ( = 1 ) .
One of the n areas is occu p ied. The nu mber of combinations
3rd case. Two of the
will be
mth
will be
case. m
of the
n
n! (n - 1 ) 1 1 !
n areas are
occu pied. The nu mber of combinations
n! (n - 2) !2 !
areas
are
occu p i ed . The
number
of combinations
n!
(n - m) !m ! F I G . 1 1 1 -1 9 : The Arabic n u m bers above the R e m a n n u m erals in the cells indicate the density in logarithmic u n its. Thus cell (1 0,1 ) will have a density of [(log 1 .3/log 3) + 5] rerts, which is 31 5.9
g rai ns/sec e n the overage.
F
1 08
nth
will
be
case.
Form alized Music
n
of the areas
are occupied. The number of the combinations n!
( n - n) !n ! The total number of combinations will be equal to the sum of all the preceding :
n!
..,. ! -, !0,.-: 0),. . ----.,... (n+
+
(n
n! I) ! 1 !
-
n!
n!
+
(n
-
[n - (n - I ) ] ! ( n - I ) !
2) !2 ! +
+ n!
( n - n) !n ! .
=
2n
The same arg um ent operates for the other two variables of the screen . Thus for the intensity, if k is the number of available regions !1G, the total number of variables g1 will be 2k ; and for the density , if r is the number of available regions !1D, the total number of variables d1 will be 2'. Consequently the total number of possi ble screens will be T
=
2(n + k + r> .
In the case of Analogique B we could o btain 206 + 4 + 7> 227 1 34,2 1 7, 728 different screens. Important comment. At the start of this chapter we would h ave accepted the richness of a musical evolution, an evol utio n based o n the method of stoch astic pro tocols of the coupled screen variables, as a function of the transformations of the entropies of these variables. From the precedin g calculation, we now see that without modifying the entropies of the ( MTPF) , (MTPG) , and ( M TPD) we may ob t ai n a su pp lementary subsidiary evo =
=
lution by utilizing the different combinations of regions (topographic criterion) .
Thus in Analogique B the (MTPF) , (MTPG) , and ( MTPD) will not On the contrary, in time th e ft , gi > d1 will have new structures, corol
vary.
laries of the changing combinations of their regions.
Com p l e menta ry Conc l u s i o n s about Screens and Their
Transform ati ons
I . Rule. To form
a
screen one may choose any combination
of regions
on F, G, and D, the };, g1, dk. 2 . Fundammtal Criterio n . Each region of one of the vari ables F, G, D
must be associable with a region corresponding to the other two variables in all the chosen couplings.
(This is accomplished by the Roman numerals. )
1 09
Markovian S tochastic Music-Appl ications
3. The preceding association is arbitrary (free choice) for two pairs, but obli g ator y for the third p air, a consequence of the first tw o . For e x ampl e ,
the
associ ations of the Roman nume rals off.. with those of g ; and with t h ose
of d,. are bo th free ; the association of the Ro man n umerals of gi with those of d,. is obli g atory, because of th e first tw o associ a tions.
4.
The co mp on e nt s f.. , g1, d,. of the screens generally have stochastic
protocols which correspond, stage by stage .
5.
The ( MTP) of these protocols will, in general, be c o u p l ed with the
help of parameters.
6. If F, G, D are the " variations " ( n u m b e r of components f.. , gh di, respectively) the 'maximum n u mb e r of cou p l i ngs between the com p onents and the p a r a m e ter s of (MTPF ) , (MTPG) , (MTPD) is the sum of the p ro duc ts GD + FG + FD . In an example from Analogique A or B : F
=
G
=
D
=
2
2 2
and th ere are
Indeed, FG
7.
(10 a n d j1 ) ( g0 and
(d0 and 12
l� +
the p a r ame t e r s of the
d1 )
couplings :
fl fa 11 ll
FD
If F, G ,
g1 )
.\
GD D a re the +
p. =
"
4 4
(MT P )
'
s are :
a,
y,
.\, p.
gl go gl de dl do dl � a >. p. a fl y ll
go
+ 4 +
=
12.
v ar ia t i o ns " (number o f components f.. ,
respectively) , the number of possi bl e screens T is the p roduct
example,
T
=
2 8.
X
if F = 2 (10 2
X
2
=
8.
and
{3
ll
j1) ,
G
=
2
( g0 and g1) ,
D
=
g1 , dk, FGD. For
2 (d0
and
d1) ,
The p ro t o co l o f the screens is stochastic (in the broad sense) and can
be summarized
(MTPZ) .
when th e
chain is ergodic (tending to regularity) , by an FGD columns.
This matrix will have FGD rows and
S P A T I A L P R OJ E C T I O N
all has been made in this chapter of the spatialization The subj ect was confined to t h e fu ndamental concept of a sonic complex and of its evo l u ti o n in itself. However n o thing would p revent broade n i ng of th e technique set o u t in this chapter and " leapi ng " i n to space . We can, for example, imagine protocols of screens a ttached to a particular p o i n t i n s p ace , with t r an s i ti o n probabil ities, space-sound co u p lings, e t c . T h e method is ready a n d the g e n e r a l application is possible, along with the recip rocal enrichments it can create . N o mention a t
of sound.
C h a pte r I V
Musical Strate gy-Strate gy, Linear Pro g rammin g , and Musical Composition Before
pa ss i n g
to the
problem of the
the use of compu ters, we shall take of games, their
theory, and
a
mechanization
of stochasti c music by
stroll in a more enjoyable realm, th at
application
in
musical
AUTO N O M O US M US I C
composition .
composer establishes a scheme o r pattern which the con ductor and the instrumentalists are called u pon to follow more or less rigor o us ly . From the final details-attacks, notes, in te nsiti es , timbres, and styles of performance-to th e fo r m of the whole work, virtually everything is written into the score . And even in the case where the com poser leaves a m argin of i mprovisation to the conductor, the instrumentalist, the machine, or to all three t o ge the r , th e u n folding o f the sonic discourse follows an open lin e withou t loops. The score-model which is presen ted to them once and for all does not give rise to any coriflict other than that between a " good " per formance in the technical sense, and its " musical expression " as desi red or suggested by the writer of the score. This opposition between the soni c realization and the symbolic schema which plot s its course migh t b e called internal conflict ; and the role of the conductors, instrumentalists, and their machines is to con trol the ou tput by feedback and comparison with the input signals, a role an alogous to that of servo-m echanisms that reproduce profiles by such m eans as gri nding machines. I n general we can s tate that The musical
1 10
Strategy,
Linear Programming, and Musical Composition
lll
techn i cal oppositions (instrumental and condu ctorial) or aesthetic logic of the musical discourse, is internal the wo r k s written until now. The tensions arc shut up in the score even
the nature of the
even those relating to the to
when more or less d e fi n ed stochastic processes
c l a ss of internal conflict
might be qualified
arc
utili zed . This traditional
as autonomous music.
F i g . IV-1 1 . Conductor 2. O r c h estra 3. Score 4. Aud ience
H ETE R O N O M O U S M U S I C
and probably v ery fruitful t o imagine another of external coriflict between, for instance, two opposing orchestras or instrumen talists . One party's move would influence and condi tion that of the o ther. The sonic discourse would then be identi fied as a very s trict, al though often stochastic, I t would b e i nteresting
class of musical discou rse, which would introduce a concept
succession of sets of acts of sonic opposition. These acts wou ld derive from
tw o ( or more) conductors as well aU in a higher dialectical harmony.
both the will of the
composer,
as
from the
will of the
112
Formalized
Mu sic
Let us imagin e a competitive situation between two o rch es tras , each having o n e
against
conductor. Each
of the condu ctors
di r e c ts sonic operations
the operations of the other. Each operation represents a
tactic and the encounter between two qualitative value which benefits
one
move
or
moves has a n u m eri cal a n d fo r
and harms the other.
This value
a
a
is
of t h e row corres p onding to v i of condu ctor A a n d the co l u m n corres p onding to movej of conductor
written in a grid or matrix at the i n tersection mo e
B . This is the p a r ti a l score iJ, re p resentin g
the paym ent one condu ctor gives two-person zero-sum game. The external conflict, or heteronomy , c a n take all sorts of forms, b u t can a l w ays be summarized by a matrix of payments ij, conforming to the m a the matical theory of games. The theory demonstra tes t h a t there is a n op t i m um way of playi ng for A , which , in the long run, guarante�s h i m a minimum advantage o r gain over B whatever B might do ; and that c on v e r s e l y there exists fo r B an op t imu m way of pl aying, which guarantees that his disad vantage or loss u n de r A whatever A migh t do will n o t exceed a certain maximum. A's minimum gain and B's maximum loss coincide in absolu te value ; this is called the game value. The in troduction of an external conflict or heteronomy into music is not entirely without precedent. In c e r t a i n traditional folk m u s i c in Europe and other c o n ti n e nt s there exist competitive forms of m u s ic in which two i nstru the other. This game, a
duel,
is defined as a
mentalists strive to confound one
a no th e r .
O ne takes the i n i ti ative
and
attempts either rhythmically or melodically to uncouple their tandem
arran gement, all the wh i l e remaining within the musical con text tradition which permits this special kind
di c to r y virtuos i ty is
of
of the
improvisation. This contra
particularly prevalen t among
the Indians, especially
among tabla and sarod (or sitar) players. th e
A musical heteronomy based on modern science
most conformist eye. But the
is thu s legitimate even to
problem is not the
historical j ustification of
a new adventure ; q u ite the contrar y , i t is the en richme nt and
forward that count. Just as stoch astic processes brough t
a
the
beautifu l
l eap
gen
co m p l e xi ty of linear polyphony and the deterministic of musical discou rse, and at the same time d is cl os e d an unsuspected opening on a tot ally asymmetric aesthetic form hitherto qualified as non sense ; in the s a m e way heteronomy introduces i n to stochastic music a c omp l e eralization to the
l ogi c
ment of dialectical s tructure .
We could equ ally well imagine setting up conflicts between two or more
instrumentalists, between one player and what we agree to call natural environ ment, or between an
orchestra or
several orches tras and the public.
But the fundamental characteristic of this situation is that there exists a gain
Strategy, and
a
L i n e ar Programming,
loss,
a
victory
and
material reward such as
for
a
a n d Musical C o m pos ition
113
a d e fe a t , which may be expressed by a m o r al or prize, m ed al , or cup for one side, and by a p e n alty
the other.
A
a more
is one in wh i ch the parties pl ay arbitrarily fol lowing or less improvised rou te, without an y condi tio n in g for conflict, and
degenerate game
compositional argument. This is a false game. A gambling device with s o u n d or lights would h ave a trivial s e n s e if it were made in a g ra t u i t o u s way, l i k e the usu al slot machines and juke boxes, that is, w i tho u t a new co mpe ti tive inner organization in s p i re d by any heteronom y . A sharp manufacturer might cash in on th i s idea and produce new so u n d and l ight d evices b ased on heteronomic principles. A less trivial u s e would be an e d u c at i o n a l apparatus which would re q uire children (or a d u l ts ) to react to sonic or l u m inou s combinations. The aesthetic i n te r es t , and h ence the rules of the game and the payments , would be determined by the players themselves by means of special input signals. In short the fundamental i n t eres t set forth above lies in the mutual con d i tioning of the two parties, a conditioning w h i ch respects the greater d ivers i ty of the m u s ic a l discourse and a certain liberty for the play e rs, but which i n v o lve s a strong i n fl u en c e by a si n gl e composer. This point of v i ew may be generalized with t h e introduction of a spatial factor in music and with the extension of the g a m e s to the art of light. In the fi e l d of calculation th e problem of games is r a pi dly b eco mi ng difficult, and not all games h ave received adequate mathematical cl ar ification, for exam p le, games for several players. We shall therefore con fine ou rs elve s to a rel atively s i m p l e case, that of the two-person ze ro - s um the re fo re without any new
game.
A N A LY S I S OF D UEL
This
1 958-59.
orchestras was co m p os e d in to relatively simple concepts : sonic constructions put into
work fo r two condu ctors and two
I t appeals
mutual correspondence
conditioned by the
Event I: A cluster
wooden part ally.
of
the
by the will of the
composer. The of
con d u ctors, who are
followi ng
eve n ts can occur :
themselves
sonic grains s u ch as pizzicati, b lo ws with the brief arco sounds distributed stochastic
bow, a n d very
Event II: Parallel sustained strings with fluctuations. Event Ill: N et wo r ks of intertwined s tri n g gl i ss a n d i . Event IV: Stochastic pe rc u s sion sou nds.
14
Formalized
Music
Event V: Stochastic wind instrument sounds. Event VI: Silence.
Each of these events is written in the score in a very precise manner and
with
sufficient len g th, so that at any moment, following h i s instantaneou s
choice, the conductor is able to cut out a slice without destroying the iden
tity
of the event.
imply an overa ll homogeneity in fluctuations.
We therefore
of each event, at the same time maintaining local
make up a list of couples of simultaneous events x, y issuing two orchestras X and Y, with our subj ective evaluations. We can
We can
from
the
the writing
also write this list in the form
of a
Table
qu alitative matrix (M1 ) .
o f Evalu ations
Couple
(x, y ) (1, I )
=
(y, x)
{ I , II ) = ( I I, I ) (1, III) = (III, I)
( I , IV) (IV, I ) (1, V) (V, I ) {II, II) (II, I II) = (III, II) (II, IV ) (IV, II) I V ( I, ) = ( V , II ) =
=
=
Evaluation
passable
(Ill, III) passable (III, IV) = (IV, I II) good + (I I I, V) = (V, II I) good (IV, IV ) passable (V good I I ( V, V) , V) passable (V, V ) =
( p)
(g) good + (g + ) passable + (p + ) very go od ( g + + ) passable (p) p as s ab l e (p) good (g) passable + (p + ) good
(p)
(g + ) (g) (p) ( g)
(p)
Strategy,
1 15
Linear Programm i ng, and Musical Composition
Conductor Y I
I
II
p
g
-
II Conductor X
g+ I--
p
p+
p
g+
g
p
p
g
p
p
p
g
-- -- --
p
p
11-
-- -- -- -
p+
IV
p
p
per row
g+ +
p+
g+
Minimum
V
IV
-- -- --
g r---
III
III
g+
g
-- -- -- -- -
g+ + p+
v
g+ +
Maximum per
column
In
(M1)
g
g
and the smallest maximum per no saddle point and no pure strategy. The introduction of the move of silence (VI) the
largest
g
colu m n do not coincide
minimum per row
(g
=f.
p) ,
and conse q u entl y the game has
modifies (M1 ) , and matrix (M2 ) results.
Conductor II
I
I
II III Conductor X
IV
p I--
g r---
g
g+
g+ +
IV g+
p
p
g
g+ p+
-- -- -- --
p
g+
g
-- -- -- --
g
g+
p
g
-- -- -- -- --
v
g+
p+
g
g
VI
V
-- -- -- --
g+ + p
r---
III
Y
p
p 1
-
1
-
p
p
1-
p
p
g+ +
g
p
p
p
p p
p
p
p
p
1-
-- -- -- -- -- -
VI
p
p-
Formalized
1 16 This
t i m e the
Music
saddle poi n ts . All tactics are possible, is still too sl ack : Conductor Y is interested i n playing tactic VI only, wh e re as conductor X can choose freely a m o ng I, I I , I I I , IV, and V. It m u s t not be forgotten that t h e rules of this matrix were established for t h e benefit of co n d u c to r X and that th e game i n
game has
several
but a closer study sh ows that the conflict
th is fo r m is n o t fair. Moreover t h e rules arc too vague. In order to pursue
o u r study we shall attempt to specify the qualita tive v a lu es by ordering them on an axis and m a k i ng them correspond to a rough numerical scale :
p- p I
If, in
p+
I 1
0
I 2
add i t i o n , we modify the value
becomes (M3) .
g
I 3
of
g+ I
g+ + I
4
5
the
c o u p l e (VI, VI) t h e matrix
Conductor Y I I
II
1 r--
3
1-
II
III IV V VI
3
5
4
4
1
I
3
2
1
3
1
3
I
1
1
1
I
3
4
4
3
---1-
III
5
1
1
4
IV
4
3
4
1
v
4
2
3
VI
I
1
1
5
3
5
Conductor X
I
- - - - 1-
3
---
1
1
1 - - 1-
( M3) has no saddle point and no recessive rows or columns. To find the solution we apply an approximat ion method, which lends i t sel f easily to c om p uter treatment but modifies the relative equilibrium of the en tries as little as possibl e . The p u rp o se of t hi s method is to find a mixed strategy ; that is to say, a weighted multiplicity of tactics of which none may be zero. I t is not possible to give all the calculations h ere [2 1 ] , but the m atrix that results from this method is (M4 ) , with the two unique strategies for X and fo r Y written in the margin of the matrix. Conductor X must therefore play
Strategy,
Li ne a r
Programming, and Musical
1 17
Composition
Conductor Y I I
II III
V
VI
2
3
4
2
3
2
18
3
2
2
3
3
2
4
4
2
1
4
3
1
5
IV
2
4
4
2
2
2
5
v
3
2
3
3
2
2
11
2
l
2
2
4
15
6
8
12
9
14
II III Conductor X
VI
---
I--
2
9 tactics
IV
1-
1-
- 1- -- -1-
(M4)
5 8 Total
in proportions 1 8/58, 4/58, 5/58, 5/58, 1 1 /58, conductor Y pl ays these six tactics in the propor tions 9/58, 6/58, 8/58, 1 2/58, 9/58, 1 4/58, respectively. The gam e value from this method is abou t 2.5 in favor of conductor X (g a me with zero-sum but still not fair) . We notice i m m e d i at e ly that the matrix is no longer symmetrical about its diagonal, which means that the t ac t i c cou p les are not commutative, e.g., (IV, II 4) =f. (II, IV 3) . There is an ori e n tati on derived from the adj ustment of the c al c u l a tio n wh ich is, in fact, an enrichment of the game. The following stage is t h e exp erimental control of the matrix. Two methods are possible : l . S i m u lat e the game, i . e . , mentally substitute on e sel f for the two conductors, X an d Y, by following the matrix entries stage by s tage , without memory and without bluff, in order to tes t the le a s t interesting case. I, II,
III,
IV, V, VI
1 5 /58, resp ectively ; wh ile
=
Game value :
=
52/20
=
2.6 points in X 's favor.
Formalized Music
1 18 2.
Choose tactics at random , b u t with frequencies proportiona:l to the
marginal
n um
b e rs
Game value :
the
in
57/2 1
(M4) .
=
2.7 points i n X's favor.
We now establish that the experimental game values are very close to
value
calcu l ated
by a p proximation.
The sonic pl"ocesses d erived from
the two experiments are, moreover, satisfactory. a
method for the definition of the optim um the value of the game by using methods of linear p arti cu lar the simplex m e th o d [22]. This method is based
We may now apply
strategies for X and programming,
on two theses :
in
rigorous
Y and
I . The fu ndamental th e o re m
is that the minimum
of game theory (the " minimax theorem ")
score (maximin) corresponding to X's opti m um
strategy is always equal to the maxim u m s c o re (minimax) corresponding to Y's
optimum strategy. 2. The calculation
of the m ax i m i n or minimax
probabili ties of the optimum s t r a te g i es of a
down to the resolution
(dual simplex method) . Here we
shall
of a
pair of dual
v al u e , j ust as the two - p erson zero-sum game, comes p r o b l e m s of li near programming
simply s t a te the s ys t e m of linear equations for the
of the minimum, Y. Let y 1 , y2, y3, y4 , y5, y6
be the
player
probabilities corresp ondin g
I, II, III, IV, V, VI of Y; y7, y 8, y9, y 1 0, y 1 1 1 y12 be the " slack " and v be the ga m e value which must be minimized. We then h ave following liaisons :
to tactics
variables ; the
2yl 3yl
2y1 3yl 2yl
4yl
+ + +
+
+
+
Y 1 + Y 2 + Ys + y,
3y 2
2y2
4y2 2y2 2y2 2y2
+ 4ya +
+ 2ya
+
+ Ys
+
2y5
+
2y4
+
+ 4y3 + 2y4 + + 3y3 + 3y4 + +
+
Y3
+ 2y4
Y3 +
4y4
+
+
3ys
2ys 2ys
3ys
1 2ys + Y7 = v v 2ys + Ya 2y6 + y9 = v 2ys + Y1o v 4ys + Y 1 1 = v
+ Ys
2y4 + 3ys
+ +
+
+
=
=
=
Ya + Y12
= v.
To arrive at a unique strategy, the c al c u la tion leads to the modification
S trategy, Linear Pro gramming, of the
score
(III, IV
=
4)
For
Tactics
(III, I V
into
ing optimum strategies :
and Musical Composition
5) . The solu tion gives For
X Tactics
Probabilities
I
6/ 1 7
II
III
0
III
3/1 7 2/ 1 7 4/ 1 7
v
VI
v
and for the game value,
42/ 1 7
=
must completely abandon tactic must avoid.
Modifying score (II, IV
strategies :
=
III
3)
2/ 1 7
2/ 1 7 1/17
2/1 7 5/ 1 7
2.47. W e have established that X of III 0) , and this we
(prob ability
to ( I I , IV
For X
Y
5/ 1 7
IV v VI
�
=
= 2 ) , we For
obtai n the following
Y
Probabilities
Tactics
Probabilities
I
1 4/56
6/56 6 /5 6 6/56
I II
1 9 /56
II
Tactics
III IV
v VI
8/56
6/ 56
III
IV
v
VI
7/56 6/56 1 /5 6
7/56
1 6/56
� 2.47 points. been modified a little, the game value has, in fact, not moved. But on the other hand the optimum strategies have varied widely. A rigorous calculation is therefore necessary , and the final matrix accompanied by its calculated strategies is (Ms) ·
and for the game value,
v
1
the follow
Probabilities
I
2/1 7
II IV
optimum
=
1 19
=
1 38/56
Although the scores have
Formalized Music
1 20
Conductor
I
3
i---
II Conductor X
II III IV V VI
2
I
3
i---
4
III
r-----
IV
2
r-----
v
3
i---
VI
Y 3
2
4
14
2
- - - - -
2
2
2
3
1
5
3
2
6
1
6
-
2
- - - -
4
4
2
2
3
2
!
-
2
6
2
8
(M5)
- -
2
3
2
2
1
19
7
6
2
2
r-----
7
16
4
16
56 Total
By applyi ng the elementary matrix operations to the rows and columns
in such a way as to make th e game fair (game value equivalent matrix (M6) with a zero game value.
=
0) ,
we obtain th e
Conductor Y I I
II
- 13
15
III
IV
43
v
- 13
15
- 13
15
VI 5 6
ll
- 13
-- -- -- -- -- -
15
II
- 13
- 13
5 6
- 13
..L
-- -- -- -- - - -
Conductor X
III
43
- 13
- 41
71
15
- 13
- 13
-- -- -- -- --
- 13
IV
43
43
- 13
- -- -- -- --
15
v
- 13
15
15
- 41
..L
- 13
�-
56
1-
1
5 6
J_ 5 6
- 13
-
- -- -- -- -- -
VI
- 13
- 13
- 41
- 13
5 6
As this matrix is
by
+ 13.
It then
d iffi cul t
5 6
1.§
43 5 6
ll
scores
- 13
(Me)
H. to read ,
it
is
simplified by dividing
becomes (M7) with a game value v
=
-
a ll the
0.07, which
Strategy, Linear
Programming, and Musical Comp osition Conductor
I
X
II
III
IV
-1
+1
+3
1-1--
V
VI
-1
+1
-1
TI
-1
1
+1
-1
_§__
-3
+5
+1
-3
_§__
+3
-1
-1
-1
_§__
+1
-1
-1
-1
- 1
-1
- --
-1
IV
-
--
+3
III
Y
-- --
+1
II
Conductor
I
+3
-- -- -- -- --
+1
v
+1
- 1
- --
-1
-1
VI
121
1-
-3
-- --
5 6
1--
56
5 6
_.!!._ 5 6
+3
TI
1 6
5 6
56
-1
1-
_L
.ll!.
14
(M7)
1 6
TI
means that at the end of the game, at the final score, condu ctor Y should give 0.07m poin ts to condu ctor X, where m is the total number of moves. If w e convert the n umerical matrix (M7) into a qualitative matrix
according to the corres p ondence :
-1
-3
T
p+
p
we obtain
+3
+1 I
I
g
T
g+
(M8) , which is not ve ry different from (lvf2) , ex ce pt for th e silence the opposite of the first value. The calculation is
couple, VI, VI, which is now
+5
finished.
p
p+
g+ +
p
g
p
p+
p
p
p
p+
p
g+ +
p
p
g+ +
p+
p
p
g+ +
g+ +
p
p
p
p+
p
p
p
g+ +
---
p+
p
p+
---
p
p
p
p
Formalized Music
1 22
Mathematical manipulation has brough t abou t a refinement of the
d u el and the emergence of total si l e nc e .
a
Silence is to be
paradox : the
co u p l e
to
avoided, b u t
VI, VI, ch arac terizing
do this it is n ecessary
to aug
ment its potentiality.
impossible to describe in these pages the fu n d a m e n t a l role of the t h i s p roblem, or the subtle arg u m e n ts we are fo rced to m ake on the way. We must be vig i la n t at every moment and over every part of the m a tr i x area. It is an i n s t a n c e of t h e kind of work where detail is dominated by the whole, an d the whole is dominated by de tail. It was to show the value of this i ntel l e ct ua l l abor that we judged it useful to set out the processes of calculation . The conductors d i r e c t with their backs to each other, using finger or light signals that are invisible to the oppos i n g orchcst;a. If the conductors use illuminated signals operated b y buttons, t he successive partial scores can be announced au tomatically on l ighted p a n els in the hall, the way th e score is d ispl ay e d at fo o tb al l games. If t h e conductors j ust usc their fingers, t h e n a referee can count the poin ts and put up the partial scores manually so they are visible in the hall. At t he end of a certain number of exchanges or minutes, as agreed up on by the conductors, one of the two is declared th e winner and is awar d ed a prize . Now that th e principle has been s e t out, we c a n envisage t h e interven tion of the public, who would be invited to evaluate the pairs of tactics of It is
mathematical treatment of
conductors X and composer
Y and
vote immediately on the make-u p of the game
music would then be the result of the conditioning of the who esta b lished the musical score, conductors X and Y, and the
matrix. The
public who construct the matrix of points.
R U LES O F TH E WO R K
The two-headed flow c h ar t
of Duel is
valid for Strategie, composed in 1 9 6 2 .
STRA TEGJE
shown in Fig.
The
IV-2. It
is equally
two orches tras are placed on
back-to-hack (Fig. IV-3), or on auditorium. They may choose and play o n e of six sonic constructions, numbered in the score from I to VI. We call them tactics and they are of stochastic structure. They were c alc u l ate d on the IBM- 7090 in Paris. In addition, each conductor can make his orchestra pl ay simultaneous combinations of two or three of these fu n d am e n ta l tactics.
either side
of the
stage, the conductors
platforms on opposite sides of the
The
six fundamental tactics
are :
1 23
S trategy, Linear Programming, and Musical Composition
I.
Winds
II . Percussion
I I I . S tring s o u n d box struck -
with the hand
IV. String pointi llistic effects
V. VI.
S tring glissandi
S u s tai n e d s tring harmonics.
Th e following are 1 3 compatible and simultaneous combinations of these tactics :
I
& II
=
VII
I & III = VIII
& IV = I X I & V = X I & VI = XI
I
II &
II &
III IV
II & V
I I & VI
XII
I
XI II
I & II &
=
=
= XIV
XV
=
I I
II
&
&
III IV II & V &
& II & VI
Thus there e x i st in all 1 9 tactics which each conductor can tra play,
The
36 1 ( 1 9
x
1 9 ) p o s si b le pairs that m ay
Game
1. Choosing tactics.
H o w will the
play ?
=
XVI
=
XVI I
=
=
XVIII XI X
make
his orches
be played simultaneously.
c o n d u ct o r s
choose which ta ct i c s t o
a. A first solution consists of ar b i tra ry choice. For example, conductor X chooses tactic I. Con d ucto r Y may then c ho o s e a ny one of the 19 tactic s including I. Conductor X, acting on Y's ch o i c e then chooses a new tactic (see Rule 7 below) . X's seco nd choice is a function of both his taste and Y's choice. I n his turn, cond uctor Y, acting on X's choice and his own t aste, either chooses a new tactic or keeps on with the old one, an d pl ays it for a certain optional length of time. And so on. We thus obtain a continuous ,
succession of couplings of the 19 structures .
b. The conductors draw lots, choosi n g a new tactic by taking one card a pack of 1 9 ; or they might make a d ra w i n g from an urn containing balls n u m b e r ed from I to XIX in d i fferent proportions. These operations can be carried out before the performance and the res u lts of the su ccessive draws set down in the form of a sequential plan which each of th": conductors will
from
h ave before him during the performance. c.
The conductors get together i n ad v a n ce and choose a fixed su ccession will direct. d. Both orchestras are directed by a s i n g l e conductor who establishes the succession of tactics according t o one of the above methods and sets
which they
them down on a mas ter plan, which
he will follow
during the performance.
Formalized Music
1 24
Fig . IV-2
1 . G a m e m a t r i x ( d y n a m ostat, d u a l reg ulator) 2 . Conductor A (dev i c e for c o m pa riso n a nd decision) 3 . Condu ctor B ( d e v i c e fo r c o m p a r i s o n a n d d e c is i o n )
4. Score A (symbolic excitati on) 5. Score B ( sym bo l i c excitatio n ) 6. O r c h estra A ( h u ma n o r e l ectro n i c tra ns fo rm i n g d evice) 7. O rc h estra 8 ( h u ma n o r e l ectro n i c t ra ns
forming
d evi ce)
8 . A u d i e n ce
B
orchestra :
1 1 1 1 1 1 1 1 2 1 2
Percussion
Bp c l a ri n et
�--
contra bassoon F rench horn
trum pet tro mb ones tu ba perc u ssion 1 vibra p h o n e
4 tom -toms 2 bongos 2 congas 5 temple b l ocks 4 wood blocks 5 bells fi rst viol i ns second viol i n s violas cellos double basses
44 instruments, or 88 players i n both o rchestras
::;;;,c [
"'
c
g" (i
� !!!. 0 0 :::l "'
:::l
I
(!)
5!:
r::r "'
(")
!e.
�
<
a· iii "'
� t:""'
s·
� "" ....,
...., 0 (lq ...., ""
I
..-------, 0 0 c:
,....
Vibraph one
'i:i
y
X
Cond uctors <
<
§.:
a· :;·
"'
"'
:;·
g: (!) ..
<
()" iii
"'
(")
!e. 0 "'
0 0 c: r::r
;r::r ., 1/) 1/) "' "'
...- Public __,.
+
3 3 s· �
Brasses
! Woodwinds
Woodwinds I
B rasses
� oq
Percussion
Mari mba phone
bassoon
1 bass drum
8 8 4 4 3
P l a cement of the O rc h estras on a S i ngle Stage
piccolo flute E[> c l a r i n et
1 m a ri m ba phone 1 m a racas 1 suspended cym ba l
Ul r+ ...., ""
Fig. I V-3 . Strategy
Composition of the
N OTE: If two stages a re used, each orch estra is arra nged in the cla ss i c m a n n er.
u
s:: -o 0> � �. C") 3 c r::r l::
0> -· -c 0 ::r "' 0 :::l (!)
"" :::1 p.
�
s:: "'
o· e:.. 0 0
3
"0 0 �. ::r. 0 :::
...... }.j U1
Formalized Music
126 e.
Actually a l l these ways const i tu te what one ma y caJI " degenerate "
competi tive situ ations. The only worthwhile setup, which adds something n ew in the c a s e of more than one orchestra, is one that in trod uces dual
conflict between the con d u ctors . In this case the pairs of tactics are per
formed sim ultaneously without i n terru ption from one choice to the next
(sec
Fig. IV-4) , an d the d ec i sions made by the conductors are conditioned
by the winnings or losses co n t a in ed in the game matrix.
X
C Oh
OR.
COlo
·oR. Y
G • INJ
"TIICTICS
tfA iNJ
TA C TICS
72
7ll
X V///
IX
.5"2
VI/
40
X IX
4IS
X V
XIV 48
XV
J& :28 v
VII
"'"
--- - -
Fig. IV-4
2.
Limiting the game. The game may be limited in several ways : a. The a ce rtain number of po i nts , and the first to reach
conductors agree to play to
it is the winner. b. The conductors agre e in a dva n ce to play n engagements. The one with more poi n ts at th e end of the nth engagement is the winner. c.
The conductors d ec id e on the du ration for the game, m seconds
(or
minutes ) , for instance. The o n e wi th more points at the end of the mth
second (or mi n u te) is the winner.
3. a.
Awarding poinLf.
One me t hod is to h ave one or t wo referees co u n t i ng the points in two columns, one for conductor X and one for conductor Y, b o th in positive numbers. The referees stop the game after the agreed limit and announce
the result to the public.
b. Another m eth od has no referees, but uses an automatic system
that
consists of an i n dividual board for each conductor. The b o ard has the
n x n ce lls of the game matrix used. Each cell has the corres p onding partial score and a p us h b utton . S u ppose that t h e game matrix is the large one of
19 x 19 cells. I f conductor X ch oos es tactiC XV against Y's IV , he presses the b utton at the in tersection of row XV a n d column IV. Corresponding to this in tersection is the cell containing the partial score of 28 points for X and the bu tton th at X m u s t push. Each b u t t o n is con n ected to a
small
adding
machine which totals up the results on an electric p a n el so that t hey can be
seen by the public as the game proceeds, just like the panels in the football
stadium, but on a smaller scale.
4. Assigning of rows or columns is made by the conductors tossing a coin. 5 . Deciding who starts the game is determined by a s e con d toss.
Strategy, Linear Programming, and Musical Composition
127
6. Reading the tactics. The orchestras perform th e tactics cyclically o n a closed loop. Thus the cessation of a tactic is made instantaneously at a bar line, at the discretion of the conductor. The subsequent eventual resumption of this tactic can be made either b y : a . reckoning from the bar line defined above, or b. reckoning from a bar line identified by a particular letter. The conductor will usually indicate the letter he wishes by displaying a large card to the orchestra. If he has a pile of cards bearing the letters A through U, he has available 22 different points of entry for each one of the tactics. In the score the tactics have a duration of at le ast two minutes. When the conductor reaches the end of a tactic he starts again at the beginning, hence the " da capo" written on the score. 7. Duration of the engagements. The du ration of each e ngage ment is optional. It is a good i d ea, however, to fix a lower limit of about l O seconds ; i.e., if a conductor engages in a tactic he must keep it up for at least 10 s econd s This limit may vary from concert t o concert. It constitutes a wish on the part of the composer rather than an obligation, and the conductors have the right to d ecide the lower limit of duration for each engagement before the game. There is no upper limit, for the game itself conditions whether to maintain or to change the tactic. 8. Result of the contest. To demonstrate the dual structure of this compo sition and to honor the conductor who more faithfully followed the con ditions imposed by the composer in the game m atrix, at the end of the combat one might a. proclaim a victor, or b. award a p rize, bouquet of flowers, cu p , or medal, whatever the concert impresario might care to donate. 9. Choice of matrix. In Strategic there exist three matrices. The large one, 19 rows x 1 9 columns (Fig. IV-5) , contains all the partial scores for pairs of the fundamental tactics I to VI and their combinations. The two smaller matrices, 3 x 3, also contain these but in the following manner : Row I and column I contain the fundamental tactics from I to V I without discrimina tion ; row 2 and column 2 contain the two-by-two compatible combinations of the fundamental ta c ti cs ; and row 3 and column 3 co n tai n the three-by three compatible combinations of these tactics. The choice between the large 19 x 1 9 m a tri x and one of the 3 x 3 matrices depends on the ease with which th e condu ctors can read a matrix. The cells with positive scores mean a gain for cond u c t or X and autom atically a symmetrical loss for conductor Y. Conversely, the cells with negative scores mean a loss for conductor X and au tom atical ly a symmetrical gain for conductor Y. The two si m ple r, 3 x 3 matri c es with differ e n t strategies are shown in Fig. IV-6. .
1 28
Formalized
Music
M AT R I X O F T H E G A M E
-
-
.r
-lib - �
%
Jt
•
,, - � -U
9'
-� 0
J ll
8 1r
�"
12
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- flo _,, -!6 Jl(
-.fl, - fJ
11 -J<;
•
liZ
'" - +8
,. -3] t
,
ID
• I - ,,
;;;; JlL
r
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r 11 6 "'
* I
.
- tl ID
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to
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f:t
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-8
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fl 88
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,
92.
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0
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- 52 - ·· - �·
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1'1' 3.Z
" -Jo
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1lC
Condu ctor Y (columns) � � � -:' "::' : .,. r ,.... . , .. =
I�
16 8
-2
4
-ft
-It
.'lo
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41
1,-1, -51
lr&
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f.l n
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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
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d ifferent tactics
. ., 11-�.d , .., ""
_ ,., -
-B -3& -.v o
• Combi nations
�
- ��
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Fig. IV-5. Strategy Two - person G ame. Va l u e of the Game
'
•
&I
_,., - .I& -2o -3.t
"
•
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- � - .36
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- ¥�
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-llo
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,
B
'
and Musical Comp osition
Strategy, Linear Programming,
I;
-6�
Ill
:. •
lH
8 �!
4' 0 Ill •
-1
3
-3
3
.... .
:. . ' *' . ' Ill • •
0
3
• y ti H Y
>< 0
'
· · · � · · · · � ... . ..
n
I�
•
Strategy
for
Y
4
A
{-,
... . 4' 0 m• H O
.. . . :. . '
*' " ' Ill e f
-3
Strategy
F i g . IV-8
Two · person Zero-sum Game. Value of the Game = 1 /1 1 . This game is not fair for Y. •
Y Wooawinas
H Strings striking sou nd - boxes • ••
Normal percussion Strings pizzic ato
ill Strings sustained
Jl-
-1
0
-5
-1
4
3
1
-3
.2 • '
�� : a:;
for
Conductor Y Icolumns) J: • : * ;; • :.t · : if!:;; x - �• • • • • • X " # 'il .. .. .. ., . . . • • . . .. .
3
� t.
.. . .
-5
•o
1 29
Y
A •
Zero-sum Game. Value of the Game = 0. This game is fair for both conductors.
Two - person
' . ' "
' "" ' m O H . :.
., :.
of two dis
Combinations
tinct tactics
' • H
; :�
' •m
Combi nations
of three dis tinct tactics
• IU
Strings glissando
. ,..
Simplification of the 1 9
x
1 9 Matrix
To make first performances easier, the conductors might use an equiva lent 3 x 3 matrix derived from the 1 9 x 1 9 matrix in the following manner: Let there b e a fragment of the matrix containing row tactics r + I, • • • , r + m and column tactics s + I, . • • , s + n with the respective probabilities 'lr+ h • • • , 'lr + .. and k, .. , • • • , k,.. •• qr + l 'lr- + f
fr + m
*• + 1
4r+ I .I + l
k, .. ,
k1 + n
Dr+ l , I + J
Dr + l . I + R
Dr + t .a + :l
a,. ... . . . + J
Or + e . a + n
a, + lll ,l + l
a, + • . • + l
Dr + m , a + n
��:a;•gy i
Formalized Music
1 30 This fragment
can
be replaced by the single score
A• + m .• + n
Li.i ::'/ � n (ar + ! ,s + J) (qr + !) (ks + J) Lr q. + , 2J ks + t
-
_
and b y the probabilities
m
Q
=
and
:L q, + , 1=1 n
K
Operating in this way matrix (the tactics will be X
X
-
25
X
1 9 m atri x we obtain the matrices i n Fig.
X
25
592 25
X
45
93 1 4
49
X
30
X
25
26
3088
1 76 1 0
-
30
25
or
x
8296
49
49
45
68 1 8
25
J=!
the same as in
1 45 2 2
25
L ks + J·
wi th the 1 9
7 704 25
=
45
X
26
2496
30
49
X
26
26
2465
- 1 354
1 82
25
- 2581
1 59 7
- 52 8
45
1818
- 1 267
640
30
25
49
26
45 30
the following IV-6) :
C h a pter V
Stochastic
Free
After this interlude , we return
Music
by Computer
to the treatment of composition
by machines.
The theory put forward by Achorripsis had to wait fo u r years before
being realized mechanically. This realization occurred thanks to
M.
Fran<_;ois Genuys of IBM-France and to M. Jacques Barraud of the Regie
Autonome des Transports Parisiens. THE A STO C HASTIC
PARA DOX :
WORK
M U S I C A N D CO M P U T E R S
EXECUTED
The general public has
a
BY
number
THE
IBM-7090
of different
reactions when faced by
the alliance of the machine with artistic creation. They fall into three categories : " It is impossible to obtain a
work cif art,
s i n ce by definition it is a handi
craft and requires moment-by-moment " creation " an
for
each detail and for
and cannot invent." " Yes, one may play games with a machine or use it for speculative purposes, but the result will not be " finished " : i t will represent o n l y an the entire structure, while a machin e is
inert thing
experiment-interesting, perhaps, but no more . "
The enthusiasts wh o at the o u tset accept witho u t flinch ing the
whole frantic brouhaha of science fiction . " The moon ? Wel l , yes,
within
our reach. Prolonged
life will
also be with
us
tomorrow-why
it's
not a
creative machine ? " These people arc among the credulous , who, in thei r i diosyncratic optimism, have replaced t h e myths of Icarus and the fairies,
which have decayed, by the scientific civil ization
partly agrees with the m In paradox nor all animism, for it progresses foreseeable at too great a distance . a n d science
.
131
of the
twentieth century,
reality, science
is
nei ther all
in limited stages that are n o t
Formalized
1 32
Music
the
arts w h a t we may c a ll ratio nalism in the etymo logi cal s e n s e : the search for proport i o n . The artist h a s always c alled upon it o u t of necessity. The r u les of construction have v a r i e d widely over the cen
There exists in all
tu ries, b u t t h e re h ave alw ays b ee n rules in every epoch because of the neeessi ly of m a k i n g oncscl f u n d e rstoo d _ Those wh o be l i eve the firs t statement
first to re fuse to a p p l y do not understand at alL
above arc the which they
the q u al i fication
Thus the musical scale is a convention
artistic to
which circumscribes
product
a
the
area
of
potentiality and perm i ts construction wi thin those li m i ts in i ts own p articu lar
symmetry.
The r u l es of Christian hymnography, of harmony, and of
coun terpoint in the various
ages
h ave al lowed artists to cons t r u c t
and
to
m a k e themselves u n d erstood by those who adopted the same constraints
through tradi tions, thro ugh col lective taste or imitation, or through sym
p athetic resonance. The rules of serial ism, for instance, those that
the t r aditiona l
octave doublings of tonal i ty,
partly new but none the less real .
banned
i m p os e d constraints which were
Now everything that is rule or repe ated constraint
is
part of the mental
machi n e . A little " i maginary machine," Philip pot would have said-a
choice, a set of decisions. A musical work can be analyzed of mental machines. A melodic theme in
a
as a
symphony is a mold,
multitude a
mental
machine, in the same way as its structure is. These mental machines
something very restrictive
i n decisive. In the last few years
really a
very general
are
and of mech a n ism is
and determ inistic, and sometimes very vague we
have seen that this idea
one. It flows through every area of h uman knowled ge
and action, from strict logic to artistic manifestations. Just as the wheel was o n c e
one
of the greatest produ cts of human
intelligence, a mechanism which allowed o n e to travel farther and faster with more luggage, so is
the compu ter,
which t o d ay allows the transforma
tion of man's ideas. Computers resolve logical problems by heu ristic methods .
But computers are not rea l ly responsible for the in trod uction of mathem atics
in t o music ; rather it
is mathematics that m akes use of the computer in if people's minds are in general ready to recognize the u s e ful ne s s of geo met r y in the plastic arts (architectu re, painting, etc.) , th ey have only one more stream to cross to be able to conceive of using more abstract, n o n vi s u al mathematics and machines as aids to musical composi tion, which is more abstract than the plastic arts.
co mpo siti o n . Yet
-
To
summarize :
l . The
creativ e thought
of
man gives birth to me n tal mechanisms,
·
1 33
Free Stochastic Music by C ompu ter
which, in the last analysis, are
merely s et s
of constraints
and choices . This
process takes place in all realms of thought, including the arts.
2. Some of these mechanisms can be expressed in mathematical terms. 3 . Some of them are physically realizable : the wheel, motors, bombs,
digital computers, analogue computers, etc.
4. C e r tain mental mechanisms may c orre spond of nature.
to certain me ch a ni sms
5. Cer ta i n mechanizable aspects of artistic creation may be s im u l a t ed by ce r tai n physical mech anisms or machines which exist or may
6. It h appens
Here then
e l ectro n ic
be cre ated .
that computers can be useful in certain ways .
is the theoretical point of d epartu re for
a utilization of
computers in musical composition .
We may further establish that the role of the living composer seems to
have e volve d , on the one hand, to one of i n v e nt in g schemes (previously forms) and exploring the limits of these schemes, an d on the other, to effect
ing the scientific
synthesis of the new
methods of construction
and of sound
emission. In a short while the se methods must comprise all the ancient and
modern me a ns of m u s i c a l instrument making, whether acoustic or electronic,
with the help, for example, of digital-to-analogue converters; these have already been used in communication studies by N. Guttman, J. R. Pierce, and M.
V.
Mathews of Bell T el e ph on e Laboratories
in New Jersey. Now
these expl ora tions necessitate impressive mathematical, logical, physical,
and psychological i m p ed im e n ta , e s pec i a l ly computers that accelerate the
mental processes necessary for clearing the way for new fields by providing immediate
experimental verifications
at all stages of musical construction .
Music, by its very abstract nature,
is the
first
of
the arts to h ave
at
tempted the conciliation of artistic creation with scientific thought. Its
industrialization is in evita b le and irreversible. Have we not already seen
attempts to industrialize serial and popular music by the Parisian team of
P.
Barbaud, P. Blanchard, and J e ani n e Ch a rb o n ni er, as well
musicological research of Hiller and Isaacson at the University In the pre c edi n g
ch apte rs we
as
by the
of Illinois ?
of musical mini m at hem a ti cs and e spe ci a lly
demonstrated some new areas
cre a tio n : Poisson, Markov processes, musical games, t he thesis of the
mum of constraints, etc. They are all based on on
the
theory of probability. They therefore lend th em selve s to b e i ng
treated and expl ore d by computers. The simplest and most meaningful
s che me is one
of minimum constraints in composition, as exemplified by Achorripsis. Thanks to my friend Georges Boudouris of th e C.N.R.S. I made the
1 34
Formalized
acq uaintance of
Jacques
director of the Ensemble
Barraud , Engineer of the Ecole
Electroniqucs de
des
Gcstion de Ia Societe
Music
Mines,
then
des Petroles
mathematics, and head of the All three arc scien tists, yet they co n s e n t e d to attempt an ex p e r i m e n t which s e e m ed at first far-fe tch ed that of a marriage of music w i th one of the m o s t powerfu l machines in the
S hell-Berre, and Fran�ois Gcn u ys, agrcgc i n
E tu d e s Scientifiq u cs Nouvelles at IBM- France .
world .
rel a t i o n s it
rarely pure lo g ic al persuasion which is is material interest. Now in t h i s case it was not l og i c much less self-in terest, that arranged the be trothal, but p u rely exp eri m e n t for experiment's sake, or game for game's sake, that indu ced collaboration. Stoch astical ly spe a k i n g my ventu re should have encoun tered failure . Yet the doors were opened, and at the end of a year and a half of contacts and h a rd work " the most u n u s u al event wit ness ed by the firm or by this musical season [in Paris] " took place on 24 May 1 962 at the h eadquarters of I BM-France. It was a live concert pre senting a work of stoch astic instrumental m u s i c entitled ST/ 1 0-1, 080262, which had been calcu lated on the IBM-7090. It was brilliantly p e rformed by the conductor C. Simonovic and his Ensemble de Musique Contem poraine de Paris. By its p assage t h rou g h the m achine, this work made tangible a stochastic method of com position, that of t he minimum of constraints and rules. In most human
is
i mportant ; usually the paramo u n t consideration ,
,
Positi o n of the Problem
w as the drawin g u p of the flow chart, i.e. , down cl e arly and in o rder the stages of the operations of the scheme of Achorripsis, 1 and adapting it to the machine stru cture. In the first chapter we set out the entire sy n th e t i c method of this minimal s tr u ctu re Since the machine is an iterative apparatus and p erfo rm s these iterations with extra o rdi n a ry speed, the thesis had to be broken down into a sequential series of op erati ons reiterated in loops. An excerpt from the first flow chart is shown in Fig. V- 1 . The statement o f the thesis o f Achorripsis receives its first machine The first working ph ase
writing
.
oriented interpretation in the following manner :
I . The work consists of a succession of sequences or movements each a1 seconds long. Their du ra ti o ns are totally independent (asymmetric) but have a fixed mean du ration, wh i ch is introduced in t h e form of a parameter. These durations and their stochastic succession are given by the formula
(See Appendix I.)
1 35
Free Stochastic Music by Computer
f li s s iHtoli.. YJL
(vic,c)
,..... rr YJI B
TGLb le. : 9(.;1) �t. .. etio .. :Z. S"o va iiA t �
9�-1
- :t- � 5 ' �-1o ot. = B t G� - Bt- i tF �F
j �
=
oJ.. ==
k'll'f = \l r c;; L = To
YUL
F i g . V-1 . E x c erpt from the F i rst Flow C h a rt of A chorripsis
Formalized Music
1 36
2. Dqinition of the mean density of the sounds during a1• During a sequence sou nds are emitted fro m several so n i c s o u rces. If the total n u m be r of these s oun ds or poi n ts during a sequence is Na, • the mean density of this point clust e r is Na.fa1 sou nds/sec. In ge ne r a l , for a given instru mental ensemble this d e nsity has limits that depend on the number of instru mentalists, the nature of their i nstruments, and the tech nical difficu l ties of performance. For a large orch estra the upper l i m i t is of the order of 1 50 sounds/sec. The lo we r limit ( V3 ) is arbitrary and positive. We choose ( V3) 0. 1 1 sounds/sec. Previous exp eriments led u s to ad o p t a logarithmic progression for the density sensation wi th a number between 2 and 3 as its base. We adopted e = 2 . 7 1 82 7 . Thus the densi ties a re i ncluded between ( V3) e0 and ( V3) eR sounds/sec. , wh ich we can draw on a line graduated logarithmically ( b ase e) •2 As o u r purpose is total independence, we attribute t o each of the sequences a1 calculated in 1 . a density re pres e n te d by a point drawn at random from the portion of the line men tioned above. However a certain concern for continuity leads us to temper the independence of the densities among seq u e n ce s a1 ; to this end we in troduce a certain " memor y " from sequence to sequence in the fo l l ow i n g manner : Let a1 _ 1 be a seq uence of duration a1 _ 1 , (DA) 1 _ 1 its d ensity, and a1 the next sequence with duration a 1 and d ens i ty (DA ) 1• Density (DA)1 will be given by the formula : =
in
which
length
x
is
a
eq u al to
segment
(R - 0) .
of line PX
and finally,
d rawn at
random
from
The probability of x is given
=
Na,
�
S
=
(1 - �) dx S
a
by
line
segment
s
of
(see Appendix I)
(DA),a,·
3. Composition Q of the orchestra during sequence a1• First the instruments are divided into r classes of timbres, e.g., flutes and clarinets, oboes and bassoons, brasses, bowed strings, pizzicati, col le g n o strokes, glissandi, wood, skin, and metal percussion instruments, etc. (See the table for Atrees.) The
composition of the orchestra is s tochastically conceived, i . e. , the distribution of the classes is not deterministic. Thus d u ri n g a seque n ce of d u rat i on a1 it may happen that we h av e 80/0 pizzicati, 1 0% percussion, 7 /0 keyboard, an d 3 /0 flute class. Unde r actual cond i ti on s the determining factor which would condition the co m p ositio n of the orchestra is density. We therefore
Free Stochastic Music
by
137
Computer
Composition of the Orchestra for Atrties Timbre classes and instruments Class
Timbre
Percussion
as
Instrument
Temple-blocks Tom-toms
Maracas
Susp. cymbal
2 3 4
Horn
5
Glissando
Flute Clarinet
Gong
French horn
Flute Clarinet B� Bass dar. B�
Violin
Trombone Tremolo or
fluttertongue
Flute Clarinet B� Bass clar. B� French horn Trumpet
Trombone
a
Trombone b
(pedal notes )
Violin Plucked
Violin
strings
Cello
Struck
Violin
strings*
Cello
9
Vibraphone
Vibraphone
Trumpet
Trumpet
12
Trombone
Bowed
Violin
strings
Cello
* collegno
I
3
I 2 3 4 5 6 7 8
2
Trombone a Trombone b
( pedal
2
1 2 I
8 10 ]]
l-5
6-9 10 11 12
9
Cello 7
Instrument No.
2
Cello
6
(ST/1 0-3,060962)
on present input data
notes)
1
2 1
2
Formalized Music
138
connect the orchestral composition with d ensi ty by means of a special diagram. An example from ST/10-1, 080262 is sh own in Fig. V-2. Fig. V-2 is expressed by the formula
Q.
=
(n - x) (e,.,. -
inwhichr thenumberofthe class,x such that n :S x :S n + I, and en.r and =
as
=
a function of n.
e,.+l,r
)
+
e,. .•
loge[(DA)1/(V3)],n = 0, 1,2, ,R, en+ 1 •• arc the probabiliti es of cfass T .
•
.
without saying that the composition of this table complexity and delicacy. Once these preliminaries have been completed, we can define, one after the other, the Na., sounds of s equence a1•
is
a
It goes
precise task of great
4. Definition of th e moment of occu"ence of the sound N within the sequence a,.
The mean density o f t he points or sounds to be distributed within
k
=
N4.fa1•
a,
is
The formula which gives the intervals separating the sound
attacks is
(See Appendix I.)
5. Attribution to the above sound of an instrument belonging to orchestra Q, which has already been calculated. First class r is drawn at random with proba bility q. from th e orchestra ensemble calculated in 3. (Consider an urn with
balls of r colors in various proportions.) Th en fro m within cl ass r the number
of the instrument is drawn according to the probability Pn given by an arbitrary table (urn with balls of n colors ). Here als o the distribution of i nstruments within a class is delicate and complex. 6. Attribution of a pitch as a function of the instrument. Taking as the zero point the lowest Bj;, of the piano, w e establish a chromatic scale in semitones of about 85 degrees . The range s of each in s trument is thus expressed by a natural n umbe r (distance). But the pitch hu o f a sound is expressed by a decimal number of which the whole number part is related to a note ofthe chromati c s ca le within the instrument's range. Just as for the density in 2., we a ccept a certai n memory of or dependence on the p receding p itch played by the same instrument, so that we have
where
z
is given
by the
probability formula
P" Pz is the pro bability
=
�
s
of the interval
(1 - -=) dz. s
z
(See Appendix I.)
t ak en at random from the ranges, and
139
Free Stochastic Music by Computer
... !',) :::r
1J �
w :I 0
Q
:I .,
r;
c Ill
Ill
i5"
:I
Fig. V-2.
�011. 3
!..
::r "'
-a
ST/10-1, 080262, Density
=
"""
!:!.
::;· �
"'
�
";DJ (J1
'9. ;;;· ... "' :I c. 0
011 01
a: 3
-..1 1J
0
i5'
Composition of the Orchestra
(DA)1
= 0.11 eu, U
=
;;;· N
()' "' .... 0
log. (DA/0.11)
()0 0
2.. co
(Q :I 0
140
Formalized
Music
sis expressed as the difference between the highest and lowest pitches that can be played on the instrument. 7.
Attribution ofa glissando speed if class r is characterized as a glissando. The I led
homogeneity hypotheses in Chap.
us
to the formula
) 2 ., • f(v = -- e-va , a.y''IT vfa =
and by the transformation
T(u)
u
to its homologue:
2 y''IT ru o e-u• du, J
=
for which there are tables.f(v) is the probability of oct:urrence ofthe
(which is expressed
portional to
in semitonesfsec. ) ; it has a parameter
a,
which
speed v is pro
the standard deviations (a = sy'2). as a function of the logarithm of the density of sequence
a is defined
by : an inversely proportional function
=
a
a,
y'1r(30 � L[(DA)d(V3)] } -
or a directly proportional function
a or a
=
v'IT( 10 � L[(DA)d( V3)] )• +
of density
function independent
a =
where
k is
a
1 7.7 +
0
random number between
35k, and I.
The constants of the preced ing formulae derive from the limits of the
speeds that string glissandi Thus for (DA),
=
may take.
145 sounds/sec. a =
and for (DA)·,
=
2s
75
semitonesfsec. semitonesfsec.,
0.13 sounds/sec. a
2s
a
=
53.2
= =
17.7 semitonesfsec.
25
semitonesfsec.
8. Attribution of a duration x to the sounds emitted.
mean duration for each instrument,
To simplify we establish
which is independent of tessitura and
Free Stochastic Music by Computer
141
nuance. Consequently we reserve the right to modify it when transcribing
in to traditional notation. The following is the list of constraints that we take
into
account for the establishment of duration
G, the
x:
maximum len gth of respiratio n
or desired duration
the density of the sequence the probability of class r Pn, the probability of the instrument (DA)i,
q,,
Then if we define z as a parameter of
the
inversely proportional to
instrument, so
z
that
sound's duration,
probability of.
the
could choose
z, ... =
G.
Instead of letting Z...ax
to freeze the growth of
z.
=
could
law
applies for any
be
of the
and in this case
G, we shall establish a logarithmic law so
This z'
given value of z.
as
G In z/ln Zmax
=
Since we admit a total independence, the distribution of the durations
will be Gaussian:
where m is and
f(x)
=
1
--- e-<x-m)•tZs•'
the arithmetic mean
s
vz:;r
of the durations,
mm
+
4.25s 4.25s
=
=
s
the standard deviation,
0
z'
the linear system which furnishes us with the constants
u
z
occurrence
will be at its maximum when (DA),pnq, is at its minimum,
we
x
a
n
=
(x - m)Jsv'2
we find the
tables.
Finally, the d u ra tion
x x
function T(u),
m
and
s.
By assuming
for which we consult the
of the sound will be given by t h e relation =
±usv'2
+
m.
We do not take into account incom patibilities between instruments, for this would needlessly burden the machine's p ro g ram and calculation.
9. Attribution of dyno.mic forms to the sounds emitted. We define four zones ppp, p, j, if. Taken three at a time they yield 43 = 64
of mean intensities:
Formalized Music
142
permutations, of which 44 are different (an urn wit h 44 colors); for example, PPP-
-p. I 0. The same operations are begun again for each sound of the cluster Na,· 11. Recalculations of the same sort are madefor the other sequences.
An extract from the sequential s t atem ent was reproduced in Fig. V-1. Now we must p roce ed to the transcription into Fortran IV, a language
"understood" by the machine (see Fig. V-3). It is not our purpose to d e scri b e the transformation of the flow chart into Fortran. However, it would be interesting to sh ow an example of the adaptation of a mathematical expression to machine meth ods . Let us consider the elementary law of probability (density function)
f(x) dx
=
ce-cx dx.
[20]
in ord er for the computer to give us length s x with the probability f(x) dx? The machine can only draw random numbers Yo with equip robability between 0 and 1. We shall "modu l ate " this prob a bility: Assume some length x0; then we have How shall we proceed
prob. (0 where F(x0)
S
x
S
x0)
=
Lzo f(x)
dx
= 1
-
e-czo
=
F(x0),
is the distribution function of x. But F(x0)
=
prob.
then
(0
S
y
S
Yo)
=
Yo
and Xo
for all
x0
�
=
ln (1 - y0) c
0.
Once the program is transcribed into language that the machine's internal organization can assimilate, a process that can take several months, we can proceed to punching the cards and setting up certain tests. Short sections are run on t he machine to detect errors of logic and orthography and to determine the values of the entry parame t ers , which are introduced in the form of variables. This is a very imp ortant phase, for it permits us to explore all parts of the program and determine the modalities of its opera tion. The final phase is the d ec oding of the re s ul ts into traditional notation, unless an a utoma tic transcriber is available.
Free Stochastic Music by Computer Table
of the
44
143
Intensity Forms
Derived from 4 Mean Intensity Values, ppp, p, f. ff
iN-----;PN
f ========-- p
/1'1
f
I
/'11 --===J ===-)'II' /1!
I
f
!II -= il'!
--==-
/
Ill
1 -===- ))!> /'!! #
IJJ -==== # ::-=--=--;9¥ .f
��
iii -===I ---==---1 .f ==---=-�� - -=/'
1 -=== /.:==- i/1 I'===- ill-==-/
ill'-===..:.
/>-=::::::
I
j'
-
!/'!
-====::
#:::. I.:::-
i
#-==-1 -)1';-==:::.f
=-
/11-==:::#
I -====.#�Ill I ==-=--=-I -==-"#
I j>
-==:::
II
1-
If'-===-I'
=#�/
I'-====/ /
I� 11P-===-/
/�!>-===/
!!! -== #'
./--======-.#
f�Ill J
} -===-) - 11 -==/ I'
j> -=== # -==- /'
If-==- t
;! -=!==- /'/' -=/-' = )>
.f
I
I -===-- I ===- 1>
I-====. # :::==- /
#
#
/
;I
� )# ;f-= #�)�/ -==::-
#:=-=- _,c --==:#' :::-
144
Formalized Music
Conclusions
A large number of compositions of the same kind as STJ 10-1, 080262
is possible for al r eady
been
a large number of orchestral combinations.
Other
by RTF (France III); soloists.
Atrles for ten soloists; and Morisma-Amorisima, for four
Al th ough this program gives a satisfactory sol ution
structure,
it is, however,
by coupling
a
works have
w ritten : STJ48-J, 240162, for large orchestra, commissioned
to
the
minimal
necessary to jump to the stage of pure composition
digital-to-analogue converter to t he computer. The
calculations would then be changed into sound, whose internal
numerical
organization
had been c onceive d beforehand. At this point one ..:ould bring to fruition and generalize the concepts described in the preceding chapters . The following are several of the advant ages of using electronic compu ters in musical composition:
l. The lon g laborious calculation made
The speed of a machin e
by hand is reduced
to nothing.
such as the IBM-7090 is tremendous-of the
order
of 500,000 elementary operations/sec. 2. Freed from tedious calculations the comp oser is able to devote him self to the gene ral problems that the new musical form poses and to explore the nooks and crannies of this form while modifying the values of the input data. For example, he may test all inst r u mental combinations from soloists to chamber orchestras, to l arge orchestras. With the aid of elec tr onic com puters t he composer becomes a sort of pilot: he presses the b utto ns , intro duc es coordinates, and s upe rvi s es the controls of a cosmic vessel sailing in the space of sound , across sonic constellations and g alaxies that he could formerly glimpse only as a distant dream. Now he can explore them at his ease, seated in an armchair. 3. The program, i.e., the list of se q uenti al operations that constitute the new musical form, is an objective manifestation of this form. The program may consequently be d isp atc hed to any point on the ear th that possesses com pute rs of the appropriate type, and may be exploited by any composer pilot.
4. Because of certain uncertainties introduced in the p rogra m , the
composer-pilot can instill his own
personality
in the sonic result he obtains.
145
Free Stochastic Music by Computer Fig. V-3. Stochastic Music Rewritten in Fortran IV c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c
C
C C C C C
C
C
C
C C C
C
C C C
C C C
C
C C
C C C
C
C
C
C C
PROGRAM
FREE
GLOSSAQV A
OF
THE
- DURATION
ALEA
ST OCHAST I C MUSIC
OF
PRINCIPAL
- PARAMET ER USEO -
AIO•A20.AI7.A35•A30 SAME
INPUT
ALFAI31
OF
-
DATA
THREE
lVI
XEN
XEN XEN
ABBREVIATIONS
S EQUENC E
EACH
!FORTRAN
IN SECONDS
NUMBERS FOR GLISSANDO CALCULATION TO ALTE R THE RESULT OF A SECONO RUN
EXPRESSIONS ENTERING
INTO
THE
THREE
SP EED
WITH
9
XEN
10
XEN XEN XEN
II
SLIDING TONES I GLISSANDI I XEN - MAXIMUM LIMIT OF SEQUENCE DURAT I ON A XEN fAMA�(Jl•I�J,KTR1 TABLE OF AN EXPRE SSIO N ENTERING INTO THE XEN CALCULATION OF THE NOTE LENGTH IN PART B XEN BF - DYNAMIC FORM NUMBER• THE LIST IS ESTABLISHED INDEPENDENTLY MEN OF THIS PROGRAM AND IS SUBJECT TO MODIFICATION DELTA - THE RECIPROCAL OF THE MEAN DENSITY OF SOUND EVENTS DURING XEN XEN A �EOUENCE OF DURATION A IEII•�I•I=I•KTRtJ=I•KTEl - PROBABILITIES OF THE KTR TIMBRE CLASS ESXEN INTRODUCED AS INPUT DATA• DEPEND IN G ON THE CLASS NUMBER I=KR AND MEN XEN ON THE PO W ER J=ll ORTAINE"D FROM V3*EXPFIUI=DA EPSI - EPSILON FOR ACCURACY JN CALCULATING PN AND £(1•J)•WHJCH �EN XEN IT IS A D V I SA BL E TO RETAIN• CGN[It�)•I=I•KTRtJ=I•�TSI - lABLE OF THE GIVEN LENGTH OF BREATH �EN FOR EACH INSTRUMENT. DEPENDING ON CLASS J ANO INSTRUMENT J XEN XEN GTNA - GREATEST NUMBER OF NOTES IN TH� SEQUENCE OF DURATION A XEN GREATEST NUMBER OF NOTES I N KW LOOPS GTNS THE
ALIM
CHAMINIJ•JJ•HAMAXII•J)•HBMINII•�)•HBMAXCI,Jl•l•l•KT�•Jcl•KTS)
TABLE OF
INSTRUMENT
COMPASS
LIMIT�•
DEPENDING
ON TIMB�F. CLASS
THE HA OR THE HB TABLE IS FOLLOWEO• THE NUMBER 7 IS A�B ITRARY. JW - ORDINAL NUMBER OF THE SEQUENCE COMPUTED• KNL - NUMBER OF LINES PER PAGE OF THE P RI NT E D RESULToKNL=50 KRJ - NUMBER IN THE CLASS KR=l USED FOR PF.RCUSS10N OR INSTRUMENTS AND
INSTRUMENT �.
TEST
INSTRUCTION
480
IN PART
6
DETERMINES
WHETHER
KTF
-
XEN
XEN
XEN XEN
XEN
XEN
XEN
XEN
POWER OF THE EXPONENTIAL COEFFICIENT E SUCH THAT
XEN
NUMBER
XEN
WITHOUT
A
DEFINITE
PITCH.
KW
-
KTR
-
CU�VE
WHICH IS USEFUL
IN CALCULATING
GLISSANDO
SPEED
13 14 15 16 17 IB 19
20
21
22
23 24 25 26 27 2B
29 30 31 32 33 34 36 37 38 35
40
39
XEN
42 43 44
MAXIMUM NUM BE R
DISTRIBUTION
12
XEN
OF TIMBRE CLASSES XEN OF ,JW KTESTJ,TAVl•ETC - EXPRESSIONS USEFUL IN CALCULATING HOW LONG THE XEN XEN VARIOUS PARTS OF THE PROGRAM WILL RUNo KT! - ZERO IF THE PROGRAM IS BEING RUNo NONZERO DURING DEBUGGING XEN KT? - NUMBER OF LOOPS• EQUAL TO 15 BV A RB IT RARV DEFINITION. XEN IMODIIIX81•IXB=7•1l AUXILIARY FUNC TI ON TO I NTERPOLATE VALUES IN XEN XEN THF. TETA I 2561 TABLE I SEE PART Tl XEN NA - NUMBER OF SOUNDS CALCULATED FOR THE SEQUENCE ACNA=OA*A1 KEN CNT(I)•I•l•KTR) NUMBER OF INSTRUMENTS A��OCATEO TO EACH OF THE XEN KTR T I MB RE CLASSES• IPNIJ•J)•I•t•KTR.J=I•KTS)•CKTS•NTII)•J=l•KTR) TABLE OF PROBABILITYXEN KEN OF F.ACH INSTRUMENT OF THE CLASS 1, IO(II•I•1•KTRI PROBABILITIES 0� THE KTR T I M B R E CLASSES• CDNSJOEREOXEN XEN AS LINEAR FUNCTIONS OF THE DENSITY DAo IS(J)•I=l•KTR) SUM OF THE �UCCESSJVE Oltl PROBABILJTJES• USEO TO XEN XEN CHOOSE THE CLASS KR BY COMPARING IT TO A �ANDOM NUMBER XI !SEE PART 3t LO O P 380 AND PART 5• LOOP 430), XEN StNA - SUM OF THE COMPUTED NOTES JN THE �W CLOUDS NA• ALWAYS LESS XEN THAN GTNS I SEE TEST IN PA�T 10 I• XEN X EN SOPI - SQUARE ROOT OF PI C 3.14159• ••) TA - SOUND ATTACK TIM� AACISSA. XEN TETAC256) - TABLE OF THE 2.S6 VALUES OF THE INTEGRAL OF THE NORMAL WEN 0AfMAX1=V34CE**IKTE-11J
8
XEN THEXEN
VALUES
6 7
XEN
41
45 46
47 48 49 50 51 52 53
54
55 56 57 58 59 60 61 62 63 64 6�
146 c c c c c c c c c c c c c c c c
Formalized Music
ANO SOUND EVENT DURATION• VIGL - GLISSANDO SPEED IVITESSE
GLISSANOO)t �HICH CAN VA�V ASt BE INVERSELY AS THE DENSITY OF THE SEQUENCE• THE ACTUAL MODE OF VARIATION EMPLOYED REMAINING THE SAME FOR THE ENTI�E SEQUENCE ISEF. PA�T 71o VITLIM - MAXIMUM LIMITING GL I SSANDO SPEED fiN SEMITONF.S/SE C )t INDEPENDENT OF•
SU��ECT
OR VARV
TO MODIFICATION.
- MINIMUM CLOUD DENSITY DA lZ1 ClleZ2CI)tlet,SJ TABLE COMPLEMENTARY TO THE
V3
CONSTANTS
�EAD
TETA
TABLE•
HC
MEN XEN
IX=lt"7
XE"N
XEN
XEN XEN
READ 20tfTETAliJ•I=1•2561
30e CZJ C I) •221
�ORMATC12F6•6) READ
XEN
XEN XEN
(),Jet t8)
FOR�ATI6CF3e2•F9,SJ/F3e2tF9.8tF.6•2•F9e8J PRINT
THE
40tTETAtZ1e22
�ORMATI*l
40
**
THE ZJ
TETA
TABLE
•
TABLE =
XEN
THE Z2
XEN
*•/•21112Fl0.6•/l•�FI0,6•/////•
*•/e7F6e2tE12,3•///t*
*•IHII
TABLE
�
*•/•8Fl4•B•/XEN
XEN
REAO 50•0ELTAeV3eAIO•A20•A17•A30•A35tBF•SQPI•EPSI•VITLIM•ALEA•
50
60 70
XEN
AXEN
*LIM FORMATCF3t0tF3.3e5F3el•F2e0tFB,7eFS,e,F4t2tFS,B•F5•21 READ
XEN
X EN XEN AXEN J<EN
60tKTI•KT2tKWtKNLtKTRtKTEtKRttGTNAtGTNS•INTCl)ti=I•KTRJ
FORMAT(5J3t21Zt2F6•0tl212)
PRINT
70tOELTAtV3•AIOtA20•A17•A30tA�5tBFtSOPitEPSI•VITLJMeALEA•
itLJM�KT1 tKTZtK\thKNL FORMATt*tOELTA
**
�
A20
*/•*
BF
=
e
tKTReKTE tKRJeGTNA•GTNSt CCI tNTC I)) •1•1 tt
V3
*•F4,0e/•* e
�
4tF6•3•/•*
*•F4el•/••
A10
�
*•F4,1•/•
*•F4el•/•*
A35
•
XEN
*•F4eltXEN
SOP! =*•F1le8t/•* EPSI =*•Fl2•8•/t* ALEA =*•Fl2e8•/•* ALIM = ••F6e2e/•* KTI • *•l3t/t KW = * • l3 t/ • * KN� = *•13•/•* KTR : *t13t/4 *•J2•/•* KRJ = *•12•/•• G TN A • *•F7•0•/•* GTNS = *•F7•0•
+tF4ele/e•
A17
A30
•
*•F3e0•/•*
VJTLIM
�
*•F5•2•/•*
**
KT2
**
KTE
=
C
90
READ
*•13•/•*
=
*/•J2l*
BO
IN
Cl.ASS
••12•*•
THERE
ARE
*•12•"*
tNSTRUMENTS•*•/l)
80•KTEST3eKTEST1eKTEST2
FORMATI5131
KTESTl
PRINT 90tKTEST3tKTESTleKTEST2 FORMATC*
KTEST3
IFIKTEST3oNEo01
R=KTE-t
=
*•13•/•*
PRINT
*tl3t/t* KTEST2
*•13)
830
A10•AID*SOPI A30=A30•SOPJ
C
C
IF ALEA IS NON-ZEROoTHE �ANDOM NUMBER IS GENERATED FROM THE TIME WHEN THE FOLLOWING JNST�UCTION tS EXECUTED• IF ALEA IS NON-ZERO EACH RUN OF THIS PROGRAM WILL PROOUCE DIFFERENT OUTPUT DATA. IFIALEAoNEoOoOI
*XEN
XEN
XEN
XEN XEN XEN XEN XEN
XEN
�N
XEN
XEN XEN
A20•A20*SOPI/� C
:teEN MEN XEN XEN
XEN JO
TO 1•1 .. 1
�0
)(F.N
i 2)(EN
MOCIIIIXF.II=I
20
XEN XEN XEN XEN
•
IXB•B-IX
c
XEN
XEN
TABLES
Q( IZJ tSf 12') •Ef 12•121 tPNt 12•'50) •SPN( 12•'50 I tNTf 1 21 • •HAMINf J 2•501 •HAMAXr 12• SO J •HBMINC 12•50) •HBMA>< C 12•50 J •GNC 12•50) *•SO I tTETAI Z56J tVIGLC 3) tM00117) tZII BJ tZZCB) tALFA( 31 tAMAXC 121 1•1
XEN
XEN AND
DTMF.NSJON
DO
XEN XI!N XEN )(EN
XEN
CALL
�ANFSETITIMEFI1)1
XEN XEN
XEN
XEN XEN XEN
66 67 68 69 70 71 72 73 74 75 76 78 77 79 so 81 82 83 84 85 86 87 BS 89 90
91
92 93 9. 95 96 97 98 99 100 114
IJS 1.26
127 128
129
130 13l
132
133 1 34
135
136
142 143 I145
141
147 148 149 ISO 151 152 153 146
Stochastic Music by Computer
Free
147
830
PRINT
DO 130
*PNC(tJ)t..J�I•KTS)
FORMATC515F2t0tF"3e:1l)
100
1
Pj:;�JNT 110
FORMAT(////•• =
F3,0t*tGN
DO
c
IF
*•F3•0•*•
1:1
•
FORMAT112Fl'o21
HAS
,,JJ
tHBMINC lt..JJ
*•12tC/t•
FOR
*•F3e0t4tHBMJN
PN : *•F6e3l)
=
INSTRUMENT NO, ••F3.0•*•HBMAX
*•12• =
*•
167
XEN
XEN
XEN 4•12•
XEN
••F 6 ,2• / 11
OF
)(EN
16.:\
166
1 68 169 170 171
172 173 1 74
175
176
XEN
177
DO 1 70 I • 1 ti
XEN
178
)(EN
179
IF
fAB5FrY-taOJ,GE.EPSIJ
?00 t=taKTR AMAX ( I l: t • 0./'E t I DO ?00 J•2•KTE
t
XEN XEN
CALL EXIT
I)
IF
PRINT
IKTt.NEeO)
190tAX
TAVl;TJMEF!ll
NLIHE•SO
1 AND 2•
KNA•O KI�O 230 Xl .. RANFI-1 l A�-DELTA *
SEQUENCE
GO
240
GO TO 250
(l
GO TO 230
TO
A�ALJM/�,o
270 UX=�*X2 GO T O 310 IF
XEN XEN XEN )(EN
187
A
SECONDS
AND CLOUD NA DURING
A
IJW.GT,J)
ISS
168 189 190
191
192
XEN
196
XEN
197
XEN
199 200
XEN
XEN
194
193
195
198
201
XEN
202
XEN
203
XE N XEN XE'N
XEN
204
206
205
XEN XEN
20A 20Q
XEN
214
XEN
GO TO 280
183
1 84
XEN XEN
XFN
J
XI=O,O 250 K2=0 260 X2=RANF ( -1 1
DEFINE
LOGFIXIl
1FfA.LE,ALIM1
IF
186
XEN
IKTeSTlsNEt01
PARTS
XEN
)(EN
SINA�O,O
IF
XEN XEN )(EN )(EN
FORMAT(IH t9E12t8l
181
l 80 182
XEN
JW:l
;?4n
165
XEN
LEVEL
162
163
XEN
XEN
DENSITY
160
XEN
XEN
200 IF IAXaGT.AMAXI IJ1 AMAXIJlcAX IF (KTt,NE,O> PntNT 210•AMAX 210 FOQMATI lH e9EI2tB1
220
XEN
XEN
AX•t.O/(E(t.�I*EXPF(AJ))
c C c
XEN
XEN
IN
1!!7 158
161
XEN
XEN
AJ=:.J-1
c
)(EN
155
tHBMA)(( I,..;) tGNC t t..J�XEN
CALL EXIT
C LAS S NUMBER *•12•/•1*
PROBABILITY
DO
190
)(EN
J:I,KTE
DO 180 Y=O,O
lBO
I
160tltfJtECit.J)t..J=t•KTEl
A
156
159
KTR
FORMAT(//////•*
PRINT
154
)(EN
)(EN
140tCECitJ)tJ=t•KTEl
140
170
=
AND
CABSFCV-J,O),GE,EPSil
00 1 50
tHAMAX(
NUMBER
lt.J)=:'f
READ
44
CLASS
J=loKTS
ll'O
SPNC
IN
160
1!150
J ,J)
** HAMIN = *•FJ,O•*•HAMAX •
120 130
t.Jt:rl •KTS)
JO•l•IJ,HAMJNC
••PNI I•.J)
)(EN
XEN
l=l•KTR
v�o.o KTS•NTCI) READ tQO,tHAMTNfltJ)tHAMAXC!tJl,HBMlN(J,J),HSMAXCJ,J)tGNCitJ)•
XEN )(EN XEN
207
211 212
210
213
Formalized Music
148 IF CRANFf-l)eGEeOe�l GO TO 2q0 ux�UP� � � • CJ,O-SORTFCX211 GO TO 300 290 UX:UPR - R * ( t,D-SQRTFCXZJI 300 IF CCUX,GEeOeOJ,ANO•fUX,LEeR)J GO TO 310 IF CK2,GE.KT21 GO TO 270 280
GO 10 310 U=U� DA=V3
K2:el<2+1
1F
260
IF
•
OA +
O,�J
+
IKN4oGEoKT21
GO TO ?.:'30
1
GO
GO TO 320
CGTNAeGTeF�OATFCNAJJ
KNA:KNA+1 320
XEN XEN XEN
TO 330
XEN XEN
)(f':N XF.N l<EN
330 UPR•U
c C c
GO
CKTI,EO.O)
TO
XEN
360
FbRMATC1Hle418•3X•4EtB.B•3Xel8)
350
FORMATCIHDtZI9tFlne21
N4=1CTI
CKTEST3,NE,01
P4RT
350
3•
PRINT
350eJWeNAeA
DEFINE CONSTITUTION
SINA•SINA
OF
ORCHESTRA DuRING SFOUENCE A
+ FL04TFINAJ *XLOGOA
XEN
XEN
)(EN XEN
N;:XINTFCXLOGOAJ IF
fCM+�J.GTeKTE)
�•KTE-2
SR:OeO
M2mM+2 00 380 t=l •KTR ALFX�EC 1eM11 BETA=E'C I ellll21 �M=M OR=f�LOCOA-XM1
*
CRF.TA-ALFXJ
SIIJ=SR
390
FORMAT(lH
TF
CKTt,NE.Ot PRINT •12F9•4J
PART 4t0EFINE
c C
c
+
ALFX
253 25ca
XEN
XEN XEN
INSTANT
XEN X EN
xEN TA
OF
EACH
POINT
IN SEQUENCE A
XEN
T=-LOGFC XJ /DA
TA•TA+T
C�TleNEeOJ
258
259
260
261
262
263 Z64
XEN XEN XEN XEN XEN
FORMATC//•1Bt3E20t8J
256
257
265
XEN
PRINT 420•N•X•TtTA
255
XEN
XEN
l<=RANFI-11
252
XEN XEN
TA=O•O GO TO 410 N=N+l
IF
251
XEN
XEN
T•O.O
410 11\20
250
XEN
N•J
11\00
XEN
XEN
390tC0111•1=1•1
246
248 249
XEN XEN
370tXMtALFXtBETA
230
231 232 233 234 235 236 237 238 23<;1 240 241 242 243 244 245
247
XEN
IF (KTleNE.OJ PRTNT 370 FORMAT(lH •3F20.8) Qll J•OR s��SR+OR
215 216 217 21B 219 220 221 222 223 224 225 226 227 228 229
XEN l<EN
Nt=M+1
380
><EN
XEN
XEN
XLOGDA=U XALOG=A20
><EN ><EN ><EN XEN ><EN ><EN
340e�WtKNAeKltK2eXI•X2tAtOAeNA
340
IF
XEN
XEN
A=OF.L TA
IF
XEN XEN
XEN
CO TO ?50
PRINT
XEN
XEN
* EXPF'CU1
NA•XINTFCA
l<EN
XEN XEN
XEN
XEN
2 66
267
268 269 270 271 272
273 274
275
Free
c C c
Stochastic Music by Computer
PART
430
CLASS AND INSTRUMENT NUMBER
�•DEFINE
XI =�ANFC-1) 00 430 t:J•KTR IF
149
CXleLE•SCJ))
l=KTR
TO
GO
TO EACH
POINT
KRct
460
PtFN•PNCK�•tNST�l
IF CKTt.NEeOl PRINT 470•Xl•SCKR)eKR•XZ•SPIENeiNSTRM 470 FORMAT! II-( •2E20,9ei6•2E20.B•I6 > PART 6t0EFINE C K R. GT,JJ
TO
GO
560
TO
GO TO
CKRtLTe7)
HN FOR
PITCH
C!NSTRMtGEeKR1)
HXsOeO IF
EACH
POINT
480 GO TO 490
OF SEQUENCE A
GO
TO
IF
CHPR.LE,O.Ol
540 �50 560
c C
c
JF
CN,GT,t)
XEN
GO TO
HXcHPR+HM
*
HX=HRR-HM
*
1F
GO
(
520
TO
IRANFr-lJeGE•Ot�)
GO TO 550
GO TO
I KeGE .� T 2l
GO TO 520
510
GO TO 560
HI KR•INSTRM):HX
JF IKTl•NE•O) PRINT 570tKtX.HX �70 FO�MATftH t16•2E20tAJ PART 7t0EFINE SPEED VJGL
GO
TO
560
TO
EACH POINT OF
304 305
30 7 306
308
311
310
XEN
3 13 314 315 316
XEN
318
XEN XEN
320
XEN
322
XEN
324 325
XEN
XEN
XEN
312
317
319
321
323
326
XE"N XEN XEN
327 328 329 330
XEN
333
XEN
XEN
XEN
XLAMBOA•O,O
XE'N
GO TO
XEN
740
303
XEN
XEN
A
297
309
XEN
lteO-SQRTF()())
298 299 300 301 302
XEN
XEN
940
294
XEN XEN
teO-SORTF(X))
292 293
XEN
XEN
JFrCHX,GEtHINF)eANOeCHX.LE•MSUP))
II(:K+l GO TO
XEN
XEN
530 520 HX=HINF•HM*X RANFC-l) GO TO 560
SJO
295 296
XEN XE N )(EN XEN )(EN
XEN
)(:RJ'iNFl-1) IF
XEN
XEN
HPR=HCKR•INSTRM)
290
XEN
XEN
XEN
HINF•HA�INCKR,JNSTRM)
28l
283
291
)(EN
XEN
500
500 HM�HSUP-HINF
510
XEN
XEN
HSuPzHAMAXIKRe!NSTRM}
1<:=0
2B4 285 286 287 288 289
XEN
4qo
HSUP=HBMA�IKR•IN5TRMl HINF=HBMIN(KRtiNSTR�)
490
282
X EN
XEN
GO TO 460
CX2.LEeSPIEN)
tNSTRM-=KTS
GO
XEN
XEN XEN
INSTRMzJ
IF
480
XEN
XEN
SPJE'NeSPNCKRtJ)
276
277
27B 279 280
XEN
450
IF
XEN XEN
XEN
X2sRANFC-I)
00 G50 J=l•KTS
IF
XEN
XEN
XEN 440
440 KTSaNT(J)
c c c
0� A
XFN
332
331
:J34 335 336
150
Formalized Music
sao -:x�t 590 XI ::RANFC 600
l= t 2A
-11
IF
CXt-O,qqQ�)
DO
630
TO
XE'N
6t0e640o6?0
XEN
XEN
630
XEN
1•1-MDDIIIXt
XEN
630
CONTIM..JE
640
XLAMBOA:FLOATFfJ-1)/lnn.n
650
XF.N
XE N
Jat+MOOJCIX, GO
620
XE N
IX=Io7
1FITF.TAIIJ-Xl)
6tn
XEN XEN
�00t6�0•6RO
I"CTETAIII-XII
XEN
XEN
670o640o660
GO TO
C720t76�1•
GO TO
1720o7601oKX
XEN
KX
XEN
XLAMAOA=?.·��
XEN
660
1•1-1
XEN
670
TXI=TETAIII
6!!0
GO TO C 720•760 00 690 I •2• 7
XEN XEN XEN XEN
690
CONTINUE
700
TXJ:t.O
XEN
TX2=ZI C I 1
XEN
GO TO
XEN
XEN
1• �X
XLAMSOA:CFLOATFCt-ll+CXt-TXJJ/CTETACI+tJ-TXIJl/lQO,O
700o71 Oo690
TXI=Z2111
I"IXI-TXII
XEN
XEN XEN
I
XLAMBDA:TX2-IITXJ-XII/ITXI-Z21 1 -I llt•ITX2-ZIII-Ill 7?0o760
710
XLAMBDA=Zll!l
720
GO TOC
720•760
I•
�·
XEN
KX
XEN
KX
ALFAili=AlO+XALOG
XEN XF.N XEN
ALFAC31=A30-XALOG
X EN
X2=RANFI-ll
ALFAC21•Al7+A35*�2 DO 730 t-=lt3
XEN
X EN
I" IVIGLII loLToOoDI V IGLI 11=-VIGLIII tF CVIGLt lltGTeVITLIMJ VIGLC II=VITLIM 730 I" IRANFI-IIoLToOo51 VIGLIIt=-VIGLIII 740 IFIKTitNEtOl PRINT 750eXI•X2eXLAMBDAeVIGL
XEN
VICLCI1=1NTFCAL�ACI1*XLAMBDA�0.SJ
c C c
750
"ORMATIIH
FOR EACH
G-=GNCIO=G/LOGF C ZMAI()
XEN
POINT
OF
A
XEN
XEN
XEN
XEN
XEN
XEN
XEN
OPNOA-=1•0/fQfi
GE:A8SFIRO*LOGFIOPNDAII
XEN
XMU=GE/2o0
XEN
SJGMA!!!!GE/4.0
XEN
KX=i'
XF.N
GO
XEN
TO �gO
X2=RANF(o-J)
TAU�SJGMA*XLAMBOA*I•4142
XOUR•XMU+TAU GO TO 790 '"
770
XEN XEN
CCKR.E0t71•0R.CKR.�0.8l> GO TO 780
ZMAX=AMAX(�R)/CV3*PIEN)
760
XEN
o6EI9o81
PART BtDEFJNE DURATION IF
XEN
XEN
IX2oGEoOo51
GO
IXDURoGEoOoOI
770
XE N
XEN
XEN
XOUA=X�U-TAU
IF
XE N
XEN
TO
GO
TO 790
XEN XEN
337 338 339 340 341 342 343 344 345 346 347 3&8 349 350 351 352 353 354 355 356 357 3!58 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390
392 393 394 395 396 397 391
Free
c: C c:
by Computer
Stochastic Music
151
7 8C XDUR=O•O 790 IFCKTleNE.OlPRINT 800•ZMAXtXMU,SIGMA,Xl•XLAMBOA•X2tXOUR BOO FORM-TCtH •SEIS•S•Eli•4•E15.8) PART
9•DEFINE
INTENSITY
FORM TO EAC� POINT OF A
CKTJ,EO,Ot GO TO 840 INLINEoLT,KNLI GO TO 610 IF (NLINEtEO,KNL) GO TO 620
IF
IF
TO
900
GO TO QOO 820 PRINT A30 830 FOQMAT r IHI) NLINE=O GO TO 900
NLINEcNLINE+I
640
IF
950 860
GO
GO TO
860
NLINE=NLINE .. 1
870
PRJNT
4),//)
PRINT
650
c: c
PART
c
IF
IOtREPEAT
lN•LT.NAI 11•
SAME
GO TO
REPEAT
(KTF�T2.�0.0J
OFFINITIONS
400
SEQUENCES
GO
TAP2=TIMEFI!I-TAV2
TO
A
910
TAP?.:TAPZ/FLOATFCNA) PRINT
c: 910 c:."ZO <>3C
941'1 c
c c c:
TCH* •6Xt
BAO PRtNT A90,N,TA,KR, J NSTRM,HX.(V(GLrll•l=1•31tXOuR•IFORM SQn FOQMATC IH ,. 17eFI2e21 t9,JB,Fllel ,FJ3•1•2FIO•l•Fl4•2•11 I) IF
407
XEN
411
414
FOq
ALL
POINTS
OF
A
402
AOA 405
406
408
409 41 0 412
liEN
413
415
XEN
416
XEN XEN )(EN
417 416
419 420 022 423 421
XEN
420
XEN
KEN
425
XEN
427
ICEN XEN ICEN X EN
428
XEN
XEN
XEN XEN X EN XEN XEN XEN
XEN
750•TAP2
IF (JW,.GEeKW1 GO TO 930 JW:..J\1'+1 IF IGTNSoGT.S!NAI GO TO ?.20 IF fKTFSTteF-OeO) CALL F.MJT
XEN XEN XEN
TAPt:TAPI/FLOATFr�WI TAP1•TIMEFI-11-TAV1
PRINT
400
XEN
JW=*•I3t4Xt*Aw*tF8e2t4Xt*NA:*•I6t4Xt*OCIJ=*•12CF4e2t*/4XEN
B�O
**GLJSSl*t4X•*GLISS2*•4X•*GL1SS3••AX•*DURATI0N*t5X••DYNAM*)
PART
liEN liEN XEN liEN
XEN
FORMATC6Xt*f'l* t8Xt*START* tSX•*CLASS* t4Xt *INSTRM*t4Xt*Pl
900
403
XEN
NLINE=I
c: C c
ICEN
XEN
XEN
R60tJ�tAtNAtl0111•f=1•KTR)
FORMATC*l
401
ICEN
CNLINEoGEoKNLI
TO
ICEN
ICEN
NLINE•l GO
398 399
XEN XEN ICEN
IFORM=XINTFCRANFC-11*BF+Oe51
810
MEN XEN XEN
7'50oTAPI
END
XEN XEN XEN XEN
ornononJtJnnn?26nno339non451onn�fi400067600n7eqn��9010010130�112500I23600
134800I4�900I'56900I660nnt79noot90ooo?oo90D2ti60022?.7no23�5ooz4A300?��ono 2657002763002B6900297400J0790031a�oo32a60033S900349100359300369400J79400 389300399200409000419700426400436000447500456900466200475500484700493700 502700511700'520500529200537900546�005'54900563300571600579600567900�95900 6C39006117006194006?70006346006420006494006�660066380o67C8006778006B4700 69140C69RI0070470071t2n0717�or723A0073000�736In�74210074AOn�753BOn7SO�nn
XEN
426 429
430
431
432 433 434
436
435
438
437
439
440 44 1
442
443 444
445
446
447 448
Formalized Music
! 52
7 6 5 1 0 0 7 7 0 7 0 0 7 7 6 1 0 f' 7 8 1 4 0 0 7 8 6 7 0 0 79 1 8 0 0 79 6 <;1 0 0 8 0 1 9 0 0 8 0 68 0 0 8 1 1 60 0 8 1 6 3 0 0 8 ?. 0 9 0 0 B 2 5 4 0 0 8 ?.9 9 0 0 8 3 4 2 0 0 8 3 R � O O B 4 2 7 0 0 8 4 6 8 0 0 8 5 0 A 0 0 8 5 4 8 � 0 8 5 8 6 0 0 8 6 2 4 0 0 86 6 1 0 0 8 6 9 B O O
B 73 3 0 0 8 76 8 0 0 8 8 0 2 0 08 8 3 5 0 0 8 B 68 0 0 8 9 C 0 0 0 8 9 3 1 0 0 8 9 6 1 0 0 89 9 1 0 0 9 0 2 0 0 0 9 0 4 8 0 0 9 0 7 6 0 0 9 1 0 3 0 0 9 1 3 0 0 0 9 1 5 5 0 0 9 1 8 1 0 0 9 2 0 5 0 0 92 2 9 0 0 92 5 20 0 9 2 7 5 0 0 9 2 9 7 0 0 9 3 1 9 0 0 9 3 4 0 0 0 93 6 1 0 0
9 3 8 1 0 0 9 4 0 0 0 0 9 4 1 9 0 0 9 4 3 8 0 0 9 4 5 7 0 0 9 4 7 3 0 0 9 4 9 0 0 0 9 5 0 7 0 0 9 5 2 3 0 0 9 5 38 0 0 9 5 5 4 0 0 9 5 6 9 0 0 9 5 8 3 0 0 9 5 <;1 7 0 0 9 6 1 1 0 0 9 6 2 4 0 0 9 6 3 7 0 0 9 6 4 9 0 0 966 1 0 0 96 7 3 0 0 9 6 8 4 0 0 9 6 9 5 0 0 97 0 6 0 0 9 7 1 60 0 9 7 2 6 0 � 9 7 3 6 0 0 9 7 4 5 0 0 9 7 5 � 0 0 9 7 6 3 0 0 9 7 7 2 0 0 9 7 B 0 0 0 9 7 8 8 0 0 9 7 9 6 0 0 9 8 0 4 0 0 98 1 1 0 0 9 B I A O O 9 8 2 5 0 0 9 8 3 ?. 0 0 Q 8 3 8 0 0 9 A 4 4 0 0 9 A � C 0 09 8 '56 0 0 9 A 6 1 0 0 98 6 7 < ' CJ 9 8 7 2 0 0 9 A 7 7 0 0 98 8 2 0 0 9 B B 6 0 0 9 8 9 1 0 0 9 8 <;1 � 0 0 9 8 9 9 0 0 9 9 0 3 0 0 9 <;1 07 0 0 9 9 1 1 0 0 99 1 �0 099 1 A 0 0 9 9 ? 2 0 0 9 02 5 0 0 9 9 2 8 0 09 9 3 1 0 0 9 9 3 4 0 0 9 9 3 7 0 0 9 9 3 9 0 0 9 9 4 2 0 0 99 4 4 0 0 99 4 7 0 0 9 9 4 9 0 0 9 9 5 1 0 0 9 9 5 3 0 0 9 95 5 0 0 9 9 5 7 0 0 99� 9 0 0 99 6 1 0 0 996 3 0 0 9 96 4 0 09 9 6 6 0 0 9 9 6 7 0 0 9 9 69 0 09 9 7 0 0 0 99 7 2 0 0 9 9 7 3 0 0 9 97 4 0 0 99 7 5 0 09 9 76 0 0 9 9 7 7 0 0 9 9 7 9 0 0 9 9 7 9 6 0 9 98 0 5 0 9 9 8 1 4 0 9 9 8 2 3 0 9 9 8 3 2 099 8 4 0 0 9 9 8 4 8 0 9 98 5 5 0 9 9 8 6 2 0 99868 0 9 9 8 7 4 0 9 9 8 8 0 0 9 9 8 8 5 0 9 98 9 1 0 9 9 8 9 6 09 9 9 0 1 0 9 9 9 0 6 0999 1 0 0 9 99 1 4 0 9 99 1 8 0 99 9 2 3 0 9 9 9 2 7 0 9 9 9 3 0 0 99 9 3 4 0 9 9 9 3 7 0 9 99 4 0 0 99 9 4 4 09994 7 0 9 9 9 5 0 099 9 5 3 0 <;1 9 9 5 6 0 9 9 95 8 0 9 9 9 6 0 0 9 9 9 6 3 0 9 <;1 0 6 5 0 <;1
Q <;I � 7 0 0 9 Q 6 0 0 9 <;1 9 7 0 0
2 5 5 0 9 9 9 7 0 0 0 0 2 6 3 0 9 99 8 0 0 0 0 ?. 7 5 0 9 99 9 0 00 0 3 1 3 0 9 9 99 9 0 0 0 3 4 6 0 99 9 9 <;19 0 0 3 7 7 099999990 4 06 0 <;1 9 <;19999<;1 1 0 0 E �O I 0 0 0 0 0 0 0 0 0 4 0 0 s O I O n 2 0 0 I 7 7 3 0 o 3 5 5 6 3 1 7 7 2 4 5 3 9 0 I O O o o o 0 7 t o o n n o o o o o n t 20 0 0 0 0 0 0 ! 5 0 5 0 0 5 0 0 1 2 0 72 0 0 0 1 6 0 0 0 2 5 0 0 0 1 2 0 1 0 1 0 2 03 0 9 0 2 02 0 1 0 1 0 2 0 2 0 1 0 1 0 0 0 0 1 0 0 7 0 0 1 0 1 0 0 0 0 1 0 0 9 0 0 1 0 1 0 0 0 0 1 0 1 2 0 0 1 0 1 0 0 00 1 0 1 1 0 0 1 0 1 0 0 0 0 1 0 0 9 0 0 1 0 1 0 0 0 0 1 0 1 2 0 0 1 0 1 0 0 0 0 ! 0 0 A0 0 1 0 1 0 0 0 0 I 0 0 80 0 1 0 1 0 n no 1 0 1 2 0 0 1 0 1 0 0 0 0 1 0 0 8 0 0 1 0 1 0 0 0 0 ! � 0 ? 0 0 1 0 1 0 0 0 0 2 0 0 ?. 0 1 7 5 5 0 0 0 0 1 09<;19 3 9 7 5 0 0 0 0 1 5999 2 9 7 1 0 0 0 0 P 06 0 0 1 7 5 4 0 0 0 0 1 0 4 0 0 3 4 8 5 0 0 0 0 1 54 0 0 1 5 6 3 0 0 0 0 1 54 0 0 1 95 3 0 0 0 0 1 0 2 0 0 3 9 75 0 0 0 0 1 5 1 5 0 2 9 7 1 0 0 0 0 1 0 0 9 0 1 7 5 4 0 0 0 0 0 7 0 9 0 1 7550 0 0 0 ! 0 0 9 0 3 3 6 3 0 0 0 0 1 0 0 9 0 1 95 3 0 0 0 0 1 0 0 70 1 0 1 3 0 0 0 0 1 0 2 0 0 3 4 8 5 0 0 0 0 1 5 2 0 0 1 5 6 30 0 0 0 1 5 0 2 0 0 00 0 3 4 6 7 0 05 0 0 0 0 0 0 1 54800500 0 000346700�000 0 0 0 1 54A00�DO 0 00 0 3 2 6 8 1 0 9 0 0 0 0 0 0 3 36 3 1 09 9<;1 0 0 0 0 1 95 3 1 0 !'1 0 0 0 0 00 1 0 1 � 07 2 0 0 0 00 0 3 4 8 7 1 5 5 0 0 0 0 0 0 1 5 7 � 1 5 5 0 0 2 5 0 8 0 4 0 8 0 1 1 3 09 0807 ! 6020 1 0 1 1 0 0 3 03 0 4 2 0 0 1 0 1 1 0 0 2 0 5 0 32 '1 0 1 0 1 1 ?.
0 3 3 5 0 3 1 � n J 1 0i0 �
0 2 1 0 0 3 D P. I 0 3 9 0 7 0 2 02 0 2 0 3 1 5 0 ? 0 7 0 2 0 2 0 2 0 2 4 1 0 207 0 3 09 0 3 1 7 04 1 60 9 0 3 1 3 2 0 0 3 ?. 0 0 50 9 0 ?. 0 5 2 8 0 1 0 3 0 4 00 4 5 0 1 1 ? 0 ?. 0 ? 0 1 06
Free Stochastic JW=
1
A=
Music
153
by Computer
9.13
N A -=
Q I I I = O . l 2 / 0 o 0 4 / 0 . 0 4 / 0 . 0 5 / 0 . 1 2 / 0 . 2 9 /0 . 0 4 / 0 . 0 4 1 0 . 1 4 / 0 o 0 6 / D . 0 6 / � . 0 3 1
S T AR T CL A S S I N S T R M 1 o . co 1 H' 0 . 10 1 6 Ooll 8 6 0.13 3 l 0.18 4 9 0.25 1 6 0 . 33 1 .. 0 . 34 1 1 (j • 4 0 l 6 0.41 q 6 C . 7 E: 7 Cl . 9 0 8 2 1 1 .00 2 1� 1 . (; 9 1 6 I . (1 9 2 6 1.23 3 6 1 .42 1 10 1.57 1 4 1.65 2 6 1 . 78 B 6 1 .92 3 1 . CJ lt 5 1 4 2.18 1 6 2.18 6 1 :t . l 9 12 9 2 . 20 l 9 2oL3 1 1 2 .32 1 4 2 .33 l 2 .54 1 b 11 2 . 57 2 .. .t. � c; z.n 1 1 3 J 2 . 80 5 5 'H 3 . 2 8 2 1 �5 3.33 7 5 3 E 3 . 38 2 l l' 3 '1 3 . 5 5 1 1 3 1! 3 . 5 6 9 c; 3 9 3 . t l) 1 12 4Q 3 .64 2 (: 4 1 3.65 5 5 lo 2 3 . 7 1 3 6 lo 3 3 . 80 B 6 "" 3.87 2 (: 4 5 3 . 89 7 5 lo t 4 o 1 5 2 5 41 4. 15 1 9 48 4.25 1 12 4 <; lo o 3 1 2 1 5 1) ' � 3 3 1n
J';
1 2 3 4 5 � 7 e c; 10 1] 12 13 14 15 H 17 18 Ic; .0: 0 21 22 23 2 lo 2� u 27 2E 2� �0 �)
P I T C H GL I S S 1 GL I S S 2 G L I S S 3 D UR A T I O N O Y NA M 3 o . r:-o 3 4 .0 o.o o .o o.o 50 �. 4 1 o.o 43 . 2 o .o o.o Oo63 21 o.o 81 . 3 o. o o.o 10 0.18 47 .0 Q.O o.o o.o 29 1 . 9(1 o.o �-" o.o o.o 35 1). 5 1 48.7 o.o 1) . 0 o.o 42 1 1 .4 o. o 0 . 37 o.o o.o 59 n . oo .3 8 . 1 o.o o.o o.o 45 o .o o. o 2 . 20 o.o o.o 0 1 .0 7 o.o 5 5 . 1) o. o (1 . 0 1 0 . 4C llo5 o.o o .o o.o 19 23.2 n . oo o.o � .o o. o 6 o.oo 26.9 o.o o .n o.o 57 lt 6 . 2 n . 32 "·" o. o tl . O 25 1) . 7 1 6 8 .5 o.o o. o (' . 0 32 46.9 1) . 6 4 Q .O o.o o.o 1 44.0 0 . 44 o.o c.n o.o 21 0. 22 36.2 ') . Q o.o (1 . (1 13 32. 5 t . o c; 0 . 1) o.o o.o 6(.\ 12.6 (I . O t o.o o.o o.o 60 o.ss o.n 3 B o CJ o.o l'! o t' 62 74 . 6 0 . 80 7 1 .0 -2 s.o -71.0 50 1 . 50 32 . 6 o.o o.n o.o 26 50 . 9 0 . 60 o. o 0 . (\ o.o 24 4 o 5 1! 0 . () o.o o.o o.o 58 0.02 49 . 3 O . CJ o.o o.o 13 o.o 0. 22 5 1 .0 o.o o.o lt 3 36 . 9 o.o (1 . 0 1) 1) . 0 1:' . 0 56 1 o 3B 31.8 1') . (') o .o o. o 14 n. 2 e I). I) (! . «) () . 0 o.o 40 1 . 6 <; 12.2 0 . (' 0 . 1) o.o 55 48.5 0.37 n.o o.o o.o 58 ll . O n .o 1 . 50 o. o o.o 1'1 . 5 2 1 5 .4 21 49 . 0 5 . (1 - 3 1 . 0 B 1 .38 O . IJ o.o o.o o.o 4 lo 7 . 3 -71.0 1 . ('1 5 46.0 - 1 7.0 24 37 . 6 0 . 14 o.o o.o o.o 0 1 o 30 o .o o.o o.o o.o 1 '3 0. 19 (: 4 . 3 o.o o.o o.o 9 3 .72 52.2 0 . (\ o.o o.o 28 59.0 0.83 o .o c .o f.' e O 38.8 11 25.0 (1 . 0 (1 z .o - 1 5 . 0 7 5 .6 0.43 11 0 . (1 ll . (l o.o 5 1 .5 57 (1 . 7 7 o.o c.o o.o 2 0 . 3 <; 12.1 o.o (! . 0 o. o 1 . 16 4 3 .0 2 -7 1 . 0 24.0 n.o eo . 3 50 0.85 3 6 . 1'1 4.0 22.0 59 .9 10 o. o (' . l(! o.o o.o 2 . 4 Cj 40 . 1 33 o.o o.n o.o o .� n . 4 E: 34 o.o c.o o.o
Fig . V-4. Provis i o nal R esu l ts of O n e Phase o f t h e Ana lysis
1 54
Formalized Music
-
0
10
I
..-
�
>
Lri D)
ii:
C h a pter VI
Symbolic we
Here
shall
atta c k the
Music
thorny p robl e m of the logic underlying musical
composition. Logic, that queen of knowledge, monopolized by m athem atics,
wavers between her own of algebra.
name,
borne
through two millennia, and the name
Let us leave the task of logically co nnecting the preceding chap ters the moment. We shall confine ourselves to following a path which may l e a d us to regions even more harmonious in the not too dis tant fu ture.
for
In
A LOG I CA L AN D ALG E B RAIC S K ETC H M U SI CA L C O M P O SITI O N
OF
we shall begin by imagining that we are suffering from We shall thus be able to reascend to the fo u ntain-head of the mental o p erations used in composi tion and attempt to extricate the general principles that are v al i d for all sorts of music. We shall not make a psycho-physiological study of perception , b u t s h a l l simply try to u n d e rstand more clearl y the phenomenon of h earing and the thought-processes involved when l is te n ing to music. In th i s way we hope to forge a t o ol for the better comprehension of the works of the past and for the constru ction of new music. We shall therefore be obliged to collect, cut up, and sol der scattered as well as organ ized enti ties an d conceptions, while u n ravel i n g the th i n thread of a l ogic, which will certainly p resen t lacunae, b u t which will a t least have the merit o f existing. this chapter
a sudden am nesia.
CASE
OF
A
SIN GLE
GENERIC
E L E M ENT
Let there b e a son i c event wh i ch i s not ! 55
endless .
I t i s see n
as a
whole,
as
Formalized
156
Music
an enti ty, and this overall pe rce p tio n is sufficient for the m o m e nt. Because of our amnesia, we d ecide that it is neuter-neither pl e asant nor unpleasant.
Postulate. We sh al l systematically refuse
every sonic event. What eve n t
or
will
a
q u alitative judgment
cou n t will be the abstract relations
on
within the
and the logical operations wh i ch m ay be of the sonic event is thus a kind of statement, sonic symbol, which may be n ot at ed gra p hically by the
betw ee n several events,
impos ed on them. The emission
in scri p tio n , or
le tte r a.
which and then disappears ; we simply have a. If it is emitted several times in s uccessi o n , the events a re compared and we co n cl ude that th ey are identical, a n d no more. Identity and tautology are t h erefo re implied by a repetition. But simultaneously an other phe nomenon, subjacent to the first, is created by reason of this very repeti tion : modulation of time. If the event were a Morse sound, th e temporal abscissa wou ld take a mea n ing external to the sou nd and independent of it. In addi tion to the deduction of tautology, then, rep e titio n causes the appearance of a new phe no me non , which is i n scribe d in time and which modulates time. To summarize : If no acco u n t is taken of the temporal element, then a s i ngle sonic event signifies only i ts statement. The sign, the symbol, the generic element a have be e n stated . A sonic event actu ally or mentally repeated signifies only an identity, a tautology : If it is emitted once it means n othing more than a single existence
appears
a v a v a
V
a · • ·
V
a
=
a.
V is an op erator that means " p ut side by side withou t regard to time." The sign means that it is the same thing. This is all that can be done with a single sonic event. =
CASE
OF T W O OR
MORE
G ENERIC
Let there b e two sonic events
and s u ch
a
and
a
ELEMENTS
and
b such that a i s not identical with b ,
that the two are distinct and e asily recognizable, l i k e the letters b, for exam pl e , which are only confused by a near-sighted person or
when they are poorly written .
If no
account is taken of the temporal element, then the two elements
are co nsidered as a pair. Consequently emitting first
then
they
a,
a
then
h,
or
fi rst b
gives us no more information about these disti nct events than when
a re
heard i n isolation after long intervals of
account is
taken of the relation of similitude
silence. And
since no
or of the time factor, we can
1 57
Symbolic Music
write for a
"#
which means
b
th a t a
b
a v
=
b
V
a,
b side by side do not
and
create a new t hi n g , having th e
same meaning as before . Therefore a commutative case
In the
of three distinguish able
of these s on i c sy m b o l s entity
in relation
events, a,
may be considered
to the
third :
(a
v
b)
This is
an
b)
v
v
v c.
c = a
v
(b
of the
the
other hand, when
t he
com b i n a tion of two
another element,
an
nothing more we may write v
c) .
time factor leads therefore to
sition outside-time-the commu tative and the are extensible to the c ase of a single event. ) On
c, a
associative law.
The exclusion
a, b,
law exists.
as forming
But since t his associational operation produces
(a
b,
two rules of compo rules
assoc iative . (These two
manifestations of the generic events
c are considered in time , then commutativity may no longer
Thus
be accepted.
a T b "# b T a, the symbol of the law of comp o s i tion which means " anterior to. " asymmetry i s the result o f our tr aditional e xperience, o f our cus tomary one-to-one correspondence between eve n ts and time instants. It is raised when we c onsider time by i tsel f without events, and the conse quent metric time which admits both the commutative and the associa tive T being
This
properties :
aTh = bTa (a T b) T c a =
CONCEPT
The
OF
D I STANCE
consideration
T
mus t penetrate the
associative law.
(INTERVAL)
of generic clements
permit much of an advance . To exploit
we
commutative law
(b T c)
internal
a,
b,
c,
. . . as en tities doe s not
and clarify w h at has j ust be e n said,
organization of the so nic sy mbols.
is modified dur life. On a primary level we perceive pitch, du ration, timbre, attack, rugosity, etc. O n another level we may dis tingu ish compl exities, degrees of
Every sonic
event is perceived as a set of qualities that
i n g its
order, variabilities, den s i ties, homogenei ties, fluctuations, thicknesses, etc.
Our
study will not attem p t
to
elu c i date
these q u estions,
which are not only
1 58
Formalized Music
difficult
but
at this moment secondary. They arc also secondary becau se q u a l i t i e s may be gra duated, even if o n l y broad ly, and may be totally o rdered . We s h a l l therefore c h oo s e one q u a l i ty and what will be said about i t will be extensible to oth ers. Let us, then, consider a series of eve n t s discernible solely by p i tc h , such as is perceived by an observer who has lost his memory. Two clements, a, b, are not enough for him to create the notion of distance or interval . We must look for a third term, c, i n order that the observer may, by successive com parisons and through his immediate sensations, form first, the c o nc ept of relative size ( h compared to a and c) , which is a primar y expression of rank ing ; and then the notion of distance, of i n terval. This mental toil will end i n the to tally ordered classification not onl y of pi tches, b u t also of melodic in tervals. Given the set of pitch intervals
and the
binary re l a t i o n S
many of the
1 . /zS/z
for al l h
E
2. h4Shb :f:. hbSh4
3.
(greater than o r
equal
)
to , we
have
H, hence reflexivity ;
except for ha
h4Shc,
ha.Shb and hbShc entail
Thus the differen t aspects of
=
hb,
hence antisymme try ;
hence transitivity.
the sensations pro d u c e d by s o n i c events
may eventually totally or partially constitute ordered sets ac c o rd in g to the
adopted. For exam ple, if we ad op te d as the unit inte rv a l of the relationship of the semitone (� 1 .059) but a relationship of 1 .0000 1 , then t h e sets of pitches a n d intervals w o u l d be very vague and would not be totally ordered because the differential sensitiviry of the human ear is inferior to this relationship. Generally for a sufficien tly large u n i t distance, many of t h e qualities of sonic e v e n t s can be to t a l ly ordered. To conform with a first-de g ree acoustic e xp e ri en ce , we shall suppose u n i t i n te rv al
pitch, not
that the ultimate asp ects of sonic
events
are frequency1 (experienced as
pitch) , intensity, and d u ra t i o n , and that every son i c evcn t may be constructed
from these
t h re e
when
duly
interwoven . In
this case the number t hree is events see
i rreducible. For other assumptions on the microstructure of sonic the Pre face and Ch apter I X .
St r u ct u re of t h e Qu a l ities of S o n i c Events*
From a naive
or d i s ta n c e
musical p ractice we
have d efined
the concept of interval
. Now let us examine sets of i nterv al s which are in fact isomorphic
to the equivalence cl asses of the N
x
N pro d u c t set
of natural numbers.
Symbolic
Music
Let
I.
1 59
there be
a
set H of pitch i n tervals (melodic) . The law of internal (h0, hb E H) a third element may be
composition states that to every couple
made to correspond . This is
hb
ha +
=
he,
such that
he
the composi te E
which we shall notate there be three so u n d s
of ha by h b,
H. For example, let
characterized by the pitches I, I I , I I I , and
let h• h(II,III> b e the intervals
in semi tones separating the cou p l es (1, I I ) and ( I I , I I I ) , res p ectively. The interval h(I .III> separating sound I and so u n d I I I will be equal to th e sum of th e semi tones of the other two . We may t h er e for e establish that internal com p osition for conj u ncted intervals is addition.
2.
The law is associative :
3.
There exists a neu tral
e l em
h 0 + ha
en t h 0 such
=
ha +
h0 =
th a t for every
that
4. For every h0
nameless ; and
there exists h� +
ha
a
=
=
h0
h�,
which returns to
the
unison ;
H,
the
zero interval ; for
it is simultaneity .
to
the inverse, such
0.
=
Corresponding to an a scen di n g melodic in te rv al h0,
ing interval
or
for duration
special element h� , called ha + h�
E
law of
h 0•
For pitch the n e u t r al element has a name, unison,
intensity the zero interval is
ha
the
there m ay be
an increasing
a descend
interval
of
in tensity (expressed in positive decibels) m ay be added a n oth er diminishing
interval
to
(in negative db) , such that it cancels
time i n te r v al there may be the sum of the two is zero, or simultaneity. 5. The law is commutative : ing
a positive
the other's effect ; correspond
a negative duration, such that
"' Following Peano, we may state an axiomatics of pitch and construct the chro matic or whole-tone scale b y m ea n s of three primary terms-origin, note, and the successor of . . . -and five primar y propositions :
the ori g in is a note; the su ccessor of a note is a note ; notes having the same successor are iden tical ; the origin is not the su ccessor of any note ; and if a property applies to t h e origin, and if when it applies to any note it also appl ies to its successor, then it applies to all n o t e s ( principal of induction). 1. 2. 3. 4. 5.
See also Chap. VII, p. 194.
Formalized
1 60
Music
been established for pitch, outside-time. But th e m to the two other fundamental factors of sonic events, and we may state that the sets H (pitch intervals) , G (intensity intervals ) , and U (durations) are fu rnished with an A belian additive group
These
the
five axioms have
examples have extended
structure.
To
sp ecify pro p erly
the
difference and the relati onship that exists
between the temporal set T and the other sets examined outside-time, and in o rd e r not to confuse, for example, set U ( d ura ti o n s characterizing a sonic event) with the time intervals chronologically se parating sonic e ve nts belon gi n g to set T, we shall summarize the su c c e s s iv e stages of our comprehension. S U M M A RY
Let there be t hre e events a, b, c emitted successively. Three e ve n ts are distinguished, and th a t is all. Second stage : A " temporal succession " is distinguished, i.e., s pondence between events and moments. There results fr o m this
First stage :
a before b '# b
before
a
a
corre
(non-commutativity) .
Third stage : Three sonic events are distinguished which divide time into two sections within the events. These two sections may be compared and then expressed i n multiples of a unit. Time b e c om es metric and the s e c ti o ns constitute generic elements of set T. They thus enjoy commutativity. According to Piaget, the concept of time among children passes through th ese three p h as es (see Bib liography for Chapter VI ) . Fourth stage : Th re e sonic events are distinguished ; the time intervals are distinguished ; and independence be tw e en the s o n i c events and the time intervals is recognized. An algebra outside-time is thus adm i tte d for sonic events, and a secondary temporal algebra exists for temporal intervals ; the two algebras are otherwise identical. (It is u seless to repeat the arguments in order to show that the tem po ral intervals between the events consti tute a set T, which is furnished with an Abelian additive group structure. ) Finally, one-to-one correspondences are admitted between algebraic func
tions outside-time and temporal an algebra in-time.
alge b r aic
functions. They may
constitute
In con cl u s i o n , most musical anal ys i s and construction may be based on : 1 . the study of an entity, the sonic event, which, ac c ord in g to our temporary assumption groups three characteristics, pitch, intensity, and duration, and which possesses a structure outside-time ; 2. the study of another si mp le r entity,
Symbolic
161
Music
time, which possesses a
temporal structure ; and
3 . the correspondence between
the str u ct u re o u tside-time an d the temporal structure : the
Vector Space
Sets H (melodic intervalsh
G (intensity intervals) ,
U
structu re in-time.
(time intervals) ,
and T ( i n terv a l s of time separating the sonic events, and i ndependent of
them)
are
totally ordered. We also assume that they may be isomorphic
numbers, and that an external law of composition for each of them may be established with set R. For every a E E (E is any one of the above sets) and for every el em ent A E R, there exists an element b = Aa such that b E E. For another approach to vector space, see the discussion of sets of intervals as a product of a group times a field, Chap . VIII, p. 2 1 0. Let X be a sequenc e o f th ree numbers x1 , x2, x 3 , corresp onding t o the under certain conditions with set R of the real
elements of the sets H, G, U, res p ectively, and arranged in a certain order :
X=
(xh x2, x3) . This sequence is
The particular c as e
of
a
the vector in
vector and
x 1 , x2, x3 are
zero vector, 0. It may also be called
th e
components. are zero is a
its
which all the components
origin of the coordinates, and by
analogy with elementary geometry, the vector with th e nu mbers
(x1, x2, x3)
as com ponents will be called point M of coordinates (xl > x 2 , x3) . Two points or vectors are said to be equal if they are defined by the same sequence :
X; = y, .
The s e t of these sequences constitutes a vector space i n three dimensions, E3• There exist two laws of composition rel ative to E3 : I . An internal law of compositi on, a d d i ti on : If X = (xh x2, x3) and Y (y1, y2, y3) , the n =
X + Y
=
(xl
+ Y1• X2 + Y2•
Xa
+
Ya) ·
The following properties are verified : a. X + Y Y + X (commutative) ; b. X + ( Y + Z) = (X + Y) + Z (associative) ; and c. Given two vectors X and Y, there exists a sin g le ve cto r Z (z1, z2, z3 ) such that X = Y + Z. We have z1 x1 - y1 ; Z is called the difference of X and Y and is nota ted Z = X - Y. In particular X + 0 0 + X = X; and each vector X may be associated with the opposi te v e c t o r ( - X ) , with components ( - x1 , =
=
=
=
- x2, - x3 ) , such that X +
( - X)
=
0.
2. An external law of composition, multipl ication by a scalar : If
p E R and X E E,
then
pX
=
(px1, px2 , px3)
The following properties are verified for
(p, q)
E
E
E3 •
R:
a.
1 ·X
=
X; b. p (qX)
Formalized
1 62
(pq)X ( associative) ; and c. (p (dis tri butive) .
BASIS
are
AND
R EFER ENT
+
OF
If it is impossible to find not all zero, such that
a1X1
+
q) X
=
pX + qX andp( X
A
VE CTOR
a
system of p
a2X2
+
· · ·
+
Y)
=
Music
pX
+ pY
SPACE
numbers
+
a'PXP
=
a11
a2,
a3,
•
•
•
, a'P
which
0,
p vectors X1 1 X2, , Xp of the space En are say that these vectors arc linearly independent. Suppose a vector of En, of which the ith co m p o n e n t is 1 , and the others are 0. This vector l1 is the ith unit vector of En. There exist then 3 unit vectors of E3, for example, li, g, ii, corres p ondi n g to t h e sets H, G, U, respectively ; and these three vectors are linearly independent, for the relation and on
not
th e condition
that the
•
•
•
zero, then we s h a l l
a/i + a2g
en t a ils a1 = a2 be written
=
a3
may
=
+ a3fi
=
0
0. Moreover, every vector
X = (x1 , x2, x3) of E
results from this that there may not exist in E3 more t h an indep endent vectors. The set li, g, il, constitutes a basis of E. By anal ogy with elementary geometry, we can say th at Oh, Og, Ou, are axes of coordinates, and that their set constitutes a referent of E3 • In such a space, all t he referents have the same origin 0. Linear vectorial multiplicity. We say that a set V of vectors of En which is non-empty constitutes a linear vectorial multiplicity if it possesses the following It immediately
3 linearly
properties :
I . If X is a vector of
scalar p may be.
V, every vector pX belongs also
to V whatever
the
2 . If X and
Y are two vectors of V, X + Y also belongs to V. From this a. all linear vectorial multiplicity con tains the vector O (O · X 0) ; and h. every linear combination a1X1 + a2 X2 + . . . + a'PXP of p vec tors of V is a vector of V.
we ded u ce that : =
REMARKS
1 . Every so ni c event may b e expressed as a vectorial multiplicity. 2. There exists only on e base, li, g, ii. Every other quality of the sounds and every other mo re complex c o m po n e n t should be analyzed as a l in e ar combination of these three unit vectors . The dimension of V is therefore 3 .
Symbolic Music
then
1 63
p, the
3. The scalars
move out of
not in
q, m ay
pr ac t i ce take
does not invalidate the g enerality o f these
example, let 0 Og, Ou, as referent, and For
all
v al u es
audible area. But this restriction o f
be
a
the
base
origi n of
/i, g, ii,
for /i, 1
=
arguments and a
trihedral of
a
,
we
for
would
practical o rd er
their applications.
reference
with
with the followi ng units :
Oh,
semi ton e ;
g, 1 = 1 0 decibels ; for ii, 1 second. for
=
The
origin
0 will be chosen arbitrarily on the " absolu te " scales the man n e r of zero on the thermometer. Thus :
established
by tradition, in
/i, 0 will be at C3 ; g, 0 will be at 50 db ; for ii, 0 will be at 1 0 sec ;
(A3
for
for
and the vectors
may be
xl
x2
the same
xl
- 3g
7/i + tg
=
written in traditional notation for
1'1'
In
sfi
=
way
+ X2
= (5
+
7)/i
+
�
( 50 -
( 1 - 3)g
+
= 440 Hz )
+ s u
1
lu
sec �
:5 0
=
J.
20
(5 - r ) u-
d.B)
=
r 2fi - 2l
+ 4u.
=
-m t
....
c s o - 2 0 = :5 0
a B)
of al l th e preced i n g pro p osi tions. We have established, thanks to vectorial algebra, a workin g language which m ay p ermit both an alyses of the works of the past and new construc tions by setting up interacting functions of the com p onents (combinations of the sets H, G, U) . Algebraic research in conjunction with ex p erimen tal
We may similarly p u rsue the verification
research by computers coupled to analogue converters migh t give us
Formalized Music
1 64
the linear relations of a vectorial multiplicity so as to obtain of existing instruments or of other kinds of son i c events . The following i s a n anal ysis o f a fragment o f SoTzata, O p . 5 7 (Appas sionata) , by Beethoven (sec Fig. VI- I ) . We do not take the timbre into account si n ce th e p i ano is considered to have only one t i m b re, homogeneous over the register of this fragment.
information on the timbres
A
A
A �----,
B,
F i g . Vl-1 I �
Assume as unit vec tors : /i, for which I � semitone ; g, for which
10
db ; and u,
for which
,
A L G E B RA
=
=
The vector X2 =
T h e vector X3
=
The vector X4 =
The vector X5
=
1 8/i
(18 (18
(18
( O P ERATI O N S +
+
+
+
the g axis, and AND
R E L A T I O N S IN S E T
O.g + 5 u corresponds to G. 3) /i + Og + 4u corresponds to B�. 6) /i + Og + 3 17 corresponds to D p . 9)/i + Og + 2 ii corresponds t o E. 12)/i + Og + 1 il corres ponds to G.
(18 + ( 1,8 + O) li
+
a l so admit the free vector v
X1 (for i = 0, I , 2, 3, 4)
origins
.
OUTSIDE-TIME
The vector X1
us
on the li axis,
ff = 60 db (invariable) on 5J on the u axis.
The vector X0
Let
I � ; . Assume for the
Og + =
J u corresponds
3!i + Og
- I u;
A)
to G.
(See Fig. VI-2 . )
then t h e vectors
are of the form x; = X0 + vi. We n o tic e that se t A consists of two vector families, X1 and iv, combined
by means of addi tion .
Symbolic Music
165
D�
5 11
Fig .
Vl-2 A second law of composition
an ari thmetic pro g ression.
Finally, the scalar
i leads to
e x i s ts
TEMPORAL
ALGEBRA
The sonic statement
( IN
SET
This
boils
on
down t o the axis
the set
Xa
T
T)
xl
T
x2
a
=
0, 1 , 2, 3, 4) ;
of the com
A is successive :
0 of the base
of A
shifting that has noth ing to
� E3
do
ii.
Thus in
the case of a simu l tanei ty (a
chord)
of the
�
of base
attacks of the
vectors d escribed for set A, the displacement wou l d be zero.
V
with the
ch ange of the base, which is in fact an operation within sp ace E3
fi, g,
is
it
T . . .
saying that the o rigin of time,
(i
variation remaining invariant.
of the vectors X1 of set
T being the operator "before."
displaced
in
an antisymmetric
ponents fi and ii of X1, the second g
is
0
six
In Fig. VI- 3 th e segments designated on the axis of t i m e by the origins 0 of X; are equal and obey the fu nction 6.t1 = 6.t1, which is an internal law
Formalized Music
1 66
of composition in se t T; or consider segment unit
equal to
l::!. t ; then /1
=
an
a
origin 0'
+ il::!. t, for i
on =
the axis of time and 1 , 2, 3, 4, 5 .
a
Vl-3
Fig .
A L G E BRA
IN-TIM E
( RELAT I O N S
AND
B E T W E E N S P A C E E3
SET T)
p on en ts H, G , U, which = il::!. t ; the values are lexicographically ordered and defined by th e increasing order i I , 2, 3, 4, 5. This constitutes an association of each of the components with the ordered set T. It is therefore an algebration of sonic events that is indepen dent of time (algebra ou tside-time) , as well as an algebration of sonic events We may say that the vectors X1 o f A have
c om
may be expressed as a function o f a parameter t1• Here 11
=
as a function of time (algebra in-time) . time
that a vector X is a fu nction of the parameter of t if its components are also a fu nction of t. This is written
In ge n er al we admit
X(t )
=
H(t)fi + G(t) g + U (t)ii.
continuous they have differentials. What is of X as a function of time t ? Suppose
When these functions are
the meaning of the variations
dX
If we
neglect
_
dH fi
dt - dt
dG +
dt
dU
_
g +
dt
_
u.
the variation of the component G, we
conditions : For
dHfdt
=
0, H
=
ch ,
and
dUfdt
be independent of the variation of t ; and for
ch
will be of invariable pitch and duration . If ch and (silence) . (See Fig. VI--4.)
will
0, U and Cu t=
=
Cu
h ave the
=
=
cu,
following
H and
U will
0, the sonic
0, there is
no
event sound
For dHfdt 0, H ch , and dUf dt Cu, U Cut + k, if '"' and cu =1- 0, we have an infinity of vectors at the unison. If cu 0, th e n we have d d a singl e vector of constant pitch '"' a n uration U = k. (See Fig. Vl-5) . =
=
=
=
=
167
Symbolic Music
For dHfdt
and dUfdt = f(t ) , U = F (t) , we have an infinite family of vectors at the unison. cht + k, and dUfdt For dHfdt ch , H Cu, if Cu < s, 0, U lim s = 0, we have a constant glissando of a single sound. If Cu > 0, then we have a chord composed of an infinity of vectors of duration cu (thick constant glissando) . (See Fig. VJ.:...5 . ) =
=
Fig.
Vl-4
0,
H
= en, =
=
=
- �-
t,
LL
Fig. Vl-5
Fig .
Vl-6
t.. , u.
Formalized Music
1 68
For dHfdt = ch, H = ch t + k, and dUfdt cu, U cut + r, we have a chord of an infinity of vectors of variable durations and p itch es . (See =
=
Fig. VI-7.)
u = c �-t � 1c U . c".t .
k
Fig. Vl-7 For dHfdt ch , H cht + k, and dUfdt f(t ) , U chord of an infinity of vectors. (See Fig. Vl-8.) =
=
=
=
F (t) ,
we have
a
h
H tJ
"
c h' t ;
+
�
� ( t)
F i g . Vl-8
Fig.
=
j(t) , H
i, G
F (t ) , and dUfdt = 0, U Cu, if Cu < e, lim e = 0, we have a thin variable glissando. If cu > 0, then we have a chord of an infinity of vectors of duration cu (thick variable glissando) . (See For dHfdt
=
=
Vl-9.)
u .. Fig. Vl-9
c"
f,u
169
Symbolic Music For dHfdt f(t ) , H = F (t ) , and dUfdt s(t ) , U chord of an infinity of vectors. ( See Fig. VI-1 0.) =
=
=
S (t ) , w e
h ave a
h
U : S lt) Fig.
Vl-1 0
In the example drawn from Beethoven, set A of the vectors X1 is not a continuous function of t. The correspondence may be written
l
X0
X1
X2
�
�
�
X3 X4 Xs �
4
�
Because of this correspondence the vectors are not commutable. Set B is analogous to set A. The fundamental difference lies in the ch ange of base in space E3 relative to the base of A. But we shall not pursue th e analysis. R emark
If our musical space has two dimensions, e. g . , pitch-time, pitch-intensity, etc., it is interesting to introduce complex variables. Let x be the time and y the pitch, plotted on the i axis. Then z = x + yi is a sound of pitch y with the attack at the instant x. Let there be a plane uv with the following equalities : u v (x, y ) , and w = u + vi. They define u (x, y ) , v a mapping which establishes a correspondence between points in the uv and xy planes. In general any w is a transformation of z. The four forms of a melodic line (or of a twelve-tone row) can be represented by the following complex mappings : pressure-time,
=
w
w
w
w
=
=
=
z,
with
u
- z, with gradation. =
=
x
u =
v =
y, which corresponds to identity (original form) and v = - y, wh i ch corresponds to inversion = - x and v = y, which corres p onds to retrogradation - x and v = - y, which corresponds to inverted retro
and
j zj 2 /z, with u I z j 2/ - z with u
=
=
x
Formalized
1 70
Music
group.2 unknown, even to present-day musicians, could be envisaged. They could be applied to any prod uct of two sets of sound characteristics . For example, w = (A z 2 + Bz + c) f(Dz2 + Ez + F) ,
These tra nsfo r m ations for m the Klein
Other
transform ations, as yet
which c an be considered as a combi n ation of two bili near tran sformations
the type p = a2• Furthermore, for a two d i mensions we can i n tro d u c e hypercomplex
separated by a transformation of musical sp a c e of more than
systems such
as the
sys t e m of quaternions.
EXT E N S I O N O F T H E T H R E E ALG EB RAS
TO
S ETS
S O N I C EVE NTS ( a n application )
OF
We have noted in the above three kinds of algebras : 1 . The algebra of the components
of
a
son ic even t, with its vector
l angu age, independent of the procession of time, therefore an
outside-time.
2. A temporal algebra,
whi ch the sonic events create on the
time, and which is independent
3.
An
algebra in-time,
space.
axis of metric
issuing from the correspondences and fu nctional
relations be tw e e n time, T,
of the vector
algebra
the clements of the set of vectors X and of the set of metric independent of the set of X.
All that has been said abo u t sonic events themselves , their components,
and about time can
be generalized for sets
of sonic events X and for sets T.
is familiar with the with the concept of the class as i t is interpreted in Boolean al gebra . We shall ad op t this specific algebra, which is i so mo rp h i c with the t he o ry of sets. I n this chapter we have assumed that the reader
concep t of the set, a nd in particular
To simplify the exposition,
we
considering the referential or universal
piano.
We shall
c on s i d e r
shall first take a co n c re te example by
set R, consisting of all
th e sounds of
a
only th e pitcl1es ; timbres, attacks, intensities, and
dura ti o ns will be u tilized in order to clarify the exposi tion of the logical
operations and relations which we shall impose on the set of pitches.
Suppose, then , a This will be set
A,
set
A of keys that have
a
characteristic property.
a subset of set R, which consists of all the keys of the
p ia n o . This subset is chosen a p rio ri and the characteri stic property is
particular choice of a certain nu mber of keys.
the
For the amnesic observer th is class may be presented by p l a yi n g the
keys one after the other, with a period of silence in between. He will deduce from this that he
h as heard a collection of so u n d s
,
or
a
listin g
of elements.
171
Symbolic Music
class, B, consisting of a certain n umber of keys , is c h o s en in the It is stated after class A by causing the e l em e n t s of B to s o u n d . The observer hearing the two cl asses, A a n d B, will note the temporal fact : A b e fore B ; A T B, ( T = before ) . Nexl he begins to notice re l ation ships be twee n the clements of the two classes . If c e rt a i n elements or keys are common to both c l a ss es the classes intersect. If no n e are common, they are disjoint. H a ll the elements of B are common to one part of A he deduces that B is a cl ass included i n A. I f all the elements of B are fo und in A, a n d all the elements of A arc fou n d in B, he deduces that the two c la ss es are indistin guishable, that they are equal. Let us ch oose A and B in such a way that they have some clements in com mon . Let the observer hear firs t A, then B, then the common part. He will deduce that : 1 . th ere was a choice of keys , A ; 2. there was a second choice of keys , B; and 3. the part common to A and B was considered. The opera tion of intersection (conj unction) has there fore been used :
Another
same way.
This
operation has
therefore
A ·B
or
B · A.
engendered a new class, which was symbolized
B. If the observer, having heard A and B, hears a m i xtu re of al l the ele ments of A and B, he will deduce that a new class is b e i ng considered, and that a logical summation has b e e n performed on the firs t two classes. This
by the sonic enumeration of th e part common to A and
operation
is
If class
the union
(disjunction) and A + B
or
is written
B
+ A.
A has been s y mbolized or played to him and he is made to hear of R except those of A , he will deduce that the complement
all the sounds
of A wi th respect to R has been chosen . This is a new operation, negation, which is written A. Hitherto we have s h o w n by an i maginary experiment that we can define and
state
classes
of sonic
events (while taking precautions for clarity
in the symbolization) ; and e ffect
three operations
tance : in tersection, union, and negatio n .
of fu ndamental
im p or
On the other h a n d , an observer m us t undertake a n intellectual task
in o r d e r to deduce from this both cl asses and o p erations. On our plane of
immediate com p rehension, we rep laced gra p hic signs by son i c events. We co n s ide r these sonic events as sy m bo l s of abstract e n tities furnished with abstract l ogical relations on which we may e ffect at least the fundamental operations of the logic of cl asses. We have not allowed special symbols fo r the statement of the classes ; only the sonic enumeration of the gen eric
Formalized Music
1 72 elements was allowed
(though
i n certai n cases, if the cl asses are already
known and if there is no ambigu ity, ment to admit
a sort of
stenosym bolization) .
s p ecial sonic symbols
We have not allowed
which are exp ressed graph ically by these operations arc
shortcuts may be taken
i n the s tate
mnem otechn ical or even psychophysiological
ex p ressed,
· ,
for the three operations
o n l y the classes resu l ting from
+, - ;
and the operations a re conseq uen tly dedu ced
mentally by the o bserver. In the
same
must deduce the
way the observer
relation of equality of the two classes, and the relation of implic ation based
The e m p ty cl ass,
be symbol i zed by a duly presented silence . In sum, then, we can only state classes, not the operations. The following is a list of corres p ondepces between the sonic symbolization an d the graphical symbolization as we have j ust defi n ed it :
on the concept of inclusion.
Gr a p h ic symbols
Classes A, B, C, . . . Intersection
( ·)
Negation ( -
)
Union ( + )
however, may
Sonic symbols Sonic
enumeration
of the
generic
elements having the properties A, B,
C,
.
•
•
(with
possible shortcuts)
Implication (---+) Membership (e)
A
Sonic
enumeration
R not i n c l u d e d
A ·B
in
of the
A
Sonic enumeration of
A + B
A·B
elements
the elem ents
Sonic enumeration of the elements
A + B
of of
of
A => B A = B This table shows that
we can reason by pinning down our thoughts by
means of sou n d . This is true even in the p r es e nt case
where, because of a remain c l o se to that immedi ate intuition from which all sciences are bui l t , we do not yet wish to propose concern for ec o no m y of means, and in o rder to
conventions symbolizing the operations · , + , - , and the relations ---+. Thus propositions of the fo rm A, E, I, 0 may not be symbolized by
sonic = ,
sounds, nor may theorems. Syllogisms and demonstrations of theorems may only
be inferred .
I 73
Sy m bo l ic Music seen
Besides these l og ic a i
that
we
relations and operations o u tside-time, may obtain t e m pora l classes ( T c l ass e s ) issuing fro m
symbolization that defi nes distances or in tervals on the axis of time.
of time is aga i n defined in a new way . It serves primarily
we
have
the sonic
The role
as a crucible, mold,
are inscribed the c l a ss e s whose relations o n e must decipher. is in some ways e q uivalent to the area of a sh e e t of paper o r a black b o a rd . It is only in a s e co nd ary sense t h a t it may b e considered as carry in g generic el e m e nt s (temporal distances) and relations or operations between these elements ( temporal algebra) . Relations and corresp ondences may be established between these temporal cl a ss e s and the outside-time classes, and we may recognize in-time operations an d relations on t h e class level . After th ese ge n e ral consid erations , we shall give an ex am p l e of m u si c a l composi tion constructed wi th the aid of the a lgebra of classes. For this we must search out a necessity, a knot of interest.
or s p ac e in which
Time
C o n struct i o n
Every Boolean e x p re ssi o n or fu nc ti o n
three c l a ss e s
A, B, C can
F (A, B, C) ,
be expressed in the form
for example, of the
called
disjunctive canonic :
where a1 0 ; I and k1 A · B · C, A · B · C, A . J1 . C, A · E · C, Jf - B · C, Jf - B - C, A- E · C, .if. J1 . C. A Boolean fun c t i on with n v ar i a b l e s can always be wri tten in such a way as to bring in a maximum of o perati ons + , . , - , equal to 3n 2n - 2 - 1 . For n 3 this number is 1 7, and is fo u n d in the function =
=
·
=
F =
A ·B·C
+
A ·B·C +
.if. B . C + Jf . J1 . c.
(I)
For three classes, e a ch of which intersects w i th
the o t h e r two, fu nction ( I ) can be represented by the Venn d ia gr a m in Fig. VI- 1 1 . The flow chart of the op era t io n s is shown in Fig. VI- 1 2. This s am e function F c a n be obtained with only ten operations : F
=
(A · B
+
A · E) · C + (A · B
+
Jf- .11) · C.
( 2)
flo w chart is gi v e n in Fig. VI-1 3. If we com p are the two expressions ofF, each of which d efi n es a different proced u re in the com posi tion of c l a ss e s A , B, C, we noti ce a more elegant Its
For m alized Music
1 74
F i g . Vl-1 1
F i g . Vl-1 2
1 75
Symbolic Music
F i g . Vl-1 3
(2 ) is more economical ( ten It is this comparison that was chosen for work for piano. Fig. VI - 1 4 shows the flow chart
symmetry in ( 1 ) than in (2) . On
the other h and
op erations as against seventeen ) .
the realization of Herma, a that directs the operations of ( 1 ) and (2) on two parallel planes, and Fig. VI- 1 5 shows the precise plan of the constru ction of Herma. The three classes A , B, C result in an appropriate set of keys of the piano. There exists a stochasti c corres p ondence between the pitch com p onen ts a n d the m om e n ts of occurrence i n set T, which themselves follow a stoch astic l aw. The intensities and densities (number of vectors/sec.) , as wei ! as the silences, hel p clarify the levels of the com position. This work was composed in 1 9 60-6 1 , and was first performed by the extraordinary Japanese pianist
Yuji Takahashi in Tokyo in
Febru ary 1 962 .
1 76
Formalized Music
In conclusion we can say that our arguments are based on relatively simple generic elements. With much more complex generic elements we could still have described the same logical relations and operations. We would simply have changed the level. An algebra on several parallel levels is therefore possible with transverse operations and relations between the various levels.
Fig . Vl-1 4
Symbolic Music
177
u:
I
I
"' ..
I
"
�i
I
..
I I I I 11
I
0
Ill
I I
li
I
1. .i ' I
�I
;: I :- I "' I I
....
I
i
... r; .
Lo
a.
I ."r
1 :-
I
I�
I
I I I I
I
I �I
:I
�I
1
,I
a! � �
.. ""
·� l
� \J 1 "" I I .. I I I
.�
v X
�
I H�
1 ""
�
�
HI
!: 1
I I I I I I I
I� .. X
1 ;.-;
, _4"1
11
\
Conclusions and Extensions for Chapters 1-VI
t h e ge n e ral framework o f an ar tis tic attitude which , for the uses mathematics i n t h ree fundamental aspects : 1 . as a philo sophical summary of the enti ty and i ts evol u tion , e.g. , Poisson 's law ; 2 . as a quali tative foundation and m e ch a n is m of the Logos, e.g., sym bolic logic, set theory, th eo r y of chain events , gam e th eory ; and 3 . as an instr u m en t of mensuration wh ich sh arp ens i nvestigation , possible realizations, a n d p er ception, e.g. , entropy calculus, m a trix calculus, vector calcul us. To make music means to express human intelligence by so ni c means. This is intel l i gence in its broadest sense, which inclu des not only the pere grin a tio n s of pure logic but also the " log ic " of e m o tions and of i n tuition. The tech nics set forth h ere, alth o u gh often rigorou s in their i n te rnal s truc tu re, leave many openings through which the most compl ex and mysterious factors of the in tel l i ge n c e may pe n e t r a te These t e ch ni c s carry on steadily between two age-old poles, which are unified by m od e r n science and ph i los o p h y : determinism and fatality on the one hand, and fre e will and uncon d i tioned choice on the oth er. Between the two poles actual everyday life goes on, p a r t ly fatalistic, p a r tl y modifiable, with the w h o l e gamut of I
have sk e t c h ed
first time,
.
i n terpenetrations and interpretations.
In r ea l i ty formalization
guide,
and axiomatization
c onstit u t e
a procedu ral
at the o u ts e t the p laci n g of sonic art on a more universal plane. Once more it can be con si d ered on the same level as the stars, the numbers, and the riches of the human brain, as it was in the gre at periods of the ancient civilizations. The better suited to modern thought. They permit,
1 78
,
Conclusions
and
Extensions for Chapters
movements of sounds that
1 79
I-VI
cause movements in
us in agreement with
the m
" procure a common pleasure for those who do not know how to reason ; and
for t hos e
who
do know, a reasoned joy
through the imitation of
harmony which they realize in perishable movem e n ts
The theses advocated
"
(Plato,
the divine
Timaeu.s) .
in this exposition arc an initial sketch, but t h ey
h a v e alr e ady been applied an d extended. Im agine th at all the hypotheses of
in Chapter II were to be of vision. Then, i nstead of acoustic grains, sup pose q u a n t a of light, i.e . , p h o t o n s . The c om p o nen t s in the atomic, quan tic hypothesis of sound-intensity, frequency, d e nsity , and lexicographic time-are then ad ap te d to the quanta of light. A single s o urce of photons, a p h oton gun, cou l d theoretically reproduce the acoustic sc re e ns described above through the e missi o n of p ho tons of a
generalized stochastic composition as described applied to the ph e n o m e n a
particular choice of frequencies, energies, and densities. In this way we could
create a luminous flow analogous to that of m u s i c issu ing from
then join to this the coordinates of space,
a
s on i c source.
we could obtain a spa ti a l music of light, a sort of space-light. I t would o n ly be necessary to activate photon guns in combination at all c o rne rs i n a gloriously illuminated area of sp a c e . It is technically possible, but p a i n t e rs would h ave to emerge from the le th a rgy of their c ra ft and forsake their brushes an d their hands, u nless a new type of visual artist were to lay hold of these new ideas, technics, and needs. A new and rich work of visual art could arise, whose evol utio n would be ruled by huge com p u ters ( to ol s vital not onl y for the calculation of bombs or p rice indexes, but a l so for the artistic life of the future) , a total a u d i ov i s u a l manifestation ruled in i ts compositional intelligence by machines serving other machines, which are, thanks to the scientific arts, directed by man .
If we
Chapter V I I
Towards
a Metamusic
Today's technocrats and their followers treat mu s ic co mp os er
( source )
sends
to a
listener (receiver) .
as a message which the In this way they believe
that the solution to the problem of the nature of music and of the arts in general lies in formulae taken from i n form ation theory. Drawing up an ac count of bits or quanta of information transmitted and received would thus seem to provide them with "obj ective" and sc i en t i fi c criteri a of aesthetic value. Yet apart from e lementary statistical recipes th is theory-which is valuable for tech nological communications-has proved incapable of giving the characteristics of aesthetic v a l u e even for a simple melody of J. S. Bach. Identifications of music with message, wi th communication, and with language are schematizations whose tendency is towards absurdities
in this exception. Hazy music c a nnot be forced into too mold. Perhaps, it will be p ossible l ater when prese nt
and desiccations . Certain Afric an tom-toms cannot be included criticism , but they are an
prec i se a theoretical
theories have been refined and new ones invented .
The followers of information theory or of cybernetics
end two groups :
extreme . At th e other divided into
re presen t one
there are th e intuitionists, who may be broadly
who exalt the grap hic symbol above the sound of make a kind of fetish of it. In this group it is the fashionable thing not to write notes, but to create any sort of design. The " music " is j u d ge d according to the beauty of the drawing. Related to this is the so-called aleatory music, which is an abuse of language, for the true term should be l . The " graphists,"
the music and
English translation or Chapter VII by G. W. Hopkins. 1 80
Towards
a Me tamu si c
th e "improvised"
181
mus i c
grandfathers
our
the fact that gr a p h ica l wri ting , whether
notation, geome tric,
is
as
faithful
k n ew . This group i s i gnor an t of
it
be symbol i c, as in traditional
or nu m e ri c a l , should be
no more than an i mage that
as possible to all the i ns tructions the c o m p o ser gives t o the
orchestra or to the machine . 1 This group is taking music outside itself. 2. Those
who add
a spectacle in the
form of extra-musical scenic acti o n
to accompany the m u sic al performance. Influenced by the " happenings"
which express the confusion
of certain artists, these composers take refuge and thus betray their very limited In fact the y concede certain defeat for their
in mimetics a n d disparate occu rrences
pure
music.
groups
share
confidence in
music in
particular.
The two
a
romantic attitude. They believe
in
immediate
abou t i ts control by the mind. But since musical act io n , unless i t is to risk falling into trivial improvisation, imprecision, and irresponsi bility, im p eri o u s l y demands reflection , these action and are not much concerned
gro u ps are in fact denying music and take it outside i tself. linear
Thought
Aristotle, that t h e mean path is the best, for in middle m e a n s co mpromise. Rather lu cidi ty and harshness of critical thought-in other words, action, reflection, an d self trans formation by the sounds themselves-is the path to follow. Thus when I shall not say, like
m us ic-as in politics-th e
scientific and mathematical thought serve music, or any human creative
activity,
it
sh o u l d amalgamate dialectically with intuition. Man is one,
i ndivisible, a nd total. He thinks with his
would like to p ropose what, to my 1 . It makes it.
2. I t
3. It
is
a
belly
is a .
fixing in sou nd . . , arg uments) .
4. It is normative, th at
doing by symp athetic drive.
his mind .
I
is,
a
of
real ization.
i m a gi n ed
unconsciously
it
virtualities
is
a
( cosmological ,
m ode l for being or
for
is catalytic : its mere p resence perm i ts i n ternal psychic or men tal
transformations
6. I t is 7. I t is
in
the sam e way as the crystal ball of the hypno tist.
the gratu itous play of a child . a
Consequently expressions arc only very limited par t icu l ar
my sti ca l (but atheistic) asceticism.
of sadness, joy, love, and dramatic situations i nstances.
and feels wi th
covers the term " music " :
sort of comportment necessary for whoever thinks it and
is an individual pleroma,
philosophical,
5. It
mi n d ,
Formalized
1 82
Musical syntax has u nd e rg o ne considerable u ph e aval
seems that i n n u merable possibili ties c oexi s t in
a
and
Music
today
state of chaos. We h ave
abu ndance of t heories , of (sometim es) i ndividual styles, of ancie n t "school s . " But how docs one make music ? What
cated b y oral teaching ? (A burn i n g q u estion, i f one is
it
an
more or less can be communi to re form musical
entire world . ) It can no t be s a i d t h a t the inform ationists or the c y berneticians-much less the i n t u itionists-have pose d the q ucstion of an ideological purge of the dross accumulated over the centuries as well as by presen t-day develop ments. In general they all remain ignorant of the su bstratu m on which they found this theory or that action. Yet this substratum exists, and it will allow us to establish for the fi rst time an axiomatic system, an d to bri ng forth a formalization which will u n i fy the ancien t past, the presen t , and the future ; m o reover it will do so on a planetary scale, co m pri sing the still separate
edu cation-a reform that is necessary in the
universes of sound in Asi a, Afri ca, etc. In
19542 I d enoun c ed linear thought (polyphony) , and demonstrated the
contradictions
of
serial music. In its p l ace I proposed
masses, vast groups characteristics such
required d efinitions tic m usic
was
was
a
world of
of sound-events,
clouds, and galaxies go vern ed
and realiz ations
using
as
sound by new
density, degree of o rder, and rate of change, which
probability theory.
Thus s tochas
numbers for it could embrace it as a particu
born. In fact this new, mass-conception with large
more general than linear polyphony,
lar instance (by No, not yet.
Today these
re d u cing
the
density of the clouds) . General harmony ?
id eas and the realizations which accompany th em
be e n around the world, and the ex pl or a tion seems to
be
have
closed for all
intents and -purposes . However the tempered d i a tonic system-our musical terra
firma
on which all our
music is
fo unded-seems
not to h ave been
breached ei ther by reflection or by music itself.3 This is where the next stage
will come. The exploration and transform ations of this system will herald a new and immensely promising era. In order to understand its determi na tive imp o rtance we must look at i ts pre-Christian origins quent development. Thus
I
shall point ou t
the
and
at
stru ctu re of the
its subse music of
anci en t Greece ; and then that of Byzantine m u sic, which has best preserved
it while developing
it, and has done so with greater fidelity tha n its sister,
the occidental plainchant.
struction in
a
After demonstrating their shall try to express in a
modern way, I
math ematical and log i c a l language what time (transverse
musicology)
and in s pace
was
abstract
logical
con
simple but u niversal
and what might be valid
(comparative
musicology) .
in
Towards a
Metamusic
1 83 a
In or de r to do this I propose to make
distinction in musical archi
t e c t u re s or categories between o u tside-time, 4 in-time, and
sc a l e , for e x am pl e , is a n
temporal. A given pitch ou tside-time architecture, for no horizontal or
vertical combination of its elements can alter it. The event in itself, that is,
its actual o cc u rr e n c e , belongs to the temporal cate gory. Finally, a me l ody or a chord on a gi ven scale is produced by relating the ou tside-time category to the temporal catego ry. Both are realizations i n - tim e of outside-time con structions. I have dealt with th i s distinction a l re a d y , but here I s ha l l show how a n ci e n t and Byzan tine music can be ana lyz ed with the aid of th e se cate gories . This approach is very general s i n c e it p e rm its b o t h a u niversal axiomatization and a formalization of many of the a s pe ct s of the various k i n ds of music of our pl a n et. St ructu re of A n c i ent M us i c
chant w a s fo u nd e d on the structure of ancient the o thers who accu s e d Hucbald of being behind th e ti m es . The rapid evolution of the mu si c of Western Europe after the ninth century simplified and smoothed out th e plainchant, and theory was Originally t h e Gregorian
music, pace
C o m b ari eu
and
left b e h i n d by practice. But shreds of the ancient theory can stil l be found
in
the sec u l ar
music of
the
fi fteenth and sixteen th centuries, w it n e s s the
Terminorum Musicae dijfinitorium
ofJohannis
Tinctoris . 5 To look at antiq u i ty
of the Gregorian ch a n t and its h ave lon g ceased to be understood . We arc only beginning to glimpse o th e r d i rections in which the modes of the p laincha n t can be ex plained . Nowadays the specialists are saying that the modes arc not in fact proto-scales, b u t that they are rather ch aracterized by melodic formulae. To the best of my kno w l ed ge only J acques C h ailley6 has introduced other concepts com plementary to that of the scale, and he would seem to be co rrect . I b e lieve we can go further and affirm that ancient music, at least up to the first centuries of C h ristianity, was not based at all on scales and m o d e s related to the o c t av e , but on tetrachords and systems. Experts on ancient music (with the above exception ) have ignored this fundamental reality, c lo u de d as their minds h ave been b y the ton al con stru ction of post-medieval music. However, this is w h a t the Greeks used in their music : a hierarchic structure whose complexity p roceeded by suc ces sive "nesting," and by inclusions and i ntersections from the particular scholars h av e been looking th rough the lens modes, which
to the general ; we can trace its main o u tline if we follow the writings of
Aristoxenos : 7
A. The primary order consists of the
tone and its subdivisions .
The whole
1 84
Formalized
Music
a m o u nt by which the i n terval of a fifth ( the penta dia pcnte) exceeds the i n terval of a fo u rth (the tetrachord, or di a tessaron) . The t o n e is divided i n t o halves, called semitones ; thi rds, calle d chromatic dieseis ; and quarters, the extremel y small enharmonic dieseis. No interval sm a l le r than the qu arter-tone was used . B. The secondary order consists of the tetrach ord. It is bounded by the interval of the dia lessaron, which is e q u a l to two an d a half tones, or thirty twelfth-tones, w h i c h we s h a l l call Aristoxenean segments. T h e two outer n o te s always maintain the same i n terval, the fourth , while the two inner notes are mobile. The pos i t i o n s of the inner notes determine the three genera of th e tetrachord (the in tervals o f the fi fth and the octave play no part in it) . The position of the notes in the tetrachord are always c o u n t ed from the lo wes t note up :
tone is defined as the
chord , or
1 . The
3 + 3
+
genus co n t a i n s two enharmonic d iese is , or If X equals t h e value of a t o n e , we can express xs12. X l i4 . X1'4 - X 2
enharmonic
24
=
30 segm ents .
th e enharmonic as
=
2. The chromatic genus co nsists of three types : a . soft, containing two chromatic dieseis, 4 + 4 + 22 30, or X113 . X1'3 - X X 5'2 ; b. h e m iolo n (ses q uialterus) , containin g two hemioloi d i es e is , 4.5 + 4 .5 + 2 1 2 = 30 segments, or X<3t2>014> X<31 ><114> . X7' 4 X5'2 ; and c. "toniaion , " con sisting of two semitones and a trihemitone, 6 + 6 + 1 8 = 3 0 segments, xs12, or Xli2 . Xtt2 . xat2 =
•
=
=
=
3 . The
di atonic
consists of: a . soft, containing a five enharmonic dieseis, 6
enharmonic dieseis, then ments,
or X1'2 - X 3 14 . X5'4 = X5'2 ; b. syntonon, tone, and a n o t h e r whole tone, 6 + 1 2
whole Xli2 · X · X
=
xst2 ,
semitone,
+ 9 + 15
then three 30 seg =
containing a semitone, +
12
= 30
segments,
a
or
or the system, is essentially a c o mbination of the elements of th e first two-tones and tetrachords eit h e r conjuncted or separated by a tone. Thus we get the p e n ta chord (ou ter interval the perfect fifth) and the o c to c h ord (ou ter interval the octave, some tim es perfect) . The subdivisions of the system follow exactly those of the tetrachord. They are also a function of connexity and of consonance. D. T h e quaternary order consists of the tropes, th e k eys , or the modes, which were probably j u st particularizations of the sys tem s, derived b y means of cadential, melodic, d o m i n a n t , registral, and other formulae, as in Byzantine music, ragas, etc. These orders account for t h e outside-time structure of Hellenic music. After Aristoxenos all the ancient texts one c a n consult on this matter give
C . The
tertiary o rder,
1 85
Towards a Metamusic this same h ierarchical
procedure . Seemi n gly
was
Aristoxenos
u s e d as
a
model. But later, traditions parallel to Aristoxenos, defective i n terpretati o ns,
and sediments d is to r te d this h i e rarchy, even in anc i e n t times. Moreover, it
seems that theoreticians like Aristides Qu inti lianos and Claudios P to le m a eos had b u t little acquaintance with m usic.
This
h ierarchical " tree " · was completed
by transition algorithms
the metabolae-from one genus to another, from one system to another, or from one mode to another.
This is
a far cry from the simple modulations or
trans p ositions of post-medieval to n al music.
Pentachords are subdivided i n to the same genera as the tetrachord they contai n . They are derived from tetrachords, but nonetheless arc used as primary concepts, on the same footing as the tetrachord, in order to define the interval of a tone. This vicious circle is accoun ted fo r by Aristoxenos'
determi n ation to rem ai n
faithful to mu sical experience (on which he insists) ,
which alone defines the s t r u c t u re of tetrachords and of
the
en tire harmonic
edifice which results combinatorially from them. His whole axiom ati cs
proceeds from there and his text i s an example of a
me th o d
to b e fo l l o wed
.
Yet the absolute (physical) value of the interval dia tcssaron is le ft undefined,
whereas the Pythagoreans defiQed i t by the
strings.
I
bel ieve this to be
a
ra ti o
3/4 of
the len gths of the
sign of Aristoxenos' wisdom ; the
c ou l d in fact be a mean value. Two Lan g u ages
Atten tion must be d r aw n to the fact that he makes use
rat io
of th e
3/4
addi tive
operation for the i ntervals, t h us foresh adowing logarithms before their time ;
this
co n t r as ts with the
practice
of the Pyth agorcans, who u sed the
geometrical (exponential) language, which is multiplicative. Here, the m e th od
of Aristoxenos is fu ndamental s i nce :
l.
it constitutes one of the two
ways in which musical theory has been expressed over the millennia ;
2.
by
u s ing addition i t i n s titute s a means of " calculation" that is m o r e economi
cal, s im pler , and better s u i ted to music ;
and 3.
te m pered sc a l e n e a rly twenty cen turies be fo r e Europe.
l ays the foundat ion o f the it was appl ied in Western
it
Over the centu ries the two languages-arithmetic
addition) and geometric
( deri ved
(operating by
from the ratios of s tring l engths, and operating by mu ltiplication) -have a l ways i ntermingled and i n terpene trated so as to create much useless confusion in the reckoning of i n tervals and co nsonances, and conseq u ently in theories. In fact they
a rc
both
ex
pressions of group structure, having two non-identical operations ; th u s they have a formal equivalence.8
Formalized Music
1 86
repeated Greeks," the y say, " had descending scales i n s t e a d of the a sc e n d i n g ones we have today." Yet there is no trac e of this in either Aristoxenos or h i s s uccessors, i ncluding Quintili anos9 and Aly pi os , who give a new and ful l e r version of the steps of many of the trop es. On the con trary, the a n ci e n t writers always begin their theoretical explana� tions and nomencl a tu re of the steps fro m the b ottom . Another bit of foolish ness is the s u pp ose d Aristoxenean scale, of wh ich no trace is to be found in his text. 10 by
There
is a h are-bra i n ed notion t h a t h as been sanctimoniously
musicologists
Structu re of
to
in
rece n t times. " Th e
Byzantine
M us i c
Now we shall look a t the structure of Byzanti n e mu s i c . I t can contribute
an i n fi n i tely better understan d i ng of ancient music, occidental plain
traditions, and the d i alectics of re ce n t Euro its wrong turns and d ead- ends . It can also serve to foresee
chant, non -Euro p ean musical
pean music, with
and constr u ct the fu ture from a view com manding the remote landscapes
of the
past as well as the
electronic
fu ture. T h u s
new
d irections of research
value. By contrast the deficiencies of serial music in certai n d o m a i n s and the d a m a ge it h as done to musical evolution by its ignora n t dogmatism w i l l be i ndirectly expose d .
w o u l d acq uire t he i r full
m e a n s of calculation, the and the Aristoxenean, the mul tiplicative and the additive . l l The fo urth is expressed by the ratio 3/4 of the monochord, or by the 30 te m p e re d segments ( 7 2 to the octave) .12 It contains three kinds of tones :
Byzantine music am algam a tes the two
Pythagorean
1 2 segments) , minor ( 1 0/9 or 1 0 se gm e n ts ) , and minim al ( 1 6( 1 5 or 8 segments) . But smaller and larger intervals are constructed and the elementary u n i ts of the primary order are m o re complex than in
major (9/B or
Aris toxenos. Byzantine music gives
diatonic scale
a
preponderant role
to
the
na tural
( th e s u pp o s ed Aristoxenean scale) whose ste ps are in the follow
note : 1 , 9/8, 5/4, 4/3, 27/ 1 6, 1 5/8, 2 (in segments 0, 54, 64, 72 ; o r 0, 1 2, 23, 30, 42, 54, 65, 72) . The degrees of this scale bear th e al p h abe ti cal names A , B, r, !1, E, Z, and H. fl. is the lowest n ote and corresponds rou ghly to G2• This scale was propounded at least as far back as the first century by Didymos, and i n the second century by P to lemy , who perm u te d one term and recorded the shift of the tetra chord (tone-tone-semitone) , which has re m a i ned unchanged ever since.1a B u t apart from this dia p as o n (octave) attraction, the musical archi tectu re is hierarchical a n d " nested " as in Aristoxenos, as follows :
ing ratios to the fi rs t
1 2, 22,
3 0, 42 ,
A. The
primary order
is b as e d on the three tones 9/8, 1 0/9, 1 6 / 1 5,
a
To w a r d s a
supermajor
187
Metamusic tone 7{6, the trihemitone
6/5 , another
major tone 1 5 / 1 4, the
s em i to n e or leima 2 56/243 , the ap ot om e of the minor tone 1 35 / 1 28, a n d fi n a l l y the comma 8 1 / 8 0 . This complexity results from the mixture of the two means of calculation .
B. The secon dary order con sis t s of the tetrachords, as defined in Aristox os, and similarly the pent achords and the octochords. The tetrachords are divided i n to th r e e genera : 3 0 seg 1 . Di atonic, subd ivided into : first scheme, 1 2 + 1 1 + 7 ments, or (9/8) ( 1 0/9) ( 1 6/ 1 5) 4/3, st art i n g on 6., H, etc ; second s ch e m e , 1 1 + 7 + 12 30 segm e n ts, or ( 1 0/9) ( 1 6/ 1 5) (9/8) 4/3, starti ng on E, A , etc ; third scheme, 7 + 1 2 + 1 1 30 segm e n ts, or ( 1 6/ 1 5) (9/8) ( 1 0/9) 4/3, starting on Z, etc. Here we notice a developed combinatorial m e t h od that is n o t evident in Aristoxenos ; only t hr e e of the six possible permu tations of the three notes a r c u s ed . 2 . Chromatic, subdivided into : 1 4 a . soft chromatic, derived from the d iat o n i c te trach ords of the first s ch e m e , 7 + 16 + 7 3 0 segments, o r ( 1 6f 1 5) (7f6) ( 1 5/ 14) 4/3 , starting on 6., H, etc. ; b . s yn tonon , or h a rd chromatic, derived from the d i a t o nic tctrachords of the seco n d scheme, 5 + 19 + 6 30 segments, or ( 2 5 6/243 ) (6/5) ( 1 35 / 1 28) = 4/3 , starting on E, A , e tc. 3. Enharmonic, d erived fro m the diatonic by alteration of the m ob i l e notes and s u bdivided into : fi rst scheme, 1 2 + 1 2 + 6 30 segments, or (9/8) (9/8) (256/243) 4/3, starting on Z, H, r, etc. ; s e con d scheme, 12 + 6 + 1 2 30 segmen ts, or (9/8) (256/243) (9/8) 4/3, starting on 6., H, A , etc . ; third scheme, 6 + 1 2 + 1 2 30 se g ments, or (2 56/243) (9/8) (9/8 ) 4/3, starting on E, A, B, etc. en
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
PARENTHESIS
sec a phe no m e no n o f abs o rp t i o n o f the a n c i e n t enharmonic h ave taken pl ace during t h e first c en t ur i es of Ch risti anity, as part of the Church fathers' struggle a gai n s t paga n ism and We c an
by t h e d i a ton i c . This must
certai n of its manifestations i n the sidered sober, severe,
and noble,
arts.
The d i ato n i c h a d always been con
u n l i ke the o t h e r types. I n fact the chro m atic
especially the enharmonic, d e m a nde d a more advanced m u sical as Aristoxcnos and t h e other th eoretic ians had a l ready po i n t e d o u t, and such a c u l ture was even scarcer amon g the m asses of the Rom a n period . Conseq u ently combinatorial specul ations on the o n e h a nd a n d pra c t i cal usage on the other must have caused t h e spec ifi c ch a racteristics of t h e en harmonic to disappe ar i n favor of th e chrom atic, a subd ivision of wh ich f"c ll genus,
and
c u l t u re ,
Formalized Music
1 88 away in Byzantine music, and of the syn tonon diatonic.
This
phenomeno n
of a bsorpti o n is com parable to that of the scales (or mod es ) of the Renais sance by the m aj o r diato n i c scale, which pe rpetu a t es th e a n c i e n t syntonon diatonic.
However, this sim p l i fi cation is curious and i t wo u l d be i nteresting to study the exact circums tances a n d c a uses. Apart fro m d i ffe re nces, or rathe r
is b u i l t strictly o n the definitions which s of the Aris toxenean ys te m s ; this was
vari a n ts of an cient interval s , Byza n t i n e typology
anci ent. 1t b u i l ds
up the
next stage w i th tetrachord s, usi ng
sin g u la rl y shed l i g ht on the
expo u n ded in s om e detail
theory
Ptolcmy. 1 5
TH E SCALES
help a:f and asso n a nce (paraphonia) . In Byzantine music the principle of iteration and j uxtaposi tion of the system leads very clearly to scales, a development which is still fairly obscure in Aristoxenos and his su ccessors, except for Ptolemy. C.
The tertiary order
by
systems havi ng the same
c o nsi st s
a n c ie n t
of t h e scales constructed with t h e
rules of co n son a n c e , dissonance,
Aristoxenos seems to have seen the system as a category and end in itself, and the co n c e p t
of the
scale did not emerge independently from
the method
which gave rise to it. In Byzantine m usic, o n the o ther hand, the system
was
It i s a s o rt of iterative operator, which starts from the lower category of tetracho rds and their derivatives, the pentachord and th e octochord, and builds up a c h ai n of more complex o rganisms , in the same manner as chromoso mes based on genes. From this point of view, system-scale co u p li n g rea ch e d a s t a ge of fulfi l lm e n t that had been unknown in ancient times. The Byzantines defined the system as the simple or multiple rep et iti o n of two, several, or all the notes of a scale. " Scale" here means a succession of notes that is already organized, such called
as
a method of constru cting scales .
the tetrachord or its derivatives. Three systems are used in Byzantine
m usic :
the octachord or dia pason
the pe n ta chord or wheel ( t rochos)
the
tetrachord
or
triphony.
(synimenon) or disjunct juxtaposition of two tetra
The system can unite elements by conjunct
(diazeugmenon)
juxtaposition. The d isj u n c t
dia paso n scale spanning a perfec t octave. pason leads to the sca l es and modes with which we arc fam iliar. T h e conj unct juxta p osi t i o n of seve r al te trachords ( tri p h o n y) prod u c es a scale in which the
chords one tone apart form the The conjunct juxtaposition
of several of these perfect octave dia
Towards a Metamusic octave is
no
1 89
longer a fixed sound in the tetrachord bu t one of
its mobile
sounds. The same applies to the conj u n c t j uxtaposition ofs eve ral pentachords ( trochos) .
The system
can
be
applied to
the three genera of tetrachords and to
each of their subdivisions, thus creating a very rich collection of scal es. Finally one may even
mix the genera of tetrachords in the same scale ( as i n i n a vast v a r i e ty. Thus the scale
t h e selidia o f Ptolemy) , which will result
order is the product of a combinatori al method-indeed, of a gigantic
are and their derivatives. The
montage (harmony)-by i te rative j uxtapositions of organisms that a l re
a dy
scale as
s trongly d i fferentiated , the tetrachords
it is
c o n ce p t i on
defined here is a richer and m o re universal
t h e impoverished
conceptions
modern
of medieval and
point of view, it is not the tempe red scale so much
as
than all
times. From t h i s
the a b so r p t i o n by the
diatonic tetrachord (and its corresponding scale) of all the o ther com binations or montages
(harmonies) of the
other tetrachords that represen ts a vast loss
of potential. (The d i a to n i c scale is derived
from
a d i sj u n ct system of two
diatonic te trachords separated by a whole tone, and
is
represe n ted by t h e
white keys on the piano.) It is t h i s potential, as much s e nso ri a l as a bstract, that we are seeking here
seen.
to
reinstate, a l b e i t
in
a
modern way, as will be
The following are examples of scales i n segmen ts of
Diatonic
scales.
1 2, 1 1 , 7 ; 1 2 ;
Diatonic t e tra cho rd s :
1 1 , 7, 1 2 , starting
starting on the lower
H or A ;
I I ; 7, 1 2, I 2, I l , starting
on
tem
on the lower fl, 1 2 ,
1 1, 7; 12;
1 2,
1 1,
7,
system by tetrachord and pen tachord , 7 , 1 2,
the lower Z; wh eel system ( trochos ) ,
1 2 ; 1 1 , 7, 1 2, 1 2 ; 1 1 , 7, 1 2, 1 2 ; etc. Chromatic scales. Soft
Byzantine
is eq ual to 30 segments) : system by disjunct terrachords,
pering (or Aristoxenean, since the perfect fourth
chromatic tctrachords :
I I , 7,
1 2,
wheel system start i n g on
If, 7 , 1 6, 7, 1 2 ; 7 , 1 6, 7, I 2 ; 7, 1 6, 7, I 2 ; e tc. Enharmonic scales. E n ha rmon i c tetrachords, second scheme : sys tem by disj u nct te trachords, s tarti ng o n fl, I 2, 6, 1 2 ; I 2 ; 1 2 , 6 , 1 2 , corres ponding to the mode produced by all the white keys starting with D. The e n h a rmonic
scales p r od u ced by the d i sj u n c t system form a l l the ecclesiastical scal es or
m od es of the Wes t, and others, for exam p le : ch romatic tctrachord , first scheme, by the triphonic s ys te m , starting on low H: 1 2 , 1 2, 6; 1 2, 1 2 , 6 ; 1 2,
I 2 , 6 ; 1 2 , 1 2 , 6.
Mixed scales. Diatonic tctrachords, fi rs t scheme disjunct sys te m , s tart i n g on l o w H, 1 2 , l I , 7 ; 1 2 ; 7, 1 6 ,
+
soft c h rom ati c ;
7 . Hard chro m a tic
te trac hord + soft chromatic ; di�junct system , s tarti ng on low If, 5 ,
12; 7,
1 6,
7;
etc. A11 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<;, bhavachakra )
animal or vegetable. I
am
citing
this mystique because i t s eems to be a very old and wid espre ad form of thought, which existed independently a b o u t the s a m e time i n the Hinduism
of India. 2
Above all, we must note that the opening taken by the Ionians has
finally surpassed all m ys t iq u e s and all religions, i n clud i n g Christianity.
English
translation of Chapter V I I I
20 1
by John
and Amber Challifour.
202
Formalized
Music
h as the spirit of this p h i l osophy been as u n i versal as today : The U . S . , C h i n a , U.S.S.R., and E u ro p e , the prese n t principal protagonists, re s t a t e i t with a homogeneity and a u n i formity that I wo u l d even dare to Never
qualify as disturbing.
Having been
Bi rth
established, the q u estion
(eAfyxos-)
embod i ed a
Wheel of
sui generis, and the various pre-Socratic schools flourished by
di ti on i n g all further developmcn t of philosophy until our time.
Two
con in
are
my opinion th e high points of t h i s period : the P yt h a go re a n concept of
numbers
and the Parmenidean d ialectics-both unique expressions
same preocc upatio n .
As it went
of the
of adaptation, u p to the fou rth century B . c . , the Pythagorean concept of n u m bers affirmed that things are numbers, or that all things are fu rn i sh ed with num bers, o r that t h i n gs are similar to num bers. This thesis developed (and this in particular i n te res ts the musician) from the study of m usical intervals in order to o b t a i n the orphic catharsis, for a ccor d i n g to Aristoxenos, the Pythagoreans used music to cleanse the soul as they used medicine to cleanse the body. This method is found in other orgia, l i ke that of Koryban tes, as co n fi r m e d by Plato in the Laws. In every way, Pythagorism has permeated a l l o ccidental though t, first of all, Greek, then Byzantine, which transmi tted it to Western Europe and to the through
its
phases
Arabs.
All musical theorists, from Aristoxenos to Hucbald, Zarlino, and Rameau, h ave returned to the same theses colored by expressions of the m om ent . But the m os t i ncredible is that all i n te ll e c t u a l activity, including the arts, is actually immersed in the world of n u mb e rs (I am omi t ti n g the few b ackward-looking or obscurantist movements ) . We are not far from the day when g en e t i c s , thanks to the geometric and combin atorial structure of DNA, will be able to metamorph ise the W h e e l of Birth at wil l , as we wish it, and as preconceived by Pythagoras . It will not be the ek-stasis (Orphic, H i n d u , or Taoist) that will have a r ri v e d at one of the su p reme goals of all
of controlling the quality of reincarn ations (hered itary rebirths 1TaAtyyevea£a) but the very force of the " th eory, " of the q u estion , wh ich is
time, that
the
essence of human action, and whose m ost stri king expression
orism .
We
are
all Pythagoreans .3
On the other
h and ,
is Py thag
Parmenides w a s a b l e to g o to the heart of the
tion of chan ge by denying it, in contrast
to
ques the
Heraklcitos. He discovered
principle of the excluded middle and logical ta u to log y , and this created such a d azzl em e n t that
evanescent
ch a n g e
he
us e d them as
a means
of c u t t i ng out, in the
which is, one, indestructible ; the
of senses, the notion of Being, of that
motionless, filling the universe, without birth and
Philosophy of Music
Towards a
203
not-Being, not exis ting, circumscribed,
not
understood) . [F]or
it will be forever impossi ble
are ; but restrain your thought us
way remains for
and
to
s p heri cal (which
Meli sso s
had
prove that things that are not
from this route of inquiry .
. . .
Only one
to speak of, namel y , that it is ; on this route there
are m an y signs i n d icating that it is uncreated and indestructible,
for
it is compl e te , u n d isturbed , and without end ; it never was, nor will
it be , for now it is all at once complete, one, con tinuous ; for w hat kind of birth a re you seeking for it ? How and from where could it
grow ?
I
will neither let you say nor think that it cam e from what is
not ; for it is unu tterable and unthinkable that a thing
is
not.
And what need would have led it to be created sooner or later if it came from nothin g ? Therefore i t must be, absolutel y , or not at
all.
-Fragmen ts 7 and 8 of Poem,
Besi d e s the ab r u pt and
by
c o m p act style of the th o u ght ,
Parmen ides4
the method of the
is o n ly that " two-faced " mor tal s accept as valid without tu rning a hair, a n d to sta t i n g that th e only truth is the n ot i on of reality i tself. But this no ti o n , substantiated with the help of a b s tra ct q u estion is absolute. It leads to denial of t he sensible world , which
made
of
contradictory ap p e a r an ce s no
logica l rules, needs
ot h er concept than that of its o p p os i t e , the not
Being, t he nothing that is immediately rendered impossible to fo rm u l ate and to conceive.
axiomatics, which su rpasses the deities and a tremendous influence o n Parmenides' contem p oraries . This was the first ab s o l u t e and c om p lete materialism. Immedi ate repercu ssions we re , in the main, the c o n ti n uity of Anaxa g oras and the atomic d i s con t i n u i ty ofLeukippos. Thus, all intellectual a ction until our time has been profoundly imbued with this strict axiom atics. The principle of the conservation of energy in physi c s is remarkabl e . En This concision
and
this
cosmogonies fu ndamental to the first elements, 5 had
ergy is that which fills the u n i verse in electromagnetic, kinetic , or material
form by virtue is
" par
of the
equivalence matter-energy . I t has become that which
excellence. " Conservation i m p l i es th at
in the
en tire u n ive rs e
and that it
it
d ocs
not vary by a single
thus t h ro ugho ut eternity. O n the other ha n d , by the same reasoning, th e logical tr u t h is tau tolo g ical : All that which is affirmed is a tru th to wh ich no al ternati ve is conceivable photon
has been
(Wi ttgenstcin) . Modern knowled ge accepts the void , but is i t truly
a
non
of an unclarified c o m pl e m en t ? After the fai l ures of th e nineteenth ce ntu ry, scie ntific thought became rather ske p tical and pragmatic. It is this fact t h at has allowed it to a d a p t Being ? O r simply the desi g nation
Formalized Music
204 and dev elop to the u tmost. " All
is positive and opti mistic. We
happens as if
.
.
.
" implies this doubt, which
pl a c e a p rovi s i on a l confidence in new theories,
but we abandon them re a d i l y for
m o re
effi c a c io u s ones p ro vi d e d th at the
suitable explanation whi ch agrees with the whol e . In fact, this attitude represents a re treat, a sort of fatalis m . This is why today's Pythagorism is r el a t i ve (exactly l i ke the Parmenidean axio matics) in all areas, i n c l u di ng the arts. Throughou t the centuries, the arts have u n d e rgone transformations that paralleled two essential cre a t i o n s of h u m an thought : the hierarchical pri n c i ple and the princi p le of numbers. In fact, these principles have domi nated mus i c, particularly sin c e the Renaissance, down to presen t-day pro ced u res of composition. In school we e m ph as i ze unity and re c o mm e n d the unity of t h em e s and of their developme n t ; but the serial system imposes a n o t h er hierarchy, with its own tau tological unity embodied in the tone row and in the pri nciple of p e r pe t u a l variatio n , which is fo u n ded on this ta utology . . . -in short, all these axiomatic princi p les that mark our lives procedures of ac ti o n h ave a
agree p e rfe ctl y with the inquiry of Being in troduced twenty-five centuries
ago by Parmenides.
It is not my in tention to show that everything has al re a d y been dis and t h at we are only p l agiarists . Th is would be obvious nonsense. There is never repetition, b u t a sort of tautological i d e n ti ty througho u t the vicissitudes of Being that might h ave mou n t ed the W h e e l of Birth . It would seem that some areas arc less m u table than others, and that some regions of the world change very slowly indeed. The Poem ofParmenides implicitly a d m i ts that necessity, n eed, causality, and j ustice identify with logic ; since Being is born fro m this logic, pure chan c e is as i mpos s i b l e as no t B e i n g This is p articularly clear in the phrase, " And what need would have led i t to be born sooner or later, if it came from nothing ? " This contradiction has domin a t e d th o u gh t throughout the millenni a. Here we ap pr o a c h another aspect of the d i a l e c t ic s perhaps the most important in the practical plan of action-determinism. If logic indeed implies the absence of chance, then one can know all and even construct ev e ry th in g with logic. The problem of choice, of decision , and of the future,
covered
-
.
,
is resolved.
We know, moreover, th at if an element of chance en ters a deterministic
construction all is undone. This is why r eli gi o ns and philosophies
every
always driven chance b ack to the limits of the universe . And what they u tilized of chance in divination practices was absolutely not con sidered as su ch but as a mysterious web of signs, sent by the divinities (who were often contradictory but who knew well what th ey wanted) , and which where have
Towards a
Philosophy
205
of Music
could be read by e l ec t sooth sayers . This web of signs can take many fo rm s the Ch i n e s e system of l - C h i ng, auguries predicting th e fu ture from the flight of bi rd s and th e en trails of sacri ficed animals, even telling fo rt u n e s from tea leaves . This i n a b i l ity to admit pure chance h as even p e rs is t e d in modern mathematical probability theory, which h as succeeded in incorporating it i nto s o m e deterministic l ogic a l laws, so that pure chance an d pure determinism are o n l y two fa ce ts of one entity, as I shall soon demonstrate with an example. To my knowled ge, there is only o n e " u nveiling" of pure chance in all of th e history of th o u ght , and it was E pi c u r us who dared to do it. Epicurus struggled aga i n s t the determin istic n e tw o r k s of the atomists, Platonists, Aristoteleans, and Stoics, who fi n ally arrived at th e negation offree will and believed t h a t man is subject to nature's will. For if all is logically ordered in the universe as well as in our bod ies, which are p r o d uc t s of it, then our will i s s ubj e c t to this logic and our freedom is nil. The Stoics admitted, for ex a m p le, that no matter how small, e v e ry action on earth h a d a re p ercussion o n the m os t distant s t ar in the u niverse ; tod ay we would say that the n e two rk of c o n ne ct i o ns is compact, sensitive, and without loss of information . This p er io d is u nj us tl y sligh ted , for it was in this time that all k i n ds of sophisms were debated, beginning with the logical calculus of the Me g a ri a n s , and it was the time in which the S toics c re a te d the logic called mo d a l , which was distinct from the Aristotelian logic of c lass e s . Moreover, S toicism , by its moral thesis, its fullness, and i ts scope, is w i thout doubt basic to the forma
it has yi el ded its place, thanks to the substitu tion of punishment in t h e person of Christ and to the myth of eternal reward at the L as t Judgment-regal solace for mortals. In o rd e r to give an axiomatic and cosmogonical foundation to the proposition of man's free will, Epicurus started with the atomic hypothesis and admitted that " in the straight line fal l that t ra nsp orts the atoms across the void, . . . at an undetermined mom e n t the a to m s deviate ever so li ttle from the vertical . . . bu t the d evi at i o n is so slight, the least possi ble, that we could not conceive of even seemingly oblique movements . " 6 This is the
tion of Chris tianity, to which
theory of ekklisis ( Lat. clinamen) set forth by Lucretius. A senseless principle
is
introduced
in to
the
grand
deterministic atomic structure. Epicurus
thus
based the structure of the u n i ve rse on determinism ( th e i nexorable and paral
lel fall
of a to m e ) and, at
the same t im e ,
on
indeterminism (ekklisis) .
It
is
striking to com pare his theory with the kinetic theory of gases first proposed
by
Daniel
Berno u l l i . I t is
founded
on the
co
rpu s c u la r nature
of m a tter
and , at the s a me time, on determi nism and indetermi nism. No one but
Epicurus
behavior.
had ever thought of utilizing chance as a principle or as
a type of
Formalized Music
206
It was
not un ti l 1 654 that
a
doctrine on the use and understanding of
chance appeared. P as cal , and es p eci a l ly Fermat, formulated it by stu dying " games of chance " -dice, c ard s , etc. Fermat stated the two primary rules
of probabilities u s i n g multiplication and ad d i tion . I n 1 7 1 3
by Jacques Bern oull i
enunciated a
was published . 7 In this
u n ive rs al law, that of Large
E. B o r e l : " Let
p
be
Ars Conjectandi fu nd am en ta l work Bernoulli
Numbers . Here it is as stated
the probability of the
ou t c o m e and
favorable
p rob a b ility of the unfavorable ou tcom e , and l e t
e
be
a
sm all posi tive
q
by the
num
ber. The probability that the d i fference between the observed ratio of
favorabl e events to unfavorable events and the theoretical ratio pfq is larger in absolute value than
e
will approach zero when the n um b e r of trials
becomes infinitely l arge." 8 C o n si d er the example of the game of heads tails. If the coin is
perfectly
symmetric, that is to
n
and
say, absol u te ly true,
we
know that the p roba b i l i ty p of heads (favorable ou tcome) and the p ro b a bility q o f tai ls (unfavorable o u tcom e ) arc each equal to 1 /2 , and the ratio pfq to I .
If we toss
the coin
n
t i m es , we will get heads P times and tails Q times,
and
the ratio PJQ will gen e r a l ly be different from l . The Law of Large Numbers states tha t the more we play, that is to say the l arge r the number n becomes, the closer the ratio PJ Q will ap p roa c h I .
Thus, Epicurus, who admits the necessity of birth at an undetermined moment, ex act contradiction to all thought, even modern , rem ains an isolated case ;* for the al e ato ry , and tr u l y stochastic e ve n t, is the result of an accepted ignorance, as H. Poincare has perfectly defined it. If probability theory ad mits an uncertainty about the out co m e of each toss, it en compasses this uncertainty in two ways. The first is hypothetical : ignorance of the tra jectory produ ces the uncertainty ; the o th e r is deterministic : the Law of Large Numbers removes the uncertainty with the help of time (or of space) . However, by ex amini n g the coin tossing closely, we will see how the sym metry is strictly bound to the unpredictability. If the coin is perfectly
in
symmetrical, that is, perfectly homogeneous and with its mass uniformly
maximum and the by redistribu ting the matter uns ymmetrically, or by re p l a c in g a little aluminum with platinum, which h as a specific weight eight tim es that of aluminum, the distribu ted, then the u n c er t ainty 9
probability for each side will be
at e ac h
1 /2.
toss will be
a
If we now alter the coin
coin will tend to land with the heavier side down. The uncertainty will
decrease and
the probabilities
for the
two faces will
be
u nequal. When
the
su b st i tuti on of m ate ri a l is pu sh ed to the limit, for example, if the aluminum is
r ep l ac e d
with
a slip
of paper and the other side is entirely
th en the uncertainty will approach zero, that is, •
Except perhaps for Heisenberg.
towa rds the
of platinum, that
certainty
Towards a the
Philosophy of Music
will land with the lighter
coin
207 side
up. Here we have sh own
the inverse
be a is noth ing more than the m a the mati cal definition of prob a bili ty : probability is the ratio o f the nu mber of favo ra bl e outcomes to the number of p oss i b l e outcomes when all o u tcomes are regarded as equally likely. Today, the axiomatic definition of probability does not remove this difficulty, it circumvents it. relation between uncertainty and symmetry. This remark seems to tautology, but it
M U S I CA L STR U CT U R ES
EX NIHIL O
at this point in the exposition, still immersed in the lines twenty-five centuries ago and which continue to reg u late the b a si s of human activity with the greatest e ffic a cy , or so it seems . It is the sou rce of those problems about which we, in the d arkness of our ignorance, c o nce rn ourselves : determinism or chance , 1 0 u nity of style or eclecticism, Thus we
are,
of force in tro d u c ed
calculated or not, intu i tion or constructivism, a priori or not, a metaphysics
or m u s i c simply as a means of e n tertainment. should ask ourselves : I . What co ns eq u e nc e does the awareness of th e Pythagorean-Parmenidean field have for musical com position ? 2 . In what ways ? To which the answers are : 1 . R e flecti o n on that which is leads us directly to the reconstruction, as much as poss i b le e x nihilo, of the ideas b asic to musical composi tion, and above all to the rejection of every idea that docs not undergo the i n q u i ry (E..\eyxos-, 8lt7Ja ts) . 2. This reconstruction will be prom p ted by modern axiomatic
of music
Actu ally, these are the questions that we
methods.
S tarti ng from certain premises we should be able to construct the in wh i ch the utterances of B ach, Beethoven, or Schonberg, for example, would be u n i q u e realizations of a gigant i c virtuality, ren dered possible by this axiomatic removal and reconstruc most g e n e r a l musical edifice
tion.
necessary to d i vi de musical construction into two parts (see and VII) : l . that which pertains to t i me, a m ap pi ng of entities or structures onto t h e ordered s tructure of time ; and 2. that which is inde pendent of temporal becomingness. T h e r e are, therefore , two categories : in-time a n d outside-time. I nc l u de d in the category ou tside-time are the du ra It is
Ch apters VI
tions and constru ctions (rel a tions and operations) that refer to elements
( poin ts ,
on the
dis t an ces, functions) that belong to and
ti m e
crea tion . In
that
can
be
expressed
axis. The temporal is then reserved to the instantaneous
Chapter V I I I made a survey of the s tru ctu re of mo n op ho n ic music,
20 8
Formal ized Music
with its rich outside-time co mb i n a to ry capability, based on the original texts of Aristoxenos of Tare n t u m and the m a n u a l s of a c tu al Byzantine m u sic . This structure ill ustra tes i n a rem arkable way t h a t which I und e rs ta nd by the category ou tside-time.
back in to the s u bconsci ous of but has not com p letel y re mov e d it ; that would h ave been impossi ble. For about three cen t u ri es after Monte Polyphony has d rive n this category
musicians of the
European
occident,
verd i , i n-time architectures, expressed chiefly by t h e
functions,
tonal (or modal)
do mi n ate d everywhere i n central and occidental Eu rope.
How that the rebi rth of ou tside-time preoccupations occu rred, with Debussy and his i n v e ntion of the whole-tone scale. Contact with three of the more conservative traditions of the Orientals was the cause of i t : the plainchant, which had vanished, but which had b
Moussorg sky ; and the Far East. "
This rebirth conti n u es ma g nificently
throu g h
Messiae n ,
with his
mod es of l i m i te d transpositions " and " non-retrogradable rhythms , " but
i t never imposes itsel f as a general n ecessity and
never
goes beyond
the
fram ework of the scales. However Messiaen himself abandoned this vein,
yielding
to
th e pressure of serial music.
In order to put things in their proper historical perspective, it is necessary to prevail upon more powerful tools such as mathematics and
go to the b ot to m of thi ngs, to the structure of musical t h o u gh t and I h ave tried to do i n C h apters VI and VII and what I a m going t o d eve lop in the analysis of Nomos alpha. Here, however, I wish to em p h asi z e the fact that i t was D e b u s sy and Messiaen11 in France who reintroduced the category outside-time in th e face of the general evolu tion that resul ted in i ts own atrophy, to the advan tage of structures in-time.1 2 In effect, atonality do e s away with scales and accepts the outside-tim e neu trali ty of the half-tone scale. W (This sit u at i on, fu rthermore, has s car ce l y changed for fifty years. ) The i n trodu ction of in-time order by Schonberg made up for this i m poverishment. Later, with the stochastic processes that I i n t ro d u c e d into musical composition, the hypertrophy of the category in-time became overwhelmin g and arrived at a dead e n d It is in this cul-de-sac that music, abusively called aleatory, improvise d , or graphic, is still stirring today. Questions of choice in the category ou tside-ti me are disregarded by m usicians as though they were unable to hear, and especially u na b le to think. In fact, they drift along unconscious, carried away by the agitations of superficial musical fashions which they u n dergo heedlessly. In depth,
logic and
composition. This is what
.
209
Towards a Philosophy of M usic however, the o u tside-time structures
do ex i s t and it is t h e privilege of man
not only to s u s ta i n them, but to construct them and to go b eyo nd them.
th e re are
Sustain them ? Certainly ;
will per mit
us
basic evidences of this order which
to inscribe ou r names i n the Pythagorean- Parmenidean field
and to l ay the platform from which our
ideas will build bridges of under aft e r all products of m i l l ions of
s tanding an d insigh t into the past (we are
years of the past) , i n t o the future (we are equally products of the future) ,
and into other sonic civilizations, so badly e xp l a i ned by the present-day
musicolog ies, for want of the o ri gi n a l
them .
tools that we
so graciously set up for
Two axiomaties will open new doors, as we shall see in the analysis
of
Nomos alpha. We sh a ll start from a naive position concernin g the perce ption of s oun d s , naive in Europe as well as in Africa, Asi a , or America. The inhabitants of all these countries learned tens or
hu ndreds of thousands of
years ago to d istinguish (if the s o u n d s were neither too long nor too short)
such c haracteristics as pitch , instants, loudness, roughness, rate of change,
color,
tim bre.
They are even a b l e to speak of the first
in terms of i n tervals.
The first axiomatics leads
us
th ree ch aracteristics
to the cons truction of all possible scales.
We will speak of pitch since it is m o re familiar, but the following ar g uments
to all characteristics which are of the same nature (instants, rou gh n ess , density, degree of disorder, ra t e of ch ange ) . We will start from the obvious as s um pti o n that wi thin certain limits men a r e able to reco g nize whether two modifications or displ acements of pit ch are identical. For example, going from C to D is the same as goi n g from F to G. We will cal l this modification elementary displacement, ELD. (It can be a comma, a halftone, an octave, etc. ) It permits us to define any Equally Tempered Chromatic Gamut as an ETCHG sieve. 1 4 By modifying the will relate
loud ness,
displacement step
ELD,
we engender a new ETCHG sieve with the same
axiomatics. With this m ate ri al we can go no farther. Here we introduce the three logical op e ra t i o ns (Aristotelean logic as seen by Boole) of conjunction
( " an d , " intersecti on, nota ted 1\ ) , di:dunction (" or," u n i o n , notated v ) , and negation (" no," complement, notated - ) , and use them to create cl asses of pitch (various ETCHG sieves) . The following
is the logi ca l expressio n with t h e conventions as indic ated
VII : The major scale (ELD
in Chapter
(Bn
where
n
1\
=
=
:! tone) :
3n+l) V ( 8n + 2 1\ 3 n + 2 ) V ( 8 n + 4 0, I , 2 , , 23, m od u lo 3 or 8 . •
.
.
1\
3 n + l) V (8n + 6
1\
Sn)
210
Formalized
(It i s po ss i b le to modify t h e step ELD by
logical function of the maj or scale with an be based on an ELD
=
1 /3
two s i e ve s , in turn, could
provi d e
be
tone or on
more c o m plex scales.
a " rational metabola."
ELD
a ny
Music
Th u s
the
equal to a quarter- tone can
o
o ther portion of a t ne . These
combined wi th the three logical o p erations to
Finally, " irrational metabolae " of ELD may
a pplied in non-instrumental music. ELD can be taken from the field of real numbers) .
be introduced, which can only be
Accordingly,
the
The scale of limited transposition no 4 1/2 tone) :
3n
A
of Olivier Messiaen15 (ELD
=
(4n + l V 4n + 3) V Sn + l A (4n V 4n + 2) V 4n + 3 V Sn + l A (4n V 4n + 2)
4n + l
n 0, 1 , . . . , modulo 3 or 4. The second axiomatics l e ads us to ve ctor spaces and graphic and n um eri c al rep resentations. 1 6 Two conjunct i n tervals a and b can be combined by a m us i c al operation to produce a new interval c. This o pe ratio n is ca lle d addition. To either an ascending or a descending interval we may add a s eco n d c o nj u n ct interval such that the result will be a u nison ; this second in terval is the symmetric interval of the first. Unison is a neutral interval ; that is, when it is added to any other interval, it does not modify it. We may a lso create intervals by association without changing the result. Finally, in composing intervals we can invert the orders of the intervals without ch anging the result. We have just shown that the naive experience of musici ans since antiqu i ty (cf. Aristoxenos) all over the earth attributes the s tr u c tu re of a commutative
where
=
group to intervals.
Now we are able to combine this group with
a
field
structure. At least
r
two fields are possible : the set of real numbers, R, and the i so m o phi c set of
to combine th e . Abelian group of intervals with the field C of com p l ex numbers or with a field of characteristic P. By definition the combination of the group of intervals with a field forms a vector space in the fo l l ow i ng manner : As we have just said, interval gro u p G possesses an internal law of composition, addition. Let a and b be two elements of the group. Thus we have :
points on
I.
a
2. a
3. a
4.
5.
a
a
a
+
straight line. I t is
b
+ b +
+
o
a
+ b
=
+ '
= =
=
c, c
cE
=
G
(a
o +
a,
b
a
o,
+
+
m o reo v er
b) + c
=
with
a
+
possible
(b
a'
=
+
c)
assoCiattvtty
G the neutral element ( u n i son ) - a t h e symmetric interval of a
with o E
=
commutativity
211
Towards a Philosophy of Music
field
in G with those in the the field of re a l numbers) then we
We notate the external composition of elements
have
C by a dot · . If >.. , p.
E
C
(where C
the foll o wi n g properti es : 6. A · a, f.L · a E G
7. l · a
=
a· I
=
a
multiplication }
(I
=
is the neutral element
m
p.·a}
8. A· (p.a) = (i\. · f.L) · a 9 . (i\ + P.) · a = i\ · a + A · (a + b) A · a + >.. · b =
C with res p ect to
associativity
or A,
p.
distributivity
M U S I C A L N OTATI O N S A N D E N C O D I N G S
The vector space structure
o f intervals o f certain
sound characteristics by
permits us to treat their elements mathematically and to express th em
the set of numbers, which is indispensable for dialogue by the set of points on a straight
convenient. The
line, graphic
two preceding axiomatics may
with
computers, or
expression often b ei ng
be applied to
all
very
sound charac m o m e n t it
teristics that possess the same structure. For example, at the
of a scale of t i m b re which might be univer the scales of pitch, instants, and intensity are. On the other hand, time, intensity, density (number of events per unit of time) , the quantity of order or diso rd e r (measured by e ntropy ) , etc . , could be put into
would not make sense to speak sally accepted
as
one - to - o ne corres p ondence with the set
points on a straight line . (See Fig.
Fig.
Vl l l-1
Pitches
I nsta nts
of real VIII-1 .)
num bers R and the
De nsities
Intensities
Mo reo ve r, the ph enome n on of soun d is
a
set of
D isord e r
correspondence of sound these axes. The simplest
characteristics and therefore a correspondence of
Formalized Music
212
by Cartesian coord i n ates ; for ex a m p le , the in Fig. VIII-2. The unique p o in t (H, T) corresponds to the sound
c o rres po n d e nce may be sh o w n two axes
that has a pitch H at the
t
r
instant
T.
· - - - - - - - - - - - -- - - - - - - - -
-
I I I I
(11, T)
I
F i g . V l l l-2
H
I must insist here on some facts that trouble many people and that are
as false guides. We are all acq uainted with the traditional perfected by thousands of years of effort, and which goes back to Ancient Greece. Here we have just represented sounds by two new methods : used by others
notation,
algebraically by a collection of numbers, and geometrically (or graphically by sketches ) .
These three types of notation are nothing more than three codes, and
indeed there is no more reason to be dismayed by a page of figures than
by a is no reason to be totemically amazed by a nicely elaborated graph . E a c h code h as its advantages and disadvantages, and the code of classical musical notation is very refined and pre cis e , a synthesis of the other two. It is absurd to think of giving an instrumentalist who knows only notes a diagram to decipher (I am neglecting here certain forms of regression-p seudomystics and mystifiers) o r pages covered with
full
m u s ic a l score, just as there
numerical notation delivered directly by a computer (u nless a special coder
mus ical nota theoretically all music can be transcribed into these three codes at the same time. The graph and table in Fig. VII I -3 are an example of this correspondence : We must not lose sight of the fact th at these three codes are only visual symbols of an auditory reality, itself consi d ered as a symboL
is added to it, which would translate the binary results into tion) . B u t
G ra p h ica l Encoding for M acrostru ctu res
At this point of this exposition , the unveiling of history as well as the have been realized in part, and it would be useless to continue. However, before concluding, I woul d like to give an example of the advantage of a diagram in studying cases of gre at complexity. axiomatic reconstruction
213
Towards a Philosophy of Music
i
A - 4 40 Cfsf f.ne.L T F i g . V l l l-3
I
/L-
.......
7..
�1 s e c
I
=J T
N 1 2
1 .00 1 .66 2.00 2.80
3
4
N H V D I
/'?
= = =
= =
p
H 1 6 6 13
note n u m ber
I
t
II
v 0 0 + 1 7. 5 0
D 0.66 0.33 0.80 ?
I 3
5
6 5
half tones with + 1 0 � A � 440 Hz slope of glissando (if it e x i sts ) in semitones/sec, positive if asce n d i n g , negative if desce nding d u ration i n seconds n u mber correspond i n g to a l ist of i ntensity forms pitch in
Let u s imagine some forms constructed with s trai ght l i n es ,
u sing string
glissandi , for exampl c . 1 7 Is it possible to distinguish some elementary forms ? S everal o f th ese elementary ruled fields are shown they
can
Moreover
constitute elements incorporated
to
into
it
wou l d be interesting
to
pass from the first to the last element
in Fig. VII I-4. In
define and usc in
sequence
the in ter
m ediary steps (continuous or discontinu ous) from one element to especi ally
fact,
larger confi g ura t ion s .
in
a
another,
more or less violen t
way. If o n e observes these sonic fields well, o n e c a n distinguish t h e following g e n era l qualities, variations of which can combine with these basic general forms :
I . Registers (medium, shrill, etc.) 2 . O verall density (large orchestra,
3 . O verall in tensity
4. Variation
s m al l
ensemble,
etc.)
of timbre (arco, sui ponticello, tremolo, etc.) (loc al variations of 1 ., 2 . , 3., 4. above)
5 . Fluctu ations
6. forms)
7.
General progress of the form (transformation into other
elemen tary
Degree of order. ( Total disorder can only make sense i fit is calculated
according to the kinetic
theory
convenient for this study.)
of gases . G raphi c representation
is
the most
2 14
Formalized Music
�.
�
��� ��
lt_ o t!t � F i g . Vl ll-4
Let
us
now su ppose the inverse, forms constructed by means of dis
continuity, by sound-points ; for example, string pizzicati. Our previous
remarks about continuity can be transferred to this case (see Fig. VIII-5) . Points 1 .-7. are identical, so very broad is the abstraction. Besides, a mixture of discontinuity and continu1ty gives us a new dimension.
Fig. Vll l-5
� �� ' .
215
Towards a Philosophy of Music G E N E RA L
O rganization Outside-Time Consider a
set U and a
CAS E
comparison of
U b y U (a product U
x
U)
.p(U, f) . Then .p(U, f) c U x U and for all pairs ( u, u1 ) E U x U such that u, u1 E U, either (u, u1) e .P(U, j) , or (u , u1) rt .P( U,f) . It is reflexive and (u ,., u1) => (u1 ,., u) ; (u ,.., u1 and u1 ,.., u') => u ,.., u' for u, u', u1 e 1/J( U, f) . d e no ted
Thus r/J( U, j) is an eq uivalence class. In particul ar if U is isomorphic to the set Q of rational numbers, the n u ,.., u1 if lu - u11 :;:; flu1 for arbitrary
ll.ul.
Now we define t/1( U,j) as the s et of weak value s of U, 1/1( U, m) as the set of average values, and r#( U, p) as the s tro ng values. We then h ave 1/J
=
1/J ( U, f)
u
.P(U, m)
u
r/J(U, p)
s;;
U
x
U
where 1/J is the qu otient set of U by 1/J. The subsets of 1/J may intersect or be disjoint, and may or may not form a partition of U x U. Here
1/J ( U, j)
--3
1/J(U, m)
-3
.p( U, p)
are ordered by the relation --3 in such a way that the elements of r/J( U,f) are smaller than those of r/J( U, m) and those of r/J( U, m) are smaller than those of 1/J(U, p) . Then
r/J(U,j) t1 1/J ( U, m) = 0, r/J( U, m) t1 1/J( U, p) = In each therefore
o f t h ese subsets
four sub-classes :
0.
we d efine four new equivalence relations and
rJli ( U, j) with u} "' (u}) ' if and only if
l u} - (u}) ' l
:;:;
flu} with u}, (u�) ' e 1/J( U, j)
for i = I , 2, 3, 4, wi th t/J1 { U, f) c r/J(U, f) and r/J1 { U, f) --3 yr { U, f) --3 rJJl ( U,j) --3 r/14 ( U,f) ordered by the same relation -3 . The same e q uiv alence relations and sub-cl asses are d e fined for 1/J ( U, m ) and r/J ( U, p) . For sim p lification
we write
u{ = {u : and the same for
u'f
and
u�.
u e .f11( U, j)},
Formalized Music
216
In
same way, e q u i vale n ce sub-cl asses are c rea ted in two other sets, U represents the set of time values, G the set of intensity values, a n d D the set of density values with the
G and D.
Here
u
{u{, uj', ue}
=
G = {g{, gj, g�} D {d{, d'j', d�} =
for i, j, k = l , 2, 3 , 4.
Take part of the triple product U
x
G
x
D composed
of the points
df) . Consider the paths VI : {uf, g'{', d{}, V2 : {u{, gf, dj'}, . . . , VS : {(uf, u�, u�, u�), (g{, g�, g6, g�) , (d'{', d�, d�, d:') } for i = 1 , 2, 3 , 4. VS will be (u�, gr,
x
a su bse t of the triple product U
G
x
D s p lit into 43
=
64 differe n t points.
In e a ch of these su bsets c h o o s e a new subset KJ defined by t h e n points KJ (j 1, 2, . . . , n and ,\ = V I , V2, . . . , VS) . These n p o i n t s are considered =
as the n vertices of a regular p olyh e d ro n . Consider the transformations
leave the polyhedron u nchanged, that is, i ts corresponding group.
To sum up, w
E
element
of U x G x D
we
h ave the
SK�
fol lo w in g chain of inclusions :
c
vertex of the poly-
Kj>..
c
c !fo s;; u
,\
set of
path ,\
of the
u
vertices
hedron Ki
(subset
p olyhedron
X
G
X
forms or complexes or sound types or a
x
X
C1 (i
x
of X
D)
C in which C is
I . The complexes C1 t r av e rs e
= 1 , 2, . . . , n) ; for example, x
H
x
X
traverse the
X
x
a
C
and thus produce group
2 . The complexes C1 are attached to corres p onding vertices
remain fixed, but the
way of of n
the set
cloud of glissandi. Map the product H
of the polyhedron Kf. the fixed vertices transformations ; we call this ope r a ti on 80•
onto the vertices
D.
X
G
Consider the two other sets H (pitch) and X (sonic material,
playing, etc. ) . Form the product H cloud of sound-points
which
vertices, also
which
producing group
is called 81• C tr ave rse s the vertices t h u s producing the gro u p transform ations of the polyhedron ; we call this operation 811 because the product can chan ge definition at each transfo rmation of the polyhedron. transformations ; this operation 3.
The p rod u c t
H
x
X
x
O r g a n izati o n I n -Time
The last mapping will be inscribed in ti m e in two possible ways in order to manifest the peculiarities of this polyhedral group or the symmetric group
21 7
Towards a Philoso p hy of Music
to which it is isomorphic : o p eration t0-the vertices of the p olyhedro n are su ccessivel y ( m o d e l of the sy m m e t ri c group) ; operation t1-the vertices are expressed simultaneously (n simu ltaneous voices) .
expressed
Product t0
x
The vertices
B0 :
Kt are expressed su ccessively with :
I . only one sonic complex C" always the same one, for example, a cloud of sound-points only, 2 . several sonic complexes, at most 1l, in one-to-one a tt a ch m e nt with
indices of vertices
Kt,
sonic complexes whose successive appearances express the operations of the polyhedral group, the vertices i ( d e fi n e d by U x G x D ) always appearin g i n the same o rd e r , 4. s ev e ral sonic complexes always in the same order w h i l e the order of 3. several
the vertices i rep ro d u ces the group transformations, 5 . several
sonic
complexes
vertices of the polyhedron. Product t0
The list
ceding
one
x
81 :
which th is
transforming
independently
from
the
product generates may be o b t a i n e d from the prex X in place of c1•
by substitu tin g H
t0 x 8 : This list may b e readily established. Case t1 and Bi is obtained from the preceding ones by analo gy .To these i n - time operational pr o d u c ts one oug h t to be able to add in-space operations when , for e x amp l e , the so n i c s o u rces are distri buted in space in significant m a n n e r , as i n Terretektorh or Nomos gamma. Product
218
Music
Formal ized
O rg a n izat i o n Outs i de-T i me
The three s ets, D (densities ) , G ( i n tensi ties) , U (durations) , are mapped vector space. The following selection (su bset) of equivalence classes, called path V I , is m a d e : D (densities) strong, G ( i n tensities) strong, U (durations) weak. Precise and o rd ered values have been given to these classes :
onto three vector spaces or o n to a single th ree-di mensional
Set
D
a
b
dl
Set G gl
1.0
0.5
I
2.0
1 .08 2.32 5.00
2 3 4
1.5
d2 da d..
2.5
second
A
c
(Elements/sec)
selection
manner : D strong,
Set D dl d2 da
G
mf f ff iff
g2
g3
g..
(subset) , called
avera ge,
0.5
with
gl g2
I 2
G
ul
2 3
Ua
4
u ..
in
Set
ul
mp
5
the following
precise
ordered a n d
p
U
values : sec
10
17
u2
mf f
g3 g..
3
sec
11 2
path V2 , is formed
U strong, Set
Elements/sec
d..
Set U
21 30
ua
u ..
Eigh t " points " of the triple product D x G x U are selected . .
For path VI : Ki
K�
r
is
= =
d1 g1u1 ;
d2 g2u 2 ;
the column For
Ks
= =
d1g4u 4 ;
d2 g3 u 3 ;
d4 g3u2 ; d4 g1u4 ;
K2
Ks
=
d3g2u1 ;
= dag2ua ;
K�
=
(sub-class) of the table of set D .
a
Ka
K7
= =
(r
K�
=
d2g4u4 ; d2gau2 ;
a,
= =
d4g 1u1 ; d3 g2u2 •
b, c. )
K..
Ka
=
=
d1 g2u3 ; d1 g4u1·
VIII-6.)
of a p oi nts , iso
cube (a m a pp ing of these eight points o n to th e ve rtic e s
The group
formed by substitutions
morphic to the symmetric gro u p
Fig.
K� d4g4 a 4 ; K"fr = d3g3u3 ;
These eight points are regarded as solidly connected to each other
so as to form
cube) .
=
K� =
path V2 :
K1
I.
K�
among these eight
P4, is taken as the organizer p rinciple.
(See
Towards
a
Ph ilosophy
A N ALYS I S OF
NOMOS
Organ ization I n - Time
I. The
219
of Music A L PHA
symmetry transformations of
a
form the hexahedral grou p isomorphic to
cube given by the elements Kf the symmetric group P4 • The
rules for in-time setting are : l . The vertices of the cube are sounded suc
each transformation
thanks to a one- to-one correspondence. 2. themselves successive (for a larger ensemble of instru m ents one could choose one of the possible simu ltaneities as in Nomos gamm a ) . They follow various graphs (kinematic d i a gra ms ) inherent in the internal str u c ture of this particular group. (Sec Figs. VIII-6, 7, 8.)
cessively at
The
transformations
arc
t9 s
2/ /
3
\
-
8
G
7
/I
,W '
v,
·�
/
I
r Q ,··
s
Q,
6
;
I
7
' § 7 4 · --- (·8 · ' " '§J t1! ' � Qe s .i�,· - · · �� �� 2� 3 2 � � , : � / � .
3
"
1 ·'· " + 1
' i' \ I f' .�
.
.
�
'
'- 8
'
'. ' � '
6
2
e :
:
6
D
o
l-li · f i g . Vll
'
:s
'
� exaM
.
.
·-
2.
6
4
.
,I
v. •
·
3
' ,'
' '
'
B
I
7
4
8
:
,
.'
s
1
� -�
.
.
..
..
. 3
5
I ' 7 D G roup ahedral) ct O ( l a dr
4
'
7
.l(
'
�
.
2
Q
,
7
"
3
"
,•
2.
E�
I 3
4
•
Qt
'
\
,
'
I
.
0
G
4
4
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I, O
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,
' '
! /' ! -- · · .' '
�I 3 � 1 �3 � �o � 2
i 3 \.
v,
0
.
:
I
. . .
- -- -
-
i ,
.
8 /
, ./ .. . / . /.. . . . .
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6/i
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C
/\
E
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I L_____Va -
7
!i
n; : - .-: :(1," �: -
Q
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7
-
. -
\j/c I
�� 3
'8
'
3 � 1 W;r 4 �2
. ·
,
I
2
6
2-
4-
Q, ,
� 2
q
8
. i .' .. . ..
·
,. -� :
/}.2. . . . . '
I
,
"
,
· ·· . · · ·
.
�2
Oo
z
. .
'
s
,
I
1
3
�
I
-..rj 0 "'
'a � ... . t-l
0
p..
�
t/) ... .
0
Towards
•
I
I
A
c
A I
('
8
G
e
A
I c$
E.:L.
,4
8 8 c r
D
pl {
8
/)l
q 'l
E {.
t
/}
B c
r
L
G
J.. :J._
A
z D Lz E
l� fj G l 91. _(/ /)'I.. f
I>
!J
1
A
Philoso p hy of Music
a
2
E
tf
!l-
:c
8
l
yt
E' L
c
D
L
D
t.l
L
L
E
t;�
I
/)
,�. t c
l..
�i£
E
E
L
z
z
L
Ll.
L
z
lF
:z
8 G2. t.Z b
4
L
I
I t?C
B
L
])
F
, IJ'
P
-4
tl 1$2.
A
L .,_
l. z
c
8
t
1$-.. L �2 I.
.4
�
II
E
fi
G l.
[)
C
6
/)
c; > /)2 9
A
E /)2. G 2. ,q G t 2 ( I.. i p ( ( E /? 8 L
�-..
D-.. F
tl-
22 1
p ?.
D
J..z
z
E'
t;l E
t;
8
lit.
G
8 A
z
� � c
c..
l>
L2
I
� c9!l lf6 cf_!_ !?,. {), t;>2 &JN fYA= t¥3
4
fR. l?.r 41.: , �- _4'.? tjJ� �" 4/q �L Qz ti>tr � 4l6 lc>i fYt �� �(I) d),, �J cfl1 � tfz � �3 q,J. q>, q.... 4J,, � 14'1 tp, � q�
(J>, Q,_ 4>>
�ll lflt if!to 4'9 Q.r ,.,. 4'" 0- lpy '?3 IP:z. <¥�
C(j
Q' I?� 1?4' �... ry,..
� � f.ro �( ti'J "'"� Qz LP, tY.r !if 41> ()� .p, D' �"' �:; <& flll cr1 (jl3 lF .P.r qJi � t;> � 0> t?cr B q9 fY.J IPil .p,., Q.t .p., .Ps- 4>, IP> 0 � f?.t I t;1 IP� � lf!r- �' LPt qJ,, tP( y>..., �4'....� (J; IA'L L tft tP1 !i>ll P6 �v 4'.r 0 4'z. &,z !4'9 (jl.;z 1 4>...,; (i L t (i != /) c ' 4J!f A J. l. ll E L I Z ti'l 4Jr J? i r fi' c L l. Jl· L F B /) .4 l /) A F · c /)2.. f f?.,. L z. 61. z I. B E
A
c
19!. � � 4>t r;., IP�o rp, �... 4>1 413 t�� IP� � c?3 0 l;r ttl-. (j)jl 4?(/ 1¥9 fl6 rp,1\ � lPv 4}� - � �u 42 tr3 I L?Il � ta-.. tr.r If$ f?.t �.. 6 l A {) ofr l t?r l t>J,o (j)� tfc!' (jl, ((; ofu tP� � "" a 4? F.,_ c .. b [= ' ((, I trAZ C?jl tPt c?t (j>., � (j>9 t;>, Ps- tllc1' 14>...1 < 4>1 � (i/� Q, c?,z �,, f¥3 1?10 t;>,., 4>, fYI 4>.r (} .:r F-z.. L
IPt 4"11 l �j fHo tYr �� (j)y � (J... th @v �# If', 4t � tn_ ry,. 1 0- @.- f'(, ltPl �It 6>� �
f.l
Example: DA
Fig.
=
Vl l l-6. I A 8 C D2 D £2 -. E
G on D
Symmetric G r o u p P 4
21 436587 341 2 7856 4 3 21 8765 2 3 1 4 6758 3 1 247568
2431 6875 41 328576
G2 G L2 L 07 02 03
:
I
!!-
"'1 c 6 t> t
'-
/.
....
z.
tl· L'l.. 9
I
L
.
�
l= j)
i'- j)l. '-2.. 4 {j t r; .8 E- z A != ((6 lJ L B r;'- I L. 2 t c L. �� tl- 8 E A- /) "2. z l t & L I y L �a_ 9 p""L C � 4 l.l �z t z c; G/), � /:J 4 B lo I?a L c .z qly � Ci tf l> F A c tf L,_ l. 1 /
the transformation of A.
1 2345678
ljl6' � .p;. � �J. 4}� �� Q, , c?, iflr �.y (f-2 t:p/J. 4'J i 4).... A, 4'p ffr (Yo, ry, (j)� �� r;, !?.to q, {j).,. t;f t;, 0
Lz. /},_
1 42 3 5867 78653421
Q8 06 01 05
7658 3 2 1 4
0 1 0 5768 1 3 2 4
421 38657
1 3425786
685724 1 3
657821 34
8756431 2 75863 1 42
a.
5876 1 4 3 2
86754231
04 85674 1 23
0 1 1 67852341
012 5687 1 243
The n u mbers in roman type
also correspo nd to G ro u p P4
=
8
A E J) c; e- c t/E 16' ti z
(Columns -+ rows)
( 1 , 2, 3 , 4 )
3 24 1 7685
�
41
Formalized
222
Outside-Time
O rg a n izat i on II.
onto a
cl c7
Eight elements from the m a c rosc o p i c sound complexes are mapped C1 in t�rcc ways, a , {3, y :
the letters
f3
cl c2
cl y
C5
=
=
Ca
Ca
C6
Cs Cs
Cs Ca
C3
=
c2
c7
C4
=
Ca
Ca
C8
c4
Music
c4
c2
=
=
ataxic cloud of sound-points
re l a t i v e ly ordered asc en d i ng or descending
points
relatively ordered cloud
of
III.
relatively ordered field of nor
c7 =
descending
atom represented
unison
ionized
a
tom
on a
sliding
ascending
sounds,
pizzicati
on
a
cello
The s e le tters are m apped one-to-one onto a
second
neither ascending
cello by i n terferen ces of
represented
accompanied by
second cube. Thus
principle.
neither
ataxic field of sliding sounds relatively ordered ascending or desc� nding field of sliding sounds
=
sound-points,
nor descending
cloud of sound
hexahed ral
group is
by
a quasi
interferences,
the eight vertices of a as the organizer
taken
Towards
a
Philosophy of Music
223
O rg a n i zation I n - Time
I I . T h e mapping
in the
order
III.
a,
{3,
of the eight forms onto the letters C1 change cyclically after each three substitutions of the cube.
y, a, . . .
The same is true for the cube of the letters C1• v,
v , v.
V, Vz
V:r. Vs
v . v, V,
Vt
' .., u· ' " Y • 1k '/i lfE. '{� 'I;
V> V, Vt Y2 Vr- V, !/� rv "� v.- 11' v, v� v.
1/:r >�r "' Ill, V3 v, v� v, V' Yv oJ.r V2. V• v,
0 �,
Fig . Vll l-8
··o�
VL
��
..
v,
"'
.
[
v, = .r + A- + B t- c.
t'.t = p +
.,3 - /)'i- f-
E- 2- -1E-
.;...
�
+
L�
li,_ f- l..
tf'6 f- '? 12.. t- 4lr t- 4'4 V.r "" ftD -f q, f- ftp'- IP s Y(, �.l. _,_ � �� + � ,�, + l?y
V�t "'-
=
Form alized Music
224 O rgan i zati o n
Outside-Ti me
IV. Take the products Kj X c, and K, X em . Then take t h e product H x X. Set H is the vector s p ace of pitch , wh ile set X is the set of ways of playing the Ci. This product is given by a table or double en tries :
set
Extremely High
Medium
High
Medium
Low
Extremely Low
...,
N
· a.
I
p izz .
=
f.c.l.
an
-
pizzicati
struck with bow no r m al arco
pizz . gl.
a trem. harm.
=
=
=
the
N N
I
wood of
.... l'
l'
!:::
..... ...c:
asp
=
sound
of playi n g
a
VI .
inter£
arco sui ponticello
tremolo area wi t h
fourth
interferences
•
A sub-space and th ird rows of
the p ro du c t set H
the moment of the
in-time setting.
a
•
C8, as and of H' is •
,
rows, extremely h igh
ce din g table, each divided into two. These fo u r parts are of th e playing range of the corresponding column.
at
LJ
tremolo
=
=
It consists of the second
The mapping of cl onto
l'
C7, Ca
are attributed to the forms C1 ,
dependent and will be determined by
.s
c:: l'
l'
l'
pitches, are reserved for path V2.
attrib uted to pa t h
v.
ci. "'
ci.. "'
c...: ..... cu .....
arco sui ponticello
asp trem.
Various methods
8
(.) .... ....
harmonic sou nd with
=
hr trem. the
normal arco with tremolo
low
8
.... .....
indicated in the table. The first and extremely
e(.)
C4, C5, C6
pizzicato-glissando
harm o ni c
8(.)
c:: l'
· a.
Cl , C2 , Ca
=
=
� l'
u
w
"EiJ
X
defined
X
the p re
in terms
is relatively
in
kinematic diagram of operations
Towards a Philosophy
of Music
225
Organ ization I n-Ti me
IV. The products K[ X c, and K! X em are the result of the product of two graphs of closed transformations of the cube in itself. The mapping of the graphs is one-to-one and sounded successively ; for example :
(See Figs. VIII-9, 1 0.)
h
"O ·
v,
�
Vr
•'Ov.,_
"'r
'a
Fig.
·� 9.
f
.. '
4,Q'
�
q,
•'
a.
"
9t qs
11.
•L
t.
• t§ t
..
O.oQ· QO:' ::o·· ·rx·
q ..
Q
1'
,.
•'
•
o-
Q,,
o,
H
x
.
4,
Q,
..
•
o,
'
•
V r
v.,
Vll l-9
V. Each
C1 is mapped onto
one of
the cells
of
X according
to
two principles : maximum expansion (minimum repetition) , and maximum contrast or maximum resemblance.
(See Fig. VIII-I I .)
Formalized Music
226
(J(
d., : /. 0 d. : /. f dJ : 2 .0
t4 : (.....
l. {rr-) [) i.(}.) p
�
A lf11 1!J) L (tt,o) l�)
t(i<)
�
QJ
2·$
P A TH E Co ) �.
'J
fljS'
22S
�.
I:J
II
,
"S'
... Jr,
k,
'�£- /
G ee 3.7.2 I If
i&j l¥� i.(k} 0
...
f.$
"7
"7
/ /1
i (§j E "i "+ i (';J L t, J!3 r \ YvL -
L
(tV)
/,,z
.,1
i &-J
k4
�
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- 1· • ""'1 ;!, = l. o- .,f·::. o -.: ·1S'$ K/ = ' •·.(51· 4. .> = .?J.r.# KI "' •·•·tPl 4. :r = '"· "•I .t/ �.�.o. ,!-.2.0 = .UJ ·.f 4 = 2-•-/.l.{.tl. =J ?r.# k/ �.u ·/'·3- H c u.r .f k/ =:J.o · R·o. �f � �. o8 .f <;/ = 3. 0 · /·:/. {2
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; ./!
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8
=2- 0- ..f - f. s-·J!I - I,!.P- ,P = 6. � -..; =-F../'t f = UI .# =le! 3?-.# "' 7.& 1
=-!' ..:': �-=� :::· G tel' 7116 J::lf /OJ.Z u· 11 .z N .I / .I ; ,f/l ../ ":1 ,
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=
.. ·- ·
r
- ., ks �
A (f, 7) .J:uUs; ') I!
j� l?J i{*J {(;
ti� e.
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: .:·:_:
"
!'.It
l��
VI
"'/ · 2 • 2 · ml k,.ot = J# . (I ·.!" • �-$'� kJ' : 2·f. #f+S � 11·<>#,1 = ;; .,f ,Kf � .:.s. .. l- ,t �:; � t.>· r - ::-tz ' = s-9J·:f k: - ;.>. � - � . u =S: tS"# kJ' "'.2-t:./· H• = &. !ff k.1 =.2..: · / • .u:; =J'". .tfl'
A;,5} L ��
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227
tr;
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ly
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t�tr, "?ljl
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Formalized Music
228
H' � ~ ~
H� [W H; � pz ''/d ., />II
::·::·>s:·:
. . .
�
H
.. ·
pz
pz
c� d ll fill
;z
cl(
pz
p:z
c{f li ,. J>fl
c/( �11 P(l
clj OJ ., Af>/
�� ~ ~ c\
)1.
pz
6eafs
=
�"
bf!ii fJ
p i z z. i c a b '
i
Fig . V l l l -1 1 . P ro d u cts H '
x
X i n the Set o f t h e C1
a )\
& e ats
Towards
229
a Philoso p h y of M usic
f' V A fl n A. >. >- f' v v v n !'- v ), A ;.-. f' f< Y v n n y f< f< V V Y ).. >J. rl f' f' A
1'
">1orWI.j� reO' fii.e .,.. ,/D
Ol n
..:l rc�
c)
d />
rc� 5 /' Ol1� '
a.
t rc� 5 }o... t.f.lle m 0 I0
Formalized
230 O rg a n i za t i o n O u ts i de -Ti me
VI. The
prod u c t s
are fo rmed.
VI I . The set
K[
X
of logical
c(
X
X
H'
X and Kj
(a)
fu n c tions
X
c,
X
Music
H•
X
X
moduli
is used in this piece. I ts
are taken from the subset formed b y the pri m e r es i d u a l classes modulo
1 8 , with multi p l i cation , and
L(m, n )
(nt
=
nj
V
V
nk V
n1)
red uction modulo II mP V
(mq
V
m,)
1 8.
II
n5
V
(n1
V
nu
V
nv )
(a)
I ts clements a r e developed :
l . From a departure fu nction : L ( l l , 1 3)
=
( 1 33
V
1 35
II 1 39 V
2.
F r om
1 37
V
( 1 30
1 39)
V
II 1 1 2 V
(114
V
v 1 3 1 v 1 36 )
l l a)
" m etabol a " of mo d u l i which i s i d e n t ic a l h e r e to
a
the
graph
coupling the cl ements of the preceding s u b s e t . This m etabol a gives th e
following
functions :
L ( l l , 1 3) ,
L ( l 7, 5 ) ,
L ( l 3, I I ) ,
L( 1 7, 7),
L( l l , 5) ,
L ( l , 5 ) , L(5, 7) , L( 1 7, I I ) , L(7, 5) , L ( l 7 , 1 3) , L(5, 1 1 ) , L( l , I I ) . (See VI I I- 1 2 , and Tab l e of the Si eve Functions and Their Mctabolae . ) 3. From three substitution r u les for indices (residual c l a sses ) : R u le a : m0 --->- n0 + 1 Rule b : If all i n d i ces w i th i n a set of parentheses are equal,
fu nction
L (m, n )
responding sieve .
(see
R ule Rule
c:
c.
m; --+ nx,
C o nv e rs i o n of i n d ices as =
next
p u ts them i n a rithmetic progression modulo the cor
Table) : x
the
Fig.
j(n/m) ;
a
conseque n ce of moduli metabol ae
for example,
74 --->-
1 1 x,
x =
4( 1 1/7)
"'
6.
(elementary displacement : o n e q u arter of a tone fo r path V2) . T h e two types of mctabolae which generate the elements of set L( m , n) be used outside-time or inscribed i n - t i m e . I n the first case, th ey give u s 4.
From
a
metabola of E L D
tone for path V l , th ree- q u arters can
the totality of the el ements ; in the second case, these elem ents appear in a temporal order. Nevertheless a
in the fi rst
case.
structure
of t e m p o r a l order is subj acent even
5. From a spe cial metabola that would s i multaneously attribute differ L(m, n) .
ent notes to t he ori gins of the s ie ve s consti tutin g the fu nction
Tow ard s a
23 1
Philoso phy of Music
O rg a n i zat i o n I n -Time
product K[ x C; x H ' x X of the path V1 interp olation of ele m en ts of the product Kj X cj X Hcxercmes X X from path V2, which are sounded in termittently . V I I . Each of the three substitutions of the two cubes K1 and Ci, the logical fu n ction L(m, n) (see Fig. Vlll-1 1 } , ch a nge s following its kinematic diagram, developed from the grou p : multiplication by pairs of residual classes and reduction modulo 1 8. (See Fig. Vlll- 1 0.) are
VI. The elements
Table
L ( 1 1 , 1 3)
of the
successively,
sou nded
=
of the
except for
Sieve F u nc tion s
and Their Metabolae
( 1 33 + 1 35 + 1 37 + 1 3 9 ) 1 1 2 + 1 30 + 1 3 1 + 1 36
+
( 1 1 4 + l l 8) 139
L ( 1 7, 5) = (5 1 + 52 + 5 3 + 54) 1 7 1 + ( 1 77 + 1 7la)54 + 51 + 5o + 52 L ( 1 3, 1 1 ) ( 1 1 2 + 1 1 4 + 1 1 7 + 1 1 9) 1 30 + ( 1 35 + 1 3 10) 1 1 9 =
L( l 7, 7)
L( I l , 5)
= =
+ 1 12 + 1 11 + 1 14 + 7 3 + 75 + 7 6 ) 1 71 (50 + 52 + 53 + 54) 1 1 0
( 71
+
+
( 1 7 6 + 1 7d 7s
71 + 7o + 73 ( 1 1 4 + l l e) 54 + 5o + 51 + 5 2 +
L ( l , 5 ) = (51 + 52 + 53 + 54) 1 1 + ( 1 1 + 1 1) 54 + 5 1 + 5 2 + 5 a L( 5 , 7) (71 + 73 + 74 + 76)50 + (50 + 51) 7s + 71 + 7 3 + 7 4 ( 1 1 2 + 1 1 5 + 1 1 6 + 1 1 9) 1 1 1 + ( 1 7 1 + 1 7 3 ) 1 1 9 L ( I 7, 1 1 ) + 1 12 + 1 15 + 1 16 =
=
L( 7 , 5) L( I 7, 1 3)
=
=
L(S, 1 1 )
=
L( l , 1 1 )
=
n
m
(51
+
52 +
( 1 3a +
53
+
54) 70 + (70
+
71) 54
l 3s + 1 38 + 1 310) 1 71 + ( 1 7 1 + 1 33 + 1 35 + 1 38
+
+
51
5 2 + 5a 1 72) 1 31 0 +
( 1 1 a + 1 1 4 + 1 1 7 + 1 1 8 ) 5 0 + (5o + 5 1) l l a + 1 1 3 + 1 1 4 + 1 1 7 ( 1 1 3 + 1 1 4 + 1 1 7 + 1 1 8 ) 1 1 + ( 1 1 + 1 o) l i s + I I a + 1 1 4 + 1 1 7 5
-
Rule c. Table
=
1
57
=
.!.!
1 .4
=
2. 2
:
5
I: = 2.6 ; 5
I
=
3 .4
7
7
11 7
13
= I =
=
7
.!2 7
=
1 .5 7
1 . 85 2.43
11
- =
II
13 .2 IT = 1 17 1. IT = 54
13 13
17
13
=
=
1 .3
Formalized Music
232
Group and sub-group of residual classes obtained by ordinary multiplication followed by reduction relative to the modulus 1 8
1
1
5
7
17
5
5
11
11
13
13
17
17
7
5
7 1
ll
13
7
7 17 13
5
1
11
11
13
11
13
l 5
13
17 7
17
7
1
17
11
13
17
7
l
11
7 5
5
7
13
1
7
13
7
13 1
13
13 1 7
1
D E T A I L E D A N A L Y S I S OF T H E B E G I N N I N G O F T HE S C O R E
(L ( 1 1 , 1 3 ) } 18
Thanks to the metabola in 5. of the outside-time organization,
origins
of the partial sieves ( 1 33
V 1 35 v 1 37 v 1 39) /1. 1 1 2 V
(1 14
V
the
1 1 8)
1\
A 3 , resp ectively, for A3 440 Hz. Hence the sieve L( l l , 1 3) will produce the following pitches : CJ, C�, D 2 , D2 t; F2, F.Ji, G2, G2#f, A2, B2t, C3, C:#lf, D3#, DJ/f, Fa f. 1 39 and 1 3 0 v 1 3 1 v 1 36 correspond to A3# and =
.
.
.
Fa#, G3#, A3t A3#, B3 , C4 t D"f. E4, E4 G4, A4, A 4 #, A'l=/11 , .
f.
The
order applied to
the sonic complexes
intensity, and duration combinations
::
· ·· · ······ . . . .. .
. . ..
· · · .
S� = �· .
(K,.)
(S,.) and to the density, for transformation fl :
are
mf
K2
=
2.25
Iff
K3
=
22.5
Iff
K"
=
10
. •
mf
233
Towards a Philosophy of Music
S5 =
�
K5
� S., = ==
K6
Se =
2.83
=
=
f
ff
3.72
ff
K7 = 7.98 K8
S = --=== a
(In this tex t C,. is rep laced by S,..)
=
First sequence (see Fig. VIII-1 3) :
f
6.08
1
2
3
4
5
6
7
8
sl Kl
s,
s.,
1
10
Sa Ke
K.,
Ss Kr,
Sa Ka
ff
ff
f
f
.j.
+
D (S,.)
=
S2
Sa
D (K,.)
=
K2
Ka 22 . 5
2.25
Iff
Iff
+
+
K,
mf
rrif
+
3.72
.j.
7.98
+
2.83
.j.
6.08
(the This part begins with a pizzicato glide on the note C, fff very sliding start s ppp) . The slope of the glid e is zero at first and then
per 2.5 seconds). consists of Of C# D struck col legno, fff (with p in the middle) . In S8 there is an introduction of beats obtained by raising
weak ( 1 /4 ton e
S3
GIJt towards A.
Second sequence, 1
Ql2(S,.) Qa (K,.)
Q12/Q3 :
beginning at
.j. Ss
= = K8 6.08
f
2
3
4
5
6
7
Ke
t Sa
K.,
+ s.,
t sl
.j. s2
+ s,
3.72
7.98
2.83
10
2.25
22.5
.j. Sa
ff
Ks
ff
f
Kt.
Ka
Ka
t Q4(S,.) = Sa
Q7(K,.) = K
a 6.08
J
QJQ7 :
2
3
4
s.,
Sa
s6 Ke
t
K.,
7 . 98
If
.j.
Ks
2.83
J
t
3.72
ff
.j.
Sa Kl
1 .0
mf
fff
Iff
mf
5
6
7
8
s,
K,
Sa Ka
Kl
sl K2
lO mf
Iff
mf
Iff
Note , as in the preceding part, the previously traction of the values of duration. sl is ataxic, lasting more than a second. Third sequence, beginning at
8
t
S2
t
22.5
calculated con-
t
1 .0
t
2.25
Formalized
'
i _,.-
Towards
a
Philosophy of Music
�
' "' i
l
235
"'
I
� I
.
�
f.
��
li
.E a; u 0
-., -9..,.,
'
�
......
I
i� .. L � ll �
CJt :<
,
pA �
.s-?
r
��
I (
0
E
'
I
;-
I
r·
'I'
�
0 � "' Ill
0> c:
.,
·c:
a.
...
0 pj
:::!:
>
li ii:
Form alized Music
236
s
In S8 the lo p e s of the glissandi i n opposite directions cancel each
other. The enlargemen t in S4 is produced by displacement of t h e
lower line a n d the inducemen t of beats . T h e cloud is introd u ced by
a
pizzi c a to on t h e C s t r i ng ; the index finger of t h e left hand is p l ace d
on the
string
at t h
e
place w here one would p l ay the note in s q u are
brackets ; then by plucking that part of the string between the nut
and the index finger w i t h the left thumb, the sound that results w i l l
be
t h e note i n parentheses.
NOMOS GAMMA-A
G E N E RALIZATI O N O F NOMOS
AL PHA
The finite combi natorial cons truction expressed by finite groups and Nomos alph a is transposed to fu l l orchestra in Nomos gamma ( 1 967 /68) . The ninety-eight musicians are sca ttered in the audience ; this scattering allows th e am plification of Nomos alpha's s t r u cture. Terretektorh ( 1 965/6 6) , which p receded Nomos gamma, innovated the scattering of the orchestra and proposed two fundamental changes : performed on one cello i n
a.
sprinkling
The quasi-stoch astic
of the orchestral musicians
among the aud ie n c e . The orchestra is i n t h e audience and the au d i ence is in the orchestra. The public should be free to
m ov e
or to si t on
camp-stools given out at the entrance to t h e h a ll . Each musician of
the orch estra should be seated on an i n d ividual , but u n rcsonan t , da'is
with his desk and instruments. The hall where t h e piece is to be per cl
formed should be
e are d
of every movable obj e c t that migh t cause
aural or v i s ual obstruction (scats, stage,
(if it were circular)
a m in
imum
default of a n e w ki nd of architecture
all types of p r
esen t
-
etc.)
A large bal l-room h avin g
d i ameter of 45 yards would
serve
in
which w i l l have t o be devised for
day music, for n e i t h e r am p hitheat res, and s t i l l l ess
normal theatres or concert-halls, arc suitable.
The scatterin g of the m usicians brin gs i n a radic a l l y new kinetic con cep tion of m u s ic which no modern electro-acoustical m eans could match . 1 9 For ifit is not possible to i m a gi n e 90
ma
g n e t ic t ape t racks re
all over the aud itori u m , on the co n t rar y it is quite p o ssib l e t o ach ieve this w i t h a cl assical orchestra of 90 musicians. The m usical composi tion w i l l th ereby be entirely lay i n g to 90 loud speakers d is se m i n a te d
enriched throughout t h e hall bot h i n spatial dimension an d in move
men t. The speeds and accelerations of the movement of the sounds
will be real ize d , an d new and powerfu l fu n ctions w i l l be able t o be
mad e usc of, s u c h
as
l ogari th m i c or Archimed ean spi ra l s , i n -time and
geome trical ly. Ordered or disordered son orous
against the other like waves
.
.
.
masses,
e tc . , will be possible.
rol l i n g one
237
Towards a Phi losophy of Music Terretektorh is thus a " Sonotron " : an ac ce l e r a t o r of sonorous particles, a disintegrator of sonorous masses, a synthesizer. It puts the
sound and the music all around t h e listener and close up to him . I t tears d ow n
t he
psychologi cal and au d i t i ve curtain t h at separates him
from the pl ayers w hen posi tioned far off on
a
pedestal, itself frequently
enough placed insi d e a box . The orchestral musician rediscovers his responsibility
as
an artis t , as an
b. The orchestral
co l o u r
individual.
is moved towards t h e spectmm of d ry
sounds, ful l of noise, in order to broaden the sound- p alette of the orchestra and to give maximum effect
above . For this effect, each
of
to
the scattering men tioned
the 90 musicians has, besides his
normal string or wind instrument , three p er cu ss ion instruments, viz. Wood-block, Maracas, and
W h ip
as well
as
small Siren-whistles,
which are of th r ee registers and give s o u n d s resembling fl am es . So if necessary,
a
shower of hail or
even
a murmurin g of pine-forests can
encompass each listener, or in fa c t any o t h e r atmos p here or linear
concept either s t at i c or in motion . F i n al l y the l istener, each one
individual ly, will find h imself either perched on top of a mountain in the middle of a storm which attacks him from all sides, or in
a
frail
barque tossin g on the open sea, or again in a u n ive rse dotted about
with l i t t l e stars of sou n d , moving in compact n ebulae or isolate d . 2 0
Now the crux o r thesis o f Nomos gamma
is a combinatori al organization
of correspondences, finite and o utsi d e the time of the s e ts of sound characteristics . Various groups are exploi t ed ; their inter dependency
their
inner s tructure and
are put i n relief musical ly : cyclic group of order 6,
groups of the rec ta n g le ( Klein) , the t ri a n gl e , the square, the pentagon, the
hexagon,
the tetrahedro n , and th e hexahedron.
The isomorphisms are established i n many ways, that is, each one of
the preceding groups is expressed by different sets and correspondences, thus o b tainin g structures set up
on
several interrelated levels. Various
grou ps are in t e rl o c k e d , i n termi ngled , and interwoven. Thus a vast sonic tapestry of non-temporal essence is fo r m e d (which incidentally includes the organ ization of time and d u r at i on s) .
The space
also con tribu tes, and is
organically treated, in the same manner as the more abstract sets of sound elements.
A
powerful deterministic and finite machinery is thus p romulgated. Is it
symmetrical to the probabilistic and stochastic machineries al ready proposed ?
The two poles, one of p u re chance , the other of pure determinacy, are
dialectically b l e n d e d in man's m ind ( a nd p e rh aps in n atu re as well, as Epicurus or Heisenberg wished i t) . The mind of man should be able to
Formalized Music
238
travel back an d forth cons tantly, with ease and elegance, through the of disarray caused by i rrationality, that separates determinacy
fan tastic wall,
from indetermin acy.
some examples. It goes without say i n g that is n o t entirely defined by group transformations. Arbitrary ranges of d e ci si o n s are disseminated into the piece, as in all my works except for th os e originated by the s toch astic program in Chap. V . However Nomos gamma represents a stage i n the me thod of mechani zation by com p uters for this category of problem. We
Nomos
w il l
gamma
Measures
now co n s id er
(three
1 -1 6
oboes.
then th ree clarinets )
O U TS I D E - T I M E S T R U C T U R E Set o f pi tche s : H
=
{H1 , H2, H3, H4, H5} . Origins : D3, G�3 , D4 ,
Set of durations : U {U1, U2, U3, U4 }. Origins : J d J , respectively, with range ± one sixteenth -note
D 5 , respectivel y , with range ± 3 semitones.
., ....• ..., ,
=
0
.......,
'-' ·
"" l sec. =
nd
>
=
'>
=
>
= >
>
P ro d u c t sets : K
set is
a
half note
>
.J}j, .IJJ, sirJ; s.fJJ} . O rigins : pp, mp, /.frj, respectively. >
J>
i
o 1 o _•._
{G1 , G2 , G3, G4 } . G 1 {ppp, ppp, pp, pp, jp, p}, > � � !; � {p, p, mp, mp, mp, mj, mj}, G3 {mj,j,j,j, .if, JJ,Jf}, G4 {Jj, sfj,fff,
Set of intensities : G
G2
a
G#4,
H
=
x
U
defined by a sieve modulo
grou p
(e.g.,
and by its
n =
X
H4 3
X
=
ELD : -! tone =
Modu li :
. . . , 30 -+ 31
un it, that is, the
H4 Kl Moduli : 2 K4
3,
ELD : ! tone
G2 2
G3 2
x
X
X
ul
ul
3
t sec
G. Each one of the points of the prod u c t
considered a s an element of an addi tive
->-
31 -+ 32 -+ 3 0 -+ 3 2 -+ 3 2 --+ 31
elementary displacement ELD :
K2
2
t sec
n
=
=
H 4
2
Ks
-! to ne
= H4
3
t tone
x
G3
x
x
G2 2
x
2
Vz 2
K3
Uz
t sec
H4 2
! tone
t sec
3
=
K6
=
H4
3
i tone
x
x
Gl 2
G3 2
-+
X
x
·
· ·)
ul 2
! sec Ua
3
i sec
are deformed by transla tions and homothetic trans H values. Let us now consider the three points K1 , K2, K3 of the product H x G x U, and map them o n e - to - o n e onto three successive moments of time . We thus define the triangle group with the following c l e m ents :
I n add ition, K2 and K3 formations of the
{/, A, A 2, B, BA, BA2} r+ { 1 23, 3 1 2, 23 1 , 1 32, 2 1 3, 32 1 }
,
a
Towards IN-TIME
Philosophy
239
of Music
STRUCTURE
vertices are s ta te d by K1, arc played successively by the oboes and the clarinets, according to the ab o ve permutation group and to the following circui t : BA, BA2, A , B, BA2, A 2• For e a ch transformation o f the tri an gle the
K2, K3, which
M easu res
1 6-22
OUTSIDE-TIME
( t h ree
o b oes
a n d three clari nets )
S T R U CT U R E
K, X c, : K l X cl > K2 X c2, K3 X c3 , in which the c, are the ways of playing. cl smooth sound without vibrato, c2 flutter tongue, c3 quilisma (i rregular oscillations of pitch). Consider now two triangles whose respective vertices are the three o bo es and the three clarinets. The K1 x C1 val u e s are the names of the vertices . All the one-to-one m a p pi n gs of the K1 x C1 names onto the three s p ace positions of the three o b oes or of the three c l ari ne ts form one trian gle Form the p rodu c t
=
=
=
group.
I N -TIM E
STRUCTURE
To each grou p transformation the nam e s K1 x C1 a re sta ted simu l taneously by the t hre e oboes, which a l t e rn a te with the three clarinets. The circuits are chosen to be I, BA, BA, I, A2, B, BA , A, BA2 and /, B, B. M easu res
404-42-A S o u n d Tapestry
The s t ri n g orchestra (sixteen fi rst violins, fourteen second v i o li ns, violas , ten cellos, and eight do u bl e basses ) is divided into two times three te am s of ei g ht instruments each : cp 1 , cfo2, cp3, tfo1 , t/J2, tfo3• The remaining twe l v e strin g s duplicate the ones sitting nearest them . In the text that fo l l ows the c/>1 and tfo1 are considered equivalent i n pairs (c/>1 - tfo1) . Therefore we shall only deal with the cp1• twelve
LEVEL
I -OUTSIDE-TIME
S T R U C T U RE
positions of th e i nstruments of each cfo1 are p u rp osely taken Onto these positions (instrumen ts) we map o n e - t o - on e e ig h t ways of p l aying d raw n from s e t X = { o n the bridge tremolo, on the brid ge tremolo and trill, sul p onticc l l o smooth, sul ponticcllo tremolo, smooth natural harmonic notes, irre g ular dense strokes wi th the wood of the bow, normal arco with tremolo, pizzicato-glissando as c e n ding or descending}. We h ave t h u s for m ed a cube : KVBOS I . Onto these same eight positions ( instruments ) of r/>1 we map o n e- to - on e eight dy n a mi c forms of intensity taken fro m the following sets : g11 The
eight
i n to consideration .
=
240
Form al ized Music
{ppp
crescendo,
ppp diminuendo, pp
crcsc, pp dim, p cresc , p dim, mp cresc, ] dim, ff cresc, ./J dim, fff cresc,
.i.tJ d i }, g� {p d i m p cresc, mp dim, mp cresc, mf dim, fcresc}. We have t h u s defined a s co n d cube : KVBOS 2. mp dim}, gu m
LEVE L
{mj cresc, mf dim, f cresc,
=
=
,
1 -I N • T I M E
mf cr e s c ,
e
f dim,
S T R U CT U R E
Each one of t h es e cu bes is transformed i nto itsclffol lowi ng the kinematic
diagrams of
KVBOS 1
the
hexahedral
fol lowing D 2 Q1 2
.
group •
•
(cf. Nomos
alpha,
p.
225) ; for exam ple, Q11 Q 7 • • • •
and KVBOS 2 fol lowi n g
2 -0UTSIDE-TIME STR U C T U R E The three p a r t i t i on s t/> 1 , t/>2, t/J 3 are now considen!d a s a tri plet o f points in space . We m a p onto them, one-to-o ne, th ree d i s t in c t pitch ranges Ha , H0, Hy in which the instru mentalists of the p r e ce di ng cubes w i l l pl ay. We have thus formed a triangle TRlA 1 . Onto these s am e three points we map one-to-one th ree elements drawn from the product (durations x intensities) , U x G {2. 5 sec g11, 0.5 sec g��. , LEVE L
=
1 .5
sec
a
g�} . We have thus defined
LEVE L
2 - I N •T I M E
second triangle TRIA 2 .
STRUCTURE
When the two cubes play
a
Level I transform ation , the two triangles
simultaneously perfo rm a transformation of the t r i a n g l e group. I f /, A, A 2, B, BA , BA 2 are the group e lements, then TRIA 1 procee ds accord ing to the kinem atic diagram A, B, BA2, A 2 , BA , BA2, and TRIA 2 proceeds simultaneously according to A, BA 2 , BA , A2, B, AB. LEVEL
3-0 UTSIDE-TJME
p ro d uc t c,
Form the
ST RUCTUR E
X
M, with
clouds of webs of pitch glissandi, c2 clouds of sounds =
A, c1
X
32 = A 3 ,
to the cyclic group LEVEL
three
c2
X
of order 6.
3-IN-TI M E
we
3o
select five elements : c1
=
A\ c3
c3
X
3 1 , c1
X
X
X
31
=
X
=
3
3o
are
=
=
=
taken :
I, cl
X
A5, which could belong
S T R U C T U RE
The nested transformation of Levels
product c,
types : cl
macroscopic
clouds of sound- p oin ts, and c3
with q u i l ism a . Three sieves w ith modulus A1
3 o , 3 1 , 3 2 . From this p r o d u c t 31
=
1
and 2
a re
plunged into th e
cl X 3z, c2 X 3o, cl X 3 1 , 3 o �� A 3 , A \ A, A 5 , I, during the corresponding arbitrary
M,, which traverses successivel y
d u rations o f 20 sec, 7 . 5 sec,
1 2.5
sec,
12.5 sec, 7.5
sec.
a
Towards
4-0 UTSI DE-TIM�
L EVEL
24 1
Philosophy of Music S T R U CTURE
The p artition o f the s t r i n g orchestra into teams cp " 1/J; i s done i n two compact and dispersed. The com p a c t mode is i t se l f divided i n to two
m od e s :
cases : Compact I and Com pact I I . For exam p le,
r/>1 II, r/> 1
i n Com p ac t I , in the
(VI;
VC6, CB4}
{V I 1 3 , VII1, VII2, V I I 1 4 , A7, VC2,
=
{VI 1 , VI7, VI6, VI9, V I 1 0 , A 8 , VC3 , CB2} dispersed m o de , r/>1 {VI2, V I 3 , V l 6 , V I I 1 , V I I 6 , Vl l w
in Compact
ith first ith
=
cello, CB; LEV E L
=
=
=
=
violin, V I I 1 do u b l e
4- I N - T I M E
ith
se c o nd
violin, A 1
=
ith
CB3 ,
viola, VC1
CB 7 } =
ith
bass.) T h ese parti tions cannot occur sim u l taneously.
STRUCTURE
A l l the mec h a n isms that sprang from Level s I , 2 , 3 arc i n turn plunged
into the
v
a riou s
above definitions of the r/> 1 and rf; teams, and su ccessively
into Compact I du ri ng the
27. 5 sec d u ration , i n to t h e d ispersed mode during
the 1 7 . 5 sec d u ration, into Compact I I duri ng 5 s e c , i n to the d i spersed mode du ring 5 sec, and i n to Compac t I d u ring 5 sec.
Thus
the
D E STI N Y ' S I N D I CAT O R S
inquiry app l i ed t o music leads
mind. Modern axiomatics disen tangle
once
us
t o the inn ermost parts of our
more, i n
a
more precise manner
now, the sign i fi c a n t grooves t h a t the past has etched o n the roc k
being.
T hes e
of
our
m e n tal premises con firm a n d j us t i fy the b i l l i ons of years of
But awaren ess of their l i m i tati o n , destroy them . of a s u dd e n it is u n thinkable that the h u m an m i nd fo rges i ts c o n of time and sp a ce in childhood and never alters i t . 2 1 Thus the
acc u m u l ation and destruction of signs.
t h e i r closure, fo rces u s to All ception
bo ttom of the cave wo u l d not reflect the beings who would b e a filtering gl ass t h a t
w o u ld
allow u s
heart of the u n iverse. I t is this bottom t h a t m u s t
Consequences :
I.
It would
realize t h i s mutation .
us
r e s o l ve
.
.
same
The
carry
a t t h e ve ry
2 . A r t , and scien ces a n nexed
to i t, s h o u l d
the d u a l i t y mo rtal-eternal : the fu ture is i n the past and
of the also two
vice- versa ; th e evanescence
the
beh i n d us, b u t
be n e c e s s a r y t o c h ange t he ord e red structu res o f
time a n d s p a c e , tho se of logi c, . Let
arc
a t w h a t is be broken u p .
to g u e s s
time ; the here is
p resen t is abolis hed , i t is everywhere at
b i l lion l i g h t-years away . . . .
space ships that ambitious tec h nology h ave p rod u ce d m ay not
u s as far as l i beration from
fan t astic perspective Parmcnid ean field.
th a t
our
art-science
mental
o pe n s
s
to
h a ck l es us
in
cou l d . This i s t h e the
Pyth agorea n
C h a pter IX
New
The
in
Proposals
Microsound
S tructure
FO U R I E R S E R I ES - BASIC I M P O RTA N C E A N D I N A D E Q U ACY
physico-mathematical apparatus
theories
of acoustics
[2, 23] is plu nged i nto the
of energy pro p ag ation i n an clastic med i u m , in which
the
a n alys i s is
cornerstone.
The
same apparatus finds in
The
prodigious
practical
the
harmonic
u n i ts of electronic circ u i t design
medium where it is realized a n d c hecke d .
the Fourier
the
development of radio a n d T V transmi ssions h a s ex p anded harmonic analysis to very broad and h e terogeneous domains.
Other theories, q u i te fa r apart, e . g. , necessary backing in Fou rier se ries.
servomechanisms and pro bability, find
In music ancient trad i tions of scales, a s wel l as those of string and p i p e resonances, also lead to circular functions and their linear comb i n ations [ 24] . In conse q uence, any attempt to produce a sound artificially could not be conceived outside t h e
apparatus,
electronic
Indeed the
framework of t h e
above physico-mathematical
which rel ics on Fou r i e r series .
long route
seemed to h ave fo und
its
traversed
by the
acousmatics of t h e
abou t the
natural
directly, i n order to tonality. Moreover, in
harmony
of
end,
on
the th eory of v i b ra t i o n of elastic bod ies
on Fourier analysis . But they
were
242
base
their
support the argument defining tonal i ty, the
20th-century dep recators of the new m usical langu ages ments
Pyth agoreans
natu ral b e d . Musical theoreticians did
theories on Fourier, more or less
and
and
based their argu that is, i n the a paradox, fo r al-
me d i a ,
thus crea ting
New
Proposals
243
in Microsound Structure
though they w an t e d to keep mu s i c in the intuitive and i nsti nctive d o m a i n , in order
to legitimatize
the tonal universe they made use of physico
m athematical arguments !
The I m p asse of H a r m o n i c A n a l ysis a n d S o m e R ea s o n s
Two m aj o r di fficulties
co m pe l u s to think i n another way :
I . The defeat by the t hrust of the new l a ngu a g es of the theory accord i n g to which harmony, cou nterpoint, e tc . , must ste m , j u s t from the basis formed by c i rc u l a r fu nctions. E.g. , how c a n we j us tily s u ch harmonic c on figurations of rece nt i nstrumental or electro-acoustic music as a c l o u d of gli ding sounds ? T h us, harmonic analysis h as been short-circuited in spite of touching a ttempts like Hindemith's expl anation of Schonberg's system [25] . Life and sou n d adve n t u res j o s t l e the tradit ional theses, w h i c h are neverth eless sti l l being taught in the conservatories ( rudimentally, of c o u rs e ) It is t h e re for e natural to think that the disruptions in m u si c i n the last 60 years tend to prove once aga i n that m u s i c and i ts " r u l es" arc socio c u l tural and h i s tor i c a l conditionings, and hence modifiable . These c o n d i t i o n s seem to be based rou g h l y on a. the abso l u t e l i m it s of o u r se nses and t h e i r deforming p o w e r (e. g . , Fl e t c h e r contours) ; b. our canv ass o f m e n t a l struc tures, some of w h i c h were treated i n the prece d i n g chapters (ordering, groups, etc. ) ; c. the means of sou n d production ( orchestral instru m e n ts, electro-aco u s t i c s o u n d synthesis, s t o r a ge and transformation analogue systems, digital soun d synthesis with c o m put e rs a n d digital to a n a l ogue converters ) . I f we modify any one of these t h r e e points, our socio-cu l t u ral cond i tioning will also tend to chan ge i n spite of an obvious inerti a i n h erent in a sort of " e n trop y of the social facts. 2. The obvious fai lure, since the b i rt h of osc i l lating c i rc u i ts in e l ec tron i cs, to reco nstit u te any sound, even the s i mp l e sounds of some orchestral .
"
instruments !
n.
The Trau toniums, Thcrcmins, and �1 artenots, a l l p re
prove it. b . S i n ce the war, al l " e lectronic " m usi c of the big hopes of t h e fi ft i es, to pull electro-aco ustic
World War II attem pts,
h as also fai l ed, in s p i te
out of its cradle o f t h e so-ca l l e d electroni c pure so u nds produced by frequency ge n e ra tors . Any e lectronic music based on such sou n d s o n l y , is marked by t h e i r si m pl istic s on o r i t y w h i c h resembles radio atmospherics o r heter·odyning. T h e serial sys t e m which h a s been u s e d s o much by electro n i c
music
,
,
m u s i c composers, c o u l d no't by any
is much
m e ans
e
i m prove t h e resu l t, s i nce i t i ts e l f
t o o e l e m entary. O n l y wh n the " pure " e l ectro n i c sou nds were
by o th e r " concrete " sounds, whi ch were m u c h r i cher and m uch more interestin g ( t h anks to E . Varcsc , Pierre Schaeffer, and PicJTc H e nry) ,
framed
244
Formal ized Music
powerfu l . c. The m o s t rece n t attempts the flower of m od e rn technology, c o m pu ters cou pled to converters , have s h ow n that i n s p i te of s o m e relative successes [26] , the so n o ro us results are even less i nteresting th a n those made ten years ago in the classic electro-acoustic s t u d i o s by means of fre q u e ncy generators , fi l ters, modula tors , a nd reverberation u n i ts.
could electronic music become really to usc
In
line
with t he se
critiques, w h a t arc the causes are some of them :
my opinion the follow i ng
1.
p
Meyer-E pler's stud ies
of t h e s e
failures ? In
[ I ] have shown tha t th e spec tral analysis
will fo r m a reference system for lines in frequency a s w ell as in amplitude . B u t these tiny ( se con d o rd e r} vari ations are am o n g those th at make the difference between a l i feless so und m a d e up of a sum of h arm on ics prod uced by a freq u ency generator and a sound of the sam e su m o f h ar m o n i cs pl ayed on an orchestral ins trument. These tiny variations, which take pl ace in the permanent, stationary p a r t of a s o u n d , would certainly require new theori es of approach, u s i n g another fu n ct i o n al basis a n d a harmonic a nalys i s on a h i ghe r leve l , e.g., stochastic p ro c es se s , l\1arkov chains, correl ated or autocorrclated rel ations, or theses of pa t te r n and form recognition . Even so, analysis theories of orchestral sounds [27] would resul t in very long a n d complex calculations, s o t h a t if w e had t o simulate such an orch estral sound from a computer and from h armonic analysis o n a first l e ve l , we would n e ed a tremendous amount of computer tim e, which is i mpossi ble for the m o me n t . 2. It se ems that the transient part of the s o un d is far more important
of even
t he
simplest orchestral
sounds (they
a l o n g time to co me ) presents variations of s p e c t r a l
than th e permane n t part i n timbre recognition a n d i n music i n general [28] .
Now, the
more
the music moves toward co mp l e x sonorities clo se
to
" noise , "
th e more n u m ero us and compl i cated t h e transients become, and t h e more t he i r syn thesis from trigonometric functions beco mes a mountain of diffi c ul ti es, even m o re u nacceptable to
a
compu ter t ha n the pe r m a ne n t states.
I t i s a s though we wanted to express a sinuous mo u n t a in silhou ette by u s i n g p o r ti o n s of c ircles. I n fa ct , it is tho usand s of tim es more complicated. The i n te l l igent car i s i n fi ni te ly d e man d i n g , and i ts voracit y for information is far fro m having b e e n
satisfied. T h is problem of a considerable amou n t o f
ca lc u l ation is comparable
to
the
that led to the kinetic gas th eor y .
3. There is no
pattern
h arm o n i c an a l ys i s or not,
s ized b y me a ns of
and
that
1 9th-century
classical
me c han i c s
problem
form recognition theory, d e pen de n t on
wou l d
enable us to in the
trigonome tric fu nctions
translate
cu rves
perception
of
synthe
forms
or
New Propos a l s in M icroso und S tructu re
245
c o n fi g urations. For i nstance, it i s i m possi b l e for us to d e fi n e e q uivalence
cl asses o f very d iversi fi ed osci l l oscope cu rves, which the
c ar
t h rows into the
s a m e bag. Furtherm ore, t h e car makes no d i s t i nction b e tween t h i n g s t h a t actu al acoustic th e o r i e s
differenti ate ( e . g . ,
phase d i fferen ces , d i ffe re n t i a l
sensit i v i ty a b i l i ty) , and v i ce vers a .
T h e W r o n g C o n cept o f J u xta posi n g F i n i t e E l ements Perh aps the u l t i m ate r e a s o n fo r s u c h
cl ifli.cul tics
l i es i n the i mp rovised
entan g l e m e n t of n o t i o n s of finity and i n fi n i ty . For example, i n sinusoidal
there i s a u n i t c l e m e n t , t h e vari ation i nc l u d ed i n 27T. Then t h i s is rep eate d e n d l essly . Seen as an economy of m e a n s , t h i s p ro c e d ur e c a n be o n e of the possi b l e o p timizations. \Ve l a bo r d u ring a l i m i t e d s p a n of t i m e ( o n e period ) , then rep eat the p ro d u c t indefinitely w i th almost no ad d i t i o n a l l a bor. Basi cally, t h erefore, we h a v e a mech a n i sm ( e .g., the sine function) engendering a fi n i te temporal obj ect, wh ich is
osc i l l a tion
finite
var i a t i o n
repeated for a s l o n g as w e w i s h . T h i s l o n g obj e c t is now con si d ered as
which
a
new
odds a rc that one can d raw any \'ariation of one v a ri a b l e ( e . g . , atmosph eric pr e s s u r e ) as a fu nction of t i me by means o f a fin i t e s u p e rposition (sum) of the preced i ng cleme n t s .
c l e ment, to
we j u x t apose si m i l ar o n e s . T h e
I n d o i n g t h i s we e x p e c t to obtain
l a r i ty as
we
l ook quite complex. If we ask on
s y m me t ries
an
i rregu lar curve, w i t h i ncreasi ng i rregu
approach " no i s es . " On the osc i l l oscope s u c h
the
eye
to
fo l l ow t h e m too fast
or
or
1 0 m i c roseco n d s because i t wou l d h ave t o
too slowly : too
a t te n t i o n , and t o o s l o w for t h e
TV
fast
fo r the eve ryd ay l i m its o f v i s u a l
l i m i ts, w h i c h p l u nge t h e i nstan taneous
j u d g m e n t i n to t h e l evel o f global perception o f
hand,
would
t h i s c u 1·ve i t w o u l d al most cc1·tai n l y be u n a b l e to m ake a n y
j udgment fro m samples last i n g say
other
c u rve
a
recogn ize particu l a r forms
for t h e s a m e sample d u ra t i o n , the
forms and colors. O n the c a r i s made to recognize
fo r ms and p a t terns, a n d therefore senses t h e correl a t i o n s between fragments of
the
pressure cu rve at vari o u s l e vels
l aws a n d rules of this abil ity o f t h e
c ases
s i ne
that
we arc
cu rves,
car
9f
u nderstanding. \Ve ignore t h e
in the more
c o m p l ex
i n terested i n . H o wever, in the case in which
we know t h a t below
a
a n d gene ral
we
su perpose
cert a i n d egree o f co m pl ex i ty t h e
car
disentangles t h e const i t u e n ts , and t h at above i t the sensation i s tran sfo r m e d
i n to t i m b re, color, p o w e r , move m e n t, roughn ess, and d egree of disord e r ;
and t h i s b r i ngs by
j u d i c iously
us
into
a
t u n n e l of ignoran ce. To
s u m �1ari ze,
we expect t h a t
p i l i n g u p s i m p l e c l e m e n t s ( p u re sou n d s , si ne fu n c t i o n s )
w i l l create any desired s o u n d s ( press u re cu rve) , even t h ose t h a t to
come
we
c l ose
very s t rong i iTCgu l ar i t i cs-al m ost stoc h astic ones. T h i s same s t a te m e n t
h old s
even
when t h e u n i t
cl em e n
t
o f t h e iteration i s taken fi·01n a fu n c t i o n
Formalized
246
Music
In general, and regard l ess of the spe c i fi c fun c t i o n o f th e proced u re c a n be cal led synthesis by finite juxtaposed elemen ts. my o p i n i o n it is from here that the deep contrad i c t i ons stem that should
other than the s i n e .
unit In
e lem e n t ,
pre vent
this
us from using i t . *
N EW P R O P OSA L I N M I C R O C O M P O S IT I O N B AS E D O N P R O B A B I LITY D I ST R I B U TI O N S
vVe
the
and
t o open a pretending . to be a b l e to simulate a l ready known sounds, will n evertheless launch m u s i c , its psychophysiology, and acoustics i n a direction t h a t i s q u ite i n ter sh a l l raise
cont radiction,
b y doing s o
we
hope
new path i n m i crosou nd synthesis research-one that withou t
esting and unexpected.
I nstead of starting fro m the unit ele m e n t conce p t and its ti reless iteration
and
su p erposition of such i terated unit ele we c a n start fro m a disord er concept a n d t h e n i ntroduce means that would increase or reduce it. This is like say i n g that we take the i nverse road : We do not wish to const ruct a com p lex sound edifice by using discontinuous unit ele ments ( bricks sine or other fu nctions) ; we wish to construct sounds with continuous variations that a r e not made out of u nit elements. This method would usc stochastic variations o f the sound pressure d i rectly. We can imagine t h e pressure vari a t i o n s produced by a particle capri ciously moving a ro u n d equili brium positions along the pressure ord inate i n a n o n deterministic way. Therefore we can im agine the use of any " random walk " or multiple combinations of them. Method 1 . Every probabi lity function is a particular stochast i c varia tion, w h i c h h as its own personality ( personal b e ha v i o r of the p a r t i c l e ) . We shall t h e n u s e any o n e of t h e m . They c a n b e discontinuous or continuou s ; e.g. , Poisson, exponential (cr "x) , normal, uniform, Cauchy (t [11 (t 2 + x 2) ] - 1 ) , fro m t h e increasing i rregular
men ts,
=
arc s i n (71 - 1 [x ( l - x) ] - 1 ' 2 ) , logi stic
Method
2. Combinations of
a
[ (ae - a:r - B) ( ! + e - a:r - 11) - 1 ]
estab l i sh e d . Exampl e : I fj(x) is the probabil ity fu nction o f X
Sn
=
with
X1 + X2 +
itself) or PK
o f the variable X.
·
·
·
+
Xn (by m e a ns
of the
n-fold
would l ike
to draw
attention
can
be
we
can form
. . .,
fu nction
convolution of
= X1 · X2 • • · XK, or a ny linear, polynom i a l ,
* I n spite of this criticism I
distr i b u t i o n s .
random variable X w i th its e l f
to the magn ificent
f(x)
manipu
V of M a x V. M a t h ews, wh ich achieves the final step in this proced ure and a u t om a t es it [ 29] . This language cert ainly represents the realization of the d ream of an electronic music composer in the fifties.
latory
language Music
247
New Proposals in Microsound Struc ture
Method 3. The ran d o m variables ( pressure, time) can be functions of ,·ariables (clas t i c forces ) , even of r a n d o m variables. Example : The pressure v a r i a b l e x is u n der the i n fl uence of a centrifugal or cen tripetal fo rc e t/J (x, t ) . For instance, if the particle ( pressu re) is influenced by a force wx ( w being a constan t ) and alsq obeys a Wiener-Levy process, then i t s dens i ty will be
other
where
X
and y
are the values of
the variable at the instants
Afethod 4 .
b arriers. E x a m p l e :
If we again ha\'e a
reflecting barriers at walk will
be
a
>
x
and
and t,
y are
'V i e n er-Lev y process with two
0 and zero, the n the d e ns i ty of t h i s random
(27Tt ) - 1 12 wh e re
0
process. ) The random variable moves b e t we e n two reflecting (clastic)
respectively. (This is a l s o known as the Ornstei n-Uhlenbc ck
.2
± 00
( ex p [ - (y -
1< = 0
+
exp
[ - (y
x
+
2ka) 2/2t]
2 ka ) 2/2t] ) ,
+ x +
the values o f the variables a t the instants 0 and
respectively, and k = 0, ± I , ± 2 , . . . . }vfethod 5. The
parameters of a
t,
prob ability fu nction can be considered
as vari ables of other probability fu nctions ( randomization , mixtures) [30] .
Examples : a.
of a Poisson d i stribution f(k) variable of th e ex p onential density
t i s the p a r a m e t e r
a n d the ran dom combination i s
which is a geo m e t r i c distri bution . b.
p and
q
arc
the probab ilities of a random
(Be rn o u lli distribution ) . The time in tervals
where
I. (x)
=
I [k ! r (k +
k=D
i s t h e m o d i fi ed Besse l function of
n
+
=
=
walk
with
bef. ( t )
jumps
j u m ps d i stribution ) . Then
between s u c cessive
random \'ariables with common d e n s i ty e - 1 (Poisson
probability of the p o s i ti o n n at instan t t w i l l
(at )"(k ! ) - le - "1, g ( t ) f3e - P1 . The
=
±
I
a rc
the
I. (2t,;pq)e - 1 (pfq)"'2,
I ) ] - l ( x/ 2 ) 2 k + n
the fi rst k i n d o f ord er
11 .
Formalized
248
� 1 us i c
Method 6. Li n e ar, polyno m i a l , . . . , c o m b i nat ions o f pro bab i l ity functions .,l; a re consid ered as well a s com p o s i t e fu nctions ( m ixt u re s of a family of distributions, transfo r m a t ions in Banach s p ace, su bord i n a t io n ,
etc.).
a n d B a re a n y pair o f i n te rTals o n the l i n e , a n d Q ( A , B) Y E B} w i t h q ( x, B) prob fX x, Y E B} ( q, u n d e r a p pro priate regularity cond i t i o n s being a proba bility d i s t r i b u t i o n in B for a given x a n d a conti nuous fu n c t i o n in x for a fixed B ; t h a t is, a cond itional probab i l i ty of the eve n t {Y E B}, gi,·cn that X x), and !-L{A } is a pro bability d is tr ib ut i on of x-€ A, the n Q (A, B) L q (x, B)!-L{dx} represe nts a m ixture a.
p ro b
If A
{X E
=
=
A,
=
=
=
of distributions q ( X, B) ,
w h i c h depends on t h e para meter servin g as the distribution of t h e ra n d o m i z e d para m e te r [30) . b. Interlocking p ro b a bility distributions ( m od u l a t io n ) . I f f� > f2, fn are the probability dist ributions of the random \·ariables X 1 , X2, X " , respectively, then we can for m of the fa m i ly
with
S�;
f..L
=
X{
+
X�
+
·
·
·
and
+ X�;
or
P:k.
=
X� · X� · ·
and
•
•
• ,
•
•
• ,
n
S"( 2 5�1) i=l
P"( n P�.) n
. x;-,_
x,
k = l
= P]1 · P'}2 • • - P;,. ,
co m b i n at i o n (functional or stochastic ) of these s u m s and prod ucts . the ai a n d yk could be generated by e i ther indepe ndent determined fu n c t i o n s , i ndependent stochastic processes, or i n terrelated de te r m i n e d or indetermined processes. In some of t h ese cases we wou l d haYe the theory o f renewal p rocesses, if, for i n stance, the a i were cons idered wai ting times Ti. From another point of view, some of these cases would also correspond to the time series a n alysis o f statistics. I n reality, t h e ear seems to realize such an analysis when i n a given sou n d i t reco g n izes the fundamental tone pitch together with tim bre , fl u c t u a tio n , o r c as u a l
or any
Furthermore,
irregu l ari t i es of t h a t sound ! In
fact, time
invented by composers, i f they had-. c. Subordination [30] . Su ppose
series analysis
s hou l d
{X(t ) }, a .l\ I arko\· i a n
have been
process with
conti nuous transition probabilit ies
Ql ( x, r) (stochastic kerne l
=
p rob
{X ( T (t
+
s) )
E
r j X ( Trs) )
independe n t of s) , and { T( t ) }, a increments. Then {X ( T (I ) ) } t s
nega t i \·e i ndepende n t
=
x}
process with non a
�farkov process
New Proposals
in :\ I i c ro so u n d S tr u cture
249
with tra nsit ion probabi l i t ies
where U1 i s the i n fi n i t e l y divi s i ble d i s t r i but i o n o f T ( t ) . This P1
subordinated to
{X (t) }, u si n g the o p erational time T ( t )
as
is said
to be
the directing process .
Method 7. T h e probability fun c t i o ns can b e filed into c l asses, that i s , i nto parent curye configurations. T h e s e c l a sses are then c o n s i d ere d as e l e me nt s of higher order sets. The c l assi ficat i o n is obtained through at least t h ree k i nds of cri teria, which can be i n terrelated : a . analytical source of derh·ed probability d istribution ; gamma, beta, , and related densi ties, s u c h as the d e n s i t y of x2 wi t h n d eg ree s of free d om ( Pe arson) ; Student's l d e n s i ty ; :\Iaxwe l l's density ; h . o t h e r mathe matical c r i t e r i a, such as .
s ta b i l ity, i n fi n i te d ivisibility ; a n d designs :
at
as such ; at
le,·el 0,
le\"el
where
c.
.
.
characteristic features of the
the Yalues of the r a n d o m \·ariable
l , where their val ues arc accumulated, etc.
curve
are accepted
M ac ro c o m position
,\Iethod 8. Further manipulations with classes of d istributions envisaged macrocomposi tion . But we w i l l not continue these speculations s i nce many things that h ave be e n ex posed in the preceding chapters could be used fru i t fu l l y in o b\"ious ways . For exa m p l e , s o u n d molecu les produced by t h e a bo Ye methods coul d be i nj e c t e d i nto the ST(ochastic) program of Chap. V , the program form i n g the macrostructure. The same c o u l d be said a b o u t Chaps. I I and I I I ( :\ l arko,·ian p rocesses a t a macrolevcl ) . A s for Chaps. V I and V I I I (sym bol ic m u s i c and group or g anization) establishing a complex microprogram is not as easy, bu t it is ful l of rich and unexpected poss i b i l ities. A l l o f the abo\"e new p ro p osa l s are b e i n g i nYCstigated at the Centers for :\fathematical and Autom ated 1\ Iusic ( C ;.. I A l\ 1 ) a t both the School of :\ I usic of Indiana U n i versity, Bloo m i n gton , Indiana, a n d the Nuclear Rese arch Ce n t e r of t h e College d e France, i n Paris. Digi tal to a n a l o g u e conYertcrs with 1 6 b i ts r e so l u t i on at a r a t e of 0.5 l 05 samples per seco n d by :.\ [e thod 7 i n trod uce us to t h e d om a i n of
·
a r e a\·a i l a b l e i n both p l aces.
1 -8 were c a l c u lated and pl o tt e d a t the Research Compu ting of I nd i a n a U n i \"e rsi ty under t h e s u p e r vis i o n of Corn e l i a Colyer. These graphs could correspond to a s ou n d d u ration of 8 m i l l iseco n d s , t h e Figs . I X ,
Center
ordinates b e i ng the sou n d p ress u r e s .
250
Formalized Music
New
25 1
Proposals in Microsound
r
� l
:;,__�
1
I I I
�-- I 1�
--
sr ')
F i g . IX-2. Time
Exponentia l
x
Cauchy
.
Densities with B a rriers a n d R a n domized
-L
IX-3
.L
IX-4
F i g . IX-3. Exponential x Cauchy Densities with B arriers and Ra ndomized
Time
F i g . IX-4. Exponential x Ca uch y Densities with Barriers and Ra ndomized
Time
IX-5
= IX-6
Fig. IX-5. H yperbolic Cosine Determined Time Fi g . IX-6. Hyperbolic Cosine Randomized Time
x Exponential x
Densities with Barriers and
Exponential Densities with B arriers and
IX-7
�� j :S::
....===== .. � ==�= -�-
...._ ,_C"
�=
·!-
·-----�
Fig . IX-7. Hyperbolic Cosine x Exponential x Cauchy Densities with B a rriers and Determi ned Time Fig. IX-8. Logistic x Exponential Densities with Barriers a n d Ra ndomized Time
Chapter X
Concerning Time , Space
WHAT I S A
and Music*
COM POSER?
A thinker and plastic artist who expresses hi msel f tluo u gh sound These two realms probably A few poi nts of sci ences
a nd
cover
convergence in
his entire
being.
relation to
mu sic:
beings .
time and space between
the
First point: In 1 954, I introduced probability tl1eory and calculus in musical
composition in o rder to control sound
ma sses both
in their invention a nd in
their evolution. This i naugurated an entirely new patl1 in music,
th a n
m ore
global
or, in general, "discrete" music. From hence came stochastic music. I wi l l come back to that. But tlte notion o f e ntropy. as formula ted by Boltzmann or Shannon, 1 b ecame fu ndamental. I ndeed , much like a god, a composer may create the reversibility of the phe nomena o f masses, and apparently, invert Eddi ngton's "arrow o f tirne." 2 Today, I use probability disn·ibutions either in computer ge ne rated sound syntltesis on a m i c ro or macrosco p ic scale, or in i nstm mental com p ositio ns. B ut tll e laws of probability tl1at I use are o ften nested and vary with time which creates a polyphony, serialism
* Excerpts of Ch apter X originally appeared in English in Perspectives of New Music , Vol . 27, N° 1 . Those excepts appea red ori g i n ;� l l y in French in Redcco uvri1· f, TemJJs, Editio11S de l'UniversiU cle Bruxelles, 1 988, Vol. 1-2. 255
256
Formalized Music
stochastic
to
dyna mics which is aesthetically interesting. This proce dure is akin
mathematical
the
tra ns for ma tio ns
of
analysis
propo se d
m ic rosc op ic entropy M exists, then M
function
or the
densi ty
eq ua tio n
Liouvill e's
essentially by =
I.
A2, where A acts
matrix. A is non-unitary
on
Prigogine;5 wh ich
on
non-unitar y
if the distribution
n a me ly, the
means that it does not
mai ntai n the size of probabilities of the states considered
d uri ng the evolutio n
of the dynamic s ys te m , although i t does maintain the average values of those
which can be observed . This implies the irreversibility of e qu ilib ri u m state; that is, it implies the irreversibility
Second
point:
This point has no obvious
ma ke
use
of
rel a tions hi p
Lorentz-Fitzgerald
and
of music ." I would to these transformations.
comments related We
all
syste m to the
to music, except that we
Ei nstein
macroscopic composi tion
the
of time.
tra ns formations
know of the speci al theory of re la ti vity and the equations o f
velocity of li gh t
.
to go from one point to
another in space, eve n
less " spatial ubiquity"-that
is,
simultaneous pres ence
of
an event or a n
obj ect in tw o sites in space. On th e contrary, o n e posits
displacement. Within signify? If the notion one
could
the
quantum
is
an
no tion
of
a local reference frame, what then does displacement
of displacement were
undoubtedly
reduce
more fundamental all
macro
and
hypothesis that
mechanics and
than that o f microcosmic
chains of displacement. Consequently freely advance), if we were to adhere to its implica tions acce pted now for decades , we wo uld
transformations to extremely short
(and this
the
moving reference frames relative to the observer. instantaneous j ump from one point to another in space, much
i f that time depends on
time,
of
From this it follows that time is not absolute. Yet time
is always there. It "takes time"
There is no
the
nevertheless like to make s o m e
Lorentz-Fitzgerald and Ei nstein, which li n k space and time because
finite
could
in
I
perhaps be forced to admit the notio n of qu a n ti fied space and its corollary,
qua ntified time. But the n, what could a quanti fied ti me and space signi fy,
a
time and space in which con tigu i ty would be abolished ? What would tl1e
p ave me n t of the universe be i f there were
gaps
between the paving stones,
i naccessible and filled with nothing? Time has already bee n proposed as
having a quantic stmcture by T.
Let
us
D. Lee of Co lumb ia Univ�rsity.
return to the notion of
time considered
as
duration. Even a fter
the experimental demonstration o f Yang and Lee which has abolished tl1e
parity
sym metry P,�; it seems that the CPT th eo re m still h olds for tl1e
symmetries of the electron (C) a nd of ti me (T), symmetries th at have not yet
Time, Space and Music
Concerning
been co m p le tely an nulled . This re mains appears to be nonreversible in
c ertai n
so
257 eve n if the
"
arrow of time "
weak interactions of
particles. We
migh t also consider the poetic interpretation of Fey n m an ,6 who holds that
when a positron (a positively charged p artide created sim u lta neo usly with an electron) co llid e s with a n electron, there is, in re al ity, o nl y one electron
rather than three elementary particles, the positron be i ng noth i ng but the
te m po ra l retrogres sio n of the first electron. Let us also not forget the theory
of retrograde ti me found in Plato's Politicos-or in the futu re co ntractio n of
the universe. Extraordi nary visions!
Quantum physics wil l have difficulty discovering
the
reversibility of
time, a theory n o t to be confused with the reversibility of Boltzmann's
of
"
arrow
e n tro py . " This difficulty is re flected i n the exp lanations that certain p hys i cists are attempting to gi ve even today for the phenomeno n ca l led the "
de layed
b ee n
choice" of the
proven on
observa tion,
in
two states---c or puscular or wave -o f a p ho to n . It has de pe nd e nti re ly on
ma n y occasions that the states
compliance
with the theses of quantum mechanics. These
explanations hint at the idea of a n i n te rv en tio n of the present i n to the past," "
contrary to the fact that casuality in quantum mecha nics c a n no t be inverted .
For, if
the
co nditio ns of observation are established to detect the particle,
the n one obtains the corpuscular state and never the wave state, a n d vice
versa. A similar discussion on non-temporality a nd the irreversibility of the notion of causality was undertake n so me time ago by Hans Re i che nbach . '
Another fun d a me n ta l experiment has to do with the cor re la tio n of the
movement of two p ho to ns emitted in op po s i te directions by a s i n gle ato m . How ca n o ne
explain
that b o th either pass through two pola ri zin g fil ms,
that bo th are blocked? It is
as
and i nstantaneously so, which is contrary to Now, tltis
experi ment could
the spec ia l
"
theory of relativity.
be a starting point for the investigatio n of
more deeply seated properties o f space, fre ed fro m this case, could the
or
if each p ho to n " k new" w ha t th e other was do i n g
the
tutelage
o f time.
In
no n loc a li ty o f quantum mec h a n ics per ha ps b e ex p lai ne d "
not by the hypothesis o f " hidden variables" in which ti me still i n te rv e n e s , but
ra th e r by the unsuspected a n d extravagent properties o f nontemporal space,
such
as
" spatial ubiquity," for e xa mp le?
Let
us
ta k e yet one more step . As s pa ce is perceptible o nl y
i n fi n i ty of chains of energy transformations,
an
it could very
a c ro ss
the
well be noth i ng but
appea rance of these chains. I n fact, le t us consider the move ment of
a
photo n . Move ment means d isplacement. Now, c o u ld this displacement be
considered an a u togene sis of the
photon by itself at each
step
of its
trajectory
Formalized Music
258
(continuous
or quantized)? This continuous auto- creation of the
co uld it not, in filet, be s pace? Third
point: Case of creating something from nothing
photon,
In musical c o m pos i ti o n , co nstruction must stem from origi nality
which
can be defined in extreme (perh a ps i nh u ma n ) cases as the creation of new rules or
l aws
,
as
far
as
that is
possib le ; as fur as
yet known or even forseeable.
without any causality.
not
possible meaning original,
Construct laws therefore
from
nothing, since
But a construction from nothing, therefore total l y engendered, totally
original, would n ec es sa ri l y call upon Such a mass
wo ul d
own. For example:
a
an i n fi nite mass of rules d ul y e ntangled.
have to cove r the laws of a universe different from our
rul es for a tonal com position have been
cons tructe d Such .
composition therefore includes , a p riori, the " tonal functio ns." It
also
i n cl udes a combi natory co n ceptio n since it acts on entities, sounds, as defi ned by the instru ments. In orde r to go beyo nd this slight degree o f originality, other functions would have to be inven ted or no functions s ho uld exist at all. One is there fore obliged to conceive of forms from thoughts bea ri ng no rela tio n to th e preceeding ones, thoughts without limits of shapes and ,
an unlimited web realm which itself excludes, by definitio n, any poss ibl e continuums of sound. However, the insertion of continuity will consequently augment the spread of this web and its compacity. Furthermore, i f one cared to engender the unengenderable in the realm o f sound, then it would be necessary to provide rules other than those for sound machines such as pipes, stri ngs , skins, etc. which is possible today thanks to computers and corresponding technologies . But technology is both but a semblance of thought and its materialisation. It is the re fore but an e p iphenomenon in this discussion. Actually, rules of sound synthesis such as those stemming from Fourier series should not be used any more as the basis
without e nd . Here, we
of
are obl iged
to progressively
weave
enta ngled rul�nd that alone in the combinatory
of construction. Others, different ones, must be formulated. A rwther
origi nality
i ndividual's or
We have see n how construction stems fro m
an
is defined by the creation of rules and laws outside of an
or even the human species' memory. However, we have left aside
the notion o f
rule
perspective:
which
rules or Jaws. Now
law sig nifies a finite or
the time has come to discuss this notion. A
infinite
procedure, always the same, applied
con ti nuous or "discrete" elements. TI1is definitio n im plies the notion
repetition, of recurrence in ti m e , or symmetry temps ). There fore , i n order for a rule
to
exist,
to
of
in rea l ms outside ti me (hors it must be applicable several
Concen1ing Time, Space and
259
Music
were to exist but once, it wou ld be point, th ere fore observable, it must be repeatable an
times in eternity's space and time. If a rule
swallowed up in this immensity and reduced to a s i ngl e
unobservable. In order for it to be infinite number of times.
(cf. Herakleitos: "It and Kratylos: "not even
Subsidiary lJUI!Slion: Can one repe at a phenomenon?
is impossible to once.")
But the
step twice i nto the same river,"
fac t remains that the universe:
a) seems, for the time being, to be made up
It is
as
of rules-procedures ;
b) that these rules-procedures are recurrent.
though the Being (in disagreement with Parme nides) , in order to con
tinue e xis ting, is obliged to die; and once again. Existe nce, the re fo re is a dotted line. ,
Can one, at las t, i ma
gi ne
engendered from nothing? Eve n
resembling this , despite
"Lamb' s
dead, is o bl iged to start his cycle
an infinitesmal microscopic rule tha t is
if
physics has
shift"
yet to
discover a n ythi n g
(which sees each point in space in our
seething in vi rtual pairs of particles and anti-particles), we can imagine such an eventuality which would, by the way, be of the same na ture as the fuct of p u re chance, detached from any cau sa lity .
u niverse as
It is necessary to depend on such a conclusion of a Universe open to the which relentlessl y would be formed or would disappear i n a truly creative whirlwind, be g inning from nothing ness and disappearing into
unprecedented
nothi ng. The sa me goes Here, below, is
for the basis of art as
the
thesis
of
well as for m a n ' s destiny.
a few astrophysicians such
as
Edward
Tryon , Alexander Vile nki n , Ala n Guth, Paul Steinhardt, adherents to the Bang theory:
Big
I f grand unified theories are correct in their prediction that baryon
number is not conserved, there is no known conservation law that
prevents the observed universe from evolving out of nothing. The infla
tionary model of the universe p rov id es
a
poss i bl e mechanism by which
the observed universe could have evolved from an infinitesimal region.
It is th en temp ting to go one step further and speculate that the en tire
universe evolved from literally nothing.
1 984)
The
also quite
(cf. ScientifiC
multiplicity o f such universes according
intriguing.
American,
Ma y,
to Linde8 fro m Moscow is
260 Here, below, is an alternative to the
Formalized Music
B ig
Bang scenario. These studies
have been followed by the physicists of the U niversity of Bruxelles ; namely R.
Brout, E. Giinzig,
F. E n gl e rt and
P. Spindc1 :
Rather tha n the U niverse being born of a n explosion, they p ropose that
it appeared ex-nihilo followin g an
instability
of the minkonskian quan
tum void, meaning that space-time was devoid
flat
or
yet-without
any
curvature."
(cf.
of any matter, Coveney,
therefore
Peter
V.,
"L'irreversibilite du temps," La Recherche, Paris, Februar y, 1 989).9
a
What is extraordinary is that both propositions, Big Bang
beginning, an origin from nothing, or nearly n o thi ng with,
or
not, admit
however, cycles
com pare, made in 1 958. At that time, I wa n ted to do away with a ll of the i nh eri te d rules of com p osition in ord er to create new ones. B u t th e qu esti o n that came to my mind a t that ti me was w he ther a music could still have mea ning ev e n if it was not built on rules of occurence. In other words, void of rules. Be l ow are the steps i n this thought process: of re - crea tio n ! With a mo s t extreme m o de s ty, I would like to
especially the last hypothesis, with a scientific-musical vision I h ad
"For it is the same thing to think and w be"
(The Poem, Par m e ni des) and my p araphrase
"For it is the same thing not to
O n tol og y :
In
be arul to be"
a Universe of Void . A brief train of w aves whose beginning and end
cide
(nil Time), perpetually triggering off.
coin
Nothingness resorbs, creates.
It is the generator of B e i n g. Time,
Causality.
This text was first p ublished in Gravesaner Bliitl.er, N° 1 1/ 1 2 , 1 958, the
great conductor, Hermann Scherchen. At that time, I temporarily resolved this problem in creating music uniquely through the help of probability distributions. I say " tem p orarily" s i n ce each probabil ity functio n has its own fi nality and the re fo re is not a nothing. * revue published by the
had
*Cf.
also page
24 for a slightly different rendition of the same material
(S .K.)
26 1
C oncerning Ti me , Space and M usic
Another q uestion
The actual state of knowledge seems to be the mani festation of the
evolution of the universe since, let us say, some fifteen billion years.10 By that,
I mean th a t k nowledge is a secretion of the history of humanity, p rodu ced by this great la p se of time. As su m i ng this hypothesis, all that which our
individual or collective brain hatches as ideas, theories or
of its me n tal structures, formed by the
the output
know-how,
is but
history of the innumerable
m ov e men ts of its cultures, in its anthropo morphic transformations, in the
e vo l uti o n
th i s i s
so,
of the
e arth , in that of the so la r system, in that of the u niverse. I f
j tivity of our
ob ec
"
knowledge and
already developing, one were to
he red ity, therefore the rules for the fu nctioning of the brain based certain premises today, on logi c or systems of lo gic , and so on , if one
and their on
as to th e "true know-how. For if, with bio-technologies transform these mental structures (our own )
th e n w e fuce a frightening, fundamental doubt
...
were to succeed in modifying them, one would gain,
as
if by sort of a miracle,
ano th er vision of o ur universe, a vision which would be built upon the or ies
a nd knowledge which are beyond the realm of our present thought. Let
us
p ursue this thought. Humanity
is,
I b el ieve , already on this pat11.
Today, humanity, it seems to me, has already taken the first step i n a new
phase o f its evolution, in which not o nl y the mutations of the brain, but also the creation of a universe very different from us,
will
that which presently surrounds
has begun. Humanity, or ge neralizing, the species which may follow it,
accomplish this
process.
Music is but a path among others for man, for
his
species, first to
exi s ti ng if man, his princ i ple of
i magine and then, after many, many generations , to entail this universe into another one, one fully created by man. I nde ed ,
species, is the image of his universe, then m a n , by virtue of the
creation fro m nothingness and dis appeara nc e into noth i ng ness (which we are forced to set), could redefine his u n ive rse in harmony w i th his creative
ess ence , such as an environment he could bestow upon himsel f.
IN MUSIC
I n the followi ng comments, the points of view o n time are taken from
gestati on or u nder observation. This is no t to say that my preceding co mme nts do not co n ce r n the m us ic ia n On the contrary, if it is i n cumbe nt o n music in
.
music to serve a s a medium for the con fron tation of philosophic o r scienti fi c
ideas o n the being, its evolution, and
composer at
their ap peara n ces, it is essential that the to th ese types of inquiry.
least give some serious thought
262
Formalized Music
Furthermore, I have deliberately not approached the psychological apprehension of time from higher levels, for example, the effects of the temporal dynamic experience while listening to a symphony or to electronic music.
What is time for a musician? What is the O ux of time which passes invisibly and impalpable? In truth, we seize it only with the help of perceptive reference-events, thus indirectly, and under the condition that these reference-events be inscribed somewhere and do not disappear without leaving a trace. It would suffice that they exist in our brain, our memory. It is fundamental that the phenomena-references leave a trace in my memory, for if not, they would not exist. Indeed, the underlying postulate is that time, in the sense of an impalpable, Heraclitian flux, has signification only i n relation to the person who observes, to me. <;Jtherwise, it would be meaningless. Even assuming the hypothesis of an objective flux of time, independant from me, its apprehension by a human subject, thus by me, must be subject to the phenomena-reference o f the flux, first perceived, then inscribed in my memory. Moreover, this inscription must satisfy the condition that it be in a manner which is well circumscribed, well detached, individualized, without possible confusion. But that does not suffice to transform a phenomenon that has left traces in me into a referential phenomenon. In order that this trace-image of the phenomenon become a reference mark, the notion o f anteriority is necessary. B u t this notion see ms to be circular and as impenetrable as the immediate notion of flux. It is a synonym . Let us alter our point of view, if only slightly. When events or phenomena are synchronic, or rather, if all imaginable events were synchronic, universal time would be abolished, for anteriority would disappear. By the same token, if events were absolutely smooth, without beginning or end, and even without modifications or "perceptible" internal roughness, time would likewise find itself abolished. It seems that the notion of separation, o f bypassing, o f di fference, o f discontinuiity, which are strongly interrelated, are prerequisite to the notion of anteriority. In order for anteriority to exist, it is necessary to be able to distinguish entities, which would then make it possible to " go" from one to the other. A smooth continuu m abolishes ti me, or rather time, in a smooth continuum, is illegible, inapproachable. Continuum is thus a unique whole filli ng both space and time. We are once again coming back to Parmenides. Why is space included among those things that are illegible? Well, because of its non-roughness. Without separability, there is no extension, no distance. The space of the universe would find itself condensed into a mathematical point without dimensions. Indeed, Parmenides' Being,
263
Concern ing Ti me, Space and Music
which fills all space and eternity, would be nothing " mathematical point." Let us get back to
the
notion of separability,
but an absol utel y smooth first
in time. At the least,
separability means non-synchronisation. We discover once again the notion o f a nteriority. It
merges with the notion of temporal orderi ng. The ordering
no e mpty spaces. It is necessary for one separable entity to be contiguous with the next, otherwise, one is subject to a con fusion of time. Two chains of contiguous events without a commmon link can be indifferently synchronous or anterior in relation to each other; time is once again abolished in the temporal relation of each of the universes represented by the two chains. On the contrary, local clocks serve as chai ns without gaps, but only locally. Our biological beings have alSo developed local clocks but
anterioity admits no holes,
they are not always effective. And memory is a spatial tra n sla tio n o f the temporal (causal) chains. We will come back to this.
I have sp oken of chains without gaps. At the moment and to my knowledge, local gaps have not yet been discovered in sub-atomic physics or
in astrophysics. And in his theory
of the relativity
of time, Einstein tacitly
accepts this postulate of time without gaps in local chai ns,
but h is
theory also
constructs special chains without gaps between sp a tia l l y separable localities.
reversibility of ti me which was of recent discoveries in sub-a to mic physics ,
Here, we a re definitely not concerned with the
partially examined above in light for
reversibility would Let us
not abolish time.
examine the
notion of separability, of discontinuity in space. Our
immediate consciousness (a me ntal category?)
all ows
us
to i magine separated
entidcs which, in turn, necessitate contiguity. A void is a u nity in this sense,
contrarily to time, i n which our inherited or acquired mental notions bar us
from
conceiving the absence
of time,
its abolitio n , as an entity sha ri ng ti me,
the primordial flux. Flux either is, or is not. We exist, there fore moment, one cannot conceive of
the
it is.
For the
halting of time. All tl1is is not a
paraphrase of Descartes or better yet, of Parmenides : it is a presently
(But certainly, by using Parmenides once more , passable : NOEIN E:£TIN TE KAI EINAI ").
i mpassable frontier.
" TO fAP AYTO
T o g e t bac k t o space, the void c a n be im agin ed
endty
as a
dwindling of t1te
(phe nomeno n) down to an infinitesimal tenuousness, havi ng no
density whatsoever. On the othe r
hand, to travel from one entity to a notl1er is result of scale. If a pe rso n who voyaged were s mall, tl1e person would not e ncompass the to ta l ity of entities, the u n iverse at once. But if t11is perso n's scale were colossal, then yes. The u nive rse would offer itself in o ne stroke, a
with hardly a scan, as when one exa mines d1e su n from a far.
264
Formalized
Music
in a de nse ne two rk of non tem p o ral contiguities, uninterrupted, extending through the entire universe. I said , in a snapshoL This is to say that i n the snapshot, the s patial relations of the entities , the forms that their co ntiguities assume, the structures, are essentially outside time (hors-tem p s). The flux o f time does n ot intervene i n any way. 1.bat is e xa ctl y what h a ppens with the traces that the phe nome nal entities have left in our memory. Their geographical map is The ennnes would
a pp ear,
as
in
a
snapshot, reunited
outside time.
Music participates both in space outside time and in the tem poral flux.
Thus, the scales o f pitch ; the scales of the church modes ; the morphologies of
h i ghe r
levels;
fugal
structures,
architectures,
en gen de ri n g sounds or pieces of music, these are
formulae
mathematical
outside time,
whether on
paper or in our memory. The necessity to cling against the current of the of time is so strong that certai n asp ects of ti me are eve n ha ul ed out of it, such as the d u ra tio n s which beco me commutable . One could say that every
river
temporal schema,
ou tsi de time
inscribed .
pre-conceived
of the temporal
Due to the
principle
or post-conceived,
is
a
re p rese ntatio n
flux in which the phenomena, the
of anteriority,
the flux
of ti me is
entities
locally
,
are
equipped
with a structu re of total order in a ma the m a tic al sense. That is to say that its i mage in our brai n, an image co nsti tute d
by the chai n of successive ev e n ts,
can be placed in a o n e - to - o n e corresp ondance with the i n tegers and even, witl1 th e aid of a useful generali zation, with real numbers (rational
irra tional).
Thus, it can be
counted. This is
and music as well, by using
same
structu re
of total
its
what the
sc i e nces
and
in ge n e ra l do ,
ow n clock, the me tron o m e . By virtue of this
order,
time
ca n
be
plac e d
in
correspondence with the p oi n ts of a line. It can thus be drawn.
a
o ne-to-one
This is done i n the sc ie nces , but also in music. O ne can now design te m p o ral architectures-rhythms-in a m o d e rn sense. Here is a tentative a x.iom a tizatio n of the te m p o ral structures l ace d outside of time : p
1 . We perce ive temporal events.
2 . Thanks to separability, these events can be assimilated to landmark
poinJs in the flux of time, poi nts which are instanta neousl y h aul e d up outside of time because of their trace in our memor y.
3 . The co m p arison of the land7111lrk poims allows us to assign to them distances, intervals, durations . A distance, translated spatially,
Concerning Time, Space and Music
265
can be considered as the displacement, the step, the ju m p from
one point to another, a non te mpo ral jump, a spatial distance.
4. It is possible to re pe at, to link toge the r these steps in a chain.
5 . There are two possible orientations, o ne by an accumulation of steps, the other by a de-accumulation.
we can construct an object which can be represe n te d by points on evenly spaced and symbolized b y the nume ral 1 with i ndex zero : 1 0 ( , 3 , 2 1 ,0, 1 , 2 , 3 ). This is the regula r rhythm, corresponding to the whole numbers. As the size of the step is not defined in the preceding propositions (recalling Bertrand Russell's observation conce rn i ng Peano's axiomatic o f natural numbers 1 1), we ca n a ffi x t o the p reced in g obj ect the fol lowi ng objects whi c h I call "sieves," by using solely proposition 4 : From
a line, ...
-
here -
,
=
,...
,-
20
=
30 = 32
From
=
{. .. 4 ,
-
,
-
2 , 0 , 2 ,4 , 6 , . . . } or 2 1
{ . . . ,-3,0,3,6,9, . . . } or 3 1
{ . . . ,-4,-1 ,2,5,8, . . . } etc . . .
=
=
{. . .
,-
3 , - 1 , 1 , 3 , 5 , . . . } or
{ . . . ,-5,-2, 1 ,4,7, . . . } or
these objects and their modular nature, and with the he l p of these th ree
logical operations:
U union, disjunction ex. 20 U n intersection,
conjunction
21
=
ex. 2o
10
n 21
· - -. .
- complementarity, negation ex. 20
=
21
=
0
L-that is to say, very complex rhythmic go as fur as a rando m-like distribution of points o n a line-if the period is sufficiently long . The interplay between com plex ity and s i m pl ic i ty is, on a higher level, a no th er way of defining the landmark points, which c e r ta i n l y plays a fu ndamental role in aesthetics, for this play is j uxtaposed with the pair release/tension. we can construct logical functions architectures which can even
Example o f a logical fu nc tio n L:
L = (M �-. n
�
n P1) U (Nr U Q , U . . . Ty) U . . .
Th e upper-case letters designate moduli and the subscripts designate shifts i n relation to a zero point o f reference. Up to this p oi nt, we have exa mi ned time perceived by means of fac ulties of atte ntio n and conscious thought-ti me o n the level of forms
our and
structures of a n order ranging from tens of minutes to a pproximatel y one twe nty-fifth of a seco nd . A stroke of the bow is a rcfcrrential eve nt that c a n define durations of a fra c t io n o f a second . Now, there a r e some subl i m i nal
Formalized Music
266
events found o n se ve ra l even lower level s . Such an examp le is that of the
envelope o"n wave form. If the duration of the note is long (about o ne minute), we perceive the rhythms of the beats as pl easant, moving vibratos. If the d u ratio n is relatively short (three seconds), the ear and the brain inte rp ret it as a timbre. That is to say that the result o f subliminal, unconscious cou nting is di ffere n t in nature and is recognized as timbre. temporal
seg
me nta tion
produced by a very choppy amplitude
the sou nd of an unvarying sinusoidal
b rie f moment to conside r the mechanism of the internal to say, the timbre-and the frequenc y of a sou n d . On the one hand , it seems that the p oi nts of deformation of the bas ilar membrane play a fu ndame ntal role in the recognitio n; but, on the other hand, a sort of temporal Morse code of electrical discharges o f neurons is taken statistically into account for the detection of tone. A remarkably com p lex subliminal cou nti n g of time is taking place. But knowled ge of acoustics in this domain is still very limited. Let us take a
ear
coupled with the brain which recognizes the wave form-that is
On th is
level, here is ano ther disconcerting phenomenon. It theory on the synthesis of computer sounds which c ircumven ts the harrr._nnic sy nthesis of Fourier, practic ed everywhere today, a theory which I i ntrod uce d now more than fifteen years ago.12 It is a question of begi n ning with any form whatsoever of an ele me ntary wave, and with each repetition, of havi n g it un d ergo small deformations a ccord i ng to certain densities o f probabilities ( Gauss , Cauchy, logistic, ... ) appropriately chosen and implemented in the form of an abstract black box. The result of these d eformatio ns is perceptible on all levels, microstructure ( timbre), is the result
subliminal
of
a new
=
ministructure (= note), mesostructure (= polyrhythm, melodic scales of
intensities), macrostructure minutes) .
(=
global evolution on the order of so me tens
I f the rate of sampling h a d
been I ,000,000
or 2,000,000
of
sampl es per
second instead o f approximately 44, 1 00 (commercial standard), one would
have had an effect impossible
to p re d ict.
of soundi ng
We see to what extent
fractals, with a sonorous effect which is
music is everywhere steeped i n time: (a) time i n
the form o f a n impalpable flux o r (b) time i n i ts froze n form, outside time,
made
possible by memory. Time is the blackboard o n which are inscribed
phenomena and
their
re a tion s
l
outside the time of the u niverse in which we
structures, rules. And, can o ne imagine a rule without rep etition? Ce rtainl y not. I have already treated this subject. Besides, a si ngle eve nt in an absolute e ter n i ty of time a nd space would make live. Rel a t ions i mply architectural
C oncerning Ti me , Space and Music
267
is uniq u e . But wait at every step, at
no sense. And yet, eac h event, like each individual on earth,
this
uniqueness is
the
equivalent of death which lies in
every moment. Now, the re petiti o n of an event,
its
reproduction
as
faithfully
as poss ib le , corresponds to this struggle against disap pearance, against nothi ngness. As if the entire universe fo u gh t despe rately to ha ng on to e xis te n ce, to b ei ng , death . The union e xam p le of perhaps
by
its own tireless renewal at every instant,
at
of Parmenides and of Herac li tus . Living spec ies
this struggle of
life o r death,
in
an inert
ever y
are a n
Universe launched
by the Big Bang (is it really i nert, that is, w i tho u t any cha nges in its
laws?). This same pri nciple of dialectical
co
m bat
is present everywhere,
veriftable everywhere. Cha n ge-fo r there is no rest-the cou pl e death and
lead the Universe, by du plic atio n, the copy being more or less exact. or less " makes the differe nce between a pe ndular, c yclic Universe, strictly determined (even a deterministic chaos), and a no nde te r m i n ed U niverse, absolutely unpredictable and chaotic. Unpredictability in thought obviously has no limits. On a first appr oach it would c orrespo n d to birth from nothi ng ness, but also to disappearance, death into nothingness. At tlte mome n t, the Universe seems to be midway between these two chasms, so m ethi ng which could be the subj ect of a nother study. This study would dea l with the pro found necessity for musical composition to be perpe tua ll y
birth
The " more
original-philosophiocally, technically, aesthetically. 1 3
In what follows a n d a s a consequence of the preceeding axioms, we will greater detail the practical questions of how to create a sieve (=
study i n
series o f points on
a line),
begin n i ng from a logical function of moduli
(periods), or inversely, from a series o f poi nts on
logical function of moduli which sh ou ld
be a bl e
a
line, how to
c rea te
a
to e n gend e r the given se ries.
This ti me, we shall use series o f " pi tchcs" taken from musical space.
Chapter XI
Sieves *
I n music, the qu esti o n o f symmetries
(spatial
identities) or
o f periodicities sample in
(identities in ti me ) plays a fundamental role at all levels: fro m a
sou nd synthesis by computers, to the architecture of a piece . It
sary
is thus
neces
to fo rmu l ate a theory permitting the construction of symmetries which
are as co m p le x as one m ig h t want, a nd inversely, to retrieve from a
given of events or objects in space or time the symmetries that constitute the
series
se ries . We shall call these series "sieves. " 1
wi ll b e said here could be a ppl ied to a ny set o f of sound o r o f well-ordered sound structures, and es pecially, to any group which entails an additive op eratio n and whose elements are multiples of a unity; that is to say that they belong to the set N of natural Everything that
characteri�tics
numbers. For example : pitches, time-points, loudnesses , densities, degrees o f
ord e r,
local timbres , etc. In th e c as e o f pitches, there must be a distinctio n between sieve (scale) and mode. I ndeed, the white keys on a piano constitute a u nique s i eve (scale) upon which are formed the " modes" of C major, D, E, G, A (natural minor), etc. J ust like I nd ia n ragas or Ol iv i e r Messiaen's modes "of li m i ted transpositions," modes are defined by cadcntial , harmonic, etc _ formulas.
But
long as
every well-ordered set can be re p resented as points on a li ne, as given for the origin and a len gth u for the u nit
a reference p oi nt is
d istance, and
this
is a sieve.
Historically,
the invention o f the well-tempered
*This chapter is sched uled to appear in a future issue of Perspectives of New Music_ to the following material is most a pp reciated
John Rahn's personal contribution
(I. X.)
268
Sieves
269
chromatic scale , attributed to the Renaissance, is of upmost importance si nc e it provided a universal standarization of the realm of pitches, as fe rtile as that which a l read y existed for rhythm. However, it shou ld be re me mb ered that the first theoretical attempt towards such a n approach which ope ne d the p ath to number theory in music dates back to Aristoxenas of Tarent. during the IVth ce ntury B . C. , in his " Harmonics."2
CONSTRUCTION OF A SIEVE
Starting from symmetries (repetitions), let us construct a sieve (scale).
As a melodic e xa mple , we shall
construct the diatonic scale formed by the
white keys of the piano. With u = o n e semitone = o n e millimeter and a zero reference point taken arbitrarily o n a no te , for example C3, we ca n notate the diatonic sieve (scale) on graph p aper scaled to the millimeter, by means of p o i n ts to the left and to the ri ght of this zero reference poi nt with successive intervals counted from left to right of 2, 2, 1 , 2, 2, 2, 1 , 2 , 2, 1 , 2, 2, 2, 1 , . . . millimeters, or we can wri te the sieve in a logical-arithmetic notati o n as L = 1 20 U 1 22 U 1 24 U 1 25 U 1 27 U 1 29 U 1 2 1 1 where 1 2 is the m o dulus of the sym m etry (period) of the octave with u for the se mitone. This notation gives all the Cs, all the Ds, . .. all the Bs, considering
that the moduli 1 2 repeat o n both sides o f the zero
reference point. The indices, 0, 2, 4, 5, 7, 9, 1 1 of tlte modulus 1 2 signify shifts to the right o f the zero of the modulus 1 2. They also represent tl1e residue classes o f congruence mod. 1 2.
With a di fferent unit dista nce n, for exa mple , a quartertone, o n e would have the same structure as the diatonic scale but the period of the s eries would no lo nger be an octave, but an augmented fourth . In a similar fashion, a pe riodic rhythm, for example
J J J •
c a n be l ogic al
•
I
J JJ •
l
J J. J
3 , 3 , 2:
I
e tc .
notated as L = 70 U 73 U 75• I n bo t h of these examples, tJ t c sign U is a u nion (and/or) of the points defined by the moduli a n d tJteir sh i ftings.
The periodicity of the diato nic sieve (scale) is external to the sieve itself and is based o n the existence of the modulus 12 (tJ1e octave ) . I ts i n ternal s y m metr y can be studied in the indices I (shi ftings, residue classes) o f tl1c
te r ms 1 21• But it
would
be i n te resti n g to
give,
whe n
it exists,
a more
h idde n
Formalized
270
Music
of the modulus 1 2 i nto simplier. 3 a nd 4, a decomposition which wou ld have the advantage of aHowi ng a comparison among di ffe re nt sieves in o rd er to stud y the de gree of th e ir difference and to be able to define a notion of distance in this wa y . symmetr y de rived from .the decomposition mod ul i (symmetries, periodicities}, such
as
Let us take the elementary s ieves 30 and 4 0• In taking the points 30 and/or the points 40, we obtain a series H 1 = ( ... , 0, 3, 4, 6, 8, 9, 1 2 , 1 5, 1 6 , 30 U 401 and if C is th e zero and u = one 1 8, 20, 2 1 , 24, 2 7 , 28, ... ) semito n e, H1 becomes ( ... C, D#, E, F#, G#, A. C, D#, E, . . . ). But if we take ( . . . , 0, 1 2, 24, 36, the poi nts common to 30 and 40, we obtain the series H2 30 n 40 where the s ig n n is the logical intersection (and) of the sets of ... ) poi nts de fined by these moduli a n d their respective shirtings . =
=
=
Hence, we observe that the series H 2 can be defined by the modulus 1 2 4 and by the logical expression L = 1 20 which gives the octaves. The number 1 2 is the smallest c o m mo n multiple of 3 and 4, which a re cop rime , meaning their largest common denominator is 1 .4 =
3
*
( Let us imagine now the elementa ry sieves 20 and 60• Then G1 = 20 U 8, 1 0, 1 2, ... ) and the common points in G2 = 20 n 60 = ( ... , 0, 6, =
. . . , 0, 2, 4, 6,
1 2, 1 8, ... ) . But h ere, the series is n o longer made into octaves preceding case.
as
in
the
To understand this, let us take another exa m p le with elementary 15 wh ic h have been adjusted to the original . We 6 and M2 mod ul i M 1 then form the pairs 60 = (Ml , l l ) and 1 50 .; (M2,I2) with 1 1 = 0 and 12 = 0 =
=
as indices.
The series of the u ni o n (M I , II ) U (M2, 12) = K l wil l be Kl { .. . , 6, 1 5, 1 8, 24, 30, 36, 42, 45, . .. } and their co mmon points (the intersection) will fo r m the s e ries (M l , I I ) n (M2, 12) = K2 where K2 = { ... , 0, 30, 60, . . . }. The period is clearly equal to 30 and the largest common denominator D of 6 and 1 5 is 3 (which is, by multiplication, the part congruent to M 1 and M2) and the sma l l es t common multiple is M3, equals 30. Now, 6 divided by the la rges t common denominator D is C l , equals 2; and 1 5 d i vi ded by the largest com mon denominator D is C2, equals 5. Generalizing, the period of the points c o m m o n to the two moduli M l a nd M2 w i ll be the smallest com mo n multiple M3 of these two moduli. So, (M 1 , I I ) n (M2, 12) (M 3 , 13) with 13 0 and M3 = D * C I * C2, where C1 = M 1 / D and C2 0, if l l = 12 M2 =
=
=
=
/ D.
=
that the operation of logical union, notated as U, o f the two elementary moduli Ml and M2 is cumulative in that it ta kes into It
will also be
noted
27 I
Sieves
periodic poi nts o f both moduli simultaneously. O n th e other operation of intersection, notated as n, is reductive since we take only the points common to both moduli.
account the
hand, the lo gi ca l
Whe n we mix the points of several moduli M I ,
a) by union, we obtain a sieve which is dense
M2, M3, M4, . . . : a nd complex
dependi ng on the elementary moduli; PI
=
(M l , l l }
U ( M 2,
12)
U
(M3, 13) U . . .
b) b y intersection, w e ob tain a sieve which i s more rarified than that
and there would even be some cases be empty of points when it lacks
of the elementary moduli, in which the sieve would
coincidences; P2 =
(M l , I l ) n (M2, 12)
n (M 3,I3) n . . .
c ) b y simultaneous co mbinations of th e tw o logical operations , we
(0)
L
obtain sieves which
{ (M i l I l l ) {( . . . )}
=
,
n
n ... } u
can
be very com p lex;
(M 12,1 1 2) ko
2:
n
... }
U
{(M 2 1 , 12 1 ) n (M22, 122)
k(i)
( II )
i= I I
of pan ;. betwee n curly b rackets should fu rnish a final pair, if it exists. The final pairs will be combined by their u nion, which will provide the desi red sieve.
The intersectio n of each set single
Now let us examine the rigorous formulation of the calculalion o f IJtc intersection of the two moduli (M l , 1 1 ) and (M2, 12) where the periods M l and M2 start from some I I and 12 respectivel y . First I I and 1 2 are red u ced by taking their moduli i n relation to M l and M 2 , I l M OD(T l , M l ) and 1 2 = MOD (I2, M2).s =
The
(1)
first coincidence will
S
=
where A. and
1 1 + A. *
a
Ml
=
eventually a pp e ar
12 + a * M2
arc ele ments of N, and i f M l
=
D*
D equal to the largest commo n denomi nator, Cl the p e rio d
follows :
M3
of the coinci dences will be:
M3
at a d is ta nce:
=
Cl and
an
d
D * Cl
- 12 (a * D * C2) - (.l. * D * C I ) and (a * C 2) - (.l. * C I ) . (I I - 12) I D
II
=
=
M2
=
D *
C2 w i th
C2 being copri me, the n *
C2. From ( 1 )
tJt c re
Form alized Music
272
the expression on the right o f the equal sign is a whole si g n should also be a whole nu mber. But, i f 1 1 - 12 is not divisible by D (for s o m e II , 12), then, there are no coi ncidences and the intersection (M I , I I ) (M2, 12) will be empty. 1 f not: Now, since
number, the expression on the left o f th e equal
(2)
(Il - 12) I D = 1Jf � N and 1Jf = 1Jf + ). * C l = a * C2.
But following Bachet de
a *
C2 - A
* C l , as wel l as :
Meziriac's theorem (I 624), in order for
two coprimes, it is necessary and sufficie n t that there exist
numbers, � and �. suc h that:
x and y to be two relative whole
� * x = � * y or �1*x=�1*y + l
(3)
1 +
where � and � come from th e recursive equatio ns : 1
MOD(� * C21 C 1 ) = I and 6
(4 )
(5)
MOD(�
I *
while letti ng � an d � C l = 1 a n d C2 = I )
Cl, 1
.
C2)
=
1
th ro u gh the successive values
run
(except i f
(6)
=
=
=
1
=
1
13 = MOD((12 + � * (I l - 1 2 ) * C2), M3) o r 13 = M OD((ll + � I * (12 - Il ) * C l ), M3) with M3 = D * Cl * C2 . Example I : Ml 60, 1 1 = 1 8, M2 42 , 12 = 48, M3 6 * 1 0 * 7 = 420, with C1 a nd C2 co p rim e . =
7,
=
=
Fro m (3) and (4) we Fro m
=
3, . . .
C l and C2 are coprime, there follows from (2) a nd (3): A I a = � , a (IJl �� ). I (-"IJl) V a n d ai (-\JI) � and if (M l , 1 1 ) n (M2, 12) (M 3, 13), then
But si n ce
=
0, I , 2,
(6) we
get:
13
get:
=
�1
=
D
=
5.
MOD(1 8 + 5 * (48 - 1 8) * I O , 420) = 258.
Example 2 : M l = 6, 11 = 31 M2 = 8 , 12 = 3, D = 21 C l 24, with C 1 a nd C2 co prime. Fro m
(4) we get: s =
An d fro m (6) we get:
the case th at I 1
=
12, then
6 , C 1 = 1 0, C2
I.
=
3, C 2
13 MOD((3 + 1 * ( 3 - 3 ) * 4), 24) = 3 ; 1 3 = I I = 12, and here M 3 = 2 4 and 1 3 =
=
=
4, M 3
that is, i n
3.
Take th e precedin g example but with 1 1 = 1 3 and 1 2 4 , s o 1 1 i s not e q ual to 12. Since 11 I D = 1 . 5 1 which is not an element of N , there are no coincidences and M 3 0 a nd 13 = 0 . B ut, i f l l = 2 a nd 1 2 1 61 and since (1 1 =
=
- 12) I D = 7 � N,
- 0) *
41
24)
=
8
we
a nd
(4) � = (M3, 13) = (24, 8). obtain from
=
l and fro m
(6) 13 = MOD(O + 1 * (2
Sieves
Computation
of the I ntersection
(M 1 ,1 1 )n (M2,12)
Are given: M l , M2 , 1 1 , 12, with Ii
D
=
the largest comm o n
=
(M3,13)
MOD(Ii, Mi) � 0 denominator of M 1 and M2
=
M3 = the smallest common multiple of M l and M 2
Cl = Ml
I D,
C2
=
M2 I D,
M3
Figure 1 .
=
D*
Cl
*
C2
273
Formalized Music
274
To compute several simultaneous intersections (coincidences) from an (0) of L, it suffices to calculate pairs in that e xp re ss io n two by two . For example:
expression between brackets in the equation the
kO
L
L=
k (i)
= 4
( IT )
i= 1
{ ( 3 , 2) n (4, 7) n (6, 1 1 ) n (8, 7)} u { (6, 9) n ( 1 5, 1 8) } u n (8, 6) n (4, 2)} u { (6, 9) n ( 1 5, 1 9)}, with kO = 4. For the first expression between brackets, we first do (3, 2) n (4, 7) = ( 1 2, 1 1 ), then, after modular reduction of the indices, (1 2, 1 1 ) n (6, 5) = (1 2, 1 1 ), then ( 1 2, I I ) n (8,7) (24, 23). We go on to the following brackets, and so on. =
{ ( 1 3, 5 )
=
Finally,
=
4,
L = (24, 2 3 ) U (30, 3) U (1 04, 70) U (0, 0) for ko k(2) 2 , k(3) 3, k (4) 2. =
=
=
=
4, k( 1 )
Through a co nvenient scanni n g, this l ogical expression will provide us with the points of a sieve constructed in the following fashion: H { ... 3, 23, 33, 47, 63, 70, 7 1 , 9 3 , 95, 1 1 9, 1 23, 1 43, 1 53, 1 67, ... 4 79, .. .J} =
a period of P = 1 560. The zero of th is sieve witlnn the s e t of pitches arbitrarily taken to be c-2 8.25 Hz and at ten octaves, (210 * 8 .25 1 6384 Hz) witl1 u equal Lo the semito ne. It will give us the n otes #D.2J B. 1 1 Ao, B � > with
=
=
can be
#Da. #As. B:v etc. For the
same
tlte series gives
+ B2, ;jlf � C3, etc.
us
zero taken to be C.2
and for u
to b e
the notes ;f/f C.2, +B_2, + E_1 , + B_ 1
equal to a qua!1:ertone, + G0, B 0 , . e//F.. A-1 , + B 1 1
Inverse case
start from a series of p oints either given or constructed deduce its s ym metries; tha t is to say, the moduli an d their shiftings (Mj, Ij), a nd construct the logical expression L describing this series of points. The steps to follow are: Let
us
intui tively a nd
a) ea ch point is considered as a po int of departure ( =
b) to
modulus.
find the modulus corresponding to this begin
point of departure, we 2 unities. If each
by ap plying a modulus o f va1ue Q
o n e of its multiples meets e ncountered and which
a
J n) of a
=
point which has not already been
belongs to tlte
given sieve, then we
S ie ves
275
forms the pair (Mn, I n). But if any one multiples happens not to corres p ond to one o f the points of the series, we abandon i t and pass on to Q + I . We proce ed so until each one of the points in the gi ven series has been taken into account. keep the modulus and it
o f its
c) if for a given Q, we garner all its points (Q, Ik) u nder another p ai r (M, I); that is, if the set (Q, Ik) is included in (M, I), then, we i gno re (Q, Ik) and pass on to the following p oi nt Ik+ J'
d) si mila rly we ig nore all the ,
(Q, I) which, while p roducing some of
the not-yet-encountered points of the given series, also produce, upstream of the index than those
of the given series.
I, some parasitical points other
series H, we will se le c t o n l y the po i n ts 1 67 inclusive. Then we could constmct the follow i ng u nion : L = (73, 70) U (30, 3) U (24, 23), with P 8760 as its period . However, if the same series H were li mited between the points 3 and 479 inclusive, (this time having 40 points), it would be generated by: L (30, 3) U (24, 23) U (1 04, 70),
An exa m pl e : fro m the preceding
between 3 a nd
,
=
=
the modu lus 30
covering 1 6 po i nts
,
the
the modulus 1 04 co ve ri ng 4 poi nts. The earlier. Its
period is 1 560.
In general, to find the
mod u l us 24 cove ri ng 20 points, and function L is identical to that give n
period o f a series of points derived from a logical is the union of moduli (Mj, Ij), it is enough
expression whose de fi nitive form
to
co mp ose the
intersection of the moduli
M1
within the parentheses two by two .
1 2, M2 6, M3 8; M l n M2 = D * C I * C2 6 * 2 * I = 1 2 = M; M n M3 D * Cl * C2 = 4 * 3 * 2 21 . And the period P 2•1. In general, one s ho u ld take i nto account as many points as possible i n order to secure a more precise logical exp re ssio n L . For example:
=
=
=
=
=
=
=
M eta bolae o f Sieves Metabolae (transformations) of sieves can come about in various ways :
a) by a change o f thc i ndi c es of the moduli . For example: L = (5, 1 ) U (3, 2 ) U (7 , 3 ) o f period P = 1 05 w i l l give t.he series: H { . , 2, 3, 4, 5, 8, 9, 1 0, 1 1 , 1 4 , 1 7, 1 9 , 20, 23, 2 4 , 26, 29, =
31,
...
.
.
} . But if a whole n u mb e r
expression L becomes for n =
n
7:
is a d d e d t o t h e i n d ices, the
276
L'
=
L'
=
(5, 1 1 )
th e indices : The series H'
(5, 1 ) =
{
U
Formalized Music
{3, 9) U (7, 1 0) a n d after modular reduction o f
U {3, 0)
...,
U {7, 3),
of the same period
P
=
1 05.
0, 1 , 3, 6, 9, 1 0, 1 1 , 1 2, 1 5, 1 6, 1 7, 1 8, 2 1 , 24, 26, 27.
. . . } derived from this last expression L', h avi n g the same intervallic structure the H series and differing from it only by its initial point, which is given by the smallest i ndex of the expression L' a nd by a shifting n of the i n tervalli c structure of H. Indeed, ifin the series H, the intervals start from 2, which is the index of the smallest modulus o f M, then the same intervals are t o be found starting from 2 + 7 9 within the series H '. This case is what musicians call "transposition" upwards and is part of the techniq ue of "variations." If, on t11e ot11er hand, we add to each index any whole number n, then the intervallic structure of the sieve changes while its period is maintained. For example: add 3, 1 , and -6 respectively to the iliree indices ofL, which becomes after their mod ular reductions: L = (5, 2) U (3, 0) U (7, 4) of period P 1 05, and which gives : H = { . . . , 0 , 2, 3, 4, 6, 7, 9, 1 1 , 1 2, 1 5, 1 7 , 1 8, 2 1 , 2 2 , 24, 2 5 , 27, 30, 3 2 , . . . }.
30,
as
=
=
b) by trans formations o f the logical o peratio ns i n some manner, using the laws of logic and mailie matics, or arb itrarily.
u nity u. For example, sing ilie natio n al anthem, which is based on ilie d iato nic scale (white keys), while tra ns formi ng th e semi tones into quartertones or into eighthtones, etc. If this metabola is used rarely melodically or harmonically, it does however occur in other ch arac te risti c s of sound such as time by changes in te mpo, and this, as far as history c an reme mber.
c) by ilie modification of its
Conclusion
I n provisional conclusion, it will be sai d that sieve theor y is t11e study of i nternal s y m me tries of a se ries of points e i ther constructed i ntui tivel y , given by observation, or invented completely fro m m od u l i of rcpetition.
the
I n what
has
been demonstrated above, the exam p les have bee n take n
from instru me ntal music. But
co
m puter
ge nera ted
sound
it is
to am p litude and/or
quite conceivable to apply iliis theory
synthesis, i m a gin i n g that ilie
the ti me of ilie sound signal ca n be ruled by sieves . The subtJe symmetries
thus created should
open a new field for ex ploration .
Chapte r XI I
Sieves :
a
User' s Guide
I wo uld like
to give credit and express m y thanks to Gerard Marino, a programmer who works with me at CEMAMu . He has adapted my own pro gram which I originally wrote in "Basic" into "C." The program i s divided into two parts:
A
Generation of points
the sieve.
B. Generation
on a straight line.
on
a strai ght line from the logical formula of
of the logical formula of the sieve from
A GENERATION O F POINTS ON A STRAIGHT
a
series
LINE
F R O M THE LOGICAL FORMUlA OF THE SIEVE Example :
D EFI NITI O N OF A SIEVE: L = [() * () * . . . * ()] + [() * () * . . . "' ()]
+
+ [ () "' () * . . . * ()]
In each parenthesis are given in order: modulus, startin g point (taken from the set of integers)
[) + [) is a union () * () is an intet·scction
Given t.hc formula of a sieve made out of unions a n d i n tersections of moduli, the program reduces the number of intersections to one and keeps only the given unions. The abscissa of the final p oints of the si eve are computed from these unions a nd displayed . N UM BER OF U N I ONS ?
=
2
union 1 : number of modules ?
=
2
modulus
start ?
1 ?
start ?
?
modulus 2 277
= = = =
3 2 4 7
of points
Formalized Music
278 union 2: nu mber of modules ?
=
2
modulus
start ?
1
?
modulus 2 ? start ?
=
6
=
9
= =
15
18
FORM U LA OF TH E SIEVE: L = [ [ 3 , 2) * ( 4, 7) ]
+ [ [ 6 , 9) * ( 1 5, 1 8) ]
REDU CTION
O F TH E I NTERSECTI O N S :
union 1
[ (3 , 2 ) * (4,7 ) ) ]
=
(1 2 , 1 1 )
decompression into prime modules ? (press 'y' for yes, any other key for no): =
(1 2 , 1 1 )
*
4,3)
u nion 2
(3 ,2)
[ (6, 9) * ( 1 5 , 18 ) ) ]
=
(30 , 3 )
decom p ression into p rime modules ? (press 'y' for yes, any other key for no) :
(30,2)
=
2, 1 )
*
(3 ,0)
S I M PLIFI ED FORM U LA OF =
L
L( 12,
POI NTS OF
rank
press
11) +
y
*
y
(5, 3 )
TH E SI EVE:
( 3 0, 3)
THE S I EV E CALCULATED WITH THIS FOPRM U LA: = 0
of first displayed point ?
Rank
< e nter >
0
10
20
30
to get a series of 10 points
3 93
95
263
273
275
287
3 63
455
8 03
875
179
11
1 83
23
543
383 46 7
633
55 1
635
647
719
723
743
813
73 1
815
827
887
899
903
911
963
97 1
993
1 053
1 055
983 1 067
1 079
995
1 083
1139
1 1 43
1 151
1 1 63
1 1 73
60
70
61 1
453 539 623
80
693
695
707
1 00
1 10 1 20 130
779 863
947
1 03 1
1 1 15
783 87 3
959 1 04 3
1 1 27
83 1 67
37 1
359
443
90
71 155
213 299
43 1 515 603
63 153
203
347
527
59
143
35
1 23
40
50
47
33
119
107 191
791
13 1 215 303 393 479
563
227 311 395 483 5 73
659
239
323 407 191
57 5 663 753
839 923 1 007 1 09 1 1 175
243
251
333 419 5 03
335
587 755
599 68 3 767
843
851
67 1
933 1 01 9 1 1 03
1 187
423
513
935
1 023 1 1 13 1 1 99
Sieves: A User' s Guide
27 9
Line# Source Line 1 2
3 4
# i nclude < stdiuooh >
#include < sldliboh >
# include < coniuooh >
--5 /* 6 ty pe d ef s truct 7 { a short mod; 9 short ini; 10 } p eriode; ----------
-
-----
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
typedef struct
34
short short
33
35 36 37 38 39 40 41 42 43
44
45 46 47 48 49 50 51
{
short
- - - -- - - - - - - - -
short short
void
---
*I *I
*I
/* n umber of terms in
the intersection /* terms in the intersection
*I *I *I *I
/* resu l ti n g period /* current point value
- - - - - - - - - - - - - - - -- - - -
Reduclnter(short
fu ncti o n p roto typ es - -
------------------
-
--------
u); /* computation of the intersections
Euclide(short m l ,short m2); /* computation of the LCD Meziriac(short cl ,short c2); /* computation of "dzeta" --
--
- -- -
/* sieve formula
'*£Crib;
-
----- - - ----- - --
-- - - -
- --- - -
/* number of unions in the formula
unb = 0;
/* current u n i o n index uO, u 1 , u = 0; i = 0; /* current intersection i ndex unsigned lo ng lastval,nO,ptnb = 0; periode CL_EMPTY = { 0, 0 } t* empty period
short
short
#define NONEM PTY flag = 0; d ecamp
=
0;
*I */ *I
*I */
Decompos(periode prr,' decomposition into prime factors
/* ------------------------------------ variables inter short
------------------ "'I */
/* intersection of several periods
clnb;
---
- - - - - - -- -----
/* modulus of the period /* starting point
unsigned long ptva l ; -- - - -
defi n i tions
/* period ( congruence class )
periode '*cl ; periode clr;
} inter; /* p eriode
types
- *I -
*/ */
*I *I *I
1
!* = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = � void main(void)
{
printf("SI EVES : user's guidc\n\n"
"A. GENERATI ON OF PO INTS ON A STRAIGHT LI N E " TH E LOGICAL FORM U LA O F TI-l E Sl EVE\n\n"
FROM\n"
"Example:\n'
"---------------------------------\n .
"DEFI N ITION OF A S I EVE :\n" " L = [() * () * o O O * () ] \n" + [() * ( ) * 0 0 0 * ()]\n" + ooo\n" + [() * () * 000 * ()]\n\n" "In each parenthesis are given in order: modulus, starti n g poi nt\n " ' (taken from th e set ofintegers)\n"
Formalized
280 Line#
52 53
54
55
Source Line "[] + [] is a union\n" " () * () is a n intersection\n\n"); prin tf(" -------- - - - - - - - - - ------------- - - -\n "
"Given the formula of a sieve made out of un i ons and\n" "intersections of mod u l i , the program red uces the number of\n" "i n tersecti o ns to one and keeps only the given u n ions.\n"
56 57
58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93
94 95 96 97 98 99 1 00 101 1 02 1 03
Music
"Th en, the abscissa of the final points of the s iev e are\n"
"computed from these unions and displayed.\n\n"); I* -------------------- get the fo rm u l a of the sieve ------------------------ *I 0) while (unb = =
{ printf("NUMBER OF UNIONS ? = " );
scanf{'%d",&unb);
}
=
(inter *)(malloc (sizeof(inter) if (fCrib = = NULL) fCrib
"'
unb));
{
p rintf("not eno u gh rne m ory\n " ) ;
exit(l); }
prin tf("
for (u
{
\n " );
---------------- - - - - - - · - ---------
=
0;
u
unb; u + + )
printf("union %d: number of modu l es ? = ", u + 1 ); scanf("%d",&£Crib[u].clnb); p ri n tf("\n " ) ; fC r ib [u ] . cl = (periode *)(malloc (sizeof(periode) * fCrib[u] .clnb));
if (fCrib[u].cl = = N U LL) { printf("not enough memory\n"); exit( 1 ) ;
}
for (i = 0; i fCrib[ u J .clnb; i + + )
{
printf("\n
modulus %d ?
sca n f("%d " , &£Crib[ u ] .cl [i ] .m od ) ; start
printf("
?
=
scanf("%d",&:£Crib[u] .clli] .ini);
}
=
•,
i + 1);
");
printf(" ---···· ··-------·------- · ··----\n ");
}
I* ········-······ reduc tion of th e formu l a ····· ------- · ·· -- ····· ······ *I printf("FORMULA OF THE SIEVE :\n\n" •
for (u
{
=
L
=
[ " );
0; u unb; u + +)
if (u l = 0) + [ "); printf(" for (i 0; i fCrib[u].clnb; i + +)
{
=
if (i ! = 0)
{
Sieves: A User's Guide
28 1
Line# Source Line
if (i % 4 = = 0) printf("\n
104 1 05
106
}
1 07 1 08
1 09
l l0 111 1 12 1 13
1 14
115
116 1 17 118
1 19 120 12 1 1 22 123 1 24 1 25 1 26
1 27 1 28
printf("(%5d,%5d)
}
134
135 136 137 138 139 140 141 1 42 1 43 1 44
145
146 147 148
fCrib[u] .cl[i] . m od, fCrib[u) .cl[i].ini);
}
printf("----------------------------\n"); printf("REDUCTION OF TH E INTERSECTIONS :\n\n"); for (u 0; u unb; u + +) =
{
printf("union %d\n [ ',u + 1); for (i = 0; i fCrib[u].clnb; i + + )
{
printf("(%d,%d) •, fCrib[u] .cl[i] .mod, fCrib[u].cl[i].ini); if (i I = fCrib[u].clnb - 1) printf("* " ) ;
}
fCrib[u] .clr = Reduclnter(u); /* reducti o n of an intersection */ p rin tf(" ] = (o/od,o/od)\n\n", fCrib[u] .clr.mod, fCrib[u).clr.ini);
decomposition into prime modules ?\n" (press 'y' for yes, any other ke y for no): " ); if (getche() = = 'y')
printf(" "
{
pr intf("\n\n (o/od,%d)", fCrib[u].clr.mod, fCrib[u].clr.ini); Decom p os(fCrib[u] .clr);
1 30 1 32 1 33
•,
printf("]\n");
1 29
131
");
printf("* " );
}
else printf("\n\n");
}
prin tf("·· ······--· · ····- ·-- ·· ······ · -··\n"); / * ······---------- displa y the s i m p li fi ed form u la ·····--······--······· *I
printf("SI MPLIFIED FORMULA OF TH E SIEVE:\n\n"); pri n tf(" L ") ; for (u 0; u unb; u + + ) =
{
=
if (u ! = 0)
{ if (u % 4
= =
0)
printf("\n
printf(" + " ) ; }
"
printf("(%5d,%5d) }
); ",
fCrib[u].cl r.mod, fCrib[u] .clr.ini);
150 151
printf("\n· ··----- - ------·-····-···- ······\n " ); {* ---·--·· · ··- --- ------ p oints o f th e sieve ·- ·· · ·-······· ·· ······· -------- ---· *I p rintf("POINTS O F THE SIEVE CALCULATED WITH THIS
152 153 154
scanf("%lu",&n 0); nO = n O - n0 % 1 0;
14 9
FORMULA:\n"); pri n tf(" ra n k of firs t displayed poi n t ?
=
"
);
Formalized Music
282 Line # Source Line 155 printf("\npress <enter> 156 "Rank I ");
for (u {
157 158
{ fCrib[u].ptval = £Crib[u] .clr.ini; flag = NONEMPTY;
161
1 62 1 63
}
else
1 61 1 65
£Crib[u] .ptval
1 66 1 67 1 68 1 69 170
return;
uO = u 1 = 0; lastv a l = OxFFFFFFFF; while ( 1 )
{
173
175 1 76 177 178 179 1 80 181 1 82
{
}
ul
(u + 1 ) % unb)
= u;
if (£Crib[u 1 ] .ptval != lastval) /* new point */
{
lastval
=
if (ptnb
{
£Crib[u1] .ptval;
=
nO)
if (ptnb % 10
= =
0)
{ getch(); /* get a cha ra cte r from the keyboard printf("\n%7lu 1 ", ptnb); } printf("%6!u fCrib[u 1] .ptval); }
1 84 185 1 86 1 87 188
*/
",
1 89 1 90 191
ptnb + + ;
}
fCrib[u 1 ] .ptval + = fCrib[u 1 ] .clr.mod;
1 92
193 1 94
uO = u 1 ;
}
1 95
206
=
if {!Crib[u ] . ptval £Crib[ u 1 ] . ptval)
1 83
204 20 5
(u = (uO + 1 ) % un b ; u ! = uO; u
for
174
202 203
= OxFFFFFFFF;
} if (flag ! = NONEM PTY)
171 172
1 97 1 98 1 99 200 201
0; u unb; u + + )
if (£Crib[u].clr.mod I= 0 I I £Crib[u].clr.ini I = 0)
159 1 60
196
=
to get a series of 10 points\n\n"
= = = = = = = = reduction of an intersection = = = = = = = = */ periode Reduclnter{short u)
/*
{
periode cl ,cl 1 ,cl2,cl3 ; short pgcd,T,n; long c l ,c2; = fCrib[u].ci[O]; for {n = 1; n fCrib[u] .clnb; n + + )
cl3
{
el l
=
cl3 ;
283
Sieves: A User's Guide
Line# Source Line 207 cl2 =
fCrib[u].cl[n];
208 209 210 21 1 212 213 214 215 216 217 21 8 219 22 0
if (el l . mod cl2. mod)
22 2
else return CL EMPTY; /* module r;;sulting from the in tersection of 2 mod u l es */
225 226 227 228 229
pgcd = Euclide(cl l . mod, cl2.mod); cl = cl l . mod I pgcd; c2 = cl 2 . mod I pgcd; if (pgcd != 1 && ( (cl l .ini • cl2.ini) % pgcd ! = 0
return CL EMPTY; i f (pgcd ! = l
{
cl = el l ; el l cl2 ; cl2 = cl; =
}
if (el l . mod ! = 0 & & cl 2 . m od ! = 0)
{
cl l .ini % = cl l .mod;
cl2.ini % = cl2.mod;
}
221
223 224
&& ((cl l .ini - cl2.ini) % pgcd = = 0) && (cl l .ini ! = cl2.in i ) && (c l = = c2) )
230 23 1
{
23 2
continue;
234 235
}
23 6
T = Meziriac((short) c l , (short) c2); cl3 . mod = (short) (cl * c2 * pgcd );
237
cl3 .ini = (short) (( cl l .ini + T * (cl2 .ini - cl l .ini) * c l ) % cl3 .mod); while (cl3 .ini cl l .ini I I cl3 .ini cl2.ini) cl3 .ini + = cl3 .mod;
238
239
240
246
247
248 249 250
251
252 253
254
255 256
257 258
=
cl3 .mod pgcd; cl3 .ini = cll .ini;
233
24 1 242 243 244 245
))
}
} return cl3;
/* = = = = = = decomposition into a n i ntersection = = = = = = = = /* of prime modules */ void Decompos (periode pr)
{
periode pf; short
fct;
if (pr.mod = = 0)
{
printf(" return;
=
(o/od,o/od)\n", pr. mod, p r . i n i ) ;
}
prin tf(" ="); for ( i = 0, fct
=
2; pr.mod
! = l; fct+ + )
*I
Formalized Music
284 Line# Source Line
{
259 260
pf.mod = 1 ; while (pr.mod % fct = = 0 & & pr.mod I =
26 1
{
262 2 63
pf. mod • = fct; p r . m od / = fct;
2 64
}
265 2 66
if (pf.mod I = 1 )
{
267 268 269
pf.ini = pr.ini % pf.mod; pr .i ni %= pr.mod; if (i ! = 0)
27 0 27 1
p rintf(" *");
printf(' (o/od,%d)', pf.mod, pf.ini);
272
273
27 4 275 276
277 278 279 280 281 282
283 284
1)
i++;
}
} } printf("\n");
/"' = = = = = = = = = = Euclide's algorithm = = = = = = = = */ short Euclide (a l , a2) /* a l = a2 0 */ short a l ; short a2;
{
short
tm p ;
while ((tmp
285
=
{
286 287
a 1 % a2)
! = 0)
al = a2;
a2 = tmp;
288
}
289 290 291
}
292
/* = = = = = = = = De Meziriac's theorem = = = = = = = = = */
293
294 295
296 297 2 98 299 300 301 302 303 304 305
return
a2;
short Meziriac short c1; short c2;
{
=
c2
/* c1
((( + + T • c l ) % c2)
!=
0 */
short T = 0; if (c2 = = T = 1;
else while '
}
(c1 , c2)
return T;
1)
1)
285
Sieves: A User's Guide B. GENERATION OF THE LOGICAL FORMULA OF
THE SIEVE FROM A SERIES OF POINTS ON A
STRAIGHT LI N E
Example:
Given a s er i es of points, find t h e starting p o ints
(periods).
=
NUMBER OF POINTS ?
o f th e poi nts point ! = 59
abscissa
point 5 point 9
=
63
= 95
0
12
p oi nt 2 = 93 p oint 6 = 1 1 point 1 0 = 7 1
poi n t 3 = 47 p oint 7 = 23 p oint 1 1 = 35
point 4 = 3 point S = 33 p o i n t 12 = 83
(ordered by their increasing abscissa) :
POINTS OF TH E S I EVE
Rank
with their moduli
3 1 1 23 3 3 3 5 47 5 9 63 7 1 83
1 0 93 95
FORMULA OF THE SIEVE:
In each
parenthesis are given in
order:
(modulus, starting po i nt, n u m be r of covered L = ( 30, 3, 4 } + ( 1 2 , 1 1 , 8)
period of the sieve: P
=
60
Line# Sourr;e Line 1 #incl u de < stdio.h > 2 # include < stdlib.h> 3 #include < strin g.h> 4 #inc l u d e < string . h > 5
6 7 8 9 10 11 12 13 14 15 16 17 18 19
20 21
22 23 24 25
I* --
- - - - - - - ---- - --------- - - - --
typedef struct
{
points)
types defini ti on s --------------- - - - - - - ---------------- *I I* period ( congruence class ) */
/* modulus of the period I* starti n g point
short mod; short ini;
r number of cov ered points
short couv;
} periode;
I*
- - - -- - - - - ---- - - - -- - - -- -- -
I*
----- - --- --------------
fun ctio n p ro totyp es
u nsigned long Euclide(unsigned long unsigned
periode"perCrib;
long
short perTotNb l on g
long
*ptCrib;
ptval;
=
- - -- - ---- - - ----- - - - - - - - ---- ---- - - --- -
m2); /" computation of the LCD
i bl es a nd con sta n ts ---- --- - --- -- -- ---- -- -- -- - --- --I* p eri o ds of the sieve /* number of p e ri od s i11 the formul a /* p oi nts of the crible /* points outside the periods /* number of points in the sieve
var a
0;
*ptReste;
short ptTotNb short p, p tnb; long
=
m l,
0;
unsi gned long percrib;
*I *I *I */
*I
*/
*/ */ */ */
286
Formalized Music
Line# Source Line
26 27 28
29 30 31 32 33 34
35
36 37 38 39 40 41 42 43
periodc;per;
#define NON REDUNDANT 0 #define REDUNDANT l # d e fine COVERED -lL
short
flag;
/* void m a i n (voi d) { printf("B. GEN ERATI O N
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
OF TH E LOGICAL FORM U LA OF TH E
SIEVE FROM\n"
" A SERIES O F POI NTS "Example:\n"
ON A STRAI GHT LI N E\n\n"
• ----------------------------
\n . "Given a series of p o i n ts , find the starting points\n"
"with their moduli (pcriods).\n\n"); --------- en try of the p o i n ts of the sieve and their sorting
/*
44 45 46
while (ptTotNb 0) { printf ("NUMBER OF POINTS
47
ptCrib
48 49 50 51
52
53 54 55
56 57 58
59 60 61 62 63 64
65 66 67 68 69 70 71 72 73 74 75
*I
= =
-------*/
? = ");
scanf("%d",&ptTotNb);
}
ptReste
= =
( l ong *)(malloc
(ptTotNb
(long *)(malloc (ptTotNb
= =
{
*
sizeof(long)));
sizeof(long)));
*)(malloc (ptTotNb * sizeof(periode))); NULL ) NULL I I perCrib NU LL I I ptRcstc
perCrib = (periode
if (ptCrib
*
= =
= =
printf("not enough memor y\n");
exit( I);
}
printf(" -------------------------------\n "abscissa of the points:\n"); 0; p p tTo t Nb ; p + + ) for (p
n
=
{ if (p % 4
= =
printf("\n
0) " );
printf("p oint %2d
=
scanf("%ld", &ptval); for (ptnb = 0; ptnb p && ptval p t nb + + )
",
p + 1 );
ptCrib(ptnb];
I
if (p tnb p) {
if (ptval ptCrib[ptnb])
/* new point 1], &ptCrib[ptnb],
memmove(& p tCrib[ptnb + sizeof(long) * (p - p tnb) ) ; else /* point already exist */
{
p -;
*I
Sieves: A User's Line#
Source Line
ptTotNb--;
76
77 78 79 80 81 82
83 84
85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
100 101
1 02
103 104 105 106 107 108 109
}
}
}
ptCrib[ptnb] = ptval;
prin tf("\n ------------------------------\11 " ) ; /* --------------------points of the sieve --- ------------------ ---------- *I printf("POINTS OF THE SIEVE ( o rd ered by their i n creas ing
abscissa):\n\n for (p
"Rank =
{
0;
--------------------------------
-------------
for (p
{
=
0; p ptTotNb; p + +)
if ( ptRcste[p)
= = COVERED ) continue; I* - -- compute a pe riod star ti ng at current point--------- *I pcr. m od = 0; - --
--
do
{
per.mod++; per.ini (short) (ptCrib[p] % (long)per.mod); per.couv = 0; for (ptnb 0, ptval = per.ini; ptCrib[ptnb); ptnb ptTotNb &Be ptval tnb++) =
=
if (ptval
{
= =
ptCrib[ptnb])
per.couv++; += per.mod;
ptval
114
120
=
r
113
121 122 123 124 125 126
==
0) printf("\n%7d I", p); printf("%6ld " , ptCrib[p]); } \11" ); prin tf("\n\n /* compu te the periods of the sieve------------------- *I memcpy(ptReste, ptCrib, ptTotNb * sizeof(long));
112
1 19
I");
p ptTotNb; p+ +)
111
118
•
if(p% 10
110
115 116 117
287
Guide
}
}
}
while (ptnb ptTotNb); /* - --- -- check the redundancy of the period - - - - --- - *I for (ptnb 0, ptval per.ini, flag = REDUNDANT; ptnb ptTotNb; ptnb++) -
-- -
--
-
=
{
if (ptval
=
= =
{ if(ptval {
ptCrib[ptnb])
==
ptReste[ptnb))
- --
--
-- -
Formalized Music
288 Line# 127 128
Source Line
129 130 131 132
ptval
}
perc:rib = perCrib(O].mod;
for (p =I; p perTotNb; p++)
{
if ((long) perCrib[p].mod = percrib) perc:ri b *= (long) perCrib[p].mod I Eudide((long)perCrib[p].mod, perc:rib); else perc:rib *= (long) perCrib[p ].mod I Eudidc(percrib, (long)perCrib[p].mod);
142
143
}
144
I*----------------- display the formula of the sieve----------------------- *I printf('FORMULA OF THE SIEVE:\n"'In each parenthesis are given in order:\n" '(modulus, starting point, number of covered points)\n\n"); printf(" L = ") ; for (p = 0; p perTotNb; p + +)
145
146 147
148 149
150
151 152
{
if(p != 0)
{
153
154
if(p %3 ==
155 156 15 7 158
167
168 169 170 171 172
173
174 175
NON_REDUNDANT) = per;
perCrib(pcrTotNb+ +]
I*----------------- compute the per i od of the sieve---------------------- *I
141
165 166
+= per.mod;
} if(flag ==
134 135 136 137 138 139 140
162 163 164
}
}
133
159 160 161
ptRcste[ptnb] = COVERED; flag = NON_REDUNDANT;
0)
printf("\n printf('+ ;
}
'
);
printf("(%5d,%5d,%5d) , perCrib[p ].mod,perCrib[p ].ini, perCrib[p].couv);
}
'
} printf("\n\n
period of the sieve: P = %lu\n', perc:rib);
I*=========== Eudide's algorithm ============= */ *I uns igned long Eudide (al, a2) I* a1 = a2 0 unsigned long a1; unsigned long a2;
{
unsigned long tmp; while (( tm p
{
=
a1 = a2; a2
}
=
tmp;
return a2;
a1 % a2) I= 0)
Chapter XIII
Dynamic Stochastic Synthesis What is
the
most economical way
to
create a
pl ane
in an amplitude all possible forms
wave
time space (atmospheric pressure-time), encompassing
From an infor ma ti cs point of view, a with only two amplitudes, ± a over n of fixed samplings. White noise is also quite simple and generated by a c ompound of stochastic functions whose samplings are dove tailed , nested, or not. from a square wave to white noise?
square wave is quite simple
But
what about
The
foundation
sounds
..
.?
waves represe nti ng of tbeir
nature
melodies, symphonies, natural
and
therefore
intelligibility is te mporal periodicity and the symmetry
brain can marvelousl y detect, w ith a fantastic precision, dynamics,
polyphonies,
of their
of
mel od ies timbres,
as well as their complex transformations
of a curve, unlike the eye whic h
fast mobility.
has
difficulty perceiving
human
the curves . The
a
,
in the
form
curve witl1 such
a
is to begin Brownian movement) constructed from varied distributions in the two dimensions, amplitude and time (a, t), all while injecting periodicities in t and symmetries in a. If the s ym me tries and periodicities are weak or infrequent, we will obtain som ethi ng close to white noise. On the otlter hand, the more numerous and complex (rich) the symmetries and perio di c ities are, the closer the resulting music will resemble a si mple held note. Fo llow i ng these principles, the whole gamut of music past and to come can be approached. Furthermore, the relationship be tween the macroscopic or microscopic levels of these injections An attempt at musical synthesis according
from a
probabilistic
to
this orientation
wave form (random walk or
plays a fundamental role.
Below, is a first approach to constructing such a wav e
.
Procedure AI. Following
the absciss
1/f seconds and f is
T
=
is
subdivided
into
a
a l ength (pe ri od ) T where start, dtis pe riod T example, n 12 (this is one
oft, we be gi n with
freely chosen frequency. At the
n equal segments ; for 289
=
Formalized Music
290
macroscopic leve l) . Every timeT is repeated, each segment 2, 3, ... ,n-1) undergoes
a
tr�.1 of (i = 0, reduces
stochastic alteration which increases or
within certain limits imposed,
for example, by elastic barriers.
1,
it
Bl. Following the amplitude axis, a value is given
to each extremity of preceding segments. These values form a polygon inscribed or enveloping a sine wave, or a rectangular form, or a fo r m born of a stochastic function such as that of Cauchy, or even a polyg o n flattened at the zero level. The Ej ordinates of these n summits undergo a stochastic alteration at each 12
the
repetitio n which is sufficiently weak and even more, compressed between two
adequate elastic barriers.
Cl. The
E
ordinates
of the
samplings
found between
T will be calculated by a linear ordinates Ei-land Ei of these extremities. extremities of a segment
A2. Abscissa of the
0
I I
I
i-t
ti-l
t
o,
I I
t;
� Figure
prec(eding)
=
�- G.1
;e
pre s (en t)
=
1.
t'1- t'1•1
i-1
t:·_,
c'
two
of the
polygon's summits
7'
o
the
interpolation
ti r' -- ---,I
Dynamic Stochastic Synthesis
291 Procedures
Construction of(J present from "logistic"
(1 ) U
distribution:
- t;- f3 �) = _a_e_a.,... -= ..-----u-a' /3) 2 (1 + e
and its distribution function,
F�)= we obtain
_ tu�)dl; = 1+EXP(�
a,-!3)
l;=- ({3 + ln(.!..::l.)) I a withy coming from the
uniform
y
distribution:
osysl.
(2)
take: l; pres
(3)
=
l; prec
+
l;
l; pres into local elastic barriers
Pass this
± 1 00 taken from {3/2, to obtain l;':
(4)
Then
() pres
do:
=
e
prcc
l;'*Rdct
+
where Rdct is
(5)
Finally
reduction factor.
pass() pres into general clastic barriers Omin and() max
obtained
a)
a
as
follows:
the minimum frequency is, say 3HZ. Then the maximum
period is T
=
i sec and each
length ofOmax
=
3
I
*
12 segments will have
12 sec.
b) The maxi mum SAMP is
ofthe
frequency cou ld be
S�P l-IZ where
the sampling rate, say 41100 HZ. T h e re fo re each of a minimum length of the period
12 segments could h av e T
12 ( 6)
=
a mean
the
1
() SAMP = min.
Repeat the above
procedures for
each of the n
=
12
segments.
Formalized Music
292
B2. Ordinates of the Polygon's summits
i
i- 1 The
Figure 2.
ith
present ordinate is o btai ne d from t11e
following manner:
ith preceding
ordinate in the
Construction of the Ei pres:
(1) Take a probability distribution W(o); ilien its distribution
function Q(W)
0
S y s 1
=
{'w (a)da. We obtain
-00
a=
V(Q, y)with
(the uniform distribution) and W(a) any distribution.
(2) Pass a ilirough local barriers (± 0.2) (3) Add this a to the Ei prec. ·
Ej pres
(4) Pass
=
Ej prec + 0
Ej pres through limitative barriers± 8 bits(±
32768) and
this is the final Ei pres.
(5) Do tltis for each of the I 0 summits within the two boundary summits of the polygon.
(6) The last boundary summit will be taken
summit of the next period.
as
t11e first boundary
C2. Construct the Ej ordinate of the sampling point t which ca n be found on the segment tj tj_1 between the ordinates Ej_1 and Ej tllrough a linear inter -
polation.
293
Dynamic Stochastic Synthesis _
E l-
Therefore,
a
(E;- Ei- 1) (tti- t;- l
li-
1)
microscopic construction.
General comment: the
distribution functions U(�) an d W(a) can be ; or more complex,
either simple, for example, sine, Cauchy, logistic, through nesting, etc.
The data given above is naturally
used
in La ligende d'Eer. This
approach to sound
synthesis
an
arbitrary
represents
...
starting point a
n
which I
o n l i ne a r dynamic -
stochastic evolution which bypasses the habitual analyses and harmonic
syntheses of Fourier since it is
app lied to the
f(t) part on the left of the equal
can be compared to current research on dyna mic systems, deterministic chaoses or fractals. Therefore, we can say that it bears the seed of future exploration.
sign of Fourier's transformation. This approach
Chapter XIV
More Thorough Stochastic Music
Introduction
This chapter deals with a generalisation of sound synthesis by using not periodic functions, but quite the opposite, non-recurring, non-linear functions. The sound space in question is one which will produce a likeness of live sounds or music, unpredictable in the short or long run, but, for example, being able to vary their timbre from pure "sine - wave" sound to noise.
Indeed, the challenge is to create music, starting, in so far as it is possible, from a minimum number of premises but which would be "interesting" from a contemporary aesthetical sensitivity, without borrowing or getting trapped in known paths. The
ontological
ideas behind tJ1is subject have
already been exposed I and V) so me 33 to this new, somewhat
in the chapters tre ating ACHORRIPSIS (cf. chapte rs
years ago, and still form the background canvas
more thorough scope, which should result in more radical experimental solutions.
If, at that time, the "waves" in the "black universe" were still produced by musical instruments and human beings, today, these "waves" would be produced mainly by probability distributions (adorned with some
restrictions)
and by
computers.
in front of an attempt, as objective as possible, of creating an automated art, without any human interference except at ilie start, only in order to give ilie initial impulse and a few premises, like in the case of the Demiourgos in Plato's Politicos, or of Therefore,
Yahweh Theory.
in
we find ourselves
the Old Testament,
or even
295
of Nothingness
in the Big Bang
Formalized Music
296 Microstructure
The fundame ntal ingredients Legende d'Eer) four in number :
used are (al most
like in
A tem poral fi c tit i o us le ngth divided i nto
a)
segments, at whose
en ds we
polygonal wave-form
(PWF);
a
the
case
of
La
given number of
draw amplitudes in order to fo r m
a
stochastic
b) As a matter of fact, this polygonc is built co n ti nu ousl y and endlessy through the help of probability distributions by cumulatively varying tem poral lengths as well as the amplitudes of the vertices; c)
In
imposed;
o rde r to avoid excessive cumulated values, e las tic barriers are
d) A linear interpolation joins the vertices.
Under certain conditions,
undeterministic, pro d uc e s
a
this
proce d ure,
relatively stable sound.
although
chaotic
and
The computation of the stochastic p o l ygo nal wa-veforms uses one
stochastic la w
that governs the amplitudes and a n oth e r one that governs the durations of the time-segments. The user chooses a m o ng several disc tinct s tochas tic laws (Bernouilli, Cauchy, Poisson, Exponential. .. ) . The sizes of the elastic-mirrors that are applied to the amplitudes a n d the dura tions can be chosen too.
Macrostructure
A)
The
duration;
pre ceedi ng pro ced u re therefore
p rod u c es
a
sound of a certain
B) A sequence (PARAG(psi%)) results from a simultaneous and temporal multiplicity of such sounds. This sequence is equally constructed through
decisions gov er n ed by probability distributions;
C)
An arbitrary chain
mus ical composition.
of
such sequences could
produce
an
interesting
DATA of the sequence
PARAG(psi%}
The end-figures of the dyn%-routes
(here, up to
16 arbitrary routes)
For each dyn%-route are defined :
arc
gJVen by dynMIN% :s; dynMAX%
1) The n um be r I max% of segments for the polygonal 2) The number of sound fields per dyn%-routc,
wave-form
(PWF),
More Thorough Stochastic Music
3)
The coefficient
of
297
the exponential distribution which stochastically
governs the sound or silence fields of this 4)
dyn%-route,
The probability (Bernoulli distribution) by which s ound field,
a
field becomes
a
5) Various digital filters,
6)
Two stochastic
laws
that govern the
amplitudes (ordinates) and the polygonal wave-forms,
intervals (durations) of the vertices of the successive (at least six d i sti nct stochastic laws are introduced),
7) If needed, two laws,
8)
a) The sizes
(ordinates),
numerical coefficients
for each
of Lhc first two elastic-mirrors that
of the
previous stochastic
are used for the amplitudes
b) The sizes of the first two elastic-mirrors that are used for the abscissa (time), c) The sizes of the second two elastic-mirrors that are used for the
amplitudes (ordinates),
d) The sizes of the second two elastic-mirrors that a re used for the abscissa (time), Proportional corrections of the mirror-sizes iu order to avoid an 9) overflow(> 16 bits) per sa�ple, 10) For all the dyn%-routes of this PARAG(psi%) sequence, a stochastic computation (through expo n en tial distribution) of the sound or silence fields is carried out, determining namely their starting points and their durations.
Formalized Music
298
TABLE of arbiu-ary su«C:S3ion of rARAG(psi%) ·sequences with: f"p%-= ordinal number of the sequence; J»i% = specific nu mber of a seq���:_ __
I
I
pso%
�
�
Q(yspMin% +
J.. ...
j'lb)
I
l
DATA ofsequ.PARAG(psi%)
1
CONSTRUCTION of .a P/I.RAG(('l"i%) sound-sequence thnnk!i to a �imult.aneous and temporal stochastic multiplichy made up with the contribution of several given dyn%-routes.
.. •
+ dJ"'lb�dynMin%
---
dJ"% � dJ"Min%+1 CONTRI BUTtON of the dyn%-routc by means of the :�ub-routinc DYNAS(dp1%), that is, slOCha.sUc comu-uct.ion or suecessive polygonal wavc:-foriil!l (PWF)
DYNAS(d]"%)
�
ABSCISSA (time) of a proenl v�tex
l�oosc. according to PARAG(psi%) the """haatic distribuuon g(x) 2. From the uniform dislr. wt
1. Choost:: , according to PARAC(p3i%} the stochastic distribution f(y). 2. From the uniform cl.istrib. we draw O:s;;Zt s 1. Then from the
F(y)_=/f(y)dy
I
�
Zl
we draw Yo an amplitude (ordinatr) inttease of this vertex. 3, Thi5 Yo is then taken through a pair of elastic mirrors. 4. The result is added to the amplitude (ordinal-=) cfthe same vertex or th� preceding PWF. 5. Thi! new value y {after it has) been uk� through a 2d pair of elastic mirron) gives the
I
y'
,.
•
I
draw d � Z2 S I. The from the dislribution function. F(x)
we draw
� �- g(x)dx
1
WF.
=
I
Z2
an inaca:Y: Xoofthe
interval (duration) th;at separ,alrS this present vertex from its predecessor. :S. This Xo is taken thr o ugh a pair of clastic: mii'TCI'S. 4. The te§Wt x'isaddcd torhe interval (duration)
amplitude (ordinate) of the resent venex of the actual
�
---
�
.. AMPLITUDE (cmlinato) of a pnsent vatcx.
di.slrib. function
---
---
that was
sep.arating same vcnia:s in the preceding PWF. 5, Titis new valUIC X (a(ter it has been taken through .a 2d pair or elastic mirror'S) gives tllc intcrv.al (duration) thatsepJor.ald the procnt vertex from the preceding on e of the actual PWF•
•
sample by sample linear intcrpob.tiom of the PWF ampli tu do (ordinate) that .are separated by this x interval (duration) 44100 samples/sec.
:
...
arbitrary digital faltering of the amplitudl5 (ordinates) and/or or the durations.
•
•
-- -
I
--
---
---
the dyn%-routcs oC' this sequence PAR.AG(psi%) and as soon as lhe above computation for just one Having w;ed all conuibutioll!l of sample is ended, then :
the
PARAG(psi%)_-scquem:�
ende::l,
we compute anew the contribution or each one of are not samples of this a. If the dynlf'e>-routo for the next sam ple, by repoung the above procedure.
h. Jfthesamples of this PARAG(psi%�uence are ended, then an"' inU:oduettl in the main programme the: DATA of the: . next PARAG(psi%)-5equence (an agreement w1th the TABLE ofthf!'..PARAG(psl%)�uentes) and we repeat rhe previous procedur e. n been computed then the tasl:. and the mUsic arc lermin.:ned. c. l(the last PARAG(psi%l·seque ce of the TABLE has
More Thorough Stochastic
---
�
Jy,.%=d]"Mox'A\-2 DI'NAS(dJ"%)
..
I
..
•
299
Music
...
d]"%•d]"MGJC- I
DI'NAS(dyn%)
•
..
..
..
...
...
•
•
�
---
I
• dyn%=4J"MGX% DI'NAS(dyn%)
....
..
I
Formalized
298
Music
fABLE of arbitrary s.ll«'C5Sion of PARAG(psi%) • �u�nccs
I
a
with: ysp% • ord i n l numbc=rofthc= sequence; p!.i'% = ���c:ific number of a seq�!_n��--
I
p>;%
�
Q(y>pM;n% +
j%)
I
r
DATAofsequ.PARAG(psi'JI&) I
CONSTRUCOON ora PARAG(p."i,;,.) �und���cr thanks to a aimuh.aneous and temporal stochast.IC mult1phc1ty made: up wilh. � conlribution or�enl glv�n dyn�rou�.
� dyn1�l,o:::::ciynMin% DI'NAS{dJ"%)
J.
dJ"% =J_,..Min%+1
I
CONTRIBUTION of�dyn%-routc= by means of the
successive polygona l wave-forms (PWF)
� AMPLITUDE (ordinate)
* ABSCISSA (time) of a proent vatcx
oC a Dec5eRt v�x.
accord ing to PAilAG(pi%) tho siOchaotic
Choose, according to PARAG(psi%) the stcdJastic
l. Choose,
distribUllon g(x)
dUtribution f(y).
2. From the unirorm distrib. we drawO:SZIS !.Then lrom the distrib. function
��
F(y) _
y)dy
=
I
Zl
we draw Yo an amplitude (ordi· nate) inc:rei\� of this VcrteK.
..
2. From t.he uniform distr. we drawdsZls 1. The from the dislrtbvtion runction. F(x)
we draw
=
j" •g(x)dx - Z2
intt:="rval (duration) that M:pa-
3. This Xo is takr:n Lhrough i!l pair of elastic mirrors. 4. The result x• is 2odded 10 the interval (duration) that was separating same vertices in the preceding PWF. 5. Thi� new value x (.altr:T it has been taken through a 2d pair of clastic mirron.) gives the interval (duration) that separates the present vertex from the preceding one of the actual PWF. ....
�
sample by sample linear interpo ti o�s of the PWF a mplitudes . (ordinate:) that are sepilratcd by thiS It mtc:rval (durauon) : 44100
samples/!.CC.
I
a�� Xo of the
r.ato: this present vertex From its pn:dcco.10r.
3. This Yo is then taken through a pair of r:la3tir: mirron. 4. The result y' is added to the: a mpl itude (ordinate) of the same vertex of the pr�ing PWF. 5. Thill n8# value y (after it has) been �ken through a 2d pair of dasdc mirron) gives the amplitudt (ordinacc)ofthe pr�nt vertex of the actual PWF.
�
---
5Ub�routine DYNAS(d)"'%). tha� ia, sUX"h:astic construction or
I.
....
---
---
---
�
..... arbitrary digital filk:ring of the amplitudes (ord inlilll$) and/or of the duration!'�. .. ......
---
---
I ---
---
- --
Having used all contributions of the dyn%-routes of this scqueno= PARAG(psi%) and as :soon as the= above computation for jUSL 0� sample is ended, th�n : If the l!oalnplo otthis PARAG(psi%)-scquence a� not cn
dyn%-routes for the next sample, by
I
bo\.'
procedure.
c. If the last PARAG(psi%)-sequena: of the TABLE has �n computed then the ta5k and the music are terminated
.1.
•
.t·or many-channel stereo music :a. compute from the start the same main programme as many times as there are channels; b. separate random·gcncrator for r2eh h nnel for tlte amplitudes and/or for the abscissa.
ca
use
299
More Thorough Stochastic Music
---
d]n%-tlyaM..-1 OI'NAS(d]ft'll>)
•
---
..
...
•
....
•
.... d]n'll>=dyaM..% DI'NAS(d]ft'll)
•
..
dyro'l!.=dytiM...,.-2 OYNAS(dyro%)
I
..
•
...
•
•
...
...
•
I
Formalized Music
300
' PROGRAMME* 'PARAG3.BAS
'AUTOMATED COMPUTATION of the SOUND-PATCHES for GENDYI.BAS ' .............................................................................----------------------------------'"''"'-·--·---'------------------------·-------------·---------··----------------------------
RANDOMIZE
'do
with -32768 n =4000
ex. '
---
then
-
- -- ---- - -----
-
--
-
n
< n <
RANDOMIZE n
- -----
' Uniform distrib.
- .. -------------------·--····-- ----------------'
--- -
32767
= 4300: RANDOMIZE n '++++++++++++++++++++++ psi%= 3 'index of this data
n
'programme.
R$
=
LTRIMS(STRS(psi%))
prtS
=
QO$
=
,
"prL" + R$: prtS "ARAGOO"
+
=
prtS
RS: QOS
=
+
".DAT"
QO$ + ".DAT"
'file for
sound-patches
'file for general data
'data file for the 13th dyn%-field: MO$ "ARAG130" + R$: MO$ MO$ + ".DAT" Ml$ "ARAG131" + R$: Ml$ Ml$ + ".DAT" M2$ "ARAGJ32" + R$: M2$ M2$ + '.DAT" =
=
=
=
=
=
'################################################ = dyn:\iin%: horiz% = 1: e% = 2: ecrvrt% = 3: convrt% = 1: mkr = 1.2
dyn%
DEBmax&(O TO 20) 20) DIM p(O TO 20) DIM ralon%(1 TO 20) DIM U2&(0 TO 20) DIM
DIM D(O TO
*This
programme has been
Marie-Helfme Serra
(I .X.).
'last sound-patch of this dyn%-field 'coefficient for the exponential distribution 'probability for the Bernoulli distribution: •0 :S p :S 1 · 'extention of the time-interval (abscissa) 'size of the upper second·elastic-mirror technically realized with the hdp of
More Thorough Stochastic
'size of the lower second-elastic-mirror
DIM V28c(O TO 20) DIM filter%(0 TO 20, 0 TO 10)
'there are ten p ossible filters per dyn%-field
'gen eral data for the sequence
OPEN QO$ FOR OUTPUT A5 # 1 "" 1: v e rtco n % ., 2
'
vertec%
N max&
=
dynMin%
301
Music
i ndexes of the am pl.-ordinate for the screen
'and the converter.
10000000 =
1
dynMax%= 16
flrt%(vertec%)
=
flrt%(vertcon%)
0
'vert.screen-filtcr
1 WRITE #1, Nmax&,
CLOSE#1
for GENDY l BA5 .
'vert.convert.-filter for GENDYl.BA5
=
dynMin%, dynMax%, flrt%(vertec%), flrt%(vertcon%)
'******** ******************************************************************
OPEN MO$ d yn% = 13
FOR OUTPUT A5 #1
113max%
=
D(dyn%) p(dyn%)
= =
'number o f divisions
13
DEBmax&(dyn%) mkr *
=
=
the
13th dyn%-field
of the
waveform
'max.number of s ou nd or silence sound-patches.
25
'proportionality factor and
coefficient for
'the exponential distribution:
.45/ (1.75
.35
ralon%(dyn%)
'as an examplc,this is
9
filter%(dyn%, horiz%)
filter%(dyn%, e%) 1 filter%(dyn%, ecrvrt%) =
=
1
*
1.25)
'the BERNOULLI distribution. 'minimal time interval extention
= 1 filtero/o(dyn%, convrt%) = 1 WRITE #1, dyn%, Il3max%, DEBmax&(dyn%), D(dyn%), p(dyn%),
ralon%(dyn%), filtero/o(dyn%, horiz%), filter%(dyn%, e%), filter%(dyn%,
filter%(dyn%, convrt%)
ecrvrt%),
CLOSE #1 OPEN M l$
FOR OUTPUT A5 #1 .01: B13 = 5: U1318c 1: VI318c = -1: U28c(dyn%) 7: 1: distrPC13 = 1 -7: R dct 13 V28c(dyn%) #1 , A13, B13, Ul318c, V1318c, U28c(dyn%), V28c(dyn%), Rdct13, dis tr PC 13 A13
WRITE
=
=
=
=
=
CLOSE #1
OPEN M2$ FOR OUTPUT A5 #1
Adl3 = 1: Bdl3 = 6: Udl31&: = 2: Vd131&: = -2: Ud132&: = 20: Vd1328c = 0: Rdcd13 = 1: distrPD13 = 2 WRITE #1,Ad13, Bdl3, Ud1318c, Vdl318c, Ud132&:, Vdl32&:, Rdcd13, distrPD13 CL OSE #1
····························•***********************************•*********
Formalized
302
Music
'#### # ## ########################## ######## ####### DIM TH&(O TO 20, 0 TO 100) 'suning point (sample) of a sound/silence patch
DIM DUR&(O TO 20, 0 TO 100) DIM THpr&{O TO 20, 0 TO 100) DIM BED&(O TO 20, 0 TO I 00) DIM sTHend&(O TO 20)
'duration of that patch 'present surting point 'variable for the cornpuution of the patches
'last sample '@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ OPEN prt$ FOR OUTPUT AS #1 'COMPUTING the sound or silence patches. FOR dyn% dynMin% TO dynMax% 'n 4000 + 100 *psi% + 10 * d yn% : RANDOMIZE n DEB& = 0: IF p(dyn%) < = 0 THEN 'ign ore this dyn%field =
=
GOT0Gp2
END IF
Gp1:
DEB&= DEB&+ 1: yl = RND: DR = -(LOG( I - y2)) I D(dyn%) DU R& ( dyn% ,
y2
=
RND
'patch-duration=EXPON.
'distrib/sec. 'same in samples.
DEB&) =DR* 44100 THpr&(dyn%, DEB&) THpr&(dyn%, DEB& - 1) + DU R &( dyn%, DE B & ) IF y1 < = p(dyn%) THEN 'the sound is in this patch! TII &(d yn%, DEB&)== THpr&(dyn%, DEB&- 1)
END IF
THDUR = THDUR + DR BED&(dyn%, DEB&) = BED&(dyn%, DEB&) + 1 DBE& == DBE& + 1
IFDEB& <
ELSE
=
DEBmax&(dyn%)THEN GOTOGpl
FOR xi% I TO DEBmax&(dyn%) THend& TH & (d yn% , xi%) + DUR&(dyn%, xi%): TELOS& = TEL O S& + DUR&(dyn%, xi%) WRITE #1, BED&(dyn%, xi%), TH&(dyn%, xi%), DUR&(dyu%, xi%), THend&:, TH&(dyn%, xi%) /44100, DUR&(dyn%, xi%) /44100, THend& /44100 'last sample of this dyn%-field =
=
'.. ________
IF THe nd& NEXT xi%
END IF
WRITE # 1, THDUR, THDUR I
DURsec
=
>=
sTHcnd&(dyn%) THEN
sTHcnd&(dyn%)
=
THend&
(TELOS& /441 00), sTHend&(dyn%)
(sTHend&:(dyn%)) /44100
303
More Thorough Stochastic Music END IF
'against the overflow
' -------
sU2& ., sU28c + U28c(dyn%) sV2& sV2& + V28c(dyn%) TI!DUR ., 0: TELOS& ., 0: D BE & ., 0 =
sTHend&(dyn%)
Gp2:
=
0
NEXTdyn% 'Proportionality for less than 16 bits amplitudes
(upper mirrors)
'++++++++++++++++++++++++++++++++++++++++++++ FOR dyn% ., dynMin% TO dynMax% IF p(dyn%) > 0 THEN IF dyn% ., 1 THEN OPEN AI$ FOR OUTPUT AS # I: U2&(dyn%) = (98 I sU2&) * U2& ( d y n % ) WRITE #1, AI, Bl, Ull&, Vll&, U2&(dyn%), V2&(dyn%), Rdctl, distrPCl CLOSE#!
ELSEIF
d yn%
�
ELSEIF dyn%
�
13 THEN
ELSEIF dyn%
=
14 THEN
2 THEN
OPEN MI$ FOR OUTPUT AS # 1: U2&(dyn%)., (98/ sU2&) * U2& (dyn%) WRITE #1, Al3, Bl3, Ul318c, Vl31&, U2&(dyn%), V2&(dyn%), Rdctl3, distrPC13 CLOSE#l
END
END IF
IF
NEXTdyn% 'Proportionality for less than 16 bits
amplitudes(lower mirrors)
'++++++++++++++++++++++++++++++++++++++++++++ FOR dyn% = dynMin% TO dynMax% IF p(dyn%) > 0 THEN
IF dyn% ., 1 THEN AI$ FOR OUTPUT AS #I: V2 & (dyn % ) (-98 I sV2&) * V2&(dyn%) WRITE #I, AI, Bl, Ull&, Vll&, U2&(dyn%), V2&(dyn%), Rdctl, distrPCI CLOSE #1 EI.SEIF dyn% = 2 THEN
OPEN
=
304
Formalized
M usic
ELS E IF d yn% = 1 3 TH EN OPEN Ml$ FOR OUTPUT AS #1: V2&(dyn%) = (-98 / sV2 &) * V2&(dyn%) WRITE #I , A l 3 , 8 1 3 , U l 3 1 &, V l 3 1 & , U2& (dyn%) , V2&:(dyn%), Rd ct l 3 , distrPC 1 3 CLOSE # l ELSEIF dyn% = 1 4 TH EN
END IF
END IF NEXT dyn%
'(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( END
'G E N D Y 1 . B A S ' = = = = = = = = = = = = = = = = = =
= =
=
=
-
'This programme controls several stochastic-d y namic sound-fields.
'A stochastic-dynamic sound-field is made out of a wave-length Tl 'divided in I max% segments (durations). Each one of these segments 'is stochastically varied by a cumulated probability-distribution . 'At the e nds of each one of these segments are comp u ted the amp l itud es
'(ordinates) that will form the waveform pol ygene. Are defined: 'for the: duration abscissa a probability distribu tion and 2 times 2 'elastic mirrors; for the a mp li tu d e ordinates a probab i l i ty distri'bution and 2 times 2 elastic mirrors. In between the vertices a linear 'interpolation of points completes the waveform polygene.
'@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ @@@@@ ' 1 st field:
'compute one sound-sample:
D ECLARE SUB DYNASl (l l max%, S M P&:, C l l &: , C l 2&: ,
fh&:, hf&, hh&:) 'compute the am p litude-ordinate: DECLARE SUB PCl (Tab l l ( ) , Tab l 2(), 1 1 %, N l & ) 'compute the time-abscissa: DECLARE S U B PD l (Tad l l (), Tad l 2 (), 1 l %, N l &:) '2d field :
t l l &, tl2&:, 1 1 %, N l &,
305
More Thorough Stochastic Music
' 1 3 th field 'compute one sound-sample: DECLARE SUB DYNAS 13 (1 1 3 max%, SM P&, C l 3 1 &:, C l 32&:, t l 3 1 &:, t l 3 2 8c , 1 1 3 % , N 1 3&: , fh &: , hf&:, hh&:) 'co mpu te the amplitude-ordinate: DECLARE SUB PC13 (Tab l3 1 () , Tab l 3 2(), 1 1 3%, N 13&) 'compute the time-abscissa :
DECLARE SUB PD 1 3 (fad 1 3 1 (), Tad l32(), ! 13%, N I 3 &) ' 1 4 th field
'@@@@@@@@@@@@@@@@@@@@@ @@@@@@@@@@@@@@@@
'Sample-file for output to the converter : OPEN "C:\SOUN D\S3 5 1 . DAT' FOR BI NARY AS #3 SON$ = "S35 1 " 'sound n umbe r o n disc '&&:&&:&&:&&&&&&&:&&:&&:&:&:&:&:&:&&&&&&&:&:&&:&:&:&:&:&:&:&:&:&:&:&:&:&:&:&:&:
'rndj initialises the random-number gene 'rator used through all this p rogram m e. '-32768 < rndj < 32767 'LEH M E R'S random-number generators are a l so used . '&&&:&&&:&&:&&:&:&:&:&:&:&:&:&:&:&:&:&:&:&:&:&:&:&:&:&:&:&:&&:&&:&&&&:&&:&:&:&:&:&:&: rndj
=
401
RANDOMIZE rndj
'for 3 2 sequen ces psi% DIM psi%(0 TO 3 1 ) 'the greatest duration-length of a sequence. DIM chD&:(O TO 3 1 ) ' &:&: &:&: & &:& &: &: &: &:&:&: &: &: &: &:&: &: &: &: &: &: &: &: &: &: &:&:&: &: &: &: &:&: &: &: &:&: &: &:&:&:& && &:&: 'psi% is the number of a given sequence. 'ysp% is an ordinal number from ysp Mi n %
to yspMax% used as an index for psi%. 'DEFI NE HERE ypsMin% and ypsMax% and the order of a freel y chosen
'succession of seq uences psi% given in the SUB ARCHSEQI (yspMin%,yspMax%)!
'For exa m p l e : yspMin% 1 =
7 OPEN "SEQSON" FOR OUTPUT AS # 1
yspMax%
=
WRITE #1 , S O N $ , yspMax%, ys p Min%
FOR ysp M i n%
=
'file to be used in the sco re routine.
0 TO ys pM ax%
CALL ARCHSEQI (yspMin%, yspMax%) WRITE # 1 , psi%
NEXT ysp%
CLOSE # ! 'dynMin% and dynMax%
(
=
mi n i mu m and
m ax
i m u m val ues of the dyn%-fields)
Fo rmalized M usic
306
'are to be found in PARAG(psi%) . '11111111111!! !!I! !IIIII!! 1 1 1 1 1 1 1 1 1 1 1 1 1 ! ! ! II!!! II!!!! II!!!I!!!!!!!!!!!!! ! I l l ! I I ! ! I ! I I ! ! ! I l l ! Ill!! II!!!!!!!!!! !II ! ! ! ! !! ! ! !
' = = = = = = = = = = = COMPUTATIO N 'S B E Gl N l NG = = = = = = = = = = = = = = = ysp%
=
ysp M in%
• • • •••• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
l bg l :
CALL ARCHSEQ l (ysp%, yspMax%)
'! l l ! l l l l l l ! ! l l l I l l ! ! I l l ! ! ! ! ! ! ! ! ! ! I ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! I l l ! ! ! l ! ! !
'$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $$$$$$$$$$$$$ $$ 'Free d i m ens io nin g of the tables '
·····-----------------------
Tables for
the ordinate values of the I l o/o segment for cum ulation.
Tabl l (l TO 2, 0 TO 9 0 ) : DIM Tab l 2( l TO 2, 0 TO 90)' K = 1 o r 2 : IjMax% 90 'Tables for the abscissa values of the II% segment for c u m u lation . DIM Ta d l l ( l TO 2, 0 TO 9 0) : D I M Tad l 2( l TO 2, 0 TO 90) DIM
•
=
'Tables for the ordinate valu es of the 12% seg ment for cumulation . DIM Tab2 l ( l TO 2, 0 TO 90) : D IM Tab22( 1 TO 2, 0 TO 90)
.
'Tables for the ordinate values of the I 13% segment for cumu lation.
Tabl3 1 ( 1 TO 2, 0 TO 90): DIM Tab l32(1 TO 2, 0 TO 90) values of the 1 1 3 % segment for cumulation. DIM Tad l 3 1 ( 1 TO 2, 0 TO 90) : DIM Tad 1 3 2 ( l TO 2, 0 TO 90) 'Tables for the ordi n a te valuo; of the I l 4% seg ment for cumulation. D I M Tab l 4 l ( l T O 2, 0 TO 90): D I M Tab l42(1 TO 2, 0 TO 90) D IM
'Tables for the abscissa
.
' * * * * ** * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * ****
'dyn% = i ndex of the
stochastic subrouti ne DYNAS(dyn%); i n d ex of the sound-patches of this routine;
'D E B & (d yn% ) = ordinal
= last sound-patch of this r o u ti n e; 'DUR& (dyn%,DEB& (dyn%)) = sound-d uration whose
'DEBmax&(dyn%)
'DEB&(dyn%);
ord i n a l number is
'TH&(dyn%,DEB&(dyn%)) = the SM P& sample at which each sound-patch
tcom me nces;
'SMP& = number of the running sample; 'Ij max%
DIM
=
number of subdivisions of a waveform time-length.
DEBmax& (O TO 20)
More Thorough Stochastic
DIM
Music
307
'max. patch numb. :dynMin%=0 TO 'dynMax%=20 'current patch numb.: 0 TO
DEB&(O TO 90)
DIM D (O
'D EBmax& (dyn%) =90
"in expon .dens.;dynMin% = 0 TO
TO 20)
'dynMax%=2 0
DIM pp(O TO 20)
" in Bernoulli dens.;dynMin%= 0 TO 'dynMax% = 2 0
DIM TH & ( O TO 20, 0 TO
'patch start:dyn% = 0 T O '20,DEB& (dyn%) = 0
90)
T0 90
DIM D U R& (O TO 20,
'patch dur.:dyn% = 0 TO '20,DEB& (dyn%) = 0
0 TO 90)
T0 90
' p atch param . : d yn% = 0 TO
DI M BED&(O TO 20, 0 TO 90) DIM U2&(0 TO
'20,DEB&(dyn%)= 0 TO 90 'upper mirror size: dynMin%=0 TO
20)
'dynMaxo/o = 2 0
'lower mirror size : dynMin% = 0 TO
DI M
V2&(0 TO 20)
DIM
sTHend&(O TO 20) fl r t% (0 TO 2 ) filter%(0 T O 20, 0 T O ralon% ( 1 TO 20)
DIM DIM DIM
'dynMax%= 2 0
'last sample of the co nsi d ered dyn%. 'final scr=n
or converter filter.
filters p er field (dyn%). 'extention of abscissa . 'ten available
10)
' * ** * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * * ** ** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
'readings of sequences' data from files wri tten by PARAG (psi%). '
R$
=
prt$
QO$
LTRI M$(STR$ (psi%))
'prt" + R$ : prt$
=
"ARAGOO"
=
AO$ =
Al$
=
A2$
=
+
R$:
=
prt$
+
" . DAT"
QO $ = QO$ +
".DAT"
"ARAG 1 2 "
BO$"ARAG20"
'general data-file for all
'seq uences.
"ARAG1 0" + R$ : AO$ = AO$ + ". DAT" "ARAG l l " + R$: Al$
'sound-patches data-files.
'specific data 'dyn%-field .
for
l st
A l $ + ".DAT" + R$ : A2$ = A2$ + ' . DAT"
+ R$ : BO$
=
=
BO$ +
MO$ +
MO$
=
'ARAG 1 30'
Ml$ M2$
= =
'ARAG 1 3 1 ' + R$ : M 1 $ M l $ + ". DAT" 'ARAG 132' + R$ : M 2$ = M2$ + '. DAT"
+
R$ : M O $
' . DAT"
NO$ = "ARAG 1 4 0' + R$ :
=
". DAT'
=
NO$ = NO$ + '.DAT"
'specific data 'dyn%-field.
for 1 3 th
Formalized M usic
308
'specific data for 14th 'd yn%-field.
'@@@@@@@@@@@@@@@@@@@@@@ @@@@@@@@@@@ @@ @@ horiz% = 1 : e% = 2 : ecrvrt% 3: convrt% 4 'filter indexes =
=
' "' ,... ,.... ,... ,... "" " ,... " " " " " ,.... "
data-files for the dyn%-fields.
'general
'****** **************************
OPEN
QO$
FOR
INPUT AS
#1
I NPUT # 1 , Nmax&:, dynM in%, d yn Max%,
flrt%( 1), flrt%(2) CLOSE # l '&:&:&:&:&:&:&&:&:&&&:&&&&&&&&&&&&&&&&&&&&&&&&:&&:&:&:&&&& &&&& 'specific data-files for each dyn% -fiel d . '
----------------------------.. ---------------- --
' * * * * * * * * * * ** * * * * * * * ** * * * * * * * * * * * ** * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * *
OPEN AO$ F O R INPUT AS # 1
dyn%
=
1
D(dyn%), pp (dyn%), ralono/o(dyn%), filtero/o(dyn%, horiz%) , filter%(d yn%, e%), filter%(dyn%, ecrvrt%), filtero/o(dyn%, convrt%)
I N PUT # 1 , dyno/o, l l max%, DEBmax&(dyn%),
CLOSE # 1
OPEN A l $ FOR IN PUT AS # 1
I N PUT # 1 , A I , B l , U l l &:, V l l &:, U 2 & (dyn%), V2&(dyn%), Rdct l , d istrPC l
CLOSE # !
OPEN A2 $ FOR IN PUT AS # 1
I N PUT # 1 , Ad l , Bd l , U d l l & , Vd l l &, Ud l2&,
CLOSE # l
Vd 1 2&:, Rdcd l , distrPD l
' * * ** * * * *** ** * * * * * * * ** * * * * * * * * * * * * * * * * ** * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * * *
BO$ F O R INPUT AS # 1 dyn% 2 I N PUT # 1 , d yn% , 12max%, DEBmax& (dyn%), D(dyn%), p p (dyn%), ralon%(dyn%), fil tero/o(d yn% , horiz%), filter%(dyn%, e%), filte ro/o (d y n% , ecrvrt%), fil ter% (d yn%, convrt%) =
' * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * ** * * * * * ** * * * * * * * ** * * * * * * * * * * * * * *
MO$ FOR I N P U T AS #1 13 INPUT # 1 , dyn%, 1 1 3 max%, DEBmax&(d yn%) , D(dyn%), pp (dyn%) , ralon%(d yn %), filter%(dyn%, horiz%), filter%(dyn%, e%), filter%(dyn%,
OPEN
dyn%
=
ecrvrt%), filter%(dyn%, convrt%)
More Thorough Stochastic Music
309
CLOSE # 1 OPEN M 1 $
FOR I NP U T AS # 1
I N PUT # 1 , A 1 3 , B 1 3 , U l 3 1 &: , V l 3 1 &: , U 2 &: (dyn%), V2&:(dyn%), Rdct 1 3 ,
distrPC 1 3
CLOSE # 1
OPEN M 2$ FOR INPUT AS # 1 INPUT # 1 , Ad 1 3 , Bd 13, Ud13 1&:, Vd 1 3 1 &:, Ud 132&:, Vd 1 32&:, Rdcd l 3 , distrPD 13 CLOSE # 1
OPEN NO$ FOR INPUT AS # 1 =
dyn% 14 I NPUT # 1 , d yn% , 1 1 4 max%, D E Bma x &: (d yn %) , D(dyn%), pp(dyn%), ralon% (dyn%) , fi!ter%(d yn%, horiz%), filter%(dyn%, e%), filter%(dyn%, ecrvrt%), filter%(dyn%, convrt%)
'************* ********* • ******** * * * * * * * * * * * * * * * * * * * * ** * ** * * * * * * * * *** **** **
'+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
' Rea d i ng of the starting sampling-points DEB&:(dyn%) of '
'sound-patches in each dyno/o-field .
··-------
.. -------------··---·----· ..................... _ _ _ _ _ _ _ _ _ _ _ _ ... .. ...... .. .. .. .. .. .. ..
OPEN prt$ FOR IN PUT AS # 1 FOR d yn% = d yn M in% TO d yn M ax% ··
IF p p (d yn %)
END IF FOR xi%
=
<
=
'loop o n the d yn%-fields
0 TH EN
'for ignored dyn%-fields.
GOTO lbg 1 0
I TO DEBmax&:(d yn%)
'loop on the sound/silent 'patches.
INPUT # 1 , BED&:(dyn%, xi%) , TH&:(dyn%, xi%) , DUR&:(dyn%, xi%), TRen d &: , THsec, DURsec, Thendsec TELOS = TELOS + DUR&(dyn%, xi%) I 44 1 00
N EXT xi%
I N PUT # 1 , TH DUR, TH D U Rpcent, sTHend&(dyn%)
TELOS = 0 '+++ ++++++++++++++++++++++++++++++++++++ +++++ ' the longest of the dyn%-field durations in this seq u en ce { psi%} is : . -----------------·-·-·-----------------------------------------
IF megDUR
END I F
<
=
sTHend&:(dyn%) TH EN megDU R sTHend& (d yn%) 'megD U R is t h e longest d y n %-field 'duration. =
Formalized Music
310 l bg l O : N EXT dyn%
CLOSE # ! '+++++++++++++++++++++ ++++++++++++++++++++++ + ch D &: ( psi %) == megD UR 'cumulation of the l on gest sDURech == sDU Rech + chD&: (psi%)
DURlept = INT(sDURech / (44 1 00 * 60)) DURsec = (sDURech / 44 1 00) MOD 60
'sequ ence -d u rations. 'duration in minutes. 'duration in seconds.
'=== == = = = = === = = = = = = = = = = = = = = = = = == = = = = = = = = = = = = = megDUR = 0
'$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
FOR dyn%
==
dynMin% TO dynMax%
DEB&:(dyn%) = l
number of the
'starti n g
'sound or
'silence patch for
NEXT dyn%
'dyno/o-field.
each
'*************************************************************************
CALL
SECN SPM&:
'screen window. 'runnin g sample.
WINDO =
=
S M P&: / 44 1 00 SMP&:
PRI NT SMP&:
MOD 4 4 1 00
'running seconds.
PRINT SECN : PRINT sam pi&: PRJ NT sec
dyn% = dyn M in%
'******** * ******************************* ********** *** * ******** ***********
SMP& = 0 fi% = 0
'sample 'a
hf& = 0 Kdyn%
number.
'screen am pl i tude of curren t sa mple.
'co n v e rter a mplitude of =
'a current sam p le.
d yn Max% + 1
'check of the d yn%-fields 'amount still availabble.
TELEN% = 0
'for testing the
en d .
music-piece
· · · ·· · · · ·· · · · · · · · · · · · · · · · · · • • • * * * * * * * * * * * *• • • · · · · · · · ·· · · · · · · · · · · · · · · · · · · · ·
'M A I N lbg 2 :
'This
PR0 GRAM M E
part concerns th e computation o f th e am plitude (ordinate ) at
'a g iv en sam p l e SMP&: by a d d i ng up the sound contributions of all 'dyn%-fields in a row from dynMin%
to d ynMax% with
their
'patches DEB&:(dyn%),their starting samples TH&: (dyn%,DEB&:(dyn%)) and
'their durations DUR8c(dyn%,DEB&:(dyn%)).This computation defines 'concurrently the amplitude and time el e men ts of the waveform pol ygones .
IF DEB&:(dyn%)
>
DEBmax&:(dyn%) TH EN
IF Kdyn% = dynMin% THEN = I : GOTO lbg5
TELEN%
More Thorou g h Stochastic
Music
31 1
ELSE
ELSEIF SMP&
<
GOTO lbgO END IF
TH & (d yn% , DEB&(dyn%)) TH EN GOTO ihgO TH& (dyn%, DEB& (dyn%)) TH EN
ELSEIF SMP& IF DUR& (dyn%, DEB&(dyn%)) =
DEB& (dyn%))
=
<>
0 AND BED& (d yn%,
1 THEN
GOTO lbg3 'begining of DYNAS[dyn%] fh& 0: hh& 0 GOTO lhg5 'no DYNAS [dyn %]
ELSE
=
END I F ELSEIF S M P &
=
TH&{dyn%, DEB& (dyn%)) + DUR&(dyn%, DEB&(dyn%)) AND BED&{dyn%, D E B& (d yn % ) ) 1 TH EN GOTO l bg4 'continuation of DYNAS[ dyn%] <
=
=
ELSEIF DEB&(dyn%)
<
=
DEBmax&(dyn%) TII EN IF DEB& (dyn%) D EBmax& (dyn%) THEN =
Kdyn%
END IF
=
Kdyn%
DEB &(d yn %)
ELSEI F dyn%
<
=
GOTO ibg2
d ynMax% THEMyn%
=
-
1
D E B & (dyn%) +
1
d yn% + 1
GOTO ibg2 GOTO lbg6
ELSE
END I F
lbgO:
IF dyn%
<
d y n M a x% THEN
dyn% + 1 GOTO lbg2 ELSE
d y n%
=
fh& 0 : hh& GOTO lbg5 END I F =
=
0
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
lbg3 :
'contribution of a d yn%-field D YNAS[dyn %]
.
-
--
-
--
-----
-
at the start:
-------- - - ----- · - -- - - -- - -------- -- - - --
IF dyn% I TH EN ClePenetr% 1 =
=
CALL DYNAS l (l lmax%, SMP&, C l 1 & , C 1 2 & , t 1 1 &, t 1 2 & , 1 1 %, N 1 & , fh&, hh&)
GOTO lbg5 ELSEI F dyn%
GOTO lbg5
=
2 THEN
hf&,
Formalized
31 2 ELSEIF dyn%
Music
= 13 THEN
=
ClePcnetr% 1 CALL DYNAS 1 3 (I l 3max%, SM P&:, C l 3 1 &: , C l 32&:, t l 3 1 &:, tl 32&:, 1 1 3%, N 13&:, fh&:, h f& , hh&:) GOTO lbg5 ELSEIF dyn% = 1 4 THEN
.
END IF --------------------------------------------------------------------
1bg4 :
'Contribution of a dyn%-field DYNAS (dyn%) after DEB&(dyn%) =
I
'This is realized with ClePenetr% -
- -- -
-- --- - -
-
-
---
--
- - -- - - - - - · - -
=
0.
- - - - - - - · -- ---- - - -
1 is ended .
------ -- - - -
I F dyn%
-=
-- - - - -
1 TH EN
CALL DYNAS I (I l max%, SMP&:, C l l &: , C I 2& ,
hh&:)
ti l &: , t 1 2&:, 1 1 % , N l &:, fh&:. hf&,
GOTO !bg5
ELSE IF dyn%
=
2 TH EN
GOTO ibg5 ELSE IF dyn% = 1 3 TH EN CALL DYNAS I 3 (I l 3max%, SMP&:, C l !l l&:, C I 32&, tl3 1 &, tl 32&:, I l 3%, N 13&:, fh&:, hf&, hh&:) GOTO !bg5
ELSE IF dyn%
.
=
14 TH EN
END IF
............ . .. .. .. .. .. .. .. .. .. .. .. .. .. ____ ,. _ _ _ _ _ _ _ _ _ _ _ _ _ _ ·----- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1bg5 : . . . . ' cumu lation of amplitudes (ordinates) at a current sample-point SAMP&.
'= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
'end-test of the usic-p iece : .
- - - - - - - - - -- - - - -
IF TELEN%
-
- · --------------
<>
0 TH EN
SOUN D 500, 500 / 200 S O U N D 2000, 2000 / 1 00
GOT0 1bg8 END I F
;===================================== ======
More Thorough Stochastic
Music
fl%
= =
hf&
313 ' to screen. ' to converter.
fl% + fh&
hf& + hh&
'incremen.of the dyn%-field . .........................................____ _ IF dyn% < dynMax% dyn% = dyn% + 1 GOTO lbg2
THEN
E N D IF
·•***** ************ * * * • • · · · · · · · · · · · · · ·· · · · · · · · · · · · · · ·· · · · · · · · · · · · · • * * * * * * *
lbg6:
'screen
ordinate
U& = 99: V& = -99: Q CALL M IRO(U&, V&)
=
fl%
fl% = Q
'mirrors : first pair.
'vertical screen fi l te r (fl rt%(vertcc% = I ) ) ' " " " " " " " " " " " " " '"' " " " " " " " " " "Ill' " "" " " " " " " '"' " " " " " " "" " " " " " '' " "
vertec% = I IF flrt%(vertec) GOTO sfltl E N D IF
Q Q
=
(trprec l %
0 TH EN
+
CI%) I 2
= (ffprec l % + ffprec2%
ffprec2%
ffprecl %
sflt l :
=
fl%
=
Q
=
=
ffp re c l % fl%
+
CI%) / 3
'filter 'filter
'fil ter
'filter
= ff% fl% = 0 'converter ordinate (for file) .
ord2%
U&
=
3 2767: V&
=
-3 2768: Q
CALL M IRO(U&, V&)
h f& = Q 'vertical converter filter
=
hf&
(veri:con%=2)
Jll lt ft ll ll l ll lll h llll l ltft l l lltl ft ft lt ll n tl ll ll ll ll ll ltltflftfthltftlthftftllftllftltllllllllhH
vertcon%
IF
..
=
2
flrt%(vertcon%) GOTO sflt2 END I F Q=
(hfprec l &
+
=
0 TH EN
h i& ) I 2
Q = (hfprec l & + h fprec2 &
hfprec2&
sflt2 :
hfprec l &
=
=
hfprec l &
h f&
'final ordinate .----- -·-- --- ..
+ hi&) I 3
' fil ter
'filter
' fi l ter
'filter
Formalized Music
3I4
Q
hf& =
'***** ********** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***** * * * * * * * * * * * * * * * * * * * * * *
I F S M P & < 4 00000000 THEN '(400000000 is an arbitrar y number.)
'for the scree n ,ifwe wish 10 show the resultant: LI N E (abs 1%, ord 1 %) - (abs2%, ord2%) abs 1 o/o = abs 2% ord 1 o/o = ord 2% SMP& S M P& + 1 sampl& = sam p l & + I 'global sampling 'point END I F =
'change
of sequence.
' * * * * ** * ** * * * * * * * * *
IF SMP&
<
=
chD Bc (psi%)
TH EN GOTO lbg9
ELSE
ysp9&
IF ysp%
<
=
ysp9&
+
1
yspMax% THEN
'yspMax9& = 'maximum number 'of sequences.
GOT0 l bg 1 ELSE
END IF
l bg 9:
GOTO !bg8
END IF
abs2%
=
sa m pl Bc M O D 639
I F abs2%
=
END I F
0 TH EN abs l9&
=
gl ob a l screen 'abscissa.
'
0
'++++++++++++++++++++++++++++++++++++++++++ + +
'every point is now written in the converter file.
t ------------- ------------------------------------------------
sample% hf&: h f&: = 0 PUT # 3 , , sam p l e9& =
'in
the
converter
'+++ + +++++++++++ + + + + + + +++++ + + + + + + + + + + + + + ++ + + +
'chronologies and beep signals. '
----
- --- ---------------------------
--
SECN
=
SMP& I 44100
seed = sam pi& I
SPM&
=
IF S P M &
EN D I F I F abs2%
S M P& MOD >
=
=
'prints
44 1 00
0 AND
0 TH E N
the seoonds
44 1 00
SPM& 2 TH E N
SOUND 1 000, 1 000 I 500
SOUND
500, 500 I 200
315
More Tho rough Stochastic Music
SOUND 2000, 2000 / 500 CALL WINDO: P RINT SMP&: PRINT S ECN PRINT sam pi&: PRINT secnd END I F '&&&&&&&&&&&&&&& &&&&&&& & &&&&&&&&&& &&&&&&&& &&& lbg7: d yn%
=
dynM in%
fl% = 0: h f& GOTO ibg2
lb g 8 :
=
0
C L OS E # 3
'the
EN D
SUB DYNAS 1 3
converter
( I l 3 max%, SM P&, C l 3 1 &, C l 3 2 &, tl 3 1 & , t l 3 2 & , 1 1 3%, N l 3 & ,
fh&, hf&, hh&) 'This is the 1 3 th d yn%-fid d subroutine of the main programme that c o m m a n d s 'the contribution of this d yn%-field to the a m p l i tude-ordinate and the time 'abscissa of the waveform polygene that arc sent both to the screen and the 'digital-to-analog sound-converter into the main p rogramme GENDY l . BAS. S HARED ClePenetr%, Ql3, Qd l 3 , dyn%, DEB&(), N max&, dynMa x%, TELEN%,
M l$, M 2 S
SHARED Tab l 3 1 (), Tab l 3 2 (), Tad l 3 1 (), Tad l 3 2 (), TH &(), D U R&(), D EBmax&(), U 2& () , V 2 & ()
SHARED ralon%(), horiz%, e%, ecrvrt%, convrt%, fi l ter%() SHARED aamp l , campi, mampl, xampl SHARED aa b sc , cabsc, mabsc, xabsc STATIC e l 3 & , p l 3 & , t l 3 & , fl 3 p rec l & , fl 3 prec2& , h 1 3 p r ec l & , STATIC A 1 3 , B l 3 , U l 3 1 & , V l 3 1 & , Rdct l 3 , distrPC 1 3
h l 3 p r ec 2 &
STATIC Ad l 3 , Bd l 3 , Ud l 3 1 & , Vd l 3 1 & , Ud l 3 2&, Vd l 3 2&, Rdcd 1 3 , distrPD 1 3
I F ClePenetr%
==
1 TH EN
' I n p u t of the stochastic-distribution coefficients ,of the elastic-
' mirror sizes,of a reduction fa c tor and o f the specific stochastic
'distribution used for comp u ti n g the a m p litude-ordinates
' wavefo r m polygone.
of the
OPEN M l $ FOR I N PUT AS # 1 INPUT # l , A 1 3 , B l 3 , U l 3 1 & , V l 3 1 &, U 2 & (dyn%), V 2 & (dyn%) , Rdct l 3 , distrPC 1 3 CLOSE # 1
'Same kind of input as above but
OPEN M2$ FOR I N PUT AS # 1
now, for the time-intervals .
INPUT # l , Ad 1 3 , Bd l 3 , Ud l 3 1 & , Vd l 3 1 & , Ud l 3 2 & , Vd l 3 2 & ,
Rdcd l 3 , distrPD 1 3 CLOSE # l
Formalized Music
316
ELSEIF ClePenetr%
= 0 THEN GOTO lbl l 3 9
E LS EIF DEB&(dyn%) I TH EN
GOTO lbl l 3 7
END IF lbl l 3 7 :
Nl3& lbl 13 1 :
'K%
=
=
2 : PSET
I F N l 3& MOD
2
(0, 0): C 13 1 & =
=
SMP&
0 TH EN
alternating switch for cumulating i n tables :preced. or present period. ELSE
K% = 2: GOTO lbl l 3 2 K%
END I F
= 1:
GOTO lbl l 3 5
lbl 1 3 2 : .
'first ordinate o f th e new period = l a st ordinate o f th e preced ing one. Tab 1 3 1 ( K% , 0) = Tab l 3 1 (K% - 1 , I l 3 ma x%) Ta b 1 3 2(K%, 0) = Tab l 3 2 (K% - 1, 1 1 3max%) Ta d 1 3 1 (K% , I ) = Tad l 3 1 ( K% - I, I l 3max%) Tad 1 3 2 (K%, 1 ) = Tad l 3 2 (K% - 1 , I l 3 max%) GOTO lbl l 36
lbl 1 3 5 : Tab 1 3 1 (K%,
0)
=
Tab 1 3 2(K%, 0)
=
Tad1 3 1 (K%, 1) Ta d l 3 2( K%, 1 )
lbl l 3 6 : 1 1 3%
=
= =
Tab l 3 1 (K% + 1 , I l 3 m ax%)
Tab l 3 2 (K% + 1 , 1 1 3 max%)
Tad l 3 l (K% + 1 , 1 1 3max%)
Tad l 3 2 (K% + 1 , 1 1 3 max%)
1
lbl l 3 3 :
p 13&
=
0
'computing the Imax ordinates. '
- ---------------------------------------
CALL PC 1 3 (Tab l 3 1(), Tab 1 3 2 () , 1 1 3%, N l 3 8c)
'computing the Imax abscissa-intervals.
'
-------------------------------------------------
CALL PD 13 (Tad l 3 1 (), Ta d l32 () , 1 1 3%, N l 38c) e 1 38c = Qd 1 3 ' horizontal abscissa filter
e 13& fl 3 lr l :
=
IF filter%(dyn%, horiz%) = 0 TH EN GOTO fl 3 lrl E N D IF (PDprc l 3 1 & + PDprc l 3 2 & + Qd 1 3) I 3 'filter
PDprc l 3 28c PDprc 1 3 1 8c
=
=
PDprc 1 3 1 8c
Qd l3
IF N 1 3 & MOD 2 = O TH E N K % = 2: GOTO lbl 1 3 4
'filter 'filter
More Thorough Stochastic Music
317
ELSE K% = 1 : GOTO lbl l34
END
IF
'= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = lbll34:
'Drawing the polygene of period
.
TI 1 3
---------------·- -----------· ---------------·
C l 32& = C 1 3 1 & sc%
=
tl3 1 &
el3&
+
639 =
Tab1 32(K%, 1 1 3% - 1 ) : t 1 3 2 & = Tab l 3 2 (K%, 1 1 3%)
t l 3 & = tl 32&
- tl 3 1 &
'====================== ======================
'LI NEAR INTERPOLATION OF ORDI NATES 'in-between the abscissa C 1 3 1 & and C l 3 2& lbl l39:
'extension of abscissa
.
IF filter%(dyn%, e%)
cl3& IF pl3&
>
GOTO lbl l 3 1 0
ELSEI F p l 3 &
=
e l 3 & AN D e 1 3 &
GOTO lbl 1 3 1 0
E LSEI F e l 3 & =
END I F
ralon%)
=
ralon%(dyn%)
<>
1 THEN
0 TH EN
0 AN D filter%(dyn%, e%) = 0 THEN 1 =
c13&
p l3 & = p l 3& +
fh&
=
END I F
e 1 3 & TH EN
(=
---------- - - - - - - - - - - - - - - - - - - - - - - - ---------·
1
= p 1 3 & * tl3& / e l3& + tl3 1 &
'Attack and decay of a so u n d -patc h . '---·-------- - - - - - - - - - · - - - - - - - - - - - - - -· - - - ------
DIAFA& S M P& - TH&(dyn%, DEB&(dyn%)) IF D I AFA & 0 AN D DIAFA < = 500 TH EN =
=
END I F
DIAFDIM& =
SMP&
fh &
=
fh&
*
TH & (d yn % , DEB&(d yn%))
I F DIAFD I M & < = l OOO TH EN
END
IF
'Acoustic Normalisation
'
-------------------------·-- -- ---
fu&
hh&
=
fh &
*
D IAFA& / 500 +
DUR&(dyn%, DEB&(dyn%))
DIAFD IM& I
1 000
fh& * 32767 I 1 00 'screen's vertical filter =
'
........ .. ....................... ......... .. .................... l F liltero/o(dyn%, ecrvrt%) TH E N
= 0
318
Formalized
ffh&
(fl3prccl &
=
ffh& flh&
END I F
+
GOTO C13Ir3
fh&) I 2
(f13prccl& + fh&) / 2 (fl3precl & + fl3prcc2& + fh&) / 3 fl3prcc2& = fl 3precl & f1 3prccl& lh& fh&
=
=
=
=
ffh&
t -------- - - - - - - - - - - - - -·------
f13Ir3:
Music
'filt.er 'filter 'filter 'filter
'lilter 'filter
'converter's vertical filter
, ... .. ..... .. .... ... ... ... ... .. .. ... ... ... . ... ... .. .. ................... ___ _
filter%(dyn%, convrl%)
IF
11-IEN
END IF hhh&
=
hhh&
=
GOTO f131r4
hh&) I 2
(h 1 3precl & +
=0
'filter
(h l 3prec l& + h 1 3prec2& + hh&) l 3 'filter
h 1 3prec2&
=
h l 3 prec l &
'filter
h l 3prcc l &
=
hh&
'filter
hh&
=
'filter
hhh&
f1 3lr4: ClePenetro/o
=
0: EXIT SUB
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
lbl l 3 10:
C l3 1 &
=
C l 3 2&
: next segment of the period 'TI 13 IF 1 1 3%
<
period .
l l 3 m ax% TH E N
ELSE I F N 13 & ELSE END I F
or next
<
1 1 3%
=
1 1 3% + 1 : GOTO lbl l33
N m a x & THEN N 1 3&
=
N 1 3 & + 1:
TELEN%
=
GOTO
lbl l 3 1
1: EXIT SUB
END SUB SUB PC 1 3 (Tab l 3 1 (), Tab l32(), 1 1 3%, N 1 3 &)
'Subroutine of the 1 3 th dyn%-field that computes the amplitude
'ordinale of the vertices SHARE D V2&()
for the waveform polygene.
dyn%, Q, SMP&, fh&, hf&, ClePcnetr%, M 1$ , prc 1 3 1 , prc 1 32, U2&(),
SHARED aampl,
campi, mampl , xampl
STATIC A 1 3 , B l 3 , U l 3 l &, V l 3 l & , Rdct l 3 , distrPC 1 3
More Thorough Stochastic M usic
IF
ClePenetr%
=
1
31 9
TH E N
'Input of the stochastic-disttibution coefficients, of the
elastic-
' mirror sizes,of a red u cti o n factor and of the specific stochastic
'distribution u sed for computing the amplitude-ordinates of the
'waveform polygone.
#1
OPEN M I S FOR I N PUT AS I N PUT
END
#I, Al3, B l 3 , U l3 1 &,
distrPC 1 3 C LOSE # !
V l 3 1 &, U 2 & (dyn%),
V2&(d yn%), Rd c t l 3 ,
IF
IF
Nl3&
MOD 2 0 TH EN K% = 1 =
E LS E K% = 2 END IF 'LE H M ER'S random-number generator: xampl ((xampl * aam pl campi) I mam p l)) * mampl z = xampl / m a m p l =
+
campi) I mampl - I NT((xampl * aampl
+
'Built-in random-number generato r :
z = RN D
pi = 3 . 1 4 1 5 92 6535 9 # : van g = 2 DO WH T LE z = 0
* pi / 4 4 1 00
z = RND LOOP
IF
"CAUCHY: Cauchy = A 1 3
*
distrPC13
=
TAN ((z - .5) • pi): Ql3 Tab l 3 l (K%, 1 1 3%) + Cauchy ELSEI F distrPC 13 = 2 TH E N =
"LOGIST. : -(LOG((l - z) / z) + B 1 3) / A l 3 :
L
1 TH E N
=
Q l 3 = Tab l 3 l (K%,
ELSEI F dimP C 1 3
=
ELSEIF distrPC 1 3
=
"HYPERBCOS.: h yp e = Al3 * LOG(TAN(z * p i / 2)): Ql3
3 TH EN =
*
(. 5 - .5
•
=
"SINUS: = A13 U&
Tab l 3 1 (K%, 1 1 3%) + hype =
•
-(LOG( l - z)) I A13 : Q 1 3
sinu
L
SIN ((.5 z) * pi)): Ql3 Tab l 3 l (K%, 1 1 3%) 5 TH EN =
+ arcsin
=
ELSEIF distrPC 13
"EXPON. : expon
+
4 THEN
"ARCSI N E : arcsin = A 1 3
1 1 3%)
Ta b l 3 l ( K% , 1 1 3%) + expon = 6 THEN
ELSEI F distrPC I 3
*
=
S I N (S M P &
*
vang *
EN D I F
B13): Ql3
U 1 3 1 &: V&
=
V13 1&: Q
Ql3
=
Q
CAL L M I RO(U&, V&)
=
=
QI3
sinu 'validate coresp.expression
Fo r m a l i zed
320 IF K%
=
ELSE END
IF
Music
1 TH EN
Tab l � l (2, I l �%) = Ql �
Tab l � l ( l , ! 1 3%)
=
Ql3
Q l 3 = Q l 3 • Rdct l 3 ' Ql 3 = Q l 3 Ql3 Tab l32(K%, ! 1 3%) + Q l 3 U2&(dyn%) : V& V 2 & (d yn %) : Q Q l 3 =
U& CAL L M IRO(U&, V&) =
=
=
'valeur fi l tree
Q
=
(prc l � l + Q) / 2
(prc l 3 1 + prc l3 2 'prc l 3 2 prc l 3 1
'Q
=
'filter +
Q) / 3
=
=
Q
Tab l 3 2 (2, 1 1 3%)
=
Ql3
Tab l 3 2 ( 1 , ! 1 3%)
=
Ql3
prc l 3 1 IF K%
Ql3 = Q
=
ELSE END IF
l TH EN
END SUB
S U B PD 13 (Tad l 3 1 (), Tad l32(), ! 1 3%, 'Subroutine of the
'between two
N l 3 8c)
1 3 th dyn%-field that computes the time-interval
vertices of the waveform polygone.
SHARED Q, Qd l 3 , l l 3max%, SMP&, fh&, h r&: , ClePenetr%, M2S SHARED aabsc, cabsc, mabsc, xabsc
STATIC Ad l 3 , B d l3, Ud l 3 1 & , Vdl 3 1 & , Ud l32&, Vd l328c, Rdcd l 3 , distrP D 1 3 I F C l eP e n e tr % 1 TH E N 'Input of the stochastic-distribution =
codlicients,of t h e d a s ti c
'mi rror sizes,of a reduction factor and of th e speci fi c slochastic
'distribution used for computing the ti me-i n terval i n -between 'two verices of the waveform polygone . O P E N M 2 $ F O R I N PUT AS # 1
IN PUT # l , Ad l 3 , Bd l 3 , Rdcd l 3 , distrPD 1 3 CLOSE # 1
END IF
I F N l 3 8c MOD 2 K%
ELSE END I F
=
= 1
K% = 2
Ud l 3 1 8c , Vd l 3 1 8c , U d l 3 2 8c , Vd l 3 2 8c,
0 TH E N
Music
More Thorou gh Stochastic
32 1
'LEHMER'S random-number generator:
'xabsc = ((xabsc"'aabsc + cabsc)lmabsc-I NT((xabsc*aabsc + cabsc)lmabsc) )*mabsc z= xabsc I mabsc 'Built-in random-n umber generator: z = RN D pi = 3 . 1 4 1 59265359 # : v a n g = 2 * pi I 44 1 00 DO WH I LE z 0 z = RND =
LOOP I F distrPD 13
=
1
TH EN
"CAU C H Y :
Cauchy = Ad l 3 "LOGIST. : L = -(LOG({l -
"'
TAN ((z - . 5 ) "' pi): Qd l 3 ELSEIF distrPD 1 3
z.) I z.)
=
"ARCS I N E :
'arcsin
=
distrPD 1 3
=
Ad l3 "'(.5 - . 5
*
SIN ((.5 - z)
*
"S I N US:
Ad l 3 "' SI N (SMP&
*
=
vang
*
=
E N D IF
I F K%
=
=
5
TH EN
6
TH EN
Bd l 3 ) : Qd l 3
Ud l 3 L & : V& Vd l 3 1 & : Q CALL M I RO(U&, V&) =
=
=
sinu 'va lidate coresp.expression
Qd l 3
Qd l 3 = Q 1 TH E N Tad l 3 1 (2 , 1 1 3%)
=
Qd l 3
Tad l 3 l ( l , 1 1 3 %)
=
Qd l 3
E LS E END I F
Qd l 3 Qd l 3
=
•
Qd l 3 * Rdcd l 3 Qd l 3
=
Qd l 3 = Tad l 3 2 (K%, 1 1 3 % ) + Qd l 3 U & = Ud l 3 2 & : V & = Vd l 3 2 & : Q = Qd l 3 CALL M I RO(U&, V&)
Qd l 3
I F K%
=
ELSE
END
E N D IF
SUB
=
+ h ype
Qd l 3 = Tad l3 1 (K%, I l 3%) + expon
ELSEI F distrPD 1 3
U&
Tad l 3 l (K%, 1 1 3%) + L
pi)) :Qd l 3 =Tad l 3 1 (K%, I l 3%) + arcsin
ELSEI F distrPD l 3
expon = -(LO G ( l - z)) I Ad 1 3 :
=
=
3 THEN
LOG(TAN(z "' pi I 2)): Qd 1 3 = Tad l 3 l (K%, 1 1 3%) E LS EI F distrPD 1 3 = 4 TH EN
"EXPO N . :
sinu
Tad l 3 l {K%, I l 3%) + Cauchy 2 TH E N
+ Bd l 3 ) I Ad l 3 : Qd l 3
ELSEIF
"HYPERBCOS . :
hype = Ad l 3 "'
=
Q
1 TH EN Ta d l 3 2(2
1 1 3% )
=
Qd l 3
Tad l 3 2 ( 1 , 1 1 3%)
=
Qd l 3
,
SON:S352; yspX: 7 : PSiY; 8 ; Pge: 3 ; dUf . Ni n : Sec: 2 : 13 ; tot . Min : Sec 128
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SOH:S352; ysy�: ? ; ps i X: 8 ; Pge: 2 ; dUP. Mi n : Sec: 2 : 13; tot . Min : Sec I " " 1 ---1----+---+---t-- • •
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189sec 1 ' ' 12 · · 13 4
' I ' ' ' ' 1
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Two pages of the "score" resu lting from the p rogramme re p roduced here.
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129.m 11 •
•
12
---4 3 · 14 · · 15 16 •
I I
17 8 19
•
•
•
• 1
•
•
•
•
•
·
I
12 1 13 114 115 116
•
•
• • 1
•
•
• •
•
•
•
•
• I • • • •
• I • • • •
JlJ
11
Appendix
I
[7] [20]
TWO LAWS O F C O NTI N U O U S P R O B A B I LITY F i rst Law
Let OA be a segment of a straight line of length l on which we place n points. Their linear density is c nfl. Suppose t h a t l and n increase in definitely while c remains constant. Suppose also that these points are numbered Al> A P , Aq, and are distributed from left to right beginning at the origin 0. Let =
.
and
.
.
The probability that x
+ dx is
Now the
given
by
th e
probability r e c u rrence
the ith segment
Pn ,
wil l
that t he re will
formula
Pn + l
have
be
n
a length
(cxf 1 )p0•
But
ex
e - cx
1
If x is very small and
=
if we
Po
Po = e - c::c and
-
ex
1!
(ex) 2
+
2!
denote i t
=
1
-
-
(ex) 3
c dx
+
3 23
+ . . .
3!
by dx, we
c2(dx) 2 �
have ·
between
points on a segm en t
p;: = n + I'
Therefore PI =
x1
·
· .
.
x,
x
is
324
Formalized 1\-lusic
l - c dx the powers of dx are infinitely small for high values, P o and p1 c dx Po c dx. Hence, the probability Px i s composed of the pro b ability P o e - cx, that there will be no point on the segment x, and the probability p1 c dx, that there will be a point in dx.
Since
=
=
=
=
=
T H E S A M E P R O B A B I L ITY
A P P R O X I M A TE C A L C U L A T I O N O F ( F O R CALC ULATI ON BY HAND )
Let there be d points to be placed on a straight line of length l. The linear density is c dfl poin ts on length l. If the lengths are expressed in av (a > 0) and cv dfa points in the unit o f length v . units v then l Then x1 iv (i 0, I, 2, 3, . . . ), and the probability, the asymptotic limit of the rel ative frequency of the segment x1, will be =
=
=
=
=
P,.,
We
e - civc�x1 •
=
shall now define the quantity �x1• The probability ( I ) is
probability P o = e - civ, that there probability p1 cl1x" that there will be e no ugh to be ignored. Set of the
=
where
n
is
a sufficien tly large natural
0 a constant
Substitute
Then equ a ti o n
and
must
(c�x,)2
<
0
( 1)
satisfy
z
�X;
<
,_
00
1=0
or
z
since
cv >
0,
e - cv <
2:
1=0
th i s
=
l fc
(e - cv) !
I = ro
•
=
expression becomes
every x1
, - l . I Q - n/2,
l e - c v , cz
1 , so that
i = co
and finally
numbe r ;
Px, = e - cfv . cz
the condition
composed
x1,
, - l . l Q - n/2.
<
Z 5 �X; < is written
no
w - n.
for �� such that for
2:
But
<
a
point on
and the point in �x1 if (c�x1) 2 is small
will
be
(1)
2
1=0
=
1,
, - ciu.
1 / ( l - e - cv)
z = --c
(2) (3)
I
Appendix Now
325
from (2) 1
1 0 - n12
r ev
_
--- < --·
c
c
Therefore then
(1
Thus, for h av e r ev
- log ( I
cv >
(I
>
-
0 -
1 0 - n12)
_
< 6 - cv <
1.
e - cv < I , and for cv < w - nl2 ) . And sinc e 0 < J O - n/ 2 <
we h ave
l Q - n/2 )
w - nl2
=
w -
+
2
and
w - n/2 In order that
e - cv
>
1
<
-
-
lo g ( 1
-
+
- log
I we
w -
+
(1
-
have
w - n1 2 )
we
l Q - (n/2)1
-...,...4- + · · ·
I O - n /2) .
w - n1 2 it is therefo re sufficient that
(4)
� w - n/2•
cv
Then w e may take
(5) and su bstitute this value in formula ( 1 ) , from which we can now set up probability t a b l es. Here is an example : Let d 1 0 points as mean value to be spread on a strai g h t l i n e seg ment of length l 1 00 em. We have to d e fi n e x1 and Px, as a fu nction of i, given I 0 - 4 is considered to be negl igi ble. that (c.6.x1) 2 From (4) , cv I 0 - 41 2 0.0 1 points i n v. Now c dfl 1 0/ 1 00 points/em, therefore c 0 . 1 p o ints/em, v = 0.0 1 /0. 1 0. 1 em, a n d x1 =
=
=
=
O. l i
em ,;, i
From
mm.
(5) ,
From ( 1 ) ,
=
_
e - o.o1
0. 1
Px,
=
=
=
=
=
=
(1
-
0.9905) 1 0 = 0.0995 ,;,
e - 0•011 . 0. 1 · 0. 1
F o r calculation b y machine
=
=
0. 1
0.01 · (0.099005) 1•
see Chapter V.
em.
Formalized Music
326
S econd Law
f(j) dj
� (I - �) dj.
=
(pitch, intensity, density, etc .) forms
an interval ( d i s Each interval is identified with a segment x taken on the axis of the variable. Le t t h e re be two points A and B on this axis corres p ondin g to the lower and upper limits of the variable. It is th e n a matter of drawing at random a segment with i n AB w h o s e length is
Each variable
tance) with its
p redecessor.
incl uded betweenj and j
event
is :
P;
+
=
dj for 0
f(j) dj
A P P R O X I M ATE D E F I N I T I O N HAND
with
=
j
s; AB. Then the probability of this
� (1 - �) dj'
( 1)
AB.
for a = BY
s;
OF
T H IS
P R O B A B I L I TY F OR C A L C U L A T I O N
B y taking dj ,a's a constant and J a s discontinuous w e set dJ v a fm fo r i 0, I, 2, 3; . . . , m. Equ ation ( I ) becomes
=
=
( iv)
Pi = ;; 1 - a 2
c.
=
c,
J
=
(2)
. .!-
But
iv
I,
whence
d.lJ On t h e
other hand P1 must
app roximation required :
2
(
P1 = -- 1 m + 1 P;
is at
becomes
i)
m
we
have
v
=
s;
c
=
a
-- ·
m + l
be taken as a
fu nction of the decimal
10-n
0, I , 2, 3, . . . )
(n
=
.
= 0, whence m 2! 2 . J on - I ; so for m = af(2 · 1 0n - l ) and dj = af (2 · I On) , and ( 1 )
a maximum when i
2 · 1 0n - 1
=
Appendix I I DEFINITION
3 27 OF THE
SAME
CALCULATION
We know
P R O B A B I L I TY F OR C O M P U TER
that the com p u ter
pro b . ( 0 where F(x0) is the =
Yo · T herefore
and
j
::5
::5
Xo
)
by rejecting the positive x0 x0
::5
a
-
root,
=
=
since
a [l
-
a[l
±
X� a
=
=
F(xo) .
pro b. (0 ::5 y ::5
=
y0)
v(l - Y o) ],
m u s t rem ain
x0
at random (of
law of density P1
smaller
than
a, we
v' ( l - Yo)]
a•
...
Appendix
II
[ 1 4]
. · · !"'4-
.
•
2 Xo
=
distribution function ofj. But F(x0)
obtain
::5
the p robability
r:ro f(j) dj Jo
=
2 Xo - xg2 = Y o and X o a a
for all 0
draw numbers y0
can only
equal probability) 0 :s; y0 :s; I . Using f(j ) dj, we h ave for some interval x0
. . " > · ·�·
Let the re be states E1, E2, E3, events necessarily occur at each •
•
•
,
·'
E, with
trial . The
r < oo ; and let o n e o f these probability that event Ek will
take place when Eh has occurred at the previous trial is . P h k.i
2. PM Pl,n,i is
k
the probability that
2: Pf.'/1 k
If for
n � oo
expressed by t h e
the interm edi ate
=
1,
with
k
=
1 , 2,
in n tri als we will =
1,
wit h
k
=
.
•
, r.
from
p ass
1 , 2,
.
state
Eh to
state
Ek ;
. . . , r.
one of the PftJ tends towards a limit Phk•
this
limit is
of all the p rod u cts Ph iPik• j bein g the index of one of s t a t e s E1 (I ::5 j ::5 r) :
sum
Phk
=
P,.lplk
+
P112P2k
+
·
·
·
+
P,.,prk•
For m a l ized ?\ I u sic
328 The s u m of all the l i m i t s P11 1c is e q u al to I : ph 1 + ph 2
ph 3 +
+
'
\Ve c a n form tables or matrices n < • >
as
p
p< n>
p< n )
rl
pen> rm
p< n )
. . ., m
Regular case .
If at
).
p
. . .,
2m>
=
follows :
. . .,
21 '
11 >
ph r
' ' +
rr
l e a s t one o f the t a bles n < • > co n ta i n s a t l e a s t o n e l i n e
o f which a l l t h e c l e m e n t s a rc posit i\'c, t h e n t h e
among the P" " t h e re e x i s t s at l e a s t i n depend e n t o f
n
one,
Ph'!} h a\·c l i mits P" "' a n d which h a s a non-zero limit
Pm •
a n d of h . T h i s is the regula r case.
Positive regular case . If at l e a s t one of the t a b l e s n<•> has all
cle m e n ts, then all the Ph k h a \·e n o n - z e ro l i m i ts P�c i n d e p e n d e n t i n dex lz . T h i s i s t h e r
The
jJositiue probabilities Pk
+ I eq uations w i t h
r
X1 X2
X3
X, But these equations yields fi rs t
r
an
regular case.
=
Xk cons t i tute the system
u n knowns :
=
X1Pn
=
X1 P 1 2
=
=
arc
+ +
X2P2 1 X2P22
+
+
X2P2 3
+
X1 P1r
+
XzP2r
+
I
= X1
+ X2 +
·
·
·
·
·
·
·
·
+
Xrf'r3
·
·
+
X,p,,
·
· · ·
+
sol u t ions of t h e
X,p,l
·
+
X1P1 3
+
of
posi ti,·c o f the initial
X,p,2
+ X,
n o t i n d e p e n d e n t , fo r the sum
of the first
r
equations
identity. A fter t h e s u bs t i t u tion of t h e l a s t equation fo r one of t h e
eq u a tions, t h e r e re m a i n s
a
system o f r e q u a tions wi t h
r
u n knowns.
:\Tow t h e re i s a d e m o n strati o n showing t h at i n the regular case the system
h as only one sol u t i o n , also that D<•>
=
D"
(nth
power o f
D) .
App endix I I I
THE N EW U P I C SYSTEM
*
Introduction UPIC (Unite Polyagogiquc I n formatiquc
du CEMAMu)1 is a machine
dedicated to the interactive co mposition of m usic al scores. The new and
version of this system runs on an AT 386 microcom p uter connected to unit. The new software offers a mouse-co ntrolled, " user-friendly" window style grap hical i n te rfac e and allows real-time drawing, e diti ng a nd p l a ying of a musical page as well as the recording of a fi nal
a real -ti m e synthesis
" performance."
Des cription The
UPIC is
a music co m p o si n g syste m which co m bi nes
a graphic (or p lay-back} d raw i n g and ed iting
s c ore editor, a voice editor and a powerfu l " performance "
system, a l l shari ng the s a m e data. Therefore, all
o perations are availab le while the music p l ays. All
m o use driven.
device
from the
me nu
the com mands are
command allows one to switch mouse to the digitizer an d vice versa.
A
th e d r awi n g
input
A UPIC score is a collecti o n og f notes that a re called " arcs . " An arc is a p itch (frequency) ve rs us time curve. The frequency variatio ns are continu ous and can cover the whole ambitus. The durations can range from 6 ms to the total d uratio n o f the musical page (1 hour maxi m u m ). 1 CEMAMu (Centre for studies in mathematics and automation of music) , fo u n ded by lannis Xenakis in 1 965 with grants from the French Cultural Ministr y .
inspi red by a similar paper published by lCMC in by Gerard Marino, Jean-M ichel Raczinski, and Marie-Helene Serra o f C EMAMu. My gratitude for their faithful dedication is herewith expressed. (I.X .) *This appendix is free ly
Glasgow, 1 990 in "Proceedings," w ri tte n
329
Formalized Music
330 Tools are provided
.
d u ra ti o n In
this
for
ob taining quantified val ue s of freque ncy
way, the notion of an arc
is
an exte nsion o f the
notio n of a note. In addition, each arc has a set of so u n d
be cha n ge d real-time,
d u ri n g playback.
Voice editing on the
UPIC i nc l u d es redraw ing
and classical
attributes that can
and redefinition of
wave forms, e nvelopes, frequency and a m p li tu de tables, modulating arc
,
sign me nt and modi fication of audio channel para meters (dynamic and
velope). All
these operations are
heard.
fcasable
as
en
during playback a nd i m mediately
Different sound interpretations of the same graphic score may be
tested
with the help
of arc groups. Grou ps contain fro m one arc to the
whole p a g e and allow instanta neous and global modi fications of sound par ameters (waveform change, tra nsposition, etc.). During performance, the musician can switch fro m one page to a noth e r and may control the te mpo and play position by moving the mouse acros s the page. The res u lti ng live interpretation may be recorded in an editable object called a "sequence." The tempo and the positio n in the sequence is controllable while the sequence is being played.
�fosition� I I : BII : II� 0 Play R e c o � d
181 P a u s e
Lil'lits
wau e f o r ll1
('..; BJ
£nul!lope
0 E HU 1
�iun I
1 [;an slit table c",) FRQ T 9 1 2 I ll.rnpl . t a b l e O .a� peat 0 r 111oo
ou : 1 2 : 115
dB !
EHIJTBDB
Channel
�
1t1 .2elt\
oo = 111 = oo . II .
Frea .
1:8] SJ!l O 0 M!!t e
Figure
D S<�v•
I I
I . Sample screen from
UPIC
1:8] ,S.t o p
( (
JI.K
I
)
Appe nd ix I I I : The
33 1
New UPIC Syste m
Page Drawing and Ed iting
A m axi m u m of four pages o f music can be op e ne d a nd d is played in
moveable and resi zeable wi n d ows .
Opening a page stored o n the disk loa ds
it into th e memory of the real-time u n i t. Therefore, all the subsequent o p erations can be car ri ed
out while
the pag e is
being played.
Arcs can be drawn by using one o f the drawing m od e s (free hand, b roke n line, etc.) I f accepted, a n arc is inserted i n the page as s o o n as its d raw ing is over; i f the l i mi t
of 64 oscillators is re ac hed , the arc will be re it is possible to mo d i fy the set of the d e fault attr ibu tes (waveform, e n ve lop e , fre q u e n c y table, a m p litude table, weight, m od ula ti n g
fused. At any ti m e ,
arc, audio chan nel). One
page
holds
a maxi m u m of
4000 arcs.
Usual editing co m mands (cut, copy, paste) a re available. For each page, fo u r groups o f any nu mber o f arcs can be created by using d i fferent typ es of selection (b l oc k , list, c r i te ri a) G e o m e tr i c o p eratio n s l i ke s ym m e try , rotation and vertical alig n m ent ca n be a pp lied to a grou p . Instantaneous modi fications
of
the attributes (waveforms, envelope,
frequency ta b le ,
amplitude table, weight, modulati ng a rc , audio channel) of
the a rcs belo ng to a group c a n be temporarily ap p l i ed and saved, if necessary. Further m ore , gro ups can be insta ntaneousl y m u te d , "solo-ed, " and/or trans p osed. Voice Edition ing
by
Each arc is associated with a n oscillator whose con figuratio n is given
the following arc attributes: waveform, e nve lo p e , modulati ng arc, audio
c h an n e l B e fore b e i n g trans mi tted to the oscillator, the graphic d a ta o f the
arc and of the e nvelope are conve rted
an am p litude table.
Waveforms and e nvelopes can
sou n d , and normalized .
respectively by a fre quency table and
be
d rawn or extracted fro m
The conte nts of the co nversion tables are defi ned ei th e r by or
a
sa m p led drawing
by a menu c o m man d and are redrawable.
the
The frequency table d e fi n ition menu c o m man d enables the
bou n dari es
o f the a mbitus
parameters (tu n i n g
(in
user
to set
hertz or hal f-tones) a n d the musical sc a l e
note a n d nu mber o f equal d i vi s i o n s in the octave). The
freq u e n c y table can be i nve rted a n d
can
be
made c o n ti n uo u s or d isc rete. I n
th e l a tte r case , the steps are the octave d ivisions.
Whe n played
w i th
a
dis
crete frequency ta b l e , the p i tch variations with i n the a rc s fo llow the
frequency steps of the tab le.
Formalized Music
332 P erform a n c e
O n l y o n e page can be played at a
ti me.
the window may be chained or not. T h e
The fo u r pages maxi m u m i n
us e r
chooses which page to
simply by clicking o n it, stops or res tarts th e progression
ance, defi nes the time li mits
of the
o f the performa nce with o p ti o n al
The tempo and play p o siti o n can be defi ned by
page or by entering their valu e s. All typ es
of
m ou se
play
p erform
loopi ng.
motions on
the
motions (forward , backward,
jumps, acceleration, slowing d ow n ) within the page are per m i tted. When not user-controlled, the page is played at a constant te mpo. A set o f channel paramete rs
(d y n a mi c and envelope)
is ass igned to
e ach page. The dynamic and envelope of the 1 6 output audio chan nels re al - ti m e controllable
sp re ad s
duri ng
perfo rmance. As
the
c h a n nel
are
e nvelope
over the whole p age , i t is therefore possible to local l y weigh t arcs
assigned to a give n cha n nel.
In the UPIC, a seque nce
is
th e record i ng, duri ng the performance
(controlled or n o t) o f all the successive positions i n the page , with a 6 ms
curacy. as
It holds
a maximum
of 12
m i nutes of performance.
a position versus time curve. Any piece
ac
It is displ ayable
of the seque nce can be overwrit
ten by a new recording or redrawn. The p erformance of a seque nce
is car ried out i nside its window with m o use moti o n co ntrols (l i k e the page itself) .
four pages are loade d ,
When work.
th e user has tw o seque nces with which to
Storage
Pages, wave fo rm s , envelopes, c o n ve rs io n tables and sequences are
stored in separate banks
(DOS files) on dis k . B an k s are user-protected.
Copying, rena ming, a n d deleting obj ec ts a n d ban ks is possible.
The user can load objects that co m e from different banks. Saving an object can be d o n e i n a n y bank.
Conclusion
This sum marizes the principal characteristics of the UPIC sys te m
today. Additio n al com mands are go in g to be integrated to the application, especially sam p lin g uti l i ti es (r ecord , play, si m p l e edition fuctions) The syn
chronization of the performance with a n ex te r n a l device as well. as the com munication between UPIC and MIDI devices is p rese n tly b e i n g studied.
Tools will be provided to allow a n o th e r application access the data of banks.
The system is being
course o f 1 9 9 1 .
UPIC
industrialized and will be commercialized in the
App e n d i x III: The
New UPIC System
333
TEC H N ICAL D E S C R IPTI O N A) Hardwa re S pecifications Host computer
PC-AT 386 with 3 Megabytes memory minimu m , hard disk, m o use MIDI board , optional digitizer tablet. All Summagraphics co m p atibl e digi ti zers are su pported (si ze AO to A4). ,
Real-time synthe s is u nit 64 oscillators
at 11. 1
(future extension t o converte r board :
kHz
1 28)
with FM
4 audio o u tput channels
2 audio i nput channels
AES/EB U i n terface (exte nsio n to
c
ap ac i ty
:
4 converter boards)
4 pages of 4000 arcs
64 wave forms (4K entries) 4 frequency tables ( 1 6K entries) 1 2 8 envelopes (4K entries)
amplitude tables ( 1 6K e ntries) 2 sequences ( 1 2 mi nutes each 6 ms accurac y)
4
,
B) Software Main Features
DOS with
E nvironment multi-applicatio n wi ndows)
enviro n m e n t
pull-down
use r- protected . Draw ing
d rawin g and is ,
and zoo mable
3.x
(graphical
and
pop-up
:
in
separate banks on disk. Banks are
every object is initialized either by a command or by a
redrawable. Obj ects are displayed in overla p ped, resizable
windows.
several types of selection (block, list, criteria) allow the creation
o f up to four gro ups
of arcs per
page . Each grou p can be mute d , solo-ed,
graphically tra nsformed and real-time controlled. Sound
menus
pages , wave forms, envelopes, freque n c y tables, amplitude
Sto ra ge
tables and seque nces are stored
Edition
Microso ft WI N DOWS
with
: :
See C (Real-time controls)
Formalized Music
334
C)
Real-Time Controls
Page controls
Tempo P l ay
time interval (with or without
Page switchi ng
Position in For each
looping)
the page
au
d io channel : dynamic, envelope
Sequence controls
Tempo Position Sequence switching
Group controls So l o
Mute
Tra nsposition Intensity
Frequency m odu la ti o n Output channel
Waveform (a mong 64)
Frequency tab le (a m on g 4) E nvel o p e (a mo n g 1 28) Am p l i tude table (among 4)
Drawing while pl a yi ng
While a page is being played , the user can mod i fy its wave forms, en
ve l opes and conversion tables.
A new arc can be heard
as soon as
its drawing is finished.
An existing arc can be redrawn within
sa m e time.
its
endpoints and heard at the
A Selected B ibliography of I annis Xenakis *
compiled by H enning Lohner lows :
The bibliography of the works of Iannis Xenakis is a rra n ge d
as
fol
I. Works by Xenakis:
1 . Books, and
2. Articles i n periodicals , booklets, and encyclopedias. I n ge neral, the source o f the translated and
first printing is given. re p ri nted many times.
The writings of
Xenakis
have been
II. Writings about Xe nakis:
l. Mo nographs and collections, the maj o rity of which are dedicated to
Xenakis, and
2. Articles in periodicals, encylopedias, and anthologies . Works in daily and weekly newspapers, with some minor exce p tions, are not i ncluded here. That applies,
in particular,
catalogs and concert programs, which frequently
to the numerous festival
have original
m a te r ial .
Likewise, record notes by or about Xenakis, rec o rd reviews and introduc tions
to works by Xena k is are excluded. It
should be mentioned here that Xe nakis, as an independent archi
tect and long-time collaborator of Le Corbusier is responsible for an exten
sive, architectural body of work, which is noted in the l ite rature . We can, at present, include only the most
important,
f
u se u l,
i nterdiscipli nary works .
The writings are organized chro nologicall y, and, within each year, ar ranged alphabetically. Articles without a specific author arc listed by the initial letter of the ti tle .
This bibliography is based on the
private
collection of the author, and
also utilizes other relevant bibliographies on the subj ect at hand. The list of abbreviations c a n be found at the end, p. 364.
*First appeared in Musik Texte No . JJ, in Musik/Konupte 54!55.
Cologne
335
1 986, in German, and
l a ter updated
Formalize d Music
336
I . Primary B i b l i ography 1 . B ooks
by Xena kis
1 . 1 . : Col l e cted Writi n gs " Les musiques formcllcs; nouveaux prindpes formcls de composition musicale. " Revue musicale 253-254, 1 963; published also as : Musiljues formelles. Paris 1 963; revised and e xpande d text in English tra nslation : FoTTIULlized Mruic. Bloo mingto n and Lo ndon, 1 97 1 . Mruique. Architecture. Collection "Mutatio ns- O rientations" Vol.
11,
edited by M. Ragon, Tournai (Belgium) 1 97 1 ; 2. expanded and re
vised edition 1 976.
1.
2 . : Dissertations
A rts!Sciences-Alliages, these de doclorat es leltres et Paris
1 955
1 979; Arts/Sciences: Alloys , 2. Articles
by Xenakis
"Provli mata ellinikis moussikis." 1 85-1 89. "I.e Couvent de la 1 955.
sciences hu1TI1Lines 1 9 76. York 1 985.
English ed ition, New
Tourette", in:
Epitheorissi technis 9 Modulor
2,
by Le
1 956 "La crise de la musique serielle. " Gravesaner Bliiuer 1
(Athens 1 955) :
C o rb us ie r ,
(July
Paris
1 956): 2-4.
"Wahrscheinlichkeitstheorie u nd Musik. " Gravesaner Blatter 6 (1 956) : 28-34.
"Brief an Hermann Scherchen." Gravesaner Blatter 6 (1 956) 35--3 6. 1 957
" I.e Corbusiers 'Elektronisches Gedicht'/Le Corbusicr's 'Electronic Poem."'
43ff/5 1 ff.
Gravesaner
Bltter/Gravesano
Review
" I.e Couvent d'erudes d e I a Tourette, oeuvre de Chretien No. 6, Paris 1 95 7 : 40-42. 1 95 8
3,
No.
9
(1 957) :
Le Corbusier, " Art
"Auf der Suche nach einer Stochastischen Musik!I n Search o f Stochas tic Music." Gravesaner BliiU er/Gravesano Review 3, No. 1 1 -1 2 (1 958) : 98ff./1 1 2 ff.
" De tre parablerna." Nutida Musik 2 (1 958-1 959) .
" Ge nese de !'architecture du pavilion: le pavilion Philips a !'Expositio n universelle de Bruxelles 1 958." Revue technique Philips 20, No . 1 ( 1 958): 1 0 p.
A Selective
337
Bibliography-Writi ngs
" Re flektioner over 'Geste
Electronique."' ("Notes
tronique"' [in Swedish]). Nmida
Musik
nal : Revue musicale 244 ( 1 959) 25-30.
sur
un Ges te Elec F re n ch origi '
1 , March 1 958;
"Le Corbusier", in: Architecture, P aris 1 958.
"The Philips Pavilio n and The Electronic Poem" (s um mary by L.C.Kalfl), A rts and Architecture 75, Nr. 1 1 1 958: 23. 1 960-6 1
Grundlage n einer stochastischcn Musik!Eiements of Stochastic Music." 4 Issues , Gravesaner Bltitter/Gravesano Review 5-6, No. 1 8 ( 1 960): 6 l f(/84ff; No. 1 9-2 0 (1 960): 1 28ff./1 40ff. ; No. 2 1 ( 1 96 1 ) : 1 02ff./ 1 1 3ff. ; No. 22 {1 96 1 ) : 1 3 1 ff./144ff. ; French original in part i n Revue d'esthitique 1 4, No. 3-4 (1 96 1 ). " Herman Scherchen." In Enzy clopMie de la Musique. Paris, 1 9 6 1 , p. 653.
" Vitruve. " In Enzyclopedie de Ia " Hermann Schcrchc n", i n :
music", in: counter, To kyo 1 961.
"Stochastic
"The riddle 1 962
" Debussy 7.
of japan",
Musique. " Paris 1 961, p.
873-874.
Enclopedie de la Musique, Paris 1 9 6 1 : 65 3
(Proceedings of the) Tokyo East- West
.
music t!*
i n : This is japan, Tokyo 1 96 1 .
a sformaliwwanie muzyki." Ruck Muzyczny
6, No. 1 6 (1 962) :
" Elements sur les procedes probabilistes (stochastiques) de com p osi tion musicale." In Panorama de /'art musical wntemporain. Edited by C. S a muel. Paris 1 9 6 2 , p. 4 1 6-425. "Un Cas; Ia
1 l ff.
m usiq ue s toch as tique
"Stochastischc Musik/Stochastic Review
."
Musica
(Chaix) 1 02
Music ." Gravesaner
6, No. 23-24, p. 1 56fiJ1 69ff.
(Sept. 1 962):
Bllitter/Gravesano
"Wer ist lannis Xenakis[Who is Iannis Xe nakis . " Gravesaner ter/Gravesatw Review No. 23-24 (1 962): 1 85/ 1 85- 1 86. 1 963
1 965
" Musiques formelles." Bulletin de souscription,
Paris 1 963,
p.
Bliit
I.
"Pierre Schaeffer." I n Die Musik in Geschichte und Gegenwart (MGG), Vol. 1 1 , Kassel l 96 3 , p . 1 535. German translation by M. Bente. " Freie Aussprache (D is kuss io n) : I . Naturgetreue Musikwiedergabe, II. Mathematik, Elektronengehirn u. Musikaliscbe Kom position/Open Disc ussio n : I. H igh Fidelity, II. Mathematics, Electro nic B rains and Musical Composition." Gravesaner Bllitter/Gravesano Review No. 26 ( 1 965): 3-4/4-5.
Formalized Music
338
" Freie stochastische
Musik
durch den Elcktronc n rcch ncr/F ree Stoch Gravesaner Bllilter/Gravesano Review
astic Music from the Computer."
7, No. 26
(1 965): 54IT.{79ff.
" Ui. voie de Ia recherche et de Ia q uestio n . " Preuves
177 (Nov . 1 965).
" Der Fall l..e Corbusicr/Co ncerning Le Corbusier. "
ter/Gravesa11JJ Review No . 27-28, p. 5-7/S- 1 0 .
" Ui. Ville Cosmique. " Paris 1 965.
In L'Urbanisrn£, utopies
Gravesaner Bllit
e t r!.alil.iis
by
F.
Choay,
Music. " Gravesaner Bliitter/Gravesano Review No. 29 (1 965- 1 966): 23ffj38ff.
" Zu einer Philosophic der Music{fowards a Philosophy of
Fre nch origi nal : " Vers un philosophic de Ia musique." Reuue d'esthl
tiqW! 2 1 , No. 2-4 ( 1 968): "Le Corbusier",
" Motsagelse n musik 1 966
"The Origins
1 73-2 1 0.
Aujourd'hui, Art et Architecture No . 5 1
och maskin " , Nutida,
of Stochastic
Musik
,
1 965.
9, No . 5-6: 23.
Music. " (translated and expanded
from "Musiques formelles"),
Tempo 78, p.
9-1 2.
excerpt
hors-temps" TN! musics of Asia. Papers read at an inter national music symposium, Manila 1 966: 1 52- 1 7 3.
"Structures 1 967
"Ad libitum." WOM 9 , No. 1 (1 967): 1 7-1 9. " Iannis Xenakis . " In
Die Musik in Geschichte
und
Gegenwart (MGG)
1 4 (1 967-1 968): 923-924; translation by D. Schmidt-Preu �.
Vol .
formalisation Fylkingen Bulletin International 2 (1 96?):
"Musikalisk axio matik och formalisering/Axiomatique et
de Ia 3 p.
composition musi ca le . "
" Vers un metamusique." La
1 968
"Ui.
Musique
Nef 29 (1 967): 24 p.
et les ordi nateurs." Edited by A Capelle. La
teraire, 1 March 1 968.
"Te moignage d'un createur" (Interview),
78-83.
1 969
Pensee el creation, Paris 1 968 :
" Problemy moj ej techniki kompozytorskiej." In
Warsaw 1 969.
" Structures universelles de tion dans le
Ia pensees mmule actuel, Paris 1 969.
Q}tinzaine lit
Horyzonty Muzyki
musicale . " In
Liberti et organisa
" Une note/E.m.a.mu." RevW! musicale 2 6 5 26 6 (1 969): 8 -
1,
p.
A Selective
1 970
3 39
Bibliography-Writings
"Short Answers to Di fficult Questions." Composer (USA) (1 970): 39ff. 1 97 1
"Den K o m is ka varldsstaden."
1 4.
2, No. 2
Nmida Musik 1 5, No. 3 (1 97 1-1 972) : 1 3-
Dossiers de l'Equipe de Mathematique et Automatique Musicales (EMAMu)." Coloquio Artes 1 3, No . 5 ( 1 97 1 ) : 40-48.
" Les
ct programmation", ITC (Inginieurs, Technicienl; et Cadres) A c-
" Musiquc
tWLLizes 2, 1 970:
55-57.
"Om Terretektorh." Nutitk Musik 1 5, No. 2 ( 1 97 1-1 972): 47. Prefuce to L'inuiation mus ica le desjeunes by M. Gagnard, Paris 1 97 1 .
"Stravinsky and Traditio n : First Though ts . " I n "Stravi nsky ( 1 8821 97 1 ), A Composers Me mori al. PNM 9, No. 2 (1 97 1 ) : 1 30. "
"Structures bars-temps. " In The Musics 1 73 .
of Asia,
Manila 1 97 1 ,
p.
1 52-
"Une erreur foss ile , Pour vous que est jesus Christ ? , Paris 1 970: 1 5 0. "
1 972
"Formalized Music Abstract."
1 974
"New Pro posals for
In RILM 6, No. 3 ( 1 972): 266.
Microsound S tructure (in Japanese). In Technology
kukan, Transonic 4, Autumn
"
1 974, p. 4 ££.
suivis de refle:xions en marge." Edited de France 48 (April 1 974): 1 30-1 33.
" Propos impromptus,
Lyon. Courier musicale
1 975
("Wisse nschaftliches De nken u nd Musik-in Greek").
46-48
( 1 975): 14
p.
"The New Music Today. " " Zur
In Inter Nationes,
Theatro
by R.
8, No.
Bonn 2 1 /1 975.
Situation", Darmsllidter Beitrlige zur Neuen Musik, Bd.XIV, Mainz
1 975: 1 6- 1 8. 1 976
"Archaioteka kai sygchrone Musik-in Greek). "
1 9 76 ) : 374-382.
mousi ke, (antike
Zeiten und
BuUetin of Critical Discography
zcitgen.
1 8-1 9 (Januar y
" Culture et creativite. " Culture 3, No. 4 ( 1 976): 4 p. 1 977
" Des U nivcrs d e
Schloezer a nd
Son." In Probllmes de la musique modeme, Scriabine. Paris 2/1 977, p . 1 9 3-200.
M.
by B. dc
" L'universo politico e soc i al e. In Musica e politica, La B iennaledi Venezia. Ve nic e 1 977, p. 548-549, translation by A Cre mo nese . "
" M usique
et architecture. " Artcurial 5 (1 977): 1 p.
" Nouvelles positions
Plastiques
sur
1, Paris 1 977.
Ia micro -structure des
s o n s , Dossiers-Arts "
Fo rmalized M usic
340 "Sur !'architecture 1 978
au Japo n" (in Jap.), Ikebana sogelsu No. 1 1 3, 1 977.
" Centre Georges Pompidou: Geste de lumicre et de son . " Le Diatope Xenakis, ed i ted by I. Xenakis. Catalo g : Paris 1 978.
" Quelques systemes diversifies en com posi ti o n musicale", Colloque de St.Huberl-Le recit et sa refrrlsentation, Paris 1 978.
1 979 " Bela Bartok." Arion (December 1 979).
of the 1 9 78 IntematioMl Computer Music Roads, Evanston 1 979.
" Opening Address", Proceedings
fe
Con rence, Vol. 1 , C.
u n h o m me ferait p l us i e u rs metiers." E ntre ticn avec Carine l.enfant. Architecture 7 (August-Sept. 1 979).
"Si la s ocie te etait 1 980
,
" Between Charybde et Scy lla." Spirali 1 1 (i n I ta li a n Dece mber 1 980); Spirales 1 (in French, February 1 98 1 ). ,
" Brief an Karl Amadeus Hartmann" ( 1 956, in English). In Amadeus llarlmann und die Musica Viva . Mai nz 1 980: 337 .
" Migrazioni ne ll a com posizione m us ical e Biennale di Venezia . Venice 1 980.
."
Karl
In Musica e ela boratore, La
" Spaces and sources of a u d i ti o ns and s p ectacles". (English summary,
M. Griech.), Proceedings offirst meeting: enlargen'TTI£11J of theatrical activi ties and architectural Jtractice, International scientific symposium, Volos 1 980: 203-2 1 2
1 981
"Dialexi." In Symbossion: synchroni port. Athe ns 1 9 8 1 , p. 1 95 fT.
" Homage to
B e la Bart6k. "
techni k a i
paradossi, co nve ntio n
Tempo 1 36 ( 1 98 1 ): 5.
" II faut se que � change!" In the series " l' I RCAM: un monopole teste par des compositeurs."
"l.e
re
Le Matin (26 January 1 98 1 ).
co n
miroir du compositeur" (Interview), Spirales No. 7, 1 98 1 : 20-2 1 .
"On the Nude Body" (statement). I n Body Print, ed i ted
Japan 1 98 1 .
by M . Tanaka.
"Temps en Musique (Time i n Music)." Co n feren ce given i n New York.
1 982
Spirales (December 1 981 ).
"Andre Schaeffner." Revue de musique 68 ( 1 9 82):
387.
tous des cervau x. " Edited b y J . N. Von der Weid. Monde de La Musique (Feb. 1 982): 66--QS.
" Demain, les compositeurs seront
"II pensiero musicale", Spirali No. 4 1 , 1 982: 44-45 . "La composition
l'evolution
musicale est a Ia fois dependente et independente de des systemes analogiques ou numer-
technologique
34 1
A Se1ective B ibliography-Writings
Son et Image, Video co nv ention report. Porte Maillot, 1 982. Cmference des journies d'Etudes (1 982). " Musica e o ri gi n a lit:a , NuTMTo e suono, La Biennale di Venezia, 1 9 82 : 4 1 42; p ubl is h ed i n French , " Musi que et originalite, Phreatirpu No. 28, iques."
"
Frii h ling 1 984 : 62-66.
" Polytopes", Festival d'Automne Paris 1 9 72- 1982, Paris 1 98 2 : 2 1 8. P re face to Informatique musicale et pedagogic, Les cahiers del'arm No. 4 (no
yea r ca. 1 98 1 -1 982): 3. ,
"Science et technologie,
cherchB et technowgie, 1 983
" II
instruments de creation", Colwque national: re
Paris 1 982.
faut se debarrasser des prej uges architecturaux. " Les Nouvelles litter
aires 23-29
(June 1 983) : 40-4 1 .
"Perspectives d e Ia musiquc contem p orai ne", Echos, No. 1 1 983: 47.
"Pour saluer Olivier Messiaen", Opera de Paris No.
1 984
"II
faut
debarasser
(25-30 May) .
1 2 , 1 9 8 3 : 6.
des p rej u ges architecturaux."
O ri gi nal i te Phreatique 28 (1 984).
"Musique et
"
Nouvelles
liueraires
in French and English, J uly-August, 1 982
.
"The Monastery o f la To u rette . I n Le Corbusier Vol. 28. Garland New "
York, 1 984.
" Notice sur Ia vie et les
travaux de Georges Auric", in: Discours pro nonces dans La seance publique tenue par l'A cadimie des Beaux-Arts No. 6, Paris 1 984: 1 3- 1 9.
" U n e xe m p l e e nviable." Revue
musicale
372-374
( 1 981): 67.
" Pour !'innovatio n cullurelle", in: Vous avez dit Fascisme? , by R. B adinter, Paris 1 9 8 4 : 275-276 .
" U n plaidoyer pour l'avant-garde?" Le Nouvel Observateur, 1 9-25 Oct.
1 984: 97.
1 985
"Alban 29.
Berg; le
" I.e pas
ber) .
dernier des
Romanti q ues",
La vie ctLllurelle, 7. 7 . 1 9 85 :
d'acier de Paul Klee. " Le Nouvel Obseroateur ( 1 5-20 Novem
" l..es conditions actuelles de Ia composition" (Zusammengetragen P. Bady), France Forum No. 223-224, 1 985: 1 0-1 2. " Music Composition Treks." I n
J
v. .
Composers and the Computer, edited by Roads . M IT Press, Cambridge, Mass, 1 985; a l ready wri tten 1 9 83, and i n French translatio n by E. Gresset published i n Musique el ordi C.
nateur".
Les Ulis, 1 983.
Formalized
342 " Voeux e n p.
musique." Harmonie/PanorarruJ,- Musique
1 986 "Avant-propos", i n : 1 1- 1 3 .
H.
Schcrchcn:
La
Qanuary
Music
1 985), 1
direction d'orchestre, Paris 1 986 :
"Briefauszug an Hermann Scherchen" (1 2.3. 1 963, frz. i m . dt. Ubs.), in: Herrma nn Scherchen Musiker, Berli n 1 9 86: 95. "Espace
musical, espace scientifique. "
1 986. " Hermann Scherchen", Le Monde de ln.
Le Courrier UNESCO
4. Paris,
Musique 89, Mai 1 9 86 : 9 1 .
"Microsystemes", Ju ne 1 986
"Originality in Music and Composition." FukttShirruJ, Symposium. Tokyo, 1 986.
1 988
"Sui tempo." In Xenakis, a ettra de Enw Restagno.
II. 1.
Torino, 1 988.
Secondary Bibliography
Books and Articles that are entirely, or for the most part, d edicated to Xenakis
1 958 Jenncrct-Gris, Ch. -E. (Le Corbusier). 1 958.
Le
Poeme e/.ectronique.
Paris,
1 960
Moles, A Les mttSiques
1 963
Richard, A, and Barraud, J., and Philippot, M. P. " Yannis Xenakis et Ia musique stochastique." Revue mttSicale 257 (1 963).
1 967
Bois, M. "Xenakis." Paris, 1 966. German translation: Jannis Xenakis, der Mensch und sein Werk. Bonn, 1 96 7-1 968. Nu merous translations into other languages.
1 968
1 969
Charles. Paris,
Barraud, J.
exp�rirnentales. Translated from the 1960.
Pour comprendre les musiq'UJ!s d'aujo urd'hui. Paris, 1 968.
Charles, D. La pemee de Xerw.kis. Diverse
English by D.
Paris, 1 968.
authors. "Varese-Xen akis-Berio--H enry; oeuvres, etudes,
perspectives." Revue musical£ 265-266 (1 969).
A Selective B ibliography-Writings
343
M., a n d Roy, J. Journees de musique cuntemporaine de Paris-25Octobre 1 968. " Revue musicale 267 {carnet critiq ue, 1 969).
Granlet,
31
Lachartre, N. "Les musiques arti ficiel les."
(April 1 969): 1 -96.
1 970
1 971
Diagrammes du monde
1 46
Bourgeois, J. Enlretien avec lannis. Paris, l 970. Rostand, C. 2/1 972.
Ewen, D .
Xenakis, biographie et
catalogue. Paris, 1 970; ex panded :
Composors of Tomorrow's Music. New York,
1 97 1 .
Coe, ] . W. " A S tudy o f Five Selected Contemporary Compositions for
Brass." Dissertatio n, India na U ni v . , 1 9 7 1 .
Potter, G. M. "The Role of Ch an ce in Contemporary Music." Disserta
tion,
1 972
Indiana U niv., 1 97 1 .
Diverse
authors. " Iannis Xe nakis. " L'Arc 5 1
Tamba,
A Soi to Sozo (Creatiun and Creative Ideas lry Coniemporary French
Fleuret, M. la nnis Xenakis. P aris , 1 972 .
( 1 952).
Composers. (I n Japanese) Tokyo, 1 97 2 ; D isse rtatio n ,
1 973
Sevrette, D. "Etude statistique
1 975
Gallaher, C.
sur
Paris,
1 974.
'Herm a , ' de Xe n a k is ." Graduate
work, Schola Cantorum. Paris, 1 973. S. " Density
diana U niv. , 1 975.
in Twentieth-Century Music." Dissertaion, I n
D. "The Musical Oeuvre of Iannis Examination work, J e rus a l e m Univ., 1 975.
Halperin,
X en a k is . "
(In Hebrew)
Revault d'Allonnes, 0. La creation artistique et les promesses de La Iiberti. 1 975. ____
Xenakis:
Les Polytopes. Paris, 1 97 5 and Olivier Revault.
Polytopes . Paris 1 975; Japanese 1 978
Fleuret, M.
78-Xenakis. Paris,
edition, To kyo 1 978.
1 978.
G. M. "Procedures fo r Analysis of Sou nd tion, Indiana, 1 978.
Roberts,
Masses." Disserta
Sato, M. I. Xenakis: sugaku ni yori sakkyoku (I. X. : Music Composed Mathematics.) (In Japanese). Dissertation, Tokyo U n iv., 1 978. 1 979
DeLio, Th . J "Structural
1 98 1
Various Authors.
Les
Pluralism."
D isse rtation, B ro w n
Regards sur Iannis Xenakis. Paris, 1 98 1 .
through
U niv. , 1 979.
Formalized Music
344
SchOnberg a Cage-Essai sur Ia notion d'espace sonore dans Ia musique contemporaine. Paris, 1 98 1 . Sward, R. L "A Comparison of the Techniques o f S tochastic and Serial Bayer, E. De
Based on a S tudy of the Th eo ries and Selected Com Iannis Xenakis and Milton Babbitt." Dissertation, Northwestern Univ., 1 98 1 . Composition
positio ns of
Vriend, J . "Nomos Alpha for Violoncello Solo (Xenakis 1 966); Analy sis and Comments . " Interface 1 0, No. 1 ( 1 98 1 ). 1 982
Varga,
B. Conversatian with Jannis Xenakis. (In Hungarian) Budapest, 1 982.
1 984
Various
Authors. 1 984-85).
Matossian,
"On Iannis Xenakis."
Nutida Musik
28, No. 3 (ca.
N. lannis Xenakis. Paris, 1 98 1 .
Solomos, G. "Aspects de Ia musique grccque co nte mp orai ne. Memoire de maitrise . " Univ.
1 9 85
Paris
Solomos, G. "Semiologi e memo ire de D.
Josep h, S. "The
IV, 1 984.
et
musiques
actuelles-Nuits de I. Xe nakis" ,
E. A U niv. Paris I V , 1 985.
Stochastic Music of Ian nis Xenakis: An Exami n atio n
of His Theory and Practice. " Dissertation, New York.
II. 2.: Magazine Articles and 1 955 Helm, E. "Do naueschingen Festival ( 1 955): 1 4 . Ste inecke,
W.
Tame Affai r. "
Musical America
Melos 22 ( 1 955): 327.
"Exotik und Mathematik." Musica 1 1 Pringshei m, H . "Miinchcn." SMZ 97 ( 1 957): 323.
Herrmann, J.
( 1 957): 206.
Schmidt-Garre, H. "Straw i nsky, Wozzeck, Jazz und tio n Melos 24 (1 957): 330. ."
Valen ti n,
1 959
junge
Genera
E. "Neues aus Miinchen." NZ 1 1 8 (1 957) 288.
Franze, J . "Bericht aus Arge ntinien." Musica
i 3 ( 1 959): 325.
Li ndlar, H. " Ge liichte r und j ubel p fi ffe . " Musica 1 3
"Scherchen 1 960
?5
" I m B rutofen des Avantgardismus: Donaueschin ge r
Musiktage 1955. "
1 957
Interviews :
Conducts
Galea, A " Ei ne gan z
Xenakis Work i n Florence. "
( 1 959): 31 0-3 1 1 .
WOM
neue Sicbe nte." NZ 1 2 1 ( 1 960) 20 .
1 (1 959) : 1 1 .
A Selective Bibliography-Wri tings
1 961
Charles, D. "lannis Xenakis." I n
p. 926.
345
Em.yclopldie de La Musiqtu. Paris, 1 96 1 ,
Davies, M. E. "A Review of the Paris Season." Musical Opinian 84 ( 1 96 1 ) : 739.
Gal ea , A
" Grenzgebiete de r Musik; dreifache Tagu n g Scherchen ." NZ 1 22 (1 96 1 ) : 42 1 -422.
v. Lewinski, W. E. " Kompositorische Spiele mit Radio B re men" Melos 2 8 ( 1 96 1 ) : 329-330.
Klang
bei
He rmann
u nd Farbei bei
Rostand, C. " Avan tgarde Season." Musical America 81 ( 1 96 1 ) : 30. 1962
Kolbe n ,
R. " Die drei Gravesaner Tagungen/The Three Gravesano Gravesaner BliiUer/Gravesano Review 2�24 {1 962): 2 fT.
Conventions."
/ 1 9 ff.
Pociej, B.
"Jesiefi
1 962-zarys problematyki." Ruch Muzyczny 6, No. 22
(1 962): 7-8.
Rozbicki, K. " Na festiwalu bez zm ia n . " Ruch Muzyczny 6, 1 0. Stepanek, V.
Galea,
22 (1 962) :
"ST/10. 1-080262." Hudelmy Roz.hledy 1 5 (1 962): 641 .
" Xenakis: u n j e u n e architecte g rec. "
1 963
No.
Guide du concert 357 (1 962): 1 456.
A " I..es caprices d 'E uterpe . " Musica (Chaix) 1 1 1
(1 963): 36.
Kness), L. "Absurdes Musiktheater und Manierismus in Ve nedig." Melos 30 (1 963) : 309. Kondracki, M. "List z USA." Ruch Mu1.yczny 7, No . 19 (1 963): 8.
Stibi lj , M.
1 964
"Varsavska jesen 1 9 62 . " Zvuk 57 (1 963): 2 1 9.
Harrison, J. "The New York Music Scene."
22.
Musical America 84 (1 964):
Kassler, M. " Musiques Formelles." (Review)
1 1 5-1 1 8 .
PNM 3,
No. I
(1 964) :
" New Works. " Musicjourn.al 22 (1 964): 55.
1 965 Charles, D. ____
"Aujourd'hui." Reml.e
d'Esthitique 18 (1 965) : 406--420.
.. "Entr'acte 'formal' or 'informal' music?." MQ 51 (1 965):
1 44-1 65 .
Finney, R . L . " Music in Greec e. " PNM 3 (1 965): 1 69-1 7 0. Galea, A, and Hambraeus, B. "Ett stycke sam tida fransk toria." Musikrevy 20 (1 96 5) : 230. ____
. "I..e s mauvais calculs de Xenakis. " Musica
( 1 965): 1 0.
musi khis
(Chaix) 1 38
Formalized
346
Kau fman, W. "To Cover Xenakis Music, Use an IBM." J une 1 9 6 5) : 2 ff.
M usic
Variety 239 (9
Hitchcock, H. W. "New York. " MQ 5 1 (1 965): 534-535.
Hopkins, B . " Paris Diary. " Composer 16 (1 965): 29. R. "A Season Strangely Flavored."
McMullen,
A merica 1 5 (1 965) : 1 38. Slonirnsky,
N.
" New M usic in Greece." MQ 5 1
High Fidelity/Musical
(1 965): 232-234.
evolutiei conceptului despre ritm in muzicase colului nostru . " (With summary in German, Italian, Russian and
Tara n u , C . "Aspecte ale
Fre nch) Lucrari de Muzicologie
1 966
1 (1 967) : 8 1 -82.
"Iannis Xenakis." Music & Musicians 1 4 (1 966): 15.
Jones, D. "The Music of Xenakis." Musical Times 1 07 ( 1 966): 495-496 . "Les methodes et les o p in io ns de Iannis Xenakis." Rev'WI musicale de La suisse romande 1 9, No. 4 (1 966): 20.
Morthenson, J. W. "Iannis
Xenakis och 'Polla 9, No . 5-6 (1 965-1 966): 22-23.
Myers, R. "From
34. Romano,
J.
ta D h i na ."' Nutida
Musik
B aroq ue to Computer." Music & Musicians 15 ( 1 966) :
" Musicos de hoy. "
350, and p. 5 in No. 35 1 .
Bue1ws Aires Musica 21 (1 966): 3 in
No.
Ruppel, K . H . "Royan will ein Ze ntru m moderner Musik werde n."
Melos 33 (1 966):
____ .
retektorh'
1 59.
"Das 'Sonotron ' wird vorgestellt-Iannis Ze na ki s beim Festival
Ro y a n
in
Kirche 4-5 (1 966) : 1 63-1 65.
'Ter uraufgefiihrt " Gotllsdiensl und
Sadie, S. " English Bach Festival." Musical Times 107 (1 966): 69 7. Schwi nger,
W. " B re mer Urauffiihrun gen Fl uch t i n die Virtuositat. "
Melos 33 (1966):
23 5.
Takahashi, Y. "Scholia 1-65."
54-55.
Nutida Musik 9, No.
1 -2 (1965-1 966) :
neboli Dalsi vymozenost?." Hudelmi Rozhledy 1 9, No. 1 0 (1 966): 3 1 7 and No. 1 1 ) : 320.
"terretektorh"
"3rd International Festival of Contemporary Art in Royan-April 1 st to 7th." (With translations in French and German)." WOM 8, No. 2-3 1 966) : 34. " Yannis Xcnakis ." San Francisco
SympMny, 1 March
1 966 p. 30. ,
A Selective
1 967
347
Bibliogra p h y-Writi ngs
Dennis, B . X ena kis 'Terretektorh' and Eo n ta . 29. '
"
U. " Die Wellen habe n
Dibelius,
zeitgenossische M us i k i n ( 1 967): 2 .
"'
Tempo 82 (1 967) : 27-
Editorial Notes. Strad 77 (1967): 329. Erhardt, L. " D ziewiec
1 3- 1 4 . Haines,
XI. F estival
sich gelegt-Das
Warschau. " Musikalische jugend
bogatych d ni." Ruch Mu:cyczny
1 1,
fii r
1 6 , No. 5
No. 21 (1 967) :
E. "Spain." MQ 53 ( 1 967)567-577 .
Kay, N. " Xenakis " Pitho p rakta"."
Luck, H.
"
Tempo 890 (1 967): 2 1-25.
l ws ki
u
n
Xe a ki s und L tos a
setzten die Mafistiibe-Eindri.icke
vom Warschauer Herbst 1 967." NZ 1 28 (1 967): 443.
" Iannis Xenakis . "
Music Events 22 ( 1 967):
1 8.
" Iannis Xen a kis verkfdrteckning. " Nutida Musik 10,
15-16. M aguire, J. " Iannis 1967):
view 50
Xe na kis : Formula fo r
(21 J une 1 9 67): 5 1 -53.
____.
" New
Orga, 39.
New Music."
"Paris Report: Contemporary Music at
Fidelity/Musical America
No.
1 7 ( 1 967): 30.
5
(1966-
Saturday Re
Full Ti lt."
High
Philharmonia C o n ce rt Music Events 22 ( 1 967 ) : 29. A
."
" Program med Music . " Music & Musicians 15 (August 1 967):
Pociej, B. "Czas od naleziony i przestrzen
1 1 , No. 22 (1 967):
5-7.
Sa n d i n L. " Reflexioner ,
musik ). De
tek n is ka
kring
rzeczywista." Ruch
ova nstaende
(=
Muzyczny
San nolikhetsteori och
forutsattni ngarna for Pithoprakta . "
Musil!. 1 0, No. 5 ( 1 966-1 967):
1 2- 1 4 .
N!aida
Schiffer, B . "Athens." (In English, German and French) WOM 9, No. 3 ( 1 967): 37ff. Smalley, R. " Oxford . " Volek, J. "Nova
(1 967): 3 1 5.
Musical Times l OB ( 1 967): 7 3 1 .
hudba
ve stare ze m i . "
Hudelmi
Wal l n er, B. " Kring centrum-tankcn." Dansk
(1 967): 1 0- 1 2 .
Walsh , S.
"'Pitho p rakta '
1 967): 43.
Falls
Down." Music
Rozhledy 20, No. 1 0
Musiktidsskrifi 42,
No.
1 -2
& Musicians 1 5, February
Formalized Music
348 Waters, E. N.
"Hatvest of the Year; Selected Acquisitions of the Music Qwtrierly journal of the Iillrary of Congress 2 14, No. 1
Divisio n . " Th4
(1 967): 46-82.
Zo i cas, L T. "Matematica in muzica si unele aspecte in creatia ro maneasca co n tem pora na . (With summaries in Russ ian, German and French.) Lucrari de Muzicologi£ 3 ( 1 967) : 57-65. "
21-
1 968 "'Akrata'-'Pithoprakta."' Buffalo Philharmonic (1 0 March 1 968), 25.
Bachmann, C . (1 9 6 8 ) : 57 1 .
H. " Zeitgenossische
"Koussevitzky
B rozen, M.
Commissio ns."
1 8 (October 1 968): 1 8-19. Butchers, C . "The Random Arts :
Tempo 85 ( 1 968) : 2-5.
High Fidelity/Musical America
High Fidelity/Musical America 1 8
(April 1 968): 27.
Drew, J. " Information, Space, and a New Music Theory 1 2, No. 1 (1 968): 98-99. R. "Xenakis, Penderecki-and America 1 8 (1 968): 20fT.
Gelatt,
Golea, A
"Eine Woche der
5 1 4.
----
··
" Grand
1 7 1 (1968) :
____ ..
254.
oeuvres
1 4-1 5 .
"Zeitgenossische
Time- Dialectic."
in Paris." NZ 1 2 9 (1 968):
Royan." Journal Musicale Fra1l{ais 1 70-
Musik in
Royan."
NZ
H amil to n , D. "Some Newer Figures in Europe. "
America 1 8 ( 1 96 8) : 55.
Hei ne ma n n , R. "Festival in
journal of
High Fidelity/Musical
Buffalo. "
Neu en Musik a
23
Xenakis, Mathe matics and Music. "
"Xenakis & Balanchi ne."
Chapin, L.
Musik in Venedig ." DMZ
Royan."
WOM 1 0, No. 3 ( 1 968): 56-57.
(Also
in
1 29 ( 1 968) :
253-
High Fidelity/Musical
German and
French)
. " Festival mit Konzept. " Mus ica 22 ( 1 968): 265. Kassler, J. C. "Report from London: Cybernetic Serendi pity." Current Musicology 7 (1 968) : 47-59. Mon nikendam, M. " Het muziekfeest van Royan . " Mens en Melodie 23
____
(1 968): 1 64-1 65.
Monteuax, J. " Egy franciaorzagy
30-3 1 .
festival."
Mustika 1 1
(August 1 968):
A Selective Biblio g raph y-Writin gs
Morrison,
M.
1 9 6 8) : 7.
319
" New Music in Europe . "
Nestev, I . "Varshavskie melodii."
Musicanada 1 6 (D ec e mbe r
Sovetskaya Muzyka 32 (1 968):
1 1 9.
Orga, A "American in London . " Music & Musicians 1 7 (December
1 968) : 49-50.
Orr,
B.
" Notes."
Composer No. 29 ( 1 968) : 44.
the Arts Today, No. 2." America 1 8 (J u ne 1 968) : 24-25 .
Plotkin, F. "Festival of
S im mo ns
,
D. " Co ncert
High Fidelity/Musical
Notes." Strad 79 (1 968): 1 1 9.
"London Music." Musical Opinion 92 (1 968) :
____ ..
Smalley R. Ro ya n . " Musical Times 1 0 9 (1 968) : 563. "
,
Souster, T. " Oxford 1 96 8 ) : 23.
____
.
Co mes to Londo n . " Music &
" Xenakis' ' Nuits."'
Terry, W . " World o f
F.
"Analyse
de 'Nomos
Alpha'
Wag ne r, K . " Komponisten aus fun f Landern im
(July
de
Ian nis Xenakis."
Hamburge r
' Neuen
(1 968): 21 5.
Walterova, E. " Xenakis a
(1968): 252-254.
zroze n
" Yannis Xenakis." Nutida Musik
1 969
16
Tempo 85 ( 1 968): 5--1 8 .
Mathematiques et sciences humaines 24 ( 1 968): 1 5 Werk."' Melos 35
Musicians
Dance (Ballet Choreographed b y Balanchine)." February 1 968): 18--1.9 .
Saturday Review 5 1 (3
Vande nbogaerde,
1 1 9.
K. " Ia nni s Xenakis' Bijutsu Techo 9 ( 1 9 69).
Akijama,
i hudebni
red . " Hwlebni
1 2, No. 2 ( 1 968-- 1 969): 1 6
Mathematical Thou gh ts . "
Alcaraz, J. A "Musica ellectro nica. " Heterofonia
Rozh�dy 2 1 .
(In Japanese)
2 , No . 9 (1 969) : 8-- 1 0 .
Archibald, B . Review of: lannis Xenakis, th£ Man and h is Music, M . Bois, ed. Notes 25, No. 3 ( 1 9 6 9 ): 4 8 1 -482. B achmann, C . H.
and German)
C h a na n ,
M.
"Alpbach-Venice-Warsaw. " (In
WOM 1 1 , No. 1 ( 1 9 69 ) : 55.
" Maxwell Davies and X e nak is . "
(August 1 969) : 50.
____
. "Orchestras at Play." Music &
1 969) : 59-60.
" Concert Notes . " Strad 80 ( 1 969): 1 81 .
Cook,
Music
English, Fre nch
& Musicians
17
Musicians 1 8 (September
N . "Shiraz." Musical Times 1 1 0 (1 969): 1 1 65.
Formalized Music
350
" Creations-1 969 (On Anaktoria)." Courrilr Musicale de France 28 ( 1 969) : 248.
" Creations-1 969 (On Eonta)." Courrier Musicale de France 26 ( 1 96 9) : 1 07.
" Creations-1 969 ( On Nomos Gamma)." 27 (1 969): 1 87 .
Courrier Musicale
de France
" Crcations-1 968 (On Nuits, Polla ta Dhi na, ST/48 and Lcs S u p pl i antes ) . " Courrier Musicale de France 25 ( 1 9 69): 2 9 .
Elliott,
C. " Ch oral Performances
Revue 1 1 ,
N o . 2 (1 969): 39.
1 9 68-1 969: France." American Clwral
'"Eonta."' New Yom Phifharnwnic (27 February 1 969) : D-F.
Evans, J. "Paris SMIP." (I n English, F re n ch and German) WOM 1 1 , No. 1 (1 969): 50-5 1 . Galea, A "Le sixieme fes tival de
Qu ne 1 969) :
---____
·
20.
Ro ya n . " journal Musicale Fra11fais
181
"Royan." NZ 1 30 { 1 9 69 ) : 228.
. " Royan: Das sechste
Festival. " Orchester 1 7 ( 1 969): 264.
Heinemann, R. " Sechstes Festival International (Ro ya n). " Musica 23 (1 969) : 356. Hiller, L "The Co m pu te r and Music, Ithaca a nd London 1 969." Within French Experiments in Computer Composition, 8 p.
J ungheinrich, H . K. "Ein Festival NMZ 1 8, No. 3 ( 1 969): 4 .
platzt aus d e n Nahten-Royan 69."
Kassler, M. "Report from Edinburgh. " PNM 7,
No . 2 (1 969): 1 75-1 76.
Lachartre, N. " Les musiques artificielles." Dio.gramTTIJ!s du Monde 1 4 6 { 1 969); within Jannis Xenakis et La musiqtu! stochastique, 1 3 p.
Maegaard, J. " Vor mand p a parnasset" Dansk Musiktidsskrij�
44
{1 969):
36-38.
" Many Wonders." The NA TS Bulletin 1 1 (January 1 969): 1 8.
Moor,
P. "Royan: Hi gh
19 (July 1 9 6 9 ) : 27.
camp, low spirits. " High Fidelity/Musical America
Noble, D. "N. Y. Philharmonic (Ozawa)." High Fidelity/Musical America 19 (May 1 969): 20-2 1 .
Perrot, M . " Entretien avec Iannis Xe nakis." Revue musicale 265-266 ( 1 9 6 9 ): 6 1-76 .
Rostand, C. " Royan besticht d urch interesante Urauffuhurungen." Melos 36 (1 969): 388.
A Selective B ibliograp hy-Writi ngs
35 1
Serres, M. "Musique et bruit de fo nd-Iannis Critique 26 1 (1 969): 1 2 p . Simmons,
D . "London
____ .
Soria,
Opinion 92 (1 969) : 6 1 9 Lo nd o n Music." Musical Opiniun 93 (1 969): 1 24. Music . " Musical
.
D . J. "Artist Li fe . " High Fidelity/Musical America 1 9 (May 1 969): 6. " Lo ndon Sin fo nietta ." Music
Thoma s, S. 34. Ull e n,
"
Xena kis; Pithoprakta."
Events
24 {December 1 9 6 9) :
0. "Krig och vetenskap-en myt for var tid. Kring Iannis
J.
Xcnakis." Nutida
Musik 13, No. 1 (196�1 970}:
1 2-15.
Zeller, H. R. " Zum Be is pi el Xenakis . " Melos 36 Zij lstra, M. " Het
(1 969): 4 1 0-4 1 6. Generale." Mens en Melodie 24
Utrechts Stud ium
Qanuary 1 969): 1 6.
1 970
Cecchi, C. "'Metastasis' y ' Pithop rakta' de Xe nakis. " Revista Musical
Chilena 24, No. 1 1 2 ( 1 9 7 0 ) :
9�1 02.
Chamfray, C. " Premieres auditions franc;aises (o n
Musicale de France 30 (1 970): 73.
____
ST/48)." Courrier
. " Premieres auditions fra n c;a ises (on Persep hassa) ."
Tier Musicale de France 29 (1 970) : 24 .
Deliege, C. " L'inve nzione musicale co ntemporanea
tesi) . "
(te ntativo disi n
Nwva Rivista Musicale ltaliana 4 ( 1 970), No. 4, p.
and No. 5 , p.
880-897 ;
Druey, P. " Ge neve . " SMZ
reprinted
from Syntheses 276
Q u ne
enjeu 1 ( 1 9 70 ) : 46-65. "Xenakis
M. "Messiaen,
67.
Xe n akis . " Musical
1 4, No. 3 (1 970): 1 3.
H.
1 5 ( 1 970) :
P. "Xe nakis-Pro file
2 1 2-2 1 4.
Musique
NZ 1 3 1 (1 970): 363.
Times 1 1 1
(Ja nuary 1 970):
Komorowska, M. "Muzyczna awa ngarda na festiwal u." Krellmann,
"
Talks." Music Events 25 (January 1 970): 6-7 .
Golea, A "Stra6burg Suche nach neuen Fo rme n . "
Harrison,
637-664;
1 969) .
1 1 0 (1 970): 38.
D urney, D. /Ja meux, D. " Rencontres avec lannis Xe na k is .
Gardiner, B .
Cour
der neuen Musik
Ruch Mmycmy
I V . " FonoForum
Ledes ma, 0. " Festival de Musica Conte mporanea . " Bwmos A ires Musi cal 25, No. 42 (1 97 0): 2.
"Le poeme electronique-Brusel duku mentace." Hudebni Rolhledy
2 3 : 508-5 1 0.
Formalized Music
352 MAche, F. B. "A propos de Xenakis." lA
Nouvelle Re:uue
"Pro Arte
ity/Musical A merica
( 1 970): 1 1 5-1 22.
Orchestra
(Cavalho)." High Fidel
1 970) : 24-25. Schaeffer, P. "La musique et
(1 970): 57-88.
Fran�aise 35
20 (May
les ordinateurs." &vue musicale 268-269
Schiffer, B. "Interessante Ne uh ei te n bereichern das Musikleben." Melos 37 ( 1 970): 22.
Londoner
Wag n e r, K. " Grenzen der Konzertsaalmusik Ductsche Erstau ffu h ru n " ge n in Hamburg. Melos 37 (1 970) : 5 1 5-5 1 6.
1 971
'"Akrata' for Sixteen
Instruments." New York Philharrrwnic (1 April
1 97 1 ) : E-F. B e n ary,
P. "Luzern
(1 97 1 ): 288.
Bowen,
Internationale
Musi kfestwochen."
SMZ 1 1 1
M. " Piano." Music & Musicians 1 9, May 1 97 1 ): 6 1 .
Chamfray, C . " Premieres auditions fra nc;aises (on Kraanerg and Synaphai). " Courrier Musicale de France 35 ( 1 971 ): 1 30. ,
Feuer, Gill,
M. " Royan 7 1 ." Muszika 1 4, June 1 9 7 1 ) : 2 1 -22.
D. "Stephen Pruslin." Musical Ti1Ms 1 1 2 (1 97 1 ): 262.
Galea, A
"
Ro ya n eine Ost-West-Begegnung. " NZ 1 32 (1 97 1 ) : 3 1 7 .
d e La Grange, H . L . " Music o f the East at
(26 J une 1 971 ): 38.
____ .
67 . ____..
Royan." Saturday R£uiew 54
"Persepolis." Music & Musicians 20 (Dece mber 1 97 1 ): 66-
"Royan 1 971 ." Music & Musicians 1 9
(J une 1 971 ): 47.
H. " Mai festspiele Wiesbaden." Oper und Konurt 9 (J une 1 97 1) : 29.
Hampel,
H ei ne mann ,
R. " Musik aus dem Osten." MUJica 25 (1 97 1 ) : 364-365.
Hohlwcg, R. "Im Trauern ( 1 97 1 ): 250-253. Konold, W.
"Kieler
gra m m . " Melos
war Royan diesmal am besten."
Saison
Melos 38
1 970/7 1 eroffnet mit vielf;Htigem Pro
38 ( 1 97 1 ) : 20-2 1 .
M�che, F . B . " Methodes linguistiques e t musicolo g ie. "
Musique enjeu
5
(November 1 97 1 ): 761f. "Ecn tri p riek der Luzerner Musikfestwoche." Mens Me/odie 26 ( 1 9 7 1 ) : 329-330.
Monni ke ndam, M. en
Paap, W.
"Vrijblijvend
musiceren-ee n wendelconcert in de Vara
studio . " Mens en Me/odie 26
( 1 97 1 ): 353.
A Selective Bibliography-Writi n gs " Palla ta
D hi na (Wo nders
353
are M a ny) . "
108.
" Premieres."
Music Educators journal 57
Boston (May
Symplwny
{1 97 1 ) : 407-
1 97 1 ): 90.
P. "'A propos des o rd i nateurs.' In De L'Experience Musica le a L'Explrience Humaine." Revue musicale 274-275 (1 97 1 ): 55-65. Schiffer, B. " Expcri me nte fur Klarinette. Klavier und Singstimmein Londoner Konzertsalen." Melos 38 ( 1 9 7 1 ) : 1 56.
Schae ffer,
Seelmann-Eggebert, U. " Luzern Einfii h rung in Ian nis Xenakis." NZ
1 32 (1 97 1 ): 6 1 8--6 1 9 .
S ne erso n , G. "Die musi kalische Ava ntgarde der sechziger Jahre. " Musik und Gesellschaft 2 1 (1 971 ): 754--7 60. '
'
____
(In
. " Serialism and Aleatoricism-An Ide n tity of
Russian) Sovetskaja Muzyka
Stuckenschmidt, H.
I (January
Contrasts." 1 97 1 ): 1 06-1 1 7 .
H. " Die Kl as si ke r der modernen West-Berlin erfolgreich . " Melos 38 (1 97 1 ) : 208.
Tru mpff,
G.
A "Wiesbaden 75 Ja hre
1 3 2 ( 1 97 1 ): 383 .
Zie li ns ki , T. A 1 8.
1 972
Bachmann,
C.
"
M us i k
Internationale Maifestspiele ." NZ
Y a n n i s Xenakis." Ruck Muzyczny 1 5 , No.
H.
"
sind i n
9 ( 1 9 7 1 ): 1 7-
de n Theater-Palasten- Pa rise r Festival 21 (December 1 972): 28.
Hera u s aus
auf neuen Wegen." NMZ
Cadieu, M. " Da Pari gi ." Nuova Revista Musicale ltaliana 6 (1 972): 6 00 601 .
Casken, J. "Poland. " Music & Musicians 2 1 , Chamfray, C. "Premieres auditions
Musicale de France
37 ( 1 972): 20.
September
1 9 72): 7 8-79 .
fran�aises, (on Orestie)." Courrier
____. "Premieres a ud itio ns fram;aises, (on Aroura) . "
Musicale de France 34 ( 1 97 1 ) : 78.
C hap m an, E. et 1 972):
East,
L.
1 0ff.
al. "English B ach Festival. "
Courrier
Music Events 27
(June
"Xenakis." Music & Musicians 2 0 (August 1 972): 6 1-62.
" First Performances."
WOM 1 4, No. 3 (1 972): 8 1 .
Fleuret, M . " Xe nakis-A Music (Ap ril 1 972): 20-27.
for t11e Future."
Music & Musicians 20
F loth u is , M. " Het muziekfeest der ISCM te Graz." Mens ( 1 972): 357-360.
en
Melodie
27
Formalized Music
354
Golea, A
7 1 4-7 1 6. Gradenwitz, P. "Greece." Musical
Gri ffiths,
franzo Herbstfestival." NZ 1 33 ( 1 972):
"SttaBburg!Paris: Deutsche Werke beherrschen
sischen Opernau ftakt-Mill gliicktes
Ti111J!S 1 1 3 (1 972): 73.
P. E n gl is h Bach." Musical Times 1 1 3 (1 972): 590. "
" New Music." Musical Ti111J!s 1 1 3 (1 972): 686.
____ ..
"Framtidens fac k1or l>ver Musik 1 5, No. 2 ( 1 97 1 -1 972) : 48-5 1 .
C.
Hoogland,
2000-ariga
ruiner."
Nutida
Krell mann, H. P . " Der Mathematiker unter den zeitgen&sische n Komponisten."
Melos 39 (1 972): 322-325.
Kiihn , H. " Xenakis, oder
(1 972): 467-468.
das Innenleben der Zahle n .
l uminis mo."
Musica
and To nietti, T. X e na k is tra medioevo QJ.ladrivium Rassegna Musicale 5 (1 972) : 45-66.
E.,
Napolitano,
"
e
"
26 il
Russcol,
H. "The liberation of Sound, an Introductio n to Electronic
Schiffer,
B.
M usic." (Englewood Cliffs, 1 972; within "Xenakis", 8 p. "Englische Orchester erteilen Aufttage an
ponisten. "
---- ··
Melos 39
(1 972): 302-306.
"Neue gricchische Musik."
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--=-=--=----=·
"Athe n : Xenakis-Au ffii hrungcn am FuBe der Akro poli s . "
--:---=:--·
" English Bach Festival : X e n a k is Exhibitio n . " Tempo 1 1 3
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___
____
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..
____
SMZ
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. "Greeks in lon do n (G reek Month
----
"
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31
Festi'iral
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1 977
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."
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----··
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Musik der Gegenwart, edited by H. 1 985): 52-59.
r.ur
Henck, Berg isch -Gladbach (1 984-
Formali zed
364
Abb revations and symbols Diss.
Dissertation
MQ No.
Musical Quarterly number
NZ
Neue Zeitschrifl filr Musik
ff.
NMZ DMZ
PNM p.
RILM
followin g
Neue Musikzeitung Osterreichen Musikzeitschrifl
Perspectives
ofNew Music
page (s) Repertoire International de Iitterature Musicale
SMZ
Schweiurische Musikuitschrift
Univ.
University
Vol. WOM
Volume(s) World of Music
Music
DISCOGRAPHY The following list
is
arranged i n alphabetical order by recording company
(a n d record number). Barri ng the various format exceptions (CD compact disk; MC = cassette) the recordi ngs are presented o n long-playing =
records. Each e ntry also
composer (with
an
i n d i ca tes the presence of works by a n o th e r asterisk [*]); the title of the compositions by Ia n n is
Xenakis; and the performers.
1.
Adda 81 042
Dame de Paris; E nsemble I n -
stru me ntal de musique
P. Zu kovsky, vi.
2.
Adda 581 224
perc.
3.
contem p oraine d e Paris ; K. Si-
CD *
Oophaa E. Chojnacka,
mo novich, dir.
clav ; S . Gualda,
Ades 14.1 22-2 C D Metastasis, Nuits
8.
9.
Herma
musique contem p orainc Paris ; K.
6.
de
Simonovich, dir.
Angel S-36655 Herma
B arclay 9202 1 7
Anaktoria,
*
Morisima-A 11Wrsima
1 1 . :Bis 256 * Psappha
1 2.
*
G . Pludemacher, p f.
7.
ij
Octuor de Paris
*
Ensemble I nstrumental d e
Babel 9054- 1 CD
Echange, Palimsest, Wa arg, Eonta dB ; Asko Ense mble, D. Por-
10.
J. Mefano, pf. A trees
Frant;e; M .
cel n , d ir.
Ades 1 6 005 *
S-36560
de
A Takahash i , p n o ; H. Sparn aay,
Couraud, dir.
Angel
Nuits
Tranchant, dir.
Rosbaud, dir. ; Soloists fro m the ORTF chorus ; M .
5.
Arion ARN 38775 * Groupe Vocal
*
Orchestra del Siidwestfu n k ; H ,
4.
of Notre-
Children's chorus
*
Mikka, Mikka S
G. Mortens e n , p e rc.
B is 338 * Keren Christian
Li ndberg, Tb .
1 3 . B is C D 4 8 2
Ple'iades
Angel S-36656
Achorripsis, Akrata, Palla ta dhina, ST/10
CD
The Kroumata semble
365
Percussion
En-
Formalized
366
1 4 . Botte a Musique
070
22. Chant du Monde LD C 278 368
*
Diarnorplwses
15.
1 6.
CD *
(electronic compositio n)
Eonta
CBS 346 1 226
Y.
Akrata The Festival C ha mbe r E n semble; R. Dufallo, dir. CBS Sony
Psappha
32 DC 673 CD
S. Joolihara,
1 7 . C BS
National Orch ORTF; M. Le
*
Raux, dir.
23. Colosseum
perc .
1 8.
CBS
Ensemble
24.
So n y 32DC69 1 CD
Makoto Aruga Perc ussion En semble
1 9 . C andide 3 1 049
Medea, Poly tope de Montreal, Syrmos
Ensemble Ars
Nova; ORTF Chorus; M. C o ns ta n t, dir.
C a ndide 3 1 000 Nomos Gamma, Terretektorth ORTF
Philharmonic
Ch . B r u ek , d.1r.
21.
*
Wakasugi, dirs.
Plelades
20.
3447253 C D
Stratigie Yomiuri Nippon S y m p h o n y Or chestra; Seij i O zawa and H.
Sony 32 DC 691 CD *
Aruga
Takahashi, pno ; Paris Con temporary Music Ense mble, K. Simonovitch, dir.
Metastasis Pithoprakta
Pllades M.
M usic
Orchestra '·
C ha n t du Monde LDX 78308
Colombia
MS-728 1
*
Akrata The Festival Chamber En semble ; R. Dufallo dir.
25. Con naisseur Musik CP 6 *
Mikka, Mikka
S
P. Zu kovsky, vi.
2 6. C yb ernetics Serendipi ty Music ICA 0 1 . 02 *
Stratlgie
Yomiuri Nippon Symphony Orchestra; S. Ozawa and E. Wakasugi, dirs .
27. Decca 4 1 1 6 1 0- 1 joncluLies
(also LDX-A 8368, C D LDC 28. Decca H EAD 1 3 (also 59 1 1 7 1 ) 278368) Antikhton, A roura, Synaphai Eonta, Metastasis, Pithoprakta G. Madge, pf.; N e w PhilharEnsemble I nstrumental de m o nia Orchestra; E. Howarth, musique contemporaine de dir. Paris ; K . S i m o no v i c h dir.; Orchcstre National d e l' ORTF; 29. Deno n CO 73 768 CD Le Roux, dir.; Y . Takahashi, Plelades Les Percussions de S tras b o u rg pf. ; L. Lo ngo P. Thibaud, tr. ; M. Chapellier, G. Moisan, ] . 30. D e n o n OX-7063 ND (also CD Toulon, trb . C0- 1 052) * ,
,
Evryali, Herma Y.
Takahashi, p f.
367
Discography E. Choj n aclc.a, c l av; Ensemble
3 1 . Deutsche Grammo p hon 2530562
Iannis Xe nakis; H. Kerste ns,
*
dir.
Nomos Alpha
41 .
S. Pal m, vel. 32.
Disques M o ntaig n e 782002
CD
Le cercle Trio (perc).
EAA-850 1 3-5
42. Erato Musicfrance 2292 45030-2
*
CD *
He'T1Tll/,
Khoai, Kombo'i E. Choj nacka,
A Takahashi, pf.
34. EMI C063 1 00 1 Achorripsis, A krata, Polla ta ST/1 0
Children's
dhina,
chorus of Notre-
Dame de Paris; Ensemble strumental de musique conte m p o raine de monovich ,
dir.
CD *
Ikhoor Paris String Trio
Okho
3 3 . EMI Angel
Erato Music fra n c e 2292 45 0 1 9-2
Par is ;
NUM 751 04 Kombo"i
43. Erato
Si- 44.
Erato
Nuits
G. Pludemacher, p f.
45. Erato STU
; Quatuor
B e rnede; E nsemble Instrumental de musique contem-
porainc de Paris ; K.
E M I Colu mbia
SC XG-55
Th . Antoniou,
dir.
38. EMI HMV CSDG-63
*
Syrnws
Ensemble Ars Nova ; Chorus o f ORTF;
Kraanerg
M. Co nstant, dir.
ORTF Philha rmonic Orch estra; Ch. B r u c
Orient-Occident
(electro nic co mposi tion)
40.
,
dir.
Bohor, Concret PH, Diamorphoses, Orient- Occident
4 9 . Erato STU 70656 (also
(Hor Zu)
PH, Medea,
k
(elec tro nic com p osition)
Antoniou, dir.
Concret
Co nsta n t, d i r.
M.
46. Erato STU 70527/28 *
48. Erato STU 70530 *
Anaktoria
E lektro la
70526 (also 9088)
Medea, Polytope de Montreal,
ST/1 0
39.
*
47. Erato STU 7 05 2 9 (also 91 1 9) Nomos Gamma, Terretektorh
Simonovich, dir.
Th.
Gualda,
M. Co uraud, dir.
36. EMI C V C -2 0 8 6 (also MC MCV-2086c) Atrles, Morsima-Amorsima, Nomos
37.
STU 70457
S.
Soloists and Cho rus o f ORTF ;
ReT1Tll/,
vel,
Gualda,
*
perc.
35. EMI CVB 2 1 90 *
Alpha, ST/4 P. Penassou,
S.
E . Choj nacka, clav. ;
In-
K.
dar;
perc.
Erato 2292-45030-2 CD A L'ile de Goree, Naama
9 1 37)
Oresteia
S.
Caillat Chorale ; Ma1 trisc de Notre-Dame de Paris ; En-
semble Ars Nova; stant,
dir.
M.
Con
Formalized Music
3 68 50.
Erato STU 7 1 1 06
Psappha
*
57.
Khoal
E.
52.
Orchestra
*
58.
j
Cho nacka, clav.
Ctndrles, Jonchaies, Nomos
Gamma 59 .
of Lisbon; Orchestre N a ti o n al
ORTF Philharmonic Or-
Gaudeamus Foundation
(3
Netherland
Gmeeoorh
de France; M. Ta b ac h n i k , dir. ;
-
Radio
CD, 1 988)
K. Hoek, Organ
60.
chestra ; Ch. Bruc k , dir.
Gramavisio n R2 79440 CD * Tetras
Arditti S tring
Erato I nterfaces (for
Q uartet
6 1 . HMV S-ASD 244 1
Hewlett-Packard)
Ars.
Spyros Sakkas , baritone;
Morsima-A morsima, Nomos Alpha,
Sylvia
ST/4
Gualda, percussion; Sym-
phonieorchester des
P. P e n as so u , vel . ;
Bay-
Quatuor
Berncde; Ensemble I nstrumental de musiq ue contemporaine de Paris; K. Simonovich, dir.
erischen Rundfunk, dir.
M�chel Tabach nik;
Kekuta.
Ktilner
Fi n l a n dia FACD 357 * Khoa"i
J. Tiensu, clav.
Erato STU 7 1 5 1 3
Gulben kian Fou ndation Ch o ru s
53.
CD *
Members of the Avanti Ch a m be r
S. Gualda, perc .
5 1 . Erato STU 7 1 266
Finlandia 1 20 366-2 Anaktoria
Rundfunk-S y mphonieor-
6 2 . H armoma · M un d"1 HMC 5 1 72 * c hester; K o . . 1 ner R un dfu nc h or, M!Sts '
dir. Michel Tabachni k .
N'shima .
Anne B a rtol lo ni, Genevieve
C. H el ffer,
Reno n , mezzo -sopranos ; En semble Instrumentale, dir. Mi
chel Tabach nik.
54. E rn s t Klett 92422
Diamorphoses
CD
Les Pe rcussio ns de Strasbourg
64. Hungaroto n 1 2569 (also CD
*
HCD 1 2569)
55. Etcetera KTC I 075 * Kraanberg Al p ha C e n tauri Ens., R. Wood
ward ,
dir.
Ete rna Stereo 827906
Dmaathen
9051 85
Plliades
Mists
(electro nic co mposition)
56 .
pf.
63. Harmo nia Mundi HMC
*
*
K. Kormendi, p f.
65.
J eugden Muziek BVHAAST 007 Eonta, Evryali, Herma G. Madge , pf. P. Eotvos , dir.
369
Discography 66. Limelight 86047
77.
*
Lukas Foss, dir.
Lyra 25 1
78. Nonesuch H-7 1 245
Eanta, Metastasis, Pitlwprakta
Bohar, Cancel PH, Diamorphoses,
Y. Takahashi, pf.; Ense mble I n stru mental de
mu
tem porai ne de Paris ; K .
Hermo.
Orient- Occident
s iq u e c o n
Si monovich, dir. 68. Mainstream 5000
(electro nic composition)
7 9 . Owl 26 *
Charisma
*
Jungerman, d . ; B a n ks, vel.
Y. Takahashi, p f.
6 9 . Mainstream MS-5008
80. Perfo rmance PER 8406 1 *
Jonchaies
*
Leicestershire Schools S ym
Achorripsis
p hony Orchestra ; P. Fletcher, dir.
Hamburger Ka m mersoliste n ; F. Travis, dir. 70. M usical Society M HS 1 1 87 Medea, Nuits
8 1 . Philips 652 1 020 (also 67 1 8040)
*
Couraud,
Les Perc ussions de Strasbourg
dir.
7 1 . Musical observations CP 2/6
*
8 2 . Philips 835485/86 (also A
0 0565/66 L, 836897 DSY) *
Orient- Occident
Mikka, Mikka S
(electro nic com position)
P. Zukovs ky, vl.
72.
Musidisc RC- 1 60 1 3 A na ktoria Morsi'T/UI,-A1TWrsima
83.
Paris ;
(electro nic 75.
84.
Black, Cb.
Mycene A
CD *
com p osition)
Nieuwe Muziek 004 Epe'i, Palimpsest,
Dmaathen,
Phleg;ra
Xenakis Ense mble ; H . Kerstens,
dir.
76.
Nippo n SFX-8683
Concret
PH
m u s iq u e c o n temp orai ne de
*
Theraps
74 Neu ma 450-74
B,
Ensemble I nstru mental de
Octuor de Paris
R.
Philips 835487
Analogique A et
,
7 3 . Neuma 450-7 1
*
Persephassa
E nsemble Ars Nova ; ORTF
Chorus ; M.
*
B u ffalo Philharmo nic Orchestra;
(electro nic composition) 67.
N on es uc h 3281 8 (also H-7 1 20 1 )
Akrata,_ Pithaprakta
Orient-Occident
*
Persepolis
(electro nic co mposition)
K.
Simonovich, dir.
Philips T 6521 045
Persipolis
(electro nic co m po s iti o n )
8 5 . P N M (Perspectives o f N ew
Music)
28 C D
Voy age ahsolu des Unaris vers Andrornlde
(electro nic co mposition com posed on UPIC at
86. R CA RS
90 0 9
(also
CEMAMu )
RE 25444)
Dikhthas, Emhellie, lkhaar, Kattas,
Mikka S, ST/4 Arditti Quartet
Mikka,
Formalized Music
370 87. RCA Victor JRZ-2501
Hibiki Hana Ma
93.
•
gie
stru mental
*
Yo miuri Nippon Symphony Or
S. Ozawa and Wa kas ugi , dirs.
89.
SaJabert Actuels
(dist.
H.
S i m o n ovic h , dir.
94. Varese Sarabande 8 1 060 * Strategie
Yomiuri Nippon Symphony Or chestra ;
SCD 8906 CD
Harmonium Mundi)
Oresteia, Kassandra U.
o f Strasbourg Chorus; Maltrise de Colmar; Anjou Vocal Ense mble; E nsemble
dir;
Sakkos, bar; S. Gualda, Sony CBS SONC- 1 0 1 63 • Akrata
tru m Ensemble; G. Protheroe ,
S.
perc .
dir.
In Preparation: 96. Disques Montaigtne 782xxx 3 CD Evryali, Mists, Herma, Dikkthas, A kea, Tetras, ST/4, Mikka, Mikka
dir.
"S", Kottos, Norrws A lpha,
90. Teldec 6.42339 AG (also C D
8 .42339 ZK)
Embellie
*
1 2 cellists fro m the Berlin
Philharmonic.
91 .
Telec/Warner Classics 2292
Quartet
97.
R. Hind, p n o ; London B rass
92.
Toshiba TA-72034
Evryali
A Takahashi, p f.
*
M FA (collection M us iq ue
Franr;ais d'Aujo urd'hui)
Charisma
4 6442-2 CD *
Eonta
Jkhoor,
C. H e l ffer, p n o ; Arditti S tring
Retours- Windungen The
dirs.
95. Wergo WER 6 1 78-2 CD Akanlhos, Dikhthas, Palimpsest, Epei I. Arditti, vln; C. Helffer, p n o ; P. de Walmsey-Clark, sop ; S p ec
The Festival Chamber En semble; R . Dufallo,
S. Ozawa a n d H .
Wa kasugi,
B asse-Normandie; D. Dehart, dir; R. Weddle , Vocal
de musi q u e c o n
te mpora i n e de Paris , K .
Strate
chestra ;
Pitlwprakta
Y. Takahashi, p f. ; Ensemble I n
(electro nic com p osition)
88. RCA Victor SJ V- 1 5 1 3
Vanguard Cardinal 1 0030
Eonta, Metastasis,
98.
A, Damie ns, cl ; P. Strauch, vic.
SaJabert Actuels SCD 9 1 02 CD (dist. Harmonium M u ndi) Bohor, La Ugende D 'Eer (electronic compositio ns)
I annis Xenakis B iographical I nformation
1 957: Ge n eva , 1 96 3 :
Athens,
European Cultural Fo u n da ti o n Award
Ma nos Hadj idakis Award
Berli n, Senate
1 9 63-64:
1 9 6 4 : Paris,
Ford Fou n d a ti o n Gra n t
Musiques ForrTIJJlles c h ose n
o f the Year . "
petition.
1 96 8 : Edinburgh, First
IFIP Co ngress
: Paris,
G ra nt
fro m the West-Be rlin
by the Perma nent Committee
French Book and Grap hic Arts
1 96 5 : Pa ris , Grand Prize awarded by
plus
Exhibits, to
be
o ne
of the
of the 50 " Books
the French Recording
Academy Com
Prize at the Com p uter-assisted Music Competition,
Gra n d Pri7..e awa rd ed b y the French Recording Academy
: Lo ndon,
Bax Society
Awards)
Prize (Harriet Cohen I n te rnational
Music
1 970 Paris Grand Prize awarde d by the French Record i n g Academy
1 97 1 : Tokyo Mod e r n M usic Award from the Ni p po n Acade m y Awards 1 972: London, Honorary M e m be r of the British Compute r Arts
Society
1 974: Paris, Gold Medal M a urice Ravel Award fro m the SACEM 1 975: Honora ry Member of the America n Ac ad e m y of Arts and Letters 1 976:
Paris,
Sorbo nne, D o c to ra t e s
: Paris, Natio nal G ra n d
Secreta ry o f State
Prize
Letters and
Humanities
in Music from the
37 1
French Cultural
Formalized Music
372 1 97 7 : Paris, Gra n d Prize, Ch arles Cros
du Presiden t de Ia Republiquc
Acade my
for
in ho nore m)
Recordings
(Grand
Prix
: Bonn, Beethove n Prize
:Amsterdam, music
1 9 8 1 : Paris,
Edison Award for the best recordi ng of conte m p orary
des Arts et des Lettrcs
Officier de l'Ord rc
1 98 2 : Paris, Chevalier de Ia Legio n d'Honneur 1 9 83: Paris, Member of the Institut de France (Academic : Berlin and Munich, Member
1 98 5 : Paris,
O ffi c ier de l O rd re '
: Athens,
1 986 : Paris,
of the Aka d e mi c der Ku nste
National
M ed al of Honor of the
Ordre National
des Beaux Arts)
du
Merite
City
du Merite
1 98 7 : Honorary Member of the Scottish Society of Com posers : Grand Prize fro m the City of Paris
1 988: Paris, Nominated to the 1 9 8 9 : Foreign
Victoires de la Musique
member of the Swedish Royal Acad e m y o f M usic
1 990 : Professor Emeritus of the U niversi tc de Pa ris I, Pan theo n-Sorbonne : Ho norary Docto r of the U niversity of Edinburgh
: Ho norary Doctor
of the U n iversity of Glasglow
Notes
1. Jcan
I . Free Stochastic
Music
L e diveloppement de la notion de temps chez l' enfant ( P aris : de France, 1 946) . Gravesaner Blatter, no. 1 ( 1 955) . Revue techn ique Philips, vol. 20, n o . 1 ( 1 958) , and Le
Piagct,
Pres ses Universitai res
2. I .
Xenakis,
3 . I. Xenakis,
Corbusier, Modulor 2 (Boulog ne-Seine : Archit ectu re d'Aujou rd'h u i , 1 955) . 4. I. Xenakis, " Wahrschein!ichkeitsth eorie u n d Musik," Gravesaner Blatter, no. 6 ( 1 956) . 5. Ibid .
6.
Ibid.
II.
M a rkovian Stoc h a stic M u s i c-Theory
1 . The description of the elemen tary structure of th e sonic symbols that is given here s erves as a point of departure for the musi cal realization, and is consequ ently only a hypothesis, ra th er than an established scientific fact. It can, nevert h eless,
be
considered as a first approxima tion to th e considera
tions introduced in information theory
matrix a sonic event is resol ved short e ffective du rations, whose
by Ga.bor
[ 1 ] . In t h e so-called Gabor
i n to elementary acoustic signals o f very amplitude
can
be
divided
equally i n to
q u anta in the sense of inlormation theory. However, these elem enta ry sig nals constitu te sinusoidal fu nctions having a Gaussian " bell " cu rve as an envelope. But one can pretty well represen t th es e signals of Gabor's by
sine
waves of short d u ration with an app roxim a tely rectangu lar en velope.
2. The choice of the logarithmic scale and of the base between 2 and 3 made in order to establ ish our i deas. In any case, it corresponds to the results of research m experimental music made by the author, e.g. , Diamorphoses.
is
373
374
Formalized Music
V. Free
Stochastic M usic by Computer
I. Sec Gravesaner B/iiUler, nos . 1 1 /1 2
c as e
(Mainz: Ars Viva Verlag, 1 957).
2. ( V3 )eR must be equal to the upper limit, e.g., to of a large orchestra .
1 50
sounds/sec. in the
VI. Symbolic Music 1 . A second-degree acoustic a nd musical ex perience makes it nec ess ar y abandon the Fourier analysis, and therefore the predominance of fre q uency in sound construction. But this problem will be treated in Chapter
to
IX.
2 . From previous edition of ForTIIIl liud Music, another wa y
same four for ms :
y
Z = x + yi f1
=
Z = x + yi = Z = f1(Z)
f2 = x
--yi = I Z I 2/ Z =
f3
=
-x
f4
=
-x
- yi
=
+ yi
f2(Z)
-Z = f3(Z)
=
=
-( I Z j 2j Z)
=
ori gi nal form
= inversion
inverted retrogradation
= f4(Z) =
retro g radation
to map these
37 5
No tes VII.
1. Cf. I.
Towards a M etamusic
Gravesaner Blatter,
Xenakis,
no. 29 (Gravesano, Tessin,
1 965).
Switzerland ,
2. Cf. I.
Xe na k i s , Gravesener BlliUer, nos. 1, 6;
the scores of Metastasis a nd
Pithoprakta (Lo nd on : B oosey and Hawkes, 1 954 and 1 956); and the recording by Lc Chant du Monde, L.D.X. A-8368 or Vanguard. do not mention here the fact that some present-day
3. I
q uarter-tones or sixth -tones because they really
diatonic field. 4. Cf. Chap. VI. 5.
J o h a n ni s
Terminorum
Tinctoris,
Richard-Masse, 1 95 1 ) .
music
uses
d o not escape from the tonal
Musicae
Diffinitorum
(Paris :
Jacques Chailley, " Le mythe des modes grecs," A c ta Musicologica, val . IV (Basel Barenreiter- Verlag, 1 956). 7. R. Westphal, Aristoxenos von Tarent, Melik und Rhythmik (Leipzig: 6.
XXVIII, fasc.
Verlag von Ambr. Abel (Arthur M ei ne r), 1 893), introduction in Ge rm a n , Greek text.
8.
G.
de France ,
Th. Guilbaud,
1 963).
MatMmatiques, Tome I (Paris: Presses Universitai res
9. Aristidou Kointiliano , Peri Mousikes Proton (Leipzig: Teubner, 1 963),
at librai rie des Meridiens, Paris. 1 0.
versions
versi o ns
a nd
=
6
+
The Aristoxenean scale seems to be one of the experimental of the an c i e n t diatonic, but does not conform to the theoretical o f either the Pythagoreans or the Aristoxeneans, X(9/8)(9/8) = 4/3 12 + 12 30 segm e nts, respectively. Archytas' version, X(7/8) (9/8) =
4/3, or Euclid's are sig nificant. On the other hand, the so -called Zarlino
sca l e is nothing but the so -c alle d Aristoxenean scale, which, in dates back to Ptolemy and Didymos.
1 1.
Avraam Evthymiadis
re al i ty
, only
EToLXE&waTJ Mrx8�p.aTa Bu�armvij� 'Mouaucij�
(Thessaloniki: O.X.A, Apostoliki Diakonia, 1 948). 1 2. I n
Quintilian
and Ptole m y the perfect fourth
equal tempered segments.
1 3. See
(Harmonics 2. 1 ,
p. 49).
into 60
for the displacement of the te tra cho rd (16/1 5) mesc (9/8) paramese ( 1 0/9 ) trite
Westphal, pp. XLVIIff.
mentioned by Ptolemy : lichanos
was divided
Formalized Music
376 1 4. In
the soft
Ptolemy the names of the ch ro mati c te trachords were p e rmuted : chromatic contained the interval 6/9, the hard or syntonon the
iuterval 7/6. Cf. Wes tp hal , p. XXXII.
1 5. Selid i o n 1 : a mixture of the s yntonon chromatic (22/2 1 , 1 2/1 1 , 7/6) toniaion d iato ni c (28/27, 7/8, 9/8 ) ; selidion 2 : a mixture of the soft diatonic (2 1/20, 1 0/9, 817) a nd the to nia i o n diatonic (28/27, 8/7, 9/8), etc. Westphal, p. XLVIII. and
1 6. E go n Wellesz, A
History of Byzo.ntine Music and Hymnography
Clarendon Press, 1 96 1 ), pp . 7 1 ff. the ancient scales
(Oxford : On p. 70 he a ga i n takes up the myth that
descended.
1 7. The s a me ne glige nce can be found among the students of ancient Helle nic culture ; for exam p le, the classic Louis Laloy in Aristoxene de Tarcnu, 1 904, p . 249.
1 8. Alain Danielou lived in India for many years and learned to play Indian i nstruments. Mantle Hood did the same with Indonesian music, and let us not forget Than Van Khe, theoretician and practici ng performer and composer of traditional Vietnamese music. Cf.
Wellcsz. Also the transcriptions by C. Hoeg, a n o the r great the proble ms of structure .
Byzantinist who neglected
the bewilderment of the "specialists" when they d iscove red that the B yza n ti ne musical notatio n is used today in traditional Romanian 20. Imagine
Rapports Complhnentaires du X!Ie Ccmgres international des Etudes byzantines, O chri d a, Yugoslavia, 1 96 1 , p. 76. These experts without doubt ign ore the fact that an i de n tic al phenomenon exists in Greece . folk music !
See
2 1 . Cf. m y text on disc L.D.X. A-8368, issued by Le Chant du Monde.
also
Gravesaner Blauer, no. 29, and C hap . VI of the present book.
22. Among themselves the elementary displacements are like the integers, that is, they are defined like ele m e nts o f the same axiomatics . 23. Alain Danielou , Northern Indian Music (Barnet, Hertfordshire : Hal cyo n Press, 1 954), vol. II, p.72.
24. This p erha ps fulfills Edw ard Van!se's wish for a spiral scale, that is, a
not lead to a p er fe ct octave. This information, unfortunately abridged, was given me by Odile Vivier.
cycle of fifths which w o u ld
25. These last stru ctu res were used in Akrata (1 964) for sixteen winds, a nd in Nomos alpha (1 965) for solo cello.
Notes
377 VI II. Towards a Philosophy of Music
1.
The " u nveiling of the his to ri cal tradition" is used h ere in E. Huss e rl's
sense; cf.
Husserliana,
transzendentale
VI. "Die Krisis
der
Europaischen Wissenschaften und
Phanomcnologie (Eine Einleitung in die phanomc nologische Philosophic) ", Pure Geometry (The Hagu e : M. Nijho ff, 1 954), pp . 21 -25, and Appendix III, pp. 379-80. die
Upanishads and Bhaga di Gila, references by Ananda K. Hinduism and Buddhism (New York: Philosophical Library,
2. Cf.
Coo maraswam y in 1 943) .
3 . " Perhaps the oddest th ing about mode r n science is its return to
The Nation, 27 September 1 924. have considered the original Greek text and the Burnet i n Early Greek Philosophy (New York: Meridian
pythagoricism." Bertrand Russell, 4. I n this translation I
translations b y John
B ooks, 1 962) and by Jean Beaufret in Le Poeme de Parminide (Paris: P.U.F. , 1 955) . 1 . Elements a re always real:
(earth, water, air)
=
(matter,
energy. Their equivalence had ready bee n forese e n by Heraclitus .
fire)
=
6. L u c re ti u s, De Ia Nature, tra ns . A Ernout (Paris, 1 924).
7. The term stochastic is used for the first time in this work. Today it is
synonymous with probability, aleatory, chance.
8 . E. B o re l ,
1 950),
p.
82.
Elimenls
de
Ia tMorie
des
probabilitis (Paris :
Albin Michel,
9. Uncertainty, measured by the entropy of i n fo rma ti on theo ry, reaches equa l .
a maximum when the probabilities p and (1 - p) are
1 0.
C£ I.
Xena ki s,
Gravesaner Bltitter,
nos . 1 , 6,
1 1/1 2 (1 955-8).
1 1 . I p rep a re d a new interpretation of Messiaen's ." modes o f limited trans positions," which was to have been publ is hed in a collection i n 1 966, but which has not yet a ppeared . 1 2. Around 1 8 7 0 A de B er th a created his " gammes
ho'TTIJ)wnes premiere et in
seconde," scales of a l te rn a ti n g whole and half tones, which would be written
our n o tation
as
(3. v 3 . +
2,
3. v 3.
)
+ t -
1 895, Loq uin, professor at the Bo rdeaux Co nservatory, had alread y preconceived the equality of the twelve tones of the octave . 1 3 . In
1 4. The fol lo wing is a new axiomati zation
tha n the o n e in C haps . V I a n d
VII.
of the
sieves, more natural
Formalized Music
378 Basic Assumpti<ms. I . The sensations create
discrete
characteristics, values,
stops (pitches, instants, i ntensities, . . . ), which can be represented as points . 2. Sensations plus comparisons of th e m create differences be tw ee n the above characteristics or
p
poin ts, which
o ne point to another. 3 . We are
steps.
the
can be described as the movem e n t,
dis lacement, or the step from one
discrete
characteristic to another, from
able to repeat.
iterate, concatenate tl1e above
4. There are two orientations in the iteratio ns-more iterations, fewer
iterations.
FoT'I1Ulliz.ation. Sets.
The bas ic assumptions ab ove engender three ll, E, resp ectivel y. From the first assumption characteristics will belo ng to various specific domains Q. From the seco nd, displacements or steps i n a specific domain Q will belong to set A, which is i ndependent of Q. From th e third, concatenations or iterations o f ele ments of !l form a se t E The two orie n tati ons in the fourth assumption can be fundamental
sets
:
Q,
represented by + and -. Product
Sets. a.
Q
X
!l
� Q (a pitch-po i n t Q x E s;;;
d is p lace ment produces a pitch-point) . b. combined with an iteratio n or
a
a
combi ned with
!l (a displacement
concatenation produces a displacement). We
can easil y identifty E as the set N of natural n u mbers pl us zero. Moreover, the fourth basic assumption leads directly to the definition the set o f i ntegers Z
of
fro m E.
We have thus bypassed
the
direct use of Peano axiomatics (introduced
i n Chaps. VI aud VII) in order to ge nerate an
EqtuJ.Uy Tempered
Chromatic
Gamm (defined as an ETCHG sieve). I ndeed it is sufficient to ch oose an y disp lacement ELD belo�gi1_1g to set 11 and form the product {ELD } x z. Set A (set of melodic intervals, e.g.), on the other hand, has a group structure. 1 5. Cf. Olivier Messiae n, Technique de mon langage musical (Paris : D urand, 1 944). _,.
• I I "' 8 c " _ _, ,. _,
-s - • I
I I I! "
-J -.z. I
,.
_,
C scale _ ,.,
f
_, I
..,_,
Messiaen Mode No 4
-tl- ::ft- 4-
M essiaen M od e No 4
•
·
·
1 •o
1
1
I
C"'J
Figure 3.
d
d
�·
"
....,
2.
Figure
� c ,, •
,� I
I
I
• • - ,, Figure
4.
1
IS'
1
•
..o
•
379
Notes
1 6. " .
the refore to nes higher tl1an needed beco me rel a xe d [lower] , as be, by curtailment of m ove m e n t; conversely those lower than needed become tens ed [higher], as they should be, by adjunction of movement. This is why it is necessary to say that tones are constituted of discrete pieces, since it is by adjunction a n d curtailment that they beco me as they sho uld be. All things composed of discrete pie ces are said to be in numerical ratio to each other. Therefore we must sa y that tones are also in numerical ratio to each o ther. But a mong numbers, some are said to be in multiplicative ratio, others in an epimorios [1 + 1/x], or o thers in an epi'TTU!ris .
they should
ratio [an integer plus a fraction havi ng a nu m e ra tor other than one] ;
therefore it is necessary to say that to n e s are also in
other .
et
.
.
" Euclid, Katatome Kan1m0s
Scripta Musica (Leipzig: B . G.
bring it in in
correspondence between
the context of this article .
1 7. Cf. my analysis of Metastasis, in (Boulogne-Seine : Architecture d'Aujourd'hui, 1 955.) a nd
same ratios to each
Teubner, 1 9 1 6). This remarkable text already
attempts to establish axiomatically the nu mbers. This is why I
these
(1 2-24), in He n ric us Menge, Phaenomena
1 8. Cf. Score by Bobsey a nd Hawkes, eds. , and
Angel.
Corbusier,
tones and
Modulor
2
record by Pathe-Marco n i
1 9. Hibiki-Harut-Ma, the electro-acoustic composition that I was commissio ned to write for the Japanese Steel Federation Pavilion at the 1 970 Osaka World Expo, used 800 loudspeakers, scattered in the ai r and in the ground. They were d i vided into approximately 1 50 i ndepende nt gro u p s The sounds we re designed to traverse these groups according to various kinematic d ia gram s After the Philips Pavilion at the 1 958 B russ e ls World's Fair, the Steel Pavilion was the most advanced attem pt at pl ac i n g sounds in space. However, o nly twelve independent magnetic tracks were available (two synchronized six-track ta pe recorders). .
.
20. Mario Bois,
Iannis Xenakis: Tlu
Boosey and Hawkes, 1 967).
Ma n
and His Music
(New Yo rk :
2 1 . Jean Piaget, u dlveloppement de la noticm de temps chez l'enfant, and La representation de l'espace cluz l'enfant (Paris . Presses Universitaires de France, 1 94 6 and 1 948).
380
Formalized X. Concerning Time,
Space and Music
1.
Sh a n n o n C . an d Weaver W. ,The Mathematical
2.
Eddington, The Nature of the
(Urbana: University of I llinois Press, 1 949).
1 929).
3.
Prigogine, 1.,
5.
Morrison,
Music
Theory of Communication
Physical World (New York:
Mac millan ,
Physique Temps et Devenir (Paris : Masson, 1 982).
4. B orn, Max, Einstein's Theory of Relativity (New York: Dover, 1 965). April, 1 957.
Philip,
"The Overthrow of Parity,"
6. Gardner, Martin, " Can Time Go
1 967, p .9 8 .
8.
Backward," Scienlifu American, Jan.
H., The Philosphy of Space and Time (New York: Dover,
7. Reichenbach,
1 958).
Scientific American,
Linde, A D., Physics Letters (1 983), 1 2 9 B 1 77. ,
9. See also Coveney, Peter V.,"The Second Law o f Thermody namics:
Entro py, Irreversibility and Dynamics," Nature N°
333 ( 1 988).
1 0. The idea of the Big Bang, a consequence
the
universe)
Stravroulakis,
et
" Solitons
propagation
generale," Annales de La Fondation
11.
Russell,
1 2.
Cf.
1 96 1 ).
Structure . "
B,
1 3. C £ Xenakis,
au tori
vari
(a
philosophie mathhnatique " New Proposals
(Paris: Payot,
in
Microsou nd
cura d i Enzo Restagno) (Torino: E.D.T.,
Earlier articles on " sieves " by Iannis Xenakis have appeared
1 968,
Paris;
Tempo
no
Nef no 29,
1 967, Paris; Revue
93, 1 970, as well as the
d'EstMtique
prev io us
in
vol .
editions
of
2. As for rhythm outside of Weste rn civUization, cf. AROM, Simha, " D u a Ia main: Les
fondcments
metriques
d Afrique Centrale ; " Analyse Musicak 1 o '
{1 987).
XI. Sieves
Formalized Music.
pied
1 2 No 4
Formalized Music,
Preuves, Nov. 1 965, Paris; La xxi ,
de Broglie
Introduction a La
chapter 9 in
1 988).
1.
of the shift (expansion of physicists. See Nikias d'actions suivant la relativite
toward the red, is not accepted by all
des musiq ues
trimestre, 1 988.
traditionelles
38 1
Notes
M
=
3.
Let there be
mk
*
n1 . . . * rt
(M, 1), with M bei ng a com posite of the form:
It is s om e ti m es necessary and . k (m , Im ) n (n 1 ,In) n . . . (rJ, Ir) =
possible to decompose it into :
(M, 1).
4. Euclid's algorithm. Let y, x be two positive whole numbers. B egi n by letting D MOD(y,x), then rep lace y with x and x with D . If D is not equal to 0, then the last is the largest common 0, the n start over. But i f D denominator. Let us call this last y, D . =
=
take two numbers : y, x 1)
L
2) 3)
D ...... M O D ( y,
y
x)
+- x, x +- D
yes -�no
$ END
exa mple : y D Y
=
30,
+-
X =
:J
21
MOD(30,2 l ) -
+- 2 1 ,
X +-
9
9
D
+-
MOD(2 1 , 9) y
D +- 9 � 0
+-
9,
=
3{
X +-
D -<- 3 � 0
3
D
+-
MOD(9,3) y
+-
3,
=
X +-
0
0
D +- 0 = 0
therefore D +- y = 3 END
a modulo b, notated MOD(a, b), is equal to the residue o f the division of a by b : a I b = e + r I b where r is this residue, i f a, b, e, and r are elements of N. 5.
6. MOD(g
� * C2 I C I
= v
*
+
C2 ,
Cl)
1 I Cl.
=
1 represents
the integer e q uatio n :
Index Achorripsis, 2 4 , 26-38,
cauchy, 266, 290, 293 See Probability
13 1 , 1 3 3 -4 3 ,
295
Alypios, 185
Analogique A , 79, 98-1 03 ,
See also Markov chain
cauchy function.
laws
1 0 5 , 109.
causa1ity, 257, 258, 259: principle,
1, 4, 8-9
Analogique B, xiv, xv (illustrations),
Chailley, Jacques, 1 83
CEMAMu (Center for Studies in
79,
1 03-9. See Markov chain 203 Anaximander, 2 0 1 Anaximenes, 20 1 Arc sine function. See Probability
Mathematics and Automation of Music), xii, xiii, 329
Anaxagoras,
Chance, 4, 3 8-3 9, 259; definition, 25 Channels (in computing), 3 29-34 Charbonnier, Jeanine, 1 3 3
laws "Arcs," 329-34; (definition), 329
Combaricu, 1 83
Aristotle, 1 8 1 Aristoxenas ofTarent, 269
Computers, 258, 266, 268, 329-3 4 :
hardware, 333 -34,software. 3 3 3 3 4 ; See also Stochastic music
Aristoxenos, 183-9, 195, 202, 208, 210 Ataxy, 63, 75-78 Atrl!es, 136-37, 144
Concret PH, 43
Coveney, Peter V., 260
Debuss y, 5, 1 93, 208 ; Debussian whole-tone scale, 196
Bachet de Meziriac, 272
Delayed
Barbaud, P., 133
Barraud, Jacques, 1 3 1 , 1 3 4 Beethoven, 1 , 1 64 , 169
Diamorphoses, 43 Didymos, 186
Bernoulli, Jacques, 206 Bessel function, 247
Distribution, random-like,
Bang theory, 259, 260, 2 95
Blanchard, P.,
257
Detenninism, 204-5
Bernoulli, Daniel, 205
Big
choice",
Descartes, 263 : discourse on method, 54
265
Duel, 1 3 -22, 1 24 . See Game theory
133
Boltzmann, 1 5, 255, 257;
theorem, 6 1
Eddington, 255
Einstein, 256, 263
Boo l ean operations, 209
Englert, F. , 260
Emile, 39, 206 Bo u dou r i s, Georges, 133 Brout, R., 260
Borel,
Entropy, 1 6, 6 1 -68, 75-78, 2 1 1 , 25 5 , 256, 257; definition, 6 1 , 1 8 6; mean e ntropy, 75
Brownian movement, 289
Envelopes (in computing), 3 29-34 383
Index
384 Epicur us , 24, 205-6, 237
Heteronomy. Sec Ga me theory
56, 67 Exp one ntial probability function. See Probability laws
Ergodism,
Flux, 266
I nteractive Isaacson,
Fourier, 258, 266, 293 : series,
242
266, 2 93
Fractals, sounding,
Frechet, Maurice, 7 9
French Cultural Ministry, 329fn Frequency and amplitude tables, 329 Fulchignoni, E . , 43
Gabor elementary
1 03
signals, 54, 58,
Game theory, 10,
133 ; a nal ysis of Duel, 1 3-22 ; a nalysis of Slrralgie, 1 22-23, 125-30; autonomous music, l l O-l l ; heteronomous music, 1 1 1 - 1 3 ; two-person zero sum, 1 1 2 ;
Gauss, 2 66
Gaussian probability distribution. See Probability l aws Genuys, FranCiQiS, 1 3 1 , 1 3 4
Glissando. See Sound
Graphic score editing,
243
co m posi tion
170
,
3 29-34
1 33
Isotropy, 1 4
Kinetic theory o f gases, 1 5 , 49, 95, 205, 2 13 , 244
261 Korybantes, 202 Kratylos, 259
Knowled g e,
La Ugende d'Eer, xii, 293 , 296 Lamb's shi ft", d efi n i ti on, 259 Landmark points, 264, 265
Law, definition , 258
Law of large numbers, 4, 8, 1 6, 3 1 , 206 Laws, 267
Le Corbusier, 10
Lee, T. D., 256
Leukippos, 203 light, velocity
329
Gregorian chant, 1 83 Groups, 210 ; Abelian additive group structure, 1 60, 199 ; Klein,
Ala n
1-Iindemith,
Hyp erbolic cosi ne function . See Probabil i ty laws
257
Fletcher-Munson diagram, 47-49, 243
' Guth,
H i ller , 133
Hucbald, 183, 202
Fermat, 20 6 Feynman,
Hibild-Hana-Ma, 269, n. J 9
of, 256
Linde, 259
p robab il i ty function . See Probability l aws liouville, equation of, 256
linear
logic, 276 ,
259
Guttman. N., 1 3 3 Gunzig, E., 260 Heisenberg, 20 6 n . , 237 Henry, Pierre,
243
Heraclitus, 267 Herakleitos, 202, 259 Herma, 175-77
log is tic
laws
function. See Probabi l ity
logistic probabilities, 266 lorenz-Fizgerald, 256 Lucretius, 205 Levy, Paul , 15, 24 Macrocom p osition, 22 ; methods, 49. See microsound structure
Index
38 5
Ma croscopi c com p [osition, 256 Markov chain, 73-7 5 . 1 3 3 , 244, 2484 9 ; analysis
of compositional ap proach, 79-98; entropy of, 86; matrix of transition probabilities, 74-75, 78, 82-4, 1 09; realizatiou of Analogique A, 98- 1 03 ; realization of Analogique B, 1 03-8; sta tio na ry dis tribu ti on , 75, 85; use of screens, 79- 1 09
Mari no, Gera rd , 277 , 329fu
Manima-Amorsima, Mathews,
144
M . V., 133 , 246n.
Matrix of transition probabilities. Sec Markov chain Maxwell, 15 ; formula, 55
Memory, 258, 262, 264, 266
Mess ia en, Olivier, 5, 8, 1 92, 2 08, 2 1 0, 268
Meltzllais, 2 -3 , 10
Mey e r-Eppler's studies, 244
Microcomposition, 22-23 , 50. See Microsound structure
Microsound structure, 24 2- 5 4 ; mi crocomposition based on prob ability distributions, 2 4 - 9;
macrocomposition, 249. See Pour
ier series MIDI devices, 3 3 2 Mod ula ting arc assignment, 329
Moduli, 267
Monteverdi, 208 Mousso rgsky, 208
Music, definition and historical back ground, 1, 4-5, 8; anc ie n t Greek, 1 82-85, 1 9 2 ; atonal, 4; Byzantine, 1 82, 1 86-92, 208; electronic, 8, 52,
243; electro-acoustic, 243; electro magnetic, S, 1 6, 43 , 1 03 . See Stochastic music, Serial music, Game theory
Nomos alpha , 208 -9, 2 1 9-36
No mos gamma, 2 17 , 2 1 9, 236-4 1
Non-linear fWJ.ctions ,
2 95
Nothingness , 259, 260, 2 6 1 , 267, 295
Orienl-Occident, 43 Originality, 258
Orp hism , 2 0 1 -2
Parity symmetry, 256
Parm enides, 24 n., 202-4, 207, 209, 259, 260, 262, 263, 2 67
Pascal, 206 Peano, 1 59
n.,
1 94-95, 265
"Performance" system, 3 2 9
Periodicities, 268
Philippot, Michel, 19-42 ; Composition pour double orchestre, 39 Philips Pavilion, 6-7, 1 0- 1 1, 4 3
Piaget, Jean,
5,
160
Pierce, ]. R., 133
Pitches, 267, 268, 269
Pilhoprakla,
15, 1 7 -2 1
Plato, 1 , 1 7 9, 202, 257 : Politicos, 257, 2 95
Poincare, H., 206
Poisson law. See Probability laws Prigogi n e, I., 256 Probabilies, 256
Probabilistic wave form, 289; Sec also Probability Probability, definition , 2 0 7 ; Cauchy, 246, 251 -52; d is tr ib u tio ns , 260; ex ponential, 12, 134, 142, 246;
Gauss i an, 14-1 5, 33, 56, 60, 140, 24 6; Interlocked densities : ex ponential X Cauch y, 2 5 1 -52; lo
gistic X exponential, 254; hyperbolic cosine X exponential, 253-54: laws, arc sine, 246; linear, 1 3 , 1 3 6; logistic, 246, 250, 291 , 293; Poisson, 1 2 , 16, 23-25, 29-32, 54, 66, 133 , 246-47; uniform, 246; Wiener-Levy, 247; th eo ry , 9, 255
Ptolemaeos, Claudios, 1 85
Ptolem y, 1 86, 1 88 Pythagoras, 1 , 207, 209; concept of numbers, 201 ; Pythagoreans, 1 85, 201 , 242; Pythagorism, 2 0 1 ,
204;
tradition, 1 93
Index
386 Quantum mechanics, Quantum
physics,
definition
256
257
Quintilianos, Aristides,
1 85-86
Raczinski, Jean-Michel, 329fn Raga, 1 98
as phase of a musical work and descri p tion, 22-25; sound-points, 1 2 - 1 3; transforma·
tion of sets of sounds, 1 6 ; white noise, 289. See Screens and Micro sound structure
Sound mass es, 255
Ragas, Indian, 268
Space, 257, 259, 2 62, 267: discon-
Ra h n John 268 n. Random walk, 289
Spatial Ubiquity", 256, 257
,
tinu ity of, 263
,
Ravel, Maurice, Bolem, 76
Real-time drawing,
Spindel, P., 260
Statistical definition, 1 6
329
Reichenbach, Hans, 25 7
Steinhardt, Paul, 259
Relativity, 257 : theory of, 256
Stochastic, 10, 39, 43-44, 1 94, 2953 2 1 ; by computer, 1 3 1-44, ma chine-oriented inter p retation of Achorripsis, 1 34 -43 ; computer
Repetition , 258, 259, 266, 267
Rhythms, 264, 265, 2 66 Rule See Law
pro- gramming and musical nota 145-54; construction, 4; definition, 4 ; dynamics, 256;
Rules, 260
Russe ll, Bertrand,
tions of, 1 95, 265
Scale, 268, 269 : diatonic, See also Sieves
music, 8,
269, 276
Scales of pitch, 264
Schaeffer,
243 Hermann, 15, 24, 260
Pierre,
Scherchen,
SchOnberg, Arnold, 2 07-8, 243 Screens, 50-79; 108-9 ; consttuction, 66-68; definition, 51; elementary operations, characteristics, 56-58; l ink i n g screens, 69-78; summary, "
56. See Markov chains
Seq ue n ce
"
(in computing), 329
Serial music, 4, 8, 182, 186, 192, 208, 255; composition, 38;
method,
52; system, 204, 243;
Vienna school , 5 , 8, 1 93 Serra, Marie-Helene, 3 29fn
displacement,
198;
elemental
theory,
1 94-
transformations of, 275 Simonovic, C., 134 Sound, clouds, 1 2; as phase of a 200;
musical work, 22-23; glissandi, 1 0, 1 3 - 1 4 , 32-36, 55, 1 40, 1 67-68, 2 13 ;
nature
Poisson
Stochastic music, 255 , Stoclws, 4, 92, 94
289 ff., 295 ff.
Stoicism, 205
Slrallgie. See Game theory Stravinsk y, Igor, 5, 192
ST/1 0-2, 080262, 1 3 4, 1 3 8-39, 1 44, 1 54 ST/48- 1, 2401 62 , 144
Symmetries, 268, 289; 269: See also Space Synnas, 80, 81 n .
(repetitions),
Systems, dyn am ic, 293
Shannon , 255
Sieves, 265, 267, 268-7 6 :
12, 145-53, 182, 255: musical composition, 5 , 43 ; Jaws, 9, 1 2-16; process, 8 1 , 244; science, 8; synthesis, dynamic, 28993; See also Probability laws: ex· ponential, Gaussian, linear,
of, 43-50; sonic entities,
Takahashi,
Yuj i , 175
Tempo, 276
See also Rhythm
Term, 69
Terreteluo rh, 2 1 7, 236-37
Thales, 20 1
Timbre, 266, 268
Index
387
1iiDe, 256, 257, 258, 259, 2 6 1 , 262, 263, 265, 266: flux of, 262, 263 ,
264: irreversability of, 256
Transformation, 69-7 5, 2 5 7 ; anthro poiDorphic, 2 6 1 : definition, 69;
stochastic transformation, 73, 75
Transition, 69, 73 -74
Transitional probabilities, 44 Transposition, 276
Uniform function . See Probability
tion), xii, 329-3 34
Voice editing (on the UPIC), 329
waveforms, 329-24
Wiener-Levy process , 24, 247 Wittgenstein, Yang, 256
Tryon, Edward, 259 laws
Variety, 61-63
Varese, 8, 243
Vilenkin, Alexander, 259
Tinctoris, Johannis, 1 83 Tonality, 2 5 8
UPIC ( U nite Polygogique
Variations, 2 7 6
Informa
Zarlino, 202
203
