Page 1

COMPUTER CANTATA: A STUDY IN COMPOSITIONAL LEJAREN A. HILLER

METHOD*

and ROBERT A. BAKER

INTRODUCTION Cantata**is the name we have given to a new composition generComputer ated by means of electronic digital computers as part of our program of studies at the Experimental Music Studio at the University of Illinois. This new example of computer music is the first large-scale computer composition completed in this studio since the Illiac Suitefor StringQuartet was written in 1957 by L. A. Hiller and L. M. Isaacson [1,2].t This ComputerCantata presents the results of a series of studies in computer music composition carried out in the spring of 1963. These studies were designed to test the efficiency and ease of use ofMUSICOMP (MUsic SImulator-Interpreter for COMpositional Procedures), a completely generalized programming scheme for musical composition intended for use with an IBM 7090 computer. Indeed, in some instances, sections of the ComputerCantatawere completed before the final version of MUSICOMP had been decided upon, and changes, additions to, and deletions from the MUSICOMP process resulted from these initial "field tests." Since our primary purpose was to demonstrate the flexibility and generality of MUSICOMP, the ComputerCantatapresents a rather wide variety of compositional procedures, some of which proved of greater esthetic value than others, and many of which could be improved by more sophisticated logic. Nevertheless, the interested composer should find these studies of significance as a concrete demonstration of the broadening of the research area of experimental composition techniques made feasible by computers and by a program such as MUSICOMP. MUSICOMP is the outgrowth of several years' experimentation with developing a logic of musical composition that is not bound to specific style parameters, historical or otherwise. We have reviewed the gradual * The authors wish to acknowledge the support of the University of Illinois Research Board which permitted them to carry out this work. ** The score of Computer Cantatais being published in the New Music Edition by Theodore Presser Co., Bryn Mawr, Pa. t Numbers in brackets refer to the bibliography at the end of this article.

62'


COMPUTER

CANTATA

evolution of our programming concepts as they have led to MUSICOMP in several recently published articles [3,4,5]. The most recent of these contains a brief description of MUSICOMP itself. As noted therein, MUSICOMP is so set up as to be readily taught to composers, since it employs much ordinary music terminology as part of its programming vocabulary. MUSICOMP is in turn written in one of the standard machine languages employed for IBM 7090 computers, namely, SCAT, or Share-Compiler-Assembly- Translator [6]. We are at present also preparing for publication a MUSICOMP programming guide that will describe this system of compositional logic in complete detail. Because of the impending existence of this manual of compositional method,we have limited the present article to a discussion of first resultsachieved by application of the method. The new ComputerCantatanot only contains results of testing MUSICOMP compositional procedure in an IBM 7090 computer, but it also includes two studies of computer sound synthesis carried out with a second computer, the CSX-1, a special-purpose computer designed and built by the University of Illinois Coordinated Science Laboratory [7]. The use of this computer for sound synthesis has recently been described by J. L. Divilbiss, the inventor of the process [8]. As will be shown later, we linked the two computers functionally by writing a special output routine for MUSICOMP that permits compositional results to be converted directly into computer generated sound. The present ComputerCantatais the same composition as the projected SecondIlliac Suitereferred to in earlier publications [3,4]. The reason for the change in title, aside from the fact that our objectives underwent continuing modification as we actually produced the composition, was that the ILLIAC, the original University of Illinois digital computer, was withdrawn from service in January, 1963. Prior to this time, we had already written some of the MUSICOMP routines for the ILLIAC. Because the ILLIAC was removed from service, we were required to rewrite the whole program for the newer computer. In the long run this has proved advantageous not only because the newer computer is more efficient and versatile, but also because it is a standard computer that is widely used. For this reason, MUSICOMP becomes immediately available to many potential users, whereas had it been written in ILLIAC programming language, it would have had a much more limited direct use. GENERAL ORGANIZATION OF THE COMPUTER CANTATA

The ComputerCantatais built up from eleven separate sections grouped into the five-movement performance plan shown in Ex. 1. *63?


PERSPECTIVES

OF NEW

MUSIC

I Prolog to Strophe I (Rhythm Study for Percussion) Random Prelude to Strophe I Strophe I (Zeroth-order Stochastic Approximation) II Prolog to Strophe II (Totally Organized Instrumental Music) Strophe II (First-order Stochastic Approximation) III Prolog to Strophe III (Polytempered Computer Sounds) Strophe III (Second-order Stochastic Approximation) Epilog to Strophe III (Polytempered Computer Sounds) IV Strophe IV (Third-order Stochastic Approximation) Epilog to Strophe IV (Totally Organized Instrumental Music) V Strophe V (Fourth-order Stochastic Approximation) Epilog to Strophe V (Rhythm Study for Percussion) Cantata Ex. 1. Performance plan of the Computer

Each movement consists of a strophe preceded by a prolog or followed by an epilog. Thus, the complete structure is at once symmetrically organized in terms of prologs and epilogs and yet encompasses a progressively unfolding plan of composition in the strophes themselves. Specifically, the strophes are all related by a common compositional technique, though not by common thematic material, while each prolog is related by a common technique to one epilog symmetrically placed in the sectional plan that is shown in more detail in Ex. 2. Thus, Prolog to Strophe I and Epilog to Strope V form complementary studies for percussion instruments only; Prologto StropheII and Epilog to StropheIV present separate studies of total organization procedures; Prolog to StropheIII and Epilog to StropheIII present allied studies for CSX-1 computer sound synthesis. In contrast to this, the five strophes employ 64?


COMPUTER

CANTATA

a stochastic compositional logic in which the stochastic order progresses from zeroth-order through fourth-order with the successive strophes. Lastly, the instrumentation plan for the various sections was designed to provide contrast between the prologs and epilogs and the strophes themselves. In general, the strophes employ a larger and more varied instrumentation than do the prologs or epilogs. The instrumentation was chosen to represent the widest selection of instrumental types, timbres, and performance techniques possible within a moderately sized chamber ensemble. As shown in Ex. 2, we designed the instrumentation to include, first, two representative members of each of the common instrumental groups of woodwinds, brass, ordinary strings, and pitched percussion played with mallets, and second, soprano voice, a group of eight unpitched percussion instruments, a typical plucked stringed instrument (a guitar), a typical electronic instrument (a theremin), and finally, various types of electronic sounds. We felt that two instruments from each of the standard instrumental groups were sufficient to demonstrate the programming of these groups in MUSICOMP. Had we so desired, full orchestra could have been employed for the ComputerCantata without exceeding the present capabilities of MUSICOMP. We included in the ComputerCantatarepresentative examples of the various significant categories of electronic sound. In each case, we deliberately selected the simplest example of each. The theremin was chosen as the electronic musical instrument. In actual fact, the theremin part turned out to be very difficult to perform on this instrument, so we have indicated in the score that this part may also be performed on an ondes Martenot, an electronic organ, or other electronic instrument. For ordinary electronic sounds, we selected, first, the three most basic periodic signals, namely, sine tone (fundamental only), square wave (fundamental plus odd upper partials), and sawtooth wave (fundamental plus all upper partials), and second, two types of noise, namely, white noise and ordinary noise. This ordinary noise, which we called for convenience "colored noise," was represented by eight characteristic recorded concrete sounds designated in the score by the mnemonic signs, CLICK, CLACK, SISS, CRACKLE, SNAP, POP, BANG, and BOOM. The third category of electronic sound was, of course, synthesized by means of the CSX-1 computer. A 15 ips performance tape for two-channel tape recorder was made up employing the newly constructed electronic music equipment in our Experimental Music Studio [9]. This tape is composed of five "cues": Cue 1 accompanies StropheI, Cue 2 accompanies StropheII, Cue 3 and Cue 4 are the Prolog to Strophe III and the Epilog to Strophe III, and Cue 5

*65?


Ex. 2. Detailed organizational scheme of the Compu III II

I Section

Primary Organizational Principle

Prolog to Strophe 1

Prolog to Strophe II

Strophe I

Percussion Zerothorder study: Density- stochastic process intensity vs. time

Strophe II

Prolog to Strophe III

Strophe III

CSXSecondCSX-1 Firstorder sound s order sound synstochastic thesis stochastic thesis of process 9 throu process 9 through 15-to 15-tone tuning tunin

Total organization study

Instruments: Voice

*

*******

*******

Flute B. Cl. Trumpet ~ ~:4:4 ~~

~~:~:~

~ ~~ ~x c c ~~:~

~~::~~

Horn Xylophone Glock. Snare Drum

~c~~l:~

~

Epilo to Strop III

~

~~I ~:~

~

~:


Tambourine Castenets Cymbal Tabor Maracas

****************~'******

Bass Drum *************

Tam-Tam Guitar Violin Viola Theremin Sine tone Sqr. wave Saw-tooth White N. Colored N. CSX-1

******


PERSPECTIVES

OF NEW

MUSIC

accompanies StropheIV. The disposition of sounds over the two channels in the five cues is given in Ex. 3. Cue1:

Cue2: Cue3: Cue4: Cue5:

Left Channel Colored Noise: CLICK SISS SNAP BANG Squarewave Squarewave CSX-1 Sound: 3 Voices CSX-1 Sound: 3 Voices

Both Channels White Noise

Sinewave Sinewave

Right Channel Colored Noise: CLACK CRACKLE POP BOOM Sawtoothwave Sawtoothwave CSX-1 Sound: 3 Voices CSX-1 Sound: 3 Voices

Sinewave

Ex. 3. Contentsof the two-channelperformancetapeforthe Computer Cantata

THE STROPHES

The texts of the five strophes are five successive stochastic approximations to spoken English derived from a synthesis by a computer of stochastic phoneme sequences. These syntheses were prepared with the ILLIAC and are based on the statistical analysis of a corpus of English text drawn at random from the publication, PLA YS, The Drama Magazine for YoungPeople, published by Plays, Inc., Boston, Massachusetts. Zeroth through fourth-order transition frequencies for the phonemic structure of this material were computed first by Professor John B. Carroll of Harvard University and then by Professors Lee S. Hultzen, Joseph Allen, Jr. and Murray Miron of the University of Illinois. These latter investigators then synthesized successive approximations to phonemic English from zeroth-order through fourth-order. With their permission, we employed samples of these approximations as the texts of the five strophes of the cantata. In Ex. 4, we reproduce the texts used. The pronunciation key is also given. Since each successive strophe imposes further constraints upon the choice of a text symbol, we decided upon a simple stochastic technique for choosing musical elements which would parallel this successive ordering. The elements to which this technique was applied were pitch, duration, dynamics, "play" or "rest" choice, and playing style. We next selected a short sample of music (Ives, Three Places in New England, second movement, mm. 14-39), and obtained frequencies of occurrence of pitches, durations, rests, dynamics, and playing style. Because of the 68 -


COMPUTER

CANTATA

limited size of the sample, the analysis could not be carried beyond second-order (digrams). Relative frequencies were then calculated and these were used as probability distributions for making choices of musical elements for the first three strophes depending on the stochastic order of the text used. Thus, for the first strophe, a random or uniform distribution was used, for the second strophe the results of first-order analysis, the probabilities p(i), were used, and for the third strophe, the results of second-order analysis, the conditional probabilities Pi(J), were used. Each of these first three strophes, then, represents an increase in the "orderedness" of the content over that of the preceding strophe. STROPHE I (Zeroth-order Approximation) shhlkachg # # mli-ln6o # eudaoaoshissn # io6zhvmpathylefa6sh6shhee6oapk#aaoozsaangep]shz6oaugaofieaossrodkefngthangfnglkbt# STROPHE II (First-orderApproximation) hlutrmkytm # tweyeh # yawa # neitbr # uunhfnw # d #lr# # r# # 1# zumhgT# uiludwodooo556yrsyr #o# tntw6 # # r# # rTw# #iavy?o ahiwd # STROPHE III (Second-order Approximation) enubr6h # sookoham # kahwu # ylaft # ay #ltiubitaw # ganet#y #jhin # dr # t # n #ykraw w #fen # hynd #ch # hr # turumenel # bayzthimay # epin # tahonehay #ingk # may # tel # fu # dwnt # kenay # mubariiy #m #w#hl #m #sit # t #ar# tui #nd #anantirkangtlhlay# pu# STROPHE IV (Third-order Approximation) in # kumensupren # bawsteyl # krawr # wiy # hav # yoow # y6hr # of #yoown # an # pebd # wan # d6hr # ntkuirz# 6wn #jhenits # hwezmn# yehrd # ubliwm # tii # pant # leng # an # intap # thu #jhan #its # taym #wk #it # g-wzf

#sho'od

# ubaywur

# eyk #zhl

# ayn #khat

# but #It

#fir# STROPHE V (Fourth-orderApproximation)

skohr# feyvd #i.zhir # perpus # ndur # gntu # hat #t weyl #skowl # ti #sey #sy # neyt # thretnitliy #an # layk # anay # kevur # ohlvyi # iay # l6wnliy # hEhr # y # d_w # hwehr # weyt # fowk6w # but # menitly # hapin # t6hwz # threy # sed # kweyndim # ay #jhlst yohru # Ex. 4. The stochastic text of the ComputerCantata with pronunciation scheme

- 69


PERSPECTIVES

OF NEW

MUSIC

Pronunciationkey

a a a a e e e i o 6

ale add ask arm

b ch d f

be chair

eve end over

g h

go hat joke (jhoke)

jh k 1 m n

pity note

oo oo

bought (bot) fooot fofod

u

up

day fill

keep late man no

ng p r s sh t 4k th

sing put red sit she to then thin

v w

very will

y z zh

yet zone vision (vizhion)

Ex. 4 (cont.)

Since we were unable to carry our analysis further, and thereby provide a basis for further increases in structural order without resorting to a priori compositional rules, we decided to follow the symmetrical plan of the composition as a whole by providing for a systematic decrease in order during the last two strophes, but with the reduction occurring more gradually than did the increase during the first three strophes. Accordingly, we devised a mathematical model which would effect this decrease by means of a weighted averaging process as follows: given any sequence of n preceding choices, the choice of a succeeding element was made dependent on each of the n foregoing elements by obtaining the weighted average of the second-order probabilities for each possible outcome that might arise from each of the preceding outcomes. For two preceding elements, this is expressed mathematically as: f(k) = cipi(k) + c2pj(k)

* 70


COMPUTER

CANTATA

where i and j are the preceding elements, and cl and c2 are arbitrary constants. This represents, of course, the probability of only one out of all possible outcomes. The probability distribution for all outcomes can be expressed in matrix notation as: Ap(kl) pj(kl) .. Pi(k2) pj(k2) ...

C1 C2

Pn(kl Pn(k2)

pi(kr) pj(kr) . . . pn(kr)

p(kl) p(k2)

Cn ,

,

\

(kr)

1

/

where the sum of each of the columns of the first matrix is equal to 1, and where the constants are chosen such that Zci = 1.2 We know from Shannon's equation: -

Hn =

Ei

log2 pi

that information content increases (and consequently structural order decreases) as the probability distribution approaches uniformity, or equal probability for all possible outcomes [10]. Conversely, as the total probability weight tends to become concentrated at a few points, information content decreases, reflecting a more ordered structure. Since an increase in the probability weight at one point must be accompanied by a decrease in the weights at other points, it is clear that the variance of the probability distribution increases with reduction of information, and vice versa. The constants, ci, in the above expression determine the relative amount of influence of an ith preceding element on the current probability distribution. If all constants were made equal, an unweighted average distribution would result; that is to say, all preceding outcomes would influence equally the choice of the current outcome. For our purposes, we arbitrarily chose the constants such that the first preceding outcome was given twice as much influence as the second, the second twice as much as the third, and so on. Thus, for the fourth-order strophe the constants were 4/7, 2/7, and 1/7. For the third-order 1 Written out

explicitly, the matrix notation above indicates

p(kl) = clPi(kl) + C2pj(kl) + *P(k2) = C1pi(k2) + C2pj(k2)+ ... P(kn) = Clpi(kn) + C2Pj(kn) +

--

+ CnPn(kl) + c,n(pk2) + cnn(kn)

Thus, the column to the left of the equal sign represents the probability distributionfor the kthchoice point, with each member of the column representing the probability of one out of the n possible outcomes. 2 The notation -ci simply means the sum of all values of c, namely cl + ** + c,; the index i thus takes the successive values, 1, 2, 3, 4 ... n.

* 71 -

C2 + c3 + c4 +


OF NEW

PERSPECTIVES

MUSIC

strophe the constants were 2/3 and 1/3. This is a not unreasonable plan in view of the finding that the influence of past choices on structural order in known communication systems appears to fall off with the distance of that choice from the choice about to be made [11]. For any set of constants in which at least two are unequal to zero, our mathematical model can only result in a decrease in the variance of the probability distribution, since any resultant p(i), representing an average of two or more values of p(i) given the preceding outcomes, must take a value between the extremes of the individual values from which p(i) is calculated. On the average, then, the variance of the probability distribution is decreased, resulting in higher information content, and lower order. However, this decrease is much more gradual than a comparable decrease due to reducing the stochastic order by one. Hence, if a plot of order vs. strophe were to be attempted, it might appear somewhat as shown in Ex. 5.

I

0

I

0

<

___~~~ ---

~ X I

I

II

III

IV

V

STROPHE Ex. 5. Hypotheticalplot of ordervs. strophein the Computer Cantata In order to illustrate the types of probability distributions resulting from this technique, let us assume that at some point in the composition a pitch is to be chosen, and that the preceding three pitches already chosen are Bb, Eb, and C, with Bb the most remote in time from the current choice choice point. point. The The uppermost uppermost three three graphs graphs in in Ex. Ex. 66 current * 72


,

0.4

0.4

PEb ( ) Pc

(i)

0.3

0.3

0.2

0.2

0.1

0.1

I C

I I,I

I

I D

E

F

G

I

II

A

C

B

0.3

E

F

0.2

0 =0

0.2

0.1 nn

0.0 D

C

E

0.2

0.1

0.1

n00 F

G

G

A

B

A

3rd order

= p(i)=

D

C

B

2/3

Pc(i)

+ 1/3

( i)

02 = 0.003612 = 0.0601

0.3

0.2

E

F

0.4

2ndorder = pc(i) a2 = 0.004184 - = 0.0646

D

A

= 0

0.1

C

I, (, 0.3

0th order =p(i) = /12 02

i

I C

D

II,llI E

F

G

A

Ex. 6. Resultant distributions,zeroth-order through fourth-order,


PERSPECTIVES

OF NEW

MUSIC

represent the probability (from our analysis) of the possible outcomes, given each of the three preceding events. Below these, the resultant distributions are shown that are used to make a choice for zeroth through fourth-order processes. As noted, the variance, a2, of the distributions increases rapidly for the first three orders, then decreases gradually for the next two. Similar results would obtain for other pitch sequences. Finally, since StropheI is the first time all of the instruments are used together, we provided a short introduction to this strophe during which the instruments enter in groups at three-measure intervals. The introduction was planned such that a progression from the most undifferentiated sound (white noise) toward purer and more ordered sounds also occurs. The band width and amplitude of the white noise are successively reduced in time intervals of 42 16th notes from initial values of 75 through 9,600 cps at triple forte to final values of "tuned" noise at approximately 900 cps at pianissimo. This is followed by tacet for "white noise" at m. 21, shortly after the voice enters for the first time. The order in which all the instruments enter is shown in Ex. 7. PROLOG TO STROPHE I and EPILOG TO STROPHE V

Since these two related sections were intended for nonpitched percussion instruments only (see Ex. 2), we decided to make the burden of structural articulation fall upon two characteristics of the music: the density of attacks per unit time, and the dynamic level of the attacks. These were coded to vary with the elapsed time from the beginnings of the sections so as to form definite perceptual contours. If we consider any one of the eight instruments used, the major tests and decisions to be made at any time point, t, in these sections were: 1. Is t an "attack point" for this instrument, i.e., has the duration for the previously chosen event completely elapsed at this time point? 2. If so, should this succeeding event be a rest or nonrest? 3. Whatever the result of decision 2, choose a duration for this event. 4. Choose a dynamic level if nonrest was chosen at decision 2. 5. Decide whether to sustain the sound by means of roll or tremolo for the entirety of the chosen duration, or to allow the sound to decay naturally according to the characteristics of the particular instrument. It will be noted that the density of attacks, meaning in this case the number of nonrest events per unit time, is a function of two parameters, namely, the density of attack points and the proportion of nonrest events from among the number of attack points. Control of the density of attack points was effected by weighting the probability distribution of 74


CANTATA

COMPUTER

Bar

o

3

9

6

18

15

12

21 toend

Voice Flute

************************

B. Cl.

*************

******************

******************************

Trumpet Horn

******************

************

Xylophone

~l*l****,.**

Glock.

******.******************************

******

******************

Snare Drum Tambourine

********x,****************

Castenets

*****x*****ac******

************************

******************

*****c*****.************

Cymbal Tabor

*****X***********************

Maracas

*****x*****X'******************************

Bass Drum

*****X******$******

Tam-Tam

*****x**********************

******

************************

Violin

******

Viola

*****************************

Guitar

******

******************

************

**********************

Theremin

***** *******************

Sine tone

***********************

Sqr. wave

******

******************

Saw-tooth

******

******************

White N. Colored N.

*****x*****x**********************

~***XS*

********

$$$*x*$$$$****~*****************x*****

Ex. 7. Order of entrance of the instruments during the prelude to StropheI

* 75'


PERSPECTIVES

OF NEW

MUSIC

choosing shorter or longer duration values as a function of time. In other words, the shorter the average duration of events is, the greater the density of events must be, and vice versa. Control of the proportion of nonrest events was effected by weighting the probability of choice of rest or nonrest as a function of time. A vocabulary of 22 different duration values formed the set of possible alternatives. These were further grouped into natural classes according to whether they formed a multiple of 1/16, 1/12, 1/10, or 1/14 of a common time measure. The complete set of possible values is shown in Ex. 8. Class I

ClassIII

Class II

rs

j1

Class IV r1 7

3-

-

1 3

J

J)

J)

JJ J.

OJ Ex. 8. Vocabulary of rhythmic choices for Prolog to StropheI and Epilogto StropheV

The choice of a duration was then divided into a two-step process whereby a class was chosen first according to a stationary probability distribution highly weighted in favor of Class I, less so for Class II, and almost negligibly for Classes III and IV. Once the class was chosen, a specific duration within that class was chosen according to a probability distribution which, for Classes I and II, varied with the time elapsed, t. Since there were only single choices within Classes III or IV, these distributions could not, of course, vary. Weighting of the probabilities of 76


COMPUTER

CANTATA

the individual alternatives within Classes I and II was accomplished by first allowing each member of the class either to accrue or pay "probability interest" compounded at each 16th-note position within the section, with different specified "interest rates" for each member, and then choosing from among the set of the first n members the sum of whose probabilities was equal to or greater than one (the probability of the last member being set as equal to one minus the sum of the first n-l members). If vnis defined as the value of the function, vo(1 + rn)t,where Voindicates the initial probability of the nth member of the set at time t = 0, and rnis the "interest rate" for this member, then the probability of this member, pn, at any time t during the Prolog to StropheI can be expressed as: n Vn

Vi <

1

i=l n-1

pn =

n-1

1 i=1

n

vi < 1 <

vi, i=1

E

vi

i=1

n-1 0

Vi >

1

i=l

By substituting P17-n (Class I) or ps-n (Class II) for Pn in the left side of the above equations, the values used during the Epilogto StropheV were obtained. The initial values, vo, for both classes were set such that each member of the class was equally probable. Thus, vo = 0.0625 for Class I, and 0.25 for Class II. The rates, rn, were chosen such that the function Vl = v0(1 + rl)t would equal 1 at t = 1,472 (92 measures), the arbitrarily chosen duration of the section exclusive of a final few measures of roll or tremolo in all instruments at the end of the Prologto StropheI. The remaining rn decreased linearly with negative values being used for n greater than or equal to 8 (Class I) or 3 (Class II). In the Prologto StropheI, then, the shortest duration values are associated with the highest rates of increase, and since the function is exponential, the density of attack points increases exponentially with time. In the Epilog to StropheV the reverse is true, since here the longest duration values are associated with the highest rates, and hence the density of attack points decreases exponentially with time. Both sections begin with equiprobable distributions for durations. From theoretical considerations, it can be shown that near the beginning of the sections the average density of attack points is around 2 per instrument per measure, or about 16 per measure. Near the end of the Prolog to StropheI, the * 77 *


PERSPECTIVES

OF NEW

MUSIC

density approaches 128 per measure, while near the end of the Epilog to StropheV the density approaches 8 per measure. The density of actual attacks is further altered by the rest or nonrest choices described directly below. Taking this into account, the density of actual attacks at the beginning of this Prolog is approximately 2 per measure and near the end about 128 per measure. At the beginning of the Epilog the density of attacks is about 16 per measure, and near the end about one per measure. Choice of rest or nonrest was made time-dependent by dividing the total duration of the section into 8 equal portions, and then allowing the probability of nonrest to vary as a simple step function of the portion within which choices were to be made. Each portion was 184 16th notes in duration since 8 X 184 = 1,472. If t/184 = Q + R, where Q is the quotient and R the remainder of the division, then the probability of nonrest at any attack point in the Prolog to StropheI was (Q + 1)/8 up to t = 1,472. During the Epilog to StropheI, the probability of nonrest was set equal to (8 - Q)/8. The step function was also used to determine dynamic levels during the sections. Dynamic symbols were associated with the integers 1 through 8 and were obtained as a function of Q. During the Prologthe level for any portion is given by the expression, D = 8 - Q. During the Epilog, the levels were arranged symmetrically about the center of the section, with D = 5.5 - IQ - 3.5 . In Ex. 9, we show the contours of the main structural parameters for the two sections that resulted from these computations. The choice of whether or not to sustain a sound by means of a roll or tremolo was then permitted to depend first on whether a duration was equal to or greater than a quarter note, and secondly on a probability distribution which did not vary with time. If the duration was less than a quarter note, the sound was not sustained. If it was equal to or greater than a quarter note, the probability of sustaining the sound was arbitrarily set to 0.25. Since the probability of choosing a duration equal to or longer than a quarter note also decreased with time during the Prolog, and increased with time during the Epilog, the proportion of sustained to nonsustained sounds of necessity varied similarly. PROLOG TO STROPHE II and EPILOG TO STROPHE IV

These sections form two parts of a single "totally organized" composition, the rationale for which was taken almost in toto from Gybrgy Ligeti's account of the compositional procedures which Pierre Boulez used for his StructureIa, from Structures, for Two Pianos, Four Hands [12]. The principal changes which we made in the organization of the piece were: (1) a different 12-tone row, (2) different instrumentation, i.e., nine instruments rather than two pianos, (3) different numbers of sections within each part of the composition. 78


CANTATA

COMPUTER

128 to

Prolog

Strophe

I

112 96 80 64 48 32 16 0

Epi log

to

Strophe

v

-f

.p I

Ip

I

I Density

16 Density

of

of

Attack

Points -

Attacks

--

Ex. 9. Structuralplan forthe controlof musicalparametersin the two percussion studies, Prologto SropheI and Epilogto StropheV

The computer was first coded to produce a 12-tone row by means of a random selection from among 12 integers without replacement. Each selection was also assigned an ordinal number which represented the first, second, .. ., twelfth member of the row. The computer then also formed the 48 permutations of these ordinal numbers which correspond to the 12 transpositions of the four forms of the row. The original form (N1) of the row together with the associated ordinal numbers are shown below, while the two "checkerboards" listing the 48 permutations are shown in Ex. 10. Since the retrograde (R) and retrograde inversion (RI) forms can be read from the tables by reading from right to left, only the two "checkerboards" shown are needed. 79


PERSPECTIVES

OF NEW

MUSIC

N and R forms 1

2

3

4

5

6

7

8

9

10

11

12

2

9

7

5

6

1

11

10

12

4

8

3

3

7

10

2

9

12

4

6

11

1

5

8

4

5

2

11

8

10

9

3

6

7

12

1

5

6

9

8

10

4

12

7

1

11

3

2

6

1

12

10

4

5

3

11

2

8

7

9

7

11

4

9

12

3

5

1

8

2

6

10

8

10

6

3

7

11

1

9

4

12

2

5

9

12

11

6

1

2

8

4

3

5

10

7

10

4

1

7

11

8

2

12

5

3

9

6

11

8

5

12

3

7

6

2

10

9

1

4

12

3

8

1

2

9

10

5

7

6

4

11

I and RI forms 1

6

10

12

9

2

8

7

5

3

11

4

6

5

8

9

2

1

11

3

4

12

7

10

10

8

3

6

5

4

12

2

11

1

9

7

12

9

6

11

7

3

5

10

2

8

4

1

9

2

5

7

3

12

4

8

1

11

10

6

2

1

4

3

12

9

10

11

6

7

8

5

8

11

12

5

4

10

9

1

7

6

2

3

7

3

2

10

8

11

1

5

12

4

6

9

5

4

11

2

1

6

7

12

10

9

3

8

3

12

1

8

11

7

6

4

9

10

5

2

11

7

9

4

10

8

2

6

3

5

1

12

4

10

7

1

6

5

3

9

8

2

12

11

Ex. 10. Transpositions of N, R, I, and RI forms of the row in Prolog to StropheII and Epilog to StropheIV

* 80'


COMPUTER

Original row (N1): Ab Ordinal number: 1

C# 2

E 3

CANTATA

F 4

Bb Eb A 5 6 7

G 8

F# 9

C D B 10 11 12

Durations, dynamic levels, and instrumental timbre were also associated with the integers 1 through 12, and choices for these parameters were made to depend on some arrangement included in the tables of Ex. 10. The integers and associated parameter "states" are shown in Ex. 11. Instrument Integer

Pitch

1

Ab

Duration

Dynamics

Group1

J

2

C#

3

E

4

F

J

p

5

Bb

J'

6

6I EL J. Eb

qp M mp

7

A

J.

mf

8

G

J

qf f

ppp

Violin

9

F#

J_

10

C

J_

11

D

12

B

' .

arco pizz

pp

.

Group2 sul pont

pppp

Viola

I

mute Trumpet

n open

Horn

Bass Clarinet

Flute

ff

Xylophone

Glockenspiel

fff

sul pont. -Guitar -normal

ffff

Ex. 11. Integers and associated parameter states in Prolog to Strophe II and Epilog to StropheIV

Each section of each part (i.e., Prologor Epilog) presents from zero to three instruments of each group, each instrument playing one permutation of the row as pitch organization, with durations for the pitches organized according to another permutation of the row. The choice of which permutation was permitted to depend upon the group and upon the part, Prolog or Epilog. These can be conveniently illustrated by Exx. 12a and 12b. Each conjunction of a specific pitch permutation with a specific duration permutation may be termed a "thread." Each thread was then assigned a constant dynamic level and a particular instrumental timbre. The succession of dynamic levels and timbres for successive threads proceeded according to one of eight arrangements obtained by moving diagonally across the tables of Ex. 10. For example, the succession of dynamic levels for the threads of Group I in the Prolog(see Ex. 13) were obtained 81 *


OF NEW

PERSPECTIVES

MUSIC

Pitch

Prolog

Epilog

Group 1:

Total of all Ni transpositions arranged according to I1.

Group 2:

Total of all Ii transpositions arranged according to N1.

Group 1:

Total of all RIi transpositions arranged according to RI1.

Group 2:

Total of all Ri transpositions arranged according to R1. Duration

Prolog

Total of all RIi transpositions Group 1: according to RI1. ,arranged _~~Prolog Total of all Ri transpositions Group 2: arranged according to R1.

Epilog

Total of all Ii transpositions Group 1: according to R1. .arranged _~~Epilog Total of all Ni transpositions Group 2: arranged according to RI1.

Ex. 12. Organization

of permutations

in Prolog to Strophe II and Epilog to Strophe

IV: (a) Pitchpermutations and (b) durationpermutations

by moving diagonally across the checkerboard of the N and R forms, starting at the lower left-hand corner. This yielded the sequence: 1 12 8 ffff qf pppp

6 mp

7 mf

3 pp

3 pp

7 6 mf mp

1 pppp

8 12 qf fff

As was mentioned earlier, this Prologand Epilog are each divided into a number of sections in which from zero to three threads of each group may play. There are seven such sections in the Prolog,eight in the Epilog. The number of instruments playing at any one time gives rise to a density contour which was symmetrically organized in each part. This was done by allowing the succession of numbers of instruments playing per section in Group 2 to be the retrograde of that in Group 1. The combined effect of these various forms of organization is summarized in Ex. 13 where in any cell the first two lines indicate the instrumental timbre choice for the thread, the third line indicates the form and transposition of the pitch permutation, the fourth line indicates the form and 82 -


Prolog to Strophe II

0

Gtr. s. pt N1 RI4

Vln. s. pt. N6 RI11

Xyl. N10 RI3

Tff

qf

pppp Gtr. norm. I1 R12 p

c&

Cx

o 0

Xyl. N12 RI5

Trpt. mute N9 RI7

mp

mf

B.C1.

Vln.

Gtr.

Vln.

N2 RI8

Ns RI2

pizz. N7 RI9

N5 RI12

N3 RIlo

p

pp

mf

mp

pppp

Hn. mute I6 R7

Hn. mute I7 R6 ff

B.C1.

Via.

Via.

Via.

Fl.

s. pt. I2 R11 mf

pizz. I3 Rio pppp

pizz. I4 Rs ppp

I5 Rs qf

f

Epilog to Strophe IV Vln.

CI4

0

0

Trpt.

B. C1.

ptizz. RI4

RIll

RI3

I12

Ill

I1o

f

pp

mf

B. C1. openopn RI5 19 fjf

Hn.

Hn.

open R12 N4 qp

open R11 N11 ff

Via. arco Rio N3 qf

Trpt.

Vln.

RI7

pizz. RI8

Vln. arco RI2

Vln. arco RI9 I5

Is

I7

I6

f

mf

mf Via. arco R9 N5 S

Trpt. open RI12 I4

ff Hn.

Hn.

G

open Rs N7 PPt

open R7 Ns qf

R N f

qf

Ex. 13. Resulting structure of the totally organized music that comprises Prolog t


PERSPECTIVES

OF NEW

MUSIC

transposition of the duration permutation, and the last line indicates the dynamic level of the thread. One further comment on the role of the computer for this type of composition is in order here. Since the computer was not used in the determination of the compositional plan itself, its function was more that of a "compiler" rather than a "composer." Once the compositional plan was assembled, the computer did no more than select the appropriate pitch, duration, timbre, and dynamic level for a specific thread, as directed by the plan. In this sense, the computer made no choices at all, since the choices were all entirely specified before the computer was set into operation. This may be contrasted with any of the other sections of the ComputerCantatawherein the computer was made to choose from among a set of alternatives, usually by means of a probability distribution. In these two totally organized sections, the total extent of composition done by the computer itself amounted to the original selection of the tone row. This points up the tautological nature of composition according to such highly predetermined compositional schemes. If future investigations of such methods of composition appear warranted, it seems to us that they should be directed toward the incorporation into them of conditional degrees of freedom such as exist in other computer music programs written to date. PROLOG TO STROPHE III and EPILOG TO STROPHE III As mentioned earlier, programs and equipment have recently been developed by Dr. J. L. Divilbiss, formerly of the University of Illinois Coordinated Science Laboratory, to enable the CSX-1 electronic digital computer to function as a sound synthesizer. In contrast to the IBM 7090 MUSICOMP programs, the CSX-1 at present carries out no compositional processes, but only accepts information concerning a composed score and converts this directly into sound. Thus, the two computers have been linked functionally, if not operationally, so that output in the form of punched cards from the IBM 7090 composition routines may be converted to paper tape and used as input information to the CSX-1, which then "performs" the IBM 7090-generated composition. A maximum of three voices may be specified for simultaneous synthesis. Specially built analog output equipment not only provides separate amplitude control for each of the three voices, but also supplies a reasonably wide choice of timbre and attack-decay shapes. Pitch is quite independent of any form of temperament, equal or unequal; hence the possibility of exploring various tuning systems other than 12-tone equal temperament immediately suggested itself. These two remaining Computer Cantata sections each make use of seven different tuning systems presented successively by partitioning each into seven sub-sections. The tuning systems used range from nine tones per octave through fifteen tones per octave in equal temperament. 84 -


COMPUTER

CANTATA

In addition, in order to make use of the stereophonic electronic music equipment in the University of Illinois Experimental Music Studio, each section is made up of six voices, three for left channel, and three for right channel. The two channels were put together after separate passes through the CSX-1. Our main objective in these two sections was to explore the various linear or vertical combinatorial possibilities which arise from the use of these tuning systems. With the familiar 12-tone tuning, we have definite a priori knowledge about the dissonance properties of most combinations of pitches. No such knowledge is available for, say, 9-tone or 15-tone tuning. Accordingly, we looked for a mathematical model which would not only fit most of our knowledge concerning 12-tone tuning, but which could also be extended to cover the other tunings. We did not expect our model to describe equally well all possible combinations in any one system since, in order to keep the model relatively simple, we ignored dissonance generated by summation and difference tones as well as dissonance associated with conflicting upper partials of separate voices. If we accept the premise that, given any three separate and distinct pitches that have been reduced to an octave span by assuming octave equivalents, the dissonance of the combination increases as the three approach one another while still remaining distinct, then a convenient means of ranking combinations on a harmonic dissonance scale, HD, can be devised. This is done by considering the sum of squares of the differences between the three. Graphically we might consider three points on a line as representing three ways of placing pitches within the space of one octave from the lowest of the three, as shown below: dl

d2

d3

dl + d2 + d3 = N For convenience of notation, let d1 = the difference between the lowest and the middle tones, d2 the difference between the middle and highest tones, and d3 the difference between the octave and the highest tone. Finally, let N = the number of tones per octave. If we then form the ratio of the sum of the squares of the differences to the square of the sum of the differences, we know that this function has a minimum value when the points are equally spaced along the given length, and approaches 1 as the points approach one another (or the octave) while remaining distinct. The function will equal 1 if the restriction of distinctness is removed. R d12 + d22 + d32

(di + d2 + d3)2 *85

d12 + d22 + d32

N2


PERSPECTIVES

OF NEW

MUSIC

Removing this restriction, however, is not realistic unless we weight the resulting value by some factor which takes into account the resulting loss of dissonance due to decreasing the number of distinct pitches. Given that k is the number of distinct pitches, we can let this weighting factor be k/3. Harmonic dissonance values of the possible combinations of three, two, or one distinct pitches is then given by the expression: HD = R(k/3)=

di2 +d22 + d32 (k/3) N2

Regardless of the number of distinct tones, this function has a minimum value of 0.333 when the points are equally spaced within the octave. The maximum value within the limits of our selected tuning systems and maximum number of voices occurs when N = 15, k = 3, and d = d2 = 1, d3 = 13, giving HD = 0.76. A similar function can be used to rank linear angularity, this being in some sense the correlative to harmonic dissonance in the linear dimension. We first let an arbitrary limit of 2N tones, for N-tone tuning, be imposed as the largest skip in any direction for a voice. We next take account of the preceding two pitches in the voice under consideration by forming the difference, dl = IPi_ - Pi-2 . A "linear dissonance" value for all possible pitches within the limits of Pi- - 2N through Pi-_ + 2N can then be computed as the ratio of the sum of the squares of the differences to the maximum possible sum of squares, 8N2 = (2N)2 + (2N)2, that is to say: LD = d12 + d22 8N2 where d2 = Pi - Pi l. The reason for again using the sum of squares function rather than a simple sum of the differences is that the latter would give the same LD value for any combination of differences which add to the same sum. The former provides distinctions between the different possible ways of adding to the same sum in that evenly spaced skips give lower LD values than skips of widely divergent magnitudes. For example, the progression C, D, B (reading up) results in a higher LD value than, for example, C, F, B, a more evenly spaced series of skips. In general, of course, the greater the sum of the intervals encompassed by the two skips, the greater the LD value. In Ex. 14, there are shown various values of LD for a few selected sums and for all ways of adding to these sums with the restriction that no skip may exceed 2N. In these particular graphs, the tuning is 14 tones per octave. In making a choice of pitch for some voice, a table of values of LD and HD for every pitch within the range, + 2N, or alternately, - 2N, from the preceding pitch of that voice was set up. Since HD values assume *86?


COMPUTER

CANTATA -=56

1.0 LD 0.9

0.8

X-49

0.7

0.6 =35

0.5

0.4

2=28 0.3

0.2

01

=7_1

0.0 0

2

4

6

8

10

12 14 d,-(d2= E-d,)

16

18

20

22

24

26

28

Ex. 14. SelectedLD valueswith N = 14 tonesper octave octave equivalents, these values were periodic over the range, whereas LD values varied continuously over the range. These values were then examined separately and tested against certain critical values which were arbitrarily chosen to indicate low, medium, or high dissonance. The values were then converted to one of three "weights" depending upon which of the three dissonance levels they represented. Consequently, we were able to sum the weights and let the ratio of any LD weight to the sum of the LD weights be considered the conditional probability of that pitch, Pi, given Pi-_ and Pi-2. Similarly, we could further consider the ratio of any HD weight to the sum of the HD weights to be the conditional probability of Pv1 given Pvi-l and Pv,_2, where the notation vi indicates the voice currently making a choice of pitch and vi-1 indicates either of the other two voices. Finally, we let the probability of any pitch be the weighted average of these two conditional probabilities, where the particular weighting constants were chosen as a function of the duration of the event and the number of nonresting voices in the event. Stated algebraically, this condition is expressed as follows: p(Pi) = cp(Pi IPi-, Pi-2) + c2p(PiIP1, Pvi-2) where the vertical bar is another notation for the conditional relation. In general, we chose the constants so that the amount contributed to the

* 87 ?


OF NEW

PERSPECTIVES

MUSIC

sum by the second addend, the HD probability values, increased with the number of nonresting tones and with the total duration of the event. The amount contributed by the first addend, LD probability values, showed, by contrast, the opposite trend. Each of the seven sub-sections for each channel was programmed as either a harmonic sub-section or a linear sub-section. In the harmonic sub-sections durations were chosen to apply to all three voices, resulting in block chord progressions. In the linear sub-sections durations were chosen for each voice separately, resulting in the opposite effect. Moreover, each sub-section was programmed to produce an over-all effect in each channel of one level of harmonic dissonance, low, medium, or high, through giving the appropriate level the highest weighting when assigning the HD weights described above. Similarly, each sub-section was programmed to produce an over-all effect in each channel of one level of linear dissonance through giving the appropriate level the highest Prologto StropheIII Tones/octave

R

12

13

14

15

14

13

12

L

15

14

13

12

11

10

9

Style (linear or harmonic) orharmonic)

R

H

L

L

H

L

L

H

L

H

L

L

H

L

L

H

LD level

R

3

2

1

3

1

2

3

L

3

1

2

1

2

1

3

R

1

2

2

3

2

2

1

L

3

3

1

1

1

3

3

HD level

Epilog to StropheIII R

9

10

11

12

13

14

15

L

12

11

10

9

10

11

12

Style (linear or harmonic) orharmonic)

R

H

L

H

L

H

L

H

L

H

H

L

L

L

H

H

LD level

R

3

1

2

1

2

1

3

L

1

2

1

3

1

2

1

R

3

3

1

1

1

3

3

L

3

2

1

1

1

2

3

Tones/octave

HD level

Ex. 15. Structural plan for the Prolog and Epilog to StropheIII

* 88?


COMPUTER

CANTATA

weight when assigning LD weights. In Ex. 15, we show the over-all plan for these two sections, Prolog to Strophe III and Epilog to Strophe III. Durations were chosen from the vocabulary shown in Ex. 8 that was used for Prolog to Strophe I and Epilog to Strophe V, with the exception that the whole-note triplet, and half-note triplet were deleted. Again, a class was first chosen by means of a probability distribution highly weighted in favor of Class I, and a specific duration from the chosen class made by a probability distribution in which the probability of any item times the duration of that item was a constant, 0.302. Theoretically, the mean duration resulting from this distribution is known to be approximately 4.5 16th notes. No further controls were placed on duration choice. Dynamic levels were not programmed but were added by means of the gain controls on the CSX-1 synthesizer during actual "performance." Timbre and attack-decay characteristics for the separate voices were also added at this time. As shown in Ex. 2, Prolog to Strophe III and Epilog to Strophe III are for CSX-1 sound only, without instrumental accompaniment. These two completed studies were inserted in the complete performance tape for two-channel tape recorder as Cue 3 and Cue 4, respectively.

BIBLIOGRAPHY 1. L. A. Hiller, Jr. and L. M. Isaacson, Illiac Suitefor String Quartet,New Music Edition, Theodore Presser Co., Bryn Mawr, Pa. 2. L. A. Hiller, Jr. and L. M. Isaacson, ExperimentalMusic, McGraw-Hill Book Co., New York, 1959. 3. L. A. Hiller, Jr. and R. A. Baker, "Computer Music," Ch. 18 in H. Borko, ed., Computer Applications in the Behavioral Sciences, Prentice-Hall,

Englewood

Cliffs, N. J., 1962, pp. 424-51. 4. L. A. Hiller, Jr., "Musical Applications of Electronic Digital Computers," GravesanerBlOtter (in preparation).

5. L. A. Hiller, Jr., "Jiingste Fortschritte in der Computer-Musik," Darmstadter Beitrage zur Neue Musik (in preparation). 6. "SCAT, SHARE-Compiler-Assembler-Translator,"Symbolic Assembly Programfor the

IBM 7090, IBM 7090/1401 Library Routine LI-UOI-SSA1-1-BX, Digital Computer Laboratory, Graduate College, University of Illinois, Urbana, Illinois. 7. R. M. Brown, R. D. Jenks, J. E. Stifle, and R. L. Tragden, Manualfor the CSX-1 Computer, Report R-136, Coordinated Science Laboratory, University of Illinois, Urbana, Illinois, 1962. 8. J. L. Divilbiss, "Real-time Generation of Music with a Digital Computer," J. Music Theory,8:99-111, 1964. 9. L. A. Hiller, Jr., "Electronic Music at the University of Illinois," J. Music Theory,7:99-126, 1963. This article describes older equipment in the studio but refersto newer equipment then under constructionthat is now in operation. 89 -


PERSPECTIVES OF NEW MUSIC 10. C. E. Shannon and W. Weaver, The MathematicalTheoryof Communication, University of Illinois Press, Urbana, Ill., 1949; see especially Weaver'sessay beginning on p. 55. 11. J. R. Pierce, Symbols,SignalsandNoise,Harper and Bros., New York, 1961, Ch. 5. 12. G. Ligeti, "Pierre Boulez," Die Reihe(Engl. Ed.), 4:36-62, 1960.

a 90 -

hiller, baker - computer cantata  
hiller, baker - computer cantata  

hiller and baker about computer cantata

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