Experimental music; composition with an electronic computer

EXPERIMENTAL MUSIC

Experimental Music

COMPOSITION WITH
AN ELECTRONIC COMPUTER

Lejaren A. Hiller, Jr.

ASSISTANT PROFESSOR OF MUSIC

SCHOOL OF MUSIC, UNIVERSITY OF ILLINOIS

Leonard M. Isaacson

MATHEMATICIAN

STANDARD OIL COMPANY OF CALIFORNIA
FORMERLY RESEARCH ASSOCIATE
UNIVERSITY OF ILLINOIS

McGRAW-HILL BOOK COMPANY, INC.

1959 NEW YORK TORONTO LONDON

320284

EXPERIMENTAL MUSIC. Copyright © 1959 by the McGraw-Hill Book Company,
Inc. Printed in the United States of America. All rights reserved. This book,
or parts thereof, may not be reproduced in any form without permission
of the publishers. Library of Congress Catalog Card Number 58-13874

WUSfC LIBRARY

PREFACE

In presenting this discussion of a series of unusual experiments in experi-
mental musical composition, the authors wish to record their appreciation
of numerous instances of invaluable assistance and encouragement they
have had during the course of carrying out this work.

In particular, we would like to acknowledge the support and interest
received from the Digital Computer Laboratory at the University of
Illinois and its director, Dr. Ralph Meagher, and to make note of the
encouragement we received to pursue this work from Dr. John P. Nash,1
who, at the time this work was carried out, was Research Professor of
Mathematics at the Digital Computer Laboratory in charge of program-
ming and mathematical research. It is perhaps of interest to mention that
the Digital Computer Laboratory at the University of Illinois is an
independent laboratory under the direct administration of the Graduate
College and among its objectives is the fostering of university research of
all types for which computers can be used. In this connection, we should
like to note the friendly interest in our project accorded us by Dr.
Frederick T. Wall, Dean of the Graduate College. Secondly, we should
like to acknowledge the assistance and interest of several members of the
University of Illinois School of Music faculty. In particular, Dr. Hubert
Kessler, of the composition and theory staff, has taken a direct interest in
this work and the many discussions we have had with him have been
invaluable in clarifying both the objectives of the work and the analysis of
the experimental results obtained. Thirdly, we are indebted to Professor
J. Robert Kelly, also of the composition and theory staff, not only for
suggesting certain experiments in regard to the production of tone rows
by means of computers, but also for helping to arrange a concert at which
the first performance of the I Iliac Suite was given. We wish to make note
of the efforts of Professor Robert Swenson and the other members of the
string quartet in preparing the Illiac Suite for the performance given at
that time. We should like to acknowledge the many helpful suggestions

J Current address: Lockheed Missile Systems Division, Sunnyvale, California.

vi PREFACE

and criticisms we have received from many other interested friends, both
within the departments mentioned and outside them. Lastly, we would
like to mention the interest we received both at the inception of this work
and later on from Professor Milton Babbitt, of the Department of Music
at Princeton University, a friend of one of the authors, who encouraged
the idea that work of this type would be of interest to contemporary
composers, and from Professor Vladimir Ussachevsky, of the Department
of Music at Columbia University. Professor Ussachevsky, in connection
with his own studies of contemporary experimental music, made a special
trip to Urbana to inquire about our work and subsequently helped arrange
the initial publication of the score of the I Iliac Suite. This has been a strong
stimulus to us to complete the present task of writing, so that we may look
forward to new projects which will extend the scope of the experiments to
be described.

Lejaren A . Hiller, Jr.

Leonard M. Isaacson

CONTENTS

Preface v

Chapter One. Nature of the Problem 1

Introduction. Chronology. Presentation of the Material.

Two. The Aesthetic Problem 10

Meaning and Form in Music. The Logic of Musical
Composition. Information Theory. Information Theory
and Music.

Three. Experimental Music 36

Experimental Music Defined. Electronic, or Synthetic,
Music. Programmed Music. Other Experiments to Gen-
erate Computer Music.

Four. The Technical Problem 58

Non-numerical Uses of Computers. Operation of Auto-
matic High-speed Digital Computers. The "Man-versus-
Machine" Problem. The Monte Carlo Method. The
Monte Carlo Method and the Generation of Music.

Five. Experimental Details 79

Outline of Experiments. Experiment One. Experiment
Two. Experiment Three. Experiment Four.

Six. Experimental Results: The llliac Suite 152

Introduction. Experiment One. Experiment Two. Ex-
periment Three. Experiment Four.

Seven. Some Future Musical Applications 165

Introduction. Applications to Music Analysis. Appli-
cations to Music Composition. Summary of Results.

Appendix 181

CHAPTER ONE

Nature of the Problem

Introduction

Upon first hearing of the idea of computer music, a person might ask:
"Why program a digital computer to generate music?" The answer to this
question is not simple, since such an undertaking immediately raises funda-
mental questions concerning the nature of musical communication and its
relation to formal musical structures. Moreover, it also raises the question
of how far it is possible to express musical and aesthetic principles in
forms suitable for computer processing. Lastly, it also brings up the
problem of what role automation of the type exemplified by high-speed
digital computers can be expected to fulfill in the creative arts.

We shall point out in Chapter 2 that the process of musical composition
can be characterized as involving a series of choices of musical elements
from an essentially limitless variety of musical raw materials. Therefore,
because the act of composing can be thought of as the extraction of order
out of a chaotic multitude of available possibilities, it can be studied at
least semiquantitatively by applying certain mathematical operations de-
riving from probability theory and certain general principles of analysis
incorporated in a new theory of communication called information theory.
It becomes possible, as a consequence, to apply computers to the study of

1

2 EXPERIMENTAL MUSIC

those aspects of the process of composition which can be formalized in
these terms.

More specifically, when we raise this question of whether it is possible
to compose music with a computer, we may note the following points:
( 1 ) Music is a sensible form. It is governed by laws of organization which
permit fairly exact codification. (As we shall later note, it has even been
claimed that the content of music is nothing but its organization.) From
this proposition, it follows that computer-produced music which is "mean-
ingful" is conceivable to the extent to which the laws of musical organiza-
tion are codifiable. (2) It is a feature of digital computers that they can be
efficiently used to "create a random universe" and to select ordered sets
of information from this random universe in accordance with imposed
rules, musical or otherwise. (3) Since the process of creative composition
can be similarly viewed as an imposition of order upon an infinite variety
of possibilities, an analogy between the two processes seems to hold, and
the opportunity is afforded for a fairly close approximation of the com-
posing process utilizing a high-speed electronic digital computer. In this
context, it should t>e noted, however, that the composer is traditionally
thought of as guided in his choices not only by certain technical rules but
also by his "aural sensibility," while the computer would be dependent
entirely upon a rationalization and codification of this "aural sensibility."

In order to examine this idea experimentally, we have carried out a
series of experiments to determine whether automatic high-speed digital
computers such as the Uliac, located at the University of Illinois, can be
used to generate music subject only to general instructions derived from
various specified "rules" of composition. In a computer, this is done by
letting the control of the musical output be limited solely by the input
instructions, and leaving factors not specifically accounted for in the input
instructions entirely to chance. In general, this appeared to be an attractive
new nonmathematical application of high-speed computer operation which
could be of interest not only as an illustration of the versatility of these
instruments but also in terms of its possible effect on the fields of musical
composition and analysis.

It is desirable to consider briefly how automatic high-speed digital com-
puters operate. These instruments function in principle much as do ordi-
nary desk calculators, but with certain significant differences. Perhaps the
most important difference is that a whole set of computing instructions,
called a code or computing program, which is entirely expressed in terms of
mathematical operations, and which is prepared from a library of instruc-

NATURE OF THE PROBLEM 3

tions called an order code, is placed into a computer prior to the actual
process of computation. This eliminates delays resulting from the slow-
ness of the human operator in handling numbers being produced during
the period of actual computation. A second important difference is that
automatic digital computers have what is called a conditional transfer proc-
ess. In the Illiac, this is essentially a yes-or-no choice operation which per-
mits the programming of decisions which depend upon whether numbers
bear positive or negative signs. Specifically, the instructions of a program
placed into the Illiac are acted upon sequentially in the normal course of
computation. However, the conditional-transfer order, whenever it is
reached in this set of instructions, may shift the sequence of operations to
another part of the program, depending upon the results of the calculations
carried out to that point. A simple example of this process is the testing
of a counting index for a change of sign set to occur at the termination of an
iterative computation cycle.

It is operations of this type used repeatedly and sequentially, perhaps
more than any other, that permit programs to be written for the computer
expressing logical processes of musical composition. This happens because
these two features of computers, namely, extremely high speed and the
ability to react differently to positive and negative numbers, permit the
practical exploitation of the so-called "Monte Carlo method" for solving
certain types of complex mathematical problems. The success of the
Monte Carlo method depends upon the generation of random integers in
great profusion, even up to the order of millions of integers. These in-
tegers, as they are produced, are examined and sorted according to the
needs of the problem to be solved, until gradually a better and better ap-
proximation to the answer to the problem is obtained.1 To do this, the
laws of probability theory are applied within the restrictions of the par-
ticular problem being studied, so that random integers not in accord with
these restrictions are discarded. The method is obviously hopelessly in-
efficient without a device such as an automatic computer, but since these
instruments have become available, the technique has been applied suc-
cessfully to a number of complex problems, both within the scientific field
and in other areas, as in the present instance.

We proposed that the composition of music could be treated by the
Monte Carlo method. We were able to act upon this proposition by re-
solving the process of generating computer music into two basic opera-

1D. D. McCracken, "The Monte Carlo Method," Sci. American, 192(5) :90,
May, 1955.

4 EXPERIMENTAL MUSIC

tions. In the first operation, the computer was instructed to generate
random sequences of integers which were equated to the notes of the
musical scale and, in certain experiments, also to rhythmic patterns, dy-
namics, and playing instructions such as arco, pizzicato, and col iegno.
These random integers, which can be generated at a rate of up to about a
thousand per second, were then processed in the second, more complex
operation in which each random integer was screened through a series of
arithmetic tests expressing various rules of composition and either used or
rejected depending on which rules were in effect. If accepted, the random
integer was used to build up a "composition" and stored in the computer
until the completed "composition" was ready to be printed out. On the
other hand, if it was rejected, a new random integer was generated and
examined. This process was repeated until a satisfactory note was found
or until it became evident that no such note existed, in which case part of
the "composition" thus far composed was automatically erased to allow
a fresh start.

The purely technical problems involved in programming the computer
to process musical information were soon seen to be the part of the total
problem which would inevitably preoccupy us during the earlier parts of
the investigation. Technical decisions of many types would necessarily
outweigh in importance subtler aesthetic considerations. Therefore, the
logical sequence of experiments seemed to be the following: (1) To select
some simple but well-known style of writing and use this as a basis to
build up an elementary technique of polyphonic writing. A simplified
version of first-species strict counterpoint was utilized for this purpose.
(2) Then, once many of the technical problems of coding had been worked
out in this way, to demonstrate that standard musical techniques could
be handled by computer programming, so that conventional musical out-
put, recognizable to musicians, is produced. The solution of the basic
problems of first-species strict counterpoint was, therefore, carried out to
produce cantus firmus settings which were academically correct in all their
most important details. (3) To demonstrate that a computer can produce
novel musical structures in a more contemporary style and to code musical
elements such as rhythm and dynamics. This was done to show that com-
puters might be used by contemporary composers to extend present com-
positional techniques. (4) To show, lastly, that computers might be used
in highly unusual ways to produce radically different species of music
based upon fundamentally new techniques of musical analysis. In this last

NATURE OF THE PROBLEM 5

experiment, a complete departure from traditional compositional practice
is illustrated.

Computer output produced as a result of carrying out these four experi-
ments was utilized to produce a four-movement piece of music we have
entitled the Illiac Suite for String Quartet. This is a transcription organized
into four movements which parallel the sequence of experiments just de-
scribed. The musical materials in these four movements were taken from
a much larger body of material by unbiased sampling procedures, so that
a representative rather than a selectively chosen musically superior group
of results would be included in the Illiac Suite. Thus, it is important to
realize when examining this score that our primary aim was not the pres-
entation of an aesthetic unity — a work of art. This music was meant to be
a research record — a laboratory notebook. The complete score of the
Illiac Suite has been recently published2 and is reproduced by permission
of the original publishers in the Appendix.

This brief introductory description we have given of certain features of
the production of computer music can now be used to provoke certain
questions relevant to the question of generating computer music. Specif-
ically, the first question we can raise is this: Are there ways to investigate
music in a quantitative way to reveal more precisely the relationship of
musical textures to conceptual frameworks which seem to be significant in
musical composition? Secondly, in a more restricted sense, can we inves-
tigate through certain types of analysis of musical forms the general logical
foundation a composer uses to build up a musical composition? Thirdly,
can we use automatic high-speed digital computers to aid a study involving
these questions? And lastly, might this also lead to new and different ways
of composing music which would interest the contemporary composer?

Chronology

Actual work on this problem was started in September, 1955, at which
time the present authors decided to collaborate to write the initial computer
programs for producing music. The initial set of instructions was designed
to cause the Illiac to generate simple cantus firmi; that is, simple diatonic
melodies to be utilized subsequently to produce simple polyphony. Both
authors had worked previously on Monte Carlo-type problems in connec-

2 L. A. Hiller, Jr. and L. M. Isaacson, Illiac Suite for String Quartet, New Music
Edition, 30:3, 1957.

6 EXPERIMENTAL MUSIC

tion with another research project and thus had had previous experience
with coding problems of this type for a computer. In fact, a sizable portion
of the basic programming techniques of the earlier music codes for the
Illiac was adapted from this earlier research. The work as previously out-
lined progressed very smoothly, so that, by the following spring, we had
accumulated enough material to begin the assembly of a musical record
of the research results in the form of the Illiac Suite. By July, 1956, the
first three movements of the suite had been completed with the exception
of what is now the Coda of the third movement. A performance of this
much of the suite was given publicly on August 9, 1956, at a concert at
the University of Illinois in Urbana, Illinois. This performance, which was
privately recorded at the same time, was by a string quartet composed of
Robert Swenson, cellist with the Walden String Quartet resident at the
University of Illinois, and three instrumentalists, Sanford Reuning, violin,
Peggy Andrix, violin, and George Andrix, viola, who were at that time
graduate students in the University of Illinois School of Music. This con-
cert attracted considerable attention because of its novelty, and representa-
tive reports of this event both prior to it and afterwards in the popular
press can be cited to indicate the extent and nature of the reaction which
occurred.3, 4

This was followed shortly thereafter by a public presentation of some of
our experimental work during a symposium on non-numerical uses for
digital computers at the 11th National Meeting of the Association for
Computing Machinery on August 28, 1956,5, 6 and since then we have
given several other oral presentations of aspects of this research.7 In the

3 Locally, P. Cahill, "Illiac — Mechanical Brain — Takes Up Composing Music,"
Champaign-Urbana News Gazette, July 8, 1956; P. Cahill, "Illiac Proves Music
Skill," ibid., Aug. 10, 1956; R. Starr, "Illiac Tackles the Arts," Champaign-Urbana
Courier, Aug. 10, 1956.

4 More generally, U. P. News release, Aug. 10, 1956, concerning concert reprinted
nationally in many newspapers; I. Wilheim, " 'Brain' Makes like Bach for Scientists,"
Washington Post, Sept. 2, 1956; Anon., "By the Numbers," Musical America, 76:13,
September, 1956; A. Carpenter, "Amazing New Uses for Robot Brains," Sci. Digest,
41(2):1, February, 1957.

5L. A. Hiller, Jr. and L. M. Isaacson, "Musical Composition with a Digital Com-
puter," Program and Abstracts for the 11th National Meeting of the Association
for Computing Machinery, University of California at Los Angeles, Aug. 27-29,
1956, p. 8 and p. 22.

6E. Cony, "Canny Computers: Machines Write Music, Play Checkers, Tackle New
Tasks in Industry," Wall St. Journal, Sept. 19, 1956. This is a general report of
the convention.

7L. A. Hiller, Jr., "The Generation of Music by High-speed Digital Computers,"
Meeting of Chicago Section, Institute of Radio Engineers, Chicago, Dec. 7, 1956; ab-

NATURE OF THE PROBLEM 7

meanwhile, during the fall months of 1956, the Coda of the third move-
ment of the Illiac Suite was finished, and the experiments which now make
up the fourth movement were carried out to complete the work included
in the Illiac Suite by the end of November, 1956. Not long thereafter,
through the interest of Vladimir Ussachevsky, Chairman of the Editorial
Board of New Music Editions, a quarterly for publishing new musical com-
positions, the publication, already referred to,1 of the complete score of the
Illiac Suite was arranged. This left one major task to carry out besides
recording the work in its entirety, which was the preparation of a published
account of the research work embodied in the Illiac Suite.

Presentation of the Material

When the time arrived to write a record of the experiments carried out
with the Illiac, it became evident that enough material had accumulated so
that a highly condensed presentation no longer seemed adequate; the variety
of problems considered during the course of the research had become quite
extensive. Moreover, it seemed important to present enough details so that
the precise extent of what was done would be clearly understood. This
research cuts across fields of interest normally considered discretely sep-
arate. Rather than break up the material and publish part of it in a form
specifically directed to a reader with a musical background and another
part directed to the reader with a background in applied mathematics, we
preferred to prepare one single discussion of the work which would be
reasonably complete and self-contained. Thus, we have included what we
feel is relevant material on experimental music, musical aesthetics, and
related topics. These discussions are required to define and differentiate
the problem in musical terms with sufficient accuracy. It seems sufficient
to point out that unless aesthetic considerations are taken into account,
projects of this type tend to turn out rather poorly. Because of this reason,
significant technical advances in the art fields too often have been over-

stract in Scanfax, 10:7, 1956; L. A. Hiller, Jr., "Some Structural Principles of Com-
puter Music," Annual meeting of the Midwestern Chapter of the American Musico-
logical Society at Michigan State University, East Lansing, Mich., May 19, 1957;
abstract to be published in J. Am. Musicological Soc; L. A. Hiller, Jr. and L. M.
Isaacson, "Musical Composition with a High-speed Digital Computer," 9th Annual
Convention of the Audio Engineering Society, New York, October 10, 1957; also
/. Audio Eng. Soc. in press; L. A. Hiller, Jr., "Musique Electronique," Encyclopedic
des sciences modernes, VIII, Editions Rene Kister, Geneva, Switzerland, 1958, pp.
110-112.

8 EXPERIMENTAL MUSIC

looked for too long a time by creative artists, since the innovators have
taken insufficient pains to evaluate what is artistically significant in their
work to present their materials in the best way. Certain contemporary ex-
periments in synthetic music discussed in Chapter 3 are cases in point.
Secondly, it is necessary that our discussion of the technical details of
coding musical problems for the computer be sufficiently detailed to be
understood by the musician who could not be expected to have acquired
a specialized knowledge of computers and how they are used. It is one of
our primary aims to demonstrate to musicians that the basic techniques
for applying these instruments to musical problems are simple in their
essence and that an extensive scientific background is not required to use
them. Lastly, we are also aware of the problem of misinformation con-
cerning computers and automation. To illustrate, we might consider a
passage from a recent book on automation:8

[In] an American public opinion poll in Detroit, people were asked to fill
up forms writing down their fears in order of magnitude. The subject which
headed the poll was fear of Russia, but to the surprise of the organizers the
second greatest fear was "fear of automation." This surprising and indeed
disturbing result shows how great a fear of the unknown can become. A fire,
started by the press in a sensational manner, fanned by science fiction writers,
poked by agitators, and with fuel added by a slight recession in the car in-
dustry, had clearly got out of hand. But there is this interesting aspect: few
of the people either writing or speaking on the subject were giving the word
"automation" the same meaning. Even so, the word became a bogy. It had
become the cliche of the year. The American man-in-the-street obviously could
not understand the welter of information and distortions that were directed at
him and naturally concluded that the whole thing was beyond his understand-
ing but that its effects were evil. He did not suspect that the writers and speakers
were of doubtful quality.

Obviously, this whole experience can easily be transcribed to other areas
in which automation is having an impact and indeed could arise also in
regard to the present work. With this point in mind, we will also discuss
in general terms how computers work and what may be expected in the
near future in their specific application to musical problems. Naturally,
if applications in the musical field are to be of significance, it is to be ex-
pected that problems will arise which will challenge the ingenuity of con-
temporary composers and musicians, but this is as it should be, since it is

8 L. L. Goodman, Man and Automation, Penguin Books, Inc., Baltimore, 1957,
p. 169.

NATURE OF THE PROBLEM 9

not in the nature of a vital art form to remain aesthetically static and in-
sensitive to technological change. Our plan in writing this book, therefore,
is not only just to present the bare outline of the particular experiments
carried out by means of the Illiac, but also to fill in enough details so that
a reasonably complete picture of the potentialities of this research might
be realized.

A reasonable organization of the material seemed to be the following:
(1) To introduce aesthetic limits to the problem, that is, to define what
can be accomplished musically with a computer at the present time, and
to define what is — at the moment at least — outside the scope of available
experimental tools. This is the purpose of Chapter 2. To clarify this prob-
lem, we have included a brief discussion of information theory, the theo-
retical basis of our method, which should serve as a bridge to the material
that follows. (2) To define the area of research musically, that is, to dis-
tinguish these experiments from and to correlate them with other types of
musical experiments both of the past and now in progress. This is the
purpose of Chapter 3. (3) To consider, in Chapter 4, technical problems.
In this discussion, we are basically concerned with two subjects; the first
being a description of how modern computers operate, and the second
being a general discussion of the mathematical methods used to set up the
problem of generating computer music. (4) The next two chapters of the
book contain detailed descriptions of the experimental techniques and
the experimental results. In Chapter 5, the programming techniques for
the various musical problems studied are reviewed in considerable detail,
while in Chapter 6, a description and an evaluation of the contents of the
Illiac Suite are given. (5) Finally, in Chapter 7, we suggest a number of
possible extensions of this work in the fields of music analysis and music
composition.

CHAPTER TWO

The Aesthetic Problem

Meaning and Form in Music

Two questions which often arise when music is discussed are, on the
one hand, the substance of musical communication, its symbolic and
semantic significance, if any, and on the other hand, the particular proc-
esses, both mental and technical, which are involved in creating and re-
sponding to musical compositions. These two basic problems have been,
of course, subjected to exhaustive discussion throughout recorded history,
to no slight extent because of the difficulties involved in demonstrating
how music acquires "significance." Because music has fewer tangible
models in nature from which artists can work, it has often been suggested
that meaning and coherence in music are achieved by more purely formal
procedures than are usually applied in either the graphic or literary arts.
It is not our purpose to examine this problem in detail, but rather only
to define musical terms so that reasonable limits for the experiments to
generate computer music may be established in musical terms.

As is well known, the nature of musical communication has been dis-
cussed by writers at least as far back as Plato and Aristotle, or the even
earlier Pythagoreans. The earliest writers saw in music an imitation of a
fundamental divine harmony of universal significance. Music was felt to
10

THE AESTHETIC PROBLEM 11

bring harmony to the body and soul through such elements as melody and
rhythm. Aristotle examined music more directly and suggested that it is
an external manifestation of inward emotional and moral states. For ex-
ample, in Problem 29,1 he suggested that music imitates "movements of
the soul" and "moral character." He asks: "Why do rhythms and tunes,
which after all are only voice, resemble moral characters (i.e., the feel-
ings) . . . ? Is it because they are movements, as actions also are?"

Ever since these earliest writers, music has been a subject of considerable
fascination to philosophers and mathematicians, even often being classified
as a form of mathematics and thus considered to reveal natural law in
terms of mathematical logic. For example, Cassiodorus (ca. 485-ca. 575)
defined the mathematical quadrivium as follows :-

Mathematical science is that science which considers abstract quantity. By
abstract quantity we mean that quantity which we treat in a purely speculative
way, separating it intellectually from its material and from its other accidents,
such as evenness, oddness, and the like. It has these divisions: arithmetic,
music, geometry, astronomy. Arithmetic is the discipline of absolute numerable
quantity. Music is the discipline which treats of numbers in their relation to
those things which are found in sound. . . .

If this attitude strikes many of us as a bit peculiar today, perhaps it is
only a result of being conditioned to think of music so completely in terms
of its supposed emotional appeal. The still-popular concept of music as a
direct emotional expression and an explicit and subjective communication
from the composer is, in fact, largely a consequence of the rather extreme
views developed during the period of the nineteenth-century Romantic tra-
dition. Nineteenth-century composers, whose works still make up a large
part of the standard repertory, often themselves felt this way about their
music. The more articulate of these composers, such as Berlioz,3 Wagner,4
and Busoni,5 to cite specific examples, were quite explicit in their attitudes

XW. D. Ross (ed.), The Works of Aristotle, vol. 7, Problemata (trans, by E. S.
Forster), Oxford University Press, New York, 1927, book XIX.

2 Cassiodorus, Institutiones, II, iii, paragraph 21, as quoted in O. Strunk, Source
Readings in Music History, W. W. Norton & Company, Inc., New York, 1950, p. 88,
footnote 6.

8 H. Berlioz, A travers Chants, etudes musicales, adorations, boutades, et critiques,
Michel Levy, Paris, 1862, and other writings.

4 R. Wagner, Das Kunstwerk der Zukunst, 1850, relevant passages in Strunk, op.
cit., pp. 874-903, in particular, p. 881.

5F. Busoni, Sketch of a New Aesthetic of Music, 1907 (trans, by T. Baker),
G. Schirmer, New York, 1911; also F. Busoni, The Essence of Music and Other

12 EXPERIMENTAL MUSIC

and spoke of music as the direct communication "from heart to heart"; of
tone as "the direct expression of feeling" (Wagner); of the need for "emo-
tional sensitivity" (Berlioz); of the role of music as dealing with the inter-
pretation of human feelings, the portrayal of "soul states"; and of the
necessity of music not being fettered by pedantic forms (Busoni). On the
other hand, it is interesting to note, by contrast, that Stravinsky, for ex-
ample, has recently restated the more traditional definition of music in
emphatic terms when he characterized "the phenomenon of music as a
form of speculation in terms of sound and time." ° It should be clearly
understood, however, that sound and time are for Stravinsky sensuous
elements, not physico-mathematical abstractions.

This speculation, moreover, is a unique language, like each significant
art form, which has validity precisely because its most significant levels of
meaning are not translatable into words. We might note that Helmholtz7
considers this problem when he reflects Aristotle's concepts of the relation
of music toward internal mental and emotional states by stating that:

Every motion is an expression of the power which produces it, and we in-
stinctively measure the motive force by the amount of motion which it pro-
duces. ... In this way melodic progression can become the expression of the
most diverse conditions of human disposition, not precisely of human feelings,
but at least of that state of sensitiveness which is produced by feelings . . .
[of] that general character temporarily shown by the motion of our concep-
tions. . . . Words can represent the cause of the frame of mind, the object
to which it refers, and the feeling which lies at its root, while music expresses
the kind of mental transition which is due to the feeling. . . . Music does not
represent feelings and situations, but only frames of mind which the hearer is
unable to describe except by adducing such outward circumstances as he has
himself noticed when experiencing the corresponding mental states. ... In
this sense, Vischer's8 rather paradoxical statement that the mechanics of mental
emotion are perhaps best studied in their expression in music may not be al-
together incorrect.

Perhaps the most significant idea contained in Helmholtz's remarks is

Papers (trans, by R. Ley), Rockliff, London, 1957, for a general collection of
Busoni's music writings.

6 1. Stravinsky, Poetics of Music, Harvard University Press, Cambridge, Mass..
1947, p. 16.

7 H. L. M. Helmholtz, On the Sensations of Tone, 2d English edition of 1885 by
A. J. Ellis, based on the 4th German edition of 1877, Dover Publications, New York,
1954, p. 250 et seq.

8 Helmholtz is referring to a passage from F. T. von Vischer, Aesthetik, Wissen-
schaft des Schonen, C. Macken, Stuttgart, 1858.

THE AESTHETIC PROBLEM 13

the definition of music as an external tonal representation of the "orderly
motion of our conceptions." Secondly, Helmholtz's comments also illus-
trate his awareness of the difficulty of trying to attach literal meanings to
musical materials. These two problems have been examined also by more
recent writers. For example, Susanne Langer,9 who believes that music has
"significant form," examines various well-known aesthetic theories such as
music being primarily pure formal design, as argued by Hanslick,10 music
being primarily self-expression, music being primarily a pleasurable expe-
rience, and so on, and rejects each as insufficient in itself to account for
the artistic merit of musical experiences. This, we might note in passing,
is rather an injustice to Hanslick and Helmholtz, both of whom anticipate
many of Langer's ideas. In any event, Langer suggests that:

If music has any merit, it is semantic, not symptomatic. ... If it has any
emotional content, it "has" it in the same sense that language "has" its con-
ceptual content — symbolically. It is not usually derived from affects nor in-
tended for them ... it is about them. Music is . . . their logical expression.
. . . For what music can actually reflect is only the morphology of feeling . . .
music conveys general forms of feelings, related to specific ones as algebraic
expressions are related to arithmetic [expressions]. . . . What most aestheti-
cians failed to see [in music] is its intellectual value, its close relation to con-
cepts, not by reason of its difficult academic laws but in virtue of its revelations.

Langer concludes that: ". . . Articulation is its life, but not assertion . . .
a significant form without conventional significance. . . ."

These definitions of meaning and form in music have been examined in
greater detail by Langer in a more recent publication11 without, however,
significant additions. Unfortunately, Langer is not particularly concerned
with musical materials as such. Therefore, a somewhat similar analysis,
more relevant to the musician's concern with his materials, is contained in
Meyer's recently published book on the aesthetic content in music.1 -
Meyer, in contrast to Langer, attempts to relate musical meaning and
musical experience to specific musical forms. He does not, as do many
writers, retain the separation between "absolute" and "referential" musical

°S. Langer, Philosophy in a New Key, New American Library, New York, 1948,
pp. 165-199.

10 E. Hanslick, The Beautiful in Music, first publ. 1854, English edition of 1891
by J. Cohen, based on 7th German edition of 1885, The Liberal Arts Press, New
York, 1957.

11 S. Langer, Feeling and Form, Philosophical Library, Inc., New York, 1953.

12 L. B. Meyer, Emotion and Meaning in Music, University of Chicago Press,
Chicago, 1956.

14 EXPERIMENTAL MUSIC

meanings — he points out that musical meaning depends upon learned re-
sponses to musical stimuli — nor does he retain the classic distinction be-
tween the ''emotional" and "intellectual" content of music, which has been
utilized as a working principle by even so recent a writer as Hindemith.13
Meyer instead proposes an "affect theory of music," based upon the concept
that emotion is evolved when a tendency to respond is inhibited. Thus, in
music, "the relationship between the tendency and its necessary resolution
is made explicit and apparent." 14 Meyer remarks that music arouses ex-
pectations, some conscious and others unconscious, which may or may not
be directly and immediately satisfied,15 and then relates meaning and emo-
tion into one unified response situation by stating that "what a musical
stimulus or a series of stimuli indicate and point to are not extramusical
concepts and objects but other musical events which are about to hap-
pen. . . . Embodied musical meaning is, in short, a product of expecta-
tion." 16 Meyer, starting from this premise, then examines in considerable
detail significant elements of musical structure, such as "structural gaps,"
the "principle of saturation," and the "principle of return," as well as more
familiar concepts such as tonality, rhythm, meter, and so on, in order to
apply this affect theory of musical response.

In so far as referential meaning is concerned, and such concepts as
Langer's "significant form" seem to fall in this category, Meyer has this
to say:17

(1) In most cultures, there is a powerful tendency to associate musical expe-
rience with extramusical experience. ... (2) No particular connotation is an
inevitable product of a given musical organization, since the association of a
specific musical organization with a particular referential experience depends
upon the beliefs and attitudes of the culture toward the experience.

Although Meyer's analysis of the psychological response to musical
forms and the dependence of musical form, conversely, upon psychological
phenomena is perhaps one of the more interesting studies of the problem
of musical content yet to come along, it is perhaps desirable, despite its
value, for us for purely practical reasons to adopt as a working premise a
somewhat more conservative and limited point of view. Therefore, if we

13 P. Hindemith, The Composer's World, Harvard University Press, Cambridge,
Mass., 1952, in particular chaps. 2 and 3.

14 Meyer, op. cit., pp. 22-23.

15 Ibid., p. 25.
"Ibid., p. 35.
17 Ibid., p. 262.

THE AESTHETIC PROBLEM 15

restrict ourselves to the general definition of music as the logical expression
of inward mental and emotional states, we may also distinguish sufficiently
for our purposes between what music communicates and how music is put
together to say effectively what it has to say. Then, since the techniques
of musical composition have been treated and codified by many theore-
ticians in discursive and practical terms, the logical treatment of musical
materials as such can be discussed in terms of language with considerable
precision. As also noted in a recent article by Hans Tischler,18 this aspect
of musical aesthetics was not really considered by Langer at all. He points
out that nowhere in Langer's discussion are criteria really set up for dis-
tinguishing "good music" from "bad music." Tischler states that an aes-
thetic appreciation of music must be based also upon a familiarity with
the medium and its technical possibilities. He notes that an entire system
of internal relations lies embedded in the medium and criticizes Langer's
preoccupation with what music appears to symbolize, stating that "to over-
look or understate this fact means pulling the basis from underneath any
aesthetic theory." He observes that Hanslick was emphatic in stressing this
point, and we might also observe that Stravinsky too complains about this
type of criticism when he says: "What is the use, in a word, of tormenting
him [the composer] with the why instead of seeking for itself the how, and
thus establishing the reasons for his failure or success?" 19

Tischler defines two species of relationships which characterize what he
calls "multirelational aesthetics," namely:

1 . Internal Relations. These change with the medium and in music con-
sist of rhythm, melody, harmony, counterpoint, tone color, expression (dy-
namics, tempo, etc.), and form or contour.

2. External Relations. These are true of all the arts and consist of
gesture, program, ethics, technical mastery, psychological drives of the
artist, function (e.g., for dance, worship, etc.), relevant historical and
sociological data, and performance.

Tischler proposes that the greater the number of relationships a work of
art reveals, the greater aesthetic significance we must attach to the particular
work of art. Whether or not this is true, for our purposes, Tischler's sep-
aration of internal relationships from external relationships is extremely
useful because it separates what we can find explicitly in a musical score
from what we must read into a score in order to become aware of its more
general referential significance. Moreover, in the long run, it may be also

18 H. Tischler, "The Aesthetic Experience," The Music Review, 17:189, 1956.

19 Stravinsky, op. cit., p. 87.

16 EXPERIMENTAL MUSIC

true that the study of musical forms will be a convenient medium for in-
vestigating the dependence of semantic meanings upon the articulation of
formal structures. It is important to note, however, that the semantic prop-
erties of music do not necessarily carry over to other forms of communica-
tion, because many of the properties of music seem to depend, in consid-
erable degree, specifically upon the medium itself.

The Logic of Musical Composition

There are five basic principles involved in musical composition which
we shall consider in the present context to be of primary significance. The
first principle is that the formation of a piece of music is an ordering process
in which specified musical' elements are selected and arranged from an in-
finite variety of possibilities, i.e., from chaos. The second principle recog-
nizes the contribution to a musical structure not only of order, but also the
relative lack thereof, and even, in certain extreme cases, of the absence of
order, namely, chaos; that is to say, the degree of imposed order is itself a
significant variable. The third principle is that the two most important
dimensions of music upon which a greater or lesser degree of order can be
imposed are pitch and time.20 There are, of course, other necessary ele-
ments of music as ordinarily considered by the composer such as dynamic
level and timbre, which also require ordering, but these will, for purposes of
simplification, be considered less significant. Next, because music exists
in time, the fourth principle is that memory, as well as instantaneous per-
ception, is required in the understanding of musical structures. Lastly, as
a fifth principle, it is proposed that tonality, a significant ordering concept,
be considered the result of establishing pitch order in terms of memory
recall.

The first principle, namely, that the process of musical composition in-
volves the choice of musical elements from an essentially limitless variety
of musical raw materials, has long received widespread recognition. In
fact, the very name composition suggests an act of arranging, of an im-
position of order, while the use of the word composer to characterize the
writer of music suggests a person who assembles and builds forms. Indeed,
the basic idea of composition as the extraction of order from chaos was

20 A more generalized picture of musical structure is that of wave-form amplitude
versus time. This concept lies at the root of experiments to synthesize musical
structures directly on film, for example. See p. 44.

THE AESTHETIC PROBLEM 17

formulated as long ago as the fourth century B.C. by Aristoxenus,-'1 who
remarked that: "The voice follows a natural law in its motion and does not
place the intervals at random." Aristoxenus also recognized the necessity
of the ordering process in both music and language when he stated that:
"The order that distinguishes the melodious from the unmelodious re-
sembles that which we find in the collocation of letters in language. For
it is not every collocation but only certain collocations of any given letters
that will produce a syllable."

This concept of opposing order and design to chaos has been a critical
issue in musical aesthetics ever since. If we limit ourselves to current
writers, we may note that Igor Stravinsky, in particular, has been most
explicit in his defense of this principle. It is, in fact, the central theme of
his Poetics of Music, previously referred to.G Several representative pas-
sages from this book can be quoted to illustrate this point. For example,
he remarks that: ". . . we feel [the necessity] to bring order out of chaos,
to extricate the straight line of our operation from the tangle of possibil-
ities";22 that: ". . . we have recourse to what we call order . . . order
and discipline." -A Stravinsky also defines art as the "... contrary of
chaos. It never gives itself up to chaos without immediately finding its
living works, its very existence threatened." 24 Stravinsky stresses the point
that: "Tonal elements become music only by virtue of their being organized
... so that to the gifts of nature are added the benefits of artifice." 2r>
Finally, he says that: ". . . to proceed by elimination — to know how to
discard . . . that is the great technique of selection." 2H

Given, therefore, that order is imposed during musical composition, the
second question immediately arises of how much order is imposed. Once
we recognize that all composition involves the selection of certain materials
out of a random environment toward order of one sort or another, we can
then ask the question of how much selection is involved in any particular
process, since it is obvious that all music falls somewhere between the two
extremes of order and chaos and that changes in musical style involve
fluctuations first toward one pole and then toward the other. Thus, "shape

21 Aristoxenus, The Harmonics (ed. and trans, by H. S. Macran), Oxford Uni-
versity Press, New York, 1902.

22 Stravinsky, op. cit., p. 5.

23 Ibid., p. 6.
2iIbid., p. 11.

26 Ibid., pp. 23-24.
26 Ibid., p. 69.

18 EXPERIMENTAL MUSIC

may, from this point of view, be regarded as a kind of stylistic 'mean' lying
between the extremes of overdifferentiation and primordial homogene-
ity." 27 It follows from this argument, as Meyer notes,28 that:

Weak, ambiguous shapes may perform a valuable and vital function ... for
the lack of distinct and tangible shapes and of well-articulated modes of pro-
gression is capable of arousing powerful desires for, and expectations of, clari-
fication and improvement. This aspect of musical structure and expression is
one which has unfortunately received but scant attention from music theorists,
aestheticians and critics who have continually construed "inevitability" to mean
unequivocal progression. . . . Yet the fact of the matter is that some of the
greatest music is great precisely because the composer has not feared to let
his music tremble on the brink of chaos, thus inspiring the listener's awe, ap-
prehension and anxiety, and, at the same time, exciting his emotions and his
intellect.

The above passages, quoted from the recent study of musical meaning
by Leonard Meyer, already referred to earlier, are a distinct contrast to
discussions of the pair of opposites, order and chaos, which tend to confuse
these terms with stylistic problems, if not also with problems of value.
Thus, order has frequently been associated with "classicism" and its equiv-
alents, and disorder with "romanticism" or "expressiveness." It is not
necessary for our purposes to assess the value of order or of chaos as such,
as Stravinsky does when he associates order with "the good," or as other
composers such as John Cage, to cite a recent example, have attempted to
do when they have set up an opposing musical aesthetic in which random-
ness or disorder is sought after as a desirable goal. The difficulty, obvi-
ously, is that few writers have attempted to define just what order is quan-
titatively in musical terms and have usually simply related this term in one
way or another to compositional procedures which satisfy their stylistic
prejudices. Moreover, few of them have ever considered it explicitly as a
quantitative variable, subject to control for expressive purposes.

In considering specific examples of how ordering processes are imposed
upon musical materials, we shall postulate that the most important involves
choices of pitch. There are necessarily many such choices in musical com-
position. In the first place, one of the most fundamental is the decision to
tune a scale to certain fixed pitches. The mere fact that most Western
music is written for a chromatic scale tuned to even temperament is in
itself a highly restrictive limitation upon random choice. The choice of a

27 Meyer, op. cit., p. 161.

28 Ibid., p. 160.

THE AESTHETIC PROBLEM 19

certain harmonic style imposes additional restrictions. Arguments in dis-
cussions of musical style, though seldom expressed explicitly as such, are
really concerned in many ways with the question of how restrictive the
selection process should be. Thus, stylistic limits in terms of pitch, for
example, are easy to distinguish. Complete disorder is characterized by the
random choice of any number of all possible pitches. On the other hand,
complete order is characterized by the arbitrary and sole choice of some
one fixed pitch.

A second basic choice process is connected with the fact that music,
like language, depends upon a series of successive selections; in other
words, that it exists in time. In fact, it is generally acknowledged that the
two most fundamental dimensions of music specifically are pitch and time.
The essential process of musical composition, therefore, consists of the
sequential selection of a series of pitches. This process is also recognized
by musicians, and again for comment, we may note that Stravinsky states
that: "Music is based on temporal succession and requires alertness of
memory. Consequently, music is a chronologic art, as painting is a spatial
art." 29 Just as the restrictions imposed upon pitch selection lead to scales
and ultimately to harmony, we find that restrictions of choice can be im-
posed upon the time dimension as well. This leads directly to the develop-
ment of meter and rhythm and ultimately to the organization of large-scale
musical structures. Moreover, the interaction of pitch selection and time-
interval selection is the basis of virtually all our known procedures for mu-
sical composition involving the internal relationships tabulated by Tischler.

To illustrate these points, we might consider the composition of a single
melodic line, restricting our argument, as we shall throughout for the sake
of simplicity, to a fixed tuning scheme, specifically, the ordinary chromatic
scale. We note initially that a melodic line is a sequence of intervals be-
tween successive notes chosen sequentially in a time scale dictated by the
choice, random or otherwise, of meter and rhythm, and that it is the se-
quence of intervals rather than of tones, or specific pitches, which gives a
melody its characteristic profile. If a mechanism is provided whereby the
successive choice of intervals can be made completely random, a random
melody is produced. On the other hand, if no choice whatever is provided,
the melody is a monotone. The imposition of a characteristic style between
these extremes involves the choice of specific rules of melodic writing which
will govern the nature of successive interval selection.

Polyphony involves the simultaneous interaction of two or more melodic

29 Stravinsky, op. cit., p. 29.

20 EXPERIMENT AL MUSIC

lines and as such is a characteristic feature of music as distinguished from
language. However, the principles of operation remain fundamentally
similar, and to handle the more complex problems involved in this inter-
action, we require the imposition of rules of harmony and counterpoint.

In the last passage we quoted from Stravinsky's Poetics of Music, there
is mentioned another significant issue that requires comment, namely, that
"music . . . requires alertness of memory." This remark points up the
fact that recognition of a musical message and, hence, the over-all organiza-
tion of a musical structure depends on its existence in time and on com-
prehending it in its totality in spite of its existence in time. Musical under-
standing has been recognized since ancient times to involve the perception
of what is going on in the immediate present, but always within the frame-
work of what has already happened and persists in the memory. For ex-
ample, Aristoxenus stated that:30 "musical cognition implies the simul-
taneous recognition of a permanent and a changeable element ... for the
apprehension of music depends upon these two faculties, sense perception
and memory; for we must perceive the sound that is present, and remember
that which is past. In no other way can we follow the phenomenon of
music. ..." St. Augustine31 also described musical communication in
these same terms when he said that listening to music depends not only on
numeri sonantes, that is, actual music which is heard, but also upon its
comparison with numeri recordabiles , that is, music which is remembered.
This process is required in order to form a musical judgment.

The consequence of this last characteristic of musical organization is of
the greatest significance, since it is at the root of our concepts and tech-
niques of thematic repetition and development, rhythmic repetition, the
need for systematic structures such as sonata form, fugue, and variation
form, and, perhaps most important of all, of our ideas of tonality. Since
certain experiments carried out with the Illiac involve investigation into
aspects of the nature of tonality, this brings up the last point we shall con-
sider at this stage, namely, a working definition for tonality. We shall
define tonality as tonal organization based on a pitch reference point for
a piece of music. A composition which uses a fixed-pitch reference point
can be said to be tonal; if it has several such fixed reference points, it may
be called polytonal. It is also presumably possible for the reference point
to shift during the course of a composition. Lastly, if no such reference

30 Aristoxenus, op. cit., pp. 27-30.

31 St. Augustine, De Musica, books I-VI (trans, by R. C. Taliaferro), The St.
John's Bookstore, Annapolis, Md., 1939.

THE AESTHETIC PROBLEM 21

point is ascertainable, the piece can be considered atonal. Again, we
shall not attempt to evaluate tonality as something "good" or "bad," but
we shall rather treat it as a parameter to be measured and to be controlled.
Moreover, if tonality in one form or another is being used to infuse co-
herence into a piece of music, tones, that is, specific pitches, acquire sig-
nificance because they are related, through specific intervals over a span
of time, to a specific tonal center. It is these long-range intervallic rela-
tionships that require memory for their recognition and which are used to
build up both small- and large-scale musical structures depending upon
tonal coherence as an organizational principle. It is important to separate
this principle from successive interval relationships which depend much
more directly only upon immediate sense perception. It is this, probably,
that Aristoxenus had in mind when he remarked:32 "Again, since intervals
are not in themselves sufficient to distinguish notes — the third part of our
science will deal with notes [and] will consider the question of whether
they are certain points of pitch, as commonly supposed, or whether they
are musical functions." In spite of this early awareness of the problem,
however, it is interesting to note that, historically, the concept of tonality
was one of the last to be formalized in terms of conscious operating prin-
ciples and can be said to be utilized consciously in its full scope perhaps
only since the time of Rameau and Bach. The rules of strict counterpoint,
for example, which are based largely on the compositional techniques of
the Renaissance and, specifically, the sixteenth century, are almost entirely
concerned with problems of successive intervals relationships and only
marginally with the question of tonality. "Sixteenth-century theorists char-
acteristically faced the problem of chord progression as if they wore
blinders that prevented them from seeing more than two chords at a time.
The extent of their scope was the passage of one consonance to the next,
the suspension and 'saving' of dissonances, and cadence patterns made up
usually of two intervals or chords." 33 Even today, in the teaching of the
theory of music, and specifically in the teaching of common practice har-
mony, the general problem of tonal organization is largely ignored. Basic
harmony in many traditional harmony textbooks is still taught largely in
terms of four-part chorale settings in the style of Bach, but with an em-
phasis solely on the rules of successive chord progression. It is in recogni-
tion of the lack of awareness of the necessity of utilizing logical processes

82 Aristoxenus, op. cit., p. 29.

:5!C. V. Palisca, "Vincenzo Galilei's Counterpoint Treatise: A Code for the Seconda
Pratica," J. Am. Musicological Soc, 4:81, 1956.

22 EXPERIMENTAL MUSIC

which depend upon and stimulate long-range recall that music analysts such
as Heinrich Schenker (1868-1935), for example, have attempted to for-
mulate more general principles of tonality and of melodic construction in
music written since 1700 (see Chapter 5, pages 133 to 134).

To summarize, (1) the process of musical composition requires the
selection of musical materials out of a random environment. This is ac-
complished by a process of elimination. The extent of order imposed de-
pends upon the nature of the restrictions imposed during the process of
selection. (2) Music is organized in terms of pitch — specifically, intervals
between notes — and in terms of time. Many possible interactions between
these two variables are expressed in terms of traditional rules of composi-
tion. (3) Musical coherence in a musical structure depends on the ex-
ploitation of memory as well as immediate sense perception. A recognition
of this principle is essential in the understanding of how proper articulation
is achieved in setting up musical structures.

Information Theory

In recent years, a new scientific theory, which has received the name
information theory, or communication theory, has been worked out in con-
siderable detail, particularly in certain practical applications in the fields of
telegraphy, telephony, and, in general, in problems of communication en-
gineering. We shall now consider certain important concepts of informa-
tion theory relevant to the general musical problems just reviewed and in
anticipation of some of the techniques applied to generate computer music.
The present discussion of information theory is abstracted primarily from
two recent and authoritative books on the subject by Shannon and Weaver34
and by Brillouin.35

Information theory depends upon a precise and limited definition of the
word information which answers the question of how to define the quantity
of information contained in a message to be transmitted. As a first step
toward an answer, it is observed that for the communications engineer the
technical problem is always the same, namely, to transmit "information"
accurately and correctly, quite without regard to the "meaning" or "value"
of the "information." It is of no concern to the engineer whether the mes-

34 C. E. Shannon and W. Weaver, The Mathematical Theory of Communication,
University of Illinois Press, Urbana, 111., 1949.

35 L. Brillouin, Science and Information Theory, Academic Press, Inc., New York,
1956.

THE AESTHETIC PROBLEM 23

sage he transmits is nonsense or of the greatest significance. Therefore, in
its current state, modern communication theory is restricted strictly to the
study of the technical problems involved in transmitting a message from
sender to receiver. Having accepted this limitation, we may then establish,
as our second premise, that every constraint imposed on freedom of choice
immediately results in a decrease of information. To help clarify this
somewhat unusual notion, it is helpful to consider how the alphabet can
be used to build up a language.30 For this purpose, let us next introduce
an additional concept of importance, namely, that we can classify com-
munication systems roughly into three main categories: discrete, contin-
uous, and mixed. A language consists of sequences of discrete symbols we
call letters; Morse code consists of sequences of dashes and dots. Other
forms of communication, however, such as paintings, photographs, or tele-
vision images, are continuous. Superficially this would seem to be the case
also with music. However, these continuous media are frequently con-
verted into discrete systems, as with the half-tone reproductions of photo-
graphs and the symbolic representation of music via musical score. As
Shannon and Weaver define it:37 "A discrete channel will mean a system
whereby a sequence of choices from a finite set of elementary symbols,
Si . . . Sn, can be transmitted from one point to another." Moreover, "It
is not required that all possible sequences of the Si be capable of transmis-
sion on the system, certain sequences only may be allowed." Thus, to
return to our consideration of language, we have twenty-seven letters in
the alphabet including the space. The simplest type of sentence might be
constructed by selecting letters sequentially with the choice of letters being
completely random, this choice being arrived at by assigning equal proba-
bilities to each letter of the alphabet. The result bears little resemblance
to an English sentence, however, except by pure chance. The situation is
one of highest potential information content: Anything might be said. We
can, however, reduce the information content of this random language in
order to achieve some higher degree of "meaning" by altering the proba-
bilities used to select the letters of the alphabet. Thus, we can first assign
probabilities based on the frequencies with which letters occur in the
English language. The next step beyond this is to assign probabilities based
on the frequencies with which letters occur one after the other. In this
way, freedom of choice is gradually reduced, and the results begin to take
on a more and more recognizable form. The decrease in information which

30 Shannon and Weaver, op. cit., pp. 13-14.
37 Ibid., p. 7.

24 EXPERIMENT AL MUSIC

occurs is said to be the consequence of introducing redundancy, which is
therefore related to order as information is related to disorder. This par-
ticular example of language construction, incidentally, is worked out in
some detail by Shannon and Weaver.

'information" is thus defined as the result of "choice" and is given a
statistical significance based upon probability theory. It is possible, there-
fore, to write algebraic expressions for the information content of a com-
munication system. In order to do this, the information content of the
system is defined purely in terms of the number of possible choices inherent
in the system itself. If we know nothing about the system, in other words,
if we are unable to define any of its properties, we must assume that the
choice is random, which is equivalent to saying that the information content
of the system is at a maximum. On the other hand, if we happen to pos-
sess some information concerning the properties of the system, it is prob-
able that we can restrict the choice process to a situation that is less than
totally random. This means that the information content of the system has
been reduced, or, in other words, we might state that the information we,
as observers of the system, have acquired concerning its properties has
been obtained at the expense of the information content of the system.
The more information about the system we acquire, the less information
the system contains. Therefore, according to Brillouin :3S

... we consider a problem involving a certain number of possible answers,
if we have no special information on the actual situation. When we happen to
be in possession of some information on the problem, the number of possible
answers is reduced, and complete information may even leave us with only
one possible answer. Information is a function of the ratio of the number of
possible answers before and after [a choice process], and we choose a logarith-
mic law in order to insure additivity of the information contained in inde-
pendent situations.

We may now follow Brillouin39 to define information algebraically:

Let us consider a situation in which P0 different possible things might happen
but with the condition that these P0 possible outcomes are equally probable
a priori. This is the initial situation, when we have no special information about
the system under consideration. If we obtain more information about the prob-
lem, we may be able to specify that only one out of the P0 outcomes is actually
realized. The greater the uncertainty in the initial problem is, the greater P0

38 Brillouin, op. tit., Introduction.

39 Ibid., pp. 1 ff.

THE AESTHETIC PROBLEM 25

will be, and the larger will be the amount of information required to make the
selection. Summarizing, we have:

Initial situation: /0 = 0 with P0 equally probable outcomes;

Final situation: /^O, with P1 = 1, i.e., one single outcome selected. The
symbol / denotes information, and the definition of the information is

I,=K]nP{) (1)

where K is a constant and "In" means the natural logarithm to the base e.

The definition of the measure of information can be generalized to cover
the case when P0 possibilities exist in the initial situation, while the final situa-
tion still contains PA possibilities;

Initially: /0 = 0 with P0 equally probable cases;

Finally: /j ^ 0 with Pj equally probable cases.

In such a case we take

/3 = K In (P0/Pi) = K In P0 - K In /\ (2)

This definition reduces to Eq. 1 when Px = 1. [Italics supplied.]

The two cases discussed by Brillouin which apply to systems narrowed
down to one choice in the first case and to a number of choices in the
second case depend on the assumption of equal a priori probabilities. This
is the simplest condition for, as we shall see, it is also possible to have un-
equal a priori probabilities, probabilities conditioned by previous choices
and many more complex situations.

At this point, it is crucial to note that Equations (1) and (2) bear a
striking resemblance to an equation, well known to physical scientists
familiar with statistical mechanics, which relates thermodynamic probability
and entropy. Thermodynamic probability, roughly, is a measure of the
number of ways in which a physical system might be arranged, and entropy
is related to this function by means of the following expression:

S = k\nW (3)

where S is the entropy of the system, W is the thermodynamic probability,
and k is Boltzmann's constant, equal to 1.36 X 10~16 erg/degree C.
We may now relate information and entropy through the ratio k/K, the
exact value of which depends on the units used to express information.

The question now arises as to the significance of the concept of entropy.
It is essentially a measure of the degree of disorder or randomness in a
physical system. Whenever a change occurs in some physical system which
results in a decrease in order, the entropy of the system is said to increase.
Conversely, an increase in order results in a decrease in entropy. For

26 EXPERIMENTAL MUSIC

example, if a sample of crystalline ice is melted to form liquid water, there
results an increase of entropy, because in the crystalline ice water molecules
are arranged in a highly ordered lattice structure, while in liquid water
these same molecules are more nearly scattered at random. Converting
liquid water to steam involves yet another entropy gain, because this
process causes the water molecules to become widely scattered and to move
about at high velocity through a large volume of space. Scrambling an egg
is an even simpler example of an entropy change. No chemical change
occurs during this process, only mixing; however, after scrambling, because
the resulting mixture is more random than before, the entropy content of
the egg has increased. Even shuffling a sorted deck of cards can in a sense
be said to bring about a change of entropy. It takes work to sort the cards
into an ordered sequence, and this work can be thought of as an extraction
of entropy. The problem of unscrambling an egg is also a problem in-
volving the extraction of entropy.

In 1929, Szilard40 recognized the close similarity between information
and entropy, but the significance of this relationship was not generally
recognized until it was rediscovered years later by Shannon to lead to the
current development of information theory. It is now recognized that
entropy is a measure of missing information. Thus, in the physical sciences,
where all systems (except perfect crystals at absolute zero, — 273.16°C)
have positive entropy content, we find that all systems except these must
of necessity be incompletely defined, this incompleteness being in direct
relation to their entropy contents. To go back to our examples, we see
that we know more about water molecules in crystalline ice than in liquid
water, because, if for no other reason, we at least know more precisely
where the molecules are.

We can now reproduce from Shannon and Weaver two useful proposi-
tions. Shannon and Weaver41 note that the entropy of a communication
system will be zero "if and only if all the P4 but one are zero, this one hav-
ing the value unity. Thus, only when we are certain of the outcome does
[the entropyl vanish. Otherwise [the entropy] is positive." Moreover, "for
a given [number of possible choices], [the entropy] n is a maximum and
equal to log n when all the Pt are equal, i.e., 1/n. This is also intuitively
the most uncertain situation." And lastly, "any change towards equaliza-
tion of the probabilities, Pi, increases [the entropy]."

40 L. Szilard, "tJber die Entropieverminderung in einem thermodynamischen System
bei Eingriffen intelligenter Wesen," Z. Physik, 53:840, 1929.

41 Shannon and Weaver, op. tit., p. 21.

THE AESTHETIC PROBLEM 27

The definition of information as a measure of a number of choices from
a random arrangement of a finite set of elements is unquestionably confus-
ing when first met with, so it is important that it be clearly understood that
by "information" we do not mean information in the everyday sense. "In-
formation" in information theory is not the same thing as "meaning," par-
ticularly semantic meaning, or "specific knowledge about," which are defi-
nitions more nearly synonymous with the common use of the word. To
clarify this point, Weaver states that:42 "The word information in commu-
nication theory relates not so much to what you do say as to what you
could say. . . . The concept of information applies not to the individual
messages (as the concept of meaning would) but rather to the situation
as a whole."

In a similar vein, Brillouin concludes:39

Our definition of information is an absolute objective definition, independent
of the observer. . . . The restrictions we have introduced enable us to give a
quantitative definition of information and to treat information as a physically
measurable quantity. . . . We define "information" as distinct from "knowl-
edge" for which we have no numerical measure. . . .

Moreover, as Weaver points out:43

The concept of information developed in this theory at first seems disappoint-
ing and bizarre — disappointing because it has nothing to do with meaning, and
bizarre because it deals not with a single message but rather with the statistical
character of a whole ensemble of messages, bizarre also because in these sta-
tistical terms the two words information and uncertainty find themselves to
be partners.

However, Weaver suggests that: "one is now, perhaps, for the first time,
ready for a real theory of meaning." He attempts to set up this problem
for future study by suggesting "three levels of communication," namely:44

Level A. How accurately can the symbols of communication be transmitted?
(The technical problem.)

Level B. How precisely do the transmitted symbols convey the desired mean-
ing? (The semantic problem.)

Level C. How effectively does the received meaning affect conduct in the
desired way? (The effectiveness problem.)

42 Ibid., p. 110.

43 Ibid., p. 116.

44 Ibid., pp. 95-96.

28 EXPERIMENT AL MUSIC

In this connection, it has been stressed by Weaver that there may be a
high degree of overlap between the three levels. He suggests that:45

A larger part of the significance [of information theory] comes from the fact
that the analysis at level A discloses that this level overlaps the other levels
more than one could possibly naively suspect. Thus, the theory of level A is,
at least to a significant degree, also the theory of levels B and C.

Brillouin similarly recognizes two areas of investigation which lie outside
current research in information theory. He points out46 that the next prob-
lem to be defined is the problem of semantic information, i.e., whether or
not a message makes sense. As noted by Brillouin, some exploratory in-
vestigations of this problem in language have apparently been carried out
by Ville47 and by Carnap and Bar-Hillel,48 who based their work on the
methods of symbolic logic, but this seems to be about the extent of current
research in this area. Lastly, Brillouin recognizes the problem of "value,"
i.e., whether or not the message is of value to the sender or receiver. Here,
he says we "invade a territory reserved for philosophy. . . . Shall we ever
be able to cross this border? . . . This is for the future to decide." This
is, of course, Weaver's level C, which in his estimation involves aesthetic
considerations in the fine arts.49

To summarize : ( 1 ) in recent years, information theory has been applied
to certain practical problems of communication engineering.50 (2) Cer-
tain authors have suggested that the concepts of information theory might
well be used more generally than just in engineering problems. (3) Infor-
mation is defined as proportional to the logarithm of the number of possible
choices available when making a decision. Information is thus analogous
to entropy. (4) Inasmuch as common communication systems utilize finite
sets of discrete symbols, these symbols can be selected sequentially by what
we will call a stochastic process to build up a "message." The information,

45 Ibid., p. 98.

46 Brillouin, op. cit., pp. 297 ff.

47 J. Ville, Actualities sci. et ind., 1145:101-114, Hermann, Paris, 1951.

48 Y. Bar-Hillel and R. Carnap, "Semantic Information," Brit. J. Phil. Sci., 4:147,
1953; see also C. Cherry, On Human Communication, John Wiley & Sons, Inc.,
New York, 1957, pp. 231-250, for a detailed discussion of this problem based on
another paper by these same authors, namely: R. Carnap and Y. Bar-Hillel, "An
Outline of a Theory of Semantic Information," M.I.T., Research Lab. Electronics
Tech. Rept. 247, 1953.

48 Shannon and Weaver, op. cit., p. 97.

50 In fact, it might be of interest to note that these applications have now become
sufficiently extensive that a technical journal, IRE Transactions on Information
Theory, which is devoted specifically to this subject, is now being published.

THE AESTHETIC PROBLEM 29

or entropy, content of a communication system is at a maximum if there
are the least number of restrictions upon the process of selecting successive
events; specifically, the largest entropy content is obtained whenever the
sequence of symbols is completely random.

Information Theory and Music

It is now necessary to examine how these concepts relate to the defini-
tions of musical meaning and form previously discussed. We can start by
noting that not only has Weaver suggested in general terms that information
theory can be applied to the study of art, but that Pinkerton,51 for example,
has proposed that the theory might be used in studies of music. Also, fairly
ambitious theoretical attempts to apply information theory to the study of
music have been published by W. Meyer-Eppler'2 and A. Moles. 5:^ More-
over, Leonard Meyer, whose concepts were reviewed in some detail earlier
in this chapter, has also recently recognized many corresponding properties
between his theories of musical meaning and information theory.54 Thus,
Meyer acknowledges the equivalence between his ideas of ambiguity and
precision of form and entropy variation and, secondly, the importance of
sequential choice processes in the building up of musical structures.

Some main points of Moles' applications of information theory to musical
communication can now be summarized. Moles postulates two determining
factors which permit a listener to build messages out of musical sounds,
namely, memory and attention, i.e., perception. This is, of course, in accord
with traditional aesthetic theory. Moles then notes that memory appears
to be divided, in terms of span, into three categories : ( 1 ) instantaneous
memory, (2) dated memory, and (3) undated memorization. However,
it is to the problem of attention that he has directed most of his studies.
He suggests that "attention" can be divided into two distinct "modes": (1)

51 R. C. Pinkerton, "Information Theory and Melody," Sci. American, 194(2) :77,
February, 1956.

52 W. Meyer-Eppler, "Statistic and Psychologic Problems of Sound," Die Reihe,
1:55 ff.; "Informationstheorie," Naturwissenschaften, 39:341, 1952. A review of some
of Meyer-Eppler's views is also given in an article by H. Le Caine, "Electronic
Music," Proc. I.R.E., 44:457, 1956.

58 A. Moles, "Informationstheorie der Musik," Nachr. Technik Fachberichte, Z'Al ,
1956; Theorie de V Information et perception esthetique, Presses Universitaires de
France, Paris, 1957; Some Basic Aspects of an Informational Theory of Music,
unpublished manuscript; and other writings. Also private conversations between Dr.
Moles and one of the present authors (L. A. H.) in Paris, June, 1957.

54 L. B. Meyer, "Meaning in Music and Information Theory, /. of Aesthetics and
Art Criticism, 15:412, 1957.

30 EXPERIMENTAL MUSIC

the semantic mode and (2) the aesthetic mode. The semantic mode is
characterized as the "language side of music — a system of organized and
standardized symbols — which can be coded 55 — i.e., translated into another
language — the score." On the other hand, the aesthetic mode "does not
appeal to intellectual faculties, but to the directly sensorial ones — even
sensual at the limit." Moles thus differentiates two types of structures as
does Tischler18 and defines the term acoustical quanta. These quanta
"make up the repertory of aesthetic symbols at a given scale of duration
[and] information rate, He, — [which] can then be computed and which
comes parallel to the semantic information rate, #s." Moles' purpose is
"to study the properties of the aesthetic message — vs. the semantic one,
both being bound into the same sequence of acoustical sets of quanta
grasped in a different manner."

One additional point made by Moles is of interest:

The fact acknowledged by many psychologists dealing with the human op-
erator that one is unable to grasp a message of more than 10-20 bits/second,56
compared with the estimated maximal capacity of some hundred bits/second)
implies that perception is a selection of definite symbols in the whole of the
message and that these symbols, these Gestalt are not picked at random, which
would simply express the utter incapacity of the listener to cope with a too
original message.57 In consequence, the structure of Music itself regarding
the color, thickness and rate of originality of the musical stuff should be di-
rectly considered by the composer. This leads to the concept of "authentic
composition" (Meyer-Eppler) which has recently found its way into experi-
mental music.

The first point to decide, if practical musical applications are to be made
for the concepts of information theory, is whether music is basically a dis-
crete, a continuous, or a mixed communication system. We should like to
propose that it is effectively a discrete system. It is thus like language, al-
though normally more complex operationally, because in language only one
symbol for an operational element is considered at a time. In music, a
number of elements are normally in operation simultaneously.

There are a variety of ways in which music operates through discrete
elements. Most importantly, as Helmholtz, for example, has noted:58

55 The idea of coding is considered in Chapter 4 in relation to computer operation.

56 A bit is a unit quantity of information and is a term used in digital-computer
theory. See Chapter 4.

57 I.e., a message with too high an entropy content.

58 Helmholtz, op. cit, pp. 250-253.

THE AESTHETIC PROBLEM 31

Alterations of pitch in melodies take place by intervals and not by continuous
transitions. The psychological basis of this fact would seem to be the same as
that which led to rhythmic subdivision periodically repeated. . . . The musical
scale is as it were the divided rod, by which we measure progression in pitch,
as rhythm measures progression in time. Hence, the analogy between the scale
of tones and rhythm occurred to musical theoreticians of ancient as well as
modern times.

We consequently find the most complete agreement among all nations that
use music at all, from the earliest to the latest times, as to the separation of
certain determinate degrees of tone from the possible mass of continuous
gradations of sound, all of which are audible, and these degrees form the scale
in which the melody moves. But in selecting the particular degrees of pitch,
deviations of national taste become immediately apparent. The number of
scales used by different nations and at different times is by no means small.

Almost all music notation, in fact, is based upon a definition of discrete
musical elements. Thus, except possibly for such recent artifacts as some
forms of electronic music (Chapter 3), we can define the basic texture of
music as an assembly of discrete symbols with the few exceptions (such as
continuous dynamics changes) being so simple that these are readily taken
care of independently.

Secondly, restricting the number of choices should tend to increase the
"meaningfulness" of messages. Thus, the most diffuse type of music is pro-
duced on the average when successive note selection is permitted to be com-
pletely random. As we shall see, music of this type is rather easily gen-
erated in a computer and forms the basic substance from which we must
fashion more characteristic structures. Thus, we note that our operational
basis is entirely in accord with Stravinsky's concepts of the logic of musical
composition discussed earlier in this chapter. Or, as noted by Helmholtz:59
"Music alone finds an infinitely rich but totally shapeless plastic material
in the tone of the human voice and artificial musical instruments which
must be shaped on purely artistic principles."

Thirdly, the problem arises as to what techniques to apply to restrict
successive choices if we desire to produce music less chaotic than random
music. It is possible, for example, to apply statistical methods and com-
pute transition probabilities for successive note selection based upon the
analysis of some known species of music. Pinkerton worked out a simple
example of how this can be done by constructing a transition-probabilities
table based upon the analysis of a set of nursery tunes. Pinkerton quite cor-

59 Ibid., p. 250.

32 EXPERIMENTAL MUSIC

rectly observed, however, that the use of such tables leads to the construc-
tion of "banal" tunes as a general rule. It is easy to see that this is bound to
occur whenever a purely statistical analysis to determine mean choices is
used as an aesthetic basis for computing transition probabilities. The diffi-
culty here is an aesthetic one; if we wish to generate something besides
banal music, other criteria must be sought.

In this connection, we should mention also that a similar but more thor-
ough study has been carried out recently by Brooks, Hopkins, Neumann,
and Wright,60 to whom we refer also in Chapter 3, since these au-
thors coded their ideas for computer processing. Like Pinkerton, these
authors subjected a sampling of simple tunes (this time hymn tunes) to
statistical analysis to form transition-probability tables. However, their
analysis was more elegant, since they carried out their calculations to the
extent of eighth-order probabilities, i.e., to include into the calculations re-
lationships as far as eight notes back. On the other hand, there is consider-
able danger in elaborating a simple eighth-order process to produce music,
since aside from simple tunes such as hymn tunes, there is little music in
which a fluctuation of transition probabilities from one part of a composi-
tion to another would not occur. This danger has been recognized, inci-
dentally, by Meyer.54 Consequently, means are required for controlling
fluctuations between randomness and order during the course of a compo-
sition.

Still another study in this same vein has been recently published by
Youngblood.61 In contrast to the above studies, Youngblood has computed
sets of transition probabilities derived from the analysis of fairly complex art
music. For his study, Youngblood has chosen a group of songs by Schubert,
Mendelssohn, and Schumann and has compared the results obtained from
the music of these three composers. He has tabulated differences as re-
flected in transition probabilities and information contents between the
individual styles of these composers, as well as similarities which one would,
of course, expect, since all three employ the same basic style of composition.

Lastly, two other recent incidents of rather simple applications of the
idea of sequential-choice processes to compose music have been also re-
ported by J. R. Pierce.62 Thus:

60 F. P. Brooks, Jr., A. L. Hopkins, Jr., P. G. Neumann, and W. V. Wright, "An
Experiment in Musical Composition," IRE Trans, on Electronic Computers, EC-
6:175, 1957.

"'J. E. Youngblood, "Style as Information," J. of Music Theory, 2:24, 1958.

62 J. R. Pierce, letter to Sci. American, 194(4): 18, April, 1956.

THE AESTHETIC PROBLEM 33

J. J. Coupling has discussed stochastic composition of music in "Science for
Art's Sake" in Astounding Science Fiction, Nov., 1950. [Similarly] Dr. D.
Slepian of Bell experimented with stochastic composition, not using statistics
but such ideas of probability as have accumulated in the minds of a group of
experimenters. Thus, he had each of a group of men add to a "composition"
after examining only one or more preceding half measures. Tape recordings
of the resulting music have been played as a part of a number of talks on
information theory.

Pierce himself, in collaboration with M. E. Shannon, has also worked out
an example of stochastic music, this music involving common chords se-
lected in random sequences. This particular example of stochastic music
is reproduced in a recent book by Pierce. 82a

It can be seen that the various experiments to produce stochastic music
thus far carried out are subject to critical limitations of one type or another.
The end products, if not banal, as Pinkerton termed his results, nevertheless
remain rather primitive. In designing our experiments, we were well aware
of the difficulty of basing experiments utilizing these new techniques on ini-
tial operating principles which might appear on first inspection to be far
removed from traditional musical procedures. An alternative procedure
was to combine relevant concepts of traditional musical experience with
the operating techniques derived from information theory and to take ad-
vantage of Weaver's suggestion that there is extensive overlap between the
three areas of investigation relevant to information theory. In this way, we
would use the stimulus provided by working with traditional music con-
cepts in terms of new operational principles as a point of departure for
formulating abstract structural bases for music synthesis.

It is interesting to note, in concluding, that attempts to apply information
theory to musical problems raise in a new guise an old issue which has
been a source of dispute in musical aesthetics many times in the past. It
is yet another attempt to codify musical aesthetics in terms of natural law.
This is, of course, an argument resorted to by many writers ever since music
was defined as an imitation of nature in ancient times. Zarlino, for example,
looked "on music as an imitation of nature and endeavored to derive his
teachings from natural law," 63 i.e., in accord with Pythagorean and Pla-
tonic theory. On the other hand, Vincenzo Galilei in attacking Zarlino's
teachings "considered numerical ratios irrelevant to the artist and the rules

02a J. R. Pierce, Electrons, Waves and Messages, Hanover House, Garden City,
N.Y., 1956, pp. 271-274.

<ia O. Strunk, Source Readings in Music History, W. W. Norton & Company, Inc.,
New York, 1959, p. 228.

34 EXPERIMENTAL MUSIC

of counterpoint a product of the demands of taste, experience and aesthetic
purpose." (54 More recently Helmholtz remarked that:65

... to furnish^ a satisfactory foundation for the elementary rules of musical
composition ... we tread on new ground, which is no longer subject to
physical laws alone. . . . Hence it follows — that the system of Scales, Modes,
and Harmonic Tissues does not rest solely upon inalterable natural laws, but
is also, at least partly, the result of aesthetical principles, which have already
changed, and will still further change, with the progressive development of
humanity. [Helmholtz's italics.]

A more subtle statement of the same basic thesis is contained in Meyer's
recent book, already referred to, when he remarks66 that three interrelated
errors have continually plagued music theory, namely, hedonism, the phi-
losophy that pleasure is the primary purpose of musical experience; atom-
ism, which is the attempt to characterize music solely by means of its dis-
crete elements; and universalism, which is "the belief that the responses ob-
tained by experiment or otherwise are universal, natural, and necessary.
This universalist approach is also related to the time-honored search for a
physical quasi-acoustical explanation of musical experience — the attempt,
that is, to account for musical communication in terms of vibration ratios
of intervals, and the like." What effect information theory will have on this
problem will be of considerable interest to watch in the future.

To summarize, music, being a nondiscursive form of communication,
operates with a semantic peculiarly dependent upon technical structure as
such. Therefore, the study of musical structure in terms of information
theory should be a significant technique for breaking through the "semantic
barrier" which seems to hamper current investigations in information theory
and should perhaps also lead to an improved delineation of the aesthetic
basis of musical composition. Specifically, in light of the apparent close
dependence of meaning upon form in music, we suggest that Weaver's over-
lap, if it exists, is particularly significant in music. The aesthetic signifi-
cance, or "value," of a musical composition depends in considerable meas-
ure upon its relationship to our inner mental and emotional transitions, but
this relationship is largely perceived in music through the articulation of
musical forms. The articulation of musical forms can be considered the
semantic content of music, and this in turn can best be understood in terms
of the technical problems of musical composition. Since the articulation of

^Palisca, op. cit.

85 Helmholtz, op. cit., pp. 250-251.

m Meyer, op. cit., p. 5.

THE AESTHETIC PROBLEM 35

musical forms is the primary problem faced by composers, it seemed most
logical to start our investigation by attempting first to restate the techniques
used by composers in terms both compatible with information theory and
translatable into computer programs utilizing sequential-choice operations
as a basis for music generation. In our investigation, as we have already
noted, therefore, we first studied the traditional craft acquired by every
composer, namely, counterpoint, harmony, rhythm, melodic construction,
and similar basic problems. Only after results were achieved in this investi-
gation did we feel that we could apply more experimental processes. Not
unexpectedly, work of this nature soon led to speculation as to whether
there exist more general principles of musical composition suitable for
computer use. It is seen that in this approach we used the differentiation of
internal and external relations suggested by Tischler, or alternately, the dis-
tinction between semantic and aesthetic quanta as suggested by Moles as a
basic operating premise. However, we did not feel that the closer depend-
ence of the relevant external meanings upon internal musical relationships
suggested by Meyer conflicted with this experimental approach, since
Meyer's analysis of musical meaning represents a broadening of these con-
cepts rather than a departure from them.

CHAPTER THREE

Experimental Music

Experimental Music Defined

It is the purpose of this chapter to summarize recent technical and artistic
developments relevant to the subject of modern experimental electronic and
synthetic music. Our primary purpose is to show how the present study of
computer-produced music is related to and yet distinguished from other
recent experiments in the production of music by unusual means. It is our
understanding that this general subject, including even a brief discussion of
our work based upon a limited amount of information, has been recently
reviewed in considerable detail in an as yet unpublished treatise by Abra-
ham Moles,1 presently associated with the Centre d'Etudes Radiophoniques
in Paris, one of the several European groups currently working in this gen-
eral area. An introductory review of the topic of recent electronic music
experiments is contained in two recent articles published in The Reporter.2
Moreover, V. Ussachevsky3 has written just recently a review article sur-

1 A. Moles, Experimental Music, in manuscript.

2R. Maren, "Music by Montage and Mixing," The Reporter, 13(5): 38, Oct. 6,
1955; R. Maren, "Electronic Music: Untouched by Human Hands," ibid., 16(8): 40,
April 18, 1957.

3 V. Ussachevsky, "The Processes of Experimental Music," ./. Audio Engineering
Soc, in press.
36

EXPERIMENT AL MUSIC 37

veying the current status of European experimentation. Lastly, Le Caine4
has written an informative historical survey of the subject of both electronic
instruments and of recent experiments in coded electronic music.

In considering this subject ourselves, we shall limit our discussion arbi-
trarily to novel means of composition and sound production, and, more-
over, we will stress the difference between attempts at rationalization of the
laws of musical composition and attempts to substitute and make use of
modern means of sound production to produce an aural effect. Thus, we
will immediately exclude from consideration forms of composition involv-
ing either a composer writing music on paper in the traditional sense or
experimentation with novel sound effects and timbres utilizing conventional
musical instruments. In so doing, we are, of course, not at all implying
that significant new music is not being written by these more traditional
procedures; we are simply restricting our discussion to the most directly re-
lated material. Secondly, given this restriction of subject matter, it seems
convenient to classify current research in experimental music into two basic
categories: (1) We can group together experimental studies of the logic
of musical composition. In terms of most recent work, this would refer
most specifically to the use of automatic high-speed digital computers to
produce musical output. For convenience, we may term such music com-
putational music, in general, and the specific type produced with com-
puters, computer music. (2) We can consider experiments involving pri-
marily the production of musical sounds by means other than the use of con-
ventional instruments played by performers. In this category, we refer to
the production of musical sounds by electronic means and by the manipula-
tion of magnetic recording tape in tape recorders and related devices. For
our purposes, it is convenient to group all these experiments under the
name of electronic, or synthetic, music, though in so doing, we refer to a
broader body of work than just the elektronische musik produced at present
in Cologne, Germany. In this second group of experiments, principles have
had to be formulated for "composing" as well as for the actual technical
production of sound, but the choice process for selecting materials to go
into a "composition" is still carried out entirely by the "composer." It is
seen that the two types of experiments are complimentary rather than com-
petitive. In fact, there are a number of ways in which the results of the
two types of experimentation might be used in the future to enhance the
effectiveness of each. This we shall consider in Chapter 7.

4H. Le Caine, "Electronic Music," Proc. I.R.E., 44:457, 1956.

38 EXPERIMENT AL MUSIC

Electronic, or Synthetic, Music

Up until the end of World War II, the applications of electronic circuits
to the production of musical sounds had been largely restricted to the de-
sign and development of a number of practical musical instruments in-
tended for live performance. Excluding from consideration the use of am-
plifiers to enlarge the sound of conventional musical instruments such as
guitars, these electrophones, as they are defined by Curt Sachs,5 seem con-
veniently to be divided into two groups of instruments. The first group in-
cludes the well-known commercial electronic organs such as the Hammond
and Baldwin. The primary purpose of these instruments is to provide a
facsimile of pipe-organ sound rather than to provide a fundamentally new
musical timbre. The technical details of how these instruments operate are
quite interesting, however, and can be found in the literature.6

Of somewhat greater interest have been several more novel electro-
phones, such as the Theremin, or Etherophone, invented by L. Theremin
in 1924, Les ondes musicales, or Les ondes Martenot, invented by M. Mar-
tenot, also in 1924, and a number of more recent instruments, such as the
Mixturtrautonium, invented in 1930 and refined by Oskar Sala. Most of
these instruments are monophonic rather than polyphonic devices. The
Hammond Solo vox attachment perhaps also falls into this group. All are
designed to supplement conventional instrumental groups and are, therefore,
new additions to the standard body of musical instruments. These instru-
ments are designed for live performance and do not represent a basic de-
parture in performance practice. However, these instruments do produce
new instrumental timbres which have been exploited to a limited extent by
contemporary composers, particularly in France.7 Recently, for example,

5 C. Sachs, The History of Musical Instruments, W. W. Norton & Company, Inc.,
New York, 1940, pp. 447-449.

6 R. H. Dorf, Electronic Musical Instruments, Radio Magazine, Inc., Mineola,
N.Y., 1954; A. Douglas, The Electronic Music Instrument Manual, 3d ed., Sir Isaac
Pitman and Sons, Ltd., London, 1957; E. G. Richardson, "Electrophonic Instruments,"
in E. Blom (ed.), Grove's Dictionary of Music and Musicians, 5th ed., The Macmillan
Company, New York, 1954, vol. II, pp. 905-908. It should be noted that Dorfs
book contains an extensive bibliography and patent list.

7 For example, several works by Oliver Messiaen, including Fete des Belles Eaux,
1938, for six ondes Martenot and Trois Petites Liturgies de la Presence Divine, 1944,
for unison soprano choir, celesta, vibraphone, maracas, Chinese cymbals, gong, piano-
forte, ondes Martenot, and strings. Also, Andre Jolivet has written a Concerto for
Ondes Martenot and Orchestra, 1947, which has recently been recorded on West-
minster XWN 18360.

EXPERIMENTAL MUSIC 39

Les ondes Martenot and the Mixturtrautonium were featured instruments
at the 1st International Congress of Electronic Music and Musique Con-
crete at Basel, Switzerland.8 Detailed descriptions of these instruments are
given in the literature references already cited.4, 6 The "electronic sackbut"
designed by Le Caine since 1945 should also be mentioned, particularly
since the sounds produced by this instrument are rather strikingly different
from those produced by most of the instruments just mentioned.4 Most
monophonic electronic instruments happen to be modified audio oscillators
and the most usual tone color they produce is simple sine-wave sound; i.e.,
pure harmonic sound consisting solely of a fundamental free of overtones.

A second technical achievement of major consequence, realized around
the end of World War II, was the commercial production of high-fidelity
magnetic tape recorders. With these instruments, for the first time, it has
become possible to record music — or for that matter, sounds of any sort —
easily, accurately, and at reasonable cost. Secondly, and of equal impor-
tance, it has become possible with these instruments to edit and to alter the
sounds once they are recorded. While it had been possible to produce,
previously, fine recordings by means of disc recording, it was also true
that adequate equipment for this purpose was expensive and elaborate.
Noncommercial disc recording was inherently unsatisfactory. Moreover,
editing and alteration of the recorded sounds were entirely impractical.

Once tape recorders became available, it was not long before their pos-
sibilities for musical "composition" were recognized. Apparently, among
the first to experiment with, tape recorders to produce a new type music
was Pierre Schaeffer, whose initial work around 1948 has led to the devel-
opment of what is now termed musique concrete. This is the first of several
types of synthetic music which we shall now briefly review. Since our dis-
cussion of this subject is meant to be illustrative rather than exhaustive,
only the most important of these studies will be mentioned.

Musique Concrete. This name is given to the musical products of a
group directed by Pierre Schaeffer working for Radiodiffusion Francaise.
In producing musique concrete, a point is made on certain obscure aesthetic
grounds of utilizing only sounds which originate in "nature" — in other
words, sounds produced by electronic means are excluded from considera-
tion. These "natural" sounds, however, are subjected to considerable alter-
ation during the course of "composition" by means of several simple but
efficient pieces of equipment. The best known of these is the Phonogene,
designed by Pierre Schaeffer, which permits direct transposition by means

8 Anon., Time, 65(23) :78, June 6, 1955.

40 EXPERIMENTAL MUSIC

of independently operated tape travelers of tape-recorded material to twelve
different pitches. Equipment for synthetic reverberation, montaging, and
variable speed control is also available.

In addition to Pierre Schaeffer, the principal composer within the mu-
sique concrete group is Pierre Henry. From time to time, however, other
composers including Milhaud, Varese, and Messiaen have used their facili-
ties. A two-record set of recording of musique concrete is available in this
country.9 Included in the records is one particularly striking piece of music,
Le Voile d'Orphee, by Pierre Henry.

At the present time, the principal research objectives of the musique
concrete group appear to be (1) the development of a notation for their
music, since up to this point the selection of materials for compositions has
been entirely empirical, and (2) the application of musique concrete to
radio, television, movies, and the theater. No major expansion of their
technical resources is apparently contemplated at the present time.

There is a growing literature on the subject. Perhaps the best sources of
information at the present time are a short mimeographed book issued by
the group which reviews their work through 1955,10 and a recent special
issue of La Revue Musicale.11

A bibliography of publications concerned with musique concrete is in-
cluded in the house publication referred to.8 There are also brief discus-
sions of musique concrete available in publications in English.12

Elektronische Musik. A second and somewhat competitive European
development is the elektronische musik prepared at the N.W.D.R. broad-
casting studios in Cologne, Germany. This laboratory for experimental
electronic music was started by Herbert Eimert, its present director, and
Werner Meyer-Eppler in the period following World War II. While there
are only two "staff composers," namely, Karlheinz Stockhausen and G. M.
Koenig, employed by the studio, other composers such as Ernst Krenek
and Pierre Boulez have used their facilities to produce elektronische musik
compositions.

9 P. Schaeffer, P. Henry, P. Arthuys, and M. Phillipot, Panorama of Musique Con-
crete, vol. I, London DTL 93090, vol. II, London DTL 93121.

10 Radiodiffusion-Television Francaise, Groupe de Recherches de Musique Concrete,
Sept Ans de Musique Concrete, 1948-1955, Centre d'fitudes Radiophoniques, Paris.

""Vers une Musique Experimental — sous la Direction de Pierre Schaeffer," La
Revue Musicale, numero speciale 236, 1957.

12 H. Searle, "Concrete Music," in E. Blom (ed.), Grove's Dictionary of Music and
Musicians, 5th ed., The Macmillan Company, New York, 1954, vol. IX, app. II, pp.
571-572; R. S. Brindle, "The Lunatic Fringe — I. Concrete Music," The Musical
Times, 97:246, 1956.

EXPERIMENT AL MUSIC 41

The efforts of this group have been widely publicized, and information
concerning their activities is readily available in both German and English.
Besides various reviews and critical articles in various German music jour-
nals,13 authoritative statements from this studio are contained in a publi-
cation called Die Reihe, distributed by Universal Edition, Vienna. Volume
1 of this journal is devoted exclusively to articles on the Cologne experi-
ments. This apparently will be the case also with volume 3, which we have
not yet seen, while volume 2 contains articles about Anton Webern, with
whom the Cologne experimenters claim close artistic affinity. Volumes 1
and 2 have also been made available in this country in English translation
by the Theodore Presser Company. Also in English, there exists a rather
extensive informal description of the activities of this studio by Allen
Forte,14 and a number of shorter articles can be cited.15

In preparing elektronische musik, in contrast to musique concrete, syn-
thetic sound sources are used. According to Forte,14 there are a number
of these in the Cologne studio, including a Bode-Melochord — an electronic
instrument with two organlike keyboards; a Monochord — not the classic
monochord, but rather a device which can produce two tones at once and
also provides for continuous variation of frequency for glissando-typQ ef-
fects; a noise generator; a buzzer; and a tone or frequency generator. This
last, in particular, seems to be the favored device because of the aesthetic
significance this group attaches to simple sine waves. Output from these
various units is recorded on magnetic tape for storage and for subsequent
processing basically similar to that employed to produce musique concrete.
In other words, the usual techniques of reverberation, variation of the fre-
quency and amplitude of the sound, montage, and so on are employed to
build up a finished piece of music.

Compositions are planned by means of a special type of score notation,

13 For example, R. Sonner, "Elektronische Musik," Z. fur Musik, 116:449, 1955;
R. Beyer, "Zur Situation der Elektronischen Musik," ibid., 452; A. L. Sieder, "Die
uberfliissige Windmachine," ibid., 456. Recent issues of Melos Magazine, published
by Der Melos Verlag, Mainz, Germany, also contain numerous articles on electronic
music and related topics.

"A. Forte, "Composing with Electrons at Cologne," High Fidelity, 6(10): 64, Oc-
tober, 1956.

15 E. Krenek, "New Development in Electronic Music," Musical America,
75(11 ):8, September, 1955; R. S. Brindle, "The Lunatic Fringe— II. Electronic Mu-
sic," The Musical Times, 97:300, 1956; R. Vlad, "Die Reihe and Electronic Music,"
The Score, 13:23, 1955; H. Searle, "Electrophonic Music," in E. Blom (ed.), Grove's
Dictionary of Music and Musicians, 5th ed., The Macmillan Company, New York,
1954, vol. IX, app. II, p. 573.

42 EXPERIMENTAL MUSIC

a sample of which is reprinted in Die Reihe.1G A compositional aesthetic
for planning the scores of these works has also been evolved. This aesthetic
is rather loosely .derived from twelve-tone-row techniques, although this
music is, of course, not limited to the usual even-tempered scale. Indeed,
a direct evolution from Webern's compositional techniques is postulated.
Serial techniques and number sequences are employed to control pitch, tone
duration, silences, and intensities of the musical elements employed in a
composition. Following an initial broadcast of seven compositions in Oc-
tober, 1954, a number of compositions by Stockhausen and other com-
posers have been presented publicly at various contemporary music festi-
vals in Europe such as at Darmstadt, Germany. Commercial recordings of
some of these pieces have been released by Deutsches Grammophon in
Europe and are now available in this country through Theodore Presser.17
Until recently, the only way to hear this music in this country had been via
privately owned tapes brought back from Germany by various people in-
terested in electronic music.

In contrast to musique concrete, much of elektronische musik sounds
simple and almost primitive. Simple sine-wave tones, relative to which the
most familiar instrumental equivalent is the recorder, lack harmonics and
are, of course, neutral and unexciting in effect. Thus, even though this mu-
sic is rather elaborately planned out, the over-all effect in the average
elektronische musik composition, in which sine tones make up a relatively
large proportion of the total sound, is curiously one of relative simplicity.
However, more recently, experiments have also been carried out to com-
bine the sounds of elektronische musik with processed sounds of recorded
human voices to synthesize a cantata-like product. At least two such pieces
have been produced, a cantata by Ernst Krenek and the Song of the Holy
Children by Stockhausen. Here, as might be expected, the superposition
of human voices upon the purely synthetic materials produces a more dra-
matic effect.

Other European Experimental Music. In Milan, Italy, a laboratory has
also been set up by a broadcasting organization, the RAI Studio di Fonolo-
gia Musicale, under the direction of Luciano Berio. In this studio, elements
of both musique concrete and elektronische musik are being utilized to pro-
duce synthetic music utilizing sounds from both natural sources and from
electronic sound-producing devices. A brief general description of the

16 Die Reihe, vol. 1, facing p. 50 in English edition.

17 H. Eimert, Etude uber Tongemische; Funf Stiicke; Glockenspiel, DG-16132;
K. Stockhausen, Studie I; Studie II; Gesang der Junglinge, DG-16133.

EXPERIMENT AL MUSIC 43

aims of this organization has been published by Berio.18 To date, however,
the activities of this group have not been publicized to an extent com-
parable to that of the older organizations.

A number of other laboratories are also springing up in Germany, in the
Netherlands, and elsewhere in Europe, which more or less duplicate the
facilities of those already described without thus far disclosing significantly
different or startling advances in technique. A typical example is the ex-
perimental studio at Philips Research Laboratories at Eindhoven, The
Netherlands. 1!) These facilities have been employed by the composer Henk
Badings to produce a twenty-minute score for a ballet entitled Cain and
Abel

American Experiments in Tape-recorder Music. A number of American
composers have also examined the possibility of producing musical struc-
tures by electronic means and by processing tape-recorded sound materials.
A review of earlier developments in this area in the United States has been
written by Vladimir Ussachevsky,-0 who, in collaboration with Otto Luen-
ing, is perhaps the best-known experimentalist in this field in the United
States. Using fairly simple equipment, these composers have thus far
achieved types of sound synthesis similar to the European experiments.
They have also been able to make some of their results available through
recently released recordings.21 In addition to demonstrating the effective-
ness of this medium for purely abstract compositions, the value of tape-
recorder music for theatrical purposes is disclosed in the Suite from King
Lear, a set of examples of music used for Orson Welles's production of
King Lear in New York in 1956.

Luening and Ussachevsky have been also concerned with the possibility
of combining musical materials on tape with live performance; in other
words, of writing compositions for tape recorder and other instruments.
Luening's Theater Piece No. 2, which received its first performance in
New York on April 20, 1956, combines tape recorder, voices, and chamber
orchestra, while Luening and Ussachevsky, as a result of a commission,

18 L. Berio, "Studio di Fonologia Musicale," The Score, 15:83, 1955.

10 H. Badings and J. W. de Bruyn, "Electronic Music," Philips Tech. Rev. 19:190,
1957-1958.

20 V. Ussachevsky, "La 'Tape Music' aux Etats-Unis," La Revue Musicale, numero
speciale 236, p. 50, 1957.

21 0. Luening and V. Ussachevsky, Tape-recorder Music, Innovations GB-1; O.
Luening and V. Ussachevsky, Rhapsodic Variations for Tape Recorder and Orchestra,
Columbia Records, Louisville series; O. Luening and V. Ussachevsky, A Poem of
Cycles and Bells, for Tape Recorder and Orchestra; Suite from King Lear; V. Ussa-
chevsky, Piece for Tape Recorder, Composers Recordings, Inc., CRI-112.

44 EXPERIMENT AL MUSIC

have written pieces for the Louisville and Los Angeles Philharmonic Or-
chestras.

Ussachevsky-0 reviews other American experiments up to 1953 as well
as his own efforts. This includes early experiments of John Cage and his
group involving the use of multichannel tape music to produce stereo-
phonic effects.

There have been other experiments in this country which should also be
mentioned. Notably, Edgar Varese has written a piece entitled Deserts
for tape recorder and chamber orchestra, in which sound effects in nature
are contrasted to orchestral sonorities12 and more recently, Poeme Elec-
tronique for Le Corbusier's Pavilion at the 1958 Brussels Exposition. Louis
and Bebe Barron have prepared the sound track for an MGM science-fiction
movie called Forbidden Planet through producing by electronic means upon
magnetic tape what they have called electronic tonalities.22

Lastly, a rather different type of tape music might be mentioned, which
is reminiscent of experiments carried out in the 1940s at Princeton Uni-
versity23 and more recently for the National Film Board of Canada by
McLaren24 and by Kendall,25 of marking film sound tracks directly with a
stylus to produce sound patterns. This is the following:26

A. H. Frisch, of New York, is now working on a mechanical means of re-
cording pitch and dynamics directly on tape. He is laying the basis for musical
composition without recourse to any instruments. The composer will work with
unmagnetized tape while sitting at his desk, and by applying to the tape specific
magnetic dyes will transfer pitch or a theme from his creative innards to the
tape. ... He will also have a means of applying amplitude or dynamics or
even vibrato. He will establish rhythm or timing by a preselected formula
shown as specific intervals on a sort of yardstick over which the tape is manip-
ulated, inch by inch, foot by foot.

The RCA Electronic Music Synthesizer. One of the most ambitious proj-
ects for the direct production of musical sound has been the construction of
the RCA Electronic Music Synthesizer by H. Olson and H. Belar27 at the

22 L. and B. Barron, "Forbidden Planet," Film Music, 15(5): 18, Summer, 1956.

23 M. Babbitt, private communication.

24 R. E. Lewis and N. McLaren, "Synthetic Sound on Film," J. Soc. Motion Picture
Engrs., 50:233, 1948.

25 A. Phillips, "Osmond Kendall's Marvellous Music Machine," MacLean's Maga-
zine, June 11, 1955, p. 22.

26 S. J. White, Letter to the Editor, High Fidelity, 7(1): 15, January, 1957.

27 H. F. Olson and H. Belar, "Electronic Music Synthesizer," J. Acoust. Soc. Am.,
27:595, 1955. .

EXPERIMENTAL MUSIC 45

RCA acoustics laboratories in Princeton, New Jersey. The Music Syn-
thesizer is an elaborate electronic device controlled by means of coded
paper input tape. The instrument builds up sounds by wave superposition
and wave-form control to simulate known instrumental or even primitive
vocal sounds, since artificial sounds can be built up by standard well-known
acoustical techniques such as, for example, Fourier synthesis of original
sound prototypes. Alternatively, fundamental tones produced by the Syn-
thesizer can be combined in novel ways, not reminiscent of conventional in-
struments. The output from the Synthesizer is recorded directly on discs
rather than on tape. A recording of selected output from this instrument
has been released by RCA.28 On the first side of this record, a step-by-step
demonstration of how sounds are built up in the Synthesizer is used to
show how a number of musical examples presented on the second side were
created. Unfortunately, many of the musical examples presented on this
second side of the record are so commonplace and of such minor artistic
significance that the potential of this instrument for more artistic purposes
has not yet been generally realized. It might also be noted in passing that
an elementary discussion of some of the principles of operation of the
Synthesizer is printed on the record jacket of this album.

This Synthesizer has been displaced by a second, improved Synthesizer
which uses electronic oscillators rather than tuning forks as primary sound
sources. It is reported that the effects produced by this newer instrument
are decidedly superior to those produced by the earlier model. If this be
so, there seems but little question that this instrument represents an im-
portant technological step beyond facilities developed in the European
studios and that it could be used not only to duplicate all the techniques of
these groups, but also to extend considerably beyond their present capacities
the study of experimental sound combinations. RCA apparently has no
direct interest in this, but they have arranged to let certain composers,
specifically, Vladimir Ussachevsky from Columbia University and Milton
Babbitt from Princeton University, work with the instrument to evaluate its
possibilities for producing experimental music.29 In this way, it may come
about that output from the Synthesizer aesthetically competitive with other
experimentation in this general field will be produced. (See also Chap-
ter?.)

28 The Sounds and Music of the RCA Electronic Music Synthesizer, RCA Victor
LM-1922.

2H M. Babbitt, private communication.

46 EXPERIMENTAL MUSIC

Programmed Music

The generation of music by means of automatic high-speed digital com-
puters is a newer technique than the foregoing types of experimentation
and, in fact, to the best of our knowledge, the inception of our experiments
with the Illiac marks the first serious study in this field. However, there
are historical precedents of a sort which should be considered in order that
the experiments be cast into correct perspective. In reviewing these prece-
dents, we emphasize, however, that our own work in no way represents the
latest development in a long tradition. There really are no direct prece-
dents, and the material we will now discuss in a way illustrates how little
historical background we can cite.

Conventional Program Music. The central body of theory and practice
of composition has largely been directly involved with structural principles
presumably existing essentially independently of referential nonmusical
"meaning." These are the principles of composition of so-called absolute
music, music which loosely can be said to exist primarily for its own sake
independent of literary or other programmatic connotations. The body of
principles so codified is reflected in our practice of writing melodic struc-
tures, counterpoint, harmony, instrumentation, and in our use of musical
forms (sonata form, variation form, etc.). However, in addition, in the his-
tory of Western music, considerable interest has been shown by composers
in structural principles superficially more nearly extramusical. One of the
most obvious of these, and one which is familiar and generally accepted by
most listeners, is the "imitation of nature," a musical tradition with a re-
spectable history, if not always given equal recognition to the so-called
higher forms of music. Techniques involving the inclusion of actual extra-
musical sounds such as cowbells to suggest pastoral scenes by Mahler, the
simulation of natural sounds with special devices such as the "wind machine"
used by Richard Strauss in his Alpine Symphony, the imitation of natural
sounds by musical instruments, for example, the simulation of bird calls and
storm sounds in Beethoven's Pastoral Symphony, the use of horn calls to sug-
gest the hunt in many examples of music, as in Berlioz's Les Troyens, for
example, all fall into this category. The imitation of visual effects and the
use of musical symbols for people and even abstract concepts has also fre-
quently been attempted in program music. Many well-known examples of
this, of course, also could be cited. In general, however, it would appear

EXPERIMENTAL MUSIC 47

that the more the thing symbolized departs from a purely musical symbol,
the more necessary it is that a literal explanation be made available to
provide that the intended meaning be properly conveyed to the listener and
the more the device employed is regarded with suspicion by musical "purists."

Program music, which is, therefore, music in which referential sound
symbols are directly employed or in which a story, a plot of one sort or an-
other, is used to connect the various parts of the composition together, re-
flects one type of utilization of extramusical concepts by composers. How-
ever, having recognized this general domain of compositional activity, let
us now consider certain special and unusual types of programmatic music
writing.

Eye Music. One persistent form of experimental program music and
one which seems at first to be of no musical significance is the technique
of writing music to conform to a visual pattern on the written or printed
page of musical score. Examples of music of this type from as early as the
fifteenth century have survived. Thus, in the period between Machaut and
Dufay, particularly at the papal court at Avignon:

. . . musicians . . . began to indulge in complicated rhythmic tricks and in
the invention of highly involved methods of notating them. . . . Here we find
compositions written in the form of a circle or a heart ... an indication of
the strong hold upon the imagination of the composer that the purely manual
business of writing exercised in those days.30

In the setting of madrigals, sixteenth-century Italian composers indulged
in word painting, not to mention even more overt forms of eye music. Word
painting denotes the use of minute melismatic melodic word images to illus-
trate musically individual words in a madrigal. Thus, on the word mare, a
rolling melody suggesting waves might be employed, while for a descent to
the underworld, a descending melody line might be used. Eye music, spe-
cifically, was an even more extreme departure from conservative practice.
This was the use of black and white notes to suggest darkness and light,
sadness and joy. Since black and white notes in notation were used then as
now to indicate rhythmic differences, it can be seen that the movement or
speed of a melody would slow down as the composer changed from black
to white notation. The relationship of music as heard to the content of the
poem would seem rather tenuous by our conventions. Luca Marenzio

30 W. Apel, The Notation of Polyphonic Music, 900-1600, The Medieval Academy
of America, Cambridge, Mass., 1942, as quoted in G. Reese, Music in the Renais-
sance, W. W. Norton & Company, Inc., New York, 1954, p. 11.

48 EXPERIMENT AL MUSIC

( 1533 1599) is perhaps the best-known madrigal composer to indulge in
word painting and eye music.31

This, as might be expected, was a practice roundly condemned by con-
servative theorists. Zarlino, in particular, denounced the practice in terms
of his theory of sensible particulars.8- However, in spite of the theorists,
the writing of eye music has persisted ever since. It was a common practice
in the Baroque period, and even Bach, for example, was not adverse to
using such devices as ascending melodies to suggest an ascent to Heaven.
In more recent times, Charles Ives has experimented with visual program
music along with his many other innovations. Thus, in the list of his com-
positions, as tabulated in the Cowells's recent biography of the composer,33
we find the following:

Yale-Princeton Game, (August, 1907). The wedge formation piece: notes set
on paper like men on the football field — one note runs around left-end for a
loss, etc. Includes kazoos. Incomplete; a sketch for a "take-off."

Giants vs. Cubs, (August, 1907). Another sketch for a "take-off," this time
a baseball game.

To cite another recent example of eye music, Villa-Lobos has written two
pieces, Melodia da Montanha {Serra da Piedade de Bello Horizonte) and
New York Sky-Line for piano,34 composed in the following way:35

These are two pieces written according to his [Villa-Lobos'] method of "milli-
metration" of curves and charts. The outline of a drawing, a photograph, or a
chart, is transferred to graph paper, so that the unit of the ordinate corresponds
to a chromatic degree, and the unit of the abscissa to a metrical note value.
The result is then set down on music paper as a melody. By this method,
Villa-Lobos drew the melody of the mountains Serra da Piedade at Bello Hori-
zonte. Villa-Lobos has similarly "millimetred" the melody of the sky-line of
New York City, and harmonized it. The whole job was done in one hour and

31 See, for example, A. Einstein, The Italian Madrigal, Princeton University Press,
Princeton, N.J., vol. I, pp. 234-244.

32 G. Zarlino, lstituzioni armoniche, section 71, as contained in O. Strunk, Source
Readings in Music History, W. W. Norton & Company, Inc., New York, 1950, pp.
248-251.

33 H. Cowell and S. Cowell, Charles Ives and His Music, Oxford University Press,
New York, 1955, p. 219.

34 H. Villa-Lobos, Melodia da Montanha and New York Sky-Line, New Music
Edition, 16(1): 15-17, 1942.

35 N. Slominsky, "Modern Composers of Brazil," New Music Edition, loc. cit., p. 4.

EXPERIMENTAL MUSIC 49

fifty minutes, between 5 and 6:50 o'clock in the afternoon of February 22,
1940.

One other form of programmed music we might mention in passing is
more or less related to the various forms of eye music just considered. This
is the use of musical anagrams such as the notes B, A, C, H (H being
the German equivalent of B flat) to symbolize "messages" such as Bach's
name in a four-note musical theme. Among well-known composers, Schu-
mann, for example, was particularly addicted to inserting alphabetical ref-
erences in his music. The Abegg Variations, the Carnaval, and numerous
others of his compositions contain such programmatic devices.86

Computational Music. If numerical sequences and patterns are used to
establish a musical structure rather than the geometrical appeal of a visual
design, we can distinguish yet another variety of programmed music. Out-
standing examples survive from as early as the fourteenth and fifteenth
centuries again, particularly in the isorhythmic motets of composers such
as Machaut and Dunstable, and in the design of mensuration and prolation
canons in which each voice is in a different rhythm. A well-known example
of this latter technique is the Prolation Mass of Ockeghem (ca. 1420-ca.
1495).37 In modern times, the use of the concept of a tone row made up
of all twelve tones of the chromatic scale, as first exploited significantly by
Schonberg, is a somewhat similar process, although, with more and more
usage, this device is being accepted as a musically "pure" procedure, even
though no particularly convincing acoustical justification for its usage seems
yet to have been formulated. However, because of its rather intriguing
properties of symmetry and its susceptibility to permutation processes such
as inversion, retrogression, and so on, the tone row as a musical concept has
stimulated experiments in musical form which offer a high degree of arith-
metical program content. The works of Schonberg largely contain permu-
tation techniques existing more or less within a context of conventional
musical operations and existing in particular reference to pitch successions.
This is also basically though somewhat less true of the works of Webern
and of contemporary composers such as Milton Babbitt. The latter has
investigated permutational techniques derived from relationships inherent
in certain highly ordered types of tone-row sequences and has tried to de-
velop compositional techniques which are independent of conventional con-

3<i R. H. StaufTer, Florestan, the Life and Work of Robert Schumann, Henry Holt
and Company, Inc., New York, 1945, pp. 290-291.
37 See, for example, Reese, op. cit., pp. 133-136.

50 EXPERIMENTAL MUSIC

siderations of tonality.3*- 89 On the other hand, there has also recently
sprung up a different type of serial writing in which not only the tones,
but also rhythms, dynamics, and other fundamental musical elements are
also controlled, and controlled, it would seem, quite arbitrarily, by means
of a "row." Since this "row" now is used to control other elements besides
pitch, it exists only as a sequence of numbers which arbitrarily denotes
many elements of a composition in addition to pitch. This new type of
writing, which has been termed totally organized music, is a combination of
rhythmic ordering, as apparently first developed by Messiaen,40- 41 with the
twelve-tone concepts of the Schonberg school. In addition to Stockhausen,
who has utilized such concepts for the production of elektronische musik,42
Boulez, in particular, has worked out composing schemes of this type.42-44
Related techniques are exploited by several contemporary Italian com-
posers, such as Nono,44 Donatoni, Togni, and Maderna.45 For our pur-
poses, one point in relation to the basic idea of totally organized music it-
self is worth noting, namely, that the very nature of this concept puts music
of this class into an extreme and limited category in terms of modern com-
munication theory. This happens quite independently of its aural effect or
of its aesthetic "value." As we pointed out in Chapter 2, both modern
aesthetic theory and modern communication theory are based upon an
awareness also of the role of lack of organization, of ambiguity or random-
ness in the structure of messages, including musical ones. This does not
exclude the possibility of tonal organization any more than it does the pos-
sibility of the total lack of organization, which we consider below. It simply
fits it in as one element within the framework of a larger picture. It should

38 M. Babbitt, "Some Aspects of Twelve-tone Composition," The Score, 12:53,
1955; M. Babbitt, The Function of Set Structure in the Twelve-tone System, unpub-
lished mimeographed manuscript.

39 G. Perle, "The Harmonic Problem in Twelve-tone Music," The Music Review,
15:257, 1954.

40 O. Messiaen, Technique de mon Langage Musicale, A. Leduc, Paris, 1948.

41 D. Drew, "Messiaen— A Provisional Study, I." The Score, 11:33, 1954; "II,"
ibid., 13:59, 1955; "III," ibid., 14:41, 1955.

42 E. Krenek, "New Development in Electronic Music," Musical America, 75(1 1):8,
September, 1955. See also articles in vol. 1 of Die Reihe.

43 M. Feldman, P. Boulez, J. Cage, and C. Wolff, Four Musicians at Work," trans-
formation, 1:168, 1952.

44 L. Pestalozza, "Post-Weberniani, concerti ed elettronici," Ricordiana, 2:333,
1956; H. Cowell, "Current Chronicle, New York," The Musical Quarterly, 38:123,
1952.

45 R. S. Brindle, "The Lunatic Fringe — III. Computational Composition," The Mu-
sical Times, 97:354, 1956.

.

EXPERIMENTAL MUSIC 51

also be noted that the term totally organized is also rather misleading, since
a totally ordered piece of music would be limited to one pitch, one rhythmic
event, one indication of dynamics, and so on.

Mathematical Aesthetics. One other development somewhat related to
the above deserves mention. This is the use of mathematical formulations
which serve as the underlying groundwork for assembling supposedly aes-
thetically significant art structures. There have been several fairly substan-
tial attempts in recent years to develop theories of aesthetic value which
can be expressed quantitatively by means of algebraic formulas, which in
turn can be utilized in arithmetical operations to produce specific "art
works." None of these have been employed in our experiments, but it
seems desirable to mention at least the more prominent of these theories,
since aspects of each might possibly be applied in future experimentation.

The first of these is Birkhoff's theory of aesthetic measure,40 which he ap-
plies to artistic objects in general, although music is singled out for par-
ticular consideration. Birkhoff defines a general function called aesthetic
measure, M, which is the feeling of value which rewards the effort to per-
ceive the complexity, C, of an aesthetic object in relation to the realization
of its degree of order, O. Birkhoff suggests that

M = O/C (4)

and starting from this premise, he attempts to evaluate the aesthetic meas-
ure of forms in geometrical objects, vases, paintings, and, in particular,
conventional music. Birkhoff devises a number of algebraic formulations
which are in excellent accord with conventional harmonic and melodic
practice and are suggestive of possible coding techniques for handling this
type of music writing. It should be noted, however, that his tendency to as-
sociate value with "pleasurable experience" is rather at variance with most
contemporary theories of musical aesthetics, such as those we have re-
viewed in Chapter 2. However, this limitation is not serious, since it affects
only the specific forms of the relationships Birkhoff has developed and not
the basic method of attack.

A second theory of this general type is the well-known Schillinger system
of musical composition,47 based upon Joseph Schillinger's theory of mathe-

46 G. D. Birkhoff, Aesthetic Measure, Harvard University Press, Cambridge, Mass.,
1933.

*7 J. Schillinger, The Schillinger System of Musical Composition, Carl Fischer, Inc.,
New York, 1946.

52 EXPERIMENTAL MUSIC

matical aesthetics.48 This is a much more ambitious attempt than Birkhoff's
to find formulations to characterize aesthetic structures, and, in addition,
a reduction to practice, particularly in the field of musical composition, has
been worked out. Schillinger goes to some pains to suggest an analogy be-
tween his techniques and engineering practice as opposed to supposedly
less "scientific" standard musical practices. Since critical analyses of Schil-
linger's ideas have been published elsewhere,49 it is perhaps important for
us at this time only to note that, to the extent that Schillinger's methods
represent a coding of musical techniques into forms convenient for further
coding for computer processing, this composing procedure might also be of
value in future experimentation with computers.

As one further example, we note the following comments about work by
A. H. Frisch, previously referred to,26 who, by "solving expressions for the
squares and cubes of binomials and trinomials, has produced a means of
timing for virtually an unlimited number of rhythmic series. . . . Frisch has
gone beyond the theoretical work of Schillinger in that he has created the
means for mechanically and simply establishing rhythm patterns."

Musical Games and Music by Chance. Hindemith50 has observed that
an English composer named William Hayes in 1751, in a satiric book en-
titled The Art of Composing Music by a Method Entirely New, Suited to
the Meanest Capacity, suggested the following compositional technique:

His advice is, to take a brush with stiff bristles (like a toothbrush), dip it into
an inkwell, and, by scraping the bristles with the finger, spatter with one sweep
a whole composition onto the staff paper. You have only to add stems, bar
lines, slurs, et cetera, to make the opus ready for immediate performance.
Whole and half notes are entirely absent, but who cares for sustained tones
anyway!

This technique, obviously dealt with by Hindemith in uncomplimentary
terms, might also be considered an extreme type of eye music. However,
if it could be carried out to avoid the building up of any visual design ex-
cept purely by accident, then pure chance would determine where the notes
would fall, in contrast to the other types of eye music previously described.
In this music, there would be a lack of any organizing principle, visual or

48 J. Schillinger, The Mathematical Basis of the Arts, Philosophical Library, Inc.,
New York, 1948.

i{' N. Slominsky, Musical Quarterly, 32:465, 1946; S. and H. Cowell, Modern
Music, 23:226, 1946; E. Carter, ibid., 23:228, 1946.

50 P. Hindemith, A Composer's World, Harvard University Press, Cambridge, Mass.,
1952, p. 123.

EXPERIMENTAL MUSIC 53

musical, and this music, if correctly prepared, would be a primitive example
of what we shall call random music, that is, music in which there is a com-
plete absence of any organizing principle except the laws of probability. 51
In this random music, moreover, we could predict that all notes would be
equally probable. Hale's suggestion, put forward in jest, actually does have
some value, since the concept of random music is very useful. As we have
already indicated in Chapter 1 , the utilization of random-number processes
is the basis of our technique for generating computer music. Moreover,
the concept also has a definite place in the study of musical aesthetics, as
we have pointed out in Chapter 2. It, of course, represents the diametri-
cally opposed musical concept to that of totally organized music just con-
sidered.

Obviously the splattering of a manuscript page with ink is an inefficient
and untrustworthy method for producing random music, and in recent
years other techniques for producing such music have been attempted,
notably by John Cage and his followers.43 Cage has employed tables of
random numbers to select musical elements such as notes. Unfortunately,
Cage's experiments are burdened with theatrical effects which tend to
mask the intrinsic interest of his innovations. Thus, "Cage has often used
the I-Ching, an old Chinese method of throwing coins or marked sticks
for chance numbers, like our use of dice ... to ensure that his composi-
tions are 'free of individual taste and memory in their order of events.' He
uses a complicated system of charts derived from coin tosses."52 This
was done in the case of the piece being reviewed, namely, Imaginary
Landscape for twelve radios "played" by twenty-four players plus a con-
ductor. Obviously, the use of the I-Ching today is an absurdly inefficient
and pointless way to produce random numbers or, even more so, random-
number sequences upon which subsequent restrictions are placed.

Cage has also employed the technique of taking a piece of music for
piano written on a number of separate pages and instructing the performer
to drop the pages before performance, so that the sequence of the pages
and the orientation of each page in the particular performance cannot be
predicted. A number of possible "arrangements" of the composition thus
become available. Moreover, there are other examples of music by chance
written by Cage and composers associated with Cage, such as Morton

51 Strictly speaking, even this music would be organized to the extent that only cer-
tain frequencies corresponding to the even-tempered scale would be admitted.

52 H. Cowell, "Current Chronicle," Musical Quarterly, 38:123, 1952.

54 EXPERIMENTAL MUSIC

Feldman, Earl Brown, and Christian Wolff. Thus, Cage himself notes
that:53

Earl Brown devised a composing procedure in which events following tables
of random numbers are written out of sequence, possibly anywhere in a total
time row and possibly anywhere else in the same total time text. I myself use
chance operations derived from the I-Ching, others from the observation of
imperfections in the paper upon which I happen to be writing.

The overlap of this last technique with eye music is obvious.

Cage's experiments with dice are by no means a total novelty for there
is a limited though definite history of musical games involving chance
operations. Of perhaps greatest interest are the musical dice games which
were quite a popular novelty in the eighteenth century.52, 54 Many com-
posers, including J. C. Bach and Mozart, became intrigued with this game.
Thus, Mozart is the author of a rather well-known artifact called the
Musikalisches Wurfelspiel. This bore the description55 "Mozart's musical
game, fitted in an elegant box, showing by an easy system to compose an
unlimited number of waltzes, rondos, hornpipes and reels — mechanical
composition of minuets." This "composition" is at present available in a
modern edition.56

Some Generalizations. Before we consider experiments involving the
generation of music by means of computers, including directly below a
review of experiments other than our own, it is perhaps worthwhile to tie
together some of the ideas we reviewed in this and the previous chapter.

1. Since modern aesthetic theory and modern communication theory
are evolving similar sets of operational concepts, the restatement in musical
terms of certain ideas of information theory should yield useful techniques
for building up a unified set of critical criteria for evaluating musical ex-
periments. For example, the experiments to produce random music and
totally organized music we have just reviewed are really both specific ap-
plications of a more generalized procedure involving the control of order-
disorder relationships in musical structure.

53 J. Cage, "Experimental Music," The Score, 12:65, 1955.

54 P. Lowenstein, "Mozart-Kuriosa," Z. fiir Musikwissenshaft, 12:342, 1930; O. E.
Deutsch, "Mit Wurfeln Komponieren," ibid., 12:595, 1930; H. Gehrigk, "Wurfelmu-
sik," ibid., 16:359, 1934.

55 A. Einstein (ed.), Kochel's Mozart — Verzeichnis, 3d ed., J. W. Edwards, Pub-
lisher, Inc., Ann Arbor, Mich., 1947, p. 909.

56 W. A. Mozart, Musikalisches Wurfelspiel, or The Dice Composer, K. anh.294d,
(ed. by A. Laszlo), Guild Publications of California, Inc., Hollywood, Calif.

EXPERIMENT AL MUSIC 55

2. Experiments in composing techniques are seen thus far to be distinct
in the most part from experiments in producing musical sounds. Some
efforts to incorporate logical processes into the production of synthetic
music have been noted, but the obviously large area of overlap of tech-
niques has yet to be studied in any detail. This is one area for future
research which should prove particularly of interest to experimental com-
posers (see Chapter 7).

3. The process itself for writing music had undergone relatively little
change for hundreds of years. Although a number of novel ways for
organizing musical information have been devised, aside possibly from
certain applications of mathematical formulas, no really significant depar-
tures from the traditional techniques of music writing have been developed.
This seemed, therefore, the area of research in which coding for computers
might really have a profound effect and was the area, therefore, toward
which we first directed our attention.

Other Experiments to Generate Computer Music

There seem to be only six other instances thus far, besides our own
experiments, of the production of music by means of high-speed electronic
digital computers, and in the first case, no actual experimental output has
apparently yet been reported. In his recent article discussing certain
aspects of information theory and music which we referred to in Chapter
2, R. C. Pinkerton57 suggested the use of computers for writing simple
melodies utilizing an elementary probability table for selecting successive
notes of a melody. However, as already indicated, no record of whether
this project was actually carried out has been published. Two other proj-
ects have both similarly been concerned with the generation of popular
or hymn tunes, again by means of probability tables for successive note
selection. The first of these experiments was done on the Datatron com-
puter at the ElectroData Division of the Burroughs Corporation in Pasa-
dena, California, by M. Klein and D. Bolitho. This work was publicized
in the press,58 and a reasonably complete semitechnical description of the
methods developed by Klein and Bolitho is contained in a company-pub-

57 R. C. Pinkerton, "Information Theory and Melody," Sci. American, 194(2) :77,
February, 1956.

58 « 'Bra{n' Computes New Tune for TV," special news release to The New York
Times, July 3, 1956, p. 16; also Associated Press news release, July 3, 1956.

56 EXPERIMENTAL MUSIC

lished pamphlet.50 The preparation of a tune called Push-Button Bertha
for broadcast and recording was carried out. The basic technique utilized
was a random-number process combined with successive selection or re-
jection of these numbers in accord with musical rules translated into arith-
metical operations. In principle, therefore, the technique developed by
Klein and Bolitho was similar to that utilized by the present authors. It is
our understanding, however, that this work has been abandoned. m

The second project of a somewhat related nature, since it was also de-
signed to deal with simple tunes, was carried out at Harvard University
by Brooks, Hopkins, Neumann, and Wright.61 The primary purpose of this
project was not to generate simple tunes by means of computers but to
analyze simple melodies in the light of certain principles of information
theory which we shall consider in detail in Chapter 5. It is important to
note, however, at this point that a computer was coded by these investi-
gators to produce melodies based upon the laws established by the analyt-
ical scheme employed.

Fourthly, according to a recent article in Computing News,62 J. W.
Granholm, editor of that magazine, and M. C. Mitchell, oboist at the
Seattle Symphony Orchestra, have become interested in possible musical
applications of computers. They have carried out preliminary studies of a
project to utilize computers for arranging and orchestration. Employing
a numbering technique similar to ours for designating the notes of the
chromatic scale, they have worked out the problem of transposition for
preparing performance parts for transposing instruments. They have also
been particularly concerned with the question of musical printout from
computers, a problem we shall also consider in Chapter 4.

Fifthly, it has been brought to our attention recently that members of
the Shell Laboratories in the Netherlands have used a computer to program
Mozart's dice-game composition discussed earlier. However, published in-
formation on these experiments was not available at the time of writing.
It is our understanding that the coding of this problem was carried out

50 "Syncopation by Automation," Data from ElectroData, August, 1956, Electro-
Data Division of Burroughs Corporation, Pasadena, Calif.

"° Private conversation between one of the present authors (L. A. H.) and Dr.
Bolitho in Los Angeles, Aug. 28, 1956.

ni F. P. Brooks, Jr., A. L. Hopkins, Jr., P. G. Neumann, and W. V. Wright, "An
Experiment in Musical Composition," IRE Trans, on Electronic Computers, EC-6: 175,
1957.

63 Anon., "Music Transcription by Computer," Computing News, 5(17) : 108-1 13,
Sept. 1, 1957.

EXPERIMENTAL MUSIC 57

by D. A. Caplin, of the Shell Petroleum Company, Ltd., in London.fiH
Lastly, a short piece of computer music has been produced by N. Gutt-
man of the Visual and Acoustics Research Department of Bell Telephone
Laboratories.64 This item is of considerable interest in that sounds were
synthetically produced on an IBM 704 computer. The composition itself
is a short piece in just intonation while the sounds produced by the com-
puter consist of triangular wave forms.

63 Letter to L. A. H. from Dr. R. J. Lundbeck, Shell Laboratories, the Hague,
Netherlands.

01 Letter to L. A. H. from Dr. J. R. Pierce, Bell Telephone Laboratories, Murray
Hill, N.J.

CHAPTER FOUR

The Technical Problem

Non-numerical Uses of Computers

Today, we do not think it unusual that machines can perform arithmetic
operations, even rather complex ones; indeed, the current viewpoint is
that there is something intrinsically mechanical about such arithmetic op-
erations. On the other hand, it is still considered somewhat unusual when-
ever the same machines are used to assist other types of thought processes.
Non-numerical applications of high-speed computers have been relatively
slightly exploited to date, but there now seems to be a rapidly growing
interest in the field. For example, at the 11th National Meeting of the
Association for Computing Machinery in August, 1956, particular atten-
tion was directed toward the new subject of non -numerical uses of high-
speed digital computers and served to focus attention on the subject. A
review of applications of this type was presented through papers on the
following topics: weather prediction,1 logistics applications,2 programming

1 P. D. Thompson, "High-Speed Computing in Weather Prediction," Program and
Abstracts for the 11th National Meeting of the Association for Computing Machinery
at the University of California at Los Angeles, Aug. 27-29, 1956, p. 6.

2W. H. Marlow, "Some Logistics Applications," ibid., p. 6, p. 12.
58

THE TECHNICAL PROBLEM 59

computers to play games such as chess and checkers,3 language transla-
tion4 and language compiling.5 At the same meeting, as noted already in
Chapter 1, a brief report of the present work was also given. In more re-
cent meetings, a trend toward the presentation of such topics has been
continued.6

It is convenient, perhaps, to classify present uses of computers into three
broad categories. The first is the obvious category of solving complex
mathematical problems in the sciences and engineering, in mathematics,
and in other fields in which specific quantitative information is required.
These direct mathematical applications are self-evident uses for computers
and require no particular comment.

Somewhat less obvious in terms of the basic information supplied the
computers is the second category of applications, namely, to business and
industrial procedures. Here the applications include, for example, the
processing of office data, the simulation and design of industrial-plant op-
erations, and the control of industrial operations through digital-computer
control of specialized analog computers and servo-mechanisms. Indeed,
these applications are now beginning to attract widespread attention and
account for a large proportion of the sales of commercial high-speed auto-
matic digital computers. From the rapidly growing literature on this sub-
ject, we can cite representative surveys.7 A condensed source of current
information on these topics is contained in the abstracts in the IRE Trans-
actions on Electronic Computers. An introductory review of this subject
recently published in Fortune might also be cited.8 Goodman's book,7

3 A. L. Samuel, "Programming a Computer to Play Games," ibid., p. 8, p. 21; see
also J. Kister, P. Stein, S. Ulan, W. Walden, and M. Wells, "Experiments in Chess,"
/. Assoc. Computing Machinery, 4:174, 1957.

* I. Wieselman, "The Potentialities of a High-capacity Store for Machine Transla-
tion," ibid., p. 8, p. 22; also typewritten copy of presentation through courtesy of
author.

5 J. Chipps, M. Koschmann, A. Pedis, S. Orgel, and J. Smith, "A Mathematical
Language Compiler," ibid., p. 9, p. 27.

6 For example, in the program of the 1957 Eastern Joint Computer Conference,
Washington, D.C., Dec. 9-13, 1957, the following titles are noted: I. Rotkin, "Mech-
anization of Letter-mail Sorting"; D. L. Gerlough, "Control of Automobile Traffic
as a Problem in Real-time Computation"; W. H. Thomas, "The Application of Elec-
tronic Computers to Air Traffic Control."

7L. L. Goodman, Man and Automation, Penguin Books, Inc., Baltimore, 1957;
Scientific American, Automatic Control, Simon and Schuster, Inc., New York, 1955;
E. M. Grabbe (ed.), Automation in Business and Industry, John Wiley & Sons, Inc.,
New York, 1955; J. Diebold, Automation, D. van Nostrand Company, Inc., Princeton,
N.J., 1952.

8W. B. Harris, "The Astonishing Computers," Fortune, 55(6): 136, June, 1957.

60 EXPERIMENTAL MUSIC

which is a useful treatment of the subjects of automation and computers
written in nontechnical language, includes detailed descriptions of actual
examples of computers now in operation in the business world. Goodman
points out that this type of application is basically of a non-numerical na-
ture when he notes that:9

In business use, it is the ability of the computer to organize and sort data
which is more important than its ability to do arithmetic; though . . . the
ability to solve complex mathematical problems is necessary when decision-
making of a managerial nature is undertaken. The office is now quickly be-
coming a "factory" for the processing of data and it is in this light that elec-
tronic data processing should be viewed.

The final category includes other types of non-numerical applications
and covers, of course, a wide range of possible fields of interest.10 In ad-
dition to some of the more important applications already mentioned, it is
interesting to note that even in the sciences auxiliary applications of a
non-numerical type are being developed. In the field of medicine, for
example, it is predicted that computers will be used in conjunction with
medical diagnosis, since these instruments will not only be able to store
known medical information and patients' case histories, but they will also
be able to sift this information to assist the actual process of diagnosis.11
In the field of chemistry, one application is the storing of formulas for
chemical compounds;12 another is their use to speed up patent searches in

9 Goodman, op. cit., p. 56.

10 The following item, which we noted in the Sept. 21, 1957, issue of The New
Yorker, gives one indication of the extent of these applications: "Systematic permu-
tations of the Tibetan alphabet with an automatic sequence computer aided lamas
high in the Himalaya mountains in discovering all the possible names for God, ap-
proximately nine billion of them. Why they undertook this problem is a matter for
metaphysical speculation. At work on the project for three centuries, the lamas fig-
ured it would require another 15 thousand years to complete the project by hand.
However, with a Mark V computer the task was accomplished in 100 days.

"Technicians who accompanied the computer to Tibet report the job was success-
fully completed. Miles of paper covered with what appeared to be gibberish were
turned out by the computer and patiently pasted in huge ledgers by the lamas. The
chief lama, in a philosophical dissertation given to the technicians, said that God's
purpose has now been achieved and the human race has finished the job it was
created to perform. What happens next?

— Automatic Control."

11 Anon., "Sarnoff Foresees Voice-controlled Systems," Research and Engineering,
3(7) :23, October, 1957.

"A. Opler and T. R. Norton, "New Speed to Structural Searches," Chem. Eng.
News, 34:2812, 1956.

THE TECHNICAL PROBLEM 61

the U.S. Patent Office;1'5 and it has even been reported that a computer has
been used to compile a list of 42,000 names with endings like -mycin to
be given to future antibiotics and drugs.14

Of the various applications cited, perhaps the use of computers for
translating foreign languages can be singled out as the one most related
to our problem. A historical survey of this problem has been written by
W. N. Locke.15 This, at the present time, is a high-priority research proj-
ect, financed primarily by the United States government, because there
are far too few translators available for the translation of contemporary
Russian technical literature into English. Technical literature, with its
relatively restricted vocabulary and relatively small use of shades of mean-
ing, idioms, and similar more difficult translation problems, turns out to
be a most suitable test medium for this type of study. It should not be
supposed, however, that all interest in this field is confined to the limited
and specific objective of translating technical Russian. The broader objec-
tives of being able to interconvert all languages and to handle nontechnical
literature are also being examined. For this purpose, an "interlingua" for
machine usage is also being examined. Each language is first translated
into interlingua and then converted into the "target" language. Interest
in the general subject of machine translation is now sufficiently great that
a monograph16 has been written on the subject and a periodical, Mechanical
Translation, devoted exclusively to this subject, now exists. There are
already available rather impressive examples of the results of machine
translation, even at this early date. The goal for the present appears to be
total conversion to the words of the target language, i.e., English, in the
specific applications discussed above, with about ninety per cent accuracy
on grammar and other problems such as multiple meanings. Texts in this
condition can be rapidly edited into literate English.

In light of these experiments, the generation of musical structures is a
rather natural non-numerical application of these instruments, since, like
mathematical or alphabetical symbols, musical symbols can be operated
upon in a wide variety of ways to produce a virtually limitless number of
combinatorial forms.

13 Anon., "Electronic Searching Moves Ahead," Chem. Eng. News, 35:98, 1957;
also Patent Office, Research and Development, Reports 4-9.

"Anon., "Drugless Names," Chem. Eng. News, 34:774, 1956.

15 W. N. Locke, "Translation by Machine," Sci. American, 194(1 ):29, January,
1956.

16 W. N. Locke and A. D. Booth (eds.), Machine Translation of Languages, John
Wiley & Sons, Inc., New York, 1955.

62 EXPERIMENTAL MUSIC

Operation of Automatic High-speed Digital Computers

Before we consider the experimental methods used to produce music
by means of the Illiac, it is desirable to describe briefly in general terms
the basic operations of automatic high-speed electronic digital computers.
Goodman7 describes their operations somewhat more completely, and de-
tailed discussions of their construction and operation are available in the
literature.17

Automatic high-speed digital computers are devices capable of perform-
ing the arithmetic operations of addition, subtraction, multiplication, and
division in time intervals of the order of milliseconds. If these instruments
were nothing more than high-speed desk calculators, we would be unable
to take advantage of the increased speed of computation since human re-
action times are much slower than computer reaction times. This is to be
contrasted with the use of ordinary desk calculators, where machine and
human reaction times are "well balanced." Consequently, in order to be
efficient, high-speed computers require instructions supplied at rates of the
order of magnitude of the rates of their arithmetic operations. This re-
quirement has led to the "automation" of high-speed computing. In the
automatic high-speed digital computer, a list of instructions is placed into
the machine and automatically selected in sequence at speeds commen-
surate with the arithmetic operations. The preparation of the list of in-
structions for the computer is referred to as coding, or programming, and
the total collection of instructions to which the machine can respond is
called the code or program. Problems to be solved by the computer
must be reduced to a pattern of instructions which are repeated over and
over again, and normally only the numbers being operated on undergo
significant change. These patterns of instructions are called loops, or
cycles.

Since most problems require decisions to be made, the list of instructions
for the computer must include, besides arithmetic orders, an order for mak-
ing decisions. Of course, the coder must foresee the decisions to be made
and plan for them accordingly. The most usual type of decision order is
one in which there are two possible results, e.g., plus or minus, zero or non-

17 See, for example, E. C. Berkeley and L. Wainwright, Computers, Their Opera-
tion and Application, Reinhold Publishing Corporation, New York, 1956; T. E.
Ivall (ed.), Electronic Computers, Principles and Applications, Philosophical Library,
Inc., New York, 1956; G. R. Stibitz and J. A. Larrivee, Mathematics and Computers,
McGraw-Hill Book Company, Inc., New York, 1957.

THE TECHNICAL PROBLEM 63

zero. When this decision order appears in the sequence of instructions, it
either leaves the sequencing unchanged, as if it were not there at all, or it
transfers, or "jumps," the sequencing to another part of the list of instruc-
tions. A common name for this type of decision order is conditional trans-
fer order.

To hold the instructions or numbers, a computer requires a storage unit,
sometimes called the memory, and a programmer must have available cor-
responding orders to store and obtain information from this unit within the
machine.18 There must also be communication between the computer and
the outside, necessitating an input and output mechanism and correspond-
ing orders. In the case of the Illiac, input and output are prepared in the
form of Teletype tape. Input is prepared in the form of coded punched
tape on Teletype machines, while output can be transcribed back into
printed numbers or letters by means of electronic-tape readers attached to
these same Teletype machines. With other instruments, input and output
is often prepared in the form of punched cards or magnetic tape.

Some of the more important units and orders for a computer are sum-
marized in Table 1. The details of coding for the Illiac in particular are

Table 1
Important Units and Orders for a Computer

Unit Order

Arithmetic Add, subtract, multiply, divide

Storage Store, read from storage

Input Read

Output Punch, print, cathode ray tube display, etc.

Decision Conditional transfer

Miscellaneous Unconditional transfer, left shift, right shift,
shift registers, etc.

discussed in a programming guide prepared for use with this machine.19
Computers like the Illiac perform arithmetic in much the same way we
do with pencil and paper except that they use the binary number system
rather than the more common everyday decimal number system. This
number system is particularly appropriate, because circuits in the com-
puter either conduct or do not conduct; that is to say, these circuits operate
through vacuum tubes used as electronic relays. These exist in either of

18 L. N. Ridenour, "Computer Memories," Sci. American, 192(6) :92, June, 1955.

19 J. P. Nash (ed.), /Iliac Programming, Digital Computer Laboratory, Graduate
College. University of Illinois, Urbana, 111., 1955, and later printings. A more general
discussion of computer programming is contained in D. D. McCracken, Digital Com-
puter Programming, John Wiley & Sons, Inc., New York, 1957.

64 EXPERIMENTAL MUSIC

two states, conducting or nonconducting. To understand the binary number
system, let us first examine the more familiar decimal notation, consisting
of ten symbols, 0, 1, 2, . . . , 9, with which we can write any number.
For example, in the decimal number 121.51, three Is appear, each repre-
senting a different absolute magnitude depending on its position with re-
spect to the decimal point.

We can also rewrite the above number in the following way:

121.51 = 1 x 102 + 2 x 101 + 1 x 10° + 5 x 10"1 + 1 x 10"2 (5)
And, in general, we can write any decimal number as

bn\0n + ^n-ilO"-1 + • • • + b1101 + b010° + fc.xlO-1 + • • • + b.m10-m

(6)
where 0 ^ bj^9. However, there is nothing unique about the number
ten as a base for a number system. Let us consider two as a base for a
number system with only the symbols 0, 1. If we impose the restriction
that ax = 0, 1, we can write numbers as

an2n + «,?_i2"-1 + • • • + ajL1 + a02° + a^2~x + • • • + a^m2~m (7)

For example, the decimal number 10 (ten) can be written in the binary
number system as 1010 which is

1 X 23 + 0 x 22 + 1 X 21 + 0 x 2° = 8 + 0 4- 2 + 0 = 10 (8)

In Table 2, we have tabulated the correspondence between decimal
and binary notation for the first sixteen integers.

Table 2

Decimal

and

Binary

Integers

Decimal number

Binary number

0

0

1

1

2

10

3

11

4

100

5

101

6

110

7

111

8

1000

9

1001

10

1010

11

1011

12

1100

13

1101

14

1110

15

1111

THE TECH NIC AL PROBLEM 65

It is this technical detail involving binary numbers of computer design
that lies behind Shannon and Weaver's comment:20 'The choice of a log-
arithmic base corresponds to the choice of a unit for measuring information.
If the base 2 is used the resulting units may be called binary digits, or more
briefly, bits, a word suggested by J. W. Tukey."

The "Man-versus-Machine" Problem

One subject which must be considered at this point is whether the new-
type computers "think." Outside the computer field, there exist many mis-
conceptions concerning what computers can and cannot do. Thus, almost
inevitably, when the subject of our work has come up, the question has
been asked: "What is going to happen to the composer?", the implication
being that the composer is going to be put out of business by an "electronic
brain."

Much of what is said about computers depends on how one defines the
word think. A commonly accepted picture of a computer is that of a ma-
chine which is an extension in certain specific functions of human intelli-
gence, much as a physical machine such as a steam engine is an extension
of human physical strength, also in certain specified directions. As we have
seen, a digital computer, or for that matter an analog computer, is capable
of doing certain mathematical tasks more rapidly and more efficiently than
a human being can. These computers respond to detailed instructions as
to what calculations to perform, carry out these instructions, and present
the results to the operator, who then has to determine whether these re-
sults are useful or meaningful depending upon the nature of the problem
being solved. The computer is unable to determine whether the answers are
correct. It can perform erroneous calculations just as efficiently as correct
ones. This is why some writers have spoken of a computer programmer
"conversing" with a computer. He feeds it certain information and tells
the computer what to do with the information. The computer carries out
the instructions, and then the programmer inspects the results.

Aids to calculation are hardly a new departure in human affairs. The
abacus, published tables of mathematical functions, the slide rule, the com-
mon adding machine, and the desk calculator are all precursors of the pres-
ent highly developed computers. All of these aids to calculation can be con-
sidered to displace the operation of human intelligence to lesser or greater

20 C. E. Shannon and W. Weaver, The Mathematical Theory of Communication,
University of Illinois Press, Urbana, 111., 1949, p. 4.

66 EXPERIMENT AL MUSIC

extent in certain specific functions. For example, long division carried out
on paper requires mental effort over a significant period of time, whereas
the same operation can be mechanized and solved with the aid of a desk
calculator in a few seconds. To the extent that this specific mental effort is
called "thinking," even an ordinary desk calculator can be said to "think."
However, with most fairly simple machines such as desk calculators, the
human functions displaced by the machine have hardly been regretted.
However, writers such as Goodman are careful to distinguish between
mechanization and automation and to point out that the effect of automa-
tion upon man should be quite the opposite to that of mechanization.
Goodman notes that:21

Charles Chaplin's Modern Times (made in 1935), by exposing the fallacy in
Henry Ford's doctrine of the "one man, one operation" concept, exemplifies
all that automation will eliminate. Such boring, repetitive, soul-breaking work
will not be required to be done in the future by human beings — a human being
possesses judgement, imagination, flexibility, and great powers of coordination
. . . whereas the machine may be better from the point of view of strength
and perception, man is irreplaceable.

With a digital computer, automation, made possible through such opera-
tions as the conditional transfer order previously described, eliminates this
"single-operation-plus-inspection" process required in the use of desk cal-
culators. Rapid routine calculations become possible. However, the total
course of a calculation must be foreseen by the operator, even though the
specific results cannot be predicted, and the method for solving the mathe-
matical problem must be worked out, even though the form of the answer
is unknown. Similarly, in a musical problem, the method of composition
must be coded, even though the final specific outcome may be unpredict-
able.

There are research efforts continually going on to improve the perform-
ance of computing machines. By simply making their internal works more
complex and by increasing their storage capacity, more complicated prob-
lems can be handled. The increased bulk of instruments of larger capacity
can be compensated to some extent by replacement of vacuum tubes by
transistors and magnetic core devices. Other more basic changes include im-
proved input and output mechanisms and high-capacity storage or mem-
ory. The advance in usefulness of computers will, therefore, rest partially
on technological improvements and the building of more complex and ver-

21 Goodman, op. cit., p. 21.

THE TECHNICAL PROBLEM 67

satile instruments. On the other hand, more complex logical operations
to which these computers might be adapted must also be anticipated. In
fact, a problem receiving considerable attention today is the analysis of the
problems of "adaptability" and "learning." The question of whether these
processes can be coded to activate computers to respond in these terms has
been considered.22 In fact, the general study of comparing men and ma-
chines, for which Wiener23 coined the word cybernetics, is based upon
analyzing the various response mechanisms of man in terms of the concepts
of automation. This includes such principles as "closed-loop control" which
are basic to automation theory. Another similarity is recognized through
the concept of "purposeful action" which suggests that present activity is
motivated in terms of future goals and which, we might also note, bears a
striking similarity to some of Meyer's concepts, considered in Chapter 2, of
directed musical motion. It is not difficult to formulate comparisons of
computer circuits and the nervous system and to attempt, by finding me-
chanical and electrical analogs to the nervous system, to analyze by these
analogies the logical foundation of the actual thought process. In fact,
some writers suggest that the essential difference between the best com-
puters and man is really just one of complexity. Kermeny24 proposes that
the two basic things which differentiate man and computers are complexity
of memory and power of reproduction. The other factors, suitable input
devices (i.e., sense organs) and output devices (i.e., equivalents to human
actions) are being developed, presumably along with sufficiently complex
programming techniques to instill learning capacity. Other writers take a
much less daring view and suggest that the problem of building a machine
of the complexity of memory equivalent to man's is not really feasible, at
least in terms of techniques we can consider today or in the near future.
Moreover, they suggest also that the qualities of human actions are far from
being understood in terms of coding. Since these are questions which can
only be decided in the future, we may conclude that, for the present, com-
puters are still definitely limited to specific tasks in which one type or an-
other of data processing are of a fairly routine nature.

The question of whether computers will ever be "creative" in the sense
that we speak of creative composing is rather similar to the problem of

22 A. G. Oettinger, "Programming a Digital Computer to Learn," Phil. Mag., 7: 1243,
1952; C. E. Shannon, "Computers and Automation," Proc. I.R.E., 41:1234, 1953.

23 N. Wiener, Cybernetics, John Wiley & Sons, Inc., New York, 1949, in particular,
Introduction and chap. 5.

2iJ. G. Kermeny, "Man Viewed as a Machine," Sci. American, 192(4) :58, April,
1955.

68 EXPERIMENT AL MUSIC

whether they ''think. " Also, we might ask: "What is meant by the term
creative?" Being "creative" would seem to depend at the very minimum,
like "thinking," on having a computer operate on a self-sustaining basis,
and to "learn from experience." If we postulate, therefore, that creative
actions involve, at the very least, a unique perception of relationships be-
tween apparently disassociated events in such a way that a new truth is dis-
closed, then creativity seems to be bound up with the question of ignorance.
Moreover, it seems that what we first consider strokes of insight and mani-
festations of "creative thought" are, once they are analyzed and codified
and, particularly, codified to the extent that they can be processed by a
computer, no longer "creative processes" in the usual sense. Seen in this
light, the pursuit of knowledge which depends on "creative thought" is a
technique of coding, of finding explicit statements of more and more com-
plex logical relationships. If "creativity" is defined in such terms, then the
question of whether machines can do this type of coding is the one which
has to be answered. The present computers, even though they may be pro-
grammed to process problems in music, for example, cannot be expected to
seek out new musical principles. These now must be fed to the machine in
explicit detail, and, at best, the uncertainty introduced by freedom of choice,
i.e., the use of the random-number processes which we have applied to
produce computer music, perhaps gives a computer a very primitive sort
of unpredictability, but this can hardly be equated to any sort of creative
process.

The Monte Carlo Method

The practical technical method employed for generating computer music
is based on the so-called "Monte Carlo method," which we shall describe
first in general terms25 and secondly in terms of its specific application to
musical problems.

The Monte Carlo method is a rather new experimental technique based,
as the name is meant to suggest, upon the operation of the laws of chance.
In essence, the method is quite simple, involving basically an examination

25 D. D. McCracken, "The Monte Carlo Method," Sci. American, 192(5): 90, May,
1955; G. R. Stibitz and J. A. Larrivee, op. cit., chap. 9; A. S. Householder (ed.),
Monte Carlo Method, Applied Mathematics Series, no. 12, National Bureau of Stand-
ards, U.S. Government Printing Office, 1951. These references, and the first two
in particular, are general introductions to the idea of the Monte Carlo method.
There are, of course, many specialized technical articles in the literature on the
method and its application.

THE TECHNICAL PROBLEM 69

of sets of randomly selected numbers which we consider to represent sig-
nificant sets of events in some model universe. The method is justified in
instances where the object of study can be assumed to be a statistically
ordered universe from which we can isolate certain elements to form a sim-
pler model universe characterized by events subject to the same statistical
order. Thus, the Monte Carlo method consists of the experimental produc-
tion of random sets which are made to conform to statistical controls. It per-
mits us, therefore, to study the laws presumably implicit in the statistical
controls chosen for the experiment. Since experimental results are obtained
in random sequence by this method, it is distinguished from more tradi-
tional laboratory experimentation in which significant environmental fac-
tors are systematically varied or controlled. An additional assumption is
often made that many events are normally made up of smaller events and
that if these smaller events can be caused to occur in the proper manner,
for example, as sequential patterns, combinations of these events can be
generated which will include the classes of larger events of interest. The
Monte Carlo method gives us a sampling from the model universe, and this
sampling may then be compared to presumably analogous or at least simi-
lar events in the natural universe. Normally, such a comparison is made
to develop an hypothesis concerning the events of the real world or else to
refine the model so that it conforms more exactly in the next experiment to
the aspects of the real universe being examined.

There is one major penalty attached to the Monte Carlo method in its
purest form. In almost any given experiment, many superfluous events oc-
cur which are of no interest unless the process under examination happens
to be truly and completely random. In a practical experiment, this means
that a screening procedure must be devised to eliminate the superfluous
events in the most efficient manner. Until recently, the inefficiency of the
Monte Carlo method limited its application to relatively simple problems.
In principle, it is possible to run a Monte Carlo experiment by hand
methods, for example, by compiling a random-number table by some simple
arithmetical process, categorizing these numbers according to the demands
of the problem under investigation, and, in due time, completing the experi-
ment. Unfortunately, hand methods are too slow for any but the simplest
sort of problems, problems for which more conventional analytical methods
are much more satisfactory. With the development of high-speed com-
puters, however, this situation has been radically changed, for these instru-
ments are able to examine and process many millions of numbers per hour.
Thus, the Monte Carlo method is finally coming into its own, since it can

70 EXPERIMENT AL MUSIC

now be applied to many complex problems of a statistical nature which
have so far resisted solution by analytical means.

A second basic feature of the Monte Carlo method involves the nature
of the procedure required for sampling. In other words, how do we go
about collecting a set of samples from our model universe? It is seen that
the only practical method for carrying out the sampling procedure itself
is the sequential one — that is, one sample at a time is analyzed for the
properties of interest and is then placed into its proper category. Practical
everyday sampling procedures, such as public opinion surveys, industrial
quality-control techniques, and so on, can all be resolved in principle to this
idea of a step-by-step process. The important next step, then, is to build up
a theory to analyze both the distribution of samples and how this distribu-
tion is affected by the particular restrictions placed upon the rules govern-
ing the sampling process itself. To do this, use is made of the theory of
probability and of classical problems of chance, such as the drawing of
colored balls from an urn. It is not necessary to go into details, but it is
essential to present at least an illustration of such an operation so that we
may define several terms useful in the analysis of sequential, or chain,
processes, as they are also often called.

Let us consider an urn containing five white, three black, and two red
balls. If we let the basic event be the drawing of a ball of a definite color,
it is possible to designate the three possible events as Ew, a white ball, Eb,
a black ball, and Er, a red ball. We may now construct a mathematical
model in which the events are represented by points. The collection of all
such points is called the event space s. There is a law of probability called
the probability distribution function, or simply the distribution function,
which defines the probability p, for each point of our event space s. For
example, p(El0) = 5/(5 + 3 + 2) = Vi. However, instead of trying to
describe a function over the points of s, it is often simpler to associate the
points of s with numbers and describe a distribution function over these
numbers. The function that associates numbers with the points of s is
called a stochastic variable, or random variable, or chance variable. We
can, for example, define the following stochastic variable x for the balls in
the urn:

t-"V0 J^b *-T

a: 0 1 2 (9)

The function that describes the probability of x is called the probability
function f{x). Thus, for balls in the urn:

THE TECHNICAL PROBLEM 71

ftO) =0.5

/(I) =0.3 (10)

/(2) =0.2

Let us now consider a sequence of drawings from the urn under the fol-
lowing conditions:

1 . After each drawing the ball will be replaced.

2. If the color of the ball is the same as that of the preceding drawing,
the ball will be returned to the urn without the event being recorded.

A process which depends on the preceding event only is an example of
a simple Markoff process, and the corresponding sequence is called a Mark-
off chain. Now, if we let pr(E) be the probability of the event E and let
prEl(E) be the probability of the event E, given the previous event was
Ei, for the above sequence, we have the following probabilities:

1. Initially:

pr(Ei() = 0.5, pr(Eb) = 0.3, pr(Er) =0.2 (11)

2. Succeeding probabilities:

pr^ (Eir) = 0, prEw (Eb) = 3/5, pr,-r (E,) = 2/5

pr*6 (E„)=5/7, pr£fe (Eb) = 0, pr*6 (E,) = 2/7 (12)

prEr (Eu) = 5/8, pvEr (Eb) = 3/8, prE). (E,) = 0

The concept of chain processes is thus a useful analytical tool for proc-
essing experimental data obtained by the Monte Carlo method. Moreover,
in terms of information theory, chain processes are also seen to have
broader implications in that the quantitative description of discrete-channel
communication systems depends upon the analysis of how discrete symbols
are chosen sequentially. It is in view of this fact that Shannon and Weaver
define their terms relevant to sequential operations and make use of the
concepts of stochastic variable and of Markoff processes already introduced.
These authors give a particularly simple definition of a stochastic process
as follows:26

We can think of a discrete source as generating the message, symbol by sym-
bol. It will choose successive symbols according to certain probabilities de-
pending, in general, on preceding choices as well as the particular symbols
in question. A physical system, or a mathematical model of a system which
produces such a sequence of symbols governed by a set of probabilities, is
known as a stochastic process.

2,i Shannon and Weaver, op. cit., p. 10.

72 EXPERIMENTAL MUSIC

Shannon and Weaver also define Markoff processes and distinguish between
zeroth-order processes in which transition probabilities are independent of
previous events, being fixed a priori; first-order processes in which there is
a dependence of the probabilities upon only the immediately preceding
event, as in the above example involving balls being drawn from an urn;
and higher-order processes in which more than just the immediately pre-
ceding event is involved.

It is now necessary to make note of at least one or two applications of
the Monte Carlo method somewhat more elegant than the drawing of col-
ored balls from an urn. This is important because the significance of the
method is obscured if we consider only simple physical situations for which
analytical solutions are readily obtained, as in the above case. A somewhat
more complex application, for example, might be the analysis of the diffu-
sion of gas molecules or of particles suspended in a liquid to derive Fick's
laws of diffusion. Since diffusion depends upon the random motions of
molecules, the method of random flights may be used to simulate this proc-
ess. A random flight is defined as the particular species of Markoff chain
in which equally weighted a priori transition probabilities are assigned to
all possible choices for each successive event, i.e., each possible choice is
equally probable. This is again a situation we have encountered previously
in the discussion of information theory and is the situation characterized in
information theory as having the maximum entropy content.

The random-flight technique can also be applied to other problems. If
restrictions are placed upon the randomness of choice either by weighting
the transition probabilities unequally or by forbidding certain choices,
other situations in nature, of lower entropy content, may also be simulated.
For example, by forbidding the return of a random flight to a point in
space previously occupied, it is possible to generate elementary models for
flexible long-chain polymer molecules such as rubber and other plastic ma-
terials. This last problem was worked on by the present authors in con-
nection with another research project.27 The Illiac was used also for this
work, and, in fact, this earlier study is mentioned here because some of the

27 F. T. Wall, L. A. Hiller, Jr., and D. J. Wheeler, "Statistical Computation of
Mean Dimensions of Macromolecules — I," /. Chem. Phys., 22:1036, 1954; F. T.
Wall, L. A. Hiller, Jr., and W. F. Atchison, "Statistical Computation of Mean
Dimensions of Macromolecules— II," ibid., 23:913, 1955; "III," ibid., 23:2314, 1955:
"IV," ibid., 26:1742, 1957; F. T. Wall, R. J. Rubin, and L. M. Isaacson, "Improved
Statistical Method for Computing Mean Dimensions of Macromolecules," ibid.,
27:186, 1957.

THE TECHNICAL PROBLEM 73

programming employed for that problem was used to initiate the work in
musical composition to be discussed. The connection seems less far-fetched
once we recall that the process of composition can be conceived as a com-
plex random flight through a tonal universe, with dimensions of pitch and
time subject to restrictions we normally associate with rules of composition.

The Monte Carlo Method and the Generation of Music

Since the Monte Carlo method generates as well as processes data, it is
the obvious technique for producing musical notes, rhythms, and other mu-
sical elements. As a first experiment, we decided to assign integer values to
the white notes of the musical scale and generate integers at random which
could be processed in turn in such a way as to build up machine representa-
tions of extremely simple musical structures. The technical problem for
this initial experiment was separated into four basic parts. The first was
the relatively simple problem of generating random sequences of integers
which, in this early experiment, were equated to the white notes from C
below middle C to C above middle C. For this scale, therefore, a range
of values 0 through. 14 were used since fifteen notes were involved. As long
as random sequences of these integers were not processed in any way and
were simply printed out as produced, we generated random white-note mu-
sic. This, as our previous discussion would indicate, is a universe of all
white-note music, both "good" and "bad." The question then arose of
how to distinguish samples which are "good" from those which are not. To
do this, we processed the randomly generated notes through a sorting proc-
ess, or sieve, accepting some notes and rejecting others. This sorting proc-
ess was the second and by far the most complex part of the problem of
generating computer music. Thirdly, the accepted notes had to be stored in
the memory of the computer and assembled step by step into a machine
representation of a finished musical "composition." Lastly, this "composi-
tion," in turn, was converted into a printed representation in number or
letter notation, which was then transcribed by hand into musical score.

Propagation of Random Sequences of Integers. An uninstructed com-
puter obviously produces nothing by itself. Therefore, even the generation
of random sequences of integers must be provided for in a set of input in-
structions. Since a computer can only operate in an orderly fashion, i.e.,
"according to rules," it cannot "run wild" and produce random integers by
accident. Hence, we had to devise an orderly method of producing random

74 EXPERIMENTAL MUSIC

integers. There are a number of procedures for propagating sequences of
random or, at least, pseudo-random integers.28

The one we have chosen is the following. Let us express a fraction /,
represented in the number system base a, in the number system base b, i.e.,
let us find the c, such that:

/ = ClbLl + c2b-2 +■■••"+ cnb-n + ■■■ (13)

where 0 ^ ct ^ b — 1. If we multiply Equation (13) by b, we have:

bf = d + C26-1 + • ■ ■ + Cnb-^1 + • • • (14)

The integer part of bf is equal to c1. After subtracting cx from Equation
(14), we may repeat this operation by another multiplication and in this
way generate the digits of yet a new representation. If / cannot be repre-
sented exactly in the new base, the process will not terminate, i.e., the
residual fraction will never be zero. If the digits of / are reasonably ran-
dom initially, the digits of the new representation will also be random. To
generate random integers in a binary machine, we can take the digits of a
transcendental, for example, such as O.Itt, or any other convenient irra-
tional number, and use the process of Equation (14) where b equals the
range of random integers desired. If b is a multiple of 2, to prevent the
process of Equation (14) from terminating, the base b + 1 is used, and
one random integer, say b, is always rejected. This process was examined
rather carefully in a previous research project27 and appears to give in-
tegers in random sequence which showed no evidence of recycling after
840,000 trials.

As a concrete illustration of how this process operates, the hand com-
putation of a random sequence of seven digits, 0, 1, 2, 3, 4, 5, and 6, is
shown in Table 3. In this example, we start with a four-digit decimal
fraction / = 0.2718 and with b — 1 . The random sequence of integers, as
shown in Table 3, is 1, 6, 2, 1, 4, . . . .

This example demonstrates that this random integer generation process
accomplishes two important things. ( 1 ) It provides a mechanism for ob-
taining a sequence of random digits. (2) It also serves to keep the ran-
dom digits in the range of a selected number system defined by the param-
eter b.

Statistical Tests and the "Try-again" Method. The first experiment at-
tempted was the writing of simple melodies, using only the following four
rules for successive melodic intervals: (1) no tritones are permitted; (2)

28 See, for example, Stibitz and Larrivee, op. cit., pp. 188-191.

THE TECH NIC AL PROBLEM 75

Th

Table 3
s Generation of Random

Integers

T

0.2718

X 7

/ .9026

X 7

6

/ .3182

X 7

2

/ .2274
X 7

I

/ .5918

X 7

4

/ .1426

Random-
1, 6,

integer sequence:
2, 1, 4, . . .

no sevenths are permitted; (3) the melody must start and end on middle
C; and (4) the range of the melody from its highest to lowest note must
not exceed one octave. This we shall term the octave-range rule. Melo-
dies varying in length from three to twelve notes were generated, the notes
being chosen at random. Whenever a rule was violated, the melody at-
tempt was terminated and the whole process was started over again. The
rules were sufficiently few in number so that it was expected that a reason-
able fraction of the attempted melodies would be completed successfully.
The statistics of this melody-generation process are shown in Figure 1,
where the ratio of successful to total (successful plus unsuccessful) melodic
attempts is plotted against the length of the melody. It is seen that the
longer the melody is required to be, the less chance there is of completing
the melody, since the probability of violating some one of the rules in-
creases as the length of the melody is increased. Since these statistics
depend on several arbitrarily chosen rules of composition, no attempt was
made to find significant correlations. It was noted, however, that with the
addition of more rules, the probability of obtaining a successful piece of
music would soon become very small, so in order to generate music more
efficiently, a try-again method was introduced instead. In this method, in-
stead of rejecting the whole of a partially completed "composition" upon
the violation of a rule, only the note which violates the rule is rejected and
another note is generated to replace it. This process is permitted to con-
tinue until a successful note is obtained or until it becomes apparent that

76

EXPERIMENTAL MUSIC

1.0*

2 34 5 67 8 9 10 II 12
LENGTH OF MELODY (NUMBER OF NOTES)
Figure 1. Statistics of elementary melody generation.

no such note exists, in which case, the "composition" is erased and a new
"composition" started. In the experiments to be discussed, the number of
trials was set at fifty by a simple counting index of the type described in
detail under Experiment Two in Chapter 5.

Four points should be emphasized in regard to the try-again method:
(1) We do not have to normalize for any unequal weighting given the
various possible melodies generated by means of any particular set of
composition rules. This is in contrast to many experiments describing
physical situations in which the statistics must be adjusted in accord with
the principle of equipartition of energy. In our case, it is sufficient only
that, in the generation of random integers, the occurrence of these integers
be equally probable in each given situation. (2) The try-again process is
a reasonably efficient procedure. (3) It is actually a closer simulation of
actual composing procedures than the "discard" method, since a composer
does precisely this. He tries again until a note fits, and if this fails to work,
only then does he go back and erase some of his completed work. This is
a major point of comparison between what the machine is capable of
doing and what a composer normally does. (4) Random search for a note
solving the difficulty that calls out the try-again process seems to us to be
a more satisfactory method than another possible alternative, the sys-

THE TECHNICAL PROBLEM 77

tematic examination of all available notes in a specified order. It is ques-
tionable how much the systematic generation of notes would speed up the
composition process, since the gain obtained by eliminating repeated tries
would be balanced at least in part by the loss suffered by never striking the
solution by good fortune ahead of sequence. There is, moreover, a more
serious objection than this to systematic note search, namely, that it leads
to the weighting of possible solutions depending on the order in which the
notes are generated. In addition, it does not yield as readily to theoretical
analysis, since it utilizes an already highly organized starting point for
further operations rather than one characterized by minimum order.

The Problem of Computer Printout. A practical problem of a different
nature is the question of computer output. There are two basic aspects of
this problem to be considered: the first being the question of direct print-
out in musical notation in score form; the second, the direct production of
sound, a problem we shall consider in Chapter 7. The practical advantage
of having a method for preparing musical printout is only too obvious,
for a deficiency of the work to be reported was the necessity of transcrib-
ing the results to standard musical notation. The printed output from the
Illiac in these experiments was strictly a makeshift in which we utilized
presently available Teletype equipment. Although a simple numerical rep-
resentation used in the earlier experiments was replaced by a letter nota-
tion and while we also managed to set up an intelligible representation of
rhythms and dynamics, these remained a basically clumsy set of symbols.
A number of solutions of this problem appear possible, such as activating
a typewriter with musical typeface and appropriate spacing controls. One
such typewriter already on the market is the Musicwriter, manufactured by
the Music Print Corporation, in Denver, Colorado. This typewriter, how-
ever, is a mechanical rather than an electrical typewriter, which causes
difficulties when attempting automatic control by means of tape readers or
punched-card readers. It should be noted, as also mentioned in Chapter 7,
that we are currently undertaking the assembly of a printout unit in cooper-
ation with Cecil Effinger, the designer of the Musicwriter. This will be a
musical typewriter controlled by five-channel punched tape of the type used
for input and output with the Illiac. In addition, we are currently doing the
coding for the Illiac which will permit this unit to be activated properly.

At the time of writing, we have also noticed in the literature that interest
in the problem of printout of musical notation is not confined to our own
project. From the recent article in Computing News previously referred

78 EXPERIMENTAL MUSIC

to in Chapter 3,29 we should like to quote the following passage to illus-
trate another approach to the same problem:

There is no technical reason that a computer cannot accept the note and
octave statements in alphabetical, or other convenient form, and that it can-
not produce printed sheet music as an output. The plotting version of the
Univac printer (see Computing News, 104) is capable of putting out acceptable
sheet music in its present version, as is the plotting version of the IBM 407
tabulator (see Computing News, 100, page 4).

29 Anon., "Music Transposition by Computer," Computing News, 5(17): 108-113,
Sept. 1, 1957.

CHAPTER FIVE

Experimental Detail

Outline of Experiments

The central body of work, exclusive of the brief preliminary examination
of the statistics of the note-selection process discussed in the last chapter,
can be grouped into four principal "Experiments." In Experiment One,
the two major objectives were (1) to develop a technique for the com-
position of a simple but recognizable type of melody and (2) to achieve
simple polyphonic writing. This second objective, in particular, seemed an
essential prerequisite to further experimentation of the type contemplated.
Strict counterpoint was selected as the musical medium for Experiment
One, which then consisted of three computer programs designed to pro-
duce successively monody, two-part writing, and, lastly, four-part writing.
However, only a limited number of counterpoint rules were employed, and
note selection was confined to the white notes of the C-major scale. This
was done to simplify the coding in this initial experiment. Therefore, in
the first part of Experiment One, a program was written to permit the
writing of simple cantus firmus melodies varying from three to twelve notes
in length. Then, two-part writing was studied in order to investigate the
coding of simple contrapuntal relationships. A number of new problems
arose — vertical or chordal relationships, contrary and parallel motion, the

79

80 EXPERIMENTAL MUSIC

rejection of dissonances, and the formation of a simple cadence in C at
the end of each cant us firmus setting. Lastly, in the third part of this ex-
periment, we were concerned primarily with three-way and four-way note
interactions and, in general, with the problem of organizing a much more
complex texture and sequence of operations.

At this juncture, it became apparent that all the important remaining
rules of counterpoint should be added to the computer code in order to
demonstrate more clearly that a recognizable type of music could be pro-
duced by computational techniques. Therefore, in what we shall call Ex-
periment Two, a completely rewritten code was prepared to permit the
generation of essentially correct four-part first-species counterpoint. This
code was so designed that the rules could be added or removed from the
set of instructions by simple alterations in the code. Musical output was
generated which included examples of settings from purely random white-
note writing, obeying no rules whatsoever, to the most restricted settings,
in which the full set of rules were applied.

One possible way to continue from this point would have been to pro-
ceed to second-species counterpoint — two-against-one writing, rhythmically
— then to third species, to fourth species, and eventually to florid counter-
point. In so doing, we would duplicate the experience of a music-theory
student, and new and important musical problems such as elementary
rhythms, weak and strong beats, and the resolution of dissonances would
be introduced in a systematic way. However, it was apparent that this
would involve a long process of code writing which, when completed,
would still only result in an imitation of one historical musical style.
Moreover, it was evident that other types of codes might provide more
effective computer methods for various technical problems of music writ-
ing and, in particular, for writing music of greater contemporary stylistic
interest than strict counterpoint. Therefore, in the first part of Experiment
Three, a method was devised for generating independently for each voice
rhythmic patterns, dynamic markings, and an index of playing instructions
for stringed instruments, such as legato, staccato, pizzicato, and sul ponti-
cello. This last is an elementary prototype for the more general problem
of orchestration. Secondly, the problem of generating the tones to be
played was reexamined. Our system was expanded to include the complete
chromatic scale rather than just white notes. Initially we generated purely
random chromatic music in order to have examples of the most chaotic
starting tonal materials, that is, materials of the highest entropy content
within the framework of the chromatic scale. Then, in the third part of

EXPERIMENTAL DETAILS 81

this third experiment, we modified the rhythm, dynamics, and playing-
instructions code so that each of these might pertain to more than one
voice at a time, and thus no longer leave the coincidence of rhythm, dy-
namics, and color effects entirely to chance. Subsequently, this code was
combined with the random-note code to provide rhythmic randomly
pitched music.

The next step was to restrict the selection of notes somewhat and to
give some coherence to the melodic lines. To this end several simple rules
were used: (1) a rather special type of octave-range rule; (2) a rule in-
volving the stepwise resolution of tritones to consonant intervals; and (3)
a "skip-stepwise" type of melodic rule. This music was generated and then
combined with the previously described code for rhythm, dynamics, and
playing instructions to produce a texture freely chromatic and dissonant
in style, and yet obviously under greater control than purely random out-
put. Since all of these musical examples turned out, as expected, to be
highly reminiscent, in terms of over-all sound, of certain styles of contem-
porary music writing, we completed this Experiment Three by developing
a simple technique for generating twelve-tone rows and certain similar
restrictive structural devices of current interest.

Experiments of this type, however, soon lead to the asking of whether
there are formal principles of organization more general than the various
rules of composition involving harmony, counterpoint, and similar tech-
niques. This question formed the basis of studies which have been grouped
together as Experiment Four. Since the technical procedure used to pro-
duce music by means of the computer was the application of the principles
of Markoff processes, it was decided to generate what can be called Mark-
off chain music. A series of studies was undertaken to produce a number
of examples of such music. To this end we defined two sets of numbers
for assigning values to both absolute and conditional transition probabilities
for successive note selection. One set of values was based on the overtone
series and permitted the assignment of probabilities for melodic intervals
related to their order of occurrence in the harmonic series and, hence, to
their relative degrees of consonance. The second set of numbers was used
to extend the idea of a leading- tone function. These values were used to
control the relative proportions of larger and smaller melodic intervals.
Transition probabilities derived from these two sets of integers were com-
bined in various ways to produce melodic output in which the proportion
and character of skips and stepwise motions, the proportion of consonant
to dissonant intervals, and the resolution of dissonant to consonant tex-

82 EXPERIMENTAL MUSIC

tures, or vice versa, were controlled by rather simple means. This last
experiment was completed by applying the concept of Markoff chain
transition probabilities to the problem of writing an extended cadence,
specifically a generalized I-IV-V-I (tonic-subdominant-dominant-tonic)
cadence in C.

An outline of the experiments carried out to produce the Illiac Suite is
given in Table 4. This is also a chronological record of work carried out
from September, 1955, through November, 1956.

Experiment One

The logic used to write a computer program differs from the logic one
might ordinarily use to explain compositional problems in strictly musical
terms. In planning a computer program, the first step is to design a block
diagram which outlines the required logical processes. This is followed by
the actual writing of the program itself. To illustrate this process, it is
convenient to consider the planning of the program for four-part first-
species counterpoint, the musical form subject to investigation in the last
part of Experiment One and in Experiment Two. This arrangement also
permits us to discuss the details of Experiment One rather sketchily since
all the significant aspects of coding developed for Experiment One were
later refined and incorporated into the more complex set of instructions
developed for Experiment Two. In both experiments the same basic mu-
sical technique, namely, strict counterpoint, was employed. The pertinent
coding details discussed under Experiment Two may, therefore, be con-
sidered to apply to Experiment One as well, though, of course, with appro-
priate simplifications.

Rules of Strict Counterpoint. The method of strict counterpoint was
formalized for pedagogical purposes in the Gradus ad Parnassum, origi-
nally published in 1725 by J. J. Fux,1 who systematically codified the com-
positional practices of Palestrina, in Fux's time the best-known composer
of the highly consonant polyphonic style of the sixteenth century. Appar-
ently, as disclosed in more recent studies,2 it seems that Fux allowed some
eighteenth-century practices to enter into his method. However, regardless

1 J. J. Fux, Steps to Parnassus (trans, and ed. by A. Mann, with the collaboration
of J. St. Edwards), W. W. Norton & Company, Inc., New York, 1943.

2K. Jeppeson, Counterpoint (trans, by G. Haydon), Prentice-Hall, Inc., Engle-
wood Cliffs, N.J., 1939. See especially ix-xvi and pp. 1-53.

Table 4
llliac Suite Experiments Summarized

Experiment One: Monody, two-part, and four-part writing

A limited selection of first-species counterpoint rules used for controlling the musical
output

(a) Monody: cantus firmi 3 to 12 notes in length

(b) Two-part cantus firmus settings 3 to 12 notes in length

(c) Four-part cantus firmus settings 3 to 12 notes in length

Experiment Two: Four-part first-species counterpoint

Counterpoint rules were added successively to random white-note music as follows:

(a) Random white-note music

(b) Skip-stepwise rule; no more than one successive repeat

(c) Opening C chord; cantus firmus begins and ends on C; cadence on C; B-F
tritone only in V1L, chord; tritone resolves to C-E

id) Octave-range rule

(e) Consonant harmonies only except for chords

(/) Dissonant melodic intervals (seconds, sevenths, tritones) forbidden
(g) No parallel unisons, octaves, fifths

(h) No parallel fourths, no chords, no repeat of climax in highest voice

Experiment Three: Experimental music

Rhythm, dynamics, playing instructions, and simple chromatic writing

(a) Basic rhythm, dynamics, and playing-instructions code

(b) Random chromatic music

(c) Random chromatic music combined with modified rhythm, dynamics, and
playing-instructions code

(d) Chromatic music controlled by an octave-range rule, a tritone-resolution rule,
and a skip-stepwise rule

(e) Controlled chromatic music combined with modified rhythm, dynamics, and
playing-instructions code

(/) Interval rows, tone rows, and restricted tone rows

Experiment Four: Markoff chain music

(a) Variation of zeroth-order harmonic probability function from complete tonal
restriction to "average" distribution

(b) Variation of zeroth-order harmonic probability function from random to
"average" distribution

(c) Zeroth-order harmonic and proximity probability functions and functions com-
bined additively

(d) First-order harmonic and proximity probability functions and functions com-
bined additively

(e) Zeroth-order harmonic and proximity functions on strong and weak beats,
respectively, and vice-versa

(/) First-order harmonic and proximity functions on strong and weak beats, re-
spectively, and vice-versa

(g) /th-order harmonic function on strong beats, first-order proximity function on
weak beats; extended cadence; simple closed form

84 EXPERIMENTAL MUSIC

of how true this is, the rules of strict counterpoint still remain a logical
abstraction of many important elements of musical structure and form a
fundamental basis for handling linear melodic flow.

Basically, the writing of strict counterpoint involves composing first a
cantus firmus — a whole-note melody of up to some twelve notes in length.
Against this cantus firmus, there are set one, two, three, and occasionally
more countermelodies. Fux divided counterpoint settings into five "spe-
cies," in order to introduce compositional problems systematically. In
first-species counterpoint, only note-against-note writing is permitted. In
second and third species, one or more of the secondary voices moves faster,
thus introducing new problems such as weak and strong rhythmic beats
and passing-note dissonances. In fourth species, syncopated writing and
the resolution of suspensions are treated; while in fifth-species, or florid,
counterpoint, all of these techniques are combined. In our discussion, how-
ever, we shall be concerned with first species only, since only this type of
counterpoint has been programmed in detail for the computer.

The rules derived from first-species counterpoint as used for computer
programming may be grouped conveniently into three categories, namely:
(1) melodic rules, (2) harmonic or vertical rules, and (3) combined
rules. It may be observed that the most important departure from the
authentic rules of first-species strict counterpoint is found in the treatment
of the cadence. The various rules are also illustrated in Figure 2.

Melodic Rules

1. No melodic line may span more than an octave, i.e., the range from
the lowest note to the highest note of a given melodic line should be an
octave or less. The limits were set as any octave from the octave C-C to
the octave C'-C", as shown in the first part of Figure 2.

2. If the melodic line is the cantus firmus itself, it must begin and end
on the tonic. C was selected arbitrarily as the tonic in our experiments.

3. If the melodic line is not the cantus firmus, it still must begin and
end on notes of the tonic chord. This was the C-major triad in our ex-
periments.

4. A melodic skip of a major or minor seventh is forbidden because
this is a dissonant melodic interval.

5. Any melodic skip, i.e., a melodic movement of a minor third or more,
must be followed by a tone repeat or by a stepwise motion with or without
a change of direction. A stepwise motion is a minor or major second. A
stepwise melodic movement may be followed by another stepwise motion

EXPERIMENTAL DETAILS

RULE (I)

OCTAVE RANGES

85

us

LOWEST

POSSIBLE

RANGE

^_ a

HIGHEST

POSSIBLE

RANGE

RULE (2)

FIRST AND LAST NOTES
OF CANTUS FIRMUS

RULE (3)

TYPICAL EXAMPLES OF
OPENING AND CLOSING
NOTES

RULE (4)

TYPICAL EXAMPLES OF
FORBIDDEN SEVENTHS

RULE (5)

TYPICAL EXAMPLES OF
SKIP-STEPWISE MOTIONS

STEP

STEP

STEP

RULE (6!

PERMITTED

FORBIDDEN THREE-NOTE
REPEATS

FORBIDDEN

RULE (7)

FORBIDDEN REPEAT
OF CLIMAX

Figure 2. The rules of first-species counterpoint illustrated by means of typical

musical examples.

86

RULE (8)

EXPERIMENTAL MUSIC

PERMITTED

TYPICAL PERMITTED,
FORBIDDEN, AND CON-
DITIONAL INTERVALS

ACCEPTED

RULE (9)

FOURTH WITH
-^- \BASS

CHORDS FORBIDDEN

FORBIDDEN

RULE (10)

YE6 PERMITTED, BUT
MUST BE RESOLVED

3zn<

I OR30:6

RULE (II)

TYPICAL EXAMPLES OF
FIRST AND LAST CHORDS

RULE (12)

TYPICAL EXAMPLE
OF CADENCE

OCTAVES

RULE (13)

FOURTHS

FORBIDDEN PARALLEL
MOTIONS

UNISONS

Figure 2. (Continued.)

EXPERIMENTAL DETAILS

RULE (14)

PERMITTED PARALLEL
MOTIONS

87

THIRDS

SIXTHS

RULE (15)

REQUIRED STEPWISE
MOTION

RULE (16)

REQUIRED CONTRARY
MOTION

Figure 2. (Concluded.)

or by a skip. It should be noted that this skip-stepwise rule prevents the
melodic line from outlining triads.

6. No more than one successive repeat of a given note is permitted.

7. It is forbidden to repeat the highest, or climax, note of a given me-
lodic line unless it is high C and the melodic line happens to move in a
tritone resolution or in the cadence up to high C. In our coding, this rule
was applied only to the top voice (Violin I).

Harmonic Rules

8. Only consonant intervals are permitted. The permissible intervals
include unisons, octaves, perfect fifths, and major and minor thirds and
sixths. Forbidden dissonant intervals include major and minor seconds and
sevenths, and the tritone, except as under Rule 10 below. By tritone, we
refer throughout to both the augmented fourth and diminished fifth, which
technically should be distinguished.

9. The perfect fourth is considered a consonance only if it does not
occur between an upper voice and the voice sounding the lowest note of a

chord. This would yield a triad in second inversion (a *? chord) which is
considered dissonant. This rule automatically screens out all perfect
fourths in two-part writing.

10. The tritone is considered a dissonant interval, but it is permitted if
it occurs in a VII6 chord in major (II6 in minor). The only tritone occur-
ring in C-major cantus firmus settings is the B-F interval, which is, there-

88 EXPERIMENTAL MUSIC

fore, permitted if D occurs below the interval. Moreover, the tritone must
resolve to the minor sixth E-C, or major third C-E.

1 1 . The first and last chords of a cantus firmus setting must be based
on the tonic triad. This is a consequence of the second and third melodic
rules. Moreover, the lowest notes of these chords must be the tonic itself.
Thus, these chords must be in root position.

12. In the cadence (the last two chords of a cantus firmus setting) the
chord just before the final chord must contain the leading tone B in any
one of the four voices, but in not more than one voice. Moreover, this
leading tone must move to the tonic. The other counterpoint rules also
apply to the cadence. It should be observed that this is a somewhat more
liberal cadence rule than that required in traditional strict counterpoint in
which the next note to the last of the cantus firmus is required to be the
supertonic D for settings in C. Actually, this more liberalized cadence for-
mula was more difficult to program than the stricter rule would have been,
but we felt that it was a more challenging problem and would produce
more interesting cadences.

Combined Rules

13. Parallel unisons, parallel perfect fifths and perfect fourths, and par-
allel octaves are forbidden.

14. Parallel thirds and sixths are permitted, including parallel major
thirds and minor sixths forbidden in the strictest counterpoint.

15. In proceeding from one chord to the next, at least one of the four
voices must move by stepwise motion or remain stationary.

16. If any two voices move in the same direction into a unison, perfect
fifth, perfect fourth, or octave, then one of the remaining voices must move
by contrary stepwise motion or stay on the same note.

Processing These Rules for Coding. Upon examination of these rules of
counterpoint, it was evident that, for setting up computer operations to be
outlined by means of a block diagram, these rules could be classified as
follows: (1) where voices are considered one at a time, (2) where voices
are considered two at a time, (3) where voices are considered four at a
time. Class 1 includes all of the melodic rules; class 2 includes all the
rules designed to exclude harmonic dissonance; and class 3 includes more
complicated rules such as having at least one voice moving in contrary
motion in respect to others.

Secondly, it was observed that the over-all selection of notes could be
divided into three classes of operation as follows: (1) initial notes, (2)

EXPERIMENT AL DETAILS 89

intermediate notes, and (3) cadence notes. Thirdly, while it was entirely
possible to generate all the intermediate-type notes randomly and reject
those which do not satisfy the rules, it was also observed that, for the sake
of efficiency, it was possible to reduce the rejection rate considerably by
establishing three basic subclasses of intermediate notes as follows: (1)
tritone-resolution notes, (2) skip-stepwise notes, and (3) random notes.
In subclass 1, the notes to be used were predetermined by the fact that a
tritone would have occurred which requires a particular resolution. In
such a circumstance, the generation of random notes would have been
wasteful, and it was far simpler to supply the required notes directly. Sub-
class 2 notes were required whenever a melodic skip occurred. Here again,
it would have been simply wasteful to generate random notes, and it was
more efficient to restrict the note selection to the few possible choices.
Since there are permitted only two possible stepwise motions plus the re-
peat of a note, it was convenient to generate randomly one of the three
increments — 1 , 0, +1 equivalent to these melodic intervals and add it to
the previous note. Subclass 3 notes, that is, all other intermediate notes,
were generated purely randomly.

The Try-again Subroutine. Whenever a note was rejected, another note
to replace it had to be generated and tested. To do this, insertion of the
try-again subroutine, described in the last chapter, into the counterpoint
program was required. A count was kept of the number of times the com-
puter tried again, and if, after a sufficient number of trials, the probability
of all the possible notes being tested at least once was very high and the
notes were still being rejected, the computer was instructed to erase from
storage the "composition" thus far completed and start all over again. As
already noted in Chapter 4, the number of trials was set at fifty by a
simple counting index.

A "Direction of Composition." If we define the term the direction of
composition as the sequence chosen for placing notes successively in a
musical composition, then we observe that a direction of composition is
required the moment two-part and four-part writing is initiated. In this
type of writing, in the experiments to be described, the direction of com-
position was set as one harmonic line, that is, one vertical line, musically,
at a time. This sequence of note selection seemed to us to be a somewhat
more practical and efficient method than the alternate procedure of com-
posing initially a complete cantus firmus and then attempting to find other
mutually compatible melodic lines which could be set against this cantus
firmus. It should be emphasized, however, that this alternate method could

90

EXPERIMENTAL MUSIC

have been employed had we so desired. Therefore, the actual "composing
procedure" adopted was the following for four-part counterpoint: A note
of the cantus firmus, or Voice 1 (cello), was generated first and then fol-
lowed successively by the generation of notes for Voice 2 (viola), for
Voice 3 (violin II), and lastly for Voice 4 (violin I), except in special
situations such as tritone resolutions and cadences in which some voices
were already determined.

Indexing the Notes of the Scale. As already noted, for the generation of
first-species counterpoint, it was possible to restrict the choice of notes to
the white notes of the scale, since C-major cantus firmus settings were be-
ing considered. This was a desirable simplification, since it reduced the
number of random notes which needed to be generated. Hence, efficiency
was increased and the complexity of the program was reduced. The notes
of the musical scale were therefore indexed in simple numerical sequences
upwards from low C, omitting the sharps and flats, as shown in Figure 3.

0 I 2 5~~4 5 6 7 8 9 10 II 12 13 14
C D E F G A B C' D' E' F* G' A' B' C"

Figure 3. Numerical index of the white notes of the musical
scale as used in Experiments One and Two.

It should be mentioned that starting the index with the number 0 rather
than 1 is a convenience for setting up arithmetical operations in the com-
puter. Otherwise, this somewhat unusual method of numbering has no
significance. It might also be noted that the output produced in both
Experiments One and Two was printed in number notation on the Tele-
type machine in accord with the index shown in Figure 3. Transcription
of this notation to standard musical notation was then carried out by hand.
In assigning integers in this way to represent the notes of the musical
scale, it should be clear that we did not adopt some completely arbitrary
procedure. In Experiments One and Two, we preserved the same order in
this numerical — or stochastic — representation as that of the musical scale.
Moreover, as we shall see later on in Experiment Three, when we revised
this representation to apply to the ordinary chromatic scale, the numerical
difference between any two of these stochastic representations corresponded

EXPERIMENTAL DETAILS 91

to the number of half-tone steps separating the notes being represented.
Thus, numerical representations became associated with particular musical
intervals. It was this fact which enabled us to add the number representing
an interval to a previous note and thereby produce the correct numerical
representation of a new note. With these basic decisions made, it was then
possible to sketch block diagrams for each of the parts of Experiment One.

Simple Monody: Coding for Cantus Firmi. In this first part of Experiment
One, we limited the control over the musical materials to the same rules
used in the preliminary tests before the try-again subroutine was developed,
namely: (1) no melodic sevenths, (2) no melodic tritones, (3) the oc-
tave-range rule, and (4) the rule that all the cantus firmi must start and
end on C. A comparison of these few rules with the counterpoint rules
listed earlier disclosed that many violations of the technique of strict coun-
terpoint could be anticipated, and indeed, as we shall see, the results
showed that all possibilities not specifically excluded by the rules employed
appeared in the musical output.

The actual computer program was similar to the "melodic subroutine"
to be described under Experiment Two and included in addition a simple
program for fixing the first and last notes as C. The lengths of the cantus
firmi were set by a simple counter. Thus for a cantus firmus containing n
notes, this counter was set at —n+1, and after each note was selected
(including the first note), unity was added to this quantity to yield the
successive values —n+2, —n+ 3, . . . , — n+n. This last value, occurring
after note n— 1 had been selected, is, of course, equal numerically to zero,
the lowest possible positive number recognized in computer calculations.
The change in sign from a series of negative numbers to a positive number
was detected by a conditional transfer order, which was then used to shift
to the instructions used to terminate the cantus firmus with the note C and
begin a new cantus firmus. The number of cantus firmi of each given
length was controlled by another counter working on the same simple
principle. Lastly, the lengths of the various cantus firmi, denoted by the
function n, were obtained by means of a series of preselected stored param-
eters used for n.

In the first run, it was decided to generate fifty samples each of cantus
firmi from three to twelve notes long to produce a total of five hundred of
these simple melodies. The actual time required to do this in the computer
was limited primarily by the time the computer required to punch the
output tape.

Simple Two-part Cantus Firmus Settings. The next step was to consider

92 EXPERIMENT AL MUSIC

simple two-part writing in which some vertical relationships between the
two voices had to be considered and yet in which the complications in-
volved in crosschecking more complex relationships between more than
two voices were still avoided. Therefore, for the new added voice, the
same melodic rules as used for the cantus firmus were put in effect, except
that the first notes were permitted to be E or G as well as C. Secondly,
vertical dissonances, specifically seconds and sevenths, the B-F tritone, and
perfect fourths, were forbidden in what was to become eventually the
"harmonic subroutine" of Experiment Two. Thirdly, as a simple cadence
subroutine, it was required that a leading tone B be inserted in either one
of the two voices in the next-to-last chord and that this B resolve upwards
to C. The selection of which voice the B was made to occur in was carried
out by means of a binary random-choice operation and was also made to
depend upon proper voice leading, in accord with the rules in effect.
Fourthly, parallel unisons, octaves, and fifths occurring between successive
chords were forbidden. The technique for accomplishing this in four-part
writing is described under Experiment Two. Again, as in the case of the
cantus firmi alone, fifty each of cantus firmus settings from three to twelve
notes long were produced.

Simple Four-part Cantus Firmus Settings. In the last part of Experiment
One, we extended our technique to include four voices, in accord with the
primary objective of this experiment. The principal problem now became
one of dealing with a set of interactions of increased complexity. In this
first attempt at four-part writing, we still kept the rules relatively simple
and added as few new rules of operation as possible beyond those used for
two-part settings. Certain operations were actually liberalized; specifically,

4 chords and the VII6 chord were permitted. As it turned out, two sets

of computations were completed in this last part of Experiment One. The
results of the first set indicated that, even at this point, the attrition because
of the restrictions already in effect was so great that certain undesirable re-
sults were occurring very frequently. Specifically, the most objectionable
result was the occurrence of many repeats of a given note in the same
voice. It turned out that many reiterations of one note became a con-
venient solution to many of the voice-leading problems which arose in this
style of counterpoint. While not necessarily unattractive in sound, this
was, for our purposes, an undesirable end result. Hence, for a second run,
an additional rule was inserted into the program to forbid more than one
successive repeat of any note.

The increase in complexity which resulted as a consequence of four-part

EXPERIMENT AL DETAILS 93

writing involved the problem that all the harmonic and parallel-motion
rules had to be cross-checked between all voices. Thus, six separate cross
checks were written into the computer instructions. These were required
for the following pairs: Voice 1-Voice 2, Voice 1-Voice 3, Voice 1-Voice
4, Voice 2- Voice 3, Voice 2-Voice 4, and Voice 3-Voice 4. This became
the entry of the "harmonic subroutine" of Experiment Two, which had to
be entered six times in order that all the voices were properly cross-checked.

Lastly, a routine for the cadence formula, listed as Rule 12, had to be
written. This turned out to be a complicated problem in itself. In this
routine, described fully under Experiment Two, all the standard voice-
leading and harmonic rules still had to be checked, but there were, as well,
additional restrictions defining acceptable note combinations. The cadence
routine was in essence a program which contained instructions embodying
these additional restrictions and also a long series of rewrite orders which
temporarily rewrote large sections of the regular counterpoint instructions
so that they might be specifically applied to the cadence formula.

As in the previous parts of Experiment One, fifty samples each of set-
tings from three to twelve notes long were produced.

Experiment Two

With the basic points considered under the topic of Experiment One in
mind, it is now possible in discussing the details of Experiment Two to
examine some block diagrams actually used in the production of musical
materials in Experiment Two. It is convenient to start with the diagram
for the main routine shown in Figure 4.

Main Routine for Four-part First-species Counterpoint. In the upper
left-hand corner of the diagram shown in Figure 4, there is a box labeled
"Initial Entry," which represents the routine instructions carried out di-
rectly after the program is read into the computer. This block includes
routine operations such as clearing storage locations, setting up indices and
counters, storing parameters, and so on. It is at this point that instructions
for the number of cantus firmus settings of various lengths to be completed
in each given computer run were included in this particular program of
instructions.

Initial Notes Subroutine.3 Following these preliminaries, the first im-

3 The term subroutine in our discussion is not restricted to the self-contained group
of orders with a formalized entry and exit, sometimes more precisely called a closed
subroutine. Both open and closed subroutines were used in this programming de-
pending upon circumstances and convenience.

94

EXPERIMENTAL MUSIC

INITIAL ENTRY

CADENCE- FINAL NOTES

i

♦

♦

SET INITIAL NOTES

v

PRINT OUT

YES

*

i

SET CADENCE?

RESET
FOR NEXT BLOCK

{no

1

1

YL5

TRITONE RESOLUTION ?

1

*

* *

1

NO,
YES
NO

|N0

1

rtb

SKIP-STEPWISE MOTION?

"try-again"
subroutine

♦NO

l"«l >

i

GENERATE RANDOM NOTE

o ^

*

■

o

THREE-NOTE REPEAT ?

YES

1

|NO

MELODIC SUBROUTINE

*

AT LEAST
ONE VOICE STEPWISE?

HARMONIC SUBROUTINE

I
1
l
1

IYES

1
1

3 OR 10 BETWEEN
LOWEST NOTES?

♦ NO

DITTO FOR VOICE 2

1

1

CONTRARY MOTION?

*

|YES

DITTO FOR VOICE 3

SET TRITONE RESOLUTION

*

♦

DITTO FOR VOICE 4

RESET-
SHIFT TO NEXT CHORD

*

i

Figure 4. Experiment Two: Block diagram for the main routine.

portant task was to set the first chord of the first cantus firmus setting in
accord with Rules 2, 3, and 11, given on pages 84 and 88. This was
done by means of a special subroutine required only at the beginning of
each cantus firmus setting. Once it had been used, this subroutine was not
entered again until after a setting had been completed and the computer
was ready to proceed with the generation of the next setting.

The basic problem was to set the chord of C major in root position into
the four voices in the most efficient manner. Since only the notes C, E, G,
C, E', G', and C", with numerical equivalents of 0, 2, 4, 7, 9, 11, and
14, respectively, were involved, it was possible to restrict the choice of
random integers to these few numbers by storing these particular numbers
in a special small table of seven entries and selecting the memory locations

EXPERIMENTAL DETAILS 95

for these numbers by random-integer generation, utilizing the technique
already described in Chapter 4. Moreover, since the initial note for the
cantus firmus (Voice 1) was limited even more to just C, C, or C",
to make the note-selection process as simple as possible, we could arbi-
trarily select C, for which the numerical equivalent is 7, for this voice.
This automatically ensured that the chord would be in root position, since
Voice 1, being for the cello, would subsequently be transposed downward
an octave.

Main Routine Continued. Since cantus firmus settings from as short as
three notes to any reasonable length might be required, the next logical
step was to ask whether the next-to-last chord had been reached and
whether the cadence routine would be required. This would happen im-
mediately, of course, only when the length of the setting had been restricted
to three notes. However, it was desirable that this decision be made at
this point because of the cyclic nature of the program. In the experiments
actually carried out with the Illiac, the longest settings generated were
twelve notes long, and all the output for Experiment Two, in particular,
was limited to this length. However, this was not the case in Experiment
One, as we have already seen. This decision operation, which not only
brought the special cadence routine into operation, but also set the lengths
of the cantus firmus settings, worked in the following way: If we wished
to write settings for cantus firmi n notes long, a cadence routine would be
required after n — 2 notes had been generated in each voice, since both the
last and next-to-last chords would be involved in the cadence. Therefore,
the number — (n — 2) could be set in a convenient storage location, and
after each chord was successfully completed, a unit increment could
be added to this quantity to obtain successively the values — (n — 3),
— (n — 4), . . . , — 1, 0, as shown in Table 5. Again since zero is con-
sidered a positive number, when the operation — (n — 2) + (n — 2) was
carried out after n — 2 chords had been formed, a change of sign took
place. This change of sign activated the conditional transfer order and
caused the sequence of operations to shift to the special cadence routine.
Until this change of sign occurred, however, the transfer order was by-
passed and the principal part of the program, which was used for the gen-
eration of intermediate notes, was entered.

The next group of six blocks of instructions bracketed as referring to
Voice 1 was repeated with minor variations also for Voices 2, 3, and 4,
taken successively. These were blocks of instructions primarily derived
from rules for the voices taken one at a time or two at a time. In fact, these

96

EXPERIMENTAL MUSIC

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EXPERIMENTAL DETAILS 97

operations took care of all of the rules for intermediate notes except some
of the more complex interactions which could only be tested for after all
four notes were selected.

Within the group of instructions for each voice, the two special types
of notes which could be set without resorting to the general random note
scheme were treated first. In the set of more general instructions — on the
right-hand side of Figure 4 — the block labeled "Set Tritone Resolution"
is a set of instructions which was activated whenever a tritone — which in
these experiments was limited to the combinations F-B, F-B', B-F',
F'-B' — had occurred between any two voices in the previous chord. The
"Set Tritone Resolution" operation also recorded between which voices
the tritone had occurred and stored this information for the "Tritone
Resolution" block for Voices 1, 2, 3, and 4. Lastly, it also eliminated tri-
tones occurring between more than two voices, which could arise only as
a consequence of doubling at the unison or octave. These doublings were
unacceptable, since they would produce forbidden parallel motions at the
unison or octave. It should also be noted that before the tritone-resolution
notes were set for the next chord, they were examined by means of the
melodic subroutine to be described below. If they were found to be un-
acceptable, the chord was erased and started over.

The tritone-resolution operation was used therefore, whenever a tritone
had occurred between two voices only, and only when one of the tritone-
interval notes had occurred in the voice for which a new note was being
generated. If the tritone note in the previous chord happened to have been
F or F' (index numbers 3 or 10) or B or B' (index numbers 6 or 13),
the notes in the new chord were automatically inserted as E or E' (2 or 9)
or C or C (7 or 14), respectively. Otherwise, the operation was bypassed.

The skip-stepwise operation was based upon Rule 3 and was set up so
as to be more efficient than testing randomly generated notes for Rule 3.
The two notes previous to the one being generated were first examined by
finding the negative absolute value of the difference of their values. Thus,
if we were about to generate note Ni9 we computed — |N*_i — Ni-2\- The
quantity 1 was then added to this value, and if the result turned out to be
positive, we knew that the previous melodic interval had been smaller
than a third, that is, it had been a stepwise motion. The positive sign, if
it occurred, was used to activate the conditional transfer order, this time
to bypass the special skip-stepwise routine and to proceed to the genera-
tion of a random note. On the other hand, if the number was found to be
negative, we simply generated at random the values +1, 0, or —1 for the

98 EXPERIMENTAL MUSIC

interval N, — Ni-i, bypassed the general random-note-generation process,
and proceeded to further testing. This calculation is illustrated in Table 6.

Table 6

Experiment Two

An Illustration of the Process for Detecting Skips in

Applying the Skip-stepwise Rule

Note Ni-i

Note Ni-2

-\Ni-i-

Ni-S\

-\Ni-i-Ni.2\ + 1

C'(7)

C'(7)

0

+ 1 ^Stepwise motion

A(5)

B(6)

— 1

Oj Generate random note

E'(9)

G'(ll)

-2

-1 '

>Skip

C"(14)

G'(ll)

-3

-2

Generate +1, 0, or — 1

etc.

•

for the interval,

Ni - Ni-i

To generate the increments +1, 0, and —1, we simply used the random-
integer-generation process previously described in Chapter 4, setting b = 3
to yield the values 0, 1, and 2 in random sequence. By subtracting 1 from
these values, we obtained in turn —1, 0, and +1 in random sequence.

The next step in the process was the generation of a random note with-
out restriction. Since there were fifteen possible notes available for selec-
tion, n was set at 17, this being the smallest prime number above 15, and
the integers 15 and 16, whenever they turned up, were rejected by means
of the process of the type shown in Table 7.

The next step was the elimination of multiple repeats of the same note,
specifically, the elimination of all repeats of the same note but one. To
do this, the quantity \N{ — iV«_i,| was first computed, and if it was found to
be unequal to zero, the test was concluded, since this meant that the pre-
vious melodic interval had not been a repeat. On the other hand, if the
difference was found to be zero, then further testing for the second interval
back was also required. The interval |M_i — iV*_2| was then computed. If
this was found to be unequal to zero, we know the second melodic interval
back had not been a repeat. In this case, the test was again concluded.
However, if this difference was also found to be zero, then two repeats in
a row had occurred. Since this is forbidden by the rules, the generated
note, in this case, had to be rejected and try-again subroutine entered.

Melodic Subroutine. The melodic subroutine, which in simplified form,

EXPERIMENTAL DETAILS

99

but with additions for selecting the first and last notes of a cantus firmus,
had been also the first music-generation code written for the Illiac as part

Table 7

Experiment Two

Process for Rejecting Certain Random integers

(Specifically 15 and 16) Out of the Set, 0, 1, . . . , 16

Random integer Random integer — 15 Conditional transfer result

14

15

16

-15 !>
-14

0

+ 1

J

Accept

Reject and recycle to repeat
random-integer generation

of Experiment One, was used both to screen out forbidden intervals be-
tween successive notes and to check longer-range interactions between the
notes of a given melodic line.

In the testing of intervals between successive notes, a simple additive
process was employed. Initially, \Nt — N*_i| was computed and the quan-
tity 7 subtracted from this absolute difference. As shown in Figure 5, each
time we tested this yielded one of a series of integers which could then be
checked to find the exact magnitude of the melodic interval. Since the
only forbidden melodic intervals were sevenths and tritones, only these
had to be screened out. Therefore, it was possible to test immediately for
whether the interval was an octave or larger, and if so, the interval was
conditionally accepted. Since intervals larger than an octave were auto-
matically eliminated by the test to follow for the octave-range rule, the
octave was, in effect, being detected by this first screening operation. Di-
rectly thereafter, it was possible to test for the seventh. If the seventh was
found, the try-again routine was entered, and the whole process started
over again. On the other hand, if the seventh was not found, then, by the
rules, all the remaining intervals were acceptable except the tritone, and
the possibility of the presence of this interval could immediately be tested
for. Therefore, a screening operation for sixths could be omitted.

100

EXPERIMENTAL MUSIC

ENTER MELODIC
SUBROUTINE

FORM 8-7

YES

YES

NO

>7,i.e.

8-7>0?

NO.ADD+I

8 = 6,i.e.,

8-7+ 1 = 0?

NO, ADD* 2

= 4.

8-7+3=0?

, NO.ADD+I

= 3,

8-7+4=0?

YES

TRITONE PRESENT?

NO

OCTAVE RANGE?

RECYCLE

YES

1_

TRY-
AGAIN"
SUBROUTINE]

YES

NO

YES

RETURN TO MAIN ROUTINE

Figure 5. Experiment Two: Block diagram for the
melodic subroutine.

It was possible to proceed directly to the detection of the presence of fifths
and fourths, that is, of whether 8 = \Ni — iV*_i| = 4 or 3. If one or the
other of these intervals was found, the presence of a tritone was possibly
indicated and additional screening was required. Thus, if the difference
8 = 4 was found, then a tritone involving the notes B-F' (indexed as 6
and 10) had to be tested for by a subtractive process. Alternatively, if the
difference 8 = 3 was found, the possibility of the existence of one of the
other two different tritone intervals, F-B and F'-B', had to be checked.
Again, subtractive tests were employed for this purpose. Moreover, the
higher interval F'— B' was first lowered by an octave, so that the same test
could be employed for both these intervals.

The next operation involved the octave-range rule, Rule 1, which was,
for us, one of the more interesting rules of counterpoint. As noted
previously, this rule states that a melodic line is limited to a span of an
octave or less, but it does not specify which particular octave span this
range must encompass. This operation is in some ways more generalized

EXPERIMENTAL DETAILS 101

than most of the others, for the specific limitations it imposes upon the
melody develop only as the melody is being written. Therefore, each time
a note was generated, track had to be kept of whether this new note was
more than an octave higher or lower than any other note already present
in the melody. If not, then the note was provisionally accepted and the
test concluded. On the other hand, if the note violated this rule by ex-
ceeding an octave, then the try-again subroutine was entered. This test
operated quite simply as follows: The note just generated, Nh was sub-
tracted from note Ni. The quantity 8, which is equivalent to the span of a
ninth, was then subtracted from the absolute value of this difference, and
if the result remained positive, we knew that a skip larger than an octave
between this most recent note and the first note of the melodic line had
been formed and the note Nt was rejected. This process was then repeated
successively for N{ — N2, N( — N3, to Nt — N,_i, and only if the condi-
tion required by Rule 1 was fulfilled for all these intervals was the note
accepted.

Finally, the repeat of a climax in the melodic line of the highest voice,
which, in accord with Rule 7, we considered to be forbidden, was tested
for. Although this routine was not a part of the melodic subroutine since
it had to be employed only in connection with the highest voice, it is con-
venient to mention it at this point. As each note in the highest voice was
accepted, a record was kept of the highest note generated up to that point,
and a specific test was eventually made of whether a repeat of this note
had occurred. This rule, however, was not applied in the cadence or in
conflict with the tritone-resolution rule.

Harmonic Subroutine. The principal function of this subroutine was to
screen out vertical dissonances and to restrict chords to perfect triads and
their first inversions, the one chord with a permissible dissonance being
VII6, which contains the tritone interval. The block diagram for this sub-
routine is shown in Figure 6. In Figure 6, we have used the symbol A to
represent a vertical interval, that is, the absolute numerical difference be-
tween the notes in two different voices on the same beat.

The subroutine has been simplified in Figure 6 to show how the testing
is carried out between any two voices, but it should be remembered that
the relationships between all possible pairs of voices had to be examined.
The six possible combinations of voices taken two at a time are V1-V2,
Vx-Va, Vx-V4, V2-V3, V2-V4, and V3-V4. The harmonic subroutine had,
therefore, to be entered a total of six times before a particular combination
of notes could be passed as acceptable. It was most efficient to design a

102

EXPERIMENTAL MUSIC

ENTER HARMONIC SUBROUTINE

\

-

VERTICAL A >7?

NO

| YES

REDUCE INTERVAL

RECYCLE

YES

1

f

A=7?

YES

| NO

YES

"TRY-AGAIN"
SUBROUTINE

A=6?

\ NO

A = 5?

T tb

YES

J NO

A =4?

YES

J NO

A =3?

\ NO

A=2?

YES

J NO

A=l?

YES ,

N p

J NO

THEN A MUST=0

1

PARALLEL MOTION?

\ NO

DOES A VOICE MOVE
AS CONTRARY STEP?

\iO

NO

\ YES

TRIT0NEIN2ff6'?

YES

RETURN TO MAIN ROUTINE

Figure 6. Experiment Two: Block diagram for the
harmonic subroutine.

recycling operation for testing all four voices by means of the melodic and
harmonic subroutines by using a set of indices which would keep track of
what was tested and which would also carry out the testing in the most
efficient manner. The order of testing is shown in Table 8.

The harmonic subroutine was used ( 1 ) to eliminate vertical sevenths and
seconds, (2) to eliminate parallel unisons, fourths, fifths, and octaves, all of
which are forbidden by the rules, (3) to test whether a unison, fourth,
fifth, or octave had been formed by the two voices moving in the same di-
rection by contrary stepwise motion, or by one or both voices remaining
stationary, and (4) to test whether a tritone had occurred. The testing of

EXPERIMENTAL DETAILS

103

Table 8

Order of Testing the Four Voices with the

Melodic and Harmonic Subroutines

Voice 1

Enter melodic subroutine for Vx

Harmonic subroutine is not entered since there is nothing to test

Voice 2

Enter melodic subroutine for V*
Enter harmonic subroutine for V\

Voice 3

V*

Enter melodic subroutine for Vx

Enter harmonic subroutine for Vx — V3, V* — Vs

Voice 4

Test for repeat of climax note

Enter melodic subroutine for Vt

Enter harmonic subroutine for Vx — Vi. Vs — Vt, V3

V>

whether D, rather than B or F, was the lowest note of the one chord which
might contain a tritone, that is, of whether the chord was VII6 and not VII
or VII6, was done later in the main routine after the whole chord being
formed had been completed.

Table 9

Experiment Two

Reducing Vertical Interval to an Octave or Less

Interval

A

A-8

Reduced

A A for testing

Double octave

14

6^

7^

0

Fourteenth

13

5

+ 1

6

Tenth

9

1

2

-7 -t

Ninth

8

0

1

. > -6

Octave

7

~n

7

0

Seventh

6

-2

6

— 1

.

► discard

.

•

and use
original

•

•

Unison

0

-8.

0

-7

104

EXPERIMENTAL MUSIC

In order to simplify the testing procedure, all vertical intervals larger
than an octave were first reduced. This was accomplished by computing
the absolute difference A = \Va — Vb\ and subtracting 8 from the result.
Tf a positive number resulted, we knew from this calculation that the orig-
inal interval was larger than an octave, and +1 was added to the result
to yield the reduced interval. On the other hand, if a negative answer was
obtained, the original interval was employed directly for the tests. This
calculation is shown in Table 9.

The next step was to eliminate sevenths and seconds and to set up the
remaining tests in the subroutine. For convenience in applying the condi-
tional transfer order, 7 was subtracted from these adjusted values of A
just computed. Sevenths and seconds were first eliminated by adding unity
successively to A— 7 and, after each addition, testing with the conditional
transfer order in order to place the interval into the proper category for
subsequent processing. The technique for doing this is shown in Table 10.

In the parallel-motion test, which was applied only when A = 0, 3, 4,
or 7, Ai was compared with A<_i to see whether A* = A«-i. Whenever
this occurred, we knew forbidden parallel motion had occurred, either
actually, as in the examples shown in Figure la, or implicitly by octave

Figure 7. Experiment Two: Examples of (a) direct parallel
motion, (b) implicit parallel motion.

displacement, as in the examples shown in Figure lb. The previous interval
for the two voices in question, A«_i, was obtained from storage and re-
duced if necessary. It was then directly tested by subtracting from it the
quantity Ai. If the difference was found to be unequal to zero, we knew
immediately that parallel motion had not occurred.

The two remaining tests included with the harmonic subroutine also de-
pended on the examination of the intervals between the voices taken two
at a time. In the test for contrary motion, again only applied whenever
A = 0, 3, 4, or 7, the signs of the differences N, — A^_i for each of the
two voices in question were determined. If the signs were unlike, then con-

EXPERIMENTAL DETAILS

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106 EXPERIMENTAL MUSIC

trary motion had occurred and the test was concluded, since contrary mo-
tion was permissible. But if like signs were found, the two voices had to
be tested for whether one or both of them had remained stationary, that
is, for whether the value Nt — N*_i = 0, which was also permissible as
such. Otherwise, it was recorded that a possible violation of Rule 16 was
indicated and that further testing in the main routine would be required to
determine whether one of the remaining voices had concurrently moved by
contrary motion. The testing for sign was done directly by means of the
conditional transfer order.

In the last test, the specific reduced interval F-B was first searched for
whenever A had been found by previous testing to be equal to 3 or 4. This
was done by determining whether any of the quantities Ni — 3, iV» — 6,
Nt — 10, or Ni — 13 equaled zero in each of the two voices to be
tested. If so, it was noted that a tritone had occurred, and the rest of the
test was then carried out. In the second part of this test, the presence of
doubling either at the unison or octave was checked, and if found, the
doubling was removed by erasing the chord and starting over again by
means of the try-again subroutine. Finally, in a later part of this test, the
presence in one of the remaining voices of a D lower than the tritone inter-
val was tested for by first finding the lowest note of the tritone interval and
then subtracting from this, successively, the notes in the two remaining
voices. If the difference was found to be positive, we knew that a note
lower than the interval existed, and we also knew that this note had to be
a D, since the notes C and A form dissonant intervals with B, and the
notes E and G form dissonant intervals with F.

Main Routine Continued. Once all the tests embodied in the melodic
and harmonic subroutines had been successfully passed, the main routine
was returned to in order that the notes now provisionally accepted could
be screened through tests expressing the rules related to combined interac-
tions between the voices. As shown in Figure 4, there were three tests
which had to be satisfied. The first, based on Rule 15, required that at
least one voice out of the four be required to move stepwise. This rule was
checked rather easily since all that was necessary was that at least one voice
be found for which \Nt — Nt-i\ = 1. As soon as this condition was found
to be satisfied, the test was concluded and the next operation started. This

was the process needed to eliminate 6 chords. The procedure used was

the following: The lowest note of the chord was first found by testing
whether Voice 1 might happen to have it by subtracting this note from the

EXPERIMENT AL DET AILS 107

notes of the chord in Voice 2, Voice 3, and Voice 4, in that order. This
process was kept up as long as the difference remained positive. If all
these differences did remain positive, it was evident that Voice 1 contained
the lowest note of the chord, and this part of the test was concluded. On
the other hand, if the difference in any of these subtractions turned out to
be negative, we knew that the other voice of the pair being examined con-
tained the lower note, and it was used to replace Voice 1 in subsequent
screening for the lowest note. This procedure was repeated until the lowest
note was found. The next step after this was to set up a check for whether
a difference of 3 or 10 existed between this lowest note and any other of

the three other notes. If not, then we knew that a 6 chord had not been

4

formed, and we were able to proceed to the next test, which was the check
for contrary motion in at least one voice whenever a unison, perfect fourth,
fifth, or octave formed by motion in the same direction had been detected
in the harmonic subroutine as described above. Two memory locations
were cleared to zero — one for positive increment and one for negative in-
crement for the four melodic intervals between the chord under examination
and the previous chord. The four melodic intervals were then computed
one at a time. If a given melodic interval was found to be zero, i.e., if it
was found to be a melodic repeat, nothing was entered into the two mem-
ory locations. On the other hand, if the interval was found to be positive
or negative, the appropriate one of the two aforementioned locations was
made nonzero. After all four intervals had been processed, both memory
locations were tested for nonzero value. The actual magnitudes contained
therein did not matter; the important thing to determine was whether only
one of the two locations was nonzero. If this condition was found, then no
contrary motion had occurred and a new chord had to be generated. On
the other hand, if both were nonzero, this showed that the requirement for
contrary motion had been satisfied, and we could then immediately proceed
to the next operation, which was the setting up of the tritone-resolution
process, described earlier, for chords containing a tritone interval.

Lastly, the two "Reset" operations, one for shifting to the next chord of
a given cantus firmus setting and the other for starting a new setting, were
both basically clearing operations for resetting indices back to original val-
ues. In addition, the "Reset: Shift to Next Chord" operation also carried
out the process of storing the chord formed into the computer memory until
the time when it was to be printed out or, alternatively, to be erased as a
consequence of the try-again subroutine.

108 EXPERIMENTAL MUSIC

Cadence Routine. As already noted in our discussion of Experiment
One, a routine had to be written for the cadence whenever chord n-2 was
reached, the length of the cantus firmus setting being set at n notes. More-
over, in this routine, as observed before, all the standard voice-leading and
harmonic rules still had to be checked. In addition, further restrictions
embodied in Rules 2, 3, 7, 11, and, in particular, Rule 12 had to be recog-
nized and properly tested for.

There were two basic procedures which were open to us. The first was
to write a complete new set of orders which would test for all the rules
applicable to the cadence as a self-contained closed subroutine, while the
second was to assemble a set of overwrite orders so that we might apply the
main routine for the major portions of the cantus firmus setting also to the
cadence. This latter procedure would require some additional testing pro-
cedures for the extra restrictions required by the cadence formula but
would eliminate duplication of various standard tests already discussed.
For this latter reason, this second procedure was adopted as the preferred
one.

In the cadence routine as written, the first step was to place the note
B (or B') in one of the four voices. Two operations were required here.
The first was to select either B or B' by two-choice random-integer genera-
tion, while the second was to determine by four-choice random-integer
generation into which of the four voices this note was to be inserted. The
operations required to check the rules of melodic writing contained in the
main routine were then called into play to determine whether the selected
note satisfied these rules. If so, the remaining notes of the chord were
selected by the ordinary note-selection processes already described and
screened for the usual restrictions. In actual fact, this eliminated all chords
but V, V6, III, III6, and VII6. In addition, a new test had also to be in-
cluded to eliminate doubling of the leading tone B, either at the unison or
octave. This test was carried out by means of simple subtractive tests of
the general type already described.

Once the leading tone chord at position n— 1 was obtained, the final
chord could be selected. Since this had to be a tonic chord in root position
just like the initial chord, the subroutine for selecting the initial chord could
readily be adapted to this purpose. It was modified, however, to ensure
that the leading tone progressed to C, and following this, the remaining
notes were generated until an acceptable combination was found. In so
doing, the restriction as to the repeat of the highest note of the highest voice

EXPERIMENTAL DETAILS 109

was relaxed to the extent that a repeat of C" occurring on this last chord
was permitted if the preceding note happened to have been B'.

The extensive set of rewrite orders required for this cadence subroutine
involved the adapting of test procedures from the other parts of the com-
puter program for Experiment Two. A complicated bookkeeping operation
was needed to set up the testing procedures for the cadence and to recon-
struct the tests in their original form for the next cantus firmus setting after
the cadence had been selected. The details of these instructions need not
be considered here since, although complex, they were entirely routine in
nature.

Printout. Printout instructions were activated in the program for Ex-
periment Two whenever a cantus firmus setting was completed. The setting
was read from the memory and printed by means of standard subroutines
available for the Illiac. After printing (actually, after output tape was
punched), the memory locations used for storage of the cantus firmus set-
ting were cleared, and the resetting of the machine for writing the next sam-
ple of counterpoint was initiated. A sample of computer output produced
by means of this program is shown in Figure 8.

07

08

05

06

09

09

12

11

09

09

11

09

09

06

07

04

05

02

03

06

05

05

02

09

09

13

12

09

09

12

12

11

12

14

11

11

07

11

12

06

05

09

08

11

12

07

06

07

Figure 8. Experiment Two: Sample of typical printed output in which some
counterpoint rules are in effect.

Varying the Number of Rules Used. As noted in the Outline of Experi-
ments and as shown in Table 4, musical output was generated by means
of this program to include examples of cantus firmus settings in which the
number of restrictions were varied from purely random white-note writing,
in which no rules whatsoever were applied, to the most restricted writing,
in which the full set of instructions was utilized. The various types of set-
tings between these two extremes produced in a series of computer runs are
tabulated in Table 4. Actually, the experiments were carried out in re-
verse order. The most complicated program, the one we have just de-
scribed, was written first. After this was in working order and producing
output, it was then a simple task to reduce its complexity by means of over-

110 EXPERIMENTAL MUSIC

write orders which inserted bypass, or "unconditional transfer orders," into
the program in front of the tests to be eliminated. When a sufficient num-
ber of these bypass orders had been inserted, the program was reduced to
the simple process of generating random white-note music.

In conclusion, it is of interest to note that this was the most complex
single music-generation program we have written so far for the Illiac. The
total number of individual arithmetical instructions required by this pro-
gram for writing strict counterpoint exceeded 1,900 individual operations.
In this respect, this program forms a considerable contrast to some of the
codes to be discussed in connection with Experiment Three and Experiment
Four, in which musical output of a less restricted nature was studied.

Experiment Three

Since the technical problems explored in this third section of work were
rather varied, it is convenient to consider each separately and then describe
how the various musical elements studied in this experiment were combined.

Rhythm. Rhythm was perhaps the most important musical element we
felt had to be treated if a fundamental compositional technique utilizing
computers was to be developed. Our objective in considering rhythm as a
musical entity to be treated by computer processing was in accord with a
recognition of this condition. It was our purpose to write a practical com-
puter program for generating rhythms so that a fundamental technique
might be demonstrated which in turn could form a basis for the further
elaboration of rhythmic devices in more complex contexts.

Just as the notes of the scale can be represented by numbers in the com-
puter, so also is it possible to symbolize rhythms numerically. In particu-
lar, the binary representation of numbers in the machine, consisting of
permutations of ones and zeros, offered a convenient set of symbols with
which we could operate. Thus, if we let 1 represent the "sounding," or
"strike," of a note and 0 a "rest," or, alternatively, the "hold" of a previ-
ously sounded note, then we can let a sequence of ones and zeros represent
a sequence of note values. For such sequences, ones and zeros, being in-
tegers, would each represent some arbitrarily chosen unit time interval,
such as a quarter note, eighth note, or some smaller value. We next ob-
serve that a rhythmic pattern is generated by the interaction of these note
values with meter, which is represented in turn by metrical patterns such as

% 1' 8' or 8* Tni,s? tne production of a rhythmic structure not only in-

EXPERIMENTAL DETAILS 111

volves the choice of note values, but also the choice between a repetitive
or a varying meter and, moreover, the selection of which particular meter
or meters to use. However, in order to keep the problem relatively simple
in these first experiments, we arbitrarily restricted our studies to a simple

metrical pattern, namely, g meter, consisting of four eighth-notes to the

measure. By further restricting the choice process to eighth notes as the
smallest possible time intervals, we could then utilize simple permutations

of four binary digits to represent the four beats of a measure in g time.

There are sixteen possible permutations of these four binary digits, per-
mutations which also happen to represent the decimal numbers 0 through

4
15. These could be used to represent the possible rhythms in a g measure,

as shown in Table 1 1 . Consequently, it was possible to propagate random
integers between 0 and 15 and to let the values of these integers set the
required rhythmic pattern for each measure. It might be noted at this
point that rhythm in other metrical systems can also be generated by en-
tirely similar processes. For triple meter, we generate random integers be-
tween 0 and 7, and for quintuple meter, random integers between 0 and 63.
Once the basic scheme for generating rhythms was set up, it was ob-
served that, like note propagation, rhythm also is normally subjected to
rules of composition and not permitted to occur entirely at random, ex-
cept for the extreme case of random music, for which an appropriate
random-integer table would have to be used. A table for this purpose
would be based upon a scale of integers sufficiently extended so that the
smallest useful time increment might be generated. It is to be noted that
our rhythm-generation scheme, even though it can hardly be considered
subject to much restriction, nevertheless represents a significant departure
from this most extreme situation since no time intervals smaller than
eighth notes were produced by it. Aside from this, however, if it were
used as is, it would still generate a new rhythmic pattern in each voice in
every measure produced. But rather than carry out this rather rudimentary
process, it seemed desirable to proceed to something somewhat more chal-
lenging and to consider factors which might yield some rhythmic order
over and above this relatively primitive situation. Since the simplest form
of rhythmic redundancy is literal repetition, the first step we utilized to
reduce further the randomness of the rhythm was a simple random repe-
tition scheme. In each voice, a rhythmic pattern was generated according

112

EXPERIMENTAL MUSIC

DECIMAL
NUMBER

Table 11
Basic Rhythmic Scheme for £ Meter

BINARY
NUMBER

RHYTHMS

CLOSED

OPEN

2
3
4

5

6

7

8

9

10

I I

12

13

14

15

0000

0001

0010

001 I

0100

0101

01 10

01 I I

1000

1001

1010

ION

MOO

1101

1 1 10

mi

-=JUb

-V7T3—

J—

n

JU J>~

J-^L

J «T3

JU^

J?J J)

m

rm

\ > J>

4 JLi

I jTJ

y $\

r i>> J>

y JD y

y rn

J^-X

fy y ^

JUlJLI

J>y n

X3-1

X3-i-jL

«m ?

j-jn

EXPERIMENTAL DETAILS 113

to the method outlined above and then a subsidiary random integer, which
was permitted in our particular experiment to have values between 1 and
12, was also generated. This subsidiary parameter was used to control
the number of measures a particular rhythmic pattern would be sustained
before a new rhythmic pattern was generated for the voice in question.
In this way, we obtained a variety of rhythmic patterns sustained for dif-
ferent lengths of time in the different voices. This was the first and simplest
of the rhythm codes we produced.

It is a common characteristic of music, however, that there exist cor-
relations of rhythms between the voices of a polyphonic piece of music.
In the above rhythmic code, this does not happen except by pure coinci-
dence. Therefore, the next step we considered was the introduction of
some vertical as well as horizontal rhythmic redundancy. In a second
rhythm code written for the Illiac, we included the generation of still a
second subsidiary random integer, this random integer taking values from
0 through 15. In this code, use was made once again of binary notation
as a type of musical symbolism. This time, in this new subsidiary random
number, ones were used to represent voices required to play the same
rhythm for the number of measures determined by the first subsidiary
random integer, the lower voices being used as the master voices, while
zeros were used to represent voices for which rhythms were generated in-
dependently. Thus, the representation 0000 meant that all voices would
have independently generated rhythms, so that vertical duplications would
occur only by coincidence. On the other hand, the representation 1111
was used to mean that all voices would have to play the same rhythm,
the rhythm being that generated for the lowest voice. In between these
two extremes, a representation such as 0101, for example, might be gen-
erated. This particular binary number indicated that Voices 2 and 4
would play the rhythm generated for Voice 2, while Voices 1 and 3 would
be rhythmically free. It should be noted, however, that this redundancy
scheme could not be rigidly applied because the integers 0001, 0010, 0100,
and 1000 were meaningless in this application, i.e., these integers would
have represented three voices free and one the same (as what?). Also,
we could not represent the cases of two different pairs being the same sep-
arately, i.e., 1-2, 3-4; 1-3, 2-4; and 1-4, 2-3. Three of the above mean-
ingless integers were therefore appropriated for these cases, leaving one
integer which was disregarded. Lastly, the complication which arose the
moment these two types of rhythmic redundancy were combined, namely,

114 EXPERIMENTAL MUSIC

which would take precedence, was settled by letting the vertical-combina-
tions rule take precedence.

The problem of notation of rhythm on the Teletype printout using
ordinary symbols was handled by means of a makeshift code, utilizing a
block of four digits to represent the rhythm in a given measure. We simply
employed the binary number notation shown in Table 11 and printed
twelve blocks, each consisting of four rows of four digits each, to represent
the rhythm in twelve measures of four-part writing. An example of this
will be illustrated in combination with other notation to be described.

This rhythm code was also arranged so that a new combination of
rhythms would be generated at the beginning of each new line. There was
no compelling reason for this; it was simply convenient to do so. This
might also be regarded as one way of blocking in an extremely elementary
longer-range structure for rhythmic patterns.

Dynamics. A second problem of considerable musical importance is the
setting of dynamics. In Experiments One and Two, during the transcrip-
tion of the experimental results to score form, an extremely simple scheme
involving a four-integer random-integer table was employed to assign
arbitrary dynamics marks to the individual voices (see Chapter 6). It
was decided, however, to effect some improvement over this elementary
technique, and not only place the generation of dynamics marks under the
control of a computer program, but also include crescendi and diminuendi
so that more than just terrace dynamics would occur.

This problem, in itself, is quite simple, and a set of operations similar
to the rhythm scheme just described was worked out. The dynamic range
was limited to pp, p, mp, mf, f, and ff, that is, to six possible values, so
that the first operation became the simple process of generating a random
integer considered equivalent to one of these dynamics marks. Secondly,
a change of dynamics was indicated by means of a second random integer
obtained from the possible choices 0, 1, or 2, which were considered
equivalent to diminuendo, no change, and crescendo. In choosing this
second random integer, restrictions had to be placed on the choice if pp
or ff, the limits of the range, had been chosen previously, since pp could
not be followed by diminuendo, and ff by crescendo. Therefore, a simple
screening operation was inserted at this point to eliminate these two un-
acceptable possibilities.

Thirdly, the duration of the dynamics indication had to be set. Since
this program was written in conjunction with the rhythm code just de-
scribed, techniques similar to those used for the rhythm code were em-

EXPERIMENTAL DETAILS 115

ployed. In the simpler and earlier of the two codes actually written, a
random integer which could take the values 0, 1, . . . , 12 was employed
to set the dynamics scheme in each voice independently. In the second
revised code, vertical correlation was used, just as in the rhythm code,
so that all possible combinations from complete independence to complete
agreement might be achieved.

The dynamics code was not correlated to the rhythm code. An entirely
independent set of parameters governed the two operations, even though
both were contained in the same computer program. Thus, the dynamic
markings in Experiment Three are at no time correlated to rhythmic pat-
terns or to changes in rhythm. Obviously, a correlation of one type or
another could have been set up had we so desired, but for the sake of
simplicity, this was not done.

Since letter notation is commonly employed to indicate dynamics, the
printing of dynamic indications offered little difficulty. The letters F, P,
and M were reserved for this purpose. To indicate crescendo, the symbol,
(, was appropriated; to indicate constant dynamic level, the symbol, =,
was used; while the symbol, ), was used for diminuendo.

Orchestration Index. This is a name we have applied to the playing
instructions which might be used to give color or timbre variation to the
sound of the music being produced. In the most general sense, this can
be thought of as the basis for a technique of instrumentation. In our ex-
periments, since we planned the output for string quartet performance and
thus had already made the choice of the particular instruments to be used,
this problem resolved to the question of the various ways of playing
stringed instruments. As is well known, there are numerous ways stringed
instruments can be played, including various manners of bowing, several
types of pizzicato, and a number of more sophisticated techniques, such
as col legno (bowing with the wooden part of the bow), sul ponticello
(playing on the bridge of the instrument), the playing of harmonics by
causing the strings to vibrate in sections, and so on. In order to prepare
a set of playing instructions, we selected sixteen of the most common
stringed-instrument playing techniques and tabulated these as shown in
Table 12. With the bowing instructions, we also included indications for
transcription purposes as to whether a zero in the rhythmic code was to
be observed as a sustained note or as a rest. These instructions for sus-
taining a note or for observing a rest were associated, as a general rule,
with the bowing instructions with which they were most conveniently
paired. Lastly, to allow for the fact that impossible combinations perform-

116 EXPERIMENTAL MUSIC

Table 12

Experiment Three

Orchestration Index

Random sexidecimal

Playing

If playing instructions

integer

instructions

are impossible, revert to

0

Bowed legato, held through
rests

F

1

Bowed detache, rests
observed

F

2

Bowed tremolo, hold
through rests

0

3

Bowed sul ponticello,
rests observed

1

4

Bowed, artificial harmonics,
hold through rests

0

5

Bowed col legno, rests
observed

1

6

Bowed sul tasto, hold
through rests

0

7

Bowed martellato, rests
observed

1

8

Bowed legato with mutes,
hold through rests

0

9

Bowed, whole tone shake,
rests observed

1

K

Bowed, glissando octave,
hold through rests

0

S

Bowed i—i

rests observed

1

N

Bowed v

rests observed

1

J

Snap pizzicato

F

Ordinary pizzicato

Rap on wooden body of
instrument with knuckles

ancewise might arise if certain playing instructions were juxtaposed, we
also permitted an alternative choice of simple legato or pizzicato to be
used during transcription, just in case a playing instruction should turn
out to be impossible. In actual fact, this was very seldom needed in tran-
scribing this music for the Illiac Suite.

Once this list was prepared, it was possible to associate a numerical
value with each instruction, and after so doing, we could use once again

EXPERIMENTAL DETAILS 117

the technique previously employed to establish rhythm and dynamics in-
dications. Thus, it was possible to generate random integers between the
values 0, 1, . . . , 15 in order to indicate particular playing instructions
for the various instruments. Moreover, as in the cases of rhythm and dy-
namics, it was also possible to establish how long and to which instruments
each playing instruction was to be assigned. Consequently, the same two
schemes for horizontal and vertical organization were incorporated into
this code as had been used for the rhythm code. In the earlier experi-
ments, a playing instruction was assigned to each instrument independently
for any random length up to twelve measures, after which time a new
playing instruction was generated for the voice in question. In the revised
code, as with the rhythm instructions, simple vertical redundancy was in-
troduced to tie the four voices together a little more closely.

Combination of Rhythm, Dynamics, and Playing Instructions. We have
already explained how the representation of rhythm on the Teletype print-
out was planned. In actual fact, instructions for controlling all three of
the above-described musical elements were written as one single computer
program so that rhythm, dynamics, and playing instructions were produced
successively. The printout for all three was arranged in the most compact
form for legibility, with the dynamics and playing instructions for each
voice being placed immediately under the rhythmic indication for the same
voice. An example of actual computer output produced by the more basic
of these two programs for these musical elements is shown in Figure 9,

0101

0101

0101

0101

0101

0101

6 FF=

6 FF=

6 FF=

6 FF=

6 FF=

6 FF=

0111

0111

0111

0111

0111

0111

S PP(

S MF(

S MF(

L MF(

7 MF(

L MF(

1010

1010

1010

1010

1010

1010

9 FF)

9 FF)

9 FF)

N F=

N F=

5 F=

1101

1101

1101

0111

0111

0111

J F)

8 F)

8 F)

8 P(

8 P(

8 P(

Figure 9. Experiment Three: Sample of computer output from the simplest
program for rhythm, dynamics, and playing instructions.

while in Figure 10, the transcription of these same results into score
form is shown.

118

VIOLIN I
VIOLIN II
VIOLA
CELLO

EXPERIMENTAL MUSIC

^H3

# sul

tr tr

| JTJ

# dim

snap pizz.

| J3*Jfc

f dim.

.JJ J

tasto

JZ3-

■mf

tr tr

j»J J>

y m

tr tr

>J jOJ ^

*

v y

;>J Jl^J JIL/TT^/TT^^JJ^,

J? cresc.

>^T3

* 4 *

martellato
V V

my

^J-^

i-CO-

col legno

etc.

etc.

etc.

etc.

■*■ rap on body of instrument

Figure 10. Experiment Three: Transcription of the computer output shown

in Figure 9.

Random Chromatic Music. The music shown in Figure 10 lacks any in-
dication of pitch; thus, the next logical step was to develop a program for
note selection which would differ from what was done previously with
strict counterpoint. For reasons already discussed, we decided to set up
a freer basis for note selection and to establish a minimum technique for
music writing which would be closer to contemporary practices than strict
counterpoint. The first step was to renumber the notes of the scale to
include all chromatic steps, black as well as white notes in terms of the
piano keyboard. Therefore, for a basic chromatic scale, we used the in-
tegers 0-30, which were considered to correspond to a range of two and
one-half octaves from C to F#", C being interpreted as the lowest play-
able C for each of the four instruments. Starting the range of notes with
C was convenient, since both the viola and cello have this as their lowest
note. The actual process of generating random music involving these
thirty-one notes was, of course, extremely simple and involved no more
than random-integer generation with the multiplier 3 1 , utilizing the process
already described in Chapter 4. At the same time, an improved printout
program was written to print the selected notes in letter rather than number
notation. In adapting the conventional Teletype machine to this purpose,
the letters A through G were used to indicate the notes themselves, the
letter X to indicate sharpened notes (F sharp being printed as FX, for
example), and primes to indicate the first and second octaves above the
lowest octave. For the sake of simplicity, notation for flattened notes was
eliminated. The printing of the results was matched to the rhythm code
described above, so that the notes and rhythms lined up in a simple four-
to-one correspondence. Thus, one line of notes became the equivalent of

EXPERIMENTAL DETAILS 119

three measures of rhythm. Actually, the two basic programs, the one for
rhythm and allied musical elements and the other for note selection, were
kept separate. This was done simply for convenience, so that each could
be revised and made more complex independently and with a minimum
amount of rewriting.

Simple Chromatic Music. It was of interest to carry out at least one ex-
periment with random chromatic music to illustrate how a degree of order
might be imposed upon this elementary material by simple means. There-
fore, in order to write a second type of chromatic music, four composi-
tional rules were imposed on the selection process. The particular rules
selected for this purpose were employed because it was felt that these
rules would impose a relatively high degree of order with a minimum of
actual instructions. The following rules were coded for the Illiac.

Rule 1. This was the melodic skip-stepwise rule already familiar in the
strict counterpoint code. This rule was used substantially unchanged ex-
cept for adjustments required in going from white-note to chromatic music.
In the present application, we permitted only melodic movements of a
whole tone or less — i.e., a whole tone or a half tone in either direction or
a repeated note — to follow a melodic motion greater than a whole tone.
On the other hand, any melodic movement obeying the rules in general
was permitted to follow a stepwise melodic movement of a whole tone
or less.

Rule 2. This was a melodic octave-range rule, also adapted from strict
counterpoint. The octave range was set to apply as far back as twenty-four
notes, i.e., six measures, maximum. After every three measures, corre-
sponding to one line of printout, had been completed, however, the back-
ward extent of scanning for the octave range was moved up three measures,
thus permitting the octave range to be changed in accord with the contents
of the last three measures only. This procedure permitted the formation
of a new octave range whenever these three last measures contained a
melodic substance spanning less than an octave. This octave-range rule,
moreover, was affected by the tritone-resolution process to be discussed
immediately below.

Rule 3. A rather complex way of resolving tritones was set up which
provided the only harmonic control over the musical material being gen-
erated. The rule is a simple example of how a reasonable compositional
rule can be devised for computer use which helps set up an appropriate
control process in a special situation.

Tritones can occur between the four voices in the following ways:

120 EXPERIMENT AL MUSIC

1. A single tritone (only two voices involved)

2. Two independent tritones (all four voices involved)

3. A single tritone with one repeated note (three voices involved)

4. A single tritone with a single note repeated twice (all four voices
involved)

5. A single tritone with both notes repeated once (all four voices
involved)

( Notes differing by a multiple of an octave were considered equivalent for
the purpose of this tritone-resolution rule.)

In the case of the tritones involving no repeated notes, specifically,
Cases 1 and 2, the resolution was required to take place by contrary half-
tone steps either inwards or outwards, depending upon the result of a
binary random choice. For Case 3, the resolution between the single note
and one of the two repeated notes was required to be the same as the
above, but the other repeated note was permitted to resolve contrary to
the first repeated note by a stepwise movement of zero to four half-tones
chosen at random. Case 4 was resolved the same way as Case 3, except
the fourth voice was permitted to move freely. Finally, Case 5 was re-
solved by requiring two stepwise motions and two motions of zero to four
half-tones chosen at random but in contrary motion to the stepwise move-
ments.

Rule 4. As a last rule, whenever the octave-range rule was violated by
a tritone resolution, the resolution was permitted, and the reference point
for the octave range was moved up timewise in the musical structure so
that the rule would again be satisfied; that is to say, a new octave range
was set by the tritone-resolution note. It should be noted that this fourth
rule is of considerable interest because it is a simple example of a built-in
rule-revision process. The applicable octave range was generated during
the course of the first three measures of a given melodic line. It then re-
stricted the range of the melody in accord with Rule 2 until a tritone
resolution occurred to force the computer to erase the octave-range limita-
tion in effect and set up a new octave range which had as one of its limits
the note farthest back in the melody which was an octave or less separated
from the note of the tritone resolution which brought about the revision
of the octave range. The operation of this process is shown in Figure 11.

In the computer, the tritone type could be determined after each chord
was generated by counting how many times the four voices entered into
tritones and then summing. This summing process is illustrated in Table
13. Only in Cases 2 and 3 was there any ambiguity resulting from this

EXPERIMENTAL DETAILS

121

FIRST OCTAVE
RANGE ( F#- F#')

THIRD OCTAVE
RANGE (B-Ab')
STILL INCOMPLETE

TRITONE
RESOLUTION

TRITONE
RESOLUTION

TRITONE
RESOLUTION

Figure 11. Experiment Three: Illustration of how the octave-range rule is
revised by tritone resolutions.

summation process, but this could be resolved by noting that one of the
counters was zero in Case 3.

The actual computing process for the application of these compositional
rules is shown in condensed form as the block diagram in Figure 12. It
is seen that the same general logical outline was required as that used for
the strict counterpoint routine previously discussed, and that the initial
step in planning the actual program for the Illiac was to decide upon an
efficient way of organizing the rules. After the initial entry, which, as in
earlier codes, involved the various routine operations required to get the
calculations started, the computation cycle was immediately entered at
the point at which tritone resolutions were carried out and the test for
skip-stepwise motion was made. Naturally, for the first chord, these rules
were bypassed, and a random note immediately generated for the first voice
and later, upon recycling, for the other three voices as well.

In general, the tritone-resolution operation was divided into three basic
operations. Further along in the process, it is observed that the new note
in each voice was tested for whether it had occurred in a vertical tritone

Table 13

Experiment Three

Detection of Tritone Combinations by Counting the Number

of Voices in Tritone Combination

Tritone

type

V1

v2

V3

V,

2Ft

Case 1

1100

1

1

0

0

2

Case 2

1111

1

1

1

1

4

Case 3

2110

2

1

1

0

4

Case 4

3111

3

1

1

1

6

Case 5

2222

2

2

2

2

8

122

EXPERIMENTAL MUSIC

relationship with any other voices already generated. If not, then the
recycling operation was carried out, unless it was the fourth voice being
produced, in which case the next part of the program was entered.

On the other hand, if a tritone was found, then a record was made of
the voices between which the interval occurred. This was done by adding
unit increments into assigned storage locations in the computer. Four
spaces in the computer memory were reserved for Voices 1 to 4, respec-
tively, as counting locations for tritone intervals. Two counters were
stepped up by one unit each by this operation, as shown in the block en-
titled "Step two counters" in Figure 12. An example of how this works
is shown in Table 14.

After the counting of all the tritones was completed, the second part
of the tritone-resolution process was carried out. This was the summation
operation already shown in Table 13. The counters in the four locations
were simply summed. If the sum %Vi was found to be zero, then we knew

*

YES
1

TRITONE RESOLUTION?

INITIAL ENTRY

-

1

♦ NO

YES

SKIP-STEPWISE MOTION?

■ .

"

♦NO

GENERATE RANDOM NOTE

*

COUNTER -VOICE 1

OCTAVE RANGE VIOLATED
BY SKIP-STEPWISE ROLE?

YES

TRY-AGAIN SUBROUTINE

f

♦ NO

COUNTER-VOICE 2

OCTAVE RANGE VIOLATED
BY TRITONE RESOLUTION ?

YES L

RESET OCTAVE RANGE

J*

♦ NO yp

// y

COUNTER-VOICE 3

SKIP-STEPWISE VIOLATED
BY TRITONE RESOLUTION 7

«•

j

♦no

COUNTER-VOICE 4

DETECT TRITONES?

YES

1

RECYCLE FOR ALL ■+ * d

VOICES /
i /

TRITONE- IIOO

♦

TRITONE-IIII

DID TRITONE OCCUR ?

YES

SWITCH FOR
DIFFERENT TRITONES

♦NO

TRIT0NE-2II0

PRINT?

YES

PRINT AND STORE
LAST 12 NOTES

♦ NO

TRIT0NE-3III

RESET: NEXT CHORD

\

TRITONE-2222

Figure 12. Experiment Three: Block diagram for chromatic writing.

EXPERIMENTAL DETAILS 123

Table 14

Experiment Three

An Example of the Operation of the Tritone Counter

Storage locations

Voice 1 Voice 2 Voice 3 Voice 4

Initially 0 0 0 0

Tritone detected between

Voices 1 and 3, add +1 0 +1 0

Result in counter +1 0 +1 0

Tritone detected between

Voices 1 and 4, add +1 0 0 +1

Result ~+2 0 +T +1

that no tritone had occurred. A bypass of the actual tritone-resolution
process was then inserted by placing an unconditional transfer order to the
skip-stepwise motion test in front of this resolution operation. On the
other hand, if this sum was found to be unequal to zero, an unconditional
transfer direct to the tritone-resolution routine was inserted immediately
before the routine. The particular tritone case was then detected by de-
termining the magnitude of the sum %Vh This was done by a simple
additive technique of the type already discussed in the descriptions of Ex-
periments One and Two. In addition, it should be noted that the counters
gave a record also of which particular voices were involved. This was
essential information, since a simple tritone of the form 1 100, for example,
can exist in a number of permutations, such as 1100, 1010, 1001, 0110,
0101, and 0011, considering the four digits to represent Voices 1 to 4,
respectively, in that order.

After detecting the tritone type by means of the sum %Vi9 and after a
small extra test in case this sum was found equal to 4, a switch was em-
ployed to direct the operations to the particular type of tritone resolution
required in accord with the rules given earlier. Cases 1 and 2 could be
treated by the same basic routine, Case 2 simply involving a repeat of
the operation, the operation itself being a simple random binary choice to
determine whether the resolution would be inwards or outwards, followed
by the actual recording of the required new notes. Cases 3 and 4 could
also be treated essentially as a pair, since in Case 4, the fourth voice was
allowed to move freely. In these cases, a random choice was first made of
which of the doubled voices would be required to move stepwise. Then,

124 EXPERIMENT AL MUSIC

a second random choice, this time a binary choice, was made to determine
the direction of the resolution. In Case 3, the remaining voice was then
required to move by contrary motion in accord with the rule described
above, while in Case 4, a further choice had to be made, namely, to decide
which of the remaining doubled voices would be required to move stepwise
and which would be permitted to move freely. Case 5, lastly, was a bit
more complex. Two random binary choices were first generated in order
to determine which voice in each of the two doubled pairs involved in the
tritone interval would be selected for stepwise resolution. Then, another
binary random choice was generated to determine whether the resolution
would occur inwards or outwards. Finally, the remaining voices were re-
quired to move in contrary motion to their doubled partners and, hence,
in respect to one another. In all five cases, the contrary-motion process
made use of a simple five-choice random-integer-choice operation followed
by the selection of the correct sign for contrary motion.

Between the test for tritones and the actual carrying out of the tritone
resolutions if tritones were found, it was necessary to insert two other
operations. As shown in Figure 12, the first of these was a set of printing
instructions which was activated by a counter after every twelve chords
had been formed. This operation caused the computer to punch tape which
would result in one line of printed output. Simultaneously, this completed
line of output was stored in the computer memory, overwriting the pre-
vious line of output, so that the octave-range rule could be applied back-
wards for three measures beyond the new line to be compared. In addi-
tion, independently of these printing instructions, counters and various
other routines were set so that the next chord would be generated.

The operation of the remaining sets of instructions is rather obvious in
the light of the detailed descriptions given of similar rules applied in Ex-
periments One and Two. The skip-stepwise-motion routine was set up
entirely similarly to the related strict counterpoint rule, with the necessary
modifications required to permit chromatic motion. The octave-range rule
likewise was adapted from the operations used for the strict counterpoint
program. The resetting of the octave-range by the tritone-resolution process
was extremely simple and involved merely the resetting of two storage
indices used for this computation. Lastly, the try-again subroutine was the
same as that used before, except for minor bookkeeping details.

Interval and Tone Rows. One simple way to initiate a study of the rela-
tionship between entropy and melody is to consider a twelve-note melody.
If we were to tabulate all possible twelve-note melodies, we would see that

EXPERIMENTAL DETAILS 125

the number of possible melodies could be expressed in terms of elementary
permutation theory. Thus, if we require only that the twelve notes be
selected from twelve possible tones with no restrictions in regard to re-
peats of tones being required or forbidden, we observe that we have defined
the condition of random music within this limited situation. If this con-
dition applies to all twelve notes, 1212 melodies are theoretically possible
— an enormously large number. This is a situation of maximum entropy,
or information, content in terms of the choice process, but it is not neces-
sarily the condition of maximum entropy content in terms of tonality. This
can be arrived at by a somewhat different procedure. Let us first note
that among all types of melodies in this random situation, the variety of
melodic types extends from the case in which all twelve tones of the
melodies are required to be the same to the case in which all twelve tones
are required to be different. At one end of this spectrum, where all the
tones are required to be the same, there exists a minimum of entropy —
i.e., a maximum of redundancy — because the total possible number of
melodies is just twelve if we permit full freedom of choice for the first
tone and just one if we restrict this choice to one specific pitch. This is
obviously totally established tonality — twelve repeats of the same tone.
At the other end of the spectrum, however, where the twelve tones are
all required to be different, each tone of the chromatic scale is sounded just
once in some specific order which may or may not be randomly produced.
This is, of course, the tone-row concept first significantly exploited by
Arnold Schonberg. The fact that the specification of a tone row is arrived
at by precisely the opposite technique as that of "ultimate tonality,"
namely, by requiring that the tones be different as contrasted to requiring
them to be the same, suggests the thought that a tone row is not just simply
an atonal device, but rather that it is a specific "antitonal" construction
which, within its own length, is deliberately devised to exclude the type of
redundancy required to set up tonality. Atonality, as a term, is therefore
better applied to the situation of random music rather than to construc-
tions such as tone rows. In this way, tone rows present rather interesting
musical properties, since they represent not only a highly restrictive and
easily managed technique for reducing the entropy content of random
music, but also an extreme condition in terms of elementary permutation
theory and a computational extreme for counteracting conventional tonal-
ity. It can be readily shown that the total number of possible tone rows
is 12!,4 which, although it is a relatively large number, is considerably less
4 12! = 12-1 1-10-9-8-7-6-5-4-3-2-1 = 479,001,600.

126

EXPERIMENTAL MUSIC

than the total possible number of twelve-note melodies. Moreover, in
everyday practice, the total possible number of such melodies is effectively
much less than this, since one type of redundancy commonly employed in
tone-row compositions is that based upon permutation techniques such as
inversion, retrogression, and the like. It is important to note also that trans-
position, of course, does not alter the basic character of tone rows any more
than it does other types of melodic lines. This is equivalent to saying that
there are really only 11! rather than 12! possible tone rows, since the par-
ticular pitch level at which the row is played is of secondary consequence
compared to the actual profile of the row. In view of what we said earlier
in Chapter 2, we may then infer that the melodic profile of a tone row de-
pends not upon a succession of the twelve tones of the chromatic scale as
such, as is often commonly supposed, but upon the mutual interrelation-
ships between the twelve notes in terms of intervals. Since in theory at
least all the twelve tones are presumed to be of equivalent importance, the
melodic profile is, therefore, a series of intervals between successive notes
in which any one or all of the twelve notes can be selected as a reference
point. Once a reference point is defined, it is observed that the tone row
consists actually of a series of all the possible melodic intervals considered
not in succession but in relation to this reference tone. To make this point
clear, we may arbitrarily choose the first note of a tone row as the reference
tone and then illustrate by an example. In Figure 13, a typical tone row is

If r t

r t t

Y T 't t 'T H1

MINOR SECOND

1 1 1 1 II

MAJOR SECOND

MINOR THIRD

l

1

MAJOR THIRD

i

PERFECT FOURTH

TRITONE

1 1 1

PFRFFCT FIFTH

MINOR SIXTH

MAJOR SIXTH

MINOR SEVENTH

MAJOR SEVENTH

Figure 13. Experiment Three: Illustration of how all the possible intervals
occur in a row of twelve different tones.

shown. It contains all the possible melodic intervals from a minor second
to a major seventh in relation to the first note of the row, C (or, for that
matter, in relation to any other note of the row). Moreover, each one of
the intervals, which we may define now as the difference /V, — Nu using the

EXPERIMENTAL DETAILS

127

notation adopted previously for notes and intervals, occurs once and only
once, but the intervals found between successive tones NL — /V,_i,
Ni+1 — Nh etc., may occur more than once. Thus, repeats of successive
melodic intervals are not forbidden.

On the other hand, we can also write rows in which repeats of successive
melodic intervals of the type Nt — Af4_i are forbidden, but repeats of ref-
erence intervals of the type Nt — N1 are permitted. This produces what
we can call an interval row as opposed to a tone row. The interval row
perhaps is a new type of melodic structure, which should present some in-
teresting musical properties, particularly in light of the experiments to be
described under Experiment Four, in which the properties of different
varieties of melodic intervals, both successive and long range, are examined
in greater detail.411 A typical interval row is shown in Figure 14. It should

-= \rxy- P=5=
I tl tl

fl_JLJLJL_Jl

Figure 14. Experiment Three: A typical interval row, not showing transposi-
tions to reduce span of row.

be noted that actual tones or their octave equivalents can be repeated in an
interval row in contrast to the tone row. It is not necessarily a situation of
greater redundancy than that of tone rows, however, since both permit the
same total number of possible choices. Redundancy in terms of tonality is
greater in the case of interval rows, but whether this is, in terms of musical
meaning, a situation of greater redundancy depends on whether successive
or referential intervals are to be considered the more significant in the con-

" It might be noted parenthetically that since working on this computer music, one
of the present authors (L. A. H.) has written an extended piano composition based
in part on this notion of an interval row.

128 EXPERIMENTAL MUSIC

struction of melodies. In any event, we may note that both tone rows and
interval rows have similar specific constraints imposed upon them in regard
to repetition of choices. In both choice processes, eleven choices of inter-
vals are permitted initially, but each time an interval is chosen, it is excluded
from the remaining choices to be made. This is analogous to the problem
of choosing colored balls from a box and not replacing them after each
choice, as opposed to a random succession which occurs when the balls are
replaced. It is by analysis of this process that the value of 11! is arrived at.
Within this particular choice process, we may also observe that the difference
between a tone row and an interval row is the following: A tone row is an
/th-order Markoff process, while an interval row is a ze roth-order Markoff
process in terms of how the intervals are evaluated. This difference is made
clear if the reference states in Figures 13 and 14 are compared.

Once we agree to the foregoing analysis of the properties of interval rows
and tone rows, it becomes simple to write a program for the computer for
generating these types of musical structures, of which the tone row, at least,
is at the present time a frequently used compositional device. The conclud-
ing part of Experiment Three, therefore, consisted of three related programs
for the Illiac for generating ( 1 ) interval rows of the type described above,
(2) tone rows, also of the type described above, and (3) tone rows of a
somewhat more restricted character. These last tone rows were written at
the suggestion of Robert Kelly, of the School of Music at the University of
Illinois, and are rather freely based upon a pedagogical approach to music
writing developed by this author for teaching counterpoint and harmony
from a contemporary viewpoint in beginning composition courses.5 The
method combines certain properties of tone rows with adaptations of a
number of the rules of strict counterpoint. The rules employed by Kelly
are given in the cited reference. Our adjustments of these rules for com-
puter processing are the following:

Rule 1 . The theme should consist of thirteen notes using all twelve tones,
with the first and last notes being C.

Rule 2. A span of a major tenth is set as the limit for the melodic range.

Rule 3. Progressions of major sevenths, minor sevenths, and tritones are
forbidden.

Rule 4. No tied-over notes are permitted. This rule was automatically
taken care of the way the tone-row generation scheme was set up (see
below ) .

5 R. Kelly, Theme and Variations, A Study of Linear Twelve-tone Composition,
Wm. C. Brown Co., Dubuque, Iowa, 1958, pp. 2-4.

EXPERIMENT AL DETAILS 129

Rule 5. Chromatic progressions in the same direction involving three
or more consecutive notes are forbidden.

Rule 6. The skip-stepwise rule is modified to permit two skips in the
same direction.

Rule 7. Consecutive sets of three notes that constitute a triad may occur
at random.

Rule 8. No tritone is permitted for N, — Ni-i unless N^i — /V,_2 =
±1, or, alternatively, 5 or 7 in the same direction as the tritone.

Rule 9. The use of sequences is prohibited. In other words, Nt — /Vi_i
cannot equal N*_2 — A^_8, and so on.

It can be seen immediately that a block diagram for a testing procedure
similar to those shown already for Experiments Two and Three could be
designed, and from this a testing program written which would include the
usual type of try-again subroutine. This testing procedure was written as
a subroutine which was grafted onto the basic interval and tone-row-genera-
tion program described below. Since this subroutine involved no basically
different programming techniques from those already described, it seems
unnecessary to consider its details, since these largely duplicate what we
have already discussed.

The Generation of Interval and Tone Rows. The production of interval
and tone rows was quite simple to carry out in the Illiac. The main routine
for this process was designed as follows: To make the note-selection proc-
ess efficient, a small table containing numerical representations from one to
twelve was stored in the computer. These were considered to be numerical
representations for all the possible intervals from a minor second to an
octave which simultaneously gave a count for each interval of the number
of half-tone steps within the interval. The unison or repeat (numerical
representation = 0) was omitted from this table, since it was decided to ap-
ply Kelly's procedure of writing cantus firmi of thirteen notes, of which the
first and, in the tone rows, the last would be set automatically as C. An
interval was then selected at random from the above-described interval
table and added to the interval or tone-row representation being formed to
obtain the first tone in the sequence of the interval or tone row. This inter-
val was then removed from the table, and all entries below it were moved
up one line and the process repeated. Thus, we started with twelve possi-
ble choices, then went successively to eleven choices, ten choices, and so
on, until we were left with only one choice for the final interval. When an
interval or tone row was completed, it was then printed out and the com-
puter reset to perform the operation all over again.

130 EXPERIMENT AL MUSIC

The way in which the intervals being selected at random were added to
the row being formed determined whether or not an interval or tone row
was formed. In view of our earlier discussion, the difference is seen to be
one of minor coding changes. To produce an interval row, the numerical
representation of the interval just chosen was added to the previous note
in the row to produce a new note in the row separated from the previous
note by the interval in question. However, since we always added new in-
tervals to a row to give upward progressions, it became necessary to em-
ploy also an octave-subtraction process, because otherwise the span of the
row would soon exceed the set limit of two and one-half octaves of the
chromatic scale. This is in accord with normal practice, however, for sub-
tracting an octave from a note does not disturb its primary functional rela-
tionship to a melodic reference point such as the first note of the melodic
line. What we actually did in the generation of interval rows was to make
a binary choice which determined whether we would simply add the in-
terval to the previous note and, therefore, move upwards in pitch or, alter-
natively, add the note and then subtract one octave to produce a note lower
than the last. Secondly, it was also necessary to test for whether the limits
of the chromatic scale being used were being exceeded in spite of this.
Therefore, a test was also made to determine whether this was so, and if
necessary, and adjustment of the movement in the proper direction, either
upwards or downwards, as the case might be, was then made. Finally, it
was observed that an interval row starting on C always of necessity ends on
F sharp.

To produce tone rows rather then interval rows, the interval randomly
selected from the table was simply added to the first note C of the row,
rather than to the previous note in the row. This automatically produced a
tone row rather than an interval row. Otherwise, the whole process was
kept exactly the same, except that the octave was removed from the table of
intervals, since the choice of this interval would lead to the repeat of a
tone (or its octave equivalent) within a tone row. This would be a viola-
tion of the requirements for such a row. In generating simple tone rows of
thirteen notes according to the chosen plan but with no restrictions, we
simply arbitrarily made this the last interval to be selected, so that the row
this time would of necessity have C or its octave equivalent as its final note.
In the case of the restricted tone rows, in addition to writing a subroutine
which would apply the rules given above, we could simply eliminate the
octave from consideration as a possible interval.

Tone- row Permutations. A basic procedure in writing tone-row music is

EXPERIMENTAL DETAILS 131

to perform geometrical permutations upon the row being utilized to com-
pose a piece of music. There are many possible geometrical operations
upon tone rows which can be used, but of these the three simplest and
most frequently used are inversion, i.e., writing each interval downwards
instead of upwards, retrogression, i.e., writing the row backwards, and
lastly, inversion of the retrograde form of the row.6- 7 In the computer,
these geometrical permutations were most easily carried out as part of the
printout procedure. In this way, they are handled rather analogously to
the process of transposition which we shall consider under Experiment
Four. We arranged the printout to contain in a block of four lines: ( 1 ) the
row itself, (2) its inversion, (3) the retrograde form, and (4) the retro-
grade inversion, each printed one under the other. The printing of these
four forms of a row was just a matter of format arrangement for the print-
out routine. The row itself was first printed a symbol at a time, each nu-
merical representation of a note being converted into its letter equivalent.
Secondly, the inversion of the row was formed by changing the signs of
the successive intervals in the row and then printing out the letter equiva-
lents of the notes of the inverted row. Lastly, the two retrograde forms were
produced by printing out these two forms of the row again, but selecting
the notes in reverse order.

Experiment Four

The objectives of Experiment Four differed from those of previous ex-
periments, since it was our purpose in this new experiment to generate
samples of music based upon certain more abstract formulations which we
believed might eventually be related to structural devices in musical com-
position such as tonality and melodic profile. In the first three experiments,

6 Strictly speaking, one should differentiate between retrograde inversions and in-
verted retrograde rows, since these two variations, although they have the same pro-
file, may not occur at the same pitch, depending upon the last note in a row. Thus,
a tone row beginning on C and ending on E, for example, will have a retrograde
inversion beginning on A flat and ending on C, while the inverted retrograde form of
the row will start on E and end on G sharp. The recognition of this difference, of
course, produces a modulatory technique in tone-row writing. In our samples, how-
ever, which begin and end on C, this problem did not arise because of symmetry
considerations. This also happens to be true for the interval rows as well, since these
must of necessity end on F sharp, which is likewise a situation of symmetry.

7 It is perhaps also desirable to note, in passing, that these basic techniques are, of
course, by no means unique to row writing. As is well known to musicians, all these,
as well as many other geometrical operations, are common procedures in many
types of well-established musical forms such as, for example, in canons and fugues.

132

EXPERIMENTAL MUSIC

and particularly the first two, we were primarily concerned with the genera-
tion of music recognizable in many respects as falling within the traditional
framework of compositional procedures. In Experiment Four, this was no
longer intended to be the case. In this last set of experiments, simpler and
perhaps more fundamental means of musical construction were investigated
than those studied previously. This was done in an attempt to find more
inclusive concepts to work with, and, in particular, concepts which might be
thought of as geometrical analogs of musical form.

The fundamental geometrical picture selected was an abstraction of the
calculating technique used in the three previous experiments. The genera-
tion of four-part musical structures was pictured as a restricted random-
flight problem in which four trajectories are traced simultaneously upon the
rather unusual coordinate system of pitch versus time. This geometrical
structure, illustrated in Figure 15, can be subjected to mathematical defini-

ETC.

ETC.

ETC.

ETC.

TIME

Figure 15. Experiment Four: A four- voiced musical
texture pictured as a random-flight problem.

tion, as we have already indicated in Chapters 2, 3, and 4. It is particularly
characterized in terms of Markoff chain processes, i.e., sequences of events
in which the choice of each new event can be made dependent upon pre-

EXPERIME NT AL DET AILS 133

vious events; or, in musical terms, the choice of each new note or interval
in a given melodic line can be made dependent upon previous notes or
intervals in the same melodic line. Utilizing this simple picture, we wrote
computer programs for generating a series of samples of what we may call
Markoff chain music.

Harmonic and Proximity Functions in Melodic Construction. If we recall
the dependences of melodic construction and tonality upon both the suc-
cessive and long-range intervallic relationships discussed in Chapter 2, we
may now note that successive melodic intervals bear three significant char-
acteristics which may be used to impart formal structure to a melody: (1)
Melodic intervals can be related to the overtone series so that we recognize
a harmonic function in melodic construction. Thus, the outlining of a
triad, C-E-G for example, imparts a sense of C major, or, at least, of the
C-major chord, and a sense of relatively high tonal order. On the other
hand, a sequence such as F-A flat-D is more ambiguous and conveys a
number of possible implications in terms of structural function, other fac-
tors being equal. (2) Melodic intervals may also be characterized by their
absolute size. Thus, stepwise melodic progressions such as half-tone or
whole-tone intervals seem to impart more order than larger skips such as
sevenths, octaves, and larger intervals. (3) We may note that many well-
constructed melodies consist of sequences of intervals which balance the
tendencies to order and disorder by balancing harmonic clarity with am-
biguity, and stepwise melodic movements with larger skips. Even in music
in which these principles seem to operate minimally, as in certain types of
contemporary music, the presence of both functions, which we may now
call harmonic and proximity functions, can be traced. In a way, this can be
said to be true even indirectly of the steps resorted to by composers in
negating tonality by devices such as twelve-tone rows.

Relation to Musical Structure. These two functions can also be examined
somewhat more broadly than just in terms of relatively short-range melodic
relationships. In terms of more recent musical styles, these two functions
of harmonic and intervallic structure also form the basis of many of the
more searching analyses of musical forms such as, to cite two representa-
tive and well-known authors, those by Hindemith8 and by Schenker,9 whose

8 P. Hindemith, Craft of Musical Composition — /. Theory, rev. ed. (trans, by A.
Mendel), Associated Music Publishers, Inc., New York, 1945.

9 H. Schenker, Neue Musikalische Theorien und Phantasien: vol. I, Harmonielehre,
1906, vol. II, Contrapunkt, 1910 and 1920, vol. Ill, Die Freie Satz, 1935, Universal
Edition, Vienna.

134 EXPERIMENTAL MUSIC

concepts have been presented in English by Katz,10 Salzer,11 and Jonas
and Borgese.1-

Both Hindemith and Schenker distinguish the harmonic and melodic
element, and both acknowledge the dependence of the harmonic elements
upon recognizable relationships to the harmonic series and of the melo-
dic element upon stepwise motions. Thus, Hindemith stresses the impor-
tance of "root-progressions," of which the strongest harmonically are shifts,
such as fifths, fourths, and thirds, in giving convincing harmonic movement
to musical passages.13 He also remarks that a smooth and convincing
melodic outline is achieved only when these important points form pro-
gressions in seconds.14

However, Hindemith confines himself almost wholly to successive mu-
sical relationships, leaving the question of larger structures relatively un-
explored; therefore, we might also consider Schenker's ideas, which are
somewhat more ambitious in scope. He suggested that elements of a musi-
cal structure could be compared to a series of structural layers, starting
from a "foreground" which is the complete work as we hear it to a "back-
ground" structure in which only a skeletal framework of the music is left.
In analyzing a piece of music, Schenker proposed that these layers can be
successively stripped away to reveal the basic framework, which he called
the ursatz and which he thought consisted of two basic elements. The first
we can interpret as the essential harmonic root movement of a whole struc-
tural unit, such as a song or sonata-allegro form. He postulated that all
satisfactory musical forms, even lengthy ones, could be simplified to one of
several simple root progressions, such as I-III-V-I and I-IV-V-I, and
was able to build quite a good argument for this idea by utilizing exclusively
examples from the works of the master composers of the eighteenth and
nineteenth centuries, the only repertory he considered admissible. The
second element of the ursatz proposed by Schenker is the urlinie. This is
a significant essential descending melodic line contained in every "accept-
able" piece of music. The urlinie is always basically simple, being a se-
quence such as, in C, G-F-E-D-C. Complex musical structures are then
built up by elaboration upon these two elements of the simple skeletal

10 A. Katz, Challenge to Musical Tradition, Alfred A. Knopf, Inc., New York, 1945.

11 F. Salzer, Structural Hearing; Tonal Coherence in Music, Albert and Charles
Boni, Inc., New York, 1952.

12 H. Schenker, Harmony, from Harmonielehre, ref. 9 above (ed. by O. Jonas and
trans, by E. M. Borgese), University of Chicago Press, Chicago, 1954.

13 Hindemith, op. cit., chap. IV in particular.
"Ibid., p. 193.

EXPERIMENT AL DETAILS 135

framework. The principles of the theory are supposed to be in operation
within sections of a piece of music in a subsidiary role as well as over the
whole musical structure. Many examples of how the Schenker system of
analysis can be applied are given in Salzer's book; it is to be noted that
Salzer goes to considerable effort to circumvent Schenker's dogmatic atti-
tudes regarding the absolute superiority of Germanic music from 1700 to
1900, which has marred the impression of his work in the eyes of some
critics.15 It is not our purpose to become involved in detailed criticisms of
methods of musical analysis, but only to extract from them several ideas
which were useful in planning the experiments to be described and which
would be of paramount importance in any future experiments in which
more elaborate musical structure might be produced. The first is the con-
cept of structural frameworks, both harmonic and melodic. This concept
is useful because it provides a differentiation between structural points of
greater or lesser significance. A working technique for structural assembly
is thus provided that permits the assigning of a greater or lesser value to
significant musical elements, particularly in relation to the time scale of the
musical structure. It is observed that this factor is almost entirely missing
in the computer music produced in the first three experiments.

The second basic point is the emphasis on a distinct harmonic element
and a distinct melodic element in the basic structure. This does seem to be
in general accord with our definitions of a harmonic and a proximity func-
tion, and suggests a method of elaboration of the techniques shortly to be
described under Experiment Four for the construction of more complex
musical structures. The particular emphasis by analysts such as Schenker
on the practices of the eighteenth and nineteenth centuries, which estab-
lished for them a standard for clarity of tonality, need not be mistaken for
an absolute necessity. If we consider musical structures antecedent to 1700
as a source of ideas, we can observe that this music was written before
the concept of tonality was clearly defined. Thus, in view of the emphasis
on contrapuntal rules rather than harmony in this earlier music, the prox-
imity-type function might be found to play a more preponderant role here.
Moreover, closeness of intervals seems to have been used as a guiding prin-
ciple, not only for simple step-by-step techniques, such as the skip-stepwise
rule of counterpoint, but also implicitly as an important structural device in,

15 M. Mann, "Schenker's Contribution to Music Theory," Music Review, 10:3-26,
1949; R. Sessions, "Heinrich Schenker's Contribution," Modern Music, 12:170-178,
1935; R. Sessions, "Escape by Theory," Modern Music, 15:192-197, 1938. This last
article in particular is a highly critical review of Schenker's Die Freie Satz.

136 EXPERIMENT AL MUSIC

for example, much primitive music, non- Western music, and in older West-
ern art music such as plain chant. Coherence is not infrequently brought
about by melismatic weaving about certain structurally important melodic
notes, or even around one fixed tonal center. This might also be true in
certain types of contemporary music as well. Thus, Hindemith,8 for ex-
ample, permits in his system of composition a structural dependence on
proximity, or neighbor-note relationships, as well as on strongly harmonic
relationships such as the fifth, fourth, and third.

Table of Weighted Probabilities. In order to institute an analytical study
of these processes, we first defined a harmonic function x, and a proximity
function y§. Then, to investigate how these functions might operate to pro-
duce melodic structures, a series of experiments was carried out in which
Markoff chain processes were used as a basic computing technique. The
first requirement was a reference table of weighted transition probabilities
stored in the computer, to be utilized as required. It should be noted that,
in essence, this had been done all along in previous experiments, but in
these earlier experiments, the transition probabilities were usually either
simple random-choice situations, i.e., all choices were weighted equally, or
else some of the choices were simply forbidden and thus given zero weight.
The counterpoint rules, for example, often amount to no more than just
this. The second requirement we imposed was that this table must contain
as few bits of basic information as possible, but that this information be of
as fundamental significance as possible. In this way, we hoped to investigate
how to avoid the problem of high-capacity information storage, which cur-
rently presents such a major difficulty in many other data-processing prob-
lems such as language translation by machine. Therefore, a table was con-
structed for arranging all successive melodic intervals from the unison
(melodically, the repeat) to the octave. This is shown in Table 15, where
the various possible melodic intervals are listed first in descending order
of consonance from the repeat to the tritone. For the purpose of machine
calculations, we associated with each interval a value of a stochastic vari-
able Vj, which runs from 0 for the unison, or repeat, to 12 for the octave.
These are the values we could add or subtract to each tone of a melody
to obtain the next tone of the melody, which resulted as a consequence of
moving through the interval represented by the variable. A melody could,
therefore, be symbolized by a sequence of values v, = Vi, v2, . . . , v„,
added successively to the first note of the melody. In the present work, the
note C was normally selected as an arbitrary starting point. Once again,
as in previous studies such as that of interval and tone rows, intervals,

EXPERIMENTAL DETAILS

137

Table 15
Table of Functions for the Generation of
Markoff Chain Music in Experiment Four

Combined

Stochastic

Harmonic

Proximity

function

Interval

variable Vj

function Xj

function y}

zj —Xj + yj

Unison

0

13

13

26

Octave

12

12

1

13

Fifth

7

11

6

17

Fourth

5

10

8

18

Major third

4

9

9

18

Minor sixth

8

8

5

13

Minor third

3

7

10

17

Major sixth

9

6

4

10

Major second

2

5

11

16

Minor seventh

10

4

3

7

Minor second

1

3

12

15

Major seventh

11

2

2

4

Tritone

6

1

7

8

Y;t, = 91 Vyj = 91 Yz,- = 182 = 2 X 91

/=0 /=0 j=0

[x} = x(vj)] [yj = v(v;)]

rather than tones, were used as the determining functions for assembling
melodic structures.

As shown in the second column of Table 15, the harmonic function Xj
was given values which run arithmetically from 1 3 to 1 in descending order
of consonance. This simple arithmetic ordering was defined as an unper-
turbed set of weighted probabilities for the harmonic function. If these
values are used directly for interval selection, this brings about, on the av-
erage, the selection of repeats thirteen times as frequently as the tritone, of
octaves twelve times as frequently as the tritone, and so on. It was be-
lieved that this simple arithmetic scale of relative weights for the harmonic
function would be an adequate representation of a neutral or mean position
in terms of the imposition of tonal order. It is upon these values that we
had to operate in order to perturb the mean harmonic texture, and thus
achieve a higher or lower average degree of tonality.

In the third column, a similar set of values is tabulated for the proximity
function y,. It is seen that these values run arithmetically from 13 down to
1, as do the values for xh but that the order of the values relative to the

138 EXPERIMENT AL MUSIC

stochastic variable v_, has been changed, so that the highest unperturbed
weight is assigned to the repeat and the next highest weights successively
to the minor second, the major second, and so on.

In the last column, values are tabulated for a third function, Zj, which is
simply the sum x, + yj. This combined function was used in certain ex-
periments to show how a simple combination of the two individual func-
tions might be carried out. It is interesting to note that the two functions
combined in this way yielded a scale of weighted probabilities that is, per-
haps, not too different from conventional melodic writing. A statistical
count of melodic intervals in appropriate examples of music might be of
interest compared to this column.

Lastly, at the bottom of Table 15, there are given values for the sum of
the weights for each function. These values were required for computer
calculations to determine the frequency with which each interval occurred
on the average. Thus, if x, only was used to generate music, the relative

frequency with which major thirds turned up, for example, was

12
xjSxj = 9/9\ =0.1

7=0

or approximately ten per cent of the time.

Generation of Integer Weights. The values given in Table 13 were en-
tered into the computer to constitute a set of weighted probabilities stored
in the memory. As shown in Table 16, in which the information given in

Table 16
Assignment of Integer Weights to a Stochastic Variable

Stochastic variable, v} Integer weight, w}

0 Wo

1 Wi

2 w2

j Wi

/= 12 in the calculations of Experiment Four

Table 15 is generalized, we could then assign any desired weights, w0, wl9
w2, . . . , M>i2, to the possible intervals from unison to tritone. Having
done this, we could then determine the sum of these weights, i.e.,

12

^Wj=W (15)

EXPERIMENTAL DETAILS 139

and generate a random integer R within the limits 0 == R == W — 1 , as de-
scribed previously under random-integer generation. The value — R was
then obtained and the weights Wj were added successively to this quantity
in sequence until the sum

R

R + £ w,

7=0

became positive. At this point, we took the value of the stochastic
variable associated with the last weight added as the desired one. By this
scheme, it is evident that the probability for obtaining Wj is Wj/W.

Changing the Weights of Transition Probabilities. The texture of music
being generated by means of transition probabilities can be varied by
changing the magnitudes of these probabilities in relation to one another.
Thus, the "reference," or unperturbed, probabilities shown in Table 15
were operated upon in several different ways in Experiment Four, in order
to see how changes in these weights would be reflected in the musical out-
put. In practice, given values such as those shown in Table 15, there were
essentially only two direct operations which could be effected, namely,
making the weights more equal or, conversely, more unequal. Taking the
harmonic function x$ as an example, it was possible to subtract unity suc-
cessively from the unperturbed probabilities to remove the less heavily
weighted probabilities one by one, the operation upon each Xj being termi-
nated whenever its value became zero. In this way, we could gradually re-
strict the successive steps in a melody to the consonant intervals and eventu-
ally to just the octave and repeat, at which point fixation upon one note in
the scale and, hence, upon a certain tonality was achieved. This technique
could, therefore, be used to generate a type of harmonic drive toward a
tonal center, while the reverse process is used to move away from a highly
organized tonal center toward a less well-defined tonal organization. Nat-
urally, it would also be possible to subtract or add larger increments than
unity for more sudden shifts in musical texture.

A shift toward random music, in which all intervals are equally probable,
was also easily achieved. This was accomplished by adding unity succes-
sively to the unperturbed transition probabilities until each one achieved
the maximum value, namely, thirteen, at which point the value was fixed.
All values for the weights, therefore, tended toward equality. Again, it
would be equally possible to reverse the process and make the weights
successively less equally weighted.

Both these operations were actually carried out, and the results thus
produced were utilized to form the first two sections of Experiment Four.

140

EXPERIMENTAL MUSIC

Moreover, some additional experiments of this type, not included in the
llliac Suite, were also completed. It was also possible to subtract integer
values from equally weighted transition probabilities — those for random
music each having a value of thirteen — in reverse order, thus reducing the
value for the melodic repeat first and the tritone last. In this way, we were
able to obtain a set of inverse transition probabilities for Xj which produced
music in which tritones occurred thirteen times as frequently as melodic
repeats. This process was continued until all the intervals except the tri-
tone were eliminated. This last was a species of music as highly restrictive
as that which consists only of repeated notes. Samples of this output were
generated by means of the llliac, but were not included in the llliac Suite
because the previous results illustrated sufficiently well the general tech-
niques of altering the weights of transition probabilities. The various opera-
tions described for altering transition probabilities are shown in Table 17.

Table 17

Changing the Weights of Transition Probabilities,

Utilizing the Harmonic Function, xj,

as an Example

Addition

*-

Subtraction

Repeat
Octave
Fifth
Fourth
Major third
Minor sixth
Minor third
Major sixth
Major second
Minor seventh
Minor second
Major seventh
Tritone

t

Fixed
tonality

4..
3..
2..
1..
0..
0..
0..
0..
0..
0..
0..
0..
0..

11
10
9
8
7
6
5
4
3
2
1
0
0

Addition

*■

Xj subtraction

=•*

13 13..

13 13..

12 13..

12 12..

10 11..

9 10..

8 9..

8..
7..
6..
5..
4..
3..

Unperturbed
distribution

13
13
13
13
13
13
13
13
13
13
13
13
12

Inverse subtraction
Inverse addition

12...
13...
13...
13...
13...
13...
13...
13...
13...
13...
13...
13...
13...

Random
music

0 0

1 0

11 10

12 11

..0

0

..0

0

..0

0

..0

0

..0

0

..0

0

..0

0

..0

0

..0

0

..0

0

..0

0

..1

0

..2

1

Inverse
unperturbed
distribution

Tritone
music

Obviously, the same types of operations can also be carried out on the
proximity function yh or for that matter on any other arbitrary selection of
transition probabilities.

EXPERIMENTAL DETAILS 141

Zeroth-order Markoff Chain Music. In a monodic pattern, if successive
notes are isolated and considered as such, it is possible to write music in
which the choice of Nn is entirely independent of its relationship to note
N„_i. As we have seen, the mathematician calls a structure of this general
type a zeroth-order Markoff chain — in particular species of random flight
in which each successive event is independent of its relationship to the
immediately preceding event. This would be zeroth-order Markoff chain
music with respect to notes. A primitive type of relationship between notes,
however, will be defined if we write zeroth-order Markoff chain music with
respect to successive melodic intervals rather than notes, i.e., if we make
In independent of In-i> Since we have already indicated in Chapter 2 that
we prefer to think of melodies in terms of successive intervals rather than
notes, let us define this as zeroth-order melodic writing. In zeroth-order
melodic writing, then, a priori transition probabilities, such as those given
in Table 13, are defined to permit moves from one note to the next and to
permit the generation of sequences of melodic intervals. As we have al-
ready noted, the values assigned these transition probabilities establish
the character of the music, but except for the unique case already consid-
ered of assigning a transition probability of unity to the unison (i.e., the
repeat melodically) — or its harmonic equivalent, the octave — and zero to
all other intervals to give rise to patterns of repeated notes, such music is
intrinsically atonal, since there is nothing in the nature of zeroth-order
transition probabilities to establish specific tonal centers. In zeroth-order
Markoff chain music, a melody starting on C, for example, can move any-
where conditioned only by transition probabilities, and if such a melody ar-
rives, for example, on F sharp, there is no compulsion to return to C or to
any other note. Moreover, even if we weight most heavily the probabilities
for consonant leaps, like rising and falling fifths, all that happens is that
the apparent harmonic movement becomes slower. Conversely, the more
we weight the more dissonant intervals, the more rapidly we tend to ex-
perience oscillations between more distantly related tonal centers. Thus,
tonality as such does not exist in such a texture except as the result of
chance. Zeroth-order Markoff chain music is the simplest type of order
placed upon random-note music, but it still contains within its species all
types of music from tonal through atonal and finally to antitonal music,
such as certain types of tone-row composition. All such more highly or-
dered systems therefore arise solely as the result of chance events and not
as a result of design. We can also conclude that the consonance or dis-
sonance of successive melodic intervals depends on the average upon how

142 EXPERIMENTAL MUSIC

transition probabilities are weighted. Ze roth-order Markoff chain music
can appear highly consonant or highly dissonant or any mixture in between.
Thus, the simplest order is imposed upon random-note music by weighting
certain zeroth-order transition probabilities more heavily than others. In
terms of information theory, random-note music is characterized by maxi-
mum entropy content, while the weighting of transition probabilities in any
direction, as in Table 17, for example, decreases entropy by increasing
redundancy. However, it should also be noted that the degree of con-
sonance or dissonance has nothing to do with tonality as such, since the es-
sential characteristic of tonality, in view of our comments in Chapter 2,
is the recall of events before note Nn-i.

It is interesting to observe that a great number of the various composi-
tional rules of traditional counterpoint and harmony are concerned with
zeroth-order Markoff chain effects. Many of the rules of first-species strict
counterpoint used to generate the music in Experiments One and Two illus-
trate this very nicely, particularly the rules forbidding certain types of
melodic intervals such as sevenths and tritones. More generally, many rules
of conventional harmony are also of this type, for example, the rules given
for the resolution of dissonant chords to certain choices among consonant
chords or possibly other dissonant chords. It is important to note that in
so far as they are usually given, these rules are stated without reference to
musical context, this being considered a separate problem.

In Experiment Four, three samples of zeroth-order melodic writing were
produced to illustrate this simple type of musical organization. The three
unperturbed transition-probability functions shown in Table 15 were used
directly for this purpose; thus, the probabilities w^ for selecting interval
1 1 were simply set equal for all values of / to the values x, of the harmonic
function, y} of the proximity function, and Zj of the combined functions,
respectively, in three successive studies.

First-order Markoff Chain Music. The simplest way to improve upon the
structural concept of zerotn-order Markoff chain music is to make the
choices of new melodic intervals dependent on what has just immediately
happened, that is, to make the choice of interval h dependent upon the
choice of interval 7,_i. This is an example of a first-order Markoff process,
and music written according to this principle may be termed first-order
Markoff chain music. In conventional music, there are examples of first-
order Markoff processes among composition rules, just as there are of
zeroth-order rules. Among the rules of counterpoint, the skip-stepwise rule
and the forbidding of certain types of parallel motion can be cited.

EXPERIMENTAL DEI AILS 143

Given the functions in Table 15, the question arose as to how we might
use these functions to produce some simple examples of first-order Markoff
chain music. From among the many possibilities, the following first-order
process was selected: A rule was employed which was a generalization of
the skip-stepwise rule previously used. This rule required that the choice
of the new interval U would be weighted most heavily against the particular
interval previously selected, /,_i, and most heavily in favor of the interval
most different from interval /*_!. Thus, for example, if the previous inter-
val //_i had been a melodic repeat, the new set of probabilities for interval
1 1 would be weighted more heavily against the choice again of a repeat than
if the previous interval had been a major third or some other intermediate
interval.

Addition is the simplest process for carrying out this type of conditional
first-order weighting, utilizing the basic functions in Table 15. It was pos-
sible simply to add the weight of the interval /,_i to all the weights of the
interval to be chosen, to obtain new weights for the actual choice process.
In actual practice, the weights w(_i — 1 rather than wt^x were added to Wj.
Examples of this process are shown in Table 18, utilizing the harmonic
function, x,. It is seen that the probability for a repeat following a repeat
was much less than for a repeat following a tritone. Conversely, the prob-
ability of a tritone following a tritone was much less than a tritone following
a repeat. In general, therefore, with this treatment of the harmonic func-
tion, dissonant melodic intervals tended on the average to follow consonant
melodic intervals, and, conversely, consonant melodic intervals tended to
follow dissonant ones. A similar treatment of the proximity function yj
would have the effect of causing small intervals to tend to follow large in-
tervals and vice versa, thus providing a generalized sort of skip-stepwise
rule.

As in the case of zeroth-order Markoff chain music, three samples of
first-order Markoff chain music were generated, again illustrating (1) the
effect of the harmonic function alone; (2) the proximity function alone;
and (3) the combined function. This function wih therefore, was set equal
in the three cases to xt + x(_i — 1, ^ + y^ — 1, and Zi + Zi-i — 1, re-
spectively, according to the calculation pattern illustrated in Table 18.

Higher-order Markoff Chain Music. A second-order Markoff process
is produced whenever a new choice is made dependent upon the previous
two events; a third-order process arises when the choice depends on the
previous three events; and so on. It is possible, therefore, to produce by
such simple reasoning complicated interactions between a new event and

144

EXPERIMENTAL MUSIC

3-s

2 5
I S

oo if, =

— = .2

o *-

Q) 1_ U

.° -a u_

* I

». X
v
-o a

? *

** FT,

u_ '£

*=>

o
X

II x

© ^

+

.■y

'm

x>

1

03

H

JD

i

O

a.

*

Mil

II "

o >

JO a>

O ON

NHOO\OOh\CVi

42 -a si

X '2 X

t/3 C/3 CO

m M in rn M

Ui S-. >-< >_i

cui-rj^OOOO

ST o ^ o 5 B B 5 w B 5 5 "G «T o

a.

EXPERIMENTAL DETAILS

145

ON Tf
On rf

o o

On Tf

O ON

a >
(* O

B

U

>

u

bo

OJ

Ih

B

O

O

03

■ —

Sh

e4 o

146 EXPERIMENTAL MUSIC

events that have preceded it. Second-order and higher-order relationships
between notes can be found in composition rules. However, a simple
cascading process of this type would not produce musical output bearing
recognizable relationships to normal musical structures as the relationships
between notes become more distant. Therefore, instead of continuing by
generating second-order Markoff music, we utilized another more relevant
concept, namely, that of defining structurally more important and less im-
portant notes. This working principle, adapted from the ideas of musical
analysis described earlier, was utilized to produce Markoff chain music in

which the notes occurring on strong beats, arbitrarily assuming time,

were made to depend on one of the generating functions, while the notes
occurring on weak beats were made to depend on another of the generating
functions. Four examples of music utilizing this principle were produced
by means of the harmonic and proximity functions given in Table 15. The
organizing principle of this music is illustrated diagrammatically in Figure
16, where it is seen that two samples each of zeroth-order and first-order
Markoff chain music were produced by these more complicated inter-
actions.

In Cases (a) and (c), the harmonic function x, was used to control the
successive choices of strong-beat notes, while the proximity function was
used for the weak-beat notes. In this elementary way, we built up simple
structures analogous to those suggested by the analysis of conventional
musical structures. Specifically, the harmonic function was used as a
longer-range structural function to block in larger tonal relationships, while
the proximity function was used to provide melodic filler inside these larger
units. In the remaining two examples, Cases (b) and (d) in Figure 16,
the roles of the two functions were simply reversed. The proximity func-
tion, Vj, was now made the more structurally significant in order to provide
musical examples in which the neighbor-note, or proximity, relationship
was predominant.

Markoff Chain Music and Tonality. In Chapter 2, we noted that tonality
in musical composition depends in its simplest form upon a recall of the
first tone of a given melodic line. It is seen, therefore, if we compare this
concept to the Markoff chain music thus far described, that none of the
musical examples can be called tonal by definition, since in none of them
was there defined a dependence of note N{ upon note Nt once i exceeded
the order of the Markoff chain process being applied. However, it is pos-
sible to define an /th-order process in which the controlled intervals are

EXPERIMENTAL DETAILS

147

(a)

(b)

(c)

Id"

ZEROTH- ORDER x,

4, 1*

O * rf> I t It I

ZEROTH -ORDER y.

ZEROTH - ORDER /y.

* — ii

ZEROTH -ORDER

FIRST- ORDER x,

"»*"

rn rn

t__±

l__±_

FIRST- ORDER /.

FIRST -ORDER y.

IV

rn m

4_^L

t_i_

FIRST -ORDER */

HARMONIC FUNCTION

-y — g

rn rri

PROXIMITY FUNCTION

PROXIMITY FUNCTION

~if— — 1»

rn m

HARMONIC FUNCTION

HARMONIC FUNCTION

.rn rn

t__L

t__L

PROXIMITY FUNCTION

PROXIMITY FUNCTION

1C

rn rn

i_L

i_L

HARMONIC FUNCTION

ON STRONG BEATS.

fn rr?

ON WEAK BEATS.

ON STRONG BEATS.

I*

■w

/t~3 m

ON WEAK BEATS.

ON STRONG BEATS.

rn rn

t_^L

t ±_

ON WEAK BEATS.

ON STRONG BEATS.

W

rn rn

|__L

ON WEAK BEATS.

etc.

etc.

.etc.

.etc.

Figure 16. Experiment Four: Illustration of strong- and weak-beat structures
utilized to generate species of Markoff chain music.

Ni — ,/Vi rather than Ni — Ni-±. Once we do this, we relate each new
note to the initial note, rather than to the immediately preceding note. We
immediately establish, as a consequence, the most elementary type of tonal
control in accord with the simple definition of tonality just given. It is in-
teresting also to note that zeroth-order and /th-order Markoff chain music
bear a relationship toward one another analogous to that between the in-
terval rows and tone rows discussed in connection with Experiment Three.

148

EXPERIMENTAL MUSIC

Therefore, in the last part of Experiment Four, a sample of /th-order
music was produced, but in combination with first-order Markoff music.

The harmonic function was used on the strong beats, assuming :? time,

and was applied as an /th-order process. Therefore, each strong beat was
related to the initial and, as we shall see, final note of this last sample of
music. These notes were arbitrarily set as C in accord with the over-all
tonality selected for the I Iliac Suite assembled out of the musical examples.
The weak beats were then assigned a secondary role, one of ornamental
or melodic filler. This type of function was best expressed by the simpler
type of Markoff process and by means of the proximity function, y'j. The
musical pattern, therefore, was the one shown in Figure 17.

ith-ORDER HARMONIC FUNCTION ON STRONG BEATS

JT3 ST!

m m

JT"3

etc.

etc.

FIRST- ORDER PROXIMITY FUNCTION FOR WEAK BEATS

Figure 17. Experiment Four: Musical organization of the last example of
Markoff chain music in Experiment Four.

Printout and Modulation. All examples of Markoff chain music were
arbitrarily printed in blocks of four lines at a time to permit transcription
of the results for the four instruments, Violins I and II, Viola, and Cello.
In this way, we were able to provide a maximum amount of musical ex-
amples in a minimum of score and playing time.

While writing the printing instructions for the Illiac for this last example,
we also incorporated a simple demonstration of how modulation might be
handled. Modulation can be defined as a shift of the reference point in an
/th-order Markoff chain process to some new tonality. It is possible to
design a simple program for resetting the reference point for the /th-order
process after any desired number of measures, or, for that matter, after a
randomly selected number of measures. Moreover, the new reference point
can be preset, or it can be chosen at random, or by some sort of more elab-
orate, restricted choice process. For example, it might be decided that after

EXPERIMENTAL DETAILS 149

m — 1 measures, a modulation from C to F was required. Therefore, on
measure m the reference point for the /th-order Markoff chain process
would be shifted from C to F. To effect this, it would be possible to re-
write the instructions for generating new notes. However, instead of re-
writing the instructions, a simpler method involving the printout routine
was employed, because it was evident that modulation is really nothing
more than a transposition of the printed results. Thus, to shift from C to
F, all we needed to do was to add the number 5 to each note while it was
still in the machine as a number. This converted a C to an F upon printing,
a C sharp to an F sharp, and so on, and yet did not disturb the basic note-
generating process. It is seen that this is effectively what is done by any
musician or composer in effecting a transposition or a modulation.

We selected the sequence I-IV-V-I, a simple cadence formula, to illus-
trate this technique and combined it with the previously described tonal
/th-order Markoff chain music to illustrate how modulation, as well as sim-
ple tonality, might be achieved. Since C was selected as the fundamental
tonality, the subsidiary tonalities became F and G, and the over-all tonal
pattern produced became C-F-G-C. The shift to F was set to occur after
six measures, the shift to G after another six measures, and the return to C
after still another six measures. The use of a preset pattern of this type was
preferred this time, since the use of more random processes for instituting
changes of this type was amply illustrated by the musical examples in Ex-
perimental Three.

Tonal Drive. Modulation is less commonly effected in a piece of music
by an abrupt transition from one key to another than by a carefully and
logically planned sequence of progressions which give the listener a sense
of having arrived at a satisfactory resting point after the modulation is
completed. Thus, for example, modulation from the tonic to dominant is a
common feature of the exposition section in conventional sonata form. In
a simple example of sonata form, the main subject solidly grounded in the
tonic key is first fully stated. It is then followed by a modulatory sequence
— often consisting of little more than broken chords — which convey the
tonal weight from the tonic to the dominant, often through use of chords
such as the dominant seventh of the dominant to produce a new resting
point upon which the second subject is grounded. This modulatory section,
to be convincing, must convey the sense of having prepared for the arrival
of the new tonal center.

In general, this type of planning ahead in building up a tonal structure,
which applies the concept of "tonal drive," can occur in modulation, as in

150 EXPERIMENT AL MUSIC

the example discussed, and also at other points in a musical structure, such
as at the close of a movement on a chosen tonal center. The chords pre-
ceding the end, therefore, would be related to the anticipated tonality, and
the process in its simplest form would be a reverse /th-order MarkofT chain,
the reference point being ahead, timewise, rather than behind.

Thus, it was possible also in the last example of MarkofT chain music
to institute a simple process of tonal drive to a final tonality of C, utilizing
a simple modification of the technique previously developed for changing
transition probabilities. All we had to do was to reverse the process pre-
viously utilized to produce the first example of MarkofT chain music as
illustrated in Table 15. This process was, therefore, incorporated also into
the instructions for this last sample of music. Unity was subtracted from
the transition probabilities after every six notes (two measures), so that
after twenty-four measures the only choice left became the melodic repeat.
However, this technique, of course, was applied to an /th-order process on
the strong beats, and, therefore, the only possible point of resolution was on
C. A zeroth-order process, by comparison, might end on any tonality. In
addition, the same process was applied to the first-order proximity func-
tion on the weak beats, so that weak-beat melodic skips were forced to be-
come smaller and smaller on the average as the end of the cadence was
approached.

A Simple Closed Structure. The combination of these techniques formed
a logical group of musical entities which illustrate how a simple closed mu-
sical structure might be generated by the Illiac. Aside from the simple
cantus firmus setting of Experiments One and Two, and the interval and
tone rows of Experiment Three, no closed structures had yet been written
to this point. Moreover, the cantus firmus settings and interval and tone
rows represented really only rudimentary materials, which in most instances
would be used for the building up of larger structures.

Therefore, the application of the harmonic function as an /th-order
MarkofT chain process on the strong beats and the proximity function as a
first-order MarkofT chain process on the weak beats, along with the process
of shifting the tonal reference point after every six measures and the shift
of the transition probabilities after every two measures by unit subtraction
until only a melodic repeat could occur, permitted the production of a
simple closed structure of twenty-six measures, which represented an ex-
tended cadence leading to a final close. This closed structure was utilized
as a coda for the last movement of the Illiac Suite to serve as a simple

EXPERIMENTAL DETAILS 151

prototype for building up more complex structures, such as conventional
musical forms.

In general, in closing our discussion of Experiment Four, it might be
noted that the actual coding of Markoff chain music was extremely simple
by comparison with the programming of strict counterpoint. It was a com-
pletely efficient process, since there was no attrition resulting from the pro-
duction of unusable notes; therefore, no try-again subroutine was required.
It is entirely possible, however, that the discarding of notes would have to
be resorted to once more complex interactions, particularly those involving
cross relationships between the voices, were devised. The speed of produc-
tion of the present examples of Markoff chain music was limited only by the
speed with which output tape could be punched by the Illiac.

CHAPTER SIX

Experimental Results:
The llliac Suite

Introduction

As noted in Chapter 1 , the most satisfactory way to present musical ideas
is in the form of actual sound. Therefore, as soon as we had planned Ex-
periment One, we also chose a musical medium through which the results
might ultimately be presented. The work described in Chapter 5 was suffi-
ciently extensive so that a four-movement suite could be assembled out
of the experimental results to parallel exactly the four major experiments.
This piece of music, which we entitled the llliac Suite in reference to the
computer used for the experiments, is therefore a chronological research
record of our experiments.

In assembling this suite, it was desirable to organize the raw results into
a playable, relatively coherent whole. Moreover, it was also desirable to
minimize the amount of arranging of the materials, so that as much as pos-
sible of the musical content of the suite could be said to be computer pro-
duced. This we have done, but in addition to the basic choices concerning
which experiments were to be carried out, certain additional decisions were
required during the transcription of the computer output to a musical me-
152

EXPERIMENTAL RESULTS: THE ILLIAC SUITE 153

dium. These will now be briefly reviewed so that if the score of the I Iliac
Suite in the Appendix is examined, it will be clear what elements in the
score result strictly from the operation of the computer and what elements
have been inserted or adjusted as a result of practical necessity. The main
problems involved in transcribing the experimental results were the fol-
lowing:

1. The first decision was the choice of a musical medium through which
the results could ultimately be heard. The use of electronic or other syn-
thetic means was eliminated in our case, since equipment of this type was
not available. A conventional instrumental medium was therefore the
only other reasonable alternative, but transcription for a keyboard instru-
ment such as the piano was also eliminated because this would introduce
the special restriction of having to have the music fit under the hands at
the keyboard. Therefore, since, even in Experiment One, the objective was
to produce a four- voiced polyphonic texture, the choice of a string quartet
medium appeared logical and convenient; the problems in transcription
would be minimized. Secondly, a string quartet has a fairly homogeneous
timbre, which was desirable from our viewpoint; and lastly, the practical
problem of having the music performed appeared less formidable than with
larger or more exotic instrumental combinations.

2. Since much more musical output was produced by means of the Illiac
than could possibly be used, it became necessary in each of the experiments
to employ some sort of unbiased screening procedure to select representa-
tive musical output. It was important that we select impartially and not on
the basis of aesthetic evaluation, since what we required were average rather
than superior results. Therefore, various selection processes were employed,
such as arbitrarily using every tenth cantus firmus setting, or arbitrarily
selecting material at the beginning or end of a sample of output, or choosing
examples at random by means of a random-integer table.

3. The choice of an over-all structure of a four-movement suite was
made. The four movements were entitled Experiments One, Two, Three,
and Four, duplicating the plan of the experiments themselves. Moreover,
the internal structures of the four movements were planned to correlate
with the details of the four experiments. The only self-contained musical
structure produced completely by the Illiac is the last section of Experi-
ment Four. This was utilized as the Coda of this last movement of the
suite.

4. The choice of tempi for the four movements and in the individual
parts of the first and third movements was made independently of com-

154 EXPERIMENTAL MUSIC

puter programming, since no programming for tempi was carried out. It
should be noted that this is a simple problem, even with the inclusion of
provisions for accelerandi and ritardandi. However, it was decided to
select reasonably contrasting tempi for the movements and to defer the
study of this musical problem in favor of other, more important projects.

5. It was desirable to transpose Voice 1, assigned to the cello, two oc-
taves downwards relative to the notation utilized for the printed computer
output; this was done to place this voice in the most favorable playing
range of the cello. For the viola, used for Voice 2, the music was trans-
posed one octave downwards, again for the corresponding reason. Voices
4 and 3 were played untransposed by Violins I and II, with the principal
exception being in the first section of Experiment One, where the part for
Violin I was transposed one octave upwards. Since the initial procedure
for Violin I tended to place the music inordinately high in the playing
range of this instrument, this transposition was subsequently abandoned.

6. Some subsidiary random-integer tables were utilized to select dy-
namic levels in Experiments One and Two and to distribute the cantus
firmi and two-voiced settings in Experiment One among the instruments.
Some doublings in these parts of this movement were also included to
provide some dynamic variety. Also, certain restrictions to only two
voices playing at a time were made in Experiment Four during transcrip-
tion. This was done to add some variety to the musical texture of Ex-
periment Four.

7. The meter for each movement or section of a movement was selected
during transcription. In terms of computer programming, this problem is
a simple one like programming tempo, and was deferred in favor of other
projects.

Outside of several other quite minor additions, the rest of the content
of the suite can be considered directly the result of computer programming.
The general structural outline of the Illiac Suite is shown in Figure 18.
The structural details of each movement of the suite will now be considered.

Experiment One

This movement is divided into three sections — presto, andante, and
allegro — which contain successively samples of computer output which
illustrate how we progressed from the writing of simple cantus firmi to
two-part and finally four-part settings. In order to show a number of
examples of each type of computer output, series of settings of different

EXPERIMENTAL RESULTS: THE ILLIAC SUITE

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lengths were strung together to form larger musical structures. Since these
first programs for the computer were designed to produce fifty samples
each of cantus firmi or cantus firmus settings from three to twelve notes
long, the structures of the three parts of this first movement were designed
to exhibit a sampling of all the output of the various different lengths. In
the presto part of Experiment One, five samples each of cantus firmi from
three notes to twelve notes long were used successively to build up this
section of the movement. To provide reasonably unbiased sampling,
every tenth sample of computer output was chosen arbitrarily for inclusion
in the suite. These are played individually by the four instruments until
near the end, where some doublings are permitted. The distribution of
the cantus firmi among the four instruments was decided by means of a
four-choice random-integer table, also generated by means of the Illiac.
The scoring was carried out by associating each instrument with one of
the numbers employed to build up the table. Successive integers in random
sequence were used one at a time to assign cantus firmi to the instruments.
It is interesting to note that this seems an entirely adequate way of scoring
this type of musical material. The selection of dynamic markings was
carried out by a similar technique. This time, the random integers were
associated with the markings ff, j, p, and pp. Finally, as previously ex-
plained, octave transpositions were employed to place the cantus firmi
into effective playing ranges for the instruments. At the close of this
section of Experiment One, a pizzicato chord was inserted to show the
transpositions used during transcription.

The second section of this movement (andante) is a sampling of two-
part cantus firmus settings. Here we put together successively individual
samples of three- to twelve-note settings and then reversed the order to
provide a change of structure and, hence, some musical contrast to the
first part of the movement. The selection of parts and dynamics was car-
ried out as in the first part of the movement, utilizing the same four-choice
random-integer table, except that, for the choice of the instruments, the
integers were used two at a time.

In the final section of the movement (allegro), a group of four-part
cantus firmus settings of increasing length make up the structure, repeating
the plan of the first part of the movement. Dynamic markings were chosen
as before. The movement ends with an arbitrarily chosen sample of the
settings, in which we allowed the voices to hold notes over many beats in
violation of the rule against successive repeats. Also, a final pizzicato

EXPERIMENTAL RESULTS: THE ILLIAC SUITE 157

chord, duplicating the ending of the first part of Experiment One, was
inserted for structural symmetry.

It is evident, upon examination of the score of Experiment One con-
tained in the complete score in the Appendix, that in these settings the
rules employed for Experiment One are not violated. However, it should
be kept in mind that these various cantus firmi and cantus firmus settings
exhibit faults one might naturally expect as a result of the absence of the
remaining strict counterpoint rules. This is not to say that many are not
without some musical interest within the limited technique given the com-
puter, but only that the strict counterpoint problem was still incompletely
solved. Obvious musical faults, in terms of strict counterpoint, include
the outlining of chords such as triads, sequences of melodic skips, six-four
chords, and a number of other imperfections. For reasons previously men-
tioned, it was decided to complete our study of the strict counterpoint
problem by writing improved settings of the type included in Experiment
Two.

The musical content of Experiment One, of course, has other obvious
limitations. Aside from the extreme simplicity of the melodic and harmonic
style, perhaps the most important limitation is the absence of harmonic
change. The whole movement rests squarely on a C-major tonality, which
yields a rather unusual static but not necessarily unpleasant effect. An
alternate procedure, which could have been carried out even at this early
point, would have been to score the cantus firmus settings in different keys
chosen perhaps randomly or by some predetermined scheme. However,
since this would have been an added musical element not directly related
to the main objective of Experiment One, the use of such a device was
not considered.

Experiment Two

In this movement, as previously explained, we wished not only to show
that we can write essentially correct first-species strict counterpoint by
means of a computer, but also how the imposition of counterpoint rules
leads to a clarification of the texture of purely random white-note music.
Therefore, in Experiment Two, the adagio of the I Iliac Suite, we have
assembled a sequence of cantus firmus settings, all arbitrarily fixed at a
length of twelve notes. After starting with two samples of purely random
music, as each section of the movement is reached, the number of rules

158 EXPERIMENTAL MUSIC

governing the choice of the notes is increased in accord with the plan
shown in Table 19. The movement ends with four extra representative

Table 19

Experiment Two

Sequence in Which Strict Counterpoint Rules Were

Successively Added to Random White-note Music

Section Added rules

[A] Random music; no rules

[B] Skip-stepwise rule; no more than one repeated note

[C] Cantus firmus starts on C with C chord for opening; cadence on C with

leading tone in one of the four voices; resolution of tritone in VII6,

F E

e-g-» B must resolve to ^

[D] Octave-range rule

[E] Only consonant chords permitted except for chords; i.e., harmonic sub-

routine added

[F] Parallel unisons, octaves, fifths, and fourths still permitted; melodic sub-

routine added

[G] Parallel fourths; chords containing tenth still permitted
[H] Best counterpoint

cadences written by means of our cadence program. Throughout, as in
the first movement, dynamics were chosen by means of a four-choice
random-integer table, and the over-all structure and octave transpositions
were worked out during the course of transcription.

Two passages from Experiment Two might be given special considera-
tion in examining this music, namely, the random white-note music used
to open the movement and the highly organized music used at the end of
the movement. A comparison of these settings illustrates how order can
be imposed upon random music by the logical processes described in detail
in Chapter 5. Experiment Two, therefore, is a simple musical illustration
of how the introduction of redundancy into a structure with a relatively
high entropy content brings about a clarification of texture. Since the
settings also became progressively more difficult for the computer to work
out, this experiment also shows how redundancy reduces the information
which may be communicated and how increasing the redundancy can only
be brought about by increasing the amount of material which must be
rejected.

This movement happens also to be one elementary example of how a
high-speed digital computer can be used to experiment with musical prob-

EXPERIMENTAL RESULTS: THE ILLIAC SUITE 159

lems in a novel way, since the writing of a sequence of textures in the
manner illustrated would be difficult to do by conventional means. The
composer unaided by a computer is not normally conditioned to maintain
an unbiased attitude toward his output, so while he could write the random
white-note music with the aid of a random-number table easily enough,
beyond this point it would be difficult for him to cope with the problem
of obeying only certain rules and letting everything else occur at random.

Experiment Three

Each of the three long fast sections of this movement is built upon the
rhythm, dynamics, and playing-instructions codes described in Chapter 5.
The first part of the movement is an example of how the simple version
of these codes operates. The only addition to the computer output here
is the insertion of pitch levels for the four instruments. These were selected
by means of an arbitrarily chosen tone row, in anticipation of the Coda of
Experiment Three in which computer-generated tone rows are presented.
All other elements in the opening measures of Experiment Three are seen
to be based upon computer output of the type shown in Figure 9. The
output for this section was transcribed directly as produced by the com-
puter, since no selection process was necessary in this experiment.

The first adagio, section [C] in the score, for which both the meter

and ff dynamic level were selected during transcription in addition to the
slow tempo, is an example of random chromatic writing. This is music
of the highest possible entropy content in terms of note selection based
upon the normal chromatic scale. It has a higher entropy content than
the white-note random music of Experiment Two, since there are five
more available choices per octave, namely, the black notes. This is truly
atonal music of the most extreme sort, as we have noted already in Chapter
5.

In the second allegro vivace section of Experiment Three, this random
chromatic writing is combined with the second, more complex rhythm,
dynamics, and playing-instructions code to produce a rather highly com-
plex dissonant musical texture. In combining these two codes together, it
should be noted that only on the Is, or "strikes," were the notes of the
random music utilized. On the Os, signifying "hold" or "rest," the notes
were skipped. In this section, each random note produced by the computer
was denoted rhythmically by an eighth note.

160 EXPERIMENTAL MUSIC

The second adagio section, section [G] in the score, illustrates what hap-
pens when the few simple rules of writing described in Chapter 5 are
imposed upon random chromatic music. It is seen that this music is much
more controlled than the music in the earlier adagio section. These re-
sults seemed to bear out our anticipation that the particular rules of writing
selected would be efficient in imposing order upon the musical output.

In the third allegro vivace, this type of material is combined with the
second rhythm, dynamics, and playing-instructions code. The same tech-
nique of combination was employed as used in the previous allegro vivace.
This music, in some ways, resembles certain kinds of contemporary music.
The continually dissonant but rather colorful musical texture, the use of
complex rhythmic ostinati, the shifting degrees of rhythmic and dynamic
independence of the four voices, the absence of conventional tonality as
a structural device, and the nature of the melodic profile all suggest ele-
ments of twentieth-century style. Aside from crucial factors such as
tonality, the major element missing is perhaps the use of imitation and
melodic repetition as a structural device. Because of this, we started at
one point writing a computer program for composing simple fugues in this
general style but later set aside this project in favor of the studies in Ex-
periment Four. It might also be noted that our choice of the tritone resolu-
tion as our one element of harmonic control happens to reflect Hindemith's
definition of two basic types of chords — those without tritone and those
with tritone.1 The two chord types in Hindemith's system bear different
structural implications. Our choice of this device happened to be arrived
at independently and came in part from our experience with the Counter-
point experiments. However, it is interesting to note in Hindemith's writ-
ing a similar emphasis on the importance of the tritone in structural
harmony.

One additional point struck us immediately when we compared these
results from Experiment Three with the output produced in Experiment
Two. In these two experiments, we have contrasted two widely different
styles — one bearing a relationship to sixteenth-century musical style, the
other to twentieth-century style. One style is highly restrictive, highly con-
sonant, but sounds quite simple, while the other style sounds dissonant and
much more complex and difficult to decipher. It is important to note, how-
ever, that simplicity of style and hence accessibility bears an inverse rela-

1 P. Hindemith, The Craft of Musical Composition. 1. Theory, rev. ed., Associated
Music Publishers, Inc., New York, 1945, pp. 94-108 in particular.

EXPERIMENTAL RESULTS: THE ILLIAC SUITE 161

tionship to the freedom of choice. The simplest style requires the severest
restrictions and has the highest degree of redundancy. On the other hand,
simpler musical styles are by no means necessarily the easiest to write,
since the difficulty of composition involved in making the best choices
from among the many available in larger structures in a less restricted
style is offset by the fact that more of the available choices are permissible
in terms of the desired end effect. This is illustrated by the comparison
of Experiments Two and Three. The work required to extract the type of
order illustrated by first-species strict counterpoint was greater than that
required to produce the last allegro vivace of Experiment Three; the pro-
gramming was more complex and more difficult. For a structure of given
dimensions, we may therefore suggest that, as a general rule, a more re-
stricted style is more difficult to write, since fewer of the interactions
between the notes can be left to chance; or conversely, more of the inter-
actions must be precisely defined. This conclusion is compatible with a
composer's experience in writing a highly restrictive sample of music, such
as a counterpoint setting, in comparison with the writing of a more loosely
organized piece of music in which the relationships are permitted to be
much more casual.

The Coda of Experiment Three, section [K], consists of one example
each of an interval row, of a tone row, and of the restricted type of tone
row described in Chapter 5. Each row utilized was arbitrarily assigned as a
solo to the cello and followed by the inversion, retrograde inversion, and
retrograde forms of the row played on the remaining three instruments.
As described in Chapter 5, these transformations of the row were easily
produced in the computer by changes in the printing instructions in the
computer code. The code written to produce these rows could, of course,
be utilized in future experiments as a basic subroutine in programs to pro-
duce more complex serial compositions.

In reviewing the musical results of Experiment Three, we have felt that
one point at which the results might have been easily improved is in the
repetition pattern of the rhythm, dynamics, and playing-instructions code.
Setting the index for this at a maximum of twelve measures made the music
seem to us somewhat overly repetitive. Resetting this factor involves
changing only one number in the set of instructions for the computer.
This, of course, is a very simple matter. An index of about six measures
rather than twelve would perhaps be more nearly the desirable value
to select.

162 EXPERIMENT AL MUSIC

Experiment Four

In preparing the score of Experiment Four, the experimental results
were transcribed directly as produced by the computer, since all samples

of output were equally good. Fast tempo and ^ meter were arbitrarily

chosen to provide a contrast to the previous movements, and dynamics
indications were inserted during transcription independently of computer
programming. As pointed out previously, results were transcribed for all
voices simultaneously to provide more experimental material in the score
in playable form. These experiments in monody were transcribed, there-
fore, as pseudopolyphony. Indeed, it might even be noted that a composi-
tional technique of this type has historical precedents of various sorts; for
example, in the part writing in late medieval music.2

The first two sections of this movement illustrate how changes in the
transition probabilities derived from the harmonic function change the
character of the music. In the first sample of zeroth-order Markoff chain
music at the opening of the movement, the transition probabilities are
adjusted after every two measures, so that by the time this initial section
is completed, a sample of every distribution from only the repeat being
permitted to the so-called average distribution has been illustrated. This
is in accord with the shifts in transition probabilities shown at the far
left-hand side of Table 15 in Chapter 5. The net effect of these shifts is
to cause the music to progress from a fixed C tonality to a rather free
texture in which the statistical distribution of melodic intervals is in accord
with the "average" or unperturbed distribution. In examining the trans-
cription of these results, it should be noted that each two measures repre-
sent the first sample of a larger amount of computer output representing
each type of distribution. Therefore, across the bar lines of every other
measure, the rules are not observed.

The second group of results, section [A] of the score, is a similar series

2 Thus, "Franco [of Cologne] states that whoever wishes to write a conductus
should first compose his own tenor . . . and then add the discantus (i.e., the du-
plum); if a third voice is wanted, care should be taken always to have it in con-
cordance with either the tenor or the discantus . . . indicating that . . . the parts
were written one at a time, rather than simultaneously." (G. Reese, Music in the
Middle Ages, W. W. Norton & Company, Inc., New York, 1940, p. 308). This type
of construction, as a matter of fact, might be of considerable interest because it
could serve as a connecting link between the Markoff chain monody produced in the
present Experiment Four and full-fledged polyphony.

EXPERIMENTAL RESULTS: THE ILLIAC SUITE 163

of extracts from computer output in which a progression from random
distribution, that is, equal occurrence, of melodic intervals to the unper-
turbed distribution is illustrated. In this example of ze roth-order Markoff
chain music, we progress, therefore, from a more random to more re-
stricted situation as opposed to the opening part of the music.

The next parts of Experiment Four contain zeroth-order Markoff chain
music first based upon the harmonic function above (section [B] of the
score), then upon the proximity function alone (section [C] of the score),
and lastly upon the combined function (section [D] of the score). This
is followed immediately by a similar sequence of samples of first-order
Markoff chain music (sections [E], [F], and [G] of the score) which serve
structurally in the movement as an answering section. In the presentation
of these samples of computer output, as noted previously, the number of
voices playing at one time was reduced arbitrarily from four to two, except
for the playing of the combined functions. This was done simply to pro-
vide some additional internal dynamic variety to this last movement. It is
observed that the character of each of these samples of music is in general
accord with the rules which govern its production. The harmonic function
music is characterized by a preponderance of skips and broken-chord pat-
terns; octaves, fifths, and other consonant harmonic intervals occur much
more frequently than stepwise motions. On the other hand, the proximity-
function music contains a higher proportion of stepwise intervals, neigh-
bor-note motions, and fragments of scales. In general, the over-all character
of this music is more melodic in texture. Finally, when the additive com-
bined function is utilized in a first-order process, the nearest approach to
recognizable melodic writing possible through utilization of this general
technique seems to have been reached. The first-order process tends to
promote a type of skip-stepwise rule, as previously explained, so that wide
skips tend somewhat more than in the zeroth-order process to be followed
by repeats and stepwise motions and consonant leaps to be followed by
dissonant leaps, and vice versa.

The next several short sections of the movement contain examples of
music in which the weak- and strong-beat functions have been differen-
tiated, firstly in a zeroth-order process and then in a first-order process.
Even within the limited set of results kept in the score, the operation of
the composing scheme outlined by means of Figure 16 in Chapter 5 can
be distinguished. Taking the two contrasting zeroth-order samples of music
from sections [H] and [I] of the score of Experiment Four, we see upon
extraction of the strong-beat notes from these samples, as shown in Figure

164

EXPERIMENTAL MUSIC

19, that the sense of the harmonic function predominates in example (a),
while the sense of the proximity function predominates in the example
(b). Moreover, an inspection of the complete examples in sections [H]
and [I] reveals that the weak-beat notes have been selected in general ac-
cord with the computer instructions.

Figure 19. Experiment Four: Extraction of the strong-beat notes from samples
of zeroth-order MarkorT chain music in sections [H] and [I] of the full score.

The Coda of Experiment Four appears to disclose in a satisfactory way
all the features expected of it as described in Chapter 5. The modulations
worked out as planned, and the harmonic and melodic simplification de-
velops measure by measure as planned until the end of the movement is
approached. Toward the end of the movement, on the strong beats, only
the notes closely related harmonically were permitted to occur, while on
the weak beats, only neighboring notes were permitted, until at the very
end only the tone C was allowed. All of this worked out in accord with
the computer programming and indicates that the design of more complex
closed musical structures might be started by using this simple prototype
as a point of departure.

CHAPTER SEVEN

Some Future Musical
Applications

Introduction

It is convenient to define two general areas of interest which might be
considered in applying computers to musical problems in the future. One
of these we may describe as the theorist's field of interest; the other, as
the composer's. The theorist analyzes music written by composers to
characterize musical forms and how they operate. On the other hand,
the composer is more of an empiricist who seeks out new musical forms
which seem to him to be satisfactory. In attempting to suggest how various
new musical projects might be started from where we have left off, we
shall arbitrarily group our suggestions into these two basic categories —
the first related in general to the analyst's problems, the second to the
composer's.

Applications to Music Analysis

The fundamental role of the music analyst is to verbalize specific musical
problems so that aspects of musical communication can in turn be defined.

165

166 EXPERIMENTAL MUSIC

In essence, the problem for the musical analyst is to find the reasons why
a composer accepts or rejects musical materials, and to this end, prin-
ciples of musical aesthetics such as those reviewed in Chapter 2 have been
gradually established. We have indicated, however, that much of the
writing on aesthetics is not yet particularly precise — Langer's "significant
form" is a case in point. We have also indicated that the investigation of
specific forms, rather than general concepts, is to be preferred for the time
being in seeking more precise definitions of musical concepts. This propo-
sition follows from the argument of significant overlap of levels of com-
munication proposed by Weaver, which we referred to in Chapter 2. The
description of musical forms perhaps ultimately can be expressed in terms
of information theory or some future equivalent, although, at the present
time, application of this theory to musical analysis is perhaps still too re-
cent to evaluate critically in any detail.

If we now tabulate various applications within this general field of in-
terest, the following suggestions come to mind as representative examples
of the large number of possible projects which might be carried out:

1. Perhaps the most obvious application of computers to musical anal-
ysis is the extension of the type of studies already illustrated by the Illiac
Suite, in which we have applied the Monte Carlo method to the problem
of musical form. As a consequence of coding aspects of this problem as
numerical information and generating experimental results by means of a
computer, a computer is made to behave as a specialized, but unbiased
composing apparatus existing in a completely isolated environment, subject
only to the controls and information the music analyst might wish to sup-
ply. In this application, a computer is an ideal instrument by means of
which analytical ideas can be tested, since the investigator starts with cer-
tain hypotheses from which he formulates operating principles; he supplies
this information to the computer; the computer then generates music based
upon these principles; and the investigator then analyzes the results to
further his investigation. This, of course, is essentially nothing more than
a standard example of experimental scientific method, but the unusual
thing is that computers provide a practical experimental technique for
carrying out such research in the musical field. It can reasonably be as-
sumed that in the future the combination of these techniques with the more
purely theoretical and speculative studies in the musical field, such as
those by Moles, referred to in Chapter 2, would be profitable.

In our work, Experiment Four, in particular, suggests a technique of
how a new type of analysis of musical forms can be brought about. This

SOME FUTURE MUSIC AL APPLICATIONS 167

experiment, the theoretical basis of which is contained in information
theory and more generally in statistical theory, could only be carried out
in a reasonable time with the aid of devices such as the Illiac. As an
initial working premise for future studies of this type, it is suggested —
with appropriate restrictions — that most musical compositions reflect a
balance between the extremes of order and disorder, and that stylistic dif-
ferences depend to a considerable extent upon fluctuations relative to these
two poles. As a second working premise, it is suggested that interest in
musical structures is achieved normally when fluctuations around some
stylistic mean between these two poles are also subject to processes of
organization and arrangement time wise within the structure. The testing
of these two general premises through the technique of generating Markoff
chain music would be a logical extension of the work of this type thus
far completed.

2. In addition to these more general studies, there are many specific
tasks of musical analysis that could also be carried out with the aid of
computers. The analysis of specific musical styles following perhaps some
of the methods cited on pages 133 and 134 is one example. For example,
estimates of the relative degrees of order and disorder of different samples
of music or different sections of given musical structures could be at-
tempted. This is suggested since entropy seems to be a more useful
variable than less well-defined concepts such as "harmonic tension." Char-
acteristic melodic profiles for different styles could also be examined and
codified. Studies of this sort would be of particular interest in musicolog-
ical research, such as finding the determining characteristics of particular
styles of historical interest. It is also possible that the results of such
analysis could be used in a practical way to identify, to sort, and to cata-
logue old music — often a tedious and laborious task. Thus, for example,
it might be possible to determine whether samples of music merely rep-
resent variants of one basic piece or fundamentally different pieces. With
adequate analytical criteria, at least a major part of such a problem could
be coded for a computer to permit automatic cross comparisons of the
samples and subsequent sorting into appropriate categories. As a specific
example, at the present time we are considering a project for transcribing,
sorting, and reproducing French lute music in a complete modern edition.1

1D. Lumsden, Un catalogue international des sources de la musique pour luth
(Les lecons d'une etude des sources anglaises), CNRS colloque, "Le Luth et sa
musique," Paris, Sept. 14, 1957; T. E. Binkley, letter to Jean Jacquot, President of
CNRS, Paris, based upon comments upon Lumsden paper just referred to.

168 EXPERIMENTAL MUSIC

Since there is a vast quantity of this music, it has been estimated that up
to ten years will be required to do the job by ordinary means. It has
been suggested, however, that instruments such as the Illiac could be used
to speed up the process. It should be pointed out in connection with this
problem that a similar application of computers has already been made in
the field of Biblical research and also in the preparation of a concordance
for the new revised standard version of the Bible.2 It has been pointed
out that it took thirty years to prepare the concordance for the St. James
version of the Bible, while for the newer Bible, the same task was carried
out in nine months with the help of a computer.

It should be stressed in this connection that such a project would include
the automatic conversion of old music into modern notation in score form,
along with performance parts, if desired. Utilizing a computer, this older
music could be worked up rather easily and after final editing could be
made available in modern copy. The first step in carrying out this project
is the design and construction of a suitable musical printout mechanism of
the types discussed in Chapter 4. Secondly, the most efficient utilization
of computers for such purposes would depend on the development of
adequate scanning devices for computers which would recognize printed
and even handwritten letters and numbers. At the present time, research
is being carried on to produce such scanning devices. This was given
recognition in a recent symposium devoted to document reading, pattern
recognition, and character synthesis.3 When these devices become generally
available, it should be possible to adapt them to the scanning of musical
notation. In the meanwhile, transcription of musical materials onto coded
tape or punched cards is an adequate substitute technique.

Other possible practical applications in musicology might include the
use of computers to realize continuo and figured bass in Baroque music
and to complete the part writing in older music where the music has either
been left incomplete or some of the parts are missing. In this last applica-
tion, as a result of informed statistical style analysis, at least a highly
probable realization of the missing parts could be produced.

3. Another practical application in this general area, namely, to peda-

2 A. Carpenter, "Amazing New Uses for Robot Brains," Sci. Digest, 41(2) : 1,
February, 1957.

3L. Cohn, R. A. Kirsch, L. C. Ray, and G. H. Urban, "Experimental Use of
Electronic Computers in Processing Pictorial Information"; T. L. Dimond, "Devices
for Reading Handwritten Characters"; A. I. Tersoff, "Automatic Registration of
High-speed Character-sensing Equipment," all given at Session IX of the 1957 Eastern
Joint Computer Conference, Washington, D.C., Dec. 9-13, 1957.

SOME FUTURE MUSICAL APPLICATIONS 169

gogical uses, should be mentioned. Several such applications have been
suggested to the authors. On pages 128 and 129 the use of restricted tone
rows as cant us firmi for counterpoint studies was discussed. Extensions
of this work, such as the generation and cataloguing of such tone rows
into different groups with characteristic properties, has been proposed by
Robert Kelly, the author of the counterpoint method for which these tone
rows are intended.4

The systematic generation of musical materials for teaching manuals for
instrumental performance would be a second application of this type. The
preparation of manuals for the guitar and the lute has also been proposed.5
In this application, the coding would be based on guitar and lute tablature
rather than standard musical notation, since tablature is in itself already a
codification of the technical limitations of these instruments.

4. A final application to music analysis, we should like to mention, is
the analysis of musical sounds themselves. This type of information would
be particularly useful in the production of synthetic music by means of
computers in conjunction with other electronic equipment. A very con-
siderable knowledge of musical sounds and their physical constitution is,
of course, available today.6 Moreover, a large amount of information on
how to process these sounds by means of electrical and electronic equip-
ment has been built up through the development of sound-reproduction
systems, broadcasting, and other commercial developments, to say nothing
of the more experimental techniques reviewed in Chapter 3.

Musical sounds are, of course, extremely varied, and the analysis of
timbre, attack, and other factors which make up these sounds becomes
quite complex and is by no means completely understood. It seems pos-
sible that a computer might be useful in improving the design of sound-
producing equipment in one significant way in particular. The analysis of
sound could be coded as digital information, using standard techniques
such as Fourier analysis and the like, and stored for computer utilization
in permanent form, perhaps, on magnetic tape or equivalent high-capacity
storage. Instruments such as the RCA Synthesizer might be extremely use-

4 R. Kelly, private communication.

5T. E. Binkley, private communication.

6 Typical books on musical acoustics are numerous and include H. L. M. Helmholtz,
On the Sensations of Tone, Dover Publications, New York, 1954; Alexander Wood,
The Physics of Music, The Sherwood Press, Cleveland, Ohio, 1944; C. A. Culver,
Musical Acoustics, 4th ed., McGraw-Hill Book Company, Inc., New York, 1956;
L. S. Lloyd, Music and Sound, Oxford University Press, New York, 1937; H. F.
Olson, Musical Engineering, McGraw-Hill Book Company, Inc., New York, 1952.

170 EXPERIMENT AL MUSIC

ful as primary sources for acquiring such stored information, or, alterna-
tively, the analysis of actual sounds might be carried out in a computer by
reversing the flow of information through a digital-to-analog type device
of the kind we will discuss in the next section. Basic programs for extract-
ing this information out of storage and building up complex sound struc-
tures could also be developed. These synthesized sound patterns could be
printed out as digital results and used to process sound by means of instru-
ments such as the Synthesizer. Or, more directly, this information could
be reconverted directly to sound, using the digital-to-analog conversion
units of the type discussed below.

Applications to Music Composition

1. There are many practical problems of composition which require
examination in order that the rather limited catalogue of compositional
techniques thus far treated might be extended. We may note, therefore,
just a few of the more urgent of these problems to illustrate the nature of
what could be done: (a) the writing of computer programs for handling
many of the traditional and contemporary harmonic practices; (b) the
writing of more complex counterpoint programs, including programs for
more contemporary counterpoint; (c) the correlation of elements such as
rhythms and dynamics to note selection; (d) the use of imitation as a
structural device; (e) the use of thematic and melodic recall and develop-
ment; (/) the coding of factors thus far neglected, such as tempo, meter,
and choice of instruments; (g) the writing of standard closed forms, such
as variation form, fugue, song form, sonata form, etc. This last is of ob-
vious major importance. Not only specific forms, but the more general
problem of form needs to be considered here. The application of ideas
such as Schenker's concept of chord prolongation and of Meyer's concept
of purposeful musical motion would undoubtedly be useful in these studies,
to say nothing of the possible results of analytical studies such as those
proposed in the previous section. In time, it is conceivable that the han-
dling of many elements involved in the writing of standard musical tex-
tures might be carried out relatively simply and efficiently with a computer.
This would depend, of course, on the ease and precision with which these
musical elements could be programmed and the information stored in
sufficiently compact form. The extent to which this may become possible
is, of course, a matter for speculation at present, so we can only suggest
that music-writing programs might be designed to produce music efficiently

SOME FUTURE MUSICAL APPLICATIONS 171

by utilizing, among other things, standardized "library subroutines" for
standard musical operations, much as today standard subroutines are uti-
lized for ordinary mathematical operations. In this connection, it is worth-
while to note that the writing of the computer programs themselves may
very well be made more efficient. Since the time consumed in computer
programming normally requires much more time than the actual time of
computation in a computer, there is intensive research today into the pos-
sibility of computers themselves generating detailed computer programs
from more general sets of instructions.7 Therefore, if these new techniques
are developed, there is no reason to suppose that they could not be adapted
to the coding of musical as well as mathematical problems.

2. The organization of more or less standard musical materials in rela-
tively novel musical textures, including combinations not easy or even
feasible by other means, might be carried out. Some possibilities which
come readily to mind include: (a) the use of different rules or even of
different styles between different voices in a polyphonic texture; (b) the
inversion of rules, forbidding what is now permitted, permitting what is
now forbidden; (c) the development of new rules of operation for handling
musical materials, such as subjecting tone rows to complex permutations
based upon the concept of these rows as arrays spaced across 12 x 12 unit
plots of pitch versus time. A project of this type could perhaps very well
be founded on some of the permutational composition techniques referred
to in Chapter 3. In this context, the totally organized music, also referred
to in Chapter 3, would appear to be a composing technique particularly
suitable for computer processing.

3. There might be developed sets of new organizing principles for
musical elements leading to basically new musical forms. In the Illiac
Suite, we have already provided a number of specific examples of how this
can be done. The production of random musical elements, either notes, or
rhythms, or scoring, is one example. This represents the opposite condi-
tion of totally organized music and, as such, becomes a formal element to
be integrated into musical structures. More generally, the control of a
precise degree of randomness and of the fluctuation of musical texture
between order and disorder would seem to be more easily controlled by
computer processing than by other means. Obviously, this development
of a composing style consciously based upon this picture of musical struc-
ture could be of significance in developing an aesthetic not only related

7 For example, see D. D. McCracken, Digital Computer Programming, John Wiley
& Sons, Inc., New York, 1957, chap. 18 in particular.

172 EXPERIMENTAL MUSIC

to concepts of information theory, on the one hand, but also, on the other
hand, of more general significance than such relatively restricted concepts
as traditional harmonic music, the tone-row technique, totally organized
music, and so on. Moreover, since the codes used for the Illiac are based
in part upon random-number processes, these codes permit the computer
a conditional sort of "freedom of choice," this "freedom" being the equiv-
alent of randomness. The extent of this freedom could be made to depend
on how much the aesthetics of music might be expressed in the most
general terms. If a large number of highly specific rules, such as the rules
of strict counterpoint, are given the machine, the freedom of choice for
the computer to select musical materials is quite limited. Not only is the
over-all aesthetic quality highly predictable, but the specific details of the
music are rarely surprising. This is equivalent to saying that the redun-
dancy is large. However, if a computer is supplied with less restrictive
rules, then neither the general aesthetic effect nor the specific musical re-
sults are necessarily so predictable. One possible consequence of this is
that the composer might no longer be preoccupied with selecting specific
notes of the scale, specific rhythms, and other such details, but rather with
more generalized sets of symbols, which would be used in turn for the
generation and arrangement of the musical details in accord with the
musical "message." This could result in a very different attitude toward
specific musical details, and they might cease to have the importance we
now attach to them. In fact, these details could easily be varied over wide
limits within the same essential "composition," just as we now permit, to
a greater or lesser extent, variability in the interpretation of music by per-
forming musicians and yet recognize that this does not destroy the unique-
ness of a musical composition. It is only fair to note, however, that if
such a development were to occur, it would be a radical departure from
the attitude prevailing at the present time.

Among the few experiments of this type not involving computers that
seem to have been carried out recently are compositions of students of
John Cage, such as Morton Feldman, who has written a composition en-
titled Intersection No. 3, for Strings, Woodwinds, and Solo Cello. In this
work, Feldman permits a high degree of improvisatory choice by the per-
formers, since the "score" is set down upon graph paper rather than in
conventional notation.8 A somewhat similar, but less extreme experiment
has also recently been carried out by Gunther Schuller in a string quartet

8H. Cowell, "Current Chronicle, New York," The Musical Quarterly, 38:123-136,
1952.

SOME FUTURE MUSICAL APPLICATIONS 173

composed for the 1957 Contemporary Arts Festival at the University of
Illinois.9 Thus far, in our experiments, this particular aspect of musical
composition has only been barely examined in direct terms. The most
conscious application of the idea of controlling the degree of chance events
occurring in a musical structure was set up in Experiment Two.

Of perhaps more immediate interest is the further extension of new
musical organizing principles related to the general concepts of information
theory, such as the Markoff chain music in Experiment Four. More ex-
periments along the line of Experiment Four could be carried out, par-
ticularly to develop cross relationships between the voices. It might be
possible to start, as previously suggested, with a musical form in which
one voice is defined as a quasl-cantus firmus to which the other voices are
related. Another possibility might be to make direct use or modifications
of the techniques suggested by Hindemith10 by means of which he classifies
all chords into six basic groups and each basic group into subgroups de-
pending on the tonal strength of each particular type of chord. This pro-
vides a basis for setting up a stochastic variable for the tonal value of
vertical note combinations. In turn, moreover, as successive chords are
considered, this variable might be used to control the "harmonic tension"
of the music as it develops in time and, more importantly, the incorporation
of harmonic factors into musical structural units. Hindemith's system has
the virtue of being simple and yet amenable to the type of musical propa-
gation developed for Experiment Four, since it is particularly applied to
what we have termed zeroth- and first-order chain processes. His ranking
of the chords, which seems reasonable as a zeroth-order evaluation, could
be valuable as an initial reference standard. Any other zeroth-order basis
for grouping chords could, of course, be used just as well, depending on
the type of experiments the coder had in mind.

4. Another project might be the systematic study of microtone music.
In the past, because of the difficulties of understanding and building sys-
tematic harmonic relationships in microtone systems such as quarter-tone
music and complex tunings employing microtone intervals to secure just
intonation in all keys, and performance difficulties, this field has largely
been neglected. Moreover, in the few examples of microtone music we have
heard, the smaller intervallic movements seem to fulfill a coloristic rather
than a functional purpose. The use of quarter-tones in Bartok's Violin

9 W. S. Goldthwaite, "Current Chronicle, Urbana, Illinois," The Musical Quarterly,
43:390, 1957.

10 Hindemith, op. cit., pp. 106-108, 115-121.

174 EXPERIMENTAL MUSIC

Concerto and in Julian Carillo's Preludio a Cristobol Colombo, to cite two
examples, seem to be cases in point. With suitable sound production
means, however, a systematic study of the harmonic and contrapuntal
relationships in microtone music could be carried out by means of com-
puters. Similar studies could be carried out also on unusual scales and
tuning systems, or even variable tuning systems in which tunings could be
changed during the course of "performance." Variable tuning, of course,
is the ideal technique for securing just intonation in all keys.

5. Perhaps the most significant application of computers would be the
combination of computers with synthetic electronic and tape music of the
various types reviewed in Chapter 3. This obviously is a natural and com-
plimentary combination, since with computer music we are primarily con-
cerned with organizing musical materials, while with synthetic music we
are concerned more directly with the production of sound. Certain specific
rather immediate results can be predicted. In future experimentation with
computer music, the advantages of being able to produce the results di-
rectly in sound as well as in score form are obvious. Not only would the
results be analyzed more efficiently, but the means would be available for
producing quickly and efficiently the final desired musical end result. More-
over, in view of certain of the other projects outlined above, the experi-
mentation would no longer have to be confined to musical materials for
conventional scales, tunings, and instruments.

However, it is not only to the advantage of future experimentation with
computer music that the combination of techniques should prove advan-
tageous. There are specific limitations to the present means of production
of electronic music of the types reviewed in Chapter 3, one of which is the
time and labor required to prepare the music. This is a tedious business
requiring painstaking effort. There is also at the present time the need for
expensive and elaborate electronic equipment, such as the RCA Electronic
Music Synthesizer, if a wide variety of sounds are to be provided. In view
of what we have already said in regard to such equipment, it is obvious
that the development of simpler equipment operated via computer pro-
gramming would serve the composer's needs as well as the music ana-
lyst's. This should be an interesting engineering problem in itself, since it
involves taking from the digital computer a signal representing some sort
of musical pulse and converting this signal to output recorded directly on
magnetic tape for playback on tape machines. Obviously, playback and
distribution of musical output not intended for live performance would be
much simplified. The problem of converting digital-computer output into

SOME FUTURE MUSICAL APPLICATIONS 175

sound is a typical example of digital-to-analog conversion, a field of in-
terest attracting considerable attention at the present time. Digital com-
puters as they are now constructed are incapable of "doing anything"; they
simply supply answers to problems. On the other hand, the second class
of computers, called analog computers, simulate by analogy, usually elec-
trical, actual processes which may be mechanical as well as purely mathe-
matical. Electrical circuits have certain properties which can be used to
simulate addition, subtraction, multiplication, and division, and even in-
tegration and differentiation of mathematical functions. It is in recognition
of this property of electrical circuits that we can build analog computers
which operate upon continuous functions rather than discrete integers as
do digital computers. Details of the principles of operation of analog com-
puters can be found in standard reference works.11 Digital-to-analog con-
version is required in the present instance, since the output desired is
recorded sound. The study of this problem could be initiated in several
ways. For example, computer output might possibly be used to activate
signals from sine-wave generators, which in turn would be fed to the re-
cording head of a magnetic tape recorder. A second procedure would be
to adapt one of the digital-to-analog converters now on the market, such
as the digital-to-voltage converter manufactured by Epsco, Inc., Boston,
Massachusetts. The combination: digital computer, digital-to-voltage con-
verter, and magnetic tape recorder seems potentially the most satisfactory
composing instrument in terms of the technology of today. In such a de-
vice, sound synthesis would be carried out mathematically in the com-
puter. Tentatively, this seems to be a technique superior to that of
activating devices such as the RCA Music Synthesizer by means of coded
taped input prepared by computer programming.

A related problem is the notation for various types of synthetic music.
As mentioned in Chapter 3, a notation has been devised for the elek-
tronische musik produced at Cologne, but it also seems apparent that a
more general notation would be desirable. This notation might, for ex-
ample, be based upon the analysis of sounds in terms of digital information.
The processing of this information in a computer by the composer to pro-
duce a written score as well as recorded sound should be highly efficient
compared with current methods of composition. Yet another related appli-

11 For example, I. A. Greenwood, Jr., J. V. Holdam, Jr., D. Macrae, Jr., Electronic
Instruments, vol. 17 of the MIT Radiation Laboratory Series, McGraw-Hill Book
Company, Inc., New York, 1958; G. A. Korn and T. M. Korn, Electronic Analog
Computers, 2d ed., McGraw-Hill Book Company, Inc., New York, 1956. Also
periodicals such as Instruments and Automation.

176 EXPERIMENTAL MUSIC

cation of considerable interest in this combined area of computer and elec-
tronic music would be the realization of music too difficult or too complex
for performance by live performers. Not an inconsiderable amount of
modern music verges on being almost too difficult to perform. Charles
Ives' Fourth Symphony, for example, is a case in point, to say nothing of
a large body of more recent works. It would seem that the threshold level
of our ability to perceive complex rhythms and tonal combinations exceeds
present performance capacities.

6. It should be mentioned, although not directly as a consequence of
the present computer experiments, that certain mechanical aids to the com-
poser might be effected by means of a computer. In particular, one great
help would be the copying of parts from scores to eliminate what is now
very tedious and time-consuming work. This could be readily accom-
plished once the scanning devices referred to earlier come into use. More-
over, these scanning devices could be used to read composer's written scores
and convert them into sound as well as into printed parts.

7. It is also necessary to take note of one less attractive possibility, but
one which must also at least be mentioned, since it is so often suggested.
This is the efficient production of banal commercial music. For example,
it is not difficult to conceive of programs for writing music of this sort to
generate songs for juke-box consumption and similar uses, probably at a
highly efficient and rapid rate. All applications of this sort, however, are
nonartistic and fall outside the area of problems of aesthetic interest. Be-
longing in a somewhat similar category is the frequently asked question
of whether synthetic Beethoven, Bartok, or Bach might also be produced
by computers. The answer to this would seem to depend on the degree to
which the elements of the styles of these composers could be verbalized,
i.e., coded in a form suitable for computer programming. Appropriate
statistical analysis of Beethoven's music might conceivably lead to the
production of synthetic average Beethoven, just as, in a sense, the applica-
tion of strict counterpoint rules can yield a reasonable simulation of aver-
age sixteenth-century style, quite independently of whether computers or
normal composing techniques are employed. The goal rather than the
means appears objectionable here, however. The conscious imitation of
other composers, by any means, novel or otherwise, is not a particularly
stimulating artistic mission. Moreover, this type of study is, in the final
analysis, a logical tautology, since it produces no information not present
initially. The statistical aspect of the problem should not be permitted to
obscure this point. Reduced to its extreme case, this process would revert

SOME FUTURE MUSIC AL APPLICATIONS 177

to coding exactly the content of a specific and particular piece of music,
feeding this information into a computer, and obtaining back from the
computer exactly the same piece of music. It is obvious that nothing is
accomplished by such an operation.

8. The preparation of computer music for other forms of communica-
tion such as the stage, films, and broadcasting might be of particular in-
terest in the use of computer music in the combinations with synthetic
music discussed later on. As noted in Chapter 3, financial support in
Europe for experiments in electronic music has been provided by govern-
ment radio stations. This support is given because of the possible exploita-
tion of this other species of experimental music in broadcasting and related
forms of communication. There is no reason to suppose that effective
means of producing musical and sound continuity for this type of end use
might not be carried out efficiently by one or another of the various means
we have suggested.

Summary of Results

If we consider the various results included in the Illiac Suite, it seems
reasonable to conclude that the basic objectives initially outlined for this
project have been achieved. It has been shown that computers provide a
novel means for studying and exploiting certain techniques of musical
composition which can be utilized to produce both conventional and un-
conventional musical structures. Secondly, by applying new concepts it
has been possible to illustrate how various musical forms and processes of
composition can be treated from a new viewpoint with the aid of com-
puters. In concluding this discussion, it is desirable to summarize what
seem to us the more significant results of these experiments.

1. The elements of musical communication have been separated into
those subject to analysis, to coding, and to transcription, namely, specific
elements of actual musical structures, and into those more intangible ele-
ments which, in general, appear to elude highly specific characterization.
It is only the first group of musical elements which were considered sub-
ject to coding and to experimentation with computers at the present time.

2. The relationship of information theory to musical problems was dis-
cussed to point out how this theory is useful in setting up a technique for
musical composition with a digital computer. The specific technique of
restricted random number processes was selected as that most desirable
for the simulation of composing procedures.

178 EXPERIMENT AL MUSIC

3. The process of musical composition was then defined in terms of the
extracting of orderly structures out of random materials by a process of
selection and rejection. Specifically, the process of musical composition
was regarded as a process of introducing redundancy into a random mu-
sical situation.

4. A technique was then devised for studying the processes of musical
composition by means of an automatic high-speed digital computer such
as the Illiac, the computer located at the University of Illinois. The tech-
nique depends upon the Monte Carlo method for generating random num-
bers. The notes of the scale are numbered in sequence upwards, and then
the computer is used to generate random numbers which can be interpreted
as the notes of random music. The effectiveness of musical rules in bring-
ing order into this musical texture can then be studied by operating upon
these numbers with mathematical processes which express rules of musical
composition.

5. It has been shown that historically well-known compositional tech-
niques, problems of current interest to composers, and even more novel
processes of composition can all be investigated by this approach. In par-
ticular, four Experiments were completed in which the following four sets
of problems were successively examined: (a) The writing of simple dia-
tonic melodies and the writing of simple two-part four-part diatonic po-
lyphony, (b) The writing of four-part first-species counterpoint. This
problem was studied to provide an example of how a well-known compo-
sitional technique can be adapted to computer processing, (c) A study
of the programming of rhythm, dynamics, playing instructions, random
and simple chromatic music, and of interval and tone rows was carried
out. This was done to demonstrate how a computer might be applied to
modern compositional problems, (d) A more abstract formulation for
musical composition, that is, the application of certain techniques of prob-
ability theory and, more particularly, of information theory to produce a
number of examples of Markoff chain music, was examined. This last
project was carried out to initiate a study of whether a more fundamental
basis than the conventional rules of composition might exist for imparting
order to musical structures.

6. Enough musical output was produced by the computer to permit
the assembly of a presentation of the experimental results in the form of
a four-movement composition for string quartet, which we have entitled
the Illiac Suite for String Quartet.

7. Lastly, some suggestions, mostly based upon the results contained in

SOME FUTURE MUSICAL APPLICATIONS 179

the /lilac Suite, have been presented to indicate how the present work
might be extended both in terms of immediate tasks and in terms of
longer-range, more speculative projects. Specific examples of applications
in the fields of music analysis and music composition were proposed.

In retrospect, it is interesting to compare the techniques applied to pro-
duce music by means of the Illiac with some of the comments of Stravinsky
in regard to musical composition quoted in Chapter 3. When Stravinsky
spoke of art as "the contrary of chaos," and said that "we feel the neces-
sity to bring order out of chaos," that "tonal elements become music only
by virtue of their being organized," and that we must "proceed by elimina-
tion— to know how to discard," he was speaking, of course, primarily in
general terms rather than in terms of detailed specific method. However,
it is pleasant to note how accurately these ideas do indeed seem to express
the logic inherent in the process of musical composition and how, perhaps
for the first time, some considerable measure of quantitative significance
can be attached to aesthetic commentary. The net result in the long run
can only lead toward a clarification of aesthetic criteria, the improvement
of musical understanding, and the production of new types of musical
compositions.

Appendix

On the pages following, the full score of the llliac Suite is reproduced
by kind permission of New Music Editions, 250 West 57th Street, New
York. In reprinting this score, we have made two changes in Experiment
Two (page 187). The first change is in the cello part at (F), measures 46
through 49 inclusive. A transcription error occurred here and the correct
notes have been inserted. The second change is at (H), also on page 187.
The parts have been rearranged to the original distribution as generated
by the computer. The distribution originally included in the llliac Suite

had been used in an attempt to eliminate 4 chords arising solely as a

result of transposition for the various instruments, but since this is a
mechanical effect leading to violations of this rule rather than intrinsic to
the problem being solved, it really serves little purpose, and hence, upon
reconsideration, we have eliminated this change.

181

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Hiller, Lejaren Arthur, 1924

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