By Michael Edwards

Communications of the ACM,

Vol. 54 No. 7, Pages 58-67
Comments (5)


In the west, the layman’s vision of the creative artist is
largely bound in romantic notions of inspiration sacred or
secular in origin. Images are plentiful; for example, a man
standing tall on a cliff top, the wind blowing through his long
hair, waiting for that particular iconoclastic idea to arrive
through the ether.
a Tales, some even
true, of genii penning whole operas in a matter of days, further
blur the reality of the usually slowly wrought process of
composition. Mozart, with his celebrated speed of writing, is a
famous example who to some extent fits the cliché, though
perhaps not quite as well as legend would have

Non-specialists may be disappointed that composition includes
seemingly arbitrary, uninspired formal methods and
calculation.c What we shall see here
is that calculation has been part of the Western composition
tradition for at least 1,000 years, This article outlines the
history of algorithmic composition from the pre- and post-digital
computer age, concentrating, but not exclusively, on how it
developed out of the avant-garde Western classical tradition in
the second half of the 20th century. This survey is more
illustrative than all-inclusive, presenting examples of
particular techniques and some of the music that has been
produced with them.

Back to Top

A Brief History

Models of musical process are arguably natural to human
musical activity. Listening involves both the enjoyment of the
sensual sonic experience and the setting up of expectations and
possibilities of what is to come: musicologist Erik Christensen
described it as follows: “Retention in short-term memory permits
the experience of coherent musical entities, comparison with
other events in the musical flow, conscious or subconscious
comparison with previous musical experience stored in long-term
memory, and the continuous formation of expectations of coming
musical events.”

This second active part of musical listening is what gives
rise to the possibility and development of musical form; composer
György Ligeti wrote, “Because we spontaneously compare any
new feature appearing in consciousness with the features already
experienced, and from this comparison draw conclusions about
coming features, we pass through the musical edifice as if its
construction were present in its totality. The interaction of
association, abstraction, memory, and prediction is the
prerequisite for the formation of the web of relations that
renders the conception of musical form

For centuries, composers have taken advantage of this property
of music cognition to formalize compositional structure. We
cannot, of course, conflate formal planning with algorithmic
techniques, but that the former should lead to the latter was, as
I argue here, an historical inevitability.

Around 1026, Guido d’Arezzo (the inventor of staff notation)
developed a formal technique to set a text to music. A pitch was
assigned to each vowel so the melody varied according to the
vowels in the text.22 The 14th and
15th centuries saw development of the quasi-algorithmic
isorhythmic technique, where rhythmic cycles (talea) are
repeated, often with melodic cycles (color) of the same or
differing lengths, potentially, though not generally in practice,
leading to very long forms before the beginning of a rhythmic and
melodic repeat coincide. Across ages and cultures, repetition,
and therefore memory (of short motifs, longer themes, and whole
sections) is central to the development of musical form. In the
Western context, this repetition is seen in various guises,
including the Classical rondo (with section structures, such as
ABACA); the Baroque fugue; and the Classical sonata form, with
its return not just of themes but to tonality, too.

Compositions based on number ratios are also found throughout
Western musical history; for example, Guillaume Dufay’s
(1400–1474) isorhythmic motet Nuper Rosarum Flores,
written for the consecration of Florence Cathedral, March 25,
1436. The temporal structure of the motet is based on the ratios
6:4:2:3, these being the proportions of the nave, the crossing,
the apse, and the height of the arch of the cathedral. A subject
of much debate is how far the use of proportional systems was
conscious on the part of various composers, especially with
regards to Fibonacci numbers and the Golden
Section.d Evidence of Fibonacci
relationships haas been found in, for instance, the music of
Schubert,19 and
Bartók,27 as well as in
various other works of the 20th

Mozart is thought to have used algorithmic techniques
explicitly at least once. His Musikalisches
Dice”)e uses musical fragments that
are to be combined randomly according to dice throws (see
Figure 1). Such formalization procedures are not
limited to religious or art music. The Quadrille Melodist,
sold by Professor J. Clinton of the Royal Conservatory of Music,
London (1865) was marketed as a set of cards that allowed a
pianist to generate quadrille music (similar to a square dance).
The system could apparently make 428 million

Right at the outset of the computer age, algorithmic
composition moved straight into the popular, kit-builder’s
domain. The Geniac Electric Brain allowed customers to build a
computer with which they could generate automatic tunes (see
Figure 2).36 Such
systems find their modern counterpart in the automatic musical
accompaniment software Band-in-a-Box

The avant-garde. After World War II, many Western
classical music composers continued to develop the
serialf technique invented by Arnold
Schönberg (1874–1951) et al. Though generally seen as
a radical break with tradition, in light of the earlier
historical examples just presented, serialism’s detailed
organization can be viewed as no more than a continuation of the
tradition of formalizing musical composition. Indeed, one of the
new generation’s criticisms of Schönberg was that he
radicalized only pitch structure, leaving other parameters (such
as rhythm, dynamic, even form) in the 19th
century.6 They looked to the music of
Schönberg’s pupil Anton von Webern for inspiration in
organizing these other parameters according to serial principles.
Hence the rise of the total serialists: Boulez, Stockhausen,
Pousseur, Nono, and others in Europe, and Milton Babbitt and his
students at Princeton.g

Several composers, notably Xenakis (1922–2001) and
Ligeti (1923–2006), offered criticism of and alternatives
to serialism, but, significantly, their music was also often
governed by complex, even algorithmic,
procedures.h The complexity of new
composition systems made their implementation in computer
programs ever more attractive. Furthermore, development of
software algorithms in other disciplines made cross-fertilization
rife. Thus some techniques are inspired by systems outside the
realm of music (such as chaos theory (Ligeti,
Désordre), neural networks (Gerhard E. Winkler,
Hybrid II “Networks”),39 and
Brownian motion (Xenakis, Eonta).

Much of the resistance to algorithmic
composition that persists to this day stems from the misguided
bias that the computer, not the composer, composes the

Back to Top

Computer-Based Algorithmic Composition

Lejaren Hiller (1924–1994) is widely recognized as the
first composer to have applied computer programs to algorithmic
composition. The use of specially designed, unique computer
hardware was common at U.S. universities in the mid-20th century.
Hiller used the Illiac computer at the University of Illinois,
Urbana-Champaign, to create experimental new music with
algorithms. His collaboration with Leonard Isaacson resulted in
1956 in the first known computer-aided composition, The Illiac
Suite for String Quartet
, programmed in binary, and using,
among other techniques, Markov
i in “random walk”
pitch-generation algorithms.38

Famous for his own random-process-influenced compositions, if
not his work with computers, composer John Cage recognized the
potential of Hiller’s systems earlier than most. The two
collaborated on HPSCHD, a piece for “7 harpsichords
playing randomly-processed music by Mozart and other composers,
51 tapes of computer-generated sounds, approximately 5,000 slides
of abstract designs and space exploration, and several
films.”16 It premiered at the
University of Illinois, Urbana-Champaign, in 1969. Summarizing
perspicaciously an essential difference between traditional and
computer-assisted composition, Cage said in an interview during
the composition of HPSCHD, “Formerly, when one worked
alone, at a given point a decision was made, and one went in one
direction rather than another; whereas, in the case of working
with another person and with computer facilities, the need to
work as though decisions were scarce—as though you had to
limit yourself to one idea—is no longer pressing. It’s a
change from the influences of scarcity or economy to the
influences of abundance and—I’d be willing to

Stochastic versus deterministic procedures. A basic
historical division in the world of algorithmic composition is
between indeterminate and determinate models, or those that use
stochastic/random procedures (such as Markov chains) and those
where results are fixed by the algorithms and remain unchanged no
matter how often the algorithms are run. Examples of the latter
are cellular automata (though they can be deterministic or
stochastic34); Lindenmayer Systems
(see the section on the deterministic versus stochastic debate in
this context); Charles Ames’s constrained search algorithms for
selecting material properties against a series of
constraints1; and the compositions of
David Cope that use his Experiments in Musical
system.10 The latter
is based on the concept of “recombinacy,” where new music is
created from existing works, thus allowing the recreation of
music in the style of various classical composers, to the shock
and delight of many.

Algorithmic composition is often viewed as a
sideline in contemporary musical activity, as opposed to a
logical application and incorporation of compositional technique
into the digital domain.

Xenakis. Known primarily for his instrumental
compositions but also as an engineer and architect, Iannis
Xenakis was a pioneer of algorithmic composition and computer
music. Using language typical of the sci-fi age, he wrote, “With
the aid of electronic computers, the composer becomes a sort of
pilot: he presses buttons, introduces coordinates, and supervises
the controls of a cosmic vessel sailing in the space of sound,
across sonic constellations and galaxies that he could formerly
glimpse only in a distant

Xenakis’s approach, which led to the Stochastic Music
(henceforth SMP) and radically new pieces (such as
Pithoprakta, 1956), used formulae originally developed by
scientists to explain the behavior of gas particles (Maxwell’s
and Boltzmann’s Kinetic Theory of
Gases).31 He saw his stochastic
compositions as clouds of sound, with individual
notesj as the analogue of gas
particles. The choice and distribution of notes was determined by
procedures involving random choice, probability tables weighing
the occurrence of specific events against those of others.
Xenakis created several works with SMP, often more than one with
the output of a single computer batch
process,k probably due to limited
access to the IBM 7090 he used. His Eonta
(1963–1964) for two trumpets, three tenor trombones, and
piano was composed with SMP. The program was applied in
particular to the creation of the massively complex opening piano

Like another algorithmic composition and computer-music
pioneer, Gottfried Michael Koenig (1926–), Xenakis had no
compunction adapting the output of his algorithms as he saw fit.
Regarding Atrées (1962), Xenakis’s biographer
Nouritza Matossian claims Xenakis used “75% computer material,
composing the remainder himself.”31
At least in Koenig’s Projekt 1
(1964)l Koenig saw transcription
(from computer output to musical score) as an important part of
the process of algorithmic composition, writing, “Neither the
histograms nor the connection algorithm contains any hints about
the envisaged, ‘unfolded’ score, which consists of instructions
for dividing the labor of the production changes mode, that is,
the division into performance parts. The histogram, unfolded to
reveal the individual time and parameter values, has to be split
up into voices.”24

Hiller, on the other hand, believed that if the output of the
algorithm is deemed insufficient, then the program should be
modified and the output
regenerated.34 Several programs that
facilitate algorithmic composition include direct connection to
their own or to third-party computer sound
generation.m This connection obviates
the need for transcription and even hinders this arguably
fruitful intervention. Furthermore, such systems allow the
traditional or even conceptual score to be redundant. Thus
algorithmic composition techniques allow a fluid and unified
relationship between macro-structural musical form and
micro-structural sound synthesis/processing, as evidenced again
by Xenakis in his Dynamic Stochastic Synthesis program
Gendy3 (1992).40

More current examples. Contemporary (late 20th century)
techniques tend to be hybrids of deterministic and stochastic
approaches. Systems using techniques from artificial intelligence
(AI) and/or linguistics are the
generative-grammarn-based system Bol
Processor software4 and expert systems
(such as Kemal Ebcioglu’s CHORAL11).
Other statistical approaches that use, say, Hidden Markov Models
(as in Jordanous and Smaill20), tend
to need a significant amount of data to train the system; they
therefore rely on and generate pastiche copies of the music of a
particular composer (that must be codified in machine-readable
form) or historical style. While naturally significant to AI
research, linguistics, and computer science, such systems tend to
be of limited use to composers writing music in a modern and
personal style that perhaps resists codification because of its
notational and sonic complexity and, more simply, its lack of
sufficient and stylistically consistent data—the so-called
sparse-data problem. But this is also to some extent indicative
of the general difficulty of modeling language and human
cognition; the software codification of the workings of a spoken
language understood by many and reasonably standardized is one
thing; the codification of the quickly developing and widely
divergent field of contemporary music is another thing
altogether. Thus we can witness a division between composers
concerned with creating new music with personalized systems and
researchers interested in developing systems for machine learning
and AI. The latter may quite understandably find it more useful
to generate music in well-known styles not only because there is
extant data but also because familiarity of material simplifies
some aspects of the assessment of results. Naturally though, more
collaboration between composers and researchers could lead to
fruitful, aesthetically progressive results.

Outside academia. Application of
algorithmic-composition techniques is not restricted to academia
or to the classical avant garde. Pop/ambient musician Brian Eno
(1948–) is known for his admiration and use of generative
systems in Music for Airports (1978) and other pieces. Eno
was inspired by the American minimalists, in particular Steve
Reich (1936–) and his tape piece It’s Gonna Rain
(1965). This is not computer music but process music, whereby a
system is devised—usually repetitive in the case of the
minimalists—and allowed to run, generating music in the
form of notation or electronic sound. Eno said about his
Discreet Music (1975), “Since I have always preferred
making plans to executing them, I have gravitated towards
situations and systems that, once set into operation, could
create music with little or no intervention on my part. That is
to say, I tend towards the roles of planner and programmer, and
then become an audience to the

Improvisation systems. Algorithmic composition
techniques are, then, clearly not limited to music of a certain
aesthetic or stylistic persuasion. Nor are they limited to a
completely fixed view of composition, where all the pitches and
rhythms are set down in advance. George Lewis’s Voyager is
a work for human improvisors and “computer-driven, interactive
‘virtual improvising orchestra.'”29
Its roots are, according to Lewis, in the African-American
tradition of multi-dominance, described by him (borrowing from
Jeff Donaldson) as involving multiple simultaneous structural
streams, these being in the case of Voyager at “both the
logical structure of the software and its performance
articulation.”29 Lewis programmed
Voyager in the Forth language popular with computer
musicians in the 1980s. Though in Voyager the computer is
used to analyze and respond to a human improviser, such input is
not essential for the program to generate music (via
MIDIo). Lewis wrote, “I conceive a
performance of Voyager as multiple parallel streams of
music generation, emanating from both the computers and the
humans—a nonhierarchical, improvisational, subject-subject
model of discourse, rather than a stimulus/response
setup.”29 A related improvisation
system, OMAX, from the Institut de Recherche et Coordination
Acoustique/Musique in Paris, is available within the now more
widely used computer-music systems Max/MSP and Open-Music. OMAX
uses AI-based machine-learning techniques to parse incoming
musical data from human musicians, then the results of analysis
to generate new material in an improvisatory

slippery chicken. In my own case, work on the
specialized algorithmic composition program slippery
13 is ongoing since 2000.
Written in Common Lisp and its object-oriented extension, the
Common Lisp Object System, it is mainly deterministic but also
has stochastic elements. It has been used to create musical
structure for pieces since its inception and is now at the stage
where it can generate, in a single pass, complete musical scores
for traditional instruments or with the same data write sound
files using samplesp or MIDI file
realizations of the instrumental
score.q The project’s main aim is to
facilitate a melding of electronic and instrumental sound worlds,
not just at the sonic but at the structural level. Hence certain
processes common in one medium (such as audio slicing and
looping) are transferred to another (such as the slicing up of
notated musical phrases and instigation of sub-phrase loops).
Also offered are techniques for innovative combination of
rhythmic and pitch data, which is, in my opinion, one of the most
difficult aspects of making convincing musical algorithms.

Lindenmayer systems. Like writing a paper, composing
music, especially with computer-based algorithms, is most often
an iterative process. Material is first set down in raw form,
only to be edited, developed, and reworked over several passes
before the final refined form is achieved. For the composer,
stochastic procedures, if not simply to be used to generate
material to be reworked by hand or in some other fashion,
represent particular problems. If an alteration of the algorithm
is deemed necessary, no matter how small, then rerunning the
procedure is essential. But rerunning will generate a different
set of randomly controlled results, perhaps now lacking some
characteristics the composer deemed musically significant after
the first pass.r

Deterministic procedures can be more apposite. For instance,
Lindenmayer Systemss (henceforth
L-Systems) whose simplicity and elegance yet resulting
self-similarity make them ideal for composition. Take a simple
example, where a set of rules is defined and associates a key
with a result of two further keys that in turn form indices for
an arbitrary number of iterations of key substitution (see
Figure 3).

Given a starting seed for the lookup and substitution
procedure (or rewriting, as it is more generally known), an
infinite number of results can be generated (see Figure

Self-similarity is clear when larger result sets are produced;
see Figure 5, noting the repetitions of sequences
(such as 2 1 1 3 and 2 3 2 3). These numbers can be applied to
any musical parameter or material, including pitch, rhythm,
dynamic, phrase, and harmony. Seen musically, the results of such
simple L-Systems tend toward stasis in that only results that are
part of the original rules are returned, and all results are
present throughout the returned sequence. However, the result is
dependent on the rules defined: subtle manipulations of more
complex/numerous rules can result in musically interesting
developments. For instance, composers have used more finessed
L-Systems—where the result of a particular rule may be
dependent on a sub-rule—leading to more organic, developing
forms. Hanspeter Kyburz’s (1960–) Cells for
saxophone and ensemble is an example. Martin
Supper38 described Kyburz’s use of
L-Systems, using results from 13 generations of L-System rewrites
to select pre-composed musical motifs. Like Hiller before him,
Kyburz uses algorithmic composition techniques to generate and
select musical material for the preparation of instrumental
scores. However, the listener is probably unaware of the
application of software in the composition of such music.

Transitioning L-Systems: Tramontana. As I tend to write
music that is concerned with development and transition, my use
of L-Systems is somewhat more convoluted. My own
Tramontana (2004) for viola and
computer14 uses L-Systems in its
concluding section. Unlike normal L-Systems, however, I employ
Transitioning L-Systems, my own invention, whereby the numbers
returned by the L-System are used as lookup indices into a table
whose result depends on transitions between related but
developing material. The transitions themselves use
Fibonacci-based “folding-in” structures where the new material is
interspersed gradually until it becomes dominant; for example, a
transition from material 0 to material 1 might look like
Figure 6.

In the case of the concluding section of Tramontana,
there is slow development from fast, repeated chords toward more
and more flageoletst on the C and G
strings. Normal pitches and half
flageoletsu begin to dominate, with a
tendency toward more of the former. At this point, flageolets on
the D string are also introduced. All these developments are
created with transitioning L-Systems. The score (see
Figure 7 for a short extract) was generated with
Bill Schottstaedt’s Common Music Notation software, taking
advantage of its ability to include algorithmically placed
nonstandard note heads and other musical signs. Perhaps worth
noting is that even before I began work with computers, I was
already composing in such a manner. Now, with slippery
algorithms, these structures can be programmed to
generate the music, test, re-work, and re-generate. A particular
advantage of working with the computer here is that it is a
simple matter to extend or shorten sections, something that
would, with pencil and paper, be so time-consuming as to be

CURTIS ROADS, 1996: It takes a good composer
to design algorithms that result in music that captures the

Musical Example: Ligeti’s Désordre
György Ligeti (1923–2006) is known to the general
public mainly through his music in several Stanley Kubrick films:
2001: A Space Odyssey, which included Lux Aeterna
and Requiem (without Ligeti’s permission, prompting a
protracted but failed lawsuit); The Shining, which
included Lontano; and Eyes Wide Shut, which
included Musica Ricercata.

After leaving his native Hungary in the late 1950s, Ligeti
worked in the same studios as Cologne electronic music pioneers
Karlheinz Stockhausen and Gottfried Michael Koenig though
produced little electronic music of his own. However, his
interest in science and mathematics led to several instrumental
pieces influenced by, for example, fractal geometry and chaos
theory. But these influences did not lead to a computer-based
algorithmic approach.v He was quoted
in Steinitz37 saying, “Somewhere
underneath, very deeply, there’s a common place in our spirit
where the beauty of mathematics and the beauty of music meet. But
they don’t meet on the level of algorithms or making music by
calculation. It’s much lower, much deeper—or much higher,
you could say.”

Nevertheless, as a further example, we shall consider the
structure of György Ligeti’s Désordre from his
first book of Piano Etudes for several reasons:

Structures. The structures of Désordre
are deceptively simple in concept yet beautifully elegant in
effect, where the clearly deterministic algorithmic thinking
lends itself quite naturally to software implementation;

Algorithmic composition. Ligeti was a major composer,
admired by experts and non-experts alike, and is generally not
associated with algorithmic composition; indeed,
Désordre was almost certainly composed
“algorithmically” by hand, with pencil and paper, as opposed to
at a computer keyboard. As such, Désordre
illustrates the clear link in the history of composition to
algorithmic/computational thinking, bringing algorithmic
composition into mainstream musical focus; and

Algorithmic models. I have implemented algorithmic
models of the first part of Désordre in the
open-source software system Pure Data, which, along with
the following discussion, is based on analyses by Tobias
Kunze,26 used here with permission,
and Hartmut Kinzler.21 It is freely
downloadable from my Web site;
tinkering with the initial data states is instructive and

Désordre’s algorithms. The main argument of
Désordre consists of foreground and background

Foreground (accented, loud). Two simultaneous instances
of the same basic process, melodic/rhythmic, one in each hand,
both doubled at the octave, and white note (righthand) and
black-notew (pentatonic, lefthand)
modes; and

Background (quiet). Continuous, generally rising quaver
(eighth-note) pulse notes, centered between the foreground
octaves, one in each hand, in the same mode as the foreground

In the first part of the piece the basic foreground process
consists of a melodic pattern cycle consisting of the scale-step
shape in Figure 8. This cycle is stated on
successively higher (right-hand, 14 times, one diatonic step
transposition) and lower (lefthand, 11 times, two diatonic steps
transposition) degrees. Thus, a global, long-term movement is
created from the middle of the piano outward, to the high and low

The foreground rhythmic process consists of slower-moving,
irregular combinations of quaver-multiples that tend to reduce in
duration over the melodic cycle repeats to create an acceleration
toward continuous quaver pulses (see Figure

The similarity between the two hands’ foreground rhythmic
structure is obvious, but the duration of seven quavers in the
right hand at the end of cycle 1a, as opposed to eight in the
left, makes for the clearly audible decoupling of the two parts.
This is the beginning of the process of disorder, or chaos, and
is reflected in the unsynchronized bar lines of the score
starting at this point (see Figure 10).

In Désordre we experience a clear, compelling,
yet not entirely predictable musical development of rhythmic
acceleration coupled with a movement from the middle piano
register to the extremes of high and low, all expressed through
two related and repeating melodic cycles with slightly differing
lengths resulting in a combination that dislocates and leads to
metrical disorder. I invite the reader to investigate this in
more detail by downloading my software

Back to Top


There has been (and still is) considerable resistance to
algorithmic composition from all sides, from musicians to the
general public. This resistance bears comparison to the reception
of the supposedly overly mathematical serial approach introduced
by the composers of the Second Viennese School of the 1920s and
1930s. Alongside the techniques of other music composed from the
beginning of the 20th century onward, the serial principle itself
is frequently considered to be the reason the
music—so-called modern music, though now close to 100 years
old—may not appeal. I propose that a more enlightened
approach to the arts in general, especially those that present a
challenge, would be a more inward-looking examination of the
individual response, a deferral of judgment and acknowledgment
that, first and foremost, a lack of familiarity with the style
and content may lead to a neutral or negative audience response.
Only after further investigation and familiarization can
deficiencies in the work be

Algorithmic composition is often viewed as a sideline in
contemporary musical activity, as opposed to a logical
application and incorporation of compositional technique into the
digital domain. Without wishing to imply that instrumental
composition is in a general state of stagnation, if the computer
is the universal tool, there is surely no doubt that not applying
it to composition would be, if not exactly an example of Luddism,
then at least to risk missing important aesthetic developments
that only the computer can facilitate, and that other artistic
fields already take advantage of. That algorithmic thinking is
present in Western composition for at least 1,000 years has been
established. That such thinking should lend itself to
formalization in software algorithms was inevitable.

However, Hiller’s work and 1959 Scientific American
article17 led to much controversy and
press attention. Hostility to his
achievementsy was such that the
Grove Dictionary of Music and
z did not include an
article on it until shortly before his death in 1994. This
hostility arose no doubt more from a misperception of
compositional practice than from anything intrinsic to Hiller’s

Much of the resistance to algorithmic composition that
persists to this day stems from the misguided bias that the
computer, not the composer, composes the music. In the vast
majority of cases where the composer is also the programmer, this
is simply not true. As composer and computer musician Curtis
Roads pointed out more than 15 years ago, it takes a good
composer to design algorithms that result in music that captures
the imagination.34

Furthermore, using algorithmic-composition techniques does not
by necessity imply less composition work or a shortcut to musical
results; rather, it is a change of focus from note-to-note
composition to a top-down formalization of compositional process.
Composition is, in fact, often slowed by the requirement that
musical ideas be expressed and their characteristics encapsulated
in a highly structured and non-musical general programming
language. Learning the discipline of programming is itself a
time-consuming and, for some composers, an insurmountable

Perhaps counterintuitively, such formalization of personal
composition technique allows the composer to proceed from
concrete musical or abstract formal ideas into realms hitherto
unimagined, sometimes impossible to achieve through any other
means than computer software. As composer Helmut Lachenmann
wrote, “A composer who knows exactly what he wants, wants only
what he knows—and that is one way or another too
little.”35 The computer can help
composers overcome recreating what they already know by aiding
more thorough investigations of the material, once procedures are
programmed, modifications and manipulations are simpler than with
pencil and paper. By “pressing buttons, introducing coordinates,
and supervising the controls,” to quote Xenakis
again,40 the composer is able to
stand back and develop compositional material en masse, applying
procedures and assessing, rejecting, accepting, or further
processing results of an often-surprising nature. Algorithmic
composition techniques clearly further individual musical and
compositional development through computer programming-enabled
voyages of musical discovery.

Back to Top


1. Ames, C. Stylistic automata in Gradient.
Computer Music Journal 7, 4 (1983), 45–56.

2. Assayag, G., Bloch, G., Chemillier, M.,
Cont, A., and Dubnov, S. OMax brothers: A dynamic topology of
agents for improvization learning. In Proceedings of the First
ACM Workshop on Audio and Music Computing Multimedia
Barbara, CA). ACM Press, New York, 2006, 125–132.

3. Austin, L., Cage, J., and Hiller, L. An
interview with John Cage and Lejaren Hiller. Computer Music
Journal 16
, 4 (1992), 15–29.

4. Bel, B. Migrating musical concepts: An
overview of the Bol processor. Computer Music Journal 22,
2 (1998), 56–64.

5. Bewley, J. Lejaren A. Hiller: Computer
Music Pioneer
. Music Library Exhibit, University of Buffalo,

6. Boulez, P. Schönberg est mort.
Score 6 (Feb. 1952), 18–22.

7. Brümmer, L. Using a digital synthesis
language in composition. Computer Music Journal 18, 4
(1994), 35–46.

8. Chomsky, N. Syntactic Structures.
Mouton, The Hague, 1957.

9. Christensen, E. The Musical Timespace,
a Theory of Music Listening
. Aalborg University Press,
Aalborg, Denmark, 1996.

10. Cope, D. Experiments in Musical
. A-R Editions, Madison, WI, 1996.

11. Ebcioglu, K. An expert system for
harmonizing four-part chorales. Computer Music Journal 12,
3 (1988), 43–51.

12. Edwards, M. A Pure Data
implementation of Ligeti’s Désordre
. Open-source music

13. Edwards, M. Slippery chicken: A
Specialized Algorithmic Composition Program
. Unpublished
object-oriented Common Lisp software;

14. Edwards, M. Tramontana. Sheet
music, Sumtone, 2004;

15. Eisen, C. and Keefe, S.P., Eds. The
Cambridge Mozart Encyclopedia
. Cambridge University Press,
Cambridge, England, 2006.

16. The Electronic Music Foundation.

17. Hiller, L. Computer music. Scientific
American 201
, 6 (Dec. 1959), 109–120.

18. Holmes, T. Electronic and
Experimental Music
. Taylor & Francis Ltd, London,

19. Howat, R. Architecture as drama in late
schubert. In Schubert Studies, B. Newbould, Ed. Ashgate
Press, London, 1998, 168–192.

20. Jordanous, A. and Smaill, A.
Investigating the role of score following in automatic musical
accompaniment. Journal of New Music Research 38, 2 (2009),

21. Kinzler, H. and Ligeti, G. Decision and
automatism in Désordre 1er étude,
premier livre. Interface, Journal of New Music Research
, 2 (1991), 89–124.

22. Kirchmeyer, H. On the historical
construction of rationalistic music. Die Reihe 8 (1962),

23. Koenig, G.M. Project 1;

24. Koenig, G.M. Aesthetic integration of
computer-composer scores. Computer Music Journal 7, 4
(1983), 27–32.

25. Kramer, J. The Fibonacci series in 20th
century music. Journal of Music Theory 17 (1973),

26. Kunze, T. Désordre
(unpublished article);

27. Lendvai, E. Bela Bartók: An
Analysis of His Music
. Kahn & Averill, London, 1971.

28. Lerdahl, F. and Jackendorff, R. A
Generative Theory of Tonal Music
. MIT Press, Cambridge, MA,

29. Lewis, G. Too many notes: Computers,
complexity, and culture in Voyager. Leonardo Music Journal
(2000), 33–39.

30. Ligeti, G. Über form in der neuen
musik. Darmstädter Beiträge zur neuen Musik 10
(1966), 23–35.

31. Matossian, N. Xenakis. Kahn &
Averill, London, 1986.

32. Norden, H. Proportions in music.
Fibonacci Quarterly 2, 3 (1964), 219–222.

33. Prusinkiewicz, P. and Lindenmayer, A.
The Algorithmic Beauty of Plants. Springer-Verlag, New
York, 1990.

34. Roads, C. The Computer Music
. MIT Press, Cambridge, MA, 1996.

35. Ryan, D. and Lachenmann, H. Composer in
interview: Helmut Lachenmann. Tempo 210 (1999),

36. Sowa, J. A Machine to Compose Music:
Instruction Manual for GENIAC
. Oliver Garfeld Co., New Haven,
CT, 1956.

37. Steinitz, R. Music, maths & chaos.
Musical Times 137, 1837 (Mar. 1996), 14–20.

38. Supper, M. A few remarks on algorithmic
composition. Computer Music Journal 25, 1 (2001),

39. Winkler, G.E. Hybrid II:
. CD recording, 2003. sumtone cd1: stryngebite;

40. Xenakis, I. Formalized Music.
Pendragon, Hillsdale, NY, 1992.

Back to Top


Michael Edwards
is a Reader in Music Technology in the School of Arts, Culture
and Environment of the University of Edinburgh, Edinburgh,

Back to Top


a. I’m thinking in particular of Caspar
David Friedrich’s painting From the Summit in the Hamburg

b. Mozart’s compositional process is complex
and often misunderstood, complicated by myth, especially
regarding his now refuted ability to compose everything in his
15 and his own statements (such
as “I must finish now, because I’ve got to write at breakneck
speed—everything’s composed—but not written yet” in a
letter to his father, Dec. 30, 1780). Mozart apparently
distinguished between composing (at the keyboard, in sketches)
and writing (preparing a full and final score), hence the
confusion about the length of time taken to write certain pieces
of music.

c. For example, in the realm of pitch:
transposition, inversion, retrogradation, intervallic expansion,
compression; and in the realm of rhythm: augmentation,
diminution, addition.

d. Fibonacci was an Italian mathematician
(c.1170–c.1250) for whom the famous number series is named.
This is a simple progression where successive numbers are the sum
of the previous two: (0), 1, 1, 2, 3, 5, 8, 13, 21…
Ascending the sequence, the ratio of two adjacent numbers gets
closer to the so-called Golden Ratio (approximately 1:1.618).

e. Attributed to Mozart though not
officially authenticated despite being designated K. Anh. 294d in
the Köchel Catalogue of his works.

f. Serialism is an organizational system in
which pitches (first of all) are organized into so-called 12-tone
rows, where each pitch in a musical octave is present and,
ideally, equally distributed throughout the piece. This technique
was developed most famously by Schönberg in the early 1920s
at least in part as a response to the difficulty of structuring
atonal music, music with no tonal center or key (such as C

g. Here, we begin to distinguish between
pieces that organize pitch only according to the series
(dodecaphony) from those extending organization into music’s
other parameters—strictly speaking serialism, also known as
integral or total serialism.

h. For a very approachable introduction to
the musical thought of Ligeti and Xenakis, see The Musical
, chapter 2,
particularly pages 36–39.

i. First presented in 1906, Markov chains
are named for the Russian mathematician Andrey Markov
(1856–1922), whose research into random processes led to
his eponymous theory, and today are among the most popular
algorithmic composition tools. Being stochastic processes, where
future states are dependent on current and perhaps past states,
they are applicable to, say, pitch selection.

j. Notes are a combination of pitch and
duration, rather than just pitch.

k. Matossian wrote, “With a single 45-minute
program on the IBM 7090, he [Xenakis] succeeded in producing not
only eight compositions that stand up as integral works but also
in leading the development of computer-aided

l. Written to test the rules of serial music
but involving random decisions.

m. Especially modern examples (such as
Common Music, Pure Data, and SuperCollider).

n. Such systems are generally inspired by
Chomsky’s grammar models
8 and
Lerdahl’s and Jackendorff’s applications of such approaches to
generative music theory.28

o. Musical Instrument Digital Interface, or
MIDI, the standard music-industry protocol for interconnecting
electronic instruments and related devices.

p. Samples are usually short digital sound
files of individual or arbitrary number of notes/sonic

q. To accomplish this, the software
interfaces with parts of the open-source software systems Common
Music, Common Lisp Music, and Common Music Notation all freely
available from

r. This is a simplistic description. Most
stochastic procedures involve encapsulation of various tendencies
over arbitrarily large data sets, the random details of which are
insignificant compared to the structure of the whole. Still, some
details may take on more musical importance than intended, and
losing them may detrimentally affect the composition. The
composer could avoid such problems by using a random number
generator with fixed and stored seed, guaranteeing the
pseudo-random numbers are generated in the same order each time
the process is restarted. Better still would be to modify the
algorithm to take these salient, though originally unforeseen
features, into account.

s. Named for biologist Aristid Lindenmayer
(1925–1989) who developed this system (or formal language,
based on grammars by Noam Chomsky
that can model various natural-growth processes (such as those of

t. Familiar to guitarists, flageolets, and
harmonics are special pitches achieved by touching the string
lightly with a left-hand finger at a nodal point in order to
bring out higher frequencies related to the fundamental of the
open string by integer multiples.

u. Half flageolets are achieved by pressing
the string, as with a full flageolet, but not at a nodal point;
the result is a darker, dead-sounding pitch.

v. Ligeti’s son, Lukas, confirmed to me that
his father was interested conceptually in computers, reading
about them over the years, but never worked with them in

w. White and black here refer to the color
of the keys on the modern piano.

x. To paraphrase Ludger Brümmer, from
information theory we know that new information is perceived as
chaotic or interesting but not expressive. New information must
be structured before it can be understood, and, in the case of
aesthetic experience, this structuring involves comparison to an
ideal, or an established notion of

y. Concerning the reaction to The Illiac
, Hiller said “There was a great [deal] of hostility,
certainly in the musical world…I was immediately
pigeonholed as an ex-chemist who had bungled into writing music
and probably wouldn’t know how to resolve a dominant seventh
chord”; interview with Vincent Plush,

z. The Grove is the English-speaking
world’s most widely used and arguably most authoritative
musicological resource.

Back to Top


F1Figure 1. First part of Mozart’s Musikalisches
(“Musical Dice”): Letters over columns refer
to eight parts of a waltz; numbers to the left of rows indicate
possible values of two thrown dice; and numbers in the matrix
refer to bar numbers of four pages of musical fragments combined
to create the algorithmic waltz.

F2Figure 2. Part of an advertisement for The Geniac
Electric Brain
, a DIY music-computer kit.

F3Figure 3. Simple L-System rules.

F4Figure 4. Step-by-step generation of results from simple
L-System rules and a seed.

F5Figure 5. Larger result set from simple L-System

F6Figure 6. Fibonacci-based transition from material 0 to
material 1. Note the first appearance of 1 is at position 13,
with the next eight positions after that, the next again five
positions after that, and so on; all these numbers are so-called
Fibonacci numbers.

F7Figure 7. Extract beginning bar 293 of the author’s
Tramontana for viola and computer.

F8Figure 8. Foreground melodic pattern (scale steps) of

F9Figure 9. Foreground rhythmic pattern
(quaver/eighth-note durations) of

F10Figure 10. Désordre. First
system of score © 1986 Schott Music GmbH & Co. KG,
Mainz, Germany. Reproduced by permission. All rights

Back to top

ACM  0001-0782/11/0700  $10.00

Permission to make digital or hard copies of part or all of
this work for personal or classroom use is granted without fee
provided that copies are not made or distributed for profit or
commercial advantage and that copies bear this notice and full
citation on the first page. Copyright for components of this work
owned by others than ACM must be honored. Abstracting with credit
is permitted. To copy otherwise, to republish, to post on
servers, or to redistribute to lists, requires prior specific
permission and/or fee. Request permission to publish from or
fax (212) 869-0481.

The Digital Library is published by the Association
for Computing Machinery. Copyright © 2011
ACM, Inc.


Michael Edwareds

For anyone interested in this topic, I thought now would be a good time to start a blog and mailing list related to algorithmic composition:


This is my Youtube video of drum exercises – counting in 3, 5, 7, and Pi. It seemed appropriate to share here.

“Intermediate – Independence Exercise

Paradiddles in feet (RLRR LRLL) with groups of 3 ( RRR LLL) on Snare drum ( same note values as feet )

When comfortable and relaxed add counting patterns on top to free your mind from what your muscles are doing. This will also help your ears to be able to focus on other rhythms, melodies and patterns ( Guitar, Bass, Piano…… ) while playing your parts no matter how complicated.

Count groups of 3’s, 5’s, 7’s. Be aware of these patterns that repeat. When you play them long enough you will realize ,notice and locate certain parts or hits that line up with other pattern you are playing. DO NOT FORCE THESE TO SYNC. Them being spot on should be a result of you playing all given parts precisely to the “Time” as it happens.

Knowing this I like practicing counting patterns that don’t repeat because it makes it a lot harder for your body to subconsciously force parts to sync. I picked Pi -3.1415926535897….”

Bartosz Dobrzelecki

I have created a collaborative playlist on Spotify that includes most of the works and composers mentioned in the text. Feel free to append more pieces relevant to the subject.

Emre Sevinc

I had great pleasure reading this article (having prepared presentations and reports on AI and music, as well as having dabbled in Common Music, I was happy to learn some surprising points such as strange and negative reactions to Hiller). But I also have to admit I’m a bit surprised to see no reference to Strasheela, “a highly expressive constraint-based music composition system”:


Nice article, some of David Cope’s more recent work is also very interesting to listen to.

There are also a series of algorithmic composition tutorials here

Displaying all 5 comments

Source link