Return to WOW/EM homepageComposers have long been fascinated by mathematical concepts in relation to music. The concept of "music of the spheres," dating back to Pythagorus, held the notion that humans were governed by the perfect proportions of the natural universe. This mathematical order may be seen in the musical interval choice and system of organization that was used by the ancient Greek culture.
Procedures that entail rules or provisions to govern the act of musical composition have been used since the Medieval period; these same principles have been applied in very specific methods to many of the recent computer programs developed for algorithmic composition.
The custom of borrowing systems of organization from outside the musical realm has been firmly established in compositional practice. However, as computer music composers have begun to appropriate highly specific technical terms from disciplines once thought to be completely unrelated to music, the definitions of these terms have become convoluted and clouded.
The term "algorithm" has been adopted from the fields of computer science and information science; however, in some cases its misappropriation has caused confusion in meaning. It is therefore necessary to clarify the underlying misconceptions that have accompanied the term "algorithmic composition" in order to eliminate the growing sense of ambiguity.
Compositions based on strict algorithmic processes without the use of computer technology have been produced. Much of the early work by composer/multimedia artist Ron Pellegrino involved some algorithmic processes. For instance, Metabiosis is an installation from 1972. The installation was set-up with lenses that sensed a change in air flow. A light source was projected onto the lenses, so that as they moved, light was cast to varying parts of the room. On the walls were photoresistors that measured differences in light intensity; the energy coming from the photocells was then sent as control voltages to several analog synthesizers. The voltages were then used to trigger changes in frequency, timbre, and amplitude. The algorithm here involved the testing and measurement of air flow patterns as each new audience member entered or left the performance space. (Pellegrino, The Electronic Arts of Sound and Light)
A more recent example of the use of non-computer algorithms occurred at the 14th Annual Bowling Green Festival of New Music and Art (1993), where Pauline Oliveros directed a piece in which each member of the audience was to fill out two cards--one with a sound production instruction and one with a listening instruction. Each person was to draw a single card and perform that action until such a time that she/he chose to exchange cards. The length of the performance was predetermined by Oliveros who served as a monitor and guide. Both Pellegrino and Oliveros used algorithmic procedures in order to obtain changes in musical material; however, neither of these examples makes use of a computer.
Historically and currently, there are three major approaches to algorithmic composition with computers: 1) algorithms for sound synthesis; 2) algorithms for compositional structure; and 3) algorithms for the correlation of sound synthesis with structure. Sound synthesis algorithms have been used in a variety of ways, from the calculation of complex waveforms (building sounds), to the evolution of timbre development over time.
Some of the algorithmic programs and compositions specify score information only. Score information includes pitch, duration, and dynamic material, whether written for acoustic and/or electronic instruments. That is, there are instances in which a composer makes use of a computer program to generate the score while the instrumental selection has been predetermined as either an electronic orchestra or a realization for acoustic instruments. Other algorithmic programs specify both score and electronic sound synthesis. In this instance, the program is used not only to generate the score, but also the electronic timbres to be used in performance.
The term algorithm has been in use for over a millennium; its etymology shall be traced in order to clarify its use in musical composition. In the ninth century, there lived an Arabic mathematician and writer named Abu 'Abdullah Muhammad ibn Musa of Khuwarizm. His most distinguished book, entitled Kitab al-Jabr wal-Muqabala (Rules of Reintegration and Reduction), was the basis for the standardization of Arabic numerals in European mathematics. Al-Jabr was later interpreted as the word, "algebra." The Latin stem of algorithm was most likely inherited from this mathematician's homeland, al-Khuwarizm (of Khuwarizm) and termed, "algorismus." The general term, algorism, means "the Arabic or decimal system of numerals;" it was used in literature dating back before 1200 B.C.E., and as may be deduced, it is one of the possible predecessors of the term algorithm.
The two more immediate sources of the term algorithm are: 1) a borrowed interpretation of the French term, "algorithme," or 2) an independent pronunciation of algorism.9 However, a comparison of these two root terms reveals that the French algorithme has a more specific relationship to "process," whereas, algorism is a more generic term applied to numerals. The distinction of process is important to the modern use of the term algorithm.
The Oxford English Dictionary presents three separate uses of the term algorithm. The first definition is used as a synonym for "algorism," previously covered. The remaining two definitions are quite interesting and require closer examination.
The second definition is related to the science of mathematics: "a process, or set of rules, usually one expressed in algebraic notation, now used especially in computing, machine translation and linguistics." When applied to the study of mathematics, the term has appeared in print since roughly 1938. It is not until ca. 1960 that it appears in reference to computer science.
Perhaps the most enlightening discovery for this research is the third definition from The Oxford English Dictionary; it is concerned with algorithm as used in the field of medicine: "a step-by-step procedure for reaching a clinical decision or diagnosis, often set out in the form of a flow chart, in which the answer to each question determines the next question to be asked." The term algorithm in this case pertains to diagnosis of illness; however, the use of a flow chart to determine the order of questions is very much aligned with a network of information from which to choose, a practice often associated with music composition, algorithmic or otherwise. Therefore, the medical use has a more specific denotation and is more directly related to use of algorithms in music than the two earlier definitions.
There are two other developmental definitions of algorithm, each belonging to a different field of study--one to mathematics and computer science, the other to medicine. The basis of both definitions is problem-solving using reductional mathematical procedures.
Early algorithmic compositions were driven exclusively by collaboration between scientists and musicians; therefore, the use of scientific terminology normally associated with computer science provided a uniform understanding of the algorithm. In the more recent past, composers have attempted to use and to apply alternative scientific and mathematical terminology to processes of musical composition. Terms such as "system" and "mapping" have also been used by composers such as Milton Babbitt, Otto Laske, Barry Truax, and Cosmin Georgescu and Mario Georgescu to describe the relationship between the rules of the "system" and the resultant musical composition. In 1976 Truax stated:
Also included in the discussion of systems for musical composition is a 1990 Interface article written by Cosmin Georgescu and Mario Georgescu. This article attempted to define a system of
To further define the term algorithm, several scientific definitions must be examined. The Encyclopedia of Computer Science and Engineering defines an algorithm as:
Given both the problem and the device, an algorithm is the precise characterization of a method of solving the problem, presented in a language comprehensible to the device. In particular, an algorithm is characterized by these properties:
Very simply, an algorithm is a process that solves a problem in a step-by-step manner either through redistribution, recursion, or branching. The solution must be found in a finite number of steps. In musical applications, algorithms may be thought of as procedures that test potential compositional material for its appropriateness within the given context. Pitch, duration, intensity, and other sound and structural qualities may be chosen according to a set of questions and answers. A basic example of an algorithm is:
Webster's College Dictionary defines algorithm as:
The five types, or processes, of algorithms listed in chapter 1--stochastic, chaotic, rule-based, grammars, and artificial intelligence--have been used by both the scientific and the musical communities. These processes are briefly defined in the following paragraphs.
Stochastic
Stochastic music and its related areas of statistics and determinism governed the earliest experiments in algorithmic composition. The first comprehensive documentation of stochastic music generated with the aid of a computer was presented by composer Iannis Xenakis in his book Musiques Formelles (Formalized Music), published in 1963, and later translated into English and reprinted. In Formalized Music, Xenakis explained the terms associated with stochastic processes:
[a stochastic process is] . . . an asymptotic evolution towards a stable state, towards a kind of goal, of stochos, whence comes the adjective "stochastic."
Statistics is the science of measuring distribution of numerical data across a broad spectrum. When data is distributed equally within the spectrum, it is said to be uniform. If, however, there are certain portions of the data that are more prevalent than others, data is weighted or biased. One manner of statistical analysis that has been frequently used in musical composition is Markov Analysis or Markov Chains. Named for the mathematician Andre Andreevich Markov (1856-1922), Markov Chains were conceived as a means by which decisions could be made based on probability. Information is linked together in a series of states based on the probability that state A will be followed by state B. The process is continually in transition because state A is then replaced by state B which continues to look at the probability of being followed by yet another state B.
The so-called orders of Markov Analysis indicate the relationship between states. For instance, zeroth-order analysis assumes that there are no relationships between states; that is, the relationship between any two states is random. First-order analysis simply counts the frequency with which specific states occur within the given data. Second-order analysis examines the relationships between any two consecutive states (e.g., what is the probability that the state B would follow state A?). Third-order analysis determines the probability of three consecutive states occurring in a row (e.g., what is the probability that state A would be followed by state B, would be followed by state C?). Fourth-order analyzes the chance of four states following each other. Composer/scientist Lejaren Hiller made use of Markov Chains, statistical analysis, and stochastic procedures in algorithmic composition beginning in the late 1950s.
The earliest composition programs were almost exclusively driven by stochastic processes. (See Lejaren Hiller, James Tenney, and Iannis Xenakis)
Since the early 1980s, developments in stochastic composition have been associated with composer Charles Ames, a student of Lejaren Hiller. Ames has documented processes of statistical analysis for distribution of numerical data. In his article, "A Catalog of Statistical Distributions: Techniques for Transforming Random, Determinate and Chaotic Sequences," Ames details a variety of distributions that have been used in automated and algorithmic music composition.
Chaotic
The second category of algorithmic process is chaos. Since the 1970s basic principles of the irregularities in nature have been studied by the scientific community, and by the 1980s chaos was the focus of a great deal of attention.
The new science has spawned its own language, an elegant shop talk of fractals and bifurcations, intermittencies and periodicities, folded-towel diffeomorphisms and smooth noodle maps. . . To some physicists chaos is a science of process rather than state, or becoming rather than being.
One subcategory of chaotic structures that has come to the forefront since ca. 1975 is fractals. Fractals are recursive and produce 'parent-child' relationships in which the offspring replicate the initial structure. Seen in visual art as smaller and smaller offshoots from the original stem, fractals were categorized by Benoit Mandelbrot in his book, The Fractal Geometry of Nature.
The following definition from Mandelbrot is offered:
The scientists who have been most associated with the study of chaos include: Benoit Mandelbrot, Edward Feigenbaum, M. Hénon, and E.N. Lorenz. The underlying principles of chaos may best be thought of in terms of natural, seemingly disorderly designs.
Composers who have applied the mathematical principles involved in studying chaos are Rick Bidlack, Tommaso Bolognesi, Charles Dodge, James Harley, Gary Lee Nelson, and Rodney Waschka II.
Rule-based
The third category of algorithmic process found in music composition programs is incorporated in rule-based systems. Many rule-based methods were first found in automatic composition applications. An elementary example of a rule-based process would center around a series of tests, or rules, through which the program progresses. These steps are usually constructed in such a way that the product of the steps leads to the next new step.
William Schottstaedt wrote about his automatic species counterpoint program in Current Directions in Computer Music Research. Schottstaedt's automatic composition programs write music based on rules from Fux' Gradus ad Parnassum. The program is built around almost seventy-five rules, such as "Parallel fifths are not allowed" and "Avoid tritones near the cadence in lydian mode."
Schottstaedt assigned a series of 'penalties' for breaking the rules. These penalties are weighted based on the fact that Fux indicated that there were some rules that could never be broken, but others did not have to be adhered to as vehemently. As penalties accumulate, the program abandons its current branch of rules and backtracks to find a new solution.
Kemal Ebcioglu wrote an automatic system called CHORAL that generates four-part chorales in the style of J.S. Bach.28 The program incorporates three hundred fifty rules, as well as an overview of principles in Schenkerian analysis, to guide the computer in its decision making. Although this is primarily a rule-based composition system, the program also makes use of 'short-cuts' to cut down on time in the process.
Grammars
A fourth category of algorithmic processes may be found in the study of linguistics and grammars. Linguistics is an attempt to identify how language functions--what are the components, how do the components function as a single unit, and how do the components function as single entities within the context of the larger unit. Beyond the linguistic use of the word grammar, composers have used modifiers such as generative, formal, and cultural. Each of the word variations coincides with a particular 'innovation' set forth by the individual programmers. Quite often the terms have identical meanings--only the phrasing differs. Marvin Minsky diagrams the process that must take place in a grammar:
The term generative grammar is most closely aligned with the original linguistic meaning of grammar. "Linguistic theory models this unconscious knowledge [of speech] by a formal system of principles or rules called a grammar, which describes (or 'generates') the possible sentences of the language." Curtis Roads has made a distinction between the specific use of generative grammars and the more open-ended field of algorithmic composition in that "Generative modeling of music can be distinguished from algorithmic composition on the basis of different goals. While algorithmic composition aims at an aesthetically satisfying new composition, generative modeling of music is a means of proposing and verifying a theory of an extant corpus of compositions or the competence that generated them."
Formal grammars have been defined by composer Kevin Jones as a ". . . set of symbols; a set of terminals or events that correspond to the event space in the schemes described. . . a set of production rules which specify ways in which symbols may be rewritten by combinations of symbols and terminals; and a starting symbol, which is used to begin the generative process."
In a third variation of the grammar algorithm, R. Douglas Reicken applied the term cultural grammars to his computer composition system, Wolfgang. Reicken defines cultural grammars as "systems that guide composition processes by way of a musical theory reflecting the music grammar of a given culture, such as, e.g. 19th [sic] century Western music."
Artificial Intelligence
The study of Artificial Intelligence in computer music warrants a comparison between Rule-based algorithms and Artificial Intelligence (AI) algorithms. Although rule-based systems make use of a series of steps to reach the logical conclusion, and may even involve back-tracking schemes in which the program is able to "back up" and begin a portion of or all of the steps again, this in no way implies that a rule-based program is learning. Computer science has "sought to find ways to embody knowledge in machines. . ." Knowledge-based, or AI, systems are those in which the computer algorithms are able to "learn" which solutions are retainable/usable by a series of comparisons with previously-stated material. This type of programming is often referred to as an "expert system" because "they're [expert systems] based on imitating the methods of particular human practitioners."
Within the boundaries of AI, there lie several areas that are closely related as subsets of the basic definition. An heuristic is one such subset. It may be thought of as an algorithmic short-cut; an heuristic solves the test based on intuition and logic. Algorithmic solutions, therefore, are used to produce goal solutions by means of a series of tests; heuristics solve a problem by intuition and anticipation of the forthcoming data. In some cases the exact difference between an algorithm and an heuristic has been debated, however, the following quotation is offered to aid in the differentiation.
Heuristic methods have sometimes been contrasted with algorithmic methods for finding problem solutions, and a certain amount of intellectual blood has been shed unnecessarily on this battlefield. Without getting into the subtleties of the disagreement, we observe that the term "algorithm" is used with considerable ambiguity in mathematics and logic. Under one commonly held definition, algorithms are decision procedures which are guaranteed to produce the solution being sought, given enough time. . . But algorithms (under this concept) are known, or practical, for only a very small subset of all interesting problems one would like to have computers solve. Over the spectrum of the broader class, heuristic methods seem to offer more general applicability.
One application of AI composition is found in a system developed by Charles Ames and Michael Domino that composes in four different popular styles: "standard" jazz, Latin jazz, ragtime, and rock. The Cybernetic Composer program works in four layers--solo, background chords, bass line, and drums. Each layer works (almost) independently of the remaining lines. Pitch, rhythm, thematic structure, and other compositional choices are calculated by a series of rules.
A second subset of AI is connectionism and has been defined by authors Peter Todd and Gareth Loy as: "systems [that] employ "brain style" computation, capitalizing on the emergent power of a large collection of individually simple interconnected processors operating and cooperating in a parallel distributed fashion."
Composer Otto Laske's research in the area of AI has been essential in the clarification of AI and music, as has the work of scientist Marvin Minsky. For a criticism and further explanation of connectionism and artificial intelligence, see Laske, "Connectionist Composition" with replies from Peter M. Todd and D. Gareth Loy, and Laske and Minsky, "Foreword: A Conversation with Marvin Minsky."
A third subcategory of AI is found in Neural Networks. As defined by D. Gareth Loy, neural networks differ from traditional AI applications because "[neural networks] . . . do not require such explicit symbolic representations to solve problems, however. Their knowledge is represented by the connection strengths between processing elements in a network, and the mutual reinforcement or inhibition of elements by other elements."
Conclusion
Because of the incorporation of scientific terms into the vocabulary of composers whose interests lie in computer-generated music, the literature and definitions have become clouded. The earlier types of algorithmic music composition that used stochastic and chaotic methods, lent themselves well to definition and categorization; however, the distinction between rule-based, grammars, and artificial intelligence algorithms in the more recent past has led to some confusion because of the lack of consistent terminology. The use of so-called intelligent programs in the creation of music that leaves many composers and programmers open to the question of what is intelligence?
The term "rule-based" does not imply a program in which learning takes place; however, several authors go so far as to apply the more generic term rule-based in place of the more specific, expert system (a term used in artificial intelligence systems), and use this as a one-for-one substitution. A program may use rules as a method for honing choice; however, the use of such rules does not constitute an expert system in which branching systems are employed to make choices based on logic and reason. The mere use of rules does not indicate intelligence. Authors Peter Desain and Henkjan Honing have stated: "Expert systems or rule-based systems accumulate knowledge in the form of a relatively large collection of rules. They describe explicitly what to do in a large collection of specific cases. Extra rules are added to model certain interaction or take care of unwanted interactions. . . a rule-based system is capable of reasoning in cases where human beings cannot oversee the consequences any more. . ."
What Desain and Honing are describing is an expert system. Their definitions incorporate aspects of rule-based systems, but are not limited to the use of rules. Instead, they are describing a system whereby the computer has knowledge and is able to solve compositional problems with reason, a system of artificial intelligence.
The need for a standardization of terminology exists. While musicians have attempted to incorporate these terms into a musical vocabulary, most have drawn upon parts of the original definitions, rather than the entire original meaning, and modified the meanings to suit a musical interpretation. One possible solution to the problem of (mis)interpretation of algorithmic uses in computer music is to develop a new set of meanings and definitions.
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Copyright September 1996,
updated February 2004.
Kristine H. Burns,
Florida International University
Questions? Contact me.