Table of Contents:
- Large-scale applications (pitch, dynamics, rhythm, instrumentation, etc.)
- Small-sacle applications (sound synthesis and timbre construction)
"Chaos" is a term adopted by the field of physics in the last 20 years or so to describe inconsistencies and nonlinearities found in many natural phenomena compared with the theories and equations that have been classically used to describe them:
"The world is nonlinear. This fact has a profound effect on the behavior of most dynamical and evolutionary systems: their future behavior cannot be predicted in the long run. The startling point is that even very simple classical systems are afflicted and not just such complex ones as the evolution of mankind or the weather. Indeed, to give an explicit example, even systems as 'simple' as the driven damped pendulum... suffer from unpredictability in certain ranges of parameter space. Deterministic systems of this kind are called chaotic, a term adapted from common language to have a notion for the stunning unexpected irregularity displayed by such systems. (Physics is full of these adaptations, e.g. force, energy, etc.). As deterministic chaotic systems (in science in the form of mathematical models) abound in nature, a new branch of physics opened up called chaos physics. It spreads through physics like the mycelia of fungi." (Lauterborn, 1990)
Chaos has been already studied and discovered in a wide range of natural phenomena such as the weather, population cycles of animals, the structure of coastlines and trees and leaves, bubble-fields and the dripping of water, biological systems such as rates of heartbeat, and also acoustical systems such as that of woodwind multiphonics.
These influences, furthermore, are starting to have an effect on music composition, particularly computer-generated composition in the last ten years or so. Discoveries of math and science usually have an effect on art and music: for example, although "we cannot say that the music of J.S. Bach is great because it is the aural equivalent of Cartesian geometry... we can hardly deny that it arises from the same Zeitgeist or whatever one chooses to call the nexus of intellectual, cultural and aesthetic currents that influence an artists" (Truax, 1990). So, too, "new music models will undoubtedly arise from the intellectual milieu that includes fractal geometry and chaotic non-linear systems" (Truax, 1990). In this paper, we will soon look at some of these "new models" that have arisen in computer-generated music.
The study of chaos is approached and modeled through the use of nonlinear dynamic systems, which are the mathematical equations whose evolution is unpredictable and whose behavior can show both orderly and/or chaotic conditions depending on the values of initial parameters. What has attracted the non-science community to these dynamic systems is that they display fractal properties and, thus, patterns of self-similarity on many levels (Truax, 1990). Already, the art community has employed such chaotic patterns, and many examples exist of nonlinear/fractal visual art created with the assistance of the computer (Pressing, 1988; Truax, 1990). So, too, musicians are starting to realize the potential of nonlinear dynamic systems, and its fractal properties are being utilized in a manner that allows musical patterns with chaotic/unpredictable variation to be automatically composed with the computer:
"Musical development or variation can be viewed as the transformation or distortion of a simple entity (a motive), often followed by some sort of return to the original motive. When certain values are chosen for the input parameters to these equations, very similar behavior can be obtained from them. Thus a series of solutions can act like a repeated group of n notes for a number of steps in the iteration process, and then break away to more unpredictable (quasi-chaotic) behavior before eventually returning to the original n-note group, perhaps somewhat altered." (Pressing, 1988)
In other words, certain areas of the nonlinear dynamic outputs can be utilized to enact quasi-'ABA' transitions on many different scales of the compositional process, from motivic variation to formal variation, etc.
Thus, the advantage of using nonlinear dynamic systems is the creation of automatically composed variation, or patterns with inexactitude. Patterns result from the fact that these systems use their outputs for successive inputs—they take the general form: y <= f(y), or y(t+1) = f(y(t)), creating the "self-similarity" and fractal properties already mentioned; and, thus, they are distinct from randomness because random numbers will not produce patterns. The chaotic regions of nonlinear dynamic systems, however, make the long-term behavior of the output unpredictable so that variation will occur within these patterns (at varying degrees depending on initial conditions); this gives the system an inherent sense of "natural" change and growth—these systems are, afterall, also used to describe changes in natural phenomena over time as already mentioned. Thus there are elements of near repetition and near periodic regularity produced by chaotic regions of nonlinear dynamic systems that produce musically interesting output since they are at once motivic (i.e. pattern producing) and yet variable in different controllable ways:
"The great attraction of nonlinear dynamical systems for compositional use is their natural affinity to the behaviors of phenomena in the real world, coupled with their mechanical efficiency of their computation and control. Chaotic systems offer a means of generating a variety of raw materials within a nonetheless globally consistent context. Chaotic sequences embody a process of transformation, the internal coherence of which is ensured by the rules encoded in the equations." (Bidlack, 1992)
Nonlinear dynamic systems can have different kinds of output behavior depending on initial conditions. As the chosen function is iterated, its value traces an orbit—a finite set of points (Gogins, 1991); also, "the point or set of points toward which the value of the function tends after an infinite number of iterations is called its attractor" (Gogins, 1991). There are three different kinds of attractors that a nonlinear dynamic system can end of having: "an attractor may consist of a single point (a single-point attractor), a number of points among which the orbit oscillates (a periodic attractor), or a complex and fractally arranged folding of space (a chaotic attractor)" (Bidlack, 1992). In this last case, after an infinite number of iterations the attractor ends up filling a definite fraction of space with points: it is, thus, fractal: "Such attractors are also called chaotic or strange. It is indeed strange, but significant, that even quite a simple function of the form z <= z2 + c will produce a fractal attractor for many values of the parameter c. This mathematical fertility of simple means is pregnant with hidden music" (Gogins, 1991). It is indeed this last type of nonlinear dynamic systems, the chaotic kind, that has attracted composers to explore and exploit its output for "hidden music."
There are different specific nonlinear dynamic equations that composers have experimented with: including, but not limited to, (a.) the logistic map—traditionally used to model a species' change in population, (b.) the Hénon map—originally introduced as a simple and efficient model of chaotic systems in general, and (c.) the three-dimensional Lorenz system—developed from a simplified model of atmospheric turbulence.
There are also two general ways in which composers have then applied these equations to their compositional process: (a.) large-scale applications in which chaos is employed as an algorithm for making choices having to do with note events—pitch, dynamic level, rhythm, and instrumentation, and (b.) small-scale applications that focus instead on applying chaos to sound synthesis and timbre construction.
The large-scale method is the area most explored thus far. It uses nonlinear dynamic systems iteratively to generate chaotic sequences of numbers that are then mapped to various note parameters (pitch, dynamic, rhythm, duration, tempo, instrument selection, attack time, etc.). Four pioneers of these methods are Jeff Pressing, Michael Gogins, Rick Bidlack, and Jeremy Leach.
Pressing (1988) would focus on regions of the nonlinear dynamic system output that were on the border between periodic and chaotic orbits, so that the behavior that he extracted from the system went back and forth between quasi-periodic and chaotic character: "The output shows unpredictability, but also traces of the nearby cyclic behavior... In musical terms, the overall effect is like a variation technique that inserts and removes material from a motive undergoing mildly erratic pitch transformations, in the style of an adventurous but development-oriented free jazz player, perhaps" (Pressing, 1988)—like Ornette Coleman, for instance, who is stylistically notorious for improvising "motivic-chain-associations" (Jost, 1974). Something else that Pressing experimented with was applying two-dimensional logistic maps, which he did in an attempt to produce interactedness and correlation between two note parameters within the nonlinear dynamic systems. However, he found these results to sound no better than the single dimension examples—and he argues that, besides, there is little musical justification for linking two different note parameters anyhow.
Gogins (1991) would create what he called an "Iterated Functions System (IFS)". This system is more powerful and general than nonlinear dynamic systems because it does not consist of a single function only, but rather is a system of several functions. At each iteration, one of the functions is chosen either in turn or at random and applied to the next value of the system, providing a tremendous degree of variability to the computations: "IFSs have proved capable of representing a wide variety of fractals with the utmost economy of means" (Gogins, 1991). Click here to listen to Gogins' piece entitled "Chaotic Squares" (1991) that employs IFSs.
Bidlack (1992) explored both traditional "dissipative" chaotic systems as well as less employed "conservative" chaotic systems. Dissipative systems "correspond to those phenomena in which friction plays a role and energy is dissipated to the surrounding environment. The overwhelming majority of natural phenomena on Earth are of this type" (Bidlack, 1992). Conservative systems "are best exemplified by the dynamics of celestial bodies, in which energy is conserved" (Bidlack, 1992). In essence, the difference is that dissipative systems have transience towards an attractor, whereas conservative systems have a constant orbit. Bidlack found, therefore, that conservative sequences exhibit much less internal consistency and "are marked by sudden changes in texture and range of values" (Bidlack, 1992). Bidlack also explored multi-dimensional mappings of nonlinear dynamic systems to various musical parameters, extending Pressings explorations of two-dimensional maps to three-dimensional Lorenz systems and four-dimensional Hénon-Heiles systems.
Leach (1995) created an automatic composition software program called XComposer that uses 1/f fractals to generate melodic material. Such 1/f, or "inverse frequency," distributions have been statistically shown to correspond to melodies of much of music (Voss and Clarke, 1975). However, since 1/f fractals entail the use of random data, they cannot create any type of repetition. Leach thus incorporates chaos into the software to generate repetition in the music: repetition of themes in the pitch, rhythm, and/or loudness of the notes. Recall that chaos is not randomness, and inherently produces patterns, and that these patterns will portray different degrees of variability, inexactitude, and long-term unpredictability.
Overall, the results of these large-scale applications are best described, I believe, by Bidlack, who describes the musical output according to the different types of behavior that can be exploited by nonlinear dynamic systems:
"Periodic attractors produce simple oscillating patterns somewhat reminiscent of classic Alberti bass figuration or of contemporary minimalist sequences. Banded chaotic attractors produce textures that at first glance appear to be composed of a regular pattern of note-events. On closer inspection, a pattern of variation within regularity is revealed, with some flexibility in the composition of the repeating sequence. Full-blown chaotic orbits produce the most complex sequences. The distribution of notes is clearly not random; rather, it verges on falling into a pattern without actually doing so. The nascent patterned quality of this texture is easily heard." (Bidlack, 1992)
And, as Pressing describes, "the produced musical examples are idiosyncratic, but show a listenable degree of structural consistency" (Pressing, 1988).
The other application of chaos is at the small-scale level, which focuses on sound synthesis and timbre construction rather than on higher-level parameters such as note sequences and rhythms, etc. Two pioneers of these methods are Barry Truax (1990) and Agostino DiScipio (1990). As Truax argues, large-scale applications of chaos to composition do not make good musical sense:
"From a more philosophical or aesthetic point of view, it is not clear than an arbitrary mapping of a non-linear function [onto the pitch of successive notes] is inherently more musical than, for instance, a random or stochastic function. The musicality may reside in the musical knowledge of the mapper more than in the source function. The audience, if suitably primed with program notes, may be convinced there is more value or interest in the result because of the technique used, but the half-life of such interest seems to be short." (Truax, 1990)
Truax believes, however, that "a more inherently micro-level implementation seems desirable if flexible timbre generation is to be performed" (Truax, 1990). What Truax and DiScipio experiment with, then, is applying chaos to granular synthesis, so that a chaotic texture is created by an unpredictable reformulation of the grains of a given sound file. Different degrees of chaos/unpredictability can be employed, furthermore, which deviate different amounts from the original sound that is being granulated: "Depending on the 'degree' of granulation, on the equation system and on the region of the relative logistic map to be explored, the output sounds are clearly derivative from the original or completely extraneous to it" (DiScipio, 1990).
In closing, it is important to note that many of the composers inspected in this paper regard the results of their nonlinear dynamic systems as merely raw materials with which to compose with. Perhaps with further development, these systems could one day automatically compose pieces that will stand on their own aesthetically. There is still much progress to be made in the application of chaos in generating music, and composers have only begun to scratch the surface of possibilities. Mere translations of the nonlinear dynamic systems into music will not, of course, achieve this end: something will have to be done with them if more convincing results are to be produced: Once again purely formalized procedures are not in themselves composition" (DiScipio, 1990). Though math and science may aid in providing ideas to the arts, their influence is by no means a secession of creativity on the part of the composer. Systems may produce convincing and beautiful music (algorithmic composition has provided many examples in the last twenty years), but somebody must still invent the systems.
It is also important to note that nonlinear dynamic systems are extremely sensitive to initial conditions, the so-called butterfly effect : "User beware! One of the most distinctive characteristics of nonlinear dynamical systems is a behavior known as sensitive dependence on initial conditions. It is because of this that relatively small quantitative differences in the input (the initial conditions) often produce very large qualitative differences in the output" (Bidlack, 1992). For this reason, different outputs will result using single-precision versus double-precision floating-point values due to round-off errors that inevitably accumulate in the iteration of the equations (this is precisely how Edward Lorenz discovered chaos in 1963). Also, identical programs compiled and run on different processors "are virtually guaranteed to produce different orbits" (Bidlack, 1992). It is thus important that the user be aware of this sensitivity.
- Bidlack, Rick (1992), Chaotic systems as simple (but complex) compositional algorithms. Computer Music Journal. 16(3): 33-47.
- DiScipio, Agostino (1990), Composition by exploration of non-linear dynamic systems. International Computer Music Conference (ICMC). Glasgow, Scotland. 1990: 324-327.
- Dodge, Charles and Thomas A. Jerse (1997), Computer Music: Synthesis, Composition, and Performance. 2nd ed. Schirmer Books: New York. 455 pp.
- Gogins, Michael (1991), Iterated functions systems music. Computer Music Journal. 15(1): 40-48.
- Jost, Ekkehard (1994), Free Jazz. 1st Da Capo Press edition. Da Capo Press: New York. 220 pp.
- Lauterborn, Werner (1990), Acoustic chaos. In Frontiers of Nonlinear Acoustics: Proceedings of 12th ISNA. eds. M.F. Hamilton and D.T. Blackstock. Elsevier Science Publishers: London.
- Leach, Jeremy and John Fitch (1995), Nature, music, and algorithmic composition. Computer Music Journal. 19(2): 23-33.
- Lorenz, Edward (1963), Deterministic nonperiodic flow. Journal of Atmospheric Sciences. 20: 130-141.
- Pressing, Jeff (1988), Nonlinear maps as generators of musical design. Computer Music Journal. 12(2): 35-46.
- Truax, Barry (1990), Chaotic non-linear systems and digital synthesis: an exploratory study. International Computer Music Conference (ICMC). Glasgow, Scotland. 1990: 100-103.
- Voss, R.F. and J. Clarke (1975), 1/f noise in music and speech. Nature. 258: 317-318.
Note: An extensive chaos composition bibliography has been compiled by Godric Wilkie, which can be found online at http://www.gozen.demon.co.uk/godric/biblio.html.
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©1999, john a. maurer iv