Page  209 ï~~Algorithms Adapted From Chaos Theory: Compositional Considerations James Harley Faculty of Music, McGill University, 555 Sherbrooke Street West Montreal, Quebec, Canada, H3A 1E3 Abstract Fundamental aspects of chaos theory are examined with the aim of clarifying the issues involved in adapting the output of nonlinear mathematical functions to music. Specific techniques for constructing compositional algorithms (based on the author's CHAOTICS software) are introduced and placed into context by means of a simple musical example. nonlinear functions to music. I Introduction Over the past 15 years, the paradigm of selfsimilarity, the central feature of nonlinear systems (chaos) theory [Schroeder, 1991], has gradually percolated throughout the music community. Musicologists [McLeod, 1992], theorists [Boon et al, 1990] and instrumental composers (e.g. Ligeti [Kinzler, 1991]) have all drawn upon various chaos- or fractal-related concepts and techniques. However, given the fundamental role that computer technology has played in the development of chaos theory, most of the work in music has been carried out by computer musicians [Harley, 1994]. Recently, applications have been developed for MIDI systems [Gogins, 1991], sound synthesis [Degazio, 1993] and formal structures [Di Scipio, 1990]. Most composers seem to agree, however, that the output of a chaotic system tends to be unusable without a strong reliance on the selective judgement of "the composer's ear" [Herman, 1993] which will most likely dictate "extensive and idiosyncratic manipulations" [Bidlack, 1992] of the raw material. However, there has been little discussion to date of the nature of such "manipulations" and the underlying principles which act to constrain them. What I would like to propose here are a number of guidelines, derived from the basic principles of chaos theory, to help clarify the compositional issues involved in adapting 2 Chaos Theory: Basic Principles 2.1 Mathematical-Geometrical-Musical Spaces Graphic representations have long been an integral aid for understanding mathematical functions. This relationship has been particularly strong for nonlinear functions, where computers, along with carrying out the enormous number of calculations necessary to plot the behaviour of each function, can also generate graphic images of those functions. However, it must be kept in mind that mathematical functions are distinct from physical, or graphical, constructs. As the philosopher Hans Reichenbach put it, "[mathematical] concepts are defined by implicit definitions and are not dependent on a unique and specific kind of visualization. Whatever visual objects we wish to coordinate to them is left to our choice" [Reichenbach, 1957]. Mathematical (as opposed to physical) geometry is a "theory of relations," and as such refers only to abstract structures. The visual (physical) space defined by Euclidean geometry is, in fact, an arbitrary frame of reference for mathematical functions. Once that is understood it can easily be seen that other non-Euclidean spaces can also be postulated. Musical, or auditory, spaces must also be defined on their own terms, and these tend to have little in common with the visual domain. The importance of the distinctiveness of ICMC Proceedings 1994 209 Composition, Composition Systems

Page  210 ï~~mathematical and musical spaces for composition will become clearer once we examine in more detail the nature of nonlinear mathematical functions. 2.2 Output Phase Space, Sets, and the Problem of Numerical Accuracy Most chaotic functions, or systems, manifest themselves within a limited range of values, usually described as an n-dimensional "phase space." Within that restricted range, however, such functions trace out infinite orbits within the phase space, never returning to the exact same position twice. That's the theory. However, in establishing relations between mathematics and real-world systems, limits must be placed on the "infiniteness" of the nonlinear functions. For example, a computer-generated graphic display of the well-known Lorenz Attractor (a system with three variables) is limited by the number of decimal places used for the calculations of the functions as well as by the number of pixels on the screen (e.g. 1024*1024). A phase space of infinite resolution is here filtered through two finite "grids" and flattened onto two dimensions. The resulting geometrical figure is nonetheless quite striking in its own right, and does convey the general behaviour of the system. Researchers have discovered, though, that this filtering process, rather than just affecting our perception of the function (in a similar way to changing lenses on a microscope), actually influences the behaviour of the system [Ford, 1986]. The global behaviour of the output remains consistent, but the detailed orbit differs markedly for different numerical resolutions of the function variables. It follows then, that in using finite symbols to represent infinity, interesting patterns (or perhaps distortions) arise which highlight both the order and disorder inherent to this process. The details of these patterns are specific to the degree of restriction (numerical resolution) placed upon the system. The more limited the phase space becomes the greater the degree of output predictability and redundance (repetition). When a nonlinear function is treated as being finite, its output can be defined as a set of elements which can then be easily studied (and manipulated) according to the axioms of set theory and topology. Viewing the output of a chaotic function as a set of discrete elements is crucial to composition based on such functions, given that music is usually viewed as being based on sets of discrete elements as well. 2.3 Nonlinear Functions: General Characteristics Nonlinear functions, while little studied until recently (due for the most part to technical limitations, i.e. the lack of computational machines), are, in fact, extremely common. There are phenomena which can be successfully modeled as linear systems, but usually must concede some degree of uncertainty, or noise (although this work should not be under-rated, as much of our technological innovations have been based on such systems). It is important to realize, however, that in spite of the vast array of nonlinear systems in existence, there are underlying characteristics which act to unite them and render their chaotic behaviour predictable (and thus manipulable), at least to a certain extent. These characteristics are: scaling (output tends to be self-similar across different scales of reference, from the local to the global), autocorrelation (previous output has a certain influence on later output, in the range between redundance and randomness often described as V/f), and metric universality (a method for predicting the behaviour of "attractor" systems, independent of the properties of particular functions [Feigenbaum, 1979]). Thus, in spite of the abundance of nonlinear systems which can be used as generative functions for music composition, these properties ensure a certain degree of consistency across the range of possibilities. On the local level of output behaviour, certain functions exhibit incremental, wave-like orbits, perhaps more conducive to certain aspects of music such as sound synthesis, while the output of other functions fluctuate rapidly within the confines of the phase space, which may be more applicable to other elements of music. In any case, in order to develop compositional algorithms or processes, it would be useful to be able to incorporate means to explore the local behaviour of chaotic functions (for this purpose, an "explore" module has been implemented in the author's CHAOTICS software, to be discussed below), as well as to be able to take advantage of the global features which unite the various functions and render them more controllable. Composition, Composition Systems 210 ICMC Proceedings 1994

Page  211 ï~~3 Compositional Tools 3.2 Defining Musical Spaces On the basis of what has been discussed above, specific techniques can be introduced from which compositional algorithms can be elaborated. These tools have been implemented in the author's CHAOTICS software [Harley, 1994], and, from 1988, a number of musical works for both acoustic instruments and synthesized sounds have been composed on the basis of these concepts and techniques. 3.1 Value Mapping As discussed above, whatever the medium used to "view" the output of a chaotic function, a filtering process must be used in order to translate values of infinite resolution into some discrete form, be it numbers of limited precision, points on a graph, digital samples or notes on scorepaper. This process can be termed value mapping. In constructing a compositional algorithm, a number of value mappings need to be carried out. To do this, it is necessary to define the musical parameters and structures that need to be generated, and then to translate these elements into numerical form over some range of values. As an example, we could build an algorithm to generate a melody for flute. This instrument has a basic range of 3 octaves, which can be translated as an array of 36 values (assuming half-steps are the basic unit of pitch). For the sake of simplicity, we will limit the rhythmic units to just three values (therefore an array of three elements), eighth-notes, quarter-notes and halfnotes. The output of the chaotic function can then be mapped onto those ranges to produce the melody. This line, depending on the degree of chaotic-ness of the function, will exhibit a certain degree of redundancy (much more so in the rhythmic organization, given a solution space of just 3 values) as well as unpredictability. Chances are good, however, that this preliminary musical output will not be judged a successful composition for flute. Why? Because, among other reasons, melodies usually do not "fill" a pitch space in the same way a nonlinear function fills a phase space, and because the range of the flute is not a uniform "space" in terms of timbre and dynamics. Just as a visual representation of a chaotic function has to be defined (assigning x-y coordinates, colours, pixel intensities, etc.), so too do musical representations. Such musical, or compositional, "spaces" should reflect the inherent (or composer-defined) characteristics of the elements under consideration, and can often assume quite elaborate configurations. Turning back to our example, a straight mapping of numerical values from lowest to highest onto the 3-octave range of the flute seems rather awkward in terms of linear continuity and harmonic consistency, and the result would likely be technically unidiomatic for the performer. There are a number of ways to ameliorate this situation, but the more interesting possibilities involve the use of additional compositional-analytical techniques. 3.3 Phase Space Analysis As noted above, as soon as the infinite values of a nonlinear function are filtered onto a discrete phase space, orbit cross-overs (redundancies, repetitions) are bound to occur. The result is that certain values from within the finite range of the output will occur more often than others. It follows then, that a "frequency" analysis (quite similar to a power spectrum analysis) over some sample range of the output, would be a useful tool for describing the global (discrete-state) behaviour of the function. This information can then be used to configure musical spaces in conjunction with compositional considerations. For our example, the 36-value pitch array could be configured along tonal lines, so that the value occuring most often would be assigned to the pitch-class C, followed by G, then E, and so on. Registral considerations could also be incorporated, so that the middle octave, being the easiest to project, might be emphasized the most, followed by the lower octave and then by the more piercing and difficult upper octave. The configuration of the "pitch space" might be ordered as follows, beginning with the most highly reiterated values: C5-G5-E5-C4-G4-E4-C6-G6-E6-FS-D5-A5S. In the same way, we may decide that the rhythmic value occuring most often should be the quarter-note, followed by the eighth-note, then ICMC Proceedings 1994 211 Composition, Composition Systems

Page  212 ï~~the half-note. Here, the musical-parametrical spaces are now being configured on the basis of compositional preferences, but are still, due to the set-like properties of the function output, directly equivalent to the solution set. At the same time, however, no direct control is being exercised over the local, value-by-value generation of these materials. 3.4 Hierarchical, Interactive Algorithms A composer may well want to implement more dynamic controls over the generative processes derived from nonlinear systems. The scaling characteristic of chaotic functions suggests a hierarchical approach to this problem. By this is meant the technique whereby a number of "value maps" are defined for each musical element, each one acting over a different scale of values. In this way, one array can affect largerscale changes in the values from another array. Turning back to our example again, we could decide to create another array such that at certain points (which could be determined by treating the output values of the chaotic function as intervals of time to be mapped over a predetermined duration) the next value of this array would trigger a transposition of the tonal center of the melody (e.g. from C to F, etc.). Another such array could be used to trigger a reconfiguration of the rhythmic values, perhaps to create a section with eighth-notes being the most reiterated value. In addition, the melodic element could be configured to interact with the rhythmic element, on the same basis. Obviously, many other configurations can be imagined and implemented. These kinds of nested processes, all of which are based on mappings from the solution orbits of chaotic systems, have the effect of creating a complex, multi-dimensional, multilayered compositional "space," which exhibits similar properties to the nonlinear system(s) used to generate the musical material. 4 Conclusion A clear understanding of the basic characteristics of nonlinear systems presented here is crucial for the elaboration of compositional strategies based on the chaotic output of such systems. It is on this premise that CHAOTICS has been developed. At the same time, it remains true that the act of composition, even algorithmic composition based on paradigms*adapted from chaos theory, is highly personal and idiosyncratic. To that end, the techniques discussed above have been implemented as modules which must be newly configured and combined for each work. The aim is not just to explore the "sounds" of chaos, but to make chaos manifest in music in compositionally substantive new ways. References [Bidlack, 1992] Rick Bidlack. "Chaotic systems as simple (but complex) compositional algorithms." Computer Music Journal, 16 (3): pp. 33-47, 1992. [Boon et al, 1990] Jean-Pierre Boon, Alain Noullez & Corinne Mommen. "Complex dynamics and musical structure." Interface, 19: pp. 3-14: 1990. [Degazio, 1993] Bruno Degazio. "Towards a chaotic musical instrument." Proceedings of the 1993 ICMC. ICMA, San Francisco: pp. 393-395, 1993. [Di Scipio, 1990]. Agostino Di Scipio. "Composition by exploration of non-linear dynamic systems." Proceedings of the 1990 ICMC. ICMA, San Francisco: pp. 324-327, 1990. [Feigenbaum, 1979] Mitchell Feigenbaum. "The universal metric properties of nonlinear transformations." Journal of Statistical Physics 21 (6): pp. 669-705, 1979. [Ford, 1986] Joseph Ford. "Chaos: solving the unsolvable, predicting the unpredictable!" In Chaotic Dynamics and Fractals. Eds. M.F. Barnsley & S.G. Demko. Academic, San Diego: pp. 1-52, 1986. [Gogins, 19911 Michael Gogins. "Iterated functions systems music." CMJ, 15 (1): pp. 40-48, 1991. [Harley, 1994] James Harley. "Generative processes in algorithmic composition: chaos and music." forthcoming in Leonardo Music Journal, 4:1994. [Herman, 1993] Martin Herman. "Deterministic chaos, iterative models, dynamical systems and their application in algorithmic composition." Proceedings of the 1993 ICMC. ICMA, San Francisco: pp. 194-197, 1993. [Kinzler, 1991] Hartmuth Kinzler. "Decision and automatism in 'Desordre', 1' Ittude, Premier Livre." Interface, 20: pp. 89-124, 1991. [McLeod, 1992] Ken McLeod. "Chaos in music criticism." unpublished, Faculty of Music, McGill University, Montreal, 1992. [Reichenbach, 1957] Hans Reichenbach. The Philosophy of Space & Time. Dover, New York, 1957 (originally published in 1928 as Philosophic Der Raum-Zeit-Lehre). [Schroeder, 1991] Manfred Schroeder. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. Freeman, New York, 1991. Composition, Composition Systems 212 ICMC Proceedings 1994