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Page 352 ï~~Further experiments with non-linear dynamic systems: composition and digital synthesis Agostino Di Scipio CSC Centro di Sonologia Computazionale, University of Padova Via S. Francesco 11, 35121 Padova, Italy Via Salaria Antica Est 33a, 67100 L'Aquila, Italy Abstract Mathematical models of chaotic non-linear systems are emloyed in many fields of science and research, and have also recently been used in computer music. This paper focuses on the musical use of particular models called one -parameter families of maps iterated in their own phase space, intended to link some properties of complex natural dynamic systems to the intimate behaviour of sound and to timbre. The methods reviewed provide microscopic (phase level) controls by means of digital synthesis and processing, including the granulation, phase modification and treatment of previously sampled or synthesized sounds as well as the digital synthesis using 2nd order-hIR cells. These micro-level applications of deterministic chaos models seem interesting inasmuch as they can suggest a variety of strategies effective in shaping the detailed morphology of the acoustic surface of music. The underlying conviction is that a revised conception is required regarding the role the composer plays when facing the sound matter in such minute details. Introduction Mathematical models of non-linear dynamic systems are employed in many fields of science and research and describe the complex behaviour of several natural phenomena of both the world "out there" - e.g. the structure of flow in fluids, gas, etc... - and the world inside of us. Chaotic structures of fractal dimension are observed even in human physiology, and the perception itself seems to be possible thanks to the essential disordered motion of neurons in the brain (Skarda & Freeman, 1987). The assumption that models of this kind can be of use for artistic purposes, namely for musical design, relies on the manifold properties of self-organization exhibited by non-linear dynamic systems (Collet & Eckmann, 1980). Various authors have presented computer music applications concerning the macro-level organization of parameters such as pitch and duration (see, for examples, Pressing, 1988; Degazio, 1986). More recently also macro-level applications have been introduced, in which digital sound synthesis is involved (Truax, 1990a; Di Scipio, 1990; Di Scipio, 1991). Since all important features of non-linear dynamics are clearly observable when the implied models are iterated for a very large number of times, it seems inherent to assume such models in conjunction with granular digital synthesis of sound, itself a technique suitable in handling huge amounts of sonic particles. With appropriate reasoning, Truax (1990a) discusses the motivations underlying this point and also refers to the essential non-linearity of some acoustical phenomena, in addition to the well know non-linearity of hearing studied by the psychoacousticians. Moreover, linking deterministic chaos to micro-level procedures of sound synthesis is suggestive in that the composer is provided with heuristic strategies operating in the intimate of the sonic matter, enabling him to employ significantly some recent and non-traditional paradigms in the representation of sound (De Poli, Piccialli & Roads, 1991). In the simplest implementation the present value of an iterated model is rescaled in order to define ICMC 352
Page 353 ï~~the frequency (and the amplitude) of a single sonic grain with sinusoidal waveform. By means of a slightly more elaborated procedure, it is also possible to process previously sampled or synthesized sounds; in other words this can be seen as reorganizing the quanta of acoustic energy of a given sound in patterns different from the original one (Di Scipio, 1990). More recent work has introduced other applications intended to control the parameters of signal processing tasks (Di Scipio, 1991). The basic principle is always to map the specific interval within which the values of the system always fall - called the phase space - in such a way as to match the range of values affordable for one or more parameters of synthesis or processing algorythms. In the author's work up to date, these methods have been implemented as batch procedures. Real time is possible, of course, but unfortunately the treatment of very long streams of samples is quite problematic to do in real time, due to the limited RAM that common DSPs support and, hence, due to the very short sounds available in that case. One-parameter families of maps The iteration of maps of an interval into itself is the simplest model of non-linear (dissipative) system. Iterations of the form x--- xn+1 = f (xn) (where f maps [-1,1] into itself) stand for discrete time versions of continuous systems advocated with success as models of biological, chemical, physical, and even social and ecologic systems (Gleick, 1987). It is also to be noted that identical qualitative results hold for a vast typology of models regardless of the number of variables they involve. Therefore (and for the sake of clarity and politeness of computer implementation) only one-parameter families of maps are taken into account, i.e. iterations of the form Xn+1 -1 - rxn2 where x is the present value of the iteration (in theory a real number, in practice a 8-byte representation) and r is the sole parameter, the one the evolution of a system depends upon. From an operating pointof view, what changes when using n-dimensional systems is the nature of the phase space which can no longer be considered as a line (in fact 1-dimensional) but as a complex plane (with 2-dimensional systems), or a space of n dimensions (with n-dimensional systems). The possible evolutions of non-linear dynamic systems are already known to the computer music community'. A hierarchy of behaviours is observed where - broadly speaking - the evolution of a non-linear system is said to be - fixed point-attracted - cycle-attracted - unpredictable (chaotic) - stable fixed point - stable limit-cycle - bounded (within the specific phase space) - unstable fixed point - periodic limit-cycle - unbounded (in particular models - non periodic limit-cycle featuring trigonometric functions) - unstable limit-cycle For a given value of r (determining the type of evolution) and unless the system immediately fades to zero or reaches a stable fixed attractor, each different value x0 causes a radically different evolution, making impossible, even grossly, to foresee the immediate future of a given iteration (this property is usually named sensitive dependance on the initial conditions). Exhaustive information about a given family of iterations is depicted in the so called logistic map, the well known scenery of period-doublings indicating the route from order to chaos with surprising phases of alternance between unpredictable evolution and unforeseen periodicity. ICMC 353
Page 354 ï~~Additional information about the ergodic behaviour of a system is achieved by means of istograms (weighted spectra) diplaying which points of the phase space are visited and the amount of times each point is visited (assuming an infinite number of iterations, all infinite points will be visited at least for a once). Two experiments with granular synthesis The first example relies on a technique of granular processing introduced in my previous work (Di Scipio, 1990), where the state x of a non-linear system is rescaled in order to point to a single sample within the digital represention of a sound, the sample considered as the first of the next grain. In the updated and extended version of this technique, each grain can be replicated before a new x is calculated. The number of identical repeats and the delay between successive grains is determined by a different mapping of the chaotic system, and, furthermore, the samples in a repeated grain are arranged back to front, their sign is inverted and their magnitude rescaled. The result of these treatments is a texture showing complex time-varying changes in the duration of each sonic fragment without substantial changes in its spectral properties. However, transformations in the timbral texture are clearly audible, depending on the state of order and organization in the modeled nonlinear system and, in part, on the discrete phase modifications introduced. This technique refers to the time stretching of sound as generated independently from the typical parallel frequency translation 2(Jones & Parks, 1988; Truax, 1990b). In addition, as all micro-level changes are controlled dynamically, it also provides a way to use in a musical sense those comb-like by-products (usually undesired) resulting from quasi-periodic repetitions of entire streams of samples (in fact a sort of dynamic group delay unit). The second example is based on the istograms referred to above. In this case the phase space of a non-linear dynamic system is mapped onto a given time interval within which a sequencex distributes unitary impulses according to the evolution of the system. Impulses (in theory flat and infinitely wide spectra) are then reduced to sounds with very limited spectra (up to a single sinusoidal component) by the action of a 2nd order-IR cell. Preferably, the Q parameter in this filter should be proportional to the density of impulses, while the center frequency can itself be dynamically controlled, non-linearly defined or generated by means of 1/f2 noise. Interestingly, the typical audible result of this technique sounds as a granular structure of variable temporal density. However, as the impulse response of 2nd order-IIR filters is somewhat rapid and noisy, a further device or even a special purpose filter must be designed in order to achieve a smoother impulse response. This, in turn, would reduce the overall procedure to a form of granular synthesis easily generalizable and implementable in real time. Discussion. A sub-symbolic approach to composition Deterministic chaos models in conjunction with digital synthesis represent appropriate means of microscopic sonic design. They are interesting in that they suggest a variety of strategies effective in sculpting the detailed morphology of the acoustic surface of music. From a practical point of view, it is important that all information relating to an entire texture is "freezed" definitely in only two values, x3 and r. Both the perceivable organization and the unpredictable evolution in the sound matter built up with the methods proposed give rise to higher level entities such as global events perceived as symbols or partial components of larger formal plans. The composer, then, developes (starting from phase level controls) the spectral and temporal shape of what once he called "a given material" (also when pre-defined by himselt), which is now, indeed, a compositional result 3. In this context, musical form can be modeled as the evolution of sound matter, its gradual "growth". ICMC 354
Page 355 ï~~Thus, such an approach to composition may possibly be thought of as a sub-symbolic one, since musical symbols - or whatever one chooses to call the homogeneous sonic percepts that cognition interprets as signs within an overall musical formal setting - are listened to as singularities growing up from the motion of sound matter, from its turbolence and ordered flow. notes 'this paper cannot be (nor it is intended to be) a complete review of all properties of non-linear dynamic systems which are conceptually interesting and useful in musical experiments; please refer to scientific literature and to the articles mentioned in the references 2 the time stretching of sound performed by means of digital granulation can be viewed as an updated version of an early analogical technique discussed in Schaeffer (1966: 425-426) 3 see Duchez (1991) for a historical perspective on the relation composer/material in contemporary and electroacoustic music. References Collet, P. & Eckmann, J.P. Iterated maps on the interval as dynamical systems, Boston, Birkhauser, 1980 Degazio, B. "Musical aspects of fractal geometry", Proceedings of the ICMC-86, Computer Music Association, 1986: 435-441 De Poli, G., Piccialli, A. & Roads, C. (eds) Representation of musical signals, Cambridge Mass., MIT Press, 1991 Di Scipio, A. "Composition by exploration of non-linear dynamic systems", Proceedings of ICMC-90, CMA, 1990: 324-328 Di Scipio, A. "Caos deterministico, composizione e sintesi del suono", Proceedings of the IX CIM, University of Genova, 1991 Duchez, M.E. "L'dvolution scientifique de la notion de materiau musical", in Barriere, J.B. (ed) Le timbre, metaphore pour la composition, Paris, C. Bourgois/IRCAM, 1991:47-81 Gleick, J. Chaos, New York, Viking Penguin, 1987 Jones, D. & Parks, T.V. "Generation and organization of grains for music synthesis", Computer Music Journal, 12(2), 1988: 27-34 Pressing, J. "Nonlinear maps as generators of musical design", C.M.J., 12(2), 1988: 35-46 Schaeffer, P. Traite des objects musicaux, Paris, Edition du Seuil, 1966 Skarda, C.A. & Freeman, W.J. "How brains make chaos in order to make sense of the world", Behavioral and Brain Sciences, 10(2), 1987 Truax, B. "Chaotic non-linear systems and digital synthesis: an exploratory study", Proceedings of ICMC-90, CMA, 1990a: 100-103 Truax, B. "Time shifting of sampled sounds with a real time granulation technique", Proceedings of ICMC-90, CMA, 1990b: 104-108. ICMC 355