The goal of this project is to produce a software musical instrument using chaotic processes as a sound synthesis method. Real-time control through MIDI note messages and continuous controllers is required. Early results are described, along with suggestions for continuing work

Degazio, Bruno

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Bruno Degazio

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Towards a Chaotic Musical Instrument

SOURCE: Sheridan Scholarly Output Undergraduate Research Creative Excellence

Sheridan CollegeSOURCE: Sheridan Scholarly Output, Research, and CreativeExcellencePublications and Scholarship Faculty of Animation, Arts & Design (FAAD)1993Towards a Chaotic Musical InstrumentBruno DegazioSheridan College, bruno.degazio@sheridancollege.caFollow this and additional works at: https://source.sheridancollege.ca/faad_publicationsPart of the Composition Commons, and the Software Engineering CommonsThis work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.This Conference Proceeding is brought to you for free and open access by the Faculty of Animation, Arts & Design (FAAD) at SOURCE: SheridanScholarly Output, Research, and Creative Excellence. It has been accepted for inclusion in Publications and Scholarship by an authorized administratorof SOURCE: Sheridan Scholarly Output, Research, and Creative Excellence. For more information, please contact source@sheridancollege.ca.SOURCE CitationDegazio, Bruno, "Towards a Chaotic Musical Instrument" (1993). Publications and Scholarship. 6.https://source.sheridancollege.ca/faad_publications/6TOWARDS A CHAOTIC MUSICAL INSTRUMENT Bruno Degazio Degazio Sound Design 192 Spadina Ave, #512 Toronto, Canada ABSTRACT The goal of this project is to produce a software musical instrument using chaotic processes as a sound synthesis method. Realtime control through MIDI note messages and continuous controllers is required. Early results are described, along with suggestions for continuing work. CUBIC OSCILLATOR WORKBENCH The driven oscillator with a cubic term, commonly known as the Duffing oscillator, is defined by the autonomous differential equations: • dx=y dy = -Ay-Bx3+G cos(wt)frequency, and spring stiffness, sliders are provided for overall amplitude and for overall pitch, which is simply the wavetable playback increment. Numerical display for all parameters is provided in machine units, except for driving oscillator frequency. which is given in cycles per second. A pair of buttons at the top right of the screen start and stop sound output. A "kickstart" button reseeds the dependent variables x and y with reasonable values, which is frequently necessary since many parameter regimes produce runaway output (more technically. there exists an attractor at positive or negative infinity). Another pair of buttons controls display of the orbit graph and the time waveform graph, which occupy the This system can be considered a simple form of physical modelling. The object modelled is a rigid beam or spring vibrnting in only one mode, driven by a sinusoidal exciting force. Parameter A models the damping force, or amount of energy lost to the system, the cubic term controlled byB models the stiffness of the of the spring, G is the driving oscillator force and w is the driving oscillator frequency. Realtime access to "'•=::-- - Cubic Osci1111tor Workbench ( Klckst11rt Osc J Qfil_J( START] ( Show Orbit ] ( Show W11ue ) figure 1 - Cubic Oscillator Workbench screenthese parameters is given in the Cubic Oscillator Workbench, a Macintosh program designed for interactive exploration of the parameter space (fig. 1). Graphic sliders for the control parameters appear along the top left of the screen. In addition to damping force, driving oscillator amplitude. driving oscillator ICMC Proceedings 1993 393 bottom half of the screen. The orbit is simply a graph of x versus y for a series of 1000 samples. The various forms of behaviour can be easily distinguished by means of the complexity of the image produced (fig. 2). The time waveform is the actual sample by sample output of the y component, displayed as y versus time. The two dependent variables, x and y, produce two 4P.08 correlated output streams which can be used for stereo sound, or for monophonic sound and a correlated control stream. a C the superposition of multiple periods in a sub-harmonic relationship. It is clearly seen in the series of orbit and waveform displays in figures 2a to 2e. figure 2 - period doubling progression in the cubic oscillatorTYPES OF BEHAVIOUR The cubic oscillator system exhibits several different behaviours which depend on the settings of the relevant parameters. Single cycle periodic behaviour is the simplest; it corresponds lo a continuous periodic waveform in the time domain and to a circle (topologically speaking) in the orbital map (fig. 2a). Periodic behaviour of order two and higher is also possible (fig. 2b-c). Quasi-periodic behaviour results from the super-position of multiple periodic behaviours, which may be incommensurate (i.e. non-rationally related). The waveform representation shows recurring similar features (which may never repeat precisely) while the orbital representation shows several crisscrossing loops ( fig. 2d). Fmally, chaotic behaviour (fig. 2e) is represented in the orbital diagram by a complicated mesh of crisscrossing orbits which grows progressively thicker with time, and in the waveform display as a continuous wave without repeating features. The route to chaos from periodic behaviour is often through a period doubling scenario. 1bis results from a) period 1 b) period 2c) period 4 d) period 8 (quasi periodic)e) chaotic f) impulse-damping behaviourUeda(l979) has experimentally derived a preliminary map of the behaviour of this system. IMPLEMENTATION In the first implementation of this system it was possible for integration to be performed (using the forward Euler method) at a sample rate of about 16khz on a 25mhz Macintosh Quadra using extended precision floating point numbers, with all parameters retained in registers. Reworking the code for 32 bit fixed point (16 bit integer.16 bit fraction) provided a remarkable speed improvement, running at about 41 khz, even without parameters being held in registers. When combined with the additional computing overhead for rcaltime sound output to the DACs and for user interface, this drops to about 22 khz, still adequate for testing. The goal of the project is to perform the integration on a Motorola 56000 DSP co-processor, which should be easily able to 4P.08 394 ICMC Proceedings 1993 handle 44.1 khz, perhaps with multiple sound channels or other synthesis "niceties" such as envelopes or filters. SUBJECTIVE EVALUATION The sounds produced by the system truly earn the name chaotic. Other terms that come to mind are irregular, unpredictable, noisy, uncontrollable, dirty and wild. A particularly interesting feature is that the behaviour for a given set of parameters can be very different depending on the prior state of the system. Technically this is due to the existence of multiple attractors at these parameter values. For a given set of parameters some attractors may be chaotic, while others may be periodic or quasi­periodic. In experimental observation with the CubicOscillator Workbench, it has been foWld that when entering a region of the parameter space with multiple attractors, the system tends to fall to the attractor most similar to its existing behaviour. The addition of a small perturbation, such as added noise or a small transient (added with the "kickstart" button) is usually sufficient for the system to jump to a different attractor. OTHER APPROACHES A less direct approach to the application of chaotic processes to sound generation is represented by the Lorenz aJtractor interface to CSound's ADSYN generator (fig.3). 1bis software module creates an ADSYN compatible control file, ADSYN.N, with chaotic trajectories for the frequencies and amplitudes of up to 64 partials. The partials need not be harmonically related; utilities arc provided to generate frequency tables in harmonic relationship, hannonic with stretched tuning (positive or negative stretching), inverse (sub) harmonic relationship and many others. LORCNZ RTIRRCTOtl ROOITIUI 1¥NTHISIS •miler of part..,. ta geaentle: � numller or llreUpotats: � pitch deut1Uan (hzl:EJ total duration (m•l: !300DG j iund•m1i,1t .. frequency (hz): � LORCNZ EQUATION PflRRMEKRS honnontc stntchlng: E:J rata(1-101� 1:j10.ooo I I n1r mouiNtY MAP I I EDITPHTIOL W£1GIIIS I I CrootumsvN rna I 1:j21.ooo j t:j2.671H I figure 3- Lorenz attractor ADSYN interface FUTURE WORK A number of improvements are projected for the current system. The most important is to offload the burden of numerical integration to a DSP in order to free up the main processor for user interface and MIDI input processing. The DSP should be able to perform the integration on multiple simultaneous channels as well as provide amplitude envelopes. Other enhancements to the basic process include the injection of small amounts of noise into the data stream to perturb the oscillator and induce a noisy transient response. 1bis will reduce its tendency to «tock into" simple periodic regimes. Driving the oscillator with a function other than sin or cos might also have this effect, since the high frequency components in less "smooth" waveforms could also induce noisy transients. OTHER SYSTEMS to investigate include: van der Pol oscillaJor dx=y dy = (1-x2y)-x double scroll oscillaJor ( Chua's circuit) dx = A(y-h(x)) dy = x-y+z dz = -By forced negaJive resistance oscillator dx=y dy = a(1-x2)y-x3+Bcos(ft) third order piecewise linear system dx = -A f(y-z) dy = -f(y-x)-z dz=y f(u)=-au+O.S(a+b)( I u+ 11-1 u-11) An extension of the workbench arrangement developed for the cubic oscillator would facilitate the investigation of these systems. A generic interface for graphic slider and MIDI control, behind which various computational 'engines' corresponding to one of the above systems could be installed, would be ideal both for rapid prototyping and for exploration. REFERENCES Rodet, X. Nonlinear Oscillator Models of Musical Instrument Excitation in Proceedi11gs of the l11tematio11a/Computer Music Conference, San Jose, CA, 1992. Ueda, Y. Steady Motions exhibited by Duffing's equation: a picture book of regular and chaotic motions. In New Approaches to Nonlinear Problems in Dynamics, ed. P.Holmes, pp. 311-22. SIAM, Philadelphia, Pa. (1979)Ueda, Y. and Akamatsu, N. Chaotically transitional phenomena in the forced negative resistance oscillator. IEEE Trans. CS 28, 217-24 1981) ICMC Proceedings 1993 395 4P.08 

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Towards a Chaotic Musical Instrument

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