Chua's Oscillator in Musical Applications
Using the sounds of chaos in music and sound design
For decades, musicians have been experimenting and working with electronic synthesizers to generate sound. In some cases the reasoning is logistical - often it is prohibitively expensive, for example, to haul a heavy and expensive piano or organ to a venue. Many times, however, the goal is creative expression in the generation of new and previously unheard or unharnessed sounds. Chaotic oscillators, including the example discovered by Leon Chua, open the door to a new universe of sonic possibilities.
Chua's oscillator is a system described by a set of three differential equations that can be realized either in digital form or in analog form using opamps and passive circuit components, simple in appearance, but extraordinarily complex in its analysis and behavior. In the video, you are listening to two of these three signals, at first from Chua's circuit acting alone, and later from this same circuit used similarly to a module in an analog synthesizer, acting through effects and with varying levels of this effected signal feeding back to further perturb the behavior of the oscillator. The video is a phase plot of the two dry (non-effected) signals from the circuit (one moving the display beam left and right, and the other up and down) on an analog oscilloscope.
The sounds from Chua's circuit are widely varied, ranging from pure sine waves to almost pure noise, with many varied behaviors within. Period doubling and intermittency effects can be particularly useful from a sonic standpoint, especially when processed through appropriate effects, such as delays, reverbs, resonant filters, ring modulators, and pitch shifters. A suitable control interface for the many parameters of the oscillator and effects system is necessary for full realization of the sonic potential of chaos.
This system found extensive use in a sound design for Marisol at Southern Illinois University at Edwardsville and was received with very positive responses from critics, audiences, cast, and crew alike. The eerie sounds generated by the combinations of chaotic oscillator with effects fit the bill perfectly to accent the dark, apocalyptic tone of the play, but certain ranges of the system's behavior and combinations of effects, and possibly even by utilizing different chaotic systems, can produce sounds with a wide and emotionally expressive variety.
Future opportunities in this area of research include mapping the system's parameter-space behavior (the complexity of which is astounding - consisting of fractal-like clouds of points where various behaviors take place), using this information to develop new methods and interfaces for control, the use of gyrator circuits to lessen the expense and simplify control of the circuit's reactive elements, investigating the effects of various feedback loop effects topologies, the use of multiple interacting systems to create further sonic variety, implementation in digital form (which could reduce cost and potentially simplify many control problems, likely at the cost of fine nuance in the behavior), and similar investigation of or exploration for other suitable chaotic oscillator systems.
Below are a few pictures of the phase plots of the Chua circuit's behavior. Click on the picture to hear an audio sample of the sounds of chaos. Note how at the end of the series, the stable attractors that appear as the variable inductor (pictured at right with circuit board) is increased decrease in pitch roughly with the musical scale. This is inherent to the behavior of the system - these four stable tones are like islands in a sea of chaos as the value of the inductor is changed - and while these islands don't always align the intervals with our musical scale, behaviors like these could still be valuable in harnessing the system musically.
Photos (click image to hear mp3 sound clip):
The next seven pictures and sound clips illustrate the musical tune of the Chua's stable orbits as the inductance is decreased, and the chaotic behavior between these orbits. The tuner software has been calbrated to F3 = 92Hz, but I believe that the base frequency of this behavior could be tuned by changing the capacitances of the circuit.
The plots below are from a MATLAB simulation of the chua oscillator system. X and Y directions each correspond to varying a particular parameter of the system and the Z direction (represented by height and/or color) is a calculated value that reflects the level of chaos in the waveform by measuring the spread of the frequency distribution of the signal using an FFT and additional mathematical processing of the frequency spectrum..
Higher values of Z (brighter colors and/or higher altitude) indicate that the output power is spread among more frequencies, which is indicative of the onset of chaos and corresponds to more harmonics and/or more chaotic noise in terms of audio output. Lower values of Z indicate simpler tones, to a minimum of 1 when the signal is at a single fundamental frequency or 0 when there is no AC signal, as in the case when the system comes to rest at a steady-state (DC) solution.
The graphs act as a map to the oscillator's behavior. Within a simulator, or with the parameters scaled to real-world component values in a hardware implementation, they should allow one to pick a range of behavior and find combinations of parameters that fulfill the requirements.