Nonlinearity and Recursion in Acoustics and Music

Perry Cook
Princeton University, USA

Nonlinearity is responsible for much of what is interesting and lively about many physical systems in the real world, and acoustical systems are a prime example of this. Modern acoustical research and simulation of musical instruments has benefited greatly from the recognition that many musical instruments can generally be characterized as linear systems, but with one localized region of nonlinearity. This allows such systems to be "taken apart", decoupling the linear and non-linear parts. Standard linear systems theory and tools can be applied to the linear parts of the instrument. The non-linear parts can be studied and modeled based on physical theory and measurements, informed by the results and theories from areas of modern non-linear mathematics study such as chaos. For example, waves traveling in a round- trip loop down and back up a trombone slide, visiting the non- linear spring of the player's lip once each time, can be expressed as a simple recursion equation exactly like some classical chaos generators. At larger time scales, fractal and chaotic note generators have been found to produce attractive melodies and musical patterns. The nature of some chaotic attractors causes them to generate patterns which seem regular enough to qualify as musical in some sense, but which also have enough variety to keep them from seeming repetitive to human listeners.