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Non-linear recursion in acoustics and music

P. Cook
Journal / Anthology

Innovation in Mathematics: Proceedings of the Second International Mathematica Symposium
Year: 1997
Page range: 3-12

Non-linearity is responsible for much of what is interesting and lively about most real-world physical systems, and acoustical systems are a prime example. Modern acoustical research and simulations of musical instruments have benefited greatly from the recognition that many musical instruments can generally be characterized as linear systems, but with one localized region of non-linearity. 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 delayed 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.

*Science > Physics > Wave Motion