Modelling the Internal Generation of Rhythm as an Extension of Nonlinear Psychophysics

Abstract

Mathematical models within nonlinear psychophysics (NPD) are applied here to the internal production of rhythms. Previously, the behaviour of convoluted cubic polynomial trajectories, after a brief period of initialization, has been explored and their local patterns encoded in symbolic form. The generation of rhythmic, short melodic and some harmonic sequences can arise spontaneously in the dynamics of those trajectories. Edge-of-chaos regimes can drive a response dynamics which then exhibits arpeggio-like subsequences, and can have a multiplicity of periodicities, some of which are not in the bifurcation series in powers of two. The symbolic dynamics are created by reducing series which can range over a closed continuum to transitions between the members of a finite set of states. The apparent degradation of information comes closer to what real subjects can achieve, or perhaps derive, from filtering their own internal neurophysiological activity. To the extent that NPD reflects (as yet unknown) neurophysiological processes, these models suggest that rhythm and tonality, while generated independently, dynamically interact in music performance, at least in experienced musicians.