Dynamic, stochastic, and topological aspects of polyrhythmic performance.

Previous research on polyrhythmic performance can be broadly summarized in terms of 2 classes of models: timekeeper models and nonlinear dynamical models. In the former approach, research has been focused on patterns of covariance among time intervals, and in the latter approach, the concentration has been on pattern (in)stability and the spatiotemporal properties of oscillating limbs. It is suggested that one can achieve a more comprehensive theory that incorporates the strengths of each of these approaches by endowing timekeeper models with nonlinear dynamics or by endowing nonlinear oscillator models with stochastic variability. Additionally, those models are complemented by a topological description of performance based on knot theory. Knot theory provides a new index of difficulty for polyrhythmic tapping, a spatial interpretation of transitions between different stable rhythms, and a possible instantiation of N. A. Bernstein's (1967a) notion of a topological motor program.