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 mod els. In the former approach, research has been focused on patterns of covar iance among time intervals, and in the latter approach, the concentration h as been on pattern (in)stability and the spatiotemporal properties of oscil lating limbs. It is suggested that one can achieve a more comprehensive the ory that incorporates the strengths of each of these approaches by endowing timekeeper models with nonlinear dynamics or by endowing nonlinear oscilla tor models with stochastic variability. Additionally, those models are comp lemented 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 topologi cal motor program.