Iterative Phase Functions on the Circle and Their Projections: Connecting Circle Maps, Waveshaping, and Phase Modulation

Abstract

In memoriam of Jean-Claude Risset’s recent passing, we revisit two of his contributions to sound synthesis, namely waveshaping and feedback modulation synthesis as starting points to develop the connection of a plenthora of oscillatory synthesis methods through iterative phase functions, motivated by the theory of circle maps, which describes any iterated function from the circle to itself. Circle maps have played an important role in developing the theory of dynamical systems with respect to such phenomena as mode-locking, parametric study of stability, and transitions to chaotic regimes. This formulation allows us to bring a wide range of oscillatory methods under one functional description and clarifies their relationship, such as showing that sine circle maps and feedback FM are near-identical synthesis methods.