Iterative method of construction for smooth rhythms

Abstract

The present article introduces the notion of smoothness of rhythm and proposes a unified method that transforms an arbitrary rhythm into a smooth one. The method employs a self-map Rav, discrete average map, on the space of rhythms of arbitrary length with a fixed number of onsets. It is shown that, for any rhythm in the space, the iterations become eventually periodic, and that the final cycle consists only of smooth rhythms. The discrete average map leads naturally to a finite directed graph, which visualizes the realm of smooth rhythms in the whole world of rhythms. This article has an Online Supplement, in which we give detailed proof of the main result.