Iterative method of construction for almost smooth rhythms

Abstract

The present article introduces the notion of marked rhythm and its almost smoothness through a certain transformation Rep, called repulsion map. A marked rhythm consists of a rhythm together with a marker, and the map Rep modifies the marked onset of the rhythm. It is shown that an iteration of the map Rep transforms an arbitrary marked rhythm into an almost smooth one. A numerical criterion for a marked rhythm to be almost smooth is given in terms of the difference of its rhythm part. Smooth rhythms are shown to be always almost smooth, and several examples of almost smooth but not smooth rhythms are constructed.