Construction of smooth rhythms through a monotone invariant measure

The present article introduces the notion of quasi-smoothness of marked rhythms through a certain transformation $Ref$, called reformation map. A marked rhythm consists of a rhythm together with a marker, and the map $Ref$ modifies the marked onset of the rhythm. It is shown that an iteration of the map $Ref$ transforms an arbitrary marked rhythm into a quasi-smooth one. A numerical criterion for a marked rhythm to be quasi-smooth is given in terms of the difference of its rhythm part. Through this criterion, the rhythm part of any quasi-smooth marked rhythm is shown to be smooth.