Mathematical Measures of Syncopation

F. Gómez, A. Melvin, D. Rappaport and G.T. Toussaint
Renaissance Banff: Mathematics, Music, Art, Culture (2005)
Pages 73–84


Music is composed of tension and resolution, and one of the most interesting resources to create rhythmic tension is syncopation. Several attempts have been made to mathematically define a measure of syncopation that captures its essence. A first approach could be to consider the rhythmic oddity property used by Simha Arom to analyze rhythms from the Aka pygmies. Although useful for other purposes, this property has its limitations as a measure of syncopation. More elaborate ideas come from the works by Michael Keith (based on combinatorial methods) and Godfried Toussaint (based on group theory). In this paper we propose a new measure, called the weighted note-to-beat distance measure, which overcomes certain drawbacks of the previous measures. We also carry out a comparison among the three measures. In order to properly compare these measures of syncopation, we have tested them on a number of rhythms taken from several musical traditions.