Godfried Toussaint
School of Computer Science, McGill University, Montréal, Québec, Canada
E-mail address: godfried@cs.mcgill.ca
(Received February 25, 2005; Accepted June 10, 2006)
Keywords: Binary Sequences, Musical Rhythm, Rhythm Dissimilarity Measures, Music Information Retrieval, Phylogenetic Analysis
Abstract. Measuring the dissimilarity between musical rhythms is a fundamental problem with many applications ranging from music information retrieval and copyright infringement resolution to computational music theory and evolutionary studies of music. A common way to represent a rhythm is as a binary sequence where a zero denotes a rest (silence) and a one represents a beat or note onset. This paper first reviews methods for visualizing and representing rhythms, and then compares various measures of rhythm dissimilarity, including the Hamming distance, the Euclidean interval-vector distance, the interval-difference distance measure of Coyle and Shmulevich, the swap distance, and the chronotonic distance measures of Gustafson and Hofmann-Engl. Traditionally, rhythmic dissimilarity measures are compared according to how well rhythms may be recognized with them, how efficiently they can be retrieved from a data base, or how well they model human perception and cognition of rhythms. In contrast, here dissimilarity measures are compared on the basis of how much insight they provide about the structural inter-relationships that exist within families of rhythms, when phylogenetic trees and graphs are computed from the distance matrices determined by these dissimilarity measures. For two fundamental collections of rhythms, namely the 4/4 time and 12/8 time clave-bell time lines used in traditional African and Afro-American music, the chronotonic and swap distances appear to be superior to the other measures, and each has its own attractive features.