Asymmetric Rhythms, Tiling Canons, and Burnside’s Lemma

A musical rhythm pattern is a sequence of note onsets. We consider repeating rhythm patterns, called rhythm cycles. Many typical rhythm cycles from Africa are asymmetric, meaning that they cannot be broken into two parts of equal duration. More precisely: if a rhythm cycle has a period of 2n beats, it is asymmetric if positions x and x + n do not both contain a note onset. We ask the questions (1) How many asymmetric rhythm cycles of period 2n are there? (2) Of these, how many have exactly r notes? We use Burnside’s Lemma to count these rhythms. Our methods can also answer analogous questions involving division of rhythm cycles of length n into equal parts. Asymmetric rhythms may be used to construct rhythmic tiling canons, in the sense of Andreatta (2003). 1. Rhythm Patterns, Rhythm Cycles, and Asymmetry Anyone who listens to rock music is familiar with the repeated drumbeat—ONE, two, THREE, four— based on a 4/4 measure. Fifteen minutes listening to a Top 40 radio station is evidence enough that most rock music has this basic beat, or its cousin: one, TWO, three, FOUR. But if we turn the radio dial, and if we’re lucky enough to live near immigrant communities, we may hear popular music with different characteristic rhythms: Latin, African, Indian—and even Macedonian. Although much of this music still is based on the 4/4 measure, some instruments play repeated patterns that do not synchronize with the “ONE, two, THREE, four” beat, creating an exciting tension between different components of the rhythm section. This article is concerned with classifying and counting rhythms that, even when shifted, cannot be synchronized with the division of a measure into two parts. In addition, we will discuss rhythms that cannot be aligned with other even divisions of the measure. Our result has a surprising application to rhythmic canons. 1.1. Rhythm patterns and cycles. A rhythm pattern is a sequence of note onsets. We will assume there is some basic, invariant unit pulse that cannot be divided; that is, every note onset occurs at the beginning of some pulse. We identify two rhythm patterns if they have the same sequence of onsets. For example,