Asymmetric Rhythms and Tiling Canons

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 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—that is, canons in which there is exactly one note onset per beat. Our results count rhythmic tiling canons where the voice entries are equally spaced. Audio recordings of all examples discussed are available at http://www.sju.edu/∼rhall/Rhythms.