Infinite Rhythmic Tiling Canons

Clifton Callender
Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture (2015)
Pages 399–402 Short Papers


Maximally-even (ME) rhythms, in which attacks (or the onsets of notes) are distributed as evenly as possible over a given number of pulses, are common in many cultures. One way to generate a ME rhythm with m attacks per n pulses is to digitize a line of slope m/n. If the slope is rational, the resulting rhythm is periodic; irrational slopes yield Sturmian sequences, which are balanced, almost periodic, self-similar and hierarchical sequences that are highly relevant in the study of musical objects possessing these same properties. Rhythmic tilings are combinations of rhythms that yield one and only one onset per pulse. Rhythmic tiling canons are tilings in which the component rhythms are version of a single tile. Sturmian rhythms can be used to create completely new kinds of aperiodic rhythmic tiling canons, in which the relations of the component rhythms are determined by the continued fraction of the slope.