Counting rhythms using combinatorics on words
Résumé
In this paper, we formalize rhythms as words over a three-letter alphabet encoding onsets, sustains, and silences. Within this framework, rhythms form a regular language, admitting both a rational expression and a deterministic finite automaton description. This formal-language perspective allows us to derive enumeration results via generating functions. In particular, the number of rhythms of length n follows the subsequence of odd-indexed Fibonacci numbers. We also introduce an equivalence relation on rhythms induced by temporal scaling, which identifies rhythms that differ only by a uniform stretching of time. We then use the Möbius inversion formula to count the number of equivalence classes.
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