Proceedings/Recueil Des Communications Année : 2026

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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hal-05582871 , version 1 (07-04-2026)

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Gonzalo Romero-García, Lama Tarsissi, Laurent Najman, Carlos Agon. Counting rhythms using combinatorics on words. Mathematics and Computation in Musics 2026 (MCM 2026), Jun 2026, Bard College, Annandale-on-Hudson, New York, USA, United States. Springer, 2026. ⟨hal-05582871v1⟩
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