Abstract
Recurrence relations have analogies to 1d tessellations, including rhythmic patterns. We elaborate the first 16 binary recurrence relations to consider such extensions of ordered partition, exploring musical instances of such analogies. The n\(^\text {th}\) term of the sequence defined by an arbitrary binary (i.e., with coefficients 0 or 1) homogeneous (i.e., with no constant term) recurrence relation counts the number of ways of partitioning (a space of size) n as an ordered sum (accumulation) of elements of a fixed set of positive integers (coalescences of unit intervals), corresponding to the non-zero precursive (“historical”) terms of the recurrence relation. Rhythmically, such recurrence relation-generated sequences count the number of ways of arranging percussive (tone-oblivious) events with particular durations determined by their respective recurrence relations’ precursive terms (antecedent summands). We take the opportunity to mention some characteristics of several of these recurrence relations and our corresponding neologisms: The 2\(^\text {nd}\) binary recurrence relation, and the first nontrivial one, the Fibonacci, is embedded within the 13\(^\text {th}\) recurrence, which we dub the “metaFibonacci.” The 5\(^\text {th}\), Narayana’s cows, is embedded within the 14\(^\text {th}\), so we call it “meta-Narayana’s cows.” The 6\(^\text {th}\), the Padovan, is embedded in the 11\(^\text {th}\), which we call “metaPadovan.”
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Acknowledgements
We thank Masanobu Kaneko for inspiring original musings about recurrence relations and number theory.
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Cohen, M., Kachi, Y. (2024). Recurrence Relations Rhythm. In: Noll, T., Montiel, M., Gómez, F., Hamido, O.C., Besada, J.L., Martins, J.O. (eds) Mathematics and Computation in Music. MCM 2024. Lecture Notes in Computer Science, vol 14639. Springer, Cham. https://doi.org/10.1007/978-3-031-60638-0_30
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