![]() ![]() We also show that essentially all Euclidean rhythms are deep: each distinct distance between onsets occurs with a unique multiplicity, and these multiplicities form an interval 1, 2, …, k − 1. Indeed, Euclidean rhythms are the unique rhythms that maximize this notion of evenness. We prove that these Euclidean rhythms have the mathematical property that their onset patterns are distributed as evenly as possible: they maximize the sum of the Euclidean distances between all pairs of onsets, viewing onsets as points on a circle. Specifically, we show how the structure of the Euclidean algorithm defines a family of rhythms which encompass over forty timelines ( ostinatos) from traditional world music. We demonstrate relationships between the classic Euclidean algorithm and many other fields of study, particularly in the context of music and distance geometry. ![]()
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