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The Distance Geometry of Music
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@MISC{Demaine_thedistance,
author = {Erik D. Demaine and Francisco Gomez-martin and Henk Meijer and Perouz Taslakian and Godfried T. Toussaint and Terry Winograd and David R. Wood},
title = {The Distance Geometry of Music},
year = {}
}
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Abstract
We demonstrate relationships between the classical Euclidean algorithm and many other fields of study, particularly in the context of music and distance geometry. Specifically, we show how the structure of the Euclidean algorithm defines a family of rhythms that encompass over forty timelines (ostinatos) from traditional world music. 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. Indeed, Euclidean rhythms are the unique rhythms that maximize this notion of evenness. 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. Finally, we characterize all deep rhythms, showing that they form a subclass of generated rhythms, which in turn proves a useful property called shelling. All of our results for musical rhythms apply equally well to musical scales. In addition, many of the problems we explore are interesting in their own right as distance geometry problems on the
Keyphrases
distance geometry euclidean rhythm classical euclidean algorithm distance geometry problem euclidean algorithm traditional world music unique rhythm forty timeline deep rhythm onset pattern generated rhythm useful property euclidean distance unique multiplicity many field distinct distance mathematical property musical rhythm musical scale
