TY - JOUR

T1 - The distance geometry of music

AU - Demaine, Erik D.

AU - Gomez-Martin, Francisco

AU - Meijer, Henk

AU - Rappaport, David

AU - Taslakian, Perouz

AU - Toussaint, Godfried T.

AU - Winograd, Terry

AU - Wood, David R.

N1 - Funding Information:
E-mail addresses: edemaine@mit.edu (E.D. Demaine), fmartin@eui.upm.es (F. Gomez-Martin), henk@cs.queensu.ca (H. Meijer), daver@cs.queensu.ca (D. Rappaport), perouz@cs.mcgill.ca (P. Taslakian), godfried@cs.mcgill.ca (G.T. Toussaint), winograd@cs.stanford.edu (T. Winograd), david.wood@upc.edu (D.R. Wood). 1 Supported by NSERC. 2 Supported by FQRNT and NSERC. 3 Supported by the Government of Spain grant MEC SB2003-0270, and by the projects MCYT-FEDER BFM2003-00368 and Gen. Cat 2001SGR00224.

PY - 2009/7

Y1 - 2009/7

N2 - We demonstrate relationships between the classic 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 which 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 circle; some of the same problems were explored by Erds in the plane.

AB - We demonstrate relationships between the classic 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 which 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 circle; some of the same problems were explored by Erds in the plane.

KW - Deep rhythms

KW - Euclidean algorithm

KW - Euclidean rhythms

KW - Generated rhythms

KW - Maximally even

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U2 - 10.1016/j.comgeo.2008.04.005

DO - 10.1016/j.comgeo.2008.04.005

M3 - Article

AN - SCOPUS:84867980912

VL - 42

SP - 429

EP - 454

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 5

ER -