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dc.contributor.authorDemaine, Erik D.
dc.contributor.authorGomez-Martin, Francisco
dc.contributor.authorMeijer, Henk
dc.contributor.authorRappaport, David
dc.contributor.authorTaslakian, Perouz
dc.contributor.authorWinograd, Terry
dc.contributor.authorWood, David R.
dc.contributor.authorToussaint, Godfried
dc.date.accessioned2015-03-25T14:35:21Z
dc.date.available2015-03-25T14:35:21Z
dc.date.issued2008-12
dc.date.submitted2008-04
dc.identifier.issn09257721
dc.identifier.urihttp://hdl.handle.net/1721.1/96166
dc.description.abstractWe 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−11,2,…,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 Erdős in the plane.en_US
dc.description.sponsorshipNatural Sciences and Engineering Research Council of Canadaen_US
dc.description.sponsorshipFonds québécois de la recherche sur la nature et les technologiesen_US
dc.language.isoen_US
dc.publisherElsevieren_US
dc.relation.isversionofhttp://dx.doi.org/10.1016/j.comgeo.2008.04.005en_US
dc.rightsArticle is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.en_US
dc.sourceElsevieren_US
dc.titleThe distance geometry of musicen_US
dc.typeArticleen_US
dc.identifier.citationDemaine, Erik D., Francisco Gomez-Martin, Henk Meijer, David Rappaport, Perouz Taslakian, Godfried T. Toussaint, Terry Winograd, and David R. Wood. “The Distance Geometry of Music.” Computational Geometry 42, no. 5 (July 2009): 429–454. © 2009 Elsevier B.V.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratoryen_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Scienceen_US
dc.contributor.mitauthorDemaine, Erik D.en_US
dc.contributor.mitauthorToussaint, Godfrieden_US
dc.relation.journalComputational Geometryen_US
dc.eprint.versionFinal published versionen_US
dc.type.urihttp://purl.org/eprint/type/JournalArticleen_US
eprint.statushttp://purl.org/eprint/status/PeerRevieweden_US
dspace.orderedauthorsDemaine, Erik D.; Gomez-Martin, Francisco; Meijer, Henk; Rappaport, David; Taslakian, Perouz; Toussaint, Godfried T.; Winograd, Terry; Wood, David R.en_US
dc.identifier.orcidhttps://orcid.org/0000-0003-3803-5703
mit.licensePUBLISHER_POLICYen_US
mit.metadata.statusComplete


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