The distance geometry of music

  • Authors:

  • Erik D. Demaine;Francisco Gomez-Martin;Henk Meijer;David Rappaport;Perouz Taslakian;Godfried T. Toussaint;Terry Winograd;David R. Wood

  • Affiliations:

  • Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA;Departament de Matemática Aplicada, Universidad Politécnica de Madrid, Madrid, Spain;School of Computing, Queen's University, Kingston, Ontario, Canada;School of Computing, Queen's University, Kingston, Ontario, Canada;School of Computer Science, McGill University, Montréal, Québec, Canada;School of Computer Science, McGill University, Montréal, Québec, Canada and Centre for Interdisciplinary Research in Music Media and Technology, The Schulich School of Music, McGill Univ ...;Department of Computer Science, Stanford University, Stanford, CA, USA;Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2009

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Abstract

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 Erdos in the plane.