Paper by Erik D. Demaine
- Erik D. Demaine, 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: Theory and Applications, volume 42, number 5, July 2009, pages 429–454. Special issue of selected papers from the 17th Canadian Conference on Computational Geometry, 2005.
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 circle; some of the same problems were explored by Erdős
in the plane.
- This paper is also available from ScienceDirect.
- The paper is 39 pages.
- The paper is available in PostScript (1930k), gzipped PostScript (809k), and PDF (453k).
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- Related papers:
- DeepRhythms_CCCG2005 (The Distance Geometry of Deep Rhythms and Scales)
See also other papers by Erik Demaine.
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