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158:, with the two long sides and one short side of the triangle corresponding to the gaps between beats. In the figure, the conventional start to a tresillo bar, the beat before the first of its two longer gaps, is at the top vertex, and the chronological progression of beats corresponds to the clockwise ordering of vertices around the polygon.
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another. The information that remains describes the beats of each bar (an evenly-spaced cyclic sequence of times) as being either on-beats (times at which a beat is emphasized in the musical performance) or off-beats (times at which it is skipped or performed only weakly). This can be represented combinatorially as a
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from mathematics and computer science for the musically inspired student". Reviewer
Russell Jay Hendel suggests that, as well as being read for pleasure, it could be a textbook for an advanced elective for a mathematics student, or a general education course in mathematics for non-mathematicians. Professionals in
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that can generate similar nearly uniformly spaced beat patterns for arbitrary numbers of beats in the rhythm and in the bar, to measure the similarity between rhythms, to cluster rhythms into related groups using their similarities, and ultimately to try to capture the suitability of a rhythm for use
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In order to study rhythms mathematically, Toussaint abstracts away many of their features that are important musically, involving the sounds or strengths of the individual beats, the phasing of the beats, hierarchically-structured rhythms, or the possibility of music that changes from one rhythm to
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Toussaint has used this book as auxiliary material in introductory computer programming courses, to provide programming tasks for the students. It is accessible to readers without much background in mathematics or music theory, and
Setheres writes that it "would make a great introduction to ideas
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have been overlooked, he concludes that "transdisciplinary efforts of this kind are necessary". Reviewer Ilhand
Izmirli calls the book "delightful, informative, and innovative". Hendel adds that the book's presentation of its material as speculative and exploratory, rather than as definitive and
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Despite concerns with some misused terminology, with "naïveté towards core music theory", and with a mismatch between the visual representation of rhythm and its aural perception, music theorist Mark Gotham calls the book "a substantial contribution to a field that still lags behind the more
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74:, but he was also a jazz drummer, held a long-term interest in the mathematics of music and musical rhythm, and since 2005 held an affiliation as a researcher in the Centre for Interdisciplinary Research in Music Media and Technology in the
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developed theoretical literature on pitch". And although reviewer Juan G. Escudero complains that the mathematical abstractions of the book misses many important aspects of music and musical rhythm, and that many rhythmic features of
165:, to analyze their mathematical properties (for instance, the fact that many of these rhythms have a spacing between their beats that, like the tresillo, is near-uniform but not exactly uniform), to devise
119:, where the vertices of the hull represent times when a beat is performed; two rhythms are considered the same if the corresponding polygons are
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62:(1944–2019) was a Belgian–Canadian computer scientist who worked as a professor of computer science for
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289:"Hunting for rhythm's DNA: Computational geometry unlocks a musical phylogeny"
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The book uses this method to study and classify existing rhythms from
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has suggested its inclusion in undergraduate mathematics libraries.
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The
Geometry of Musical Rhythm: What Makes a "Good" Rhythm Good?
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rhythm, in which three beats are hit out of an eight-beat
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230:Hendel, Russell Jay (May 2013),
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31:is a book on the mathematics of
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395:The Geometry of Musical Rhythm
393:Izmirli, Ilhan M., "Review of
352:The Geometry of Musical Rhythm
312:The Geometry of Musical Rhythm
310:Escudero, Juan G., "Review of
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199:may also find it of interest.
78:at McGill. In 2009 he visited
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426:Gotham, Mark (June 2013),
350:(April 2014), "Review of
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39:. It was written by
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111:of a subset of the
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43:, and published by
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80:Harvard University
60:Godfried Toussaint
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428:"Review of
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232:"Review of
163:world music
109:convex hull
456:Categories
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271:2020-05-24
212:References
167:algorithms
156:octahedron
37:drum beats
378:122974584
264:Biography
121:congruent
101:rotations
144:tresillo
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113:vertices
97:necklace
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317:zbMATH
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