The Geometry of Musical Rhythm: What Makes a "Good" Rhythm Good? is a book on the mathematics of rhythms and drum beats. It was written by Godfried Toussaint, and published by Chapman & Hall/CRC in 2013 and in an expanded second edition in 2020. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries. [1]
Godfried Toussaint (1944–2019) was a Belgian–Canadian computer scientist who worked as a professor of computer science for McGill University and New York University. His main professional expertise was in computational geometry, [2] but he was also a jazz drummer, [3] 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 Schulich School of Music at McGill. [2] In 2009 he visited Harvard University as a Radcliffe Fellow in advancement of his research in musical rhythm. [2] [3]
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 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 necklace, an equivalence class of binary sequences under rotations, with true binary values representing on-beats and false representing off-beats. Alternatively, Toussaint uses a geometric representation as a convex polygon, the convex hull of a subset of the vertices of a regular polygon, where the vertices of the hull represent times when a beat is performed; two rhythms are considered the same if the corresponding polygons are congruent. [4] [5]
As an example, reviewer William Sethares (himself a music theorist and engineer) presents a representation of this type for the tresillo rhythm, in which three beats are hit out of an eight-beat bar, with two long gaps and one short gap between each beat. The tresillo may be represented geometrically as an isosceles triangle, formed from three vertices of a regular octahedron, 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. [5]
The book uses this method to study and classify existing rhythms from world music, 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 algorithms 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 in music by a mathematical formula. [5] [6]
Toussaint has used this book as auxiliary material in introductory computer programming courses, to provide programming tasks for the students. [5] It is accessible to readers without much background in mathematics or music theory, [4] [7] and Setheres writes that it "would make a great introduction to ideas from mathematics and computer science for the musically inspired student". [5] 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. [1] Professionals in ethnomusicology, music history, the psychology of music, music theory, and musical composition may also find it of interest. [7]
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 developed theoretical literature on pitch". [7] 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 contemporary classical music have been overlooked, he concludes that "transdisciplinary efforts of this kind are necessary". [4] Reviewer Ilhand Izmirli calls the book "delightful, informative, and innovative". [6] Hendel adds that the book's presentation of its material as speculative and exploratory, rather than as definitive and completed, is "exactly what [mathematics] students need". [1]
In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron.
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The clave is a rhythmic pattern used as a tool for temporal organization in Afro-Cuban music. In Spanish, clave literally means key, clef, code, or keystone. It is present in a variety of genres such as Abakuá music, rumba, conga, son, mambo, salsa, songo, timba and Afro-Cuban jazz. The five-stroke clave pattern represents the structural core of many Afro-Cuban rhythms.
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
In geometry, a polytope is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces.
A simple polygon that is not convex is called concave, non-convex or reentrant. A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive.
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Algorithms that construct convex hulls of various objects have a broad range of applications in mathematics and computer science.
Godfried Theodore Patrick Toussaint was a Canadian Computer Scientist, a Professor of Computer Science, and the Head of the Computer Science Program at New York University Abu Dhabi (NYUAD) in Abu Dhabi, United Arab Emirates. He is considered to be the father of computational geometry in Canada. He did research on various aspects of computational geometry, discrete geometry, and their applications: pattern recognition, motion planning, visualization, knot theory, linkage (mechanical) reconfiguration, the art gallery problem, polygon triangulation, the largest empty circle problem, unimodality, and others. Other interests included meander (art), compass and straightedge constructions, instance-based learning, music information retrieval, and computational music theory.
In computational geometry, the method of rotating calipers is an algorithm design technique that can be used to solve optimization problems including finding the width or diameter of a set of points.
William A. Sethares is an American music theorist and professor of electrical engineering at the University of Wisconsin. In music, he has contributed to the theory of Dynamic Tonality and provided a formalization of consonance.
A bell pattern is a rhythmic pattern of striking a hand-held bell or other instrument of the idiophone family, to make it emit a sound at desired intervals. It is often a key pattern, in most cases it is a metal bell, such as an agogô, gankoqui, or cowbell, or a hollowed piece of wood, or wooden claves. In band music, bell patterns are also played on the metal shell of the timbales, and drum kit cymbals.
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The Euclidean rhythm in music was discovered by Godfried Toussaint in 2004 and is described in a 2005 paper "The Euclidean Algorithm Generates Traditional Musical Rhythms". The greatest common divisor of two numbers is used rhythmically giving the number of beats and silences, generating almost all of the most important world music rhythms, except Indian. The beats in the resulting rhythms are as equidistant as possible; the same results can be obtained from the Bresenham algorithm.
William Oscar "Willie" Anku was a Ghanaian music theorist, ethnomusicologist, composer, and performer. His work combined Western set theory with computer programming and experience in working with performers of various West African musical traditions to create a comprehensive theory of African rhythm. He was "unique among Africa-based music theorists in attracting the attention of the US-based Society for Music Theory," being invited to give plenary lectures and receiving tributes from prominent US-based theorists.
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In discrete geometry and computational geometry, the convex hull of a simple polygon is the polygon of minimum perimeter that contains a given simple polygon. It is a special case of the more general concept of a convex hull. It can be computed in linear time, faster than algorithms for convex hulls of point sets.
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