Emphasizing a visual geometric treatment of musical rhythm and its underlying structures, the author—an eminent computer scientist and music theory researcher—presents new symbolic geometric approaches and often compares them to existing methods. He shows how distance geometry and phylogenetic analysis can be used in comparative musicology, ethnomusicology, and evolutionary musicology research. The book also strengthens the bridge between these disciplines and mathematical music theory. Many concepts are illustrated with examples using a group of six distinguished rhythms that feature prominently in world music, including the clave son.
Exploring the mathematical properties of good rhythms, this book offers an original computational geometric approach for analyzing musical rhythm and its underlying structures. With numerous figures to complement the explanations, it is suitable for a wide audience, from musicians, composers, and electronic music programmers to music theorists and psychologists to computer scientists and mathematicians. It can also be used in an undergraduate course on music technology, music and computers, or music and mathematics.
Toussaint leads us across the globe on the track of some of music’s most powerful and resilient rhythmic patterns. Along the way, we’re treated to many marvels — musical and mathematical. This is a book full of ideas and suggestions for further experiments in thinking. It is an invitation for mathematicians to try their hands at unraveling the puzzles of musical time patterns and an invitation for musicologists to explore the widest range of current quantitative methods.
—Christopher Hasty, Walter W. Naumburg Professor of Music Theory, Harvard University
In The Geometry of Musical Rhythm, Godfried Toussaint takes the reader on a dazzling tour of rhythmic and mathematical structures, from the beat of the Cuban Clave Son to the music of Steve Reich, and from the atomic structure of crystal lattices to the arrangement of radio telescopes.
Grounded in his hands-on experience as a percussionist and training as a computer scientist, and using six ‘distinguished timelines’ from African and Afro-Cuban music as musical touchstones throughout the book, Toussaint explores a wide range of mathematical and geometrical tools and concepts to unpack what makes these rhythms distinctive, ubiquitous, and beautiful. Along the way we learn just what makes a rhythm ‘rhythmic,’
Toussaint serves up his observations and analyses in bite-sized chapters, making both the mathematics and the music accessible to a wide readership, whether they are professional musicologists and mathematicians or simply anyone who has ever been fascinated by the swing and syncopation of rhythms from around the world.
—Justin London, affiliated researcher, Centre for Music and Science, University of Cambridge, UK, and professor of music, Carleton College, Northfield, Minnesota, USA
The Geometry of Musical Rhythm sets new standards for scholarship on musical rhythm. Ranging across numerous world repertories, Godfried Toussaint captures the geometrical underpinnings of a variety of rhythmic patterns and in so doing, helps us to see, hear and feel them more intensely. No other book brings the properties of rhythm into such sharp focus while encouraging debate about what makes one rhythm good, another bad; what underlies perceived similarities between rhythmic patterns; and why one rhythm might be complex while another is simple. Toussaint’s fascinating book shows why rhythm is at once the most fundamental and yet elusive of parameters.
—Kofi Agawu, Princeton University and author of Representing African Music
The Geometry of Musical Rhythm is a refreshingly accessible exploration of rhythms that have been performed around the world and throughout history. Toussaint, a renowned mathematician and an amateur percussionist himself, has read broadly and reflected thoughtfully about ways in which rhythms can be described, categorized, and compared.
The centrepiece of his discussion is son clave, which Toussaint traces from 13th-century Baghdad to its recent role as a truly global rhythm that has been prominent in such popular idioms as rock and salsa. Starting from very slim premises, Toussaint explains dozens of mathematical tools that quantify selected aspects of rhythmic simplicity and similarity. These measures result in phylogenetic trees that specify the extent to which certain rhythms are structurally close to one another. Especially valuable in this regard is the precision with which Toussaint applies contrasting formulations of ways in which music can change.
On one hand, those whose mathematical experience is limited to what they learned in high school will be able to follow Toussaint’s graceful exposition quite readily; on the other hand, specialists in adjacent areas of the sciences, psychology, and ethnology will discern suggestive avenues of research far beyond the fundamental approach to musical time that he unfolds in many short, easily digested chapters.
—Jay Rahn, Department of Music, York University
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ISBN 9781466512023, January 2012, 365 pp