Geometry of Musical Rhythm

Godfried T. Toussaint is a Canadian computer scientist born in Belgium. Presently, he is a Research Professor of Computer Science at the University of New York, Abu Dhabi, United Arab Emirates, a Research Affiliate at the Massachusetts Institute of Technology in the Computer Science and Artificial Intelligence Laboratory, a Researcher in the Center for Interdisciplinary Research in Music Media and Technology (CIRMMT) in the Schulich School of Music at McGill University, and an Emeritus Professor of Computer Science at McGill. After receiving a PhD in electrical engineering from the University of British Columbia in Vancouver, Canada, he taught and did research at the School of Computer Science at McGill University, in the areas of information theory, pattern recognition, pattern analysis and design, computational geometry, instance-based learning, music information retrieval, and computational music theory. In 1978, he received the Pattern Recognition Society’s Best Paper of the Year Award and in 1985 he was awarded a Senior Killam Research Fellowship by the Canada Council. In May 2001, he was awarded the David Thomson Award for excellence in graduate supervision and teaching at McGill University. He is a founder and cofounder of several international conferences and workshops on computational geometry. He is an editor of several journals, has appeared on television programs to explain his research on the mathematical analysis of flamenco rhythms, and has published more than 360 papers. In 2009, he was awarded a Radcliffe Fellowship by the Radcliffe Institute for Advanced Study at Harvard University, for the 2009–2010 academic year, to carry out a research project on the phylogenetic analysis of the musical rhythms of the world. After spending an additional year at Harvard University, in the music department, he moved in August 2011 to New York University in Abu Dhabi.

"The late Godfried Toussaint studied the rhythms of the world like a gold panner, collecting with meticulousness and passion all the motifs that different cultures have given birth to. Thanks to his skill as a mathematician, he extracted fascinating properties from them. There is no doubt that this unique book will survive for a very long time."—Marc Chemillier, Directeur d'études, École des Hautes Études en Sciences Sociales"Through the original use of distance geometry for analyzing musical rhythm and the visualization of rhythms as cyclic polygons, Gottfried Tousssaint’s fascinating book will be extremely valuable to any researcher involved in in the field of rhythm."—Simha Arom, Ethnomusicologist"The new edition of The Geometry of Musical Rhythm takes us further along Godfried Toussaint’s journey through the world’s rhythms. There are new discussions of metric complexity, rhythm visualization, rhythmic performance, and the evolution of rhythmic patterns. Almost every chapter has been expanded and informed by the latest scholarship in music theory, music psychology, ethnomusicology, and music informatics. Specialists and lay readers alike will find this edition even more engaging and valuable than the first, giving us even more reasons to delight in what makes a "good" rhythm good."— Professor Justin London, Carleton College "A unique and seminal work of original and meticulously detailed scholarship, this newly published second edition of "The Geometry of Musical Rhythm : What Makes a "Good" Rhythm Good?" is unreservedly recommended as a core addition to both college and university library collections."—Midwest Book ReviewPraise for the previous edition"Toussaint’s Geometry presents a whirlwind tour of the world’s rhythms … For a reader interested in musical rhythm, Geometry is a great introduction to the computer science and mathematics of rhythm. For a reader interested in algorithms and mathematical reasoning, the musical focus provides compelling examples lying at the intersection of the arts and the sciences."—William A. Sethares, Journal of Mathematics and the Arts, 2014"… a delightful, informative, and innovative study in the geometric interpretation of rhythm. … It is a pleasure to find an author who has such good command of mathematics and music and who can explain their interconnections with such literary skill. I recommend this book wholeheartedly to every serious student of geometry and music."—Ilhan M. Izmirli, Mathematical Reviews, March 2014"This dynamic fluid presentation of mathematics is exactly what our undergraduate and graduate students need. … I would highly recommend this book for everyone."—Russell Jay Hendel, MAA Reviews, May 2013

• What Is Rhythm?. A Steady Beat. Timelines, Ostinatos, and Meter. The Wooden Claves. The Iron Bells. The Clave Son. Six Distinguished Rhythm Timelines. The Distance Geometry of Rhythm. Classification of Rhythms. Binary and Ternary Rhythms. The Isomorphism of Rhythm and Scale. Binarization, Ternarization, and Quantization of Rhythms. Syncopated Rhythms. Necklaces and Bracelets. Rhythmic Oddity. Off-Beat Rhythms. Rhythm Complexity. Dispersion Problems and Maximally Even Rhythms. Euclidean Rhythms. Leap Years: The Rhythm of the Stars. Approximately Even Rhythms. Rhythms and Crystallography. Complementary Rhythms. Radio Astronomy and Flat Rhythms. Deep Rhythms. Shelling Rhythms. Phantom Rhythms. Reflection Rhythms and Rhythmic Canons. Toggle Rhythms. Symmetric Rhythms. Odd Rhythms. Other Representations of Rhythm. Rhythmic Similarity and Dissimilarity. Regular and Irregular Rhythms. Evolution and Phylogenesis of Musical Rhythm. Rhythmic Combinatorics. What Makes the Clave Son Such a Good Rhythm?. The Origin, Evolution, and Migration of the Clave Son. Epilogue. References. Index.