2nd Edition

The Geometry of Musical Rhythm What Makes a "Good" Rhythm Good?, Second Edition

By Godfried T. Toussaint Copyright 2020
    370 Pages 300 B/W Illustrations
    by Chapman & Hall

    370 Pages 300 B/W Illustrations
    by Chapman & Hall

    370 Pages 300 B/W Illustrations
    by Chapman & Hall

    The original edition of The Geometry of Musical Rhythm was the first book to provide a systematic and accessible computational geometric analysis of the musical rhythms of the world. It explained how the study of the mathematical properties of musical rhythm generates common mathematical problems that arise in a variety of seemingly disparate fields. The book also introduced the distance approach to phylogenetic analysis and illustrated its application to the study of musical rhythm. The new edition retains all of this, while also adding 100 pages, 93 figures, 225 new references, and six new chapters covering topics such as meter and metric complexity, rhythmic grouping, expressive timbre and timing in rhythmic performance, and evolution phylogenetic analysis of ancient Greek paeonic rhythms. In addition, further context is provided to give the reader a fuller and richer insight into the historical connections between music and mathematics.

    Preface to First Edition

    Preface to Second Edition

    1. What is Rhythm

    2. Isochrony, Tempo, and Performance

    • Isochronous Rhythms
    • Tempo
    • Performance

    3. Timelines, Ostinatos and Meter

    4. The Wooden Claves

    5. The Iron Bells

    6. The Clave Son: A Ubiquitous Rhythm

    7. Six Distinguished Rhythm Timelines

    • The Shiko Timeline
    • The Clave Rumba
    • The Soukous Timeline
    • The Gahu Timeline
    • The Bossa-Nova Timeline

    8. The Distance Geometry of Rhythm

    9. Classification of Rhythms

    10. Binary and Ternary Rhythms

    11. The Isomorphism Between Rhythms and Scales

    12. Binarization, Ternarization, and Quantization of Rhythms

    13. Syncopated Rhythms

    • Metrical Complexity
    • Keith’s Measure of Syncopation

    14. Necklaces and Bracelets

    15. Rhythmic Oddity

    16. Off-Beat Rhythms

    17. Rhythm Complexity

    • Objective, Cognitive and Performance Complexities
    • The Lempel-Ziv Complexity
    • The Cognitive Complexity of Rhythms
    • Irregularity and the Normalized Pairwise Variability Index

    18. Meter and Metric Complexity

    • What is Meter?
    • Does African Rhythm Possess Meter?
    • Pulse Saliency Histograms in Renaissance and Common Practice Music
    • African Rhythm Timelines and Western Music
    • Keith’s Mathematical Measure of Meter Complexity
    • The Interaction Between Meter and Rhythm Perception

    19. Rhythmic Grouping

    20. Dispersion Problems: Perfectly Even, Maximally Even, and Balanced Rhythms

    21. Euclidean Rhythms, Euclidean Strings, and Well-Formed Rhythms

    22. Lunisolar Rhythms: Leap Year Patterns

    23. Almost Maximally Even Rhythms

    24. Homometric Rhythms and Crystallography

    25. Complementary Rhythms

    26. Flat Rhythms and Radio Astronomy

    27. Deep Rhythms

    28. Shelling Rhythms

    29. Phase Rhythms: The "Good," the "Bad," and the "Ugly"

    30. Phantom Rhythms

    31. Reflection Rhythms, Elastic Rhythms, and Rhythmic Canons

    • Reflection Rhythms
    • Paradiddle Method
    • Alternating-Hands Method
    • Elastic Rhythms
    • Rhythmic Canons

    32. Toggle Rhythms

    33. Symmetric Rhythms

    • Hourglass Rhythms
    • Sub-Symmetries

    34. Rhythms with an Odd Number of Pulses

    35. Visualization and Representation of Rhythms

    • Alternating-Hands Box Notation
    • Spectral Notation
    • TEDAS Notation
    • Chronotonic Notation
    • Phase Space Plots
    • Tangle Diagrams

    36. Rhythm Similarity and Dissimilarity

    • Hamming Distance
    • Swap Distance
    • Directed Swap Distance
    • Many-to-Many Assignment Distance
    • Geometric Distance
    • Edit Distance

    37. Grouping and Meter as Features of Rhythm Similarity

      • The Edit Distance with Grouping Information
      • The Edit Distance with Meter Information

      38. Regular and Irregular Rhythms

      39. Evolution and Phylogenetic Analysis of Musical Rhythms

      • Phylogenetic Analysis of Cultural Objects
      • Phylogenetic Analysis of Flamenco Compás
      • The Guajira
      • Phylogenetic Analysis of Ancient Greek Paeonic Rhythms

      40. Rhythm Combinatorics

      41. What Makes the Clave Son Such a "Good" Rhythm?

        • Evenness and Balance
        • Rhythmic Oddity
        • Off-Beatness
        • Weighted Off-Beatness
        • Metrical Complexity and Keith’s Measure of Syncopation
        • Main-Beat Onsets and Closure
        • Cardinality of Distinct Durations
        • Cardinality of Distinct Adjacent Durations
        • Onset-Complexity and Sum of Distinct Distances
        • Deep Rhythms, Deepness, and Shallowness
        • Tallness
        • Prototypicality and Phylogenetic Tree Centrality
        • Mirror Symmetry
        • Pulse Sub-Symmetries
        • IOI Duration Sub-Symmetries
        • Area of Phase Space Plots
        • Fractal Metric Hierarchy
        • Shadow Contour Isomorphism

        42. On the Origin, Evolution, and Migration of the Clave Son

        43. Epilogue


        • Books, Book Chapters and Theses
        • Journal Articles and Conference Proceedings


        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 author of this book, the Canadian professor of computer science Godfried Toussaint, who died in 2019, describes music as one of his passions. This is apparent throughout this work, which draws on his wide knowledge of different musical styles from all cultures.

        The book is very informative about a wide range of styles of music and musical instruments and their history. If you have an interest in music and in a range of areas of mathematics, you are likely to find plenty of topics of interest."
        —Mathematical Gazette

        "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 Review

        Praise 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