The Geometry of Musical Rhythm: What Makes a

The Geometry of Musical Rhythm

What Makes a "Good" Rhythm Good?, Second Edition, 2nd Edition

By Godfried T. Toussaint

Chapman and Hall/CRC

312 pages | 300 B/W Illus.

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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 Evolutionary 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.


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

Table of Contents

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

About the Author

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.

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Recreations & Games
SCIENCE / Physics