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The Geometry of Musical Rhythm: What Makes a “Good” Rhythm Good?

Godfried T. Toussaint
Chapman & Hall/CRC
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Russell Jay Hendel
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This book is written in a leisurely exploratory style and can be used in the following wide range of undergraduate and graduate settings.

  • As the sole book (or an accompanying book) in an introductory liberal arts math course fulfilling a general-education university requirement
  • As the sole book in an upper-level, undergraduate, or first-year, graduate, discrete, mathematics elective

The book can also be used

  • If your field overlaps with discrete mathematics and you wish to explore some unsolved real-world problems
  • If you just want to read an enjoyable mathematical text.

In the remainder of the review I will examine in detail each of the above bulleted items as well as discuss the book’s style and its lack of exercises.

For an Introductory Liberal Arts Math Course Fulfilling a Gen Ed Requirement: What do we want when we design an introductory-liberal arts math course for students who will never see a math course again? Presumably, we want topics that will show i) the beauty of mathematics ii) its democratic plurality, cutting across and applying to many cultures, iii) the applicability of mathematics to many facets of our daily lives, iv) that computationally simple math can give rise to deep applications.

One successful textbook along those lines was For All Practical Purposes, by the Consortium for Mathematics and its Applications (COMAP). The initial edition of For All Practical Purposes had no or inadequate exercises. But its message was so powerful that many institutions used it. Some instructors developed their own exercises while others wrote similar textbooks.

In the late 90s, it was discovered that group theory could succinctly account for the diverse symmetry patterns found on pottery and clothing in a wide variety of cultures. Math textbooks, including For All Practical Purposes, soon became filled with pictures of pottery and clothing as well as symmetry classification systems.

Toussaint has done with rhythm what For All Practical Purposes did with symmetry patterns. The book is filled with rhythms and dance patterns from several dozen cultures, classical and popular. The Geometry of Musical Rhythm shows the universality of mathematical applicability. Its math is simple, its applications deep, and it shares the beauty of music. Despite the lack of exercises this book is ideal for a one-time general education math course.

For an Upper-Level, Undergraduate, or 1st-Year, Graduate, Discrete, Math Elective: The Geometry of Musical Rhythm is filled with simple, advanced and cutting-edge applications of discrete mathematics. Rhythms are represented using a circular clock representation of Z modulo n; rhythms are compared using hamming and other distance measures, complexity measures, entropy, decision trees, and a host of graphical methods. The measures are current and borrowed from a variety of fields such as crystallography. Here are two simple examples.

  • The normalized pairwise variability index, nPVI, is a measure similar to, but with more sensitivities than, the variance. The nPVI is currently used in linguistics but its application to music is only recent.  
  • Toussaint skillfully shows how the use of phylogenetic trees and proximity graphs can model distances between rhythms that are captured in a directed-swap matrix; computer exploration can then reconstruct a probable ancestor rhythm to a collection of current rhythms.

If this book is used in an upper-level undergraduate or first year graduate course then there would have to be supplementation with more technical mathematical papers but Toussaint supplies ample footnotes and references. The instructor would have to supply exercises. Alternatively, the course could be conducted using a seminar style where students present referenced papers and explain underlying ideas and details.

Style: The book is written in an exploratory style rather than a definition-theorem-elegant-proof style. Here are two examples.

  • The chapter What makes the Clave Son such a Good Rhythm presents 19 possible measures of “goodness” and compares six rhythms using these measures. We are not told the answer to “what is a good rhythm?” Rather we explore a variety of reasonable approaches.
  • The chapter on Complementary Rhythms delays the proof of the main theorem, leaving it to the final paragraphs. The main focus of the chapter is on i) first, motivating the main result with simple rhythmic examples where special symmetries make the main result intuitive, ii) then exploring the discovery of the main theorem in the crystallography literature, iii) tracing the historical discovery of proofs over several decades in crystallography and music and finally iv) indicating an extremely simple and elegant proof which was communicated orally.

Throughout the chapter and book the emphasis is not on clean, polished and finished results but rather the focus is on exploration, evolution and inter-disciplinary discovery. This dynamic fluid presentation of mathematics is exactly what our undergraduates and graduate students need.

Bibliography: There is a 25 page bibliography with over 700 references. Papers referenced are mostly current papers written within the past 10 years. The subject areas include musical, ethnographic, computer and mathematical subject matter.

Enjoyable reading: The book is written in a leisurely pace, with more prose than technical material. Each chapter is short. But the book only appears non-technical; it has a wealth of current mathematics. I would highly recommend this book for everyone.

Russell Jay Hendel, RHendel@Towson.Edu, holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art and poetry.

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
Keith’s Measure of Syncopation

Necklaces and Bracelets

Rhythmic Oddity

Off-Beat Rhythms

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

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
Hourglass Drums and Hourglass Rhythms

Odd Rhythms

Other Representations of Rhythm
Alternating-Hands Box Notation
Spectral Notation
TEDAS and Chronotonic Notation

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?
Maximal Evenness
Rhythmic Oddity
Weighted Off-Beatness
Metrical Complexity
Main-Beat Onsets and Closure
Distinct Durations
Distinct Adjacent Durations
Onset-Complexity and Distinct Distances
Deep Rhythms, Deepness, and Shallowness
Phylogenetic Tree Centrality
Mirror Symmetry
Shadow Contour Isomorphism

The Origin, Evolution, and Migration of the Clave Son