The Geometry of Musical Rhythm: What Makes a "Good" Rhythm Good?

357 Pages·2013·7.69 MB·English

by  Godfried T. Toussaint

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“Toussaint leads us across the globe on the track of some of music’s most E
powerful and resilient rhythmic patterns. Along the way, we’re treated to many O
marvels—musical and mathematical. … It is an invitation for mathematicians to
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try their hands at unraveling the puzzles of musical time patterns and an invitation MUSICAL
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for musicologists to explore the widest range of current quantitative methods.”
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—Christopher Hasty, Harvard University
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“Godfried Toussaint takes the reader on a dazzling tour of rhythmic and Y RHYTHM
mathematical structures, from the beat of the Cuban Clave Son to the music of
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Steve Reich, and from the atomic structure of crystal lattices to the arrangement
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of radio telescopes. Grounded in his hands-on experience as a percussionist M
What Makes a “Good” Rhythm Good?
and training as a computer scientist … Toussaint explores a wide range of
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mathematical and geometrical tools and concepts to unpack what makes these
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rhythms distinctive, ubiquitous, and beautiful. …”
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—Justin London, University of Cambridge and Carleton College C
“No other book brings the properties of rhythm into such sharp focus while A
encouraging debate about what makes one rhythm good, another bad; what L

underlies perceived similarities between rhythmic patterns; and why one rhythm R
might be complex while another is simple. Toussaint’s fascinating book shows
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why rhythm is at once the most fundamental and yet elusive of parameters.”
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—Kofi Agawu, Princeton University and author of Representing African Music
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“… a refreshingly accessible exploration of rhythms that have been performed H
around the world and throughout history. … On one hand, those whose M
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 T
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musical time that he unfolds in many short, easily digested chapters.”
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—Jay Rahn, York University s
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Godfried T. Toussaint
K14867
K14867_Cover.indd 1 12/4/12 9:52 AM

The
GEOMETRY
MUSICAL
of
RHYTHM
What Makes a “Good” Rhythm Good?

The
GEOMETRY
MUSICAL
of
RHYTHM
What Makes a “Good” Rhythm Good?
Godfried T. Toussaint
Illustrative drawings by Yang Liu
CRC Press
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Version Date: 20121203
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For the love of my life—
Eva Rosalie Toussaint

Contents
Prolegomenon, xi
Author, xvii
Chapter 1 ■ What Is Rhythm? 1
Chapter 2 ■ A Steady Beat 9
Chapter 3 ■ Timelines, Ostinatos, and Meter 13
Chapter 4 ■ The Wooden Claves 17
Chapter 5 ■ The Iron Bells 19
Chapter 6 ■ The Clave Son 23
Chapter 7 ■ Six Distinguished Rhythm Timelines 27
Chapter 8 ■ The Distance Geometry of Rhythm 33
Chapter 9 ■ Classification of Rhythms 39
Chapter 10 ■ Binary and Ternary Rhythms 45
Chapter 11 ■ The Isomorphism of Rhythm and Scale 51
Chapter 12 ■ Binarization, Ternarization, and Quantization of Rhythms 57
Chapter 13 ■ Syncopated Rhythms 67
KEITH’S MEASURE OF SYNCOPATION 70
Chapter 14 ■ Necklaces and Bracelets 73
vii
viii   ◾   Contents
Chapter 15 ■ Rhythmic Oddity 85
Chapter 16 ■ Off-Beat Rhythms 99
Chapter 17 ■ Rhythm Complexity 107
OBJECTIVE, COGNITIVE, AND PERFORMANCE COMPLEXITIES 109
LEMPEL–ZIV COMPLEXITY 117
COGNITIVE COMPLEXITY OF RHYTHMS 119
IRREGULARITY AND THE NORMALIZED PAIRWISE VARIABILITY INDEX 120
Chapter 18 ■ Dispersion Problems and Maximally Even Rhythms 121
Chapter 19 ■ Euclidean Rhythms 129
Chapter 20 ■ Leap Years: The Rhythm of the Stars 137
Chapter 21 ■ Approximately Even Rhythms 143
Chapter 22 ■ Rhythms and Crystallography 151
Chapter 23 ■ Complementary Rhythms 157
Chapter 24 ■ Radio Astronomy and Flat Rhythms 165
Chapter 25 ■ Deep Rhythms 175
Chapter 26 ■ Shelling Rhythms 187
Chapter 27 ■ Phantom Rhythms 195
Chapter 28 ■ Reflection Rhythms and Rhythmic Canons 207
Chapter 29 ■ Toggle Rhythms 217
Chapter 30 ■ Symmetric Rhythms 225
HOURGLASS DRUMS AND HOURGLASS RHYTHMS 233
Chapter 31 ■ Odd Rhythms 237
Chapter 32 ■ Other Representations of Rhythm 243
ALTERNATING-HANDS BOX NOTATION 243