The Geometry of Musical Rhythm

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

Godfried T. Toussaint

• Publisher: Chapman and Hall/CRC
• Year: 2019
• Print ISBN: 9780815350385
• eBook ISBN: 9781351247757
• Topic: Mathematics
• Subtopic: Games in Mathematics
• Index: Mathematics

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.

CHAPTER 1

What is Rhythm?

RHYTHM IS A FUNDAMENTAL FEATURE of all aspects of life.¹ Creating music, listening to music, and dancing to the rhythms of music are practices cherished in cultures all over the world. Although the function of music as a survival strategy in the evolution of human species is a hotly debated topic, there is little doubt that music satisfies a deep human need.² To the ancient philosopher Confucius, good music symbolized the harmony between heaven and earth.³ The nineteenth-century philosopher Friedrich Nietzsche puts it this way: “without music life would be a mistake.”⁴ And the Blackfoot people roaming the North American prairies “traditionally believed that they could not live without their songs.”⁵

Of the many components that make up music, two stand tall above all others: rhythm and melody. Rhythm is associated with time and the horizontal direction in a typical Western music score. Melody, on the other hand, is associated with pitch and the vertical direction. Rhythm can do very well without melody, but melody cannot exist without rhythm. Although rhythm and melody may be studied independently, in music, they generally interact together and influence each other in complex ways.⁶ Experimental results have shown that melody and rhythm (pitch and time) can be encoded in the human brain, either independently or in a combined manner, which depends on the structure of the melody as well as the experience of the listener.⁷ Of these two properties, rhythm is considered by many scholars to be the most fundamental of the two, and it has been argued that the development of rhythm predates that of melody in evolutionary terms.⁸ “Rhythm is music’s central organizing structure.”⁹ The ancient Greeks maintained that without rhythm, melody lacked strength and form. Martin L. West writes: “rhythm is the vital soul of music,”¹⁰ the philosopher Andy Hamilton notes that “rhythm is the one indispensable element of all music,”¹¹ and Ton de Leeuw considers that “rhythm is the highest and most autonomous expression of time-conciousness.”¹² Joseph Schillinger writes: “The temporal flow of music is primarily a matter of rhythm.”¹³ Christopher Hasty offers a concise universal definition of music as the “rhythmization of sound.”¹⁴ From the scientific perspective, psychological experiments designed to assess the dimensional features of the music space, based on similarity judgments of pairs of melodic fragments, suggest that the major dimensions are rhythmic rather than melodic.¹⁵ The American composer George Gershwin believed that the public loved his music because of its rhythm, and in analyzing his rhythms, Isabel Morse Jones writes: “Gershwin has found definite laws of rhythm as mathematical and precise as any science.”¹⁶

Curt Sachs asks the question: “What is rhythm?” and replies: “The answer, I am afraid is, so far just—a word: a word without a generally accepted meaning. Everybody believes himself entitled to usurp it for an arbitrary definition of his own. The confusion is terrifying indeed.”¹⁷ In other words, there is no simple answer to this question. Christopher Hasty cautions that “rhythm is often regarded as one of the most problematic and least understood aspects of music.”¹⁸ James Beament echoes this sentiment when he writes: “Rhythm is often considered the most difficult feature of music to understand.”¹⁹ For Robert Kauffman “The difficulties of dealing with rhythm are immense.”²⁰ Wallace Berry writes: “The awesome complexity of problems of rhythmic structure and analysis can be seen when one appreciates that rhythm is a generic factor.”²¹ Berry goes on to note that another consideration that makes studying rhythm difficult is the fact that meanings ascribed to terms such as “rhythm,” “meter,” “accent,” “duration,” and “syncopation” are vague and used inconsistently. Elsewhere he writes more concisely: “Rhythm is: everything.”²² In spite of some of these difficulties, or perhaps because of them, many definitions of rhythm have been offered throughout the centuries. Already in 1973, Kolinski wrote that more than 50 definitions of rhythm could be found in the music literature.²³ Before diving into the geometric intricacies of rhythm that are explored in this book, it is instructive to review a few examples of definitions and characterizations of rhythm, both ancient and modern.

Plato: “An order of movement.”²⁴

Baccheios the Elder: “A measuring of time by means of some kind of movement.”²⁵

Phaedrus: “Some measured thesis of syllables, placed together in certain ways.”²⁶

Aristoxenus: “Time, divided by any of those things that are capable of being rhythmed.”²⁷

Nichomacus: “Well marked movement of ‘times’.”²⁸

Leophantus: “Putting together of ‘times’ in due proportion, considered with regard to symmetry amongst them.”²⁹

Didymus: “A schematic arrangement of sounds.”³⁰

Aristides Quintilianus: “Rhythm is a scale of chronoi compounded according to some order, and the conditions of these we call arsis and thesis, noise and quietude.”³¹

Vincent d’Indy: “Rhythm is the primordial element. One must consider it as anterior to all other elements of music.”³²

S. Hollos and J. R. Hollos: “In its most general form rhythm is simply a recurring sequence of events.”³³

S. K. Langer: Rhythm is “The setting-up of new tensions by the resolution of former ones.”³⁴

H. W. Percival: “The character and meaning of thought expressed through the measure or movement in sound or form, or by written signs or words.”³⁵

D. Wright: “Rhythm is the way in which time is organized within measures.”³⁶

A. C. Lewis: “Rhythm is the language of time.”³⁷

J. Martineau: “Rhythm is the component of music that punctuates time, carrying us from one beat to the next, and it subdivides into simple ratios.”³⁸

A. C. Hall: “Rhythm is made by durations of sound and silence and by accent.”³⁹

T. H. Garland and C. V. Kahn: “Rhythm is created whenever the time continuum is split up into pieces by some sound or movement.”⁴⁰

J. Bamberger: “The many different ways in which time is organized in music.”⁴¹

J. Clough, J. Conley, and C. Boge: “Patterns of duration and accent of musical sounds moving through time.”⁴²

G. Cooper and L. B. Meyer: “Rhythm may be defined as the way in which one or more unaccented beats are grouped in relation to an accented one.”⁴³

D. J. Levitin: “Rhythm refers to the durations of a series of notes, and to the way that they group together into units.”⁴⁴

P. Vuust and M. A. G. Witek: “Rhythm is a pattern of discrete durations and is largely thought to depend on the underlying perceptual mechanisms of grouping.”⁴⁵

A. D. Patel: “The systematic patterning of sound in terms of timing, accent, and grouping.”⁴⁶

R. Parncutt: “A musical rhythm is an acoustic sequence evoking a sensation of pulse.”⁴⁷

C. B. Monahan, and E. C. Carterette: “Rhythm is the perception of both regular and ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication Page
• Table of Contents
• Preface to the First Edition
• Preface to the Second Edition
• Author
• CHAPTER 1 ▪ What is Rhythm?
• CHAPTER 2 ▪ Isochrony, Tempo, and Performance
• CHAPTER 3 ▪ Timelines, Ostinatos, and Meter
• CHAPTER 4 ▪ The Wooden Claves
• CHAPTER 5 ▪ The Iron Bells
• CHAPTER 6 ▪ The Clave Son: A Ubiquitous Rhythm
• CHAPTER 7 ▪ Six Distinguished Rhythm Timelines
• CHAPTER 8 ▪ The Distance Geometry of Rhythm
• CHAPTER 9 ▪ Classification of Rhythms
• CHAPTER 10 ▪ Binary and Ternary Rhythms
• CHAPTER 11 ▪ The Isomorphism Between Rhythms and Scales
• CHAPTER 12 ▪ Binarization, Ternarization, and Quantization of Rhythms
• CHAPTER 13 ▪ Syncopated Rhythms
• CHAPTER 14 ▪ Necklaces and Bracelets
• CHAPTER 15 ▪ Rhythmic Oddity
• CHAPTER 16 ▪ Offbeat Rhythms
• CHAPTER 17 ▪ Rhythm Complexity
• CHAPTER 18 ▪ Meter and Metric Complexity
• CHAPTER 19 ▪ Rhythmic Grouping
• CHAPTER 20 ▪ Dispersion Problems: Perfectly Even, Maximally Even, and Balanced Rhythms
• CHAPTER 21 ▪ Euclidean Rhythms, Euclidean Strings, and Well-Formed Rhythms
• CHAPTER 22 ▪ Lunisolar Rhythms: Leap Year Patterns
• CHAPTER 23 ▪ Almost Maximally Even Rhythms
• CHAPTER 24 ▪ Homometric Rhythms and Crystallography
• CHAPTER 25 ▪ Complementary Rhythms
• CHAPTER 26 ▪ Flat Rhythms and Radio Astronomy
• CHAPTER 27 ▪ Deep Rhythms
• CHAPTER 28 ▪ Shelling Rhythms
• CHAPTER 29 ▪ Phase Rhythms: The “Good,” the “Bad,” and the “Ugly”
• CHAPTER 30 ▪ Phantom Rhythms
• CHAPTER 31 ▪ Reflection Rhythms, Elastic Rhythms, and Rhythmic Canons
• CHAPTER 32 ▪ Toggle Rhythms
• CHAPTER 33 ▪ Symmetric Rhythms
• CHAPTER 34 ▪ Rhythms with an Odd Number of Pulses
• CHAPTER 35 ▪ Visualization and Representation of Rhythms
• CHAPTER 36 ▪ Rhythmic Similarity and Dissimilarity
• CHAPTER 37 ▪ Grouping and Meter as Features of Rhythm Similarity
• CHAPTER 38 ▪ Regular and Irregular Rhythms
• CHAPTER 39 ▪ Evolution and Phylogenetic Analysis of Musical Rhythms
• CHAPTER 40 ▪ Rhythm Combinatorics
• CHAPTER 41 ▪ What Makes the Clave Son Such a Good Rhythm?
• CHAPTER 42 ▪ On the Origin, Evolution, and Migration of the Clave Son
• CHAPTER 43 ▪ Epilogue
• REFERENCES
• INDEX