From Music to Mathematics: Exploring the Connections

From Music to Mathematics: Exploring the Connections

by Gareth E. Roberts

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Overview

Taking a "music first" approach, Gareth E. Roberts's From Music to Mathematics will inspire students to learn important, interesting, and at times advanced mathematics. Ranging from a discussion of the geometric sequences and series found in the rhythmic structure of music to the phase-shifting techniques of composer Steve Reich, the musical concepts and examples in the book motivate a deeper study of mathematics.

Comprehensive and clearly written, From Music to Mathematics is designed to appeal to readers without specialized knowledge of mathematics or music. Students are taught the relevant concepts from music theory (notation, scales, intervals, the circle of fifths, tonality, etc.), with the pertinent mathematics developed alongside the related musical topic. The mathematics advances in level of difficulty from calculating with fractions, to manipulating trigonometric formulas, to constructing group multiplication tables and proving a number is irrational.

Topics discussed in the book include

Rhythm
Introductory music theory
The science of sound
Tuning and temperament
Symmetry in music
The Bartók controversy
Change ringing
Twelve-tone music
Mathematical modern music
The Hemachandra–Fibonacci numbers and the golden ratio
Magic squares
Phase shifting

Featuring numerous musical excerpts, including several from jazz and popular music, each topic is presented in a clear and in-depth fashion. Sample problems are included as part of the exposition, with carefully written solutions provided to assist the reader. The book also contains more than 200 exercises designed to help develop students' analytical skills and reinforce the material in the text. From the first chapter through the last, readers eager to learn more about the connections between mathematics and music will find a comprehensive textbook designed to satisfy their natural curiosity.

Product Details

ISBN-13: 9781421419183
Publisher: Johns Hopkins University Press
Publication date: 02/15/2016
Pages: 320
Sales rank: 630,307
Product dimensions: 6.90(w) x 10.00(h) x 1.00(d)
Age Range: 18 Years

About the Author

Gareth E. Roberts is an associate professor of mathematics at the College of the Holy Cross.

Table of Contents

Preface xi

Acknowledgments xv

Introduction xvii

1 Rhythm 1

1.1 Musical Notation and a Geometric Property 1

1.1.1 Duration. Geometric sequences 2

1.1.2 Dots: Geometric series 4

1.2 Tune Signatures 9

1.2.1 Musical examples 10

1.2.2 Rhythmic repetition 13

1.3 Polyrhythmic Music 17

1.3.1 The least common multiple 19

1.3.2 Musical examples 22

1.4 A Connection with Indian Classical Music 28

References for Chapter 1 31

2 Introduction to Music Theory 33

2.1 Musical Notation 34

2.1.1 The common clefs 34

2.1.2 The piano keyboard 37

2.2 Scales 41

2.2.1 Chromatic scale 42

2.2.2 Whole-tone scale 44

2.2.3 Major scales 45

2.2.4 Minor scales 49

2.2.5 Why are there 12 major scales? 50

2.3 Intervals and Chords 55

2.3.1 Major and perfect intervals 56

2.3.2 Minor intervals and the tritone 57

2.3.3 Chords 59

2.4 Tonality, Key Signatures, and the Circle of Fifths 64

2.4.1 The critical tonic-dominant relationship 65

2.4.2 Key signatures 67

2.4.3 The circle of fifths 69

2.4.4 Transposition 72

2.4.5 The evolution of polyphony 74

References for Chapter 2 79

3 The Science of Sound 81

3.1 How We Hear 81

3.1.1 The magnificent ear-brain system 82

3.2 Attributes of Sound 84

3.2.1 Loudness and decibels 84

3.2.2 Frequency 86

3.3 Sine Waves 88

3.3.1 The sine function 89

3.3.2 Graphing sinusoids 91

3.3.3 The harmonic oscillator 94

3.4 Understanding Pitch 97

3.4.1 Residue pitch 98

3.4.2 A vibrating string 104

3.4.3 The overtone series 105

3.4.4 The starting transient 107

3.4.5 Resonance and beats 108

3.5 The Monochord Lab: Length versus Pitch 115

References for Chapter 3 118

4 Tuning and Temperament 119

4.1 The Pythagorean Scale 119

4.1.1 Consonance and integer ratios 120

4.1.2 The spiral of fifths 122

4.1.3 The overtone series revisited 124

4.2 Just Intonation 127

4.2.1 Problems with just intonation: The syntonic comma 129

4.2.2 Major versus minor 131

4.3 Equal Temperament 133

4.3.1 A conundrum and a compromise 133

4.3.2 Rational and irrational numbers 135

4.3.3 Cents 138

4.4 Comparing the Three Systems 141

4.5 Strähle's Guitar 144

4.5.1 An ingenious construction 145

4.5.2 Continued fractions 149

4.5.3 On the accuracy of Strähle's method 155

4.6 Alternative Tuning Systems 158

4.6.1 The significance of log2(3/2) 158

4.6.2 Meantone scales 159

4.6.3 Other equally tempered scales 161

References for Chapter 4 163

5 Musical Group Theory

5.1 Symmetry in Music 165

5.1.1 Symmetric transformations 166

5.1.2 Inversions 169

5.1.3 Other examples 173

5.2 The Bartok Controversy 182

5.2.1 The Fibonacci numbers and nature 183

5.2.2 The golden ratio 184

5.2.3 Music for Strings, Percussion and Celesta 185

5.3 Group Theory 191

5.3.1 Some examples of groups 192

5.3.2 Multiplication tables 193

5.3.3 Symmetries of the square 195

5.3.4 The musical subgroup of D4 197

References for Chapter 5 201

6 Change Ringing 203

6.1 Basic Theory, Practice, and Examples 203

6.1.1 Nomenclature 204

6.1.2 Rules of an extent 205

6.1.3 Three bells 208

6.1.4 The number of permissible moves 210

6.1.5 Example: Plain Bob Minimus 211

6.1.6 Example: Canterbury Minimus 213

6.2 Group Theory Revisited 216

6.2.1 The symmetric group Sn 216

6.2.2 The dihedral group revisited 218

6.2.3 Ringing the cosets 221

6.2.4 Example: Plain Bob Doubles 223

References for Chapter 6 227

7 Twelve-Tone Music 229

7.1 Schoenberg's Twelve-Tone Method of Composition 229

7.1.1 Notation and terminology 230

7.1.2 The tone row matrix 233

7.2 Schoenberg's Suite für Klavier, Op. 25 235

7.3 Tone Row Invariance 238

7.3.1 Using numbers instead of pitches 241

7.3.2 Further analysis: The symmetric interval property 242

7.3.3 Tritone symmetry 245

7.3.4 The number of distinct tone rows 248

7.3.5 Twelve-tone music and group theory 249

References for Chapter 7 252

8 Mathematical Modern Music 253

8.1 Sir Peter Maxwell Davies: Magic Squares 253

8.1.1 Magic squares 255

8.1.2 Some examples 256

8.1.3 The magic constant 258

8.1.4 A Mirror of Whitening Light 260

8.2 Steve Reich: Phase Shifting 268

8.2.1 Clapping Music 271

8.2.2 Phase shifts 276

8.3 Xenakis: Stochastic Music 278

8.3.1 A Greek architect 278

8.3.2 Metastasis and the Philips Pavilion 279

8.3.3 Pithoprakta: Continuity versus discontinuity 280

8.4 Final Project: A Mathematical Composition 283

References for Chapter 8 287

Credits 289

Index 293

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