From Music to Mathematics: Exploring the Connections

by Gareth E. Roberts

Overview

A guided tour of the mathematical principles inherent in music.

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:	9781421419190
Publisher:	Johns Hopkins University Press
Publication date:	02/15/2016
Sold by:	Barnes & Noble
Format:	eBook
Pages:	320
File size:	17 MB
Note:	This product may take a few minutes to download.
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
Acknowledgments
Introduction
1. Rhythm
1.1. Musical Notation and a Geometric Property
1.1.1. Duration
1.1.2. Dots
1.2. Time Signatures
1.2.1. Musical examples
1.2.2. Rhythmic repetition
1.3. Polyrhythmic Music
1.3.1. The least common multiple
1.3.2. Musical examples
1.4. A Connection with Sanskrit Poetry
References for Chapter 1
2. Introduction to Music Theory
2.1. Musical Notation
2.1.1. The common clefs
2.1.2. The piano keyboard
2.2. Scales
2.2.1. Chromatic scale
2.2.2. Whole-tone scale
2.2.3. Major scales
2.2.4. Minor scales
2.2.5. Why are there 12 major scales?
2.3. Intervals and Chords
2.3.1. Major and perfect intervals
2.3.2. Minor intervals and the tritone
2.3.3. Chords
2.4. Tonality, Key Signatures, and the Circle of Fifths
2.4.1. The critical tonic-dominant relationship
2.4.2. Key signatures
2.4.3. The circle of fifths
2.4.4. Transposition
2.4.5. The evolution of polyphony
References for Chapter 2
3. The Science of Sound
3.1. How We Hear
3.1.1. The magnificent ear-brain system
3.2. Attributes of Sound
3.2.1. Loudness and decibels
3.2.2. Frequency
3.3. Sine Waves
3.3.1. The sine function
3.3.2. Graphing sinusoids
3.3.3. The harmonic oscillator
3.4. Understanding Pitch
3.4.1. Residue pitch
3.4.2. A vibrating string
3.4.3. The overtone series
3.4.4. The starting transient
3.4.5. Resonance and beats
3.5. The Monochord Lab
References for Chapter 3
4. Tuning and Temperament
4.1. The Pythagorean Scale
4.1.1. Consonance and integer ratios
4.1.2. The spiral of fifths
4.1.3. The overtone series revisited
4.2. Just Intonation
4.2.1. Problems with just intonation
4.2.2. Major versus minor
4.3. Equal Temperament
4.3.1. A conundrum and a compromise
4.3.2. Rational and irrational numbers
4.3.3. Cents
4.4. Comparing the Three Systems
4.5. Strähle's Guitar
4.5.1. An ingenious construction
4.5.2. Continued fractions
4.5.3. On the accuracy of Strähle's method
4.6. Alternative Tuning Systems
4.6.1. The significance of log2(3/2)
4.6.2. Meantone scales
4.6.3. Other equally tempered scales
References for Chapter 4
5. Musical Group Theory
5.1. Symmetry in Music
5.1.1. Symmetric transformations
5.1.2. Inversions
5.1.3. Other examples
5.2. The Bartók Controversy
5.2.1. The Fibonacci numbers and nature
5.2.2. The golden ratio
5.2.3. Music for Strings, Percussion and Celesta
5.3. Group Theory
5.3.1. Some examples of groups
5.3.2. Multiplication tables
5.3.3. Symmetries of the square
5.3.4. The musical subgroup of D4
References for Chapter 5
6. Change Ringing
6.1. Basic Theory, Practice, and Examples
6.1.1. Nomenclature
6.1.2. Rules of an extent
6.1.3. Three bells
6.1.4. The number of permissible moves
6.1.5. Example
6.1.6. Example
6.2. Group Theory Revisited
6.2.1. The symmetric group Sn
6.2.2. The dihedral group revisited
6.2.3. Ringing the cosets
6.2.4. Example
References for Chapter 6
7. Twelve-Tone Music
7.1. Schoenberg's Twelve-Tone Method of Composition
7.1.1. Notation and terminology
7.1.2. The tone row matrix
7.2. Schoenberg's Suite für Klavier, Op. 25
7.3. Tone Row Invariance
7.3.1. Using numbers instead of pitches
7.3.2. Further analysis
7.3.3. Tritone symmetry
7.3.4. The number of distinct tone rows
7.3.5. Twelve-tone music and group theory
References for Chapter 7
8. Mathematical Modern Music
8.1. Sir Peter Maxwell Davies
8.1.1. Magic squares
8.1.2. Some examples
8.1.3. The magic constant
8.1.4. A Mirror of Whitening Light
8.2. Steve Reich
8.2.1. Clapping Music
8.2.2. Phase shifts
8.3. Xenakis
8.3.1. A Greek architect
8.3.2. Metastasis and the Philips Pavilion
8.3.3. Pithoprakta
8.4. Final Project
8.4. References for Chapter 8
Credits
Index

Editorial Reviews

Overall, From Music to Mathematics is a pleasing and well-written book that is accessible for everyone who wants to explore the connections between music and mathematics. Gareth Roberts does a great job of making numerous suggestions on how music can be used to illuminate mathematical concepts. . . From Music to Mathematics is very enjoyable to read – not only for students, but for anyone who loves music and mathematics.
—Musicae Scientiae

Overall, I strongly recommend this as an excellent basis for teaching.
—MathSciNet

The fact that most of the aspects of music discussed seem to be relatively peripheral to the immediate experience of music emphasises that nobody has yet provided a convincing justification for the universal belief that “mathematics and music go together”. Recommended to anyone with an interest in the precise details of both the music and the mathematics.
—The Mathematical Gazette

From the Publisher

Overall, I strongly recommend this as an excellent basis for teaching.

MathSciNet

The fact that most of the aspects of music discussed seem to be relatively peripheral to the immediate experience of music emphasises that nobody has yet provided a convincing justification for the universal belief that “mathematics and music go together”. Recommended to anyone with an interest in the precise details of both the music and the mathematics.

The Mathematical Gazette

Overall, From Music to Mathematics is a pleasing and well-written book that is accessible for everyone who wants to explore the connections between music and mathematics. Gareth Roberts does a great job of making numerous suggestions on how music can be used to illuminate mathematical concepts. . . From Music to Mathematics is very enjoyable to read – not only for students, but for anyone who loves music and mathematics.

Musicae Scientiae