From Music to Mathematics: Exploring the Connections

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 12.
• 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 23. 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 34. 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. Strahle's Guitar
• 4.5.1. An ingenious construction
• 4.5.2. Continued fractions
• 4.5.3. On the accuracy of Strahle'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 45. 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 Bartok 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 56. 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 67. 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 fur 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 78. 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