The Mathematics of Music and Art

Preface
Introduction: Let’s Start at the Beginning…
Contents
List of Figures
1 Preliminaries
	1.1 Rational and Irrational Numbers
	1.2 Logarithms and Exponentials
		1.2.1 Exponential Function
		1.2.2 Exponential Equations
		1.2.3 Logarithmic Function
	1.3 Pitch, the Musical Scale, and the Piano
	1.4 Musical Notation, Rhythm
	1.5 Consonance and Dissonance
	1.6 Key, Melody, Chords, and Harmony
2 The Pigeonhole Principle
	2.1 Pigeons and Pigeonholes
	2.2 Rational Approximations
3 Continued Fractions
	3.1 Finite Continued Fractions
	3.2 Infinite Continued Fractions
	3.3 The Golden Ratio
	3.4 Convergents
4 Pythagoras and Music
	4.1 The Monochord
	4.2 The Circle of Fifths and the Musical Scale
	4.3 Why Does the Occidental Chromatic Scale Have 12 Notes?
5 Temperaments of the Scale
	5.1 The Pythagorean Scale
	5.2 Just Intonation
	5.3 The Equally Tempered Scale
	5.4 Comparison
	5.5 Project
6 Patterns in Music
	6.1 Rhythmic Patterns
	6.2 Patterns in Melody and Harmony
		6.2.1 The Diatonic Major Scale
		6.2.2 The Relative Minor Scale
		6.2.3 Triads and Chords
		6.2.4 Chord Inversions
		6.2.5 Modular Arithmetic in the Scale
	6.3 Patterns in Musical Composition
		6.3.1 Transposition
		6.3.2 Composition
	6.4 Symmetry in Harmony
	6.5 Symmetry in Composition
	6.6 Project
7 Analysis and Synthesis of Music
	7.1 Harmonic Oscillations
	7.2 Fourier Series
	7.3 Analysis of Sound
	7.4 Sound Synthesis
	7.5 The Acoustic Envelope
8 Beats, Resonance, and Tuning
	8.1 Beats
	8.2 Resonance
	8.3 Tuning
9 Digital Music and Information
	9.1 Binary Representation of Numbers
	9.2 Analog Versus Digital
	9.3 Digitalization of Sound and Music
10 Last Thoughts
A Mathematical Induction
B ¿Do Convergents Actually Converge?
C Fibonacci and the Golden Ratio
Further Reading
Index