This book explores the relationships between music, the sciences, and mathematics, both ancient and modern, with a focus on the big picture for a general audience as opposed to delving into very technical details. The language of music is deciphered through the language of mathematics. Readers are shown how apparently unrelated areas of knowledge complement each other and in fact propel each other’s advancement. The presentation as well as the collection of topics covered throughout is unique and serves to encourage exploration and also, very concretely, illustrates the cross- and multidisciplinary nature of knowledge. Inspired by an introductory, multidisciplinary course, the author explores the relationships between the arts, sciences, and mathematics in the realm of music. The book has no prerequisites; rather it aims to give a broad overview and achieve the integration of the three presented themes. Mathematical tools are introduced and used to explain various aspects of music theory, and the author illustrates how, without mathematics, music could not have been developed.
Author(s): Arturo Portnoy
Series: Synthesis Lectures on Mathematics & Statistics
Publisher: Springer Nature
Year: 2023
Language: English
Pages: 145
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
