The purpose of this book is to provide a concise introduction to the mathematical theory of music, opening each chapter to the most recent research. Despite the complexity of some sections, the book can be read by a large audience. Many examples illustrate the concepts introduced. The book is divided into 9 chapters. In the first chapter, we tackle the question of the classification of chords and scales. Chapter 2 is a mathematical presentation of David Lewin's Generalized Interval Systems. Chapter 3 offers a new theory of diatonicity in equal-tempered universes. Chapter 4 presents the Neo-Riemannian theories based on the work of David Lewin, Richard Cohn and Henry Klumpenhouwer. Chapter 5 is devoted to the application of word combinatorics to music. Chapter 6 studies the rhythmic canons and the tessellation of the line. Chapter 7 is devoted to serial knots. Chapter 8 presents combinatorial designs and their applications to music. The last chapter, chapter 9, is dedicated to the study of tuning systems.
Franck Jedrzejewski is a researcher in mathematical physics at INSTN/CEA (Paris-Saclay University). He has also a PhD in music and musicology and in philosophy. In 2013, he was elected Director of Program at College International de Philosophie (CIPh), an institute founded in 1983 by Jacques Derrida, François Châtelet and Dominique Lecourt. He has pub lished more than 20 books, most of them in French: Dictionnaire des musiques microtonales (Dictionary of Microtonal Music), Hétérotopies musicales (Modèles mathématiques de la musique), La musique dodécaphonique et sérielle: une nouvelle histoire, or in English Looking at Numbers with the composer Tom Johnson. His research is highly transdisciplinary (music, philosophy, mathematics and Polish and Russian avantgardes). He currently teaches at Paris-Saclay University and the National Institute for Nuclear Science and Technology (INSTN).
Author(s): Franck Jedrzejewski
Publisher: World Scientific Publishing
Year: 2024
Language: English
Pages: 269
City: New Jersey
Contents
About the Author
Introduction
References
1. Musical Set Theories
1.1 Pitch Classes
1.2 Chords and Scales
1.3 Sets of Limited Transposition
1.4 Enumeration of Chords and Scales
1.5 Exercises
References
2. Generalized Interval Systems
2.1 Generalized Interval System
2.2 Interval Function
2.3 Injection Number
2.4 Babbitt’s Hexachord Theorem
2.5 Interval Sum
2.6 Indicator Function
2.7 Homometric Sets
2.8 Exercises
References
3. Generalized Diatonic Scales
3.1 Sets of Progressive Transposition
3.2 Well-Formed Scales
3.3 Generalized Diatonic Scales
3.4 Generalized Major and Minor Scales
3.5 Exercises
References
4. Voice Leading and Neo-Riemannian Transformations
4.1 Isographic Networks
4.2 Automorphisms of the T/I Group
4.3 Automorphisms of the T/M Group
4.4 PLR Transformations
4.5 JQZ Transformations
4.6 Neo-Riemannian Groups
4.7 Atonal Triads
4.8 Seventh Chords
4.9 Hierarchy of Rameau Groups
4.10 Exercises
References
5. Combinatorics on Musical Words
5.1 Musical Words
5.2 Syntactic Monoids
5.3 Formal Grammars
5.4 Words and Rhythms
5.4.1 Lyndon words
5.4.2 Euclidean rhythms
5.4.3 Maximally even rhythms
5.4.4 Deep rhythms
5.4.5 Rhythmic oddity
5.5 Words and Scales
5.5.1 Dyck words
5.5.2 Infinite words and complexity
5.5.3 Sturmian words
5.5.4 Christoffel words
5.5.5 Sturmian morphisms
5.6 Plactic Congruences
5.6.1 Robinson–Schensted–Knuth correspondence
5.6.2 Plactic modal classes
5.6.3 Pentatonic modes
5.6.4 Hexatonic modes
5.6.5 Heptatonic modes
5.6.6 Octatonic modes
5.7 Rational Associahedra
5.8 Exercises
References
6. Rhythmic Canons
6.1 Tilings
6.1.1 Mask polynomials
6.1.2 Cyclotomic polynomials
6.1.3 Basic properties of tilings
6.1.4 Perfect rhythmic tilings
6.2 Tijdeman’s Theorem
6.3 Hajós Groups
6.4 Coven–Meyerowitz Conjecture
6.5 Fuglede Conjecture
6.6 Vuza Canons
6.7 Exercises
References
7. Serial Knots
7.1 Chord Diagrams
7.2 Enumeration of Tone Rows
7.3 All-Interval 12-Tone Rows
7.4 Types of Tone Rows
7.5 Combinatoriality
7.6 Similarity Measures
7.7 Serial Groups
7.8 Exercises
References
8. Combinatorial Designs
8.1 Difference Sets
8.2 Block Designs
8.3 Resolvable Designs
8.4 Kirkman’s Ladies
8.5 Block Design Drawings
8.6 Tom Johnson’s Graphs
8.7 Exercises
References
9. Tuning Systems
9.1 Cents and Beats
9.2 Some Commas
9.3 Historical Temperaments
9.4 Harmonic Metrics
9.5 Continued Fractions
9.6 Best Approx
9.7 Musical Scale Construction
9.7.1 Euler–Fokker genera
9.7.2 Farey tunings
9.7.3 Partch odd-limit
9.8 Three-Gap Theorem and Cyclic Tunings
9.8.1 Cyclic tunings
9.8.2 Ervin Wilson’s CPS
9.9 Tuning Theory
9.9.1 Hellegouarch’s commas
9.9.2 Tuning systems
9.10 Exercises
References
Solutions to Exercises
Index
