A Compendium of Musical Mathematics

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. Sample Chapter(s) Introduction Chapter 1: Musical Set Theories Contents: About the Author Introduction Musical Set Theories: Pitch Classes Chords and Scales Sets of Limited Transposition Enumeration of Chords and Scales Exercises References Generalized Interval Systems: Generalized Interval System Interval Function Injection Number Babbitt's Hexachord Theorem Interval Sum Indicator Function Homometric Sets Exercises References Generalized Diatonic Scales: Sets of Progressive Transposition Well-Formed Scales Generalized Diatonic Scales Generalized Major and Minor Scales Exercises References Voice Leading and Neo-Riemannian Transformations: Isographic Networks Automorphisms of the T/I Group Automorphisms of the T/M Group PLR Transformations JQZ Transformations Neo-Riemannian Groups Atonal Triads Seventh Chords Hierarchy of Rameau Groups Exercises References Combinatorics on Musical Words: Musical Words Syntactic Monoids Formal Grammars Words and Rhythms Words and Scales Plactic Congruences Rational Associahedra Exercises References Rhythmic Canons: Tilings Tijdeman's Theorem Hajós Groups Coven–Meyerowitz Conjecture Fuglede Conjecture Vuza Canons Exercises References Serial Knots: Chord Diagrams Enumeration of Tone Rows All-Interval 12-Tone Rows Types of Tone Rows Combinatoriality Similarity Measures Serial Groups Exercises References Combinatorial Designs: Difference Sets Block Design Resolvable Designs Kirkman's Ladies Block Designs Drawings Tom Johnson's Graphs Exercises References Tuning Systems: Cents and Beats Some Commas Historical Temperaments Harmonic Metrics Continued Fractions Best Approximations Musical Scale Construction Three-Gap Theorem and Cyclic Tunings Tuning Theory Exercises References Solutions to Exercises Index Readership: The book can be used as a support for a course at the graduate level. It will also be of interest to mathematics teachers and some musicians, who have a background in music. The book can be read by musicians or music lovers, students of music conservatories who want to understand the mathematical structures that arise in music theory.