A A Compendium Of Musical Mathematics eBook

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.

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 theT/I Group
  • Automorphisms of theT/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...