Franck Jedrzejewski 
COMPENDIUM OF MUSICAL MATHEMATICS, A [EPUB 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 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.

Key Features:

• The concise readability of this little book is a definite advantage that will be appreciated by mathematicians


• The novelties proposed in this book (which cannot be found elsewhere): new definition of neo-Riemannian transformations, new classification of dodecaphonic series, new classification of modes via the plactic monoid, etc. will attract many readers


• The opening towards the world of research as one can perceive it through the articles published by the Journal of mathematics and music