The Mathematics Behind the Music: Musical Enumerations, Asymmetric Rhythms, & Crab Canons

Abstract

The overall goal of this thesis is to provide a glimpse into the inner life of music that can be seen through the use of mathematics. Chapter 1 provides the reader with some background knowledge needed including necessary definitions for intervals and transformations that will be used in subsequent chapters. The main focus of the thesis will be on musical canons, specifically rhythmic canons and crab canons. Chapter 2 is a summary of work by Hall and Klingsberg on counting the number of asymmetrical rhythms which is then applied to rhythmic canons. Chapter 3 includes a brief summary of musical canons, an analysis of Bach's "2 Canon a 2" from Musical Offering, as well as a definition for isomorphic crab canons essential for the work in Chapter 4. Chapter 4 gives two separate algorithms for creating crab canons suggesting that a certain mathematical structure can be used to compose musical crab canons. Chapter 5 provides the reader with possibilities for further research. A brief musical background and prior knowledge of combinatorics techniques are not necessary, but of course helpful, in the reading of this thesis.