The Mathematics of Musical Set Theory

Musical set theory is a set of mathematical tools and algorithms that can be used to analyse atonal (and other) music. As well as the fundamental problems of musical analysis, musicians and musical theorists coming to musical set theory for the first time also have the very significant problem of understanding the mathematics involved. That is where this book comes in. We explore the mathematics underpinning musical set theory, and and provide a textbook in which you can find everything you need without having to search through other mathematical texts. We assume very little basic knowledge of mathematics, and all concepts and notation are defined as clearly as possible as we go along. As we introduce mathematical concepts such as sets, groups, permutations etc. we illustrate how they apply to musical set theory, and other fields of music. Our aim is to take you on an adventure.This book is an interactive text, because everything in it is supported by code in the elegant, expressive and immensely powerful Wolfram Language (the language of Mathematica). For example, we don't just define what a, "prime form", is conceptually and mathematically, we show you how to calculate it, and provide algorithms and code to do that. Similarly for other mathematical objects such as interval vectors and set complexes. Everything that can be calculated is calculated. This approach allows us to make even very abstract ideas concrete and easy to understand via code and interactive demonstration programs. As with mathematical knowledge, we assume only a basic knowledge of coding, and the book is amply supported by the rich Mathematica ecosystem of online documentation, tutorials and books. Even very experienced programmers will find much of great interest in the Wolfram Language and the way we use it here.In order to achieve interactivity, this book comes in three formats, a paperback, an electronic book and a Mathematica Notebook. The notebook is fully executable and can be downloaded free of charge from the Wolfram Notebook Archive. It brings to life the wealth of algorithms and programs illustrated in the text. The code also provides a powerful and complete toolkit for the theorist who needs to apply musical set theory. In computing terms, it constitutes a "domain specific language" for musical set theory. As such, it provides a firm foundation for constructing larger programs.