This book has been a long time in the making. My interest in the connections between mathematics and music started in earnest in the early nineties, when I bought a second-hand synthesizer. This beast used a simple frequency modulation model to produce its sounds, and I was fascinated at how interesting and seemingly complex the results were. Trying to understand what was going on led me on a long journey through the nature of sound and music and its relations with mathematics, a journey that soon outgrew these origins.

Eventually, I had so much material that I decided it would be fun to try to teach a course on the subject. This ran twice as an undergraduate mathematics course in 2000 and 2001, and then again in 2003 as a Freshman Seminar. The responses of the students were interesting: each seemed to latch onto certain aspects of the subject and find others less interesting; but which parts were interesting varied radically from student to student.

With this in mind, I have tried to put together this book in such a way that different sections can be read more or less independently. Nevertheless, there is a thread of argument running through the book; it is described in the introduction. I strongly recommend the reader not to try to read this book sequentially, but at least to read the introduction first for orientation before dipping in.

The mathematical level of different parts of the book varies tremendously.