This book is the result of material that the author compiled while teaching an undergraduate course on the subject of sound and music and their relationship with mathematics. The mathematical level of different parts of the book varies tremendously from algebra to partial differential equations. Chapter 1 begins with the fundamental question "What is special about sine waves that we consider them to be the pure sound of a given frequency?" Chapter 2 deals with the mathematical subject that answers the question "To what extent can sound be broken into sine waves?". The answer is, of course, Fourier analysis. The mathematics of Bessel functions is also developed in Chapter 2. Chapter 3 goes on to describe the mathematics associated with musical instruments, which are divided into five categories depending upon the mathematical description of the sound they produce . This is done in terms of the Fourier theory developed in chapter 2, but it is really only necessary to have a vague understanding of Fourier theory for this purpose.
Chapter 4 is where the theory of consonance and dissonance is discussed along with the simple integer ratios of frequencies. Consonance and dissonance are musical terms describing whether combinations of notes sound good together or not. This is a preparation for the discussion of scales and temperaments in Chapters 5 and 6. The emphasis in these two chapters is on the relationship between rational numbers and musical intervals. The fundamental question here is "Why does the modern western scale consist of 12 equally spaced notes to an octave?" Has it always been this way? Are there other possibilities? After the discussion of scales, the book breaks off of its main thread to consider a couple of other subjects where mathematics is involved in music, the first being computers and digital music. Chapter 7 discusses how to represent sound and music as a sequence of zeroes and ones, and again Fourier theory is used to understand the result. Also described is the closely related Z-transform for representing digital sounds, and this is then used to discuss signal processing, both as a method of manipulating sounds and producing them. This leads to a discussion of digital synthesizers in Chapter 8, where we are again confronted with the questionof what it is that makes musical instruments sound the way that they do. The discussion is based around FM synthesis. Although this is an old-fashioned method of sound synthesis, it is simple enough to understand many of the salient features before taking on more complex synthesis methods.
Chapter 9 changes the subject completely and examines the role of symmetry in music. The area of mathematics concerned with symmetry is group theory, and the reader is introduced to some of the elementary ideas from group theory that can be applied to music. The book contains numerous exercises, and the answers to almost all of them are included in the book. It should be noted that the author assumes the reader can read music, as this subject is not approached with the exception of a few entries in the appendices. Thus this book is more of mathematics for musicians rather than vice versa. There is an online version of the book available if you want to browse it before deciding to buy. To me, this is one of the clearest books on the relationship of mathematics to music I have read. The text is accessible and clear, there is a good use of graphics, and the exercises emphasize the understanding of the mathematics presented. I highly recommend it.