Music and mathematics have crossed paths in many ways over the years. In particular, the use of mathematical techniques to create pieces of music can be traced back several centuries. Such work has especially proliferated with the advent of computers. *Algorithmic Composition* provides a panoramic view of math- and computer-based methods of writing music, with emphasis on work of the last few decades.

The author has organized the book thematically, according to mathematical approaches that have, at one time or another, gained currency among composers and musicologists. Thus there are chapters devoted to Markov models, generative grammars, chaos and self-similarity, genetic algorithms, cellular automata, neural networks, and so on. This list of topics carries a hint of bandwagon-chasing, and as can be expected, the ways these ideas have been harnessed musically have varied along a continuum between sophistication and superficiality.

Typically, a chapter of *Algorithmic Composition* begins with a general description of the underlying area of math or computer science. Then follow synopses of different people’s applications of that area to musical composition. Finally, a chapter wrap-up discusses the strengths and weaknesses of the approach under consideration. In particular, the author considers certain techniques to be better or worse suited to two distinguishable aims of work in this field: “creating genuine, original compositions” versus “style imitation, where musical material is generated according to a given style or represents an attempt to verify a musical analysis by resynthesis.” Each chapter carries its own list of references to the work under discussion; besides expected sources such as David Cope’s book *Virtual Music* or papers from *Computer Music Journal,* one also finds dissertations, articles from proceedings of AI conferences, and more.

The author is a composer who works with computer music. Although Springer classifies *Algorithmic Composition* in its mathematics line, the author seems to have envisioned an audience containing at least as many of his fellow musicians as mathematicians. The mathematical descriptions do not aim to provide a detailed working knowledge, but rather an introductory taste, of the various topics.

Unfortunately, at times accounts of the mathematics go awry. Rather than representing a benign laxity deliberately adopted in consideration of the book’s mixed audience, some of the inaccuracies seem to reflect misconceptions on the part of the author. For example, the statement, “Given an infinite grid and a correspondingly infinite memory in a computer, a particular CA [cellular automaton] could naturally and continually generate its patterns anew without ever lapsing into a cycle,” (so far, so good) carries the footnote, “But due to the fact that even space is curved, this innovative CA will, unfortunately, also fall into a periodical cycle.” In addition, this old computability theorist definitely winced at the brief but quite muddled summary of Turing’s work.

Nor are the errors limited to mathematics. On a single page the author managed to get wrong the names of all three inventors of the transistor and both founders of Apple Computer. Likewise, according to this book, Grace Hopper “after 1945 was promoted to admiral to become the highest ranking woman of the US Army.” In fact, Hopper became Rear Admiral — *naval,* of course — in 1985.

So at the level of detail, take *Algorithmic Composition* with salt to taste. Despite its shortcomings, however, the book brings together an often scattered literature, systematizing and summarizing it to provide a picture of many of the ways in which composers have adopted and adapted mathematical ideas.

Leon Harkleroad is the author of The Math Behind the Music*,* published in the MAA’s Outlooks book series, and has presented several MAA Minicourses on math and music. He is an old enough computability theorist to remember when people in the field were called recursion theorists.