The relationship between mathematics and music has been a long-lasting one, going back to the Greeks in antiquity; various authors, artists, and mathematicians (Euler comes to mind here) have provided significant contributions to a better understanding of this strong link. Cool math for hot music, published in Springer’s “Computational Music Science” series, offers the personal view of Guerino Mazzola, a seasoned researcher in computer music who also happens to be one of the two series editors. Motivated by the questioning of two of Mazzola’s PhD students, who ended up being his coauthors, this 320-page monograph targets musicians with a desire to see how mathematics can relate to their practice. Despite its informal and almost kitschy title, the book’s intent is to introduce the reader to a significant chunk of formal mathematics, while suggesting how such material can be exploited in a variety of musical situations, from sheet music analysis to composition to performance. The focus of the book is mostly symbolic: music is viewed via its description in scores, and sound and its continuous aspects are barely mentioned (although tempo and gestures are discussed near the end of the book).

The structure of this eight-part book is linear, building mathematical knowledge from the ground up. A short historical introduction is presented in Part 1, where the first author’s musical evolution is put into perspective with the whole history of the interactions between music and mathematics. Parts 2 to 6 are dedicated to algebra, while Part 7 looks at some analytical notions pertinent to music performances, such as continuity and differentiability. The book ends with Part 8, which provides solutions to exercises, references, and an index. Each part includes a few rather short chapters, for a total of 34 chapters in the whole book; each chapter introduces a specific mathematical notion, with comments, exercises, and audio examples (available online).

The algebraic approach to music theory informs the core of the book. Part 2 introduces sets, functions, and relations, and shows how they constitute the foundations of mathematics. Peano’s axiomatics for numbers, the subject of Part 3, is built on ordinals, which use Zermelo and Fraenkel’s theory of sets. The whole range of number types is then presented, from rationals to reals to complex numbers; recursive definitions and related proving techniques (induction) are also introduced. Each time, musical examples are provided: for instance, an excerpt of Ferneyhough’s “Third String Quartet” score illustrates involved time structures, modeled as rationals. Part 4 moves to graphs, putting geometry into play, and nerves, a rather obscure topological notion related to coverings; this last notion is used in one of Mazzola’s pieces, “L’Essence du Bleu,” which is detailed in chapter 16. Part 5 focuses on the basis of algebra, moving from monoids to groups to actions. This part is perhaps the most formal one in the book, but also the one most heavily illustrated with musical examples: counterpoints, Coltrane’s Giant Steps analysis, and modulations in Beethoven’s Op. 106 “Hammerklavier” sonata are here on the agenda. More advanced algebraic concepts such as rings, fields, matrices, and modules are addressed in Part 6, also heavy in formalism. One major application of this part deals with tuning systems: in particular, the Pythagorean, Just, 12-tempered, and Chinese “Thirds-divide subtract-add” approaches are introduced and compared. This part ends with, cherry on the cake, a brief introduction to category theory and Yoneda’s lemma. The few final chapters, in Part 8, move to analytical matters, namely continuity and differentiability, with applications related to continuous processes such as performances and gestures.

Even though I enjoyed reading this book, which is well written and has very few typographical errors, I kept having doubts about its possible readership. With no mathematical background, I don’t think a musician will glean much from the formal parts of the material, at least without outside help; in particular, keeping one’s motivation alive to absorb the wealth of provided abstract definitions will probably prove difficult. However, more mathematically versed readers might appreciate the journey, with its wide range of issues and interesting bits of culture, in both mathematics and music theory. On the minus side, I have to admit that one might be somewhat put off by what can sometimes seem to be an overuse of abstract formalism (Yoneda’s lemma to compare Gould’s and Horowitz’s interpretations of Beethoven’s “Appassionata” sonata... Really?), self-serving comments (for example, Section 2.15, or the number of references to the main author’s work), far-fetched analogies (for example, relating in Section 33.7 performance gestures and the string theory of particle physics), and somewhat aggressive comments (for example, post-World War II European musicology is characterized as “dialectic mumbo-jumbo” (p. 28)). Yet, overall, I recommend this book to mathematicians interested in symbolic music examples, mathematically inclined musicians, and computer music researchers.

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