The mathematics of music is climbing on the charts. In recent years, math-and-music special sessions, minicourses, and presentations have proliferated at national and regional meetings of the MAA and AMS. The program of the 2004 Joint Mathematics Meetings in Phoenix included:

- 22 half-hour papers in the AMS-MAA Special Session on Mathematical Techniques in Musical Analysis;
- 4 twenty-minute music-related papers in the MAA Session on Mathematics and the Arts;
- Erich Neuwirth's hour-long MAA Musical Presentation: The Mathematics of Acoustic Paradoxes.

And in Europe, the continent where the Pythagoreans started the math-and-music ball rolling? Five years ago, the European Mathematical Society sponsored a two-day, three-city conference on the subject. Sessions in Lisbon, Paris, and Vienna — linked by teleconferencing — each focused on a different aspect of the relationship between math and music. The sixteen papers in the book under review provide a sampling of the presentations at the conference.

The editors begin with contributions from the Lisbon session, which concentrated on historical perspectives. Several nice papers came out of this session. Eberhard Knobloch, a noted historian of mathematics, traced the interplay of combinatorial principles with music in the work of Mersenne, Leibniz, Euler, and others. Benedetto Scimemi described various geometric and numerical methods that 18th-century musicians applied to address equal temperament. In Jean Dhombres' paper, Lagrange occupies center stage. The article by Wilfrid Hodges and Robin Wilson provides musical examples of various mathematically-flavored compositional devices, such as retrogrades and inversions. While this paper may not contain much new to aficionados of math and music, it is very welcome to have a good selection of such examples gathered in one place. (By the way, Hodges has also contributed an article "The Geometry of Music" to the recent Oxford volume *Music and Mathematics: From Pythagoras to Fractals*, co-edited by Wilson.) The other Lisbon paper, which opens the book, does not live up to the caliber of the offerings described above. But overall, the Lisbon essays supply much interesting, and often unfamiliar, information on past explorations of mathematics in music.

The Paris session dealt with mathematical logic and music, and the seven articles from there are of much more uneven quality. In fact, musicologist Laurent Fichet's contribution refers to "those musicians who used mathematics to give the impression that their theories were scientific," and his criticism could justly be leveled at some of the chapters here. Although Fichet's quote suggests deliberate obfuscation, often misapplications of math in music result honestly from the perpetrator's own confusion. At any rate, the literature contains many theories in which vague analogies between music and (the author's conception of) some area of mathematics become elevated to the status of full-blown explications. Here I will not dissect the more marginal papers in this book. Suffice it to mention one article which attempts to meld category theory with semiotics and manages to achieve a quite high jargon-to-insights ratio. And unaccountably, the editors have included a 24-page paper that does not deal with music at all.

Still, the Paris session produced some papers worthy of note. Fichet's article briefly examines some mathematical approaches used by music theorists in the past century. Shlomo Dubnov and Gérard Assayag developed a general prediction/generation model, used to create music based on the styles of different bodies of music, such as the Bach *Two-Part Inventions*. And Marc Chemillier explored the interface between ethnomathematics and music, albeit with a lengthy detour into visual art. Citing Daniel Andler, Chemillier warns that "there may exist a gap between the formal properties of traditional objects..., which are discovered by ethnomathematicians and expressed in their own mathematical language, and the cognitive processes of peoples who produced these objects." I would add that this point applies equally to *post facto* mathematical analysis of musical compositions.

The Viennese session emphasized mathematical aspects of sound and its production. Three of the papers survey this field and share an engineering flavor. Of these three, the essay by Giovanni De Poli and Davide Rocchesso is both the longest and the most mathematically detailed. They discuss various models of sound generation — spectral, cellular-automata, finite-difference, and many others. The remaining paper printed in this section was actually presented by Erich Neuwirth in Lisbon. But his remarks on several tunings, along with *Mathematica* code for demonstrating how they sound, reflect the theme of Vienna. The *Journal of New Music Research* has separately published additional contributions to this session.

It is disappointing to note that Springer seems not to have bothered with copy-editing, all the more so given the price of this volume. The book abounds with typos and unidiomatic English. Despite this and despite the variable quality of the articles, *Mathematics and Music* does furnish much interesting material about some of the points of contact between the two fields. Of course, many more connections exist than the book could address. For one such connection, I recommend an unlikely source: *The Mathematics of Juggling* by Burkard Polster. (See the MAA Online review posted 12/3/03.) That book, incongruously but happily, contains an excellent description of the mathematics of ringing the changes on bells. That such a chapter can find its way into a book on juggling gives yet another indication of the current math-and-music boom. But boom or bust, the math-music link will perennially attract interest — it's one of the classics.

Leon Harkleroad has taught several MAA Minicourses on mathematics and music and is writing a book on the subject, to be published jointly by the MAA and Cambridge University Press in the Outlooks series. His other specialties include computability (née recursion) theory and history of mathematics. Musically, he is a decades-long pianist, budding percussionist, and current member of five performing groups.