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Mathematics has been used for centuries to describe, analyze, and create music. In this book, Leon Harkleroad explores the math related aspects of music from its acoustical bases to compositional techniques to music criticism, touching on * overtones, scales, and tuning systems * the musical dice game attributed to Mozart and Haydn * the several-hundred-year-old style of bell-playing known as ringing the changes * the twelve-tone school of composition that strongly influenced music throughout the twentieth century and many other topics involving mathematical ideas from probability theory to Fourier series to group theory. He also relates some cautionary tales of misguided attempts to mix music and mathematics. Both the mathematical and the musical concepts are described in an elementary way, making the book accessible to general readers as well as to mathematicians and musicians of all levels. The book is accompanied by an audio CD of musical examples.

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• Preface(p. xi)
• Acknowledgments(p. xiii)
• 1 Mathematics and Music, a Duet(p. 1)
• 2 Pitch: The Ground of Music(p. 5)
• 3 Tuning Up(p. 21)
• 4 How to Vary a Theme Mathematically(p. 33)
• 5 Bells and Groups(p. 55)
• 6 Music by Chance(p. 71)
• 7 Pattern, Pattern, Pattern(p. 93)
• 8 Sight Meets Sound(p. 105)
• 9 How NOT to Mix Mathematics and Music(p. 117)
• Bibliography(p. 129)
• Contents of the CD(p. 135)
• Illustration Credits(p. 137)
• Index(p. 139)

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CHOICE Review

The extent and complexion of the connection one finds between music and mathematics depends sensitively on what one means by "music" and "mathematics." Certainly, the received conception of (even just Western classical) "music" has seen dramatic change from the era of Bach or Mozart to that of Cage, say, and mathematics no less so from Euler to Cantor or Grothendieck. As seemingly uncontroversial a characterization of music as "the art of sound" will attract attacks as both too broad and too narrow; likewise, any attempt at reducing mathematics to a "science of number." These ambiguities open the possibility of quite diverse treatments, and recently many have indeed found their way to print. Edward Rothstein's Emblems of Mind (CH, Oct'95, 33-0842) perhaps first brought this old subject to the attention of a wide public.Besides the works under review, there are now popular or undergraduate-level treatments: Trudi Hammel Garland's Math and Music: Harmonious Connections (1995); Music and Mathematics: From Pythagoras to Fractals, ed. by John Fauvel et al. (CH, Nov'04, 42-1621); Anthony Ashton's Harmonograph: A Visual Guide to the Mathematics of Music (2003); and Gareth Loy's Musimathics (CH, Nov'07, 45-1511). At the research level, Guerino Mazzola's The Topos of Music (CH, Oct'03, 41-0842) and Jan Haluska's more specialized The Mathematical Theory of Tone Systems (2003) deserve note, and all this ignores so many books on computers and music. Against this background, Harkleroad is distinctive for seemingly allowing the unifying power of mathematics, rather than the fundamental structures of music, to drive the choice of topics. For example, group theory makes several appearances, in contexts as diverse as twelve-tone music, change ringing, and square dancing. This learned but unsystematic book aims to whet especially musically inclined reader appetites for learning more mathematics by hitting some high points of the interaction from an elementary viewpoint.Benson (Univ. of Aberdeen, UK) ranges widely, but his book's heart is a mathematical exposition of the physics of the sounds of ordinary music, all at a level suitable for advanced undergraduates. Benson goes beyond merely celebrating the two subjects' connection, or exploiting one to vaunt the study of the other. Rather, he has crafted an essential companion for the modern creative musician possessing a technical flair. At the same time, by dint of its range and depth, his book might form the basis of an exciting capstone course for undergraduate mathematics majors. One hopes that someone will write a companion volume of comparable length exploring the detailed musicological implications of these mathematical insights in the context of specific works, styles, or historical shifts. Summing Up: Recommended. Both books. Upper-division undergraduates through faculty. D. V. Feldman University of New Hampshire