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Music and Mathematics: From Pythagoras to Fractals

John Fauvel, Raymond Flood and Robin Wilson, editors
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Bonnie Shulman
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It’s an unusually warm November weekend and sounds of street musicians waft my way as I sit in a café in New York City. I’m attending the 20th annual conference of the Society for Literature, Science and the Arts (SLSA), embedded in the Science + Art Festival 2006, taking place at various venues in SoHo, Greenwich Village, and museums and theaters across the city. It seems like the perfect setting to review the book I just finished reading, Music and Mathematics: From Pythagoras to Fractals.

Mathematics and music have been linked together as arts and sciences since ancient Greece. This wide-ranging collection of papers includes contributions by scholars whose expertise spans the disciplines of mathematics, music, computer science, history and physics. However the book is not (as often happens with such collections) just a fortuitous juxtaposition of chapters related by some connection to a common topic, but rather a truly interdisciplinary meeting of minds. The authors often cite each others work, and the essays are deftly woven together in an overview by music historian Susan Wollenberg, The text is complemented by images, including diagrams, photographs and reproductions of ancient, medieval and modern manuscripts, all sprinkled generously throughout the text. Although the chapters do build on each other, it is also possible to dip into this book at any point. I thought of assigning some of the essays in various classes (more on this below) and will definitely refer a colleague of mine who studies fractals and is also an amateur organist to read Robert Sherlaw Johnsons’ final chapter, “Composing with Fractals.”

All the authors have engaging, though different, writing styles, often with a refreshing sense of humor. In his chapter on “Musical cosmology: Kepler and his readers,” J.V. Field, discussing a treatise on music written by Athanasius Kircher says “In his own day Kircher was certainly taken very seriously, but in ours short extracts, punctuated with some of his many elegant illustrations, can easily give the impression that he was, to put it bluntly, a weirdo.” (p. 37) Jonathan Cross, in his chapter “Composing with numbers: sets, rows and magic squares” cannot resist a pun at the expense of composer Alban Berg who is obsessed with the numbers 23 (his own “fateful” number) and 10 “the fateful number of Mrs. Hanna Fuchs-Robettin with whom, it transpires, Berg had become passionately involved.” After this revelation Cross remarks wryly “Thus the intertwining of 10 and 23 has not only structural implications for the composer but strongly extra-musical (extra marital?) ones too.” (p. 135)

The book has appeal for a wide audience. Professional mathematicians and musicians will not be disappointed in the level of exposition, nor will they be bored — there is something for everyone to learn. On the other hand, technical details will not overwhelm or alienate the layperson without formal training in either field. I was particularly satisfied with the chapter on “Tuning and temperament: closing the spiral” by Neil Bibby. I thought I knew what the “equally tempered” scale was, but found myself unable to explain it satisfactorily to a student one day. This essay not only filled in the holes in my own understanding, but it is rich enough to provide inspiration for several student projects on the subject. Adding a note of poignancy, Bibby is one of the four authors (along with John Fauvel, Charles Taylor and Robert Sherlaw Johnson) to whose memory the book is dedicated, “whose untimely deaths occurred while this book was being completed.” (Preface)

Once we have an equal tempered scale, the problem becomes how to build the instruments to reflect these fixed intervals. The historian in me was tickled by Ian Stewart’s “humbling tale” of how “an outstanding triumph of a practical man [is] nullified by a professional mathematician’s carelessness.” (p. 68) I realize that this narrative of “Faggot’s fretful fiasco” is familiar in mathematical musical circles, but it was new to me. In 1743 a talented Swedish instrument maker, Daniel Strahle, published a simple and clever method for locating the frets on a guitar (which amounts to constructing the twelfth root of 2). Jacob Faggot, a founding member of the Swedish Academy, performed some (faulty) geometric calculations to check out the method, and found a maximum error more than ten times the actual error (1.7% instead of 0.15%). It was not until 1957 that another mathematician, J.M. Barbour, found Faggot’s mistake. (See: J.M. Barbour. A geometrical approximation to the roots of numbers. American Mathematical Monthly 64 (1957) 1-9.) As Stewart laments, “if only Faggot had bothered to measure the angle” instead of merely calculating it!

Other favorites of mine in the ten chapters were “Ringing the changes: bells and mathematics” by Dermot Roaf and Arthur White, and “The science of musical sound” by Charles Taylor. But I found all the chapters in this book thoroughly delightful reading. It’s a great resource for optional readings or projects for students in many different classes (mathematics for the liberal arts, musical theory, history of mathematics). Buy it just for the pleasure of reading well crafted essays on fascinating topics.

Bonnie Shulman is Associate Professor of Mathematics at Bates College in Lewiston, ME. In a previous incarnation (1975-1981), she was a founding editor of Bombay Gin, a journal of poetry and writing of Naropa University’s Jack Kerouac School of Disembodied Poetics, and she has long been interested in the connections between mathematics and the arts.

Music and mathematics: an overview , Susan Wollenberg
Part I: Music and mathematics through history
1. Tuning and temperament: closing the spiral , Neil Bibby
2. Musical cosmology: Kepler and his readers , J.V. Field
Part II: The mathematics of musical sound
3. The science of musical sound , Charles Taylor
4. Faggot's fretful blunder , Ian Stewart
5. Helmholtz: combinational tones and consonance , David Fowler
Part III: Mathematical structure in music
6. Musical frieze patterns , Wilfrid Hodges
7. Ringing the changes: bells and mathematics , Dermot Roaf and Arthur White
8. Composing with numbers: sets, rows and magic squares , Jonathan Cross
Part IV: The composer speaks
9. Microtones and projective planes , Carlton Gamer & Robin Wilson
10. Composing with fractals , Robert Sherlaw Johnson
Notes on contributors
Notes, references, and further reading