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The Math Behind the Music

Leon Harkleroad
Cambridge University Press/Mathematical Association of America
Publication Date: 
Number of Pages: 
[Reviewed by
Sarah Boslaugh
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In the early 1980s many of my former classmates, and even some of our professors, from the New England Conservatory in Boston began reinventing themselves as computer programmers. Granted, the computer field was burgeoning in those years, but there was still plenty of competition to get into training programs and to land good jobs, and time and time again former musicians jumped to the proverbial head of the class. Of course they were disciplined and hardworking (try earning your living as a musician for a few years and almost anything else will seem easy by comparison), But whether or not musical training is good preparation for mathematical endeavors and whether or not there is a common set of aptitudes which could be expressed in either discipline, there certainly seemed to be a close relationship between musical ability and success as a programmer.

Of course, I'm not the first to note the connections between music and mathematics; in fact these observations usually take one of two forms. One is that musical pitches can be described numerically in terms of frequencies and combinations of frequencies, which are related to physical properties of the sounding instrument, such as the length of a vibrating string. This observation, in the Western world at least, dates back to Pythagoras. The second type of observation is that music consists largely of patterns and that a composer's skill depends on how well he creates and manipulates patterns using a limited materials (for instance the 7 pitches of the diatonic scale commonly used in the West). Many people share an interest in both music and mathematics, and although a number of books have been written on the relationships between the two subjects, they have often tended to be academic, esoteric, or both.

The Math Behind the Music avoids those extremes and introduces, at a level appropriate to amateur musicians and mathematicians alike, examples of some of the connections between music and mathematics. It is not a comprehensive exposition of the mathematical nature of music but rather 1) a presentation of some interesting and sometimes offbeat musical topics, such as change-ringing and contra dancing,  which may be used to illustrate mathematical concepts, and 2) an explication of some of the mathematics behind musical sound, such as the derivation of several different tuning systems used at different historical times in Western music. The Math Behind the Music does not require advanced understanding of either mathematics or music: in fact most of the text could be easily understood by an interested high school student. The text includes musical and graphic illustrations, and the accompanying CD includes performance of many of the musical examples cited in the text.

Perhaps not surprisingly for a book covering many topics, the quality varies among the individual chapters and sections. The third chapter, on tuning systems, is one of the best. It explains, with admirable clarity, the different frequencies that may be assigned to notes in the Western scale, depending on the system of tuning or temperament used, and why those differences matter to the music performed. On the other hand, Harkleroad's comments about twelve-tone composition in the fourth chapter betray either a lack of sympathy for the music or a complete misunderstanding of the compositional process. But disagreements about individual topics are not really important: The Math Behind the Music is a text which relates music and mathematics in a manner accessible to many individuals, and provides a bibliography of further readings on each chapter's topics, so the best use of this book may be as a jumping-off point for further examinations of whichever topic interests an individual reader.

The Math Behind the Music is part of the Outlooks series, co-published by Cambridge University Press and the Mathematical Association of America. This series explores the interplay between mathematics and other disciplines: previous volumes are When Topology Meets Chemistry, by Erica Flapan (2000), Calculated Bets: Computers, Gambling and Mathematical Modeling to Win , by Steven S. Skiena (2001), and Social Choice and the Math of Manipulation  by Alan D. Taylor (2005).

Leon Harkleroad has taught several MAA Minicourses on mathematics and music and has published in many journals: his specialties include compatibility theory and the history of mathematics. Harkleroad's 1997 article "How Mathematicians Know What Computers Can't Do" won a George Polya award from the MAA. He is also a pianist and percussionist and performs regularly with several ensembles.

Sarah Boslaugh, PhD, MPH, is a Performance Analyst for BJH HealthCare in Saint Louis, Missouri. She published An Intermediate Guide to SPSS Programming with Sage in 2005 and is currently editing The Encyclopedia of Epidemiology for Sage (forthcoming, 2007) and writing Secondary Data Sources for Public Health (forthcoming, 2007) for Cambridge University Press. She holds a bachelor's degree in music from the University of Nebraska and regularly performs in the Saint Louis area on fiddle and viola da gamba, and with the Invera'an Pipe Corps.