The Pythagoreans, as is well-known, explored mathematical aspects of music. But since then, what mathematicians have worked in this area? How have musicians and other non-mathematicians contributed? How has or has not such work related to contemporaneous trends in mathematics and in musical practice (not to mention other relevant fields such as physics and anatomy)?
Wardhaugh’s book focuses on the math-and-music scene in England during the later seventeenth century. Who was involved? Rather, who wasn’t? Newton, Wallis, Barrow, Pell, Brouncker, Nicolaus Mercator, Boyle, and Hooke all participated to varying extents. So did professional musician John Birchensha and parson/music theorist Thomas Salmon, both of whom gave presentations to the Royal Society. Besides discussing the work of these figures, Wardhaugh expands his scope, both geographically and chronologically, to detail contributions by Descartes, Mengoli, and others.
The investigations of these people reflected the mathematical developments of the time. The recently-developed tool of the logarithm brought new perspectives on tuning systems and, Wardhaugh argues, on the nature of musical pitch itself. And, of course, results from the theory of vibrating strings and oscillations in general had many musical implications.
The subject is interesting, and the material is nicely presented. Although scholarly and equipped with all the requisite citations and footnotes, this book is not at all the dry tome that its title might suggest. Occasional lapses do occur, such as the statement, “the pressure and volume of a gas are directly proportional to one another at a constant temperature.” But this confusion of direct and inverse proportionality seems more of a slip than a sign of mathematical cluelessness. Indeed, the vita posted on Wardhaugh’s web site sports a bachelor’s degree in mathematics, in addition to a master’s in music and a doctorate in history. Wardhaugh has drawn on his background in all three areas to produce an intriguing and informative snapshot of an active period in the longstanding dialog between mathematics and music.
Leon Harkleroad is the author of The Math Behind the Music, published in the MAA’s Outlooks book series, and has presented several MAA Minicourses on math and music. He fondly recalls Michael Flanders’ direct-proportion version of Boyle’s Law for the political arena: “The greater the external pressure, the greater the volume of hot air.”