Peter Pesic reviews Music: A Mathematical Offering by David J. Benson in American Scientist.
“Music is a science,” wrote the great composer and theorist Jean-Philippe Rameau in 1722, “which should have definite rules; these rules should be drawn from an evident principle; and this principle cannot really be known to us without the aid of mathematics.” David J. Benson’s book Music: A Mathematical Offering gives the latest and fullest view of music in the light of mathematics. A professor of mathematics at the University of Aberdeen as well as a keen amateur singer, Benson has assembled a fascinating variety of topics that make his book a uniquely rich source, whether for classroom use, reference or self-study. He has constructed the different sections that flow from his general introduction to be independent, allowing readers to follow their own interests and predilections.
This book goes into mathematical details that many general accounts avoid, and here Benson deserves special praise for his skill and clarity. He does expect his readers to be familiar with standard college calculus, but he always presents his arguments with enough helpful explanation, good examples and exercises to put his audience at ease. Some sections do go into more depth (including a few references to complex variables), but the reader can skip them if so inclined.
Benson begins with the physics of sound, not neglecting the perceptual subtleties of human hearing, which he outlines clearly. After covering the basics of vibrating strings and their wave motions, he turns to a nice exposition of Fourier’s theory of harmonic analysis, readily accessible to someone who has studied basic calculus.