Review of Music by the Numbers: From Pythagoras to Schoenberg
Princeton University Press, Princeton and Oxford, 2018
176 pp., illus. 43 b/w. Trade, $24.95
For anyone who ponders the role mathematics has in music, this short and delightful book is a joy to read from cover to cover. It is full of anecdotes and interesting facts and, being by Eli Maor is intensely readable. The book itself demands no more mathematical knowledge than that possessed by any intelligent reader, and indeed no more musical knowledge than that possessed by anyone who has at least tried to play an instrument. The prologue is a historical scene setter; then there is a chapter about the different intervals and the role of the Pythagoreans of ancient Greece. Once more, the historical background is immaculate, and this reviewer is pleased to see confirmed that the Pythagorean scale cannot work as 2ᵐ can never be equal to 3ⁿ for any integers m, n something I have quoted more than once to my own students. Chapter 3 moves us forward to the 18th century and the development of acoustics, harmony and the phenomenon of beats when two notes are played that are very close in frequency. There follows an account of how vibrating strings are analysed and how they produce sounds. Technicalities are avoided of course, but there is enough here in the notes to Chapter 4 for anyone who wants to research the more mathematical developments. We then move to waveforms and timbre. The difference between music and noise has in recent times been more a question of taste than precise definition. Two chapters on musical gadgets like the metronome and tuning fork, and musical rhythms get us to the link between the revolution in music in the early 20th century (Stravinsky and Schoenberg) and the revolutions in physics (mainly Einstein’s relativity; it is not easy to see any link to quantum mechanics). The last part of the book hints at a more personal viewpoint. His lack of appreciation of Schoenberg’s music is shared by most. I liked his description of it being locally based as it does not reference standard scales like the classical music of old. Locally based mathematics is similarly esoteric. There is also a personal acknowledgment to Michael Sterling. In particular his instrument the Bernoulli that is based on equal scales and made by attaching strings to a logarithmic spiral. The logarithmic nature of the spiral together with the 300 angles between the strings ensuring that any particular radial string has the correct interval with the next and is an octave apart as it intersects the spiral. I have no idea either how it sounds or whether it is easy to play! Judging from the photo on page 141, a harpist might be the best equipped to try his newest version, the Bernoulli Involute.
Do not expect a complete chronological history of the interaction between music and mathematics, this book is here to entice you into the field by presenting snippets from history and hopefully encourage you to explore the relationships further. This easy to read book hits the right notes to do this.