Music by the Numbers: From Pythagoras to Schoenberg

The prolific author Eli Maor has released yet another very readable and enjoyable book on the history of mathematics from Princeton University Press. Like his other books, it is not intended to be comprehensive (Trigonometric Delights, for instance, presents “selected topics in trigonometry from a historic point of view”) and at times is intensely personal. Music by the Numbers is essentially an homage to some of the relations between two subjects, mathematics and music, which the author has often mentioned as twin loves in previous work.

Maor’s books certainly rely more on the secondary literature and may be a little more apt to repeat enjoyable-but-likely-false stories than history of mathematics textbooks. On the other hand, they also have a quite different intended audience, and I enjoy using excerpts from his books with non-major mathematics students. So here we see the harmonious blacksmith of Pythagoras with only a short later remark that we have almost no primary information about Pythagoras. There is a well-written description of how Helmholtz showed that overtones exist (at about the same time as spectral analysis was happening with light as well, Maor points out) using an ingenious method you’ll have to read about on page 67. It is interesting to read about the non-discovery of Fourier series, seeing Euler get annoyed at his contemporaries, and Maor’s discussion of the Slinky and waves and the life story of the metronome. Some will like his comparison between transposing key and coordinate transformations.

Unfortunately, this effort is not as convincing as many of his other books. There are three main reasons for this. First, it is simply far less mathematical than his other work, while also trying to be accessible to those with no previous exposure to musical notation or theory. It’s a good thing to reach many audiences, but assuming a little more of the reader would have enabled more interesting treatments of many topics.

Second, there are musicological errors, particularly around the issue of temperament (tuning of the instrument to avoid bad-sounding intervals). For instance, while Maor rightly notes that with modern temperament the key we play a piece in may not matter much, this is manifestly not the case with older instruments, and so one rightly could talk about the character of a key given (say) a given temperament. In fact, musicologists use this to analyze what sort of temperament Bach’s organs might have been tuned in. While Bach’ Well-Tempered Clavier was perhaps “said to” be about equal temperament, it almost certainly wasn’t that specific. Since Maor mentions Mersenne and Vincenzo Galilei a fair amount, discussing keyboards with split keys for D sharp and E flat, or how tuning of fretted instruments evolved differently than for keyboards (respectively) would have been very much in line with the book.

Finally, the book has as an additional underlying leitmotif the comparison between the revolutions in Western common-practice music and mathematical physics in the 20th century. While there are certainly interesting parallels between Albert Einstein’s life and that of a pioneer of non-tonal composition, Arnold Schoenberg, I find the analogy between local frames of reference and metrics in relativity, on the one hand, and the extremely local meter in Stravinsky’s “Rite of Spring” and the lack of tonal center in twelve-tone music, on the other, entertaining but fairly forced. This focus, especially at the end of the book, is strangely ill-fitting giving the concreteness of the rest of the book.

Instead, it would have been wonderful to have such a good storyteller give us some of the updates in the world of music theory, where groups and graphs have been taken in as important pieces of the story for understanding harmony of the Romantic era. Do a search on the internet for Beethoven’s 9th Symphony and walks on a torus, or on the use of orbifolds to track chord motion, or for how to use Burnside’s lemma to count the number of symmetric twelve tone rows. This would have played very well as a counterpoint to Maor’s skepticism about string theory and its multiverses at the end of the book.

As with all of Maor’s books, this one belongs in your library so that leading students can learn about unknowns like Joseph Sauveur in the fascinating story of how mathematics and music intersect. But while you’re using it to prepare to tell your calculus class that your ear does Fourier analysis, just be sure to put a few of the more in-depth books in your prep pile as well.

Karl-Dieter Crisman teaches mathematics at Gordon College in Massachusetts, where he also gets to work on open source software, the mathematics of voting, and examining connections between all of these and issues of belief and faith.