Music by the Numbers: From Pythagoras to Schoenberg
By Eli Maor
Princeton Press, 176pp | Buy on Amazon

Music is often said to be deeply connected to mathematics. Indeed, many of the world’s foremost classical composers such as Bach or Stravinsky claimed that music must have some math-like logic, but not much has been said about the influence music has had on mathematics. In his latest book, Eli Maor argues that the two have equally influenced each other, despite each advancing on their distinct, separate paths.

A former professor of the history of mathematics at Loyola University Chicago, Maor charmingly writes about how music has inspired mathematicians for centuries. The famous Greek philosopher and mathematician Pythagoras once said, “There is geometry in the humming of the strings, there is music in the spacing of the spheres.” It’s no surprise that Maor begins his book with Pythagoras, who was the first to establish the octave as a fundamental musical interval.

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Noticing that certain ratios of string length produced pleasant combinations of sounds (consonances), whereas ratios of larger numbers produced dissonances, Pythagoras saw this as a sign that nature is governed by simple numerical ratios. For instance, by extrapolating his findings of vibrating strings, Pythagoras claimed that planets in motion also produce sounds and their ratios corresponded to tonal musical intervals in the Pythagorean scale. His idea would dominate scientific thought for thousands of years.

Perhaps the most famous example of how music has influenced science is the vibrating string problem, a 50-year-old debate that pitted Bernoulli, Euler, D’Alembert, and Lagrange — some of the greatest mathematicians of all time — against each other. Although they came close, the four just couldn’t settle the debate. However, Maor writes that their work “spearheaded the techniques needed to deal with the continuum, of which the vibrating string was but the simplest example.”

Maor goes on to weave further interesting connections between music and mathematics, ending with the simultaneous emergence of Einstein’s theory of relativity and Arnold Schoenberg’s twelve-tone music. Schoenberg’s twelve-tone or series scale enables composers to write atonal music whose rules dictate that no note is bound to any home key whatsoever. Instead, the notes are played relative to its predecessor — this way all systems of reference are equivalent to one another, just like in Einstein’s theory.

You don’t necessarily need either mathematical or music training to enjoy Music by the Numbers, as the author does a great job at explaining musical scales or linear algebraic equations. All in all, this was a great book that anyone with an interested in music, mathematics, or both will find engaging and useful.