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Mathematics and Music

When James Stewart spoke on “Mathematics and Music” at the MAA Carriage House on April 27, he did not explain to the capacity crowd how permutation groups apply to Steve Reich’s “Clapping Music.” Nor did he make good on his threat to draft a member of the audience to join him in a performance of this percussive piece. Stewart didn’t dwell on George Birkhoff’s book-length attempt to quantify beauty or Norwegian composer Rolf Wallin’s use of fractal algorithms to construct harmonies.

No. Stewart’s topic for the evening was as much psychological as musical or mathematical. He offered insight into why the late geometer H.S.M. Coxeter took the scores of Bruckner symphonies to bed with him as reading material. 

Stewart doesn’t care just about Coxeter’s bedtime rituals. They are an instance of a broader trend: the tendency of the mathematically inclined to gravitate toward music. Himself a Ph.D. mathematician and violinist, a bestselling textbook author and sometime concertmaster, James Stewart devoted his contribution to MAA’s Distinguished Lecture Series to exploring possible reasons for this frequent pairing of proclivities.

The intersection of math and music is rife with diversions: curious commonalities and illuminating applications to ponder. What to make, for instance, of the fact that in math and music alone among fields of human endeavor child prodigies crop up on occasion? (Think Mozart or Mendelssohn, Gauss or Galois.)

“One never hears of 9-year-old poets,” Stewart observed. “Or historians.” A young mathematician or composer need not rely on accumulated experience, he explained, but can draw instead on inner resources. Only abstract arts suffer kid geniuses.

It is tempting, too, in theorizing about what draws mathematicians to music, to cite the mathematics tied up in such fundamentals as rhythm, harmony, and melody. One can—and Stewart did—talk about time signatures; the formula for determining a string’s frequency of vibration from its length, tension, and linear density; the translational symmetry of the Beatles’ “Being for the Benefit of Mr. Kite.”

And there’s the steady pulse and sudden phase shifts of minimalist music; the continued fractions argument for why octaves should comprise 29, 41, or 53 notes instead of a measly 12; the fact (unknown to Stewart before an audience member brought it to his attention) that Art Garfunkel has a master’s in math from Columbia. “Interesting,” Stewart conceded, of all of this, “but it doesn’t provide the explanation.”

MAA Distinguished Lecture: James Stewart

So what does? Why have mathematicians from Pythagoras and Euclid to Riemann and Poincaré played instruments, written treatises on music theory? In a word, “form.” Both math and music have it. Igor Stravinsky noted that musical form is “far closer to mathematics than to literature, not perhaps to mathematics itself, but certainly to something like mathematical thinking and mathematical relationships.”

And Stewart would go even further. “I think that if Stravinsky had known a bit more about mathematics,” he said, “he would have omitted the qualifying phrase.” Mathematics is very much concerned, Stewart argued, with structure, “the way mathematical objects fit together and relate to each other.” Structure, said Stewart, is the mathematical analog of form.

To illustrate what he argues is the essential similarity between math and music, Stewart outlined a bijection between six geometric transformations—translation, reflection in x, reflection in y, rotation, dilation with scale factor greater than one, and dilation with scale factor less than one—and their equivalents in music. Rotation, in Stewart’s scheme, corresponds to retrograde inversion, for example. A musical theme is repeated, but “upside down and backwards,” he noted.

Composers use such structure-preserving transformations to achieve the balance of unity and variation that makes for an aesthetically pleasing piece. “Listen closely to a Bach fugue,” Stewart said, “and you can hear the geometry.”

Along with structure, logic also plays a part. A well-constructed fugue or symphony flows with the satisfying inevitability of a mathematical proof, Stewart said.

Mathematicians recognize in music something familiar, something they strive for in their mathematical pursuits. An affinity for music is, according to Stewart, mathematicians’ response to this recognition. It explains, for one thing, Coxeter’s habit of perusing Bruckner scores. “He was fascinated,” said Stewart, “by the geometric patterns.”

Itself fascinating, James Stewart’s lecture left listeners keen to explore further the interconnectedness of math and music, to dance a minuet, perhaps, or track down the music-related entries mathematician Jean le Rond d’Alembert contributed to Diderot’s famous encyclopedia. Unfortunately, logistics prevented the audience from accepting en masse the speaker’s invitation to the April 28 premiere at Integral House of a Steve Reich marimba composition. —Katharine Merow

Audio and video editing by Laura McHugh

 Listen to the full lecture (mp3)

 Ivars Peterson interviews James Stewart about math and music (mp3) (April 27, 2010)


This MAA Distinguished Lecture was funded by the National Security Agency.

Related Links:

2009 Interview with James Stewart

"James Stewart and the House That Calculus Built" (pdf) from MAA FOCUS (August/September 2009)