Bach on a Möbius Strip: Marcus du Sautoy Visualizes How Bach Used Math to Compose His Music




“A mathematician’s favorite composer? Top of the list probably comes Bach.” Thus speaks a reliable source on the matter: Oxford mathematician Marcus du Sautoy in the Numberphile video above. “Bach uses a lot of mathematical tricks as a way of generating music, so his music is highly complex,” but at its heart is “the use of mathematics as a kind of shortcut to generate extraordinarily complex music.” As a first example du Sautoy takes up the “Musical Offering,” and in particular its “crab canon,” the genius of which has previously been featured here on Open Culture.

Written out, Bach’s crab canon “looks like just one line of music.” But “what’s curious is that when you get to the end of the music, there’s the little symbol you usually begin a piece of music with.” This means that Bach wants the player of the piece to “play this forwards and backwards; he’s asking you to start at the end and play it backwards at the same time.” His composition thus becomes a two-voice piece made out of just one line of music going in both directions. It’s the underlying mathematics that make this, when played, more than just a trick but “something beautifully harmonic and complex.”




To understand the crab canon or Bach’s other mathematically shaped pieces, it helps to visualize them in unconventional ways such as on a twisting Möbius strip, whose ends connect directly to one another. “You can make a Möbius strip out of any piece of music,” says du Sautoy as he does so in the video. “The stunning thing is that when you then look at this piece of music” — that is the fifth canon from Bach’s Goldberg Variations — “the notes that are on one side are exactly the same notes as if this thing were see-through.” (Naturally, he’s also prepared a see-through Bach Möbius strip for his viewing audience.)

In 2017 du Sautoy gave an Oxford Mathematics Public Lecture on “the Sound of Symmetry and the Symmetry of Sound.” In it he discusses symmetry as present in not just the Goldberg Variations but the twelve-tone rows composed in the 20th century by Arnold Schoenberg and even the very sound waves made by musical instruments themselves. Just this year, he collaborated with the Oxford Philharmonic Orchestra to deliver “Music & Maths: Baroque & Beyond,” a presentation that draws mathematical connections between the music, art, architecture, and science going on in the 17th and 18th centuries. Bach has been dead for more than a quarter of a millennium, but the connections embodied in his music still hold revelations for listeners willing to hear them — or see them.

Related Content:

Take an Intellectual Odyssey with a Free MIT Course on Douglas Hofstadter’s Pulitzer Prize-Winning Book Gödel, Escher, Bach: An Eternal Golden Braid

The Genius of J.S. Bach’s “Crab Canon” Visualized on a Möbius Strip

Visualizing Bach: Alexander Chen’s Impossible Harp

How a Bach Canon Works. Brilliant

The Math Behind Beethoven’s Music

Based in Seoul, Colin Marshall writes and broadcasts on cities, language, and culture. His projects include the Substack newsletter Books on Cities, the book The Stateless City: a Walk through 21st-Century Los Angeles and the video series The City in Cinema. Follow him on Twitter at @colinmarshall or on Facebook.


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