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.