###### by Claire Rubio

What is your favorite counting number (or, natural number: 1,2,3, …)? My favorite counting number has always been 20. I can list a few reasons why I like this number. In addition to the 20^{th} being the date of my birth, the digits that make up this number are both even, and two is the only even prime number. I like that 20 is divisible by both even and odd numbers, including half of the first ten counting numbers.

One musician who seems to have a favorite number (though he is not around today to confirm this) is Johann Sebastian Bach. His favorite number must be 14. And, while 14 is similar to 20 in its parity (i.e., they are both even), Bach took advantage of a certain musical beauty in the number 14.

Another fact of Bach’s life that supports my argument for his favorite number is his initiation of a society simply for musicians to discuss their application of mathematics within their compositions. This group called the *Korrespondierenden Sozietät der Musicalischen Wissenschaften* (Corresponding Society of the Musical Sciences) met to raise ideas about using mathematics in their music, and Bach insisted on being the 14^{th} member to join. His attachment to 14 may have come from the sum of the letters in his name. If you replace each letter with its position in the English alphabet, assuming that a is the first, then the sum of digits that spell out Bach is 2+1+3+8 = 14. What a logical rationale to enjoy this two-digit number, and it all came from a man who did not formally learn any mathematics beyond elementary arithmetic.

He wielded its mathematical properties to take a line of simple music and encode directions to generate a complex piece: from one line of music comes an infinite loop of perfect, overlapping harmony. Bach took a mathematical approach to efficiently writing an entire work out of one line of music. To demonstrate this infinite looping of decoded music, first consider the crab canon by Bach.

Canons use a unique device, a staggered start, to create a lovely harmony: one musician begins, and a measure or two later, a second musician begins the same melody, and so on. Familiar canons include *Frère Jacques* or *Three Blind Mice*. Bach took this a step further and included an additional layer of mellifluousness. In canons, the foundation or what the composer wants to be the main thing going on in a piece, is called the subject while whatever accompanies this is called the countersubject. The crab canon looks like a single line of music until you arrive at the very end.

The fancy B-like symbol (it’s actually a clef!) that Bach placed at the end of this canon indicates that it is the *beginning* of the piece. Instead of finishing the piece with a double bar, Bach decided to write a backwards clef. In fact, not only the clef, but the key signature and also the time signature appear at the end written backwards, as if the musician really should arrive there, make a 180-degree turn and proceed to begin the piece again at the end. The end now becomes the beginning of the piece.

Bach stamped the end of the last bar with this symbol which instructs the decoder – or astute musician – to then play this piece from finish to start. Ah, but this is a canon *backwards *too! In fact, Bach’s delightful mathematical musicality in this canon produces a forwards and backwards round which, when played simultaneously, creates a harmony all on its own. The crab canon, and other works of Bach, contain this self-harmonizing element which gives an added symmetry to the piece. So, if we were to play the canon forwards and backwards at the same time to hear the harmony, we would read sheet music something like this. Notice too that upon returning to the beginning of the piece, the voices might switch lines. If the subject takes the countersubject on the second play and the countersubject takes the subject, that is the same as changing direction of play with one line.

A mathematical way to consider this is to take the end of the piece and glue it to the beginning. Suppose we printed out the crab canon twice on a long strip of paper, with one strip going backwards to account for the harmony within itself. Then, take the two lines of music and glue the start and finish to each other. This would produce a kind of ring or cylinder. Traveling along this object would produce an infinite canon that continues without a clear end, and, upon arriving at the “end” would promptly begin again.

While a good starting place to geometrically consider Bach’s canon, this construction does not quite fit all his canons. It does work for the never-ending round of the crab canon, but Bach’s other canons require a more complex surface to complement his encrypted music.

To construct this more complex surface, let’s focus on another canon by Bach – this is where 14 is important. Bach composed a set of 14 canons embedded in another manuscript which he titled in German “Verschiedene canones über die ersteren acht Fundamental Noten vorheriger Arie aus den Goldberg-Variationen,” or “Various Canons on the First Eight Bass Notes of the Preceding Aria.” Bach created a puzzle in each canon for the aware musician to decode, sometimes with only one, brief line of music. There is not the same “forwards is backwards” nature as in the crab canon; Bach gave different instructions here. The fifth canon ought to be flipped upside down to create its own harmony, and then shifted over a few bars. So, when the forward direction begins, the harmony is slightly offset from the horizontal, upside-down melody.

Let’s make things a little more interesting with the cylinder. If, before the beginning and end get glued, we instead *twist* the strip of music and then glue it together,, you get a slightly different ring. This is a mathematical object called a *Möbius strip*.

In the previous examples with a cylinder, there is only music on the outer surface – and there *is *an inner side to the cylinder. On this new ring of music, there is *not *an inside or outside of the strip. Rather, the twist before gluing allowed the outer surface and inner surface of this ring to become one and the same. Now, if we were to trace our finger along the music from beginning to end, we could continue playing this seamlessly in an infinite loop without changing *sides *of the surface. Why stop there? Turns out, Bach’s canon 5 of the 14 can be printed on transparent paper and glued to form a* *Möbius strip where the “inner” and “outer sides” are the same notes. Due to the spacing of the canon, the exact same notes would align perfectly on either “side”!

I wonder – what other mathematical objects allow music to unveil its complexity? Are there any scores of music waiting to be decoded by projection onto a cube or sphere? Or, better yet, why not project scores onto *any* Platonic solid, like a dodecahedron?

Bach’s 14 canons contain too much mathematical coding to suppose their self-harmonization is a coincidence. If the musician understands how to fit all the pieces of the mathematical puzzle within the canon together, then they can unlock its beauty.

As with the Möbius strip, let’s finish by returning to the beginning. By the same logic that made Bach’s favorite number 14, let’s return to my favorite number. If you replace each letter in my last name with its position in the English alphabet with A = 1, B = 2, etc., you arrive at this sum: R+U+B+I+O = 18+21+2+9+15 = 65. I suppose I should consider changing my favorite number to 65.

**Links to explore further:**

(AMS about Bach’s canons) http://www.ams.org/publicoutreach/feature-column/fcarc-canons

(Math Scholar on Bach) https://mathscholar.org/2021/06/bach-as-mathematician/

(Article on Society of Musical Science) https://www.bach-cantatas.com/Lib/Mizler-Lorenz-Christoph.htm

(AMS paper) Surface Topology in Bach canons: http://www.ams.org/publicoutreach/feature-column/fc-2016-10

Music on a Clear Möbius Strip: https://www.youtube.com/watch?v=sToqbqP0tFk

Claire Rubio spends most days in the year either arguing about mathematics or the moon in St. Paul, MN. When not thinking about the roots of unity, she enjoys exploring new features of Minnesota geology, cranking out homemade pasta, and reading Shakespeare.

* Header Image: *Elias Gottlob Haussmann, Public domain, via Wikimedia Commons

## 2 Comments

## Henry Lewis

Fabulous post. Bach has always intrigued me. This may help explain some of his mystery. Thanks for the post.

## Mary Radtke

Wow! How did he figure out how to do that? How did you or anyone else figure out that he did it?