Variations on a theme

It's Thanksgiving here in the US, and so we thought we'd illustrate our usual subject -- complexity --  much more pleasantly today, while we eat turkey rather than complain about the ones that get published.

In that holiday spirit, here's a video depicting Johan Sebastian Bach's "Crab Canon."  We stumbled across it on YouTube; it's attributed to Jos Leys and Xantox.



The mechanics of this canon are beautifully illustrated in this video rendering.  Musical canons come in a variety of forms but are essentially compositions in two or more voices, in which each voice can stand alone but they also weave together harmoniously.  Bach's Crab Canon, with its double counterpoint, is an example of one of the most challenging of canonical structures to compose.  And when it came to canons, Bach was the mad genius at following rules so intricate that it makes our ordinary heads spin.  As the video shows, this piece is Möbius strip-like in the way the two voices interweave.  (Indeed, here are instructions for making your own Möbius strip of this piece.)

But we wanted to understand more about canons, and specifically Bach's canons, so we asked our violinist daughter, Amie, some further questions, and she in turn sent us on to her polymath musician friend, Ben Grow.  He's a conductor, composer/arranger, teacher and multi-instrumentalist in New York City.

Ben writes:

"Bach is s a composer who, even during improvisations, could squeeze canons out of even the most unlikely-sounding musical material. Much of his music is imitative even if it isn't described as a canon, so when he set out to write a proper canon his powers were huge. That said, the "Crab Canon" is unusual even for Bach.

"A defining characteristic of canons is that their voices enter at less predictable time intervals than simple rounds and they often enter at intervals besides the unison. Also, canons can be imposed over an independent bass line. The canons of the Goldberg Variations, for example, occur at every diatonic interval through the major 9th and all but one have a separate bass that doesn't participate in the canon (the canon at the 9th is only two voices, so both participate; the bass line outlines the same chord progression as the aria that is used in every movement).



"The 'Crab Canon' is unusual because all the dissonances must be functional in both directions. Because we expect dissonances to resolve a certain way, it's difficult to give them a double role – it's even more difficult to make the thing sound like real music!

"The way a canon works is built on certain laws of tonality (the function of notes in dissonance and consonance), which I guess are slightly different between musical styles/eras, but are generally the same. Certainly, Bach's artistry is what's most impressive about the canon because the form seems academic and dry, but I don't think he stretched the rules. Canons either work or they don't, but the real test is if they sound like music or or something similar. This is where most others would follow the rules of tonality and counterpoint but create something awkward. Bach's mastery is that his ear and taste are completely connected to the more mathematical elements of music – I think his filters only showed him the beautiful options. Since he was such a master improviser the mere time it took to write music down was more than enough to edit the work to perfection. Canons are also rare, and I can't think of other composers of the time who wrote so many separately in addition to fragments nested within bigger works."

So, there are strict rules of musical form but whether or not a piece of music works, and beautifully does not depend on the rules being followed.  Indeed, following the rules is easy.  As Ben says:

"If you'd like to try to write a canon, just write a measure of music into one voice and then copy it a measure later into the second voice. The second measure of the first voice will need to fit over the second voice's first measure (which is the same as the first voice's first measure). You can trial-and-error stitch together a canon this way if you don't have the rules of harmony to guide you – you can just use your ear!"

So I did just that.  I have never composed music of any sort before, so this really was just following the rules to see what happened.  This was not, of course, the experience true composers are said to have, of simply transcribing the music they hear in their heads!  To make it easier, and in keeping with the genetic-y theme of this blog, Amie suggested I use just the letters A, C, G, and T -- though, there not being a note T, I substituted E instead (to stretch this analogy to the breaking point, we could say that's rather like RNA using U, uracil, instead of T.)

The result?  The ACGT Canon, composed by me and recorded by Amie.  She decided to play it pizzicato, and here, if you'll forgive the hiss in the recording because she recorded it on her computer, is what we got.



You'll agree that it follows the rules.  But, ahem, you'll also agree that clearly not just anyone could produce the kind of beauty that Bach did, over and over and over; music that transcends time and touches souls.  But if his greatness did not come from stretching the rules in ways that others hadn't thought of but instead he wrote within the rules in ways that others didn't or couldn't do, what made him great?  Perhaps that is an emergent property made up of, but not explained by, an assemblage of parts: in Bach's case, the result of a combination of instruments or voices, conception, inspiration and genius, the whole being greater than the sum of these parts.

That is, just like life, it's complex, it's irreducible.  But achingly beautiful.  An organism is an assemblage of atoms rigidly following the universal rules of chemical interaction.  Slap some nucleotides together, following the rules, and you'll get something, but the rules themselves don't tell you what it will be.

These atoms make up gargantuan molecules, like DNA, that, too, interact rigidly following chemical rules.  In turn, massive complexes of DNA, RNA copies, and proteins literally by the thousands interact following physical rules but in locally organized regions, and surrounded by millions of other molecules that make a cell membrane -- again, each rigidly following the rules of molecular interaction.  And if that weren't enough, a human is made of countless billions of cells that interact mechanically.

In all this, there is a chaotic buzz of chance factors, variations on the theme. But it is not chance that organizes these elements into the emergent, unpredictable, awesome, beautifully organized trait that is you -- you who, using your assemblage of countless molecules, can appreciate beautiful music.

Despite our hubris, whether we will ever truly understand how this happens is quite a thought to chew thankfully on.  So, whether it's a holiday for you today or not, we hope you simply enjoy this venture into the beauty of Bach.  Thanks, Ben.