Thursday, April 23, 2015

How Bach's music is like Escher's drawings and Gödel's theorem

Years ago I remember reading Douglas R. Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braidwhich compares Bach's music to Escher's drawings and Gödel's mathematical reasoning. You may remember Echer's drawings, like the one below, where the water at the bottom of the waterfall flows "downstream" until it reaches the top of the falls again:

What he does with the mathematics of Gödel may be of particular interest to mathematicians and philosophers of mathematics and meta-theories of theories, but my particular interest was in what Hofstadter did with Bach's "Musical Offering" (Das Musikalische Opfer), or that part of it known as the Crab Canon (canon cancrizans). From a mathematical point of view, what Bach does is simply breath-taking. First, he plays through the melody in the trebel clef, then plays it backwards, then he plays the same sequence simultaneously forwards and backwards, then transposes the trebel and base clefs, and so on. What is remarkable is how everything fits together. As I visualize it, it must have been like playing a chess game and anticipating not one or two moves of your opponent alongside your responses, but perhaps one or two hundred. Bach might have made a hell of a chess player.

But just today I discovered this video of Bach's "Crab Canon" visualized on a Möbius Strip such as Escher might have drawn. Watch it. Follow the notes and what Bach does at each step. It's simply mind-numbing. And ethereally beautiful.

Here's what Colin Marshall says about it in his post, "The Genius of J.S. Bach’s “Crab Canon” Visualized on a Möbius Strip" (Open Culture, February 2013):
The most impressive of Johann Sebastian Bach’s pieces, musicophiles may have told you, will knock you over with their ingeniousness, or at least their sheer complexity. Indeed, the music of Bach has, over the past two and a half centuries, provided meat and drink to both professional and amateur students of the relationship between ingeniousness and complexity. It’s no mistake, for instance, that the composer has offered such a rich source of intellectual inspiration to Gödel, Escher, Bach author Douglas R. Hofstadter, well beyond having given him a word to fill out the book’s title. Listen to the first canon from Bach’s Musical Offering, and you’ll hear what sounds like a simple beginning develop into what sounds like quite a complex middle. You may hear it and instinctively understand what’s going on; you may hear it and have no idea what’s going on beyond your suspicion that something is happening.
If you process things more visually than you do aurally, pay attention to the video above, a visualization of the piece by mathematical image-maker Jos Leys. You can follow the score, note for note, and then watch as the piece reverses itself, running back across the staff in the other direction. So far, so easy, but another layer appears: Bach wrote the piece to then be played simultaneously backwards as well as forwards. But prepare yourself for the mind-blowing coup de grâce when Leys shows us at a stroke just what the impossible shape of the Möbius strip has to do with the form of this “crab canon,” meaning a canon made of two complementary, reversed musical lines. Hofstadter had a great deal of fun with that term in Gödel, Escher, Bach, but then, he has one of those brains — you’ll notice many Bach enthusiasts do — that explodes with connections, transpositions, and permutations, even in its unaltered state. Alternatively, if you consider yourself a consciousness-bending psychonaut, feel free get into your preferred frame of mind, watch Bach’s crab canon visualized, and call me in the morning.

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