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.
Related content:
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The Open Goldberg Variations: J.S. Bach’s Masterpiece Free to Download
Glenn Gould Explains the Genius of Johann Sebastian Bach (1962)
Colin Marshall hosts and produces Notebook on Cities and Culture and writes essays on literature, film, cities, Asia, and aesthetics. He’s at work on a book about Los Angeles, A Los Angeles Primer. Follow him on Twitter at @colinmarshall.
I can’t see the image here. How else can I view the canon imposed on a mobius strip?
This is the direct link to the video on youtube: https://www.youtube.com/watch?v=xUHQ2ybTejU
Perhaps I am not of sufficient mind to appreciate this, but I fail to see the association of the Möbius to this piece. I certainly understand the complimentarity of the musical halves, and to reverse each simultaneously is genus, but this to my mind likens to reflection, rather than any association with a möbius form. What does one perceive as the added dimension to enlist the möbius?
I agree with Dave Webb.
If you listen to the musical extracts beginning at 1’14″ and at 2’04″, you will hear that they are exactly the same. Obviously the two halves start and end at the same time and hence implicitly could continue forever (like a Möbius strip) – but that’s not the clever bit.
A “mind-blowing coup de grâce” only for those who “process things more visually than [they] do aurally”!
Why is such fluff and hyperbole needed in writing about a simple exercise in counterpoint? Any experienced composer can write a crab cannon. While I agree that Bach did compose some pieces of great complexity, this cannon is not one of them. I guess that’s what you get when you reduce the music in our culture to its lowest common denominator…people going gaga over a cannon.
I must assume you didn’t try to construct a Möbius strip, or search for a solution, because the Möbius strip is very simple to construct. In fact, watching the video itself shows one such construction.
cute little animation! and of course, it is always good to be reminded of the shoulders we stand on. thank you!
Eye-hand coordination. That was my problem from age 6 on.