The Genius of J.S. Bach’s “Crab Canon” Visualized on a Möbius Strip

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:

A Big Bach Download: All of Bach’s Organ Works for Free

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.

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Comments (52)
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  • M. Hyatt says:

    I can’t see the image here. How else can I view the canon imposed on a mobius strip?

  • Matthew says:

    This is the direct link to the video on youtube:

  • Dave Webb says:

    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?

  • Thomas Mansell says:

    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”!

  • Richard says:

    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.

    • M Sage says:

      Come on. Spell it right to get the meaning right. nnA “cannon” is a war gun designed to kill and destroy.nnA “canon” is, according to Merriam-Webster, a contrapuntal musical composition in which each successively entering voice presents the initial theme usually transformed in a strictly consistent way.nnnnSome people who are exposed to canons for the first time are entranced. Something that appeals to their fancy can lead to more explorations and their own flights of fancy, whether based on mathematical equations or not. Don’t detract from their experience.

  • jc morrison says:

    cute little animation! and of course, it is always good to be reminded of the shoulders we stand on. thank you!

  • Billie Hunter says:

    Eye-hand coordination. That was my problem from age 6 on.

  • David DePriest says:

    It’s OK, but not that amazing….He was just playing around.]

  • ALW says:

    Why all of the ‘this is so simple/who cares/I covered this in music theory 101’ attitudes? Someone took a foundational piece of music and presented it in a new way… in a way that a broader audience can appreciate it. Why can’t we cheer for that instead of saying how unimpressive it is?

  • Hal Portner says:

    The thing about a Möbius strip that relates so well to Bach’s Crab Cannon is that once a strip with two surfaces written on both sides is given a half twist, there results only one surface that repeats in either direction to infinity. Apply this visual model to an auditory crab cannon – voila!

  • Ben in Seattle says:

    Nice, but when I saw the description, I had assumed the genius of the Möbius strip representation would be that there would only be one marker going around the strip, but it would be playing both “sides” at once.

    Given that Leys used two markers in the animation, he might as well have used an ordinary circle and it would have been more illuminating.

    I don’t fault Leys for this; the music doesn’t seem to actually fit a Möbius strip. I blame Bach for his lack of foresight into a future where steam-powered fortepiano automatons read music from sheets folded into strange geometries.

  • Gordon Geise says:

    Jesus Mortimer Christ.

    No, James, a Möbius strip isn’t “impossible”; the word was ill chosen. Still, it is a one-sided figure on which you can have musical markers (or ants) on opposite sides, right across the paper (or whatever medium) from each other. That makes it pretty weird, no matter how easy it is to construct. By the way, it was super-awesome of you to imply the author here didn’t even know what a Möbius strip was and didn’t even watch the video he was describing.

    Yes, Richard, this canon is a fairly dry exercise and many of Bach’s other canons ad fugues are more ingenious and more complex. Please submit a better crab canon and I’ll be happy to listen to you whine about boring Bach. Ah well, I guess that’s what you get when you reduce music education to a bare minimum: people who can’t even spell “canon”.

    ALL Y’ALL: The Möbius strip works for this canon only when the “back” of the strip, before the half-twist and connection, is printed upside down (necessarily running in the same direction). The canon thereby flows in either direction and ends up “right side up” again by the end. You cannot do that with a simple loop–or rather, you could, but it would entail printing the canon twice, once in full on each side, in opposite directions, and thus not upside-down. In THAT scenario, a single musical marker could be shown to play both sides to the same effect; but like I said, you’re printing the canon twice and the strip of paper (or whatever medium) is twice as long. The Möbius strip is thus more elegant.

  • John and Elena says:

    We showed our daughter (violinist/singer) the video and she was impressed.

    Also blown away by Crab Canon were M.C.Escher (his 1965 work “Crab Canon”) and Doug Hofstadter (his pulitzer prize winning Godel, Escher, Bach). Hofstadter uses Crab Canon as a jumping off point to explain number theory.

  • It’s got a good beat. And you can dance to it.

  • Vicki Carr says:

    Gordon Geise gets me thinking…with a Möbius strip wouldn’t the line, at some point, begin playing upside down, perhaps being readable as bass staff?

    • Marie says:

      Nope. Notice how in the movie they take the second half and flip it upside-down first before making the strip? Try it yourself and you’ll see you’d never be reading it upside down!

  • Olivier S says:

    The human mind is amazing. It always needs to criticize, assert itself over another. Looking at the video was like looking at a flower opening into the sun. Enjoy it. Dance with it. We live only once.

  • enlightened says:

    The artistic mind will constantly be an amazing thing to watch. Thank You to whomever took the time and generously presented something so complicated, to someone like myself, as something simple. Which, it isn’t. Not by a longshot.

  • Nalrus says:

    As a longtime fan of Escher, Mu00f3bius and All Things Animated (as well as JSB) I was delighted to see & hear this. Honi soit mal y pense.n

  • Maurice Depestre says:

    this is simply amazing and now i know why i’ve always been a fan of both, Bach and Escher….

  • Maurice Depestre says:

    this is simply amazing and now i know why i’ve always been a fan of both, Bach and Escher….

  • Maurice Depestre says:

    this is simply amazing and now i know why i’ve always been a fan of both, Bach and Escher….

  • sjorsca says:

    Folding space and time by vihartn

  • sjorsca says:

    Folding space and time by vihartn

  • sjorsca says:

    Folding space and time by vihartn

  • jansumi says:

    Okay, Bach wins again.

  • Phil Estine says:

    I’d be more interested in a crab sandwich on a minibus trip.

  • mmabq says:

    “ingenuity” not “ingeniousness”

  • george atwood says:

    this piece exemplfies Bach’s pervasive theme of unifying dualities, reaching for the healing of his own divided soul nn

  • spandrel says:

    I think Frederick the Great should get some of the credit. It’s his theme, a challenge to Bach. The maestro immediately sat at the keyboard and knocked out a four part fugue. Now that’s gotta be annoying!

    • Inspector Fu says:

      Four parts? Fugue? It’s a canon, and it’s two parts. Unless you’re talking about another piece entirely.

  • spandrel says:

    Mt apologies, it was a three part fugue. Bach went on to write ‘A Musical Offering’, a suite of canons and fugues using the theme, from which the crab canon comes.nnnnThe point of the crab canon is that it can be played backward and upside down. In answer to Vicki Carr and Marie, I initially thought the pins flipped round but they don’t. The second time round they are both playing the theme upside down, one of them backwards. Thus the reason for putting it on a Mobius strip.nnnThere are two more ways of playing it against itself and I was just trying to figure out a way of visualising it when my head exploded.

  • spandrel says:

    Sorry about that, it was a three part fugue. Bach went on to write ‘A Musical Offering’ using the theme and that is where the crab canon comes from.nnnI deleted a previous post as I got it wrong again. On studying the film slowed down it is clear that the pins reverse directions so they are effectively playing it twice. The problem is it doesn’t sound like that. Heeelp! At first I though they carried on round thus playing it upside down as this is also possible, but I don’t think the video shows this. Can anyone play the keyboard? Make a Mobius strip and play it round twice.?

  • robot433 says:

    Let’s also not overlook the fact that this is amazing music first and can be enjoyed without understanding Bach’s organizational methods. The master’s ear for melody and harmony can sometimes be overshadowed by his (admittedly) remarkable scaffolding. Thanks for sharing!

  • spandrel says:

    Weird! I deleted that guest post. Oh well!nAt school I was annoyed that we de-constructed the double violin concerto (AABA) but found that it didn’t lessen the enjoyment. Bach showed everyone what was possible using the new equal tempered tuning rather than natural tuning. They say the devil has all the best tunes but it’s Bach really. He did it all before anyone else had a chance.nnnThere are four more versions of the Crab Canon two-part counterpoint and it would be nice to see and hear these. This video shows two but it could show the others by turning the pins upside down. In short, and if you fancy laying out a table, there are sixteen possible arrangements but four of those have both parts playing the same thing. Half the others are reflections which leaves six. Perhaps Jos Leys can be persuaded to do two more videos showing the other four.

  • Hannere van Wyk says:

    Amazing – thank you!

  • Fr Chip Johnson says:

    Amazingly SILENT!

  • Abra Cadabra says:

    I am watching this and I am just like ” I can play this with some practice” then the two balls show up so I am like ” I will have to do this with a friend”. After that when it started to twist, I am just thinking”…..OH MY GOODNESS…”, with my face like :ODDDD

  • Trevor J Dowling says:

    The best comment has to be preferring a ‘Crab sandwich on a Minibus trip’

  • Rutiger Farnsworth says:

    I wish I had a “crab cannon.” I’d just go around all day, shooting at ignorant people with shellfish.

  • Alastair Macfadyen says:

    It’s known that Bach was interested in mathematics, seeing this has just gone a bit further to prove what a wonderful man he was!

  • Rob Speer says:

    This video isn’t genius, it’s just bullshit masquerading as genius. It cheapens both Hofstadter and Bach to make this tenuous analogy. We already know the structure of the Crab Canon, and it has nothing to do with Mobius strips.

    If it’s okay for me to post a link, here’s an explanation why:

  • dave julian says:

    The added dimension is time. The two halves are not just connected in sequence, they are played over each other forming the two sides of the Mobius.

  • Moritz says:

    “I blame Bach for his lack of foresight into a future where steam-powered fortepiano automatons read music from sheets folded into strange geometries.”

    This just made my day!!!

  • Toad says:

    I’m guessing that some parts of it haven’t aged well at all, but Gödel Escher Bach was a seriously great book. Back in the day, if you took the journey seriously, worked out the problems and played along with all the thought experiments and got all the way through the damn thing, that book could rewire your thinking. I speak from experience; it’s one of the handful of books that literally changed my life.

  • Alex says:

    Yes, ALW, thanks.

  • Perry Otherly says:

    Well, fluff and hyperbole indeed, stuff and nonsense! If you fail to see the kernektion you’ve lost your erection.

  • Perry Otherly says:

    Hey all y’all, have ye ever heard Moebius Grape? They’s a really gut rockandrolling band. I’s bet they can twist the bars around mebbe rebbe even to 3-D, done you know.

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