Bach canon played as a moebius strip

This video of Canon 1 à 2 from J. S. Bach's Musical Offering (1747) being turned into a Moebius strip, then played in two directions at the same time would have been good to watch and listen to while I was reading the mind-bending Gödel, Escher, Bach: An Eternal Golden Braid many years ago. (via cgr 2.0)


  1. I really wished I understood the genius of this, and I do appreciate the geekiness and coolness of the exercise, and at the same time its so far beyond me. I think I first realized this when I had to google Moebius strip.

  2. Ah, Godel, Escher, and Bach. I loved that book when I was a kid. (My dad had it for some college course, and I started reading it because I liked the pictures. Then I kept on reading it because I found the adventures of Achilles and the Tortoise entertaining. And then I kept on reading it because the ideas made my head wrap around on itself.)

  3. Really lovely.

    Also, for the Gödel, Escher, Bach fans out there I highly recommend a more recent book also by Douglas Hofstadter – “I Am a Strange Loop”. It’s basically a more direct explanation of the central theme of GEB, written because Hofstadter felt that so many people missed that point. I was one of those that missed it, and I found the later book very provocative, fascinating and enlightening.

  4. Shhh…here’s the secret — how to write a crab canon: write it two measures at a time as if it were a 2 voice piece. First and last, 2nd and penultimate, 3rd and antipenultimate working towards the middle. Make sure that each set works out harmonically, iron out the wrinkles in the middle measure, line up the two lines into one line, voila.

    We had to do this in counterpoint class as a basic exercise (good enough for Bach, damn well good enough for us). We also had to write them so that the second voice worked out when played backwards and upside down. Ugh.

  5. simply marvelous. i’m not much of a fan of the right hand through the first two measures, but combining them forward and backward, upside down, etc, is simply marvelous…

  6. Killer. I recall having to write these kinds of exercises in school as well.

    The trick is being able to play both parts (forwards and backwards/flipped) simultaneously, while only looking at the one line of music. That will hurt your brain something fierce.

  7. It does seem like the mobius strip is a surefire way to make this look more complicated than it is, though…

  8. Isn’t the same exact thing happening in the moebius strip version that we already saw when it was just playing forward and backward at the same time?

    Since the moebius gradually turns the staff upside down, it might be cool if the music played this way, the notes smushing together where the strip goes flat, and then inverting high and low notes when the staff is upside down.

    Shouldn’t that be antepenultimate?

    Yes, yes it should be, but I figured the spello didn’t need it’s own corrective reply. Sorry for any misunderstanding. I didn’t mean one should pair up the 3rd measure with the measure was most opposed to the next-to-last, I did mean that one should pair the 3rd with the one that is last but two.

  10. JS Bach’s music has been studied by many mathematicians for its symmetry. This is but one of many examples.

    I wish musicians of today would pay more attention to the craft that Bach used to create these works of art. Like Leonardo Da Vinci, he embedded all sorts of mathematical symmetries in his creations.

    Maybe art and music schools should study more math.

  11. My sheer and utter coindidence, I JUST picked up ‘Strange Loop’ from the bookstore, and I’m about to start it. Perhaps, after reading it, and once my brain has reassembled itself back inside my skull, I’ll go back and reread GEB.

    While listening to Wendy Carlos. Looped. Backwards.

  12. #1

    I really wished I understood the genius of this, and I do appreciate the geekiness and coolness of the exercise, and at the same time its so far beyond me. I think I first realized this when I had to google Moebius strip.

    I understand perfectly well what a Moebius strip is, and I still don’t get it. It just sounds like some kid plinking randomly away on a harpsichord to me, with alternating vague trends up and down the keyboard.

    It’s possible that you need an IQ of at least 160 to ‘get’ Bach.

  13. Bach’s piece is great, but not because it is great math, but because it is great music. Mathematicians attach to art and music to suggest that what they do is inherent in nature, but it is not. Math is only a language among other languages. The point of a Moebius strip is that it has no and cannot have any ‘translation’ in art or music. You can call a Moebius strip artistic (in the sense of a concept) but it has no representation in art. So the video is simply misleading.

  14. This is the best thing I’ve seen in a while.
    I’m not sure if it’s inspiring or if it makes me want to give up music forever.

  15. Onigorom @26:

    Mathematicians don’t need to attach to art to suggest that what we do is inherent in nature. Unlike art and music, what mathematicians do is inherent in nature. The best maths always feels like discovery, not invention.

    Mathematicians attach to art and music because they’re often interesting. And mathematicians are geeks first; we can’t let a technically interesting topic drop.

    I’m not aware of any art with a representation of a Moebius strip, but I suspect that’s just because I don’t know enough art. Either way that says nothing about the validity – mathematical or artistic – of Moebius strips.

    The single greatest word of praise a mathematician uses for work in pure maths is not “clever” or “useful”. It is “elegant”.

    (Like most concepts of beauty, it’s very hard to describe to anyone else what constitutes an elegant theorem. But we know it when we see it.)

  16. As the counterpoint student above describes, this was a well-known technique in that era. What’s really cool is that this wasn’t just an abstract trick that only the composer knew about: it was sometimes actually meant to be performed like the moebius strip in the video!

    Telemann wrote quite a few wind “duets” in this manner: they were meant to be played from a single sheet of paper on a table between the two performers (standing opposite each other). So each plays the upside-down inverse of the other.

    It was pretty mind-blowing the first time I played an oboe duet this way. (“You mean we can play a duet from the single sheet? Even though my part is upside-down? And backwards?”)

  17. @DAVIDSD:

    It is not a computer-generated harpsichord. You can hear the action rising and falling under the sound of the plucked strings. Thud thud thud. It’s possible that it is an electric harpsichord with sampled sounds (sampled from a real harpsichord), but I would bet a lot of money that the sound is not computer-generated.

  18. @Tynan — I’ve actually done a Mobius strip quilt, but alas I have no electronic picture of it to post. It was a pretty cool project and required much more planning than a flat quilt.

  19. Yeah, Bach was pretty much the man.

    Shame that his genious wasn’t realized until after his death. Although, really, the idea of musical fame or a “start” is relatively new (starting around Beethoven).

    Bach worked as a conduit of (what her perceived as) God and devine inspiration.

    He also kinda when nuts towards the end and tried to make a piece using his name as the harmonic structure (In his time “B” was seen as B-flat and “H” was seen as B natural).

  20. @Dagyofou: Ooops, really didn’t write what I meant there. I meant “I’m not aware of any art with a representation of a Moebius strip, aside from the obvious…”

    To make this more embarrassing, the Escher is behind me on a poster on my wardrobe. Within touching distance as I wrote.

    @Takuan: As always, you have said what I meant, but with greater elegance.

  21. A word of clarification: antepenult is the noun; antepenultimate is the adjective. When you say the measure is the antepenult, you use the word correctly. Or, you can say it is the antepenultimate measure. In that case, “measure” is the noun. Comes from Greek linguistics and the terms are fair use in music.

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