3 ms·

The Crab Canon is not literally a Möbius strip, of course. But the Möbius strip is a fabulous analogy that goes a long way toward helping people understand wha

The Crab Canon is not literally a Möbius strip, of course. But the Möbius strip is a fabulous analogy that goes a long way toward helping people understand what is going on. I've used this in my teaching on that topic: http://www2.nau.edu/tas3/MOcancrizans.pdf http://www2.nau.edu/tas3/MOcancrizans.pdf

rspeer 10y ago
You made that page? Interesting. I'd like you to explain. I believe that the Crab Canon is in no way analogous to a Möbius strip, and the topology is in fact quite different. When you go around a Möbius strip once, you end up in the same place you were, flipped upside down. The Crab Canon is not a Möbius strip, it's a palindrome. It is not flipped upside down, it's flipped left to right, in the direction of time. This is already illustrated in GEB, with the clefs at both ends of the page, as well as by Bach himself. The video bamboozles its viewers with poor graphics. They can't see that the music on both "sides" of the Möbius strip is completely unrelated. A Möbius strip does have an analogue in music: it would be a canon by inversion. Bach wrote some of those. It would take a particularly short canon by inversion to actually be represented by a Möbius strip; the gap between the entries of the parts would have to be half the length of the song. In your notes, what role does the half twist play? Why would it not be the same without the twist?
- Keyserlingk 10y ago
  >>the music on both "sides" of the Möbius strip is completely unrelated. Glad to see the "enquotion" of the word "sides." The point of the analogy is that a Möbius strip has but ONE side. The analogy illustrates thereby that while there may seem to be TWO melodies (different on each "side"), it is actually ONE and the same melody that counterpoints itself. The problem with the YouTube video is that its TWO playheads move in opposite directions. There should actually be but ONE playhead, wrapping around both "sides" of the Möbius strip, playing BOTH of its "sides" simultaneously, thereby showing ONE melody in counterpoint with itself (as in the pdf link) All analogies break down at some point, as your observation that the melody inverts at the midpoint is valid. But is not the Möbius strip itself the visual representation of a mathematic? As such, would there not be inconsistencies there too? I'm not a mathematician so cannot think of any. But as a musician, I believe the better analogy would be a simple canon at the octave or unison, like the E minor "fugue" of the Well-Tempered Clavier, book 1. http://bach.nau.edu/clavier/nature/fugues/Fugue10.html http://bach.nau.edu/clavier/nature/fugues/Fugue10.html Toward the bottom right there is a link to another Möbius pdf. One might again object that it inverts the canon follower, but that is easily solved by inverting the follower line of the graphic (which misrepresents the canon in two dimensions as traditional music notation, making it a canon in contrary motion). To be sure, all analogies break down at some point, and never more so than in the representation of non-dimensional sound in three dimensions. To my way of thinking, the “twist” of the Möbius strip (with its resulting melodic inversion) is best understood by thinking of the thumb/forefinger "playhead" as having but one clockwise orientation, in which case the melodic inversion evaporates, but the contrapuntal inversion is retained.
  - rspeer 10y ago
    It is indeed true that a Möbius strip should have one side. The best use of the mathematical idea of a Möbius strip does not involve a piece of paper you look at from one direction or the other, and see different notes; instead, you should be seeing the same notes, but see them right-side-up or upside-down depending on how many times you have gone around the strip. The Crab Canon analogy breaks down completely, because the notes are never upside-down. There are many canons that could apply to in the Musical Offering, like the ones that Bach illustrates by putting a second, upside-down clef on them. > The problem with the YouTube video is that its TWO playheads move in opposite directions. The PDF says: "To "perform" your model, one musician would start at each clef, read around the strip in the prescribed direction to the other clef, and then reverse directions." I believe the two playheads the only thing that gets you the actual structure of the Crab Canon, which is a palindrome. When you model it as a Möbius strip, then ignore the twist as "oh well the analogy doesn't quite work", and gradually change everything about it to say "oh, it's a palindrome; we just made it a Möbius strip to look cool" -- then it's not a Möbius strip. My current belief is that various people hear that the most famously nicknamed canon in the Musical Offering is a Möbius strip, and then contort their minds to believe that it is true, and tell other people the same thing. You could make the analogy work with the right canon! I think one of the canons in BWV 1072-1078 would do the trick. But they don't have catchy nicknames or references in GEB.

rspeer 10y ago

You made that page? Interesting. I'd like you to explain. I believe that the Crab Canon is in no way analogous to a Möbius strip, and the topology is in fact quite different. When you go around a Möbius strip once, you end up in the same place you were, flipped upside down. The Crab Canon is not a Möbius strip, it's a palindrome. It is not flipped upside down, it's flipped left to right, in the direction of time. This is already illustrated in GEB, with the clefs at both ends of the page, as well as by Bach himself. The video bamboozles its viewers with poor graphics. They can't see that the music on both "sides" of the Möbius strip is completely unrelated. A Möbius strip does have an analogue in music: it would be a canon by inversion. Bach wrote some of those. It would take a particularly short canon by inversion to actually be represented by a Möbius strip; the gap between the entries of the parts would have to be half the length of the song. In your notes, what role does the half twist play? Why would it not be the same without the twist?

Keyserlingk 10y ago
>>the music on both "sides" of the Möbius strip is completely unrelated. Glad to see the "enquotion" of the word "sides." The point of the analogy is that a Möbius strip has but ONE side. The analogy illustrates thereby that while there may seem to be TWO melodies (different on each "side"), it is actually ONE and the same melody that counterpoints itself. The problem with the YouTube video is that its TWO playheads move in opposite directions. There should actually be but ONE playhead, wrapping around both "sides" of the Möbius strip, playing BOTH of its "sides" simultaneously, thereby showing ONE melody in counterpoint with itself (as in the pdf link) All analogies break down at some point, as your observation that the melody inverts at the midpoint is valid. But is not the Möbius strip itself the visual representation of a mathematic? As such, would there not be inconsistencies there too? I'm not a mathematician so cannot think of any. But as a musician, I believe the better analogy would be a simple canon at the octave or unison, like the E minor "fugue" of the Well-Tempered Clavier, book 1. http://bach.nau.edu/clavier/nature/fugues/Fugue10.html http://bach.nau.edu/clavier/nature/fugues/Fugue10.html Toward the bottom right there is a link to another Möbius pdf. One might again object that it inverts the canon follower, but that is easily solved by inverting the follower line of the graphic (which misrepresents the canon in two dimensions as traditional music notation, making it a canon in contrary motion). To be sure, all analogies break down at some point, and never more so than in the representation of non-dimensional sound in three dimensions. To my way of thinking, the “twist” of the Möbius strip (with its resulting melodic inversion) is best understood by thinking of the thumb/forefinger "playhead" as having but one clockwise orientation, in which case the melodic inversion evaporates, but the contrapuntal inversion is retained.
- rspeer 10y ago
  It is indeed true that a Möbius strip should have one side. The best use of the mathematical idea of a Möbius strip does not involve a piece of paper you look at from one direction or the other, and see different notes; instead, you should be seeing the same notes, but see them right-side-up or upside-down depending on how many times you have gone around the strip. The Crab Canon analogy breaks down completely, because the notes are never upside-down. There are many canons that could apply to in the Musical Offering, like the ones that Bach illustrates by putting a second, upside-down clef on them. > The problem with the YouTube video is that its TWO playheads move in opposite directions. The PDF says: "To "perform" your model, one musician would start at each clef, read around the strip in the prescribed direction to the other clef, and then reverse directions." I believe the two playheads the only thing that gets you the actual structure of the Crab Canon, which is a palindrome. When you model it as a Möbius strip, then ignore the twist as "oh well the analogy doesn't quite work", and gradually change everything about it to say "oh, it's a palindrome; we just made it a Möbius strip to look cool" -- then it's not a Möbius strip. My current belief is that various people hear that the most famously nicknamed canon in the Musical Offering is a Möbius strip, and then contort their minds to believe that it is true, and tell other people the same thing. You could make the analogy work with the right canon! I think one of the canons in BWV 1072-1078 would do the trick. But they don't have catchy nicknames or references in GEB.