6 ms·

>>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 M

>>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

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.