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grouchymusicologist comments on A discarded review of 'Godel, Escher Bach: an Eternal Golden Braid' - Less Wrong
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Forgive my ignorance, not having read GEB, but I can't help being underwhelmed by the Bach example. This Youtube video plays the Bach canon in question. The canon begins in C minor and modulates up in whole tones until it arrives at C minor again an octave higher (the Youtube recording returns to the same octave, but it does so using trickery--notice that in Bb minor, the sixth time through, it ends on a D notated a ninth above the next C but sounding only a step above it).
Unless I'm missing something, this is rather like saying that if you walk for ten hundred-metre lengths, you'll end up a kilometre from where you started. Yes, you will, but so what?
Complete traversals of a set of whole-tone-related keys are incredibly rare in music this early; I don't know of another example and would not be that surprised if one doesn't exist. You're right that such a piece seems easy in principle, but that's from a current viewpoint. So I think that Hofstadter may in part be implicitly relying on some knowledge of the early-eighteenth-century context when he stresses how unusual the piece is.
(There is actually also a technical reason why this loop is so strange -- it has to do with what music theorists call "crossing the enharmonic seam" -- but Hofstadter doesn't appear to be referring to that.)
It is a neat trick, and not something that happens often, but I would guess that's because it's not useful as anything other than a neat trick. I'm not seeing the eternal golden braid in it, is all.
Actually, if Bach had kept the pattern intact without "crossing the enharmonic seam" it wouldn't be much of a loop at all; the piece would end up in B# minor after six repetitions.
(edit: sp.)
Yeah, I'm in agreement with you and others that it isn't the most compelling example he could have chosen.
As to the enharmonic seam thing, that is indeed the point: you either have to cross the enharmonic seam by spelling two identical-sounding intervals differently (in this case, one of the major seconds has to be spelled as a diminished third) or else you have to deny the seeming aural fact of octave equivalence by spelling the return of C as B-sharp. Since composers are extremely reluctant to do the latter, they have no choice but to do the former -- a commonplace in the nineteenth century, a bit of a special trick in the mid-eighteenth.