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tingram comments on A discarded review of 'Godel, Escher Bach: an Eternal Golden Braid' - Less Wrong

6 Post author: lukeprog 16 December 2011 05:51AM

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Comment author: tingram 16 December 2011 06:34:23AM *  5 points [-]

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?

Comment author: grouchymusicologist 16 December 2011 02:43:36PM 4 points [-]

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.)

Comment author: tingram 16 December 2011 03:33:25PM *  0 points [-]

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.)

Comment author: grouchymusicologist 16 December 2011 04:24:19PM 0 points [-]

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.

Comment author: Luke_A_Somers 16 December 2011 03:17:21PM 2 points [-]

I believe this is one of the weakest possible examples from the book... though it may be one of the few that can actually be extracted from context without requiring dragging in a great deal more. I would have gone with one of the Escher examples, as it's quicker to grok than Gödel, and the Bach examples are, well, a bit of a stretch.

Comment author: Nymogenous 16 December 2011 06:53:21AM 0 points [-]

If I recall correctly, he focuses on the fact that the piece may be played in a cyclical fashion, allowing an infinite loop of sorts.

Comment author: DanArmak 16 December 2011 12:12:56PM 1 point [-]

What's special or interesting about a musical piece that can be played cyclically? Such a piece is easy to compose by editing the two ends to align, even without reference to whatever is in the middle of the piece.

In the Bach example, if you go up an octave on every loop, you can't play forever anyway (within human hearing).

Comment author: Nymogenous 16 December 2011 03:03:43PM 1 point [-]

It's not that hard to do (somewhat harder if you want to keep your Bach-style harmonies intact), and I don't think anyone claimed it was that hard, simply that it induces an interesting self-referential cycle. There's something rather amusing (at least to me) about a piece of music that can be played an infinite number of times without repeating a musical phrase more than the few times it occurs in a single cycle.

As for the quick rise out of the human range of hearing, it's just a small side effect that prevents musicians from getting caught in an infinite loop.

Comment author: DanArmak 16 December 2011 03:18:43PM 2 points [-]

So GEB's entire point here is that some infinite sequences of similar-but-different objects have self-referential formulations?

Just like these are equivalent: a(n+1)=a(n)+2; a(0)=0 vs: a(n)=2n for all n

Each element has the structure of "an even integer", but the first form is self referential while the second one isn't.

I fail to see a deep meaning in this, or any similarity with consciousness. Can someone enlighten me? Did I merely take the book's example out of context?

Comment author: Nymogenous 16 December 2011 03:26:08PM 2 points [-]

Nope, you've got the right idea about his example. It occurs early on in the book, while he's trying to explain simple concepts to readers through non-technical analogy; sort of the way he explained complement spaces to readers by first asking which word contains the sequence "ADAC" in order (headache), and then asked what word contains the sequence "HEHE" in order; nothing particularly special about that either, but it teaches the reader a useful trick without presenting it mathematically.