Incidentally, 13.5(base 10) = 11.6(base 12)

]]>Why is it impossible for the universe to have created a perfect circle (or a perfect sphere)? Is it because pi is an irrational number? That reasoning is backward.

As for my statement that math would be totally different if we had been born with 12 fingers and ended up with a base-12 mathematical system: The physical world is what it is, but the math we use to describe it evolved over time (remember when there was no zero?). My contention (wholly unproveable though it is) is that math would have evolved in a completely different way if we had started at 12, or any other number.

In the same way that you can't accurately translate between languages by simple word replacement with a bilingual dictionary, the evolved base-12 system wouldn't simply look like a direct base-12 translation of today's base-10 system.

Maybe I'm just underinformed, though. Admittedly, I haven't the slightest idea how to convert 13.5 into base 12, much less what pi looks like in base 12.

]]>Shh. Remember the legendary fate of Hippasus.

]]>Now I'm imagining the Pythagoreans coming up with Kripke semantics and trying to invent a possible world without the Pythagorean Comma…

It's not just a matter of the 12-tone scale – the equation (3/2)^j = 2^k has *no* positive integer solutions j,k. (Or equivalently, 3^j = 2^(j+k).) The fundamental problem is that 2 and 3 are relatively prime, which means a power of one can't be written as a power of the other. And they're relatively prime because they *are* prime. So to avoid the PC and have a perfect cycle of fifths, you have to live in *a universe where 3 is a power of 2.*

The amount of inharmonicity in a plucked string depends on its stiffness — at least that's what I've always read — and steel piano wires are a lot stiffer than gut harpstrings. In the case of the guitar, for example, Järveläinen et al., "Perceptibility of Inharmonicity in the Acoustic Guitar", Acta Acustica 92(5), 2006, found that in listening tests conducted using "test tones resynthesized from real recordings using high-resolution parametric modeling", listeners could easily detect inharmonicity in the lowest notes for steel strings, but "mean thresholds were close to, though above, typical amounts of inharmonicity in the [nylon-stringed] guitar".

The thresholds involved, as I understand the paper, are for telling the difference between synthetic sounds with inharmonicity and those without. I don't know what results you'd get if you looked at the effects of tuning octaves by listening for beats — that might well be a more sensitive test. But maybe not — the amount of inharmonicity in piano notes can be pretty large: Fletcher and Rossing (The Physics of Musical Instruments, p. 363) say that "a typical value for the inharmonicity coefficient in the middle register (B=.0004)" will "shift the 17th partial one 'partial position' to the frequency of the 18th partial of an ideal string without stiffness", or a factor of about 1.014. This is in the same range as the the Pythagorean Comma of (3/2)^12/2^7, or about 1.013643. I believe that the inharmonicity coefficient for nylon strings is typically an order of magnitude lower, though this also depends on length and tension.

But anyhow, the Pythagorean Comma is a mathematical fact, and the inharmonicity of partials in plucked strings is a physical fact. Neither one has anything to do with the imperfections of human hearing.

]]>Fair enough, but the Ancient Greeks did have harps and lyres, and if I understand the physical basis of scale stretching on a piano correctly – the fact that the overtone series of a stiff vibrating string runs slightly sharp – then they ought to have run into the same problem. At the very least, the practical problems of tuning a harp or a lyre may have helped confuse Pythagoras's thinking about the issue.

That said, however, I've now read the original posting on Right Wing Nation, and I agree that scale stretching doesn't seem to be what the writer is talking about.

]]>In a word, no. In his own words:

However, the ear is imperfect, so we have the Pythagorean Comma, a serious problem identified by Pythagoras that was not solved until the 18th Century. In ancient Greece (until the 18th Century) perfect fifths were used to tune instruments, and this led to a problem that limited what could be played.

Aside from his explicit claim that the Pythagorean Comma is due to the ear being imperfect rather than to the fact that (3/2)^12 != 2^7, there's also the fact that pianos were thin on the ground in ancient Greece.

]]>That's a particularly un-apt sentiment given the long and tragic history of cephalopods and writing.

]]>I see, I see. I guess I was thrown off because I would never use the word "pleasing" in that context.

On the other hand it sounds sort of like the arguments for equal temperment: it may be ugly but you can change it to any size (or key in the case of equal temperment.)

]]>It wasn't my assertion, it was someone else's. Although, I suppose I implicitly approved of it by attempting to provide a reference for it. In any case, I thought I remembered reading something about that. Isn't there also a chapter on the ability to recognize tunes, regardless of what pitch they're played at? I'm almost certain of it. Perhaps I only thought to myself while reading this chapter, that you could also slightly alter the size of various intervals and the tune would still be recognizable. That is what I took correction by the brain to mean.

[(myl) Though it was someone else's assertion, you're so far the only one who has offered any evidence for it. And I believe that your suggested evidence is based on a misunderstanding of Levitan's book. You may be remembering the discussion (p. 133 and following) of experiments by Benjamin White, showing that people could recognize tunes not only in transposition (where intervals are exactly retained), but also "in almost every case … more often than chance could account for" when the melodic contour is retained in a more qualitative way, with interval sizes shrunk or stretched in various ways. The modified tunes always still stayed within the standard tempered chromatic scale, so that the question of how people hear slightly mis-tuned intervals didn't come up. And Levitan doesn't put this experiment forward as evidence that the brain hears intervals in general in a non-veridical way, or more specifically that fifths are heard as tempered rather than exact intervals, but rather as evidence for a gestalt sense of overall melodic contour, an "abstract generalization" independent of particular intervallic sequences. (The citation is Benjamin White, "Recognition of Distorted Melodies", *American Journal of Psychology*, 73(1): 100-107, 1960. Levitan's conclusions are probably true, but there are some problems with White's experiment, for example the fact that the procedure involved forced choice among ten tunes that were identified to the subjects at the beginning, and heard repeatedly in various modified versions during the course of the experiment.)

Again, I'm very skeptical of the claim that human interval perception hears tempered rather than exact fifths — and I'm somewhat skeptical of the claim that our judgment of fifths is systematically different from the mathematical 3/2 ratio, though I don't know the literature on this subject well enough to be certain about it. If anyone has any credible evidence one way or the other, I'd be interesting in seeing it.]

Nick: Explains A series

Thanks for the explanation.

I simply took pleasing to mean "easy on the eyes" or something like that. I don't typically want to print things two-up, or have need to. There are too many subscripts in my work to make that an attractive option.

]]>What? How do you not see it? Compare the alternative (it wouldn't matter if you picked another one, but there basically aren't any in common use) US Letter. What we can we do with that ratio? Nothing. No-one even really seems to know how the ratio came about.

Some posters seem to have assumed I meant that the shape had a mystical aesthetic charm. Not at all, a variety of rectangles are sort of vaguely appealing to humans, there's no reason to pick a specific ratio just for that (enthusiasm for golden rectangles in architecture not withstanding) and no reason to think an octopus would have the same aesthetic. But if the octopus can appreciate the ratio itself it will see how ingeniously chosen it is.

In practical terms its pleasing to anyone who has to actually print things. Europeans are used to it not mattering how you print something, if you print something intended for A5 at A4 it will simply be larger, and if printed at A5 when intended for A4 it will be smaller, the proportions are unaltered. It's also trivial to automatically "two up" a document for printing two pages per side to save paper. European photocopiers almost universally offer to shrink or enlarge documents for you. With other ratios, including US Letter such ordinary everyday practices become cumbersome and difficult.

In purely mathematical terms you'd have as good a claim for the golden ratio, but it turns out that nobody's everyday business requirements involve cutting squares out of rectangles.

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