Well ... perhaps I wasn't writing so clearly as I'd hoped. Some of the responses didn't seem to follow from what I thought I'd said !
I shall amplify only what was, I think, my most interesting point, which was about a remark someone else in Tuning quoted from Beethoven, that he could hear a difference between C# and Db. I don't know the source; I'm taking it at face value.
When someone of Beethoven's competence makes a remark that seems nonsensical, I suggest that the best thing to do is to assume that he was saying something reasonable, but that he was saying it with a shorthand, or that he was making unmentioned assumptions and corollaries, or something of that sort. I mean, it seems far more likely that Beethoven would make an error of verbal communication than that he'd make a musical error.
Now, surely everybody who reads this list is aware that C# and Db are physically identical on a piano. Since Beethoven probably also knew this, what did he mean when he said that they were different ? I am only guessing, of course, but perhaps he was referring to the expectations set up by different harmonic contexts. V in F# Major (which would be C# Major) would perhaps be reasonably expected to feel more brilliant than flat VI in F Major (which would be Db Major). Any good tuner could favor one way or the other. (Someone else also mentioned something like this in the last Tuning.)
Also, the piano itself could slip to a different flavor from that at which it had been set if, for example, the pin block were holding unevenly. A fortepiano (which is what Beethoven was often playing) can slip quite a bit if you play it hard, and Beethoven played so hard that he broke strings sometimes. In which case, as another possibility, it would be understandable for him to talk about a difference between C# and Db.
But I don't pretend to know what Beethoven meant in this case. I think it's interesting food for thought.
Yours,
Will Grant
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Charles Lucy asked: >If irrationals were treated differently to rationals, how >did the ancient Greeks handle Pi?
The first real proof of Pi's irrationality was not until the 18th century - I think you'll find it was either by Euler or Lambert. The proof pi is transcendental didn't appear till even later. But the way the Greeks attacked the problem of finding the ratio between a circle's diameter and area suggests they at least had doubts about pi's rationality. This is perhaps best exemplified in the way Archimedes, among others, made ever-finer polygonal approximations (of known area) to a circle, from both inside and outside, converging towards the "true" ratio which could never be completely attained. I believe he even says something to the effect that while it is false to believe that in this way you can "exhaust" the whole area of the circle, you can in fact come arbitrarily close.
Persistent failure to properly "square the circle" -- definitely one of antiquity's most popular problems -- could also have sown seeds of doubt among the Greeks as to pi's rationality. One would imagine that this would have created some consternation for early Pythagoreans, who had rather hoped that the cosmos and everything in it worked according to numbers -- in fact in some sense that everything *was* number. But if the ratio of a circle's radius to its circumference was irrational, then the lengths of the celestial bodies' circular paths around the "Central Fire" would be incommensurable with their distances from it, i.e. have no common number. (This particular cosmological model, in favour with early Pythagoreans, had a Central Fire around which were placed the earth, moon, and Sun; the five planets known to the ancients; a fixed sphere of stars and a "counter-earth", perpetually invisible to us since it remained diametrically opposed to ours on its orbit around the central fire, and apparently incorporated to fudge the number of "things" out there to add up to ten, the so-called perfect number...)
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