Beethoven and C#/Db

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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