On 2002-12-31 11:19, Zubivka wrote:
Unfretted strings–OK, a C# is tiny bit sharper than a Bb, correct me if I’m wrong.
Now I thought this tuning would change intonation when going downscale as compared to upscale. . .
I assume that you mean that C# is a tiny bit sharper than Db, which would be correct. I think C# is sharp to Bb no matter how you tune. 
As for the scale flattening as you go down and sharpening on the way up . . . to follow Zoob’s question and Ronaldo’s response, actually, this is the case.
If you were to tune a piano “justly,” this is how you’d do it. Take a pitchfork, but only one, and tune that note. A=440Hz, for example. Then tune all the other A’s to that A. You’ll notice, if you use an analyzer, that to sound “correct,” your A’s will get a little sharp to the predicted values the further you go up the scale. Meaning, while the next A might come out to 880Hz, the following A might be 1762Hz or sharper. Likewise, as you went down the scale, the notes would get a little flatter. I’m not sure exactly why it is, but the phenomenon is well-described.
There are more tuning systems. Meantone tuning is a system which keeps the ratios such that the major thirds (C-E, D-F#) are the most in tune, at the expense of making the fifths (C-G, D-A) slightly out of tune. Meantone was more common at the beginning of polyphony, I think, when Western music moved from simple organum (which refers, I think, to fixed intervals between lines, at say, a fifth) to more complicated harmonies which needed in-tune thirds. There’s even a “golden meantone” tuning based on the constant phi which is the mean of the quotients of adjacent elements of the Fibonacci sequence. Phi is also the ratio of width::height of a lot of Greek temples because the ratio was thought pleasant to the eye.
Down with equal temperament! JUSTice or BAN!
Stuart
