Re: Adaptative JI

Kami ROUSSEAU wrote:

> The WTC idea could be applied to 22TET as well

> -Kami

You mean well-temperament? Well, not literally. Traditional
well-temperament, using 12 tones, must have an average fifth of 700
cents. Smaller-than-average errors will yield meantone, where 5-limit
harmonies are improved up to the point where the average size of the
fifths constructing them is 697" or 696". Larger-than-average errors
will then virtually always make their fifths closer to just intonation,
yielding harsh 5-limit harmonies but approximating the Pythagorean, or
pure 3-limit, at 702". If six fifths, enough for one diatonic scale, are
tuned to an average of 698", the other six must average 702", giving
three major thirds ~22" sharp and four minor thirds ~22" flat -- harsh,
but borderline acceptable -- and fifths at the Pythagorean/pure 3-limit
ideal.

A 22-tone well-temperament would have an average fifth of 709.091 cents.
This is 12" away from 697", and on the other side of just intonation
(thus harmonic waste is unavoidable). If only six fifths are only
tempered to an average of 699", the other sixteen share a total of
60.5454 cents of error, so each is 712.875". This is borderline
acceptable, and trying to get more than one diatonic key into meantone
(by tempering more fifths to 699") or trying to get a better meantone in
the one diatonic key would make at least one of the other fifths totally
unacceptable. The sixteen 712.875" fifths yield thirteen major thirds
~65.186 cents sharp and fourteen minor thirds ~54.266 cents flat --
TOTALLY UNACCEPTABLE!

However, my paper (I'm sure you know the one by now) describes a 22-tone
temperament with two different sizes of fifth, where some (decatonic)
key areas give acceptable 7-limit harmony while others give near-just
5-limit harmony, with subsets conforming to the Hindu diatonic scales in
both 22-degree/sruti numbering and (necessarily) consonance/dissonance
differentiation.