Re: Bravo, Paul! (106/212-tet tour de force)

Hello, there, Paul, and everyone.

Thank you immensely for an incredible demonstration that treating an n-tet
as a meantone [in the narrower sense of a tuning where two regular major
seconds yield or approximate a 5:4 major third -- MS] can be more
complicated than I might have suspected. Your presentation was very
engaging, as well as having a certain mathematical humor, and helping me
to see the humor of my own predicament.

My problem, as you nicely showed, was that I was looking only at the
problem of dividing a near-5:4 major third into two "mean-tones,"
without asking about the _source_ of those mean-tones, namely fifths.

Of all people, I should be aware that two fifths make a major ninth,
having focused on the 4:6:9 sonority in medieval music for around 20
years. I'm likewise aware, for example, that one of the complications
of a 7:4 is that it can't be divided into two concordant fourths,
unlike 16:9. In Xeno-Gothic, one uses two intervals a Pythagorean
comma apart, a ~7:4 for special cadences where a narrow minor seventh
contracts to a fifth, and 16:9 for a 9:12:16. An interval like the
22-tet minor seventh is precisely a meantone compromise, filling both
roles -- and, the point of your superb presentation, involves
compromising _the fourths themselves_!

That was the critical point I missed, getting all caught up in,
"Here's a solution to the syntonic diatonic-style complication of
53-tet as 5-limit -- just make the scale twice as large, so that the
~5:4 major third is a nice even number of steps." The point is that to
emulate meantone, we need to emulate _tempering the fifth by
1/4-comma_, just as you show.

Of course, this means that (as in a usual Renaissance meantone) the
fifths will indeed be compromised -- the usual virtually-pure fifth of
53-tet (or, in this case, 212-tet) will be 1/4 of a 53-tet step
narrower than pure -- thus four times as many steps needed to bring
about this tempering.

Curiously, I sometimes tend to use "meantone" in a much more general
sense to mean any regular tuning or temperament, with 53-tet thus an
instance of "1/315-comma meantone." Taken as a "meantone" in that
sense, of course, 53-tet succeeds without any problem <grin>. Since
the scale has an odd number of steps, the "mean-tone" (9 steps) must
also have an odd number, so that as you show they will add up to a
major ninth evenly divisible into two fifths. Likewise, with 22-tet,
the "mean-tone" (in a Pythagorean interpretation) of 4 steps has an
even number of steps.

Anyway, your presentation was great teaching and made me aware that
the topic of emulating tempered meantones with scales such as
multiples of 53-tet can be more complicated than I might have
guessed. Also, I learned that a successful division of one interval
(the 106-tet ~5:4 into two equal major seconds) does not a tested
system make.

While I could wish I had been more cautious in testing my 106-tet
solution before proposing it here, I treasure your correction as a
wonderful piece of music theory literature and also a jumping-off
point for your discussion of 212-tet and 76-tet with many rich
themes.

Most appreciatively,

Margo Schulter
mschulter@value.net