Re: Digest 323 -- Wolves and commas in 12 of 31-tet

> Message: 12
> Date: Mon, 20 Sep 1999 15:25:38 +0930
> From: "McDougall, Darren Scott - MCDDS001" <MCDDS001@students.unisa.edu.au>
> Subject: 12 from 31, wolves and commas
>

> When I tune my synth to QC mean-tone, there are four intervals that look
> like a third but sound awful (not surprising). When tuned to Pythag,
> there are four intervals (those dim4ths) that look and sound like thirds
> (nice surprise) : I guess I wouldn't call *them* wolves.

Hello, there, and curiously the diminished fourths of Pythagorean _can_
sound "odd" or even a bit "Wolvish" if one is accustomed for a given
piece or set of pieces to major thirds at a consistent 81:64 (~408 cents),
the usual Pythagorean interval. Around the early 15th century, as Paul
Erlich mentions, this narrow 8192:6561 major third (only a schisma or
~1.95 cents from 5:4) came into vogue, and evidently led by around
1450-1480 to meantone temperament as an effort to obtain such thirds at as
many places in the gamut as possible.

In 1/4-comma meantone for only 12 notes per octave -- very close to
31-tet, or vice-versa, since the fifth is tempered ~5.38 cents in
1/4-comma and ~5.18 cents in 31-tet -- the wide diminished fourth _is_ a
"Wolf" (~427 cents) in a usual Renaissance setting, but makes a fair
approximation of a "supermajor third" of 9:7 (~435 cents) in "Neo-Gothic"
cadences where a major third expands to a fifth, for example. Marchettus
of Padua _may_ have endorsed such a tuning in his treatise of 1318, as the
Monz and I have discussed here on various occasions.

> Concerning 12 from 31, there must be intervals that on a keyboard sound
> different in different places -- like the way QCMT has 8 good thirds and
> 4 bad. What I am now wondering (and I hope this is true) is that in the
> places on the keyboard where the intervals sound unlike what is
> expected, do we get another approximation of a just interval? -- like
> those lucky dim4ths in Pythag?

Curiously enough, the very wide "Wolf" fifth in 1/4-comma meantone (~737
cents) is an almost perfect 49:32, with a couple of hundredths of a cent.
One can actually use this ratio to derive a rational approximation for the
1/4-comma meantone fifth, dividing 49:32 by 128:125 (the lesser diesis of
about 41 cents) and getting 6125:4096.

As for the diminished fourth, this is indeed an integer ratio on a
1/4-comma keyboard: 32:25, the difference between a pure 2:1 octave and
two pure major thirds at 5:4 each (making an augmented fifth of 25:16).

In 31-tet, these intervals would be slightly different -- but very close
to 1/4-comma, I might guess.

Most respectfully,

Margo Schulter
mschulter@value.net