Re: Meantone-JI connection?

Hello, there, is this is a very brief attempt at a possible
interpretation on the remark that 1/4-comma and 1/3-comma meantone are
the forms which permit "just intonation" (JI).

I'd interpret this possibly to be pointing out that in 1/4-comma
meantone (very close to 31-tone equal temperament or 31-tet), major
thirds are a pure 5:4; in 1/3-comma (very close to 19-tet), minor
thirds are a pure 6:5.

While it seems likely that 1/4-comma meantone was very popular in the
16th and early 17th centuries, I suspect that 1/3-comma may not have
been quite on par with it, because of the narrow fifths (~7.17 cents
from 3:2). Another contributor suggested 2/7-comma meantone (Zarlino's
proposal) as a pleasing tuning for the diatonic scale in this
historical setting, and in fact Mark Lindley remarks that it can be
very pleasing for music with lots of what Zarlino would call
"arithmetic divisions" of the fifth -- that is, sonorities with the
minor third below and major third above (string-ratio 6:5:4,
frequency-ratio 10:12:15).

Also, much in agreement with Paul Erlich, I'd like the emphasize that
indeed it is possible to use various temperaments (including n-tet's
and meantone) with an awareness of the JI intervals being
approximated, and also of the compromises being made.

Most respectfully,

Margo Schulter
mschulter@value.net