Re: Are 19-tet and 31-tet meantone tunings?

Hello, there, and in response to the "Monz," I'd like to state my view
that whether 19-tet and 31-tet are regarded as "meantone" tunings
depends on the definition, but I would say "Yes" under most historical
definitions which don't limit "meantone" to the one special case of
1/4-comma (i.e. syntonic comma) meantone.

What I suspect Monz means to say is that 19-tet is not equal to
_exactly_ 1/3-comma (pure 6:5 minor thirds), and that 31-tet is not
equal to _exactly_ 1/4-comma (pure 5:4 major thirds). If "meantone"
were customarily defined as a tuning with all regular fifths equal
where _either_ the major or minor thirds are pure, then both 1/3-comma
and 1/4-comma would uniquely qualify, but not 19-tet or 31-tet, their
very close approximations.

However, such a definition of "meantone" seems rather uncommon to me,
because temperaments where all regular fifths have the same size but
neither major nor minor thirds are pure also routinely get included,
for example Zarlino's 2/7-comma tuning. Here I might propose three
ways in which "meantone" is typically used, the first two being more
commonly used maybe in historical contexts and the third more
typically among contemporary theorists (e.g. Paul Erlich):

(1) In the narrowest definition -- literal "mean-tone" -- meantone
refers exclusively to the 1/4-comma temperament with pure major thirds
at 5:4, where a regular major second (~193.155 cents) is equal to
precisely the _mean_ of the 9:8 and 10:9 whole-tones of tertian just
intonation which together form a 5:4 third. This definition would
exclude 1/3-comma, of course, as well as 19-tet and 31-tet, etc.

(2) In a broader concept -- what I might call "characteristic
meantone," -- the term refers to a tuning where all regular fifths
have the same size, and these fifths are tempered by a quantity
somewhere around the range of 1/3-1/6 syntonic comma. This is the
range typical for Renaissance and early 17th-century practice, where
thirds are pure or quite close to pure (within 1/3-comma). Also, it is
a range where "Wolves" occur, both diminished sixths and diminished
fourths regarded as 'unplayable' by 16th-17th century standards.

(3) In the broadest concept, meantone says simply that all regular
fifths have the same size. Thus Pythagorean (0-comma meantone), 12-tet
(~1/11-comma), and 17-tet (in effect ~-2/11-comma, with fifths this
around _wider_ than just) are all meantone tunings, although not
"characteristic meantone tunings" with thirds within about 1/3-comma
of a pure 5:4 or 6:5. Incidentally, under this definition, Pythagorean
is a meaning tuning but not, of course, a _temperament_ -- maybe we
could call it a meantone untemperament <grin>.

Applying these criteria, we see that 19-tet and 31-tet are indeed
"meantone" temperaments under definitions (2) and (3). These tunings
have all regular fifths equal, and are within the range of
"characteristic meantone."

Strictly speaking, since 19-tet fifths (~7.22 cents narrower than 3:2)
are tempered a tidge _more_ than 1/3-comma (~7.17 cents), not only
these intervals but also the major thirds fall a hair (about 0.05
cents) outside the "within 1/3-comma of pure" test -- but if we say
"not more than _about_ 1/3-comma from pure," this solves the problem.

In the case of 31-tet, of course (~5.18 cents of temperament, as
opposed to ~5.38 cents for 1/4-comma with pure 5:4 thirds), we're near
the center of the historical meantone range, and I don't see how we
can say that this tuning is not "meantone" -- unless we follow
definition (1) above, reserving the term for the unique case of
1/4-comma and only exactly 1/4-comma.

Psychologically, I might guess, there could be a tendency somehow to
regard a meantone tuning expressible as some neat fraction of a
syntonic comma (e.g. not only 1/3-comma or 1/4-comma, but also
2/7-comma or 1/5-comma, etc.) as more "truly meantone" than something
like 19-tet or 31-tet, which can't be expressed so neatly. Could this
be an example of how, even with such tunings as meantone where
irrational ratios are unavoidable, integer ratios for the division of
the syntonic comma may make a tuning look more "legitimately"
meantone?

Most respectfully,

Margo Schulter
mschulter@value.net

Margo wrote,

>However, such a definition of "meantone" seems rather uncommon to me,
>because temperaments where all regular fifths have the same size but
>neither major nor minor thirds are pure also routinely get included,
>for example Zarlino's 2/7-comma tuning. Here I might propose three
>ways in which "meantone" is typically used, the first two being more
>commonly used maybe in historical contexts and the third more
>typically among contemporary theorists (e.g. Paul Erlich):

>[...]

>(3) In the broadest concept, meantone says simply that all regular
>fifths have the same size. Thus Pythagorean (0-comma meantone), 12-tet
>(~1/11-comma), and 17-tet (in effect ~-2/11-comma, with fifths this
>around _wider_ than just) are all meantone tunings, although not
>"characteristic meantone tunings" with thirds within about 1/3-comma
>of a pure 5:4 or 6:5. Incidentally, under this definition, Pythagorean
>is a meaning tuning but not, of course, a _temperament_ -- maybe we
>could call it a meantone untemperament <grin>.

Margo, I would never call Pythagorean or 17-tET meantone tunings.
A meantone tuning is one where the fifth is diminished by a small
amount in order to improve the thirds.

However, I have seen this third definition used by a few, who speak
of such oxymorons as "positive meantones" (implying that the fifth
is augmented rather than diminished). I would certainly lean much closer
to your second definition.