Repost (TD 723:6) -- Hypermeantones, meantones, hypomeantones

[Please note that this message is a repost of an article appearing in TD
723:6. I am reposting because of a guess that a special character I
quoted in my first post, which appeared as an "o-acute" on my display but
is apparently a "cents" symbol in some other character set scheme, may
have caused a reported problem in nondigest mode in which my text was
treated as an attachment in a nonstandard character set. Here I am
replying to a response by Paul Erlich in TD 719.]

Hello, there, Paul, and thank you very much for some very interesting
and stimulating comments, as well as your correction of my
misassociation of 7-tet with Javanese/Balinese pelog (of which more in
another reply I'm posting along with this).

> Since you're willing, under special circumstances, to extend the
> hypermeantone range to include 27-tET, with its fifths ~9.15 cents wider
> than pure, might you also be willing to extend the "stylistic
> meantone" range to include, under special circumstances, 26-tET with
> fifths ~9.65 cents flatter than pure? As Dave Keenan and I have
> discussed, you can get very interesting effects from such a tuning,
> which I've been able to experiment with to a limited extent on my
> keyboard.

This is a very interesting question, and before suggesting two
possible ways of extending the definition of meantone in this fashion,
I would emphasize that definitions of the kind we are discussing and
refining -- and especially ones that I must take responsibility for
offering -- should never stand in the way of making good music.

While 26-tet is clearly outside the range of _historical_ European
meantone (19-tet or ~1/3-comma being the narrowest regular fifth of
which I'm aware in this tradition), this doesn't necessarily mean that
our "meantone" range couldn't be broadened to include it. Especially
if 26-tet proves an attractive tuning for some compositions with a
"meantone-like" tertian style -- maybe a "neo-Renaissance" or
"neo-Baroque" outlook, as it were (compare neo-Gothic), or also a
"neo-Classic" or "neo-Romantic" outlook -- then we might have a very
strong musical motivation for doing so.

Dave Keenan has proposed one definition of meantone which would
include 26-tet: a tuning where the regular major third is the best
approximation of 5:4. This gives us a meantone range of tunings with
fifths from ~691.50 cents to 700 cents.[1]

While, as mentioned in a footnote in my paper citing our e-mail
correspondence, we might take it as a _necessary_ condition that any
meantone have a regular major second with a ratio between 9:8 and
10:9, suppose we also take this as a _sufficient_ condition. Then our
lower limit for meantone would be the point where the diatonic major
second is 10:9 (~182.40 cents), and thus the fifth at ~691.20 cents
(~11.75 cents, or precisely 1/2 syntonic comma). This is the point
where our regular major third will be exactly one syntonic comma
narrower than pure (100:81).

Musically, I guess that calling 26-tet a meantone rather than a
hypomeantone would emphasize its utility for at least some
"meantone-like" styles. Calling it a hypomeantone might emphasize that
it is outside the usual range of historical meantones.

You raise a very interesting parallel with 27-tet -- and a kind of
apparent paradox, if one chooses to accept 27-tet as a neo-Gothic
variety of hypermeantone while not extending the range of "stylistic
meantones" to include 26-tet.

One fine point: while "hypermeantone" itself, as defined in my paper,
extends all the way out to 5-tet, my inclusion of 27-tet also as a
"_neo-Gothic_ hypermeantone" under certain special circumstances
despite its tempering of the fifth by ~9.15 cents indeed raises
exactly the question you have presented.

In other words, if 27-tet can be stylistically acceptable under
certain circumstances for Gothic or neo-Gothic music, despite the
prime concords in this style being tempered by 9.15 cents, might not
26-tet likewise be recognized as an acceptable "stylistic meantone"
under certain circumstances also.

If a rough symmetry between meantone and neo-Gothic hypermeantone
seems to make sense, then one solution might be to recognize a
"peripheral meantone" zone roughly from 26-tet to 19-tet, say,
analogous to a "peripheral neo-Gothic hypermeantone" zone from 22-tet
to 27-tet, say.

In effect, this scheme would say, "Usual meantones are tempered up to
a bit more than 7 cents in the narrow direction (19-tet), and usual
neo-Gothic hypermeantones up to about the same amount in the wide
direction (22-tet). However, under some conditions meantones can have
fifths narrowed by up to ~9.65 cents (26-tet), and neo-Gothic
hypermeantones can have fifths widened by up to ~9.15 cents (27-tet)."

A graphical representation of such a proposal, with dotted regions
showing the peripheral zones from 26-tet to 19-tet, and from 22-tet to
27-tet:

~-16.25 ~-9.65 ~-7.22 0 ~+7.14 ~+9.15 ~18.04
~685.71 ~692.31 ~694.74 ~701.96 ~709.09 ~711.11 720
|------------|------|-------------|----------|------|-----------|
7-tet 26-tet 19-tet Pyth 22-tet 27-tet 5-tet
|------------|......--------------|-----------------------------|
hypomeantone meantone hypermeantone
|-----------.......
neo-Gothic range

Your remarks open various kinds of questions to what may be quite a
running dialogue. For example, while meantone has a lot of historical
associations (however definitionally relevant), "hypermeantone" and
"neo-Gothic" may be pretty much new concepts. How might this affect
definitional symmetry or asymmetry for the "stylistic meantone" and
"neo-Gothic" categories?

Also, is there is a parallel between the syntonic comma in the
meantone realm and the septimal comma in the outer portion of the
neo-Gothic hypermeantone range? A necessary condition that meantones
have regular major seconds between 9:8 and 10:9, and neo-Gothic
hypermeantones between 9:8 and 8:7 -- rather noncontroversial, it
seems to me, at least as a _necessary_ condition -- maybe suggests
this parallel.

If we say that stylistic meantones typically temper the fifth up to
1/3-(syntonic)-comma, and neo-Gothic hypermeantones up to
1/4-septimal-comma, but that in special circumstances we can go up to
around 1/2-(syntonic)-comma or 1/3-septimal-comma, this might be
stylistically apt while also approaching absolute symmetry as to the
amount of tempering of the fifth involved in either direction.

> Certainly in the case of 9:12:16, I would say that, for most
> musically useful timbres and registers, the 16:9 minor seventh is
> already too complex a ratio to be perceptibly worsened by
> mistuning. Instead, bringing it any distance (in this case, halfway)
> towards 7:4 only improves its concordance. Meanwhile, the two
> perfect fourths are only tempered by 1/4 of a septimal comma. As a
> result, this two-stacked-fourths chord (and for similar reasons,
> chords of three stacked fourths) sounds very pleasant in "pure 9:7
> hypermeantone tuning", or the virtually identical 22-tET.

This is another good question: does the pleasant effect of 9:12:16 in
Pythagorean derive in part from the pure 16:9 minor seventh, or simply
from the two pure fourths? How about the role of the 9:8 in 4:6:9?
Do stylistic expectations or settings influence these questions?

In a 7-limit setting with stable tetrads, sonorities built from three
stacked fourths in 22-tet would (as you point out in your article
also) have a special appeal because of the near-pure 7:6 (or its
octave) as the outer interval of D3-G3-C4-F4, or example.

In a complex 3-limit setting, I wonder how this would compare with the
Pythagorean 27:36:48:64, which I have found quite pleasant. If
Japanese gagaku uses basically pure fifths for its vertical
sonorities, then such sonorities would occur quite often.

Anyway, thank you for raising some questions inviting a lot more
dialogue.

Most appreciatively,

Margo Schulter
mschulter@value.net

----
Note
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1. Note also Keenan's ingenious definition of a Wolf fifth less
accurate as an approximation of 3:2 than some other interval produced
by a reasonably short chain of such fifths. This gives a range of
"non-Wolf" fifths from around 12.7 cents narrow to 12.9 cents wide,
not too far from one traditional limit of 1/2 Pythagorean comma or
~11.73 cents in either direction.