Re: Hypermeantones [...]

Before I come to the point, a couple of quick comments.
Sorry for the misatribution, Paul, the result of a moment of distraction.
I have caught the tail end of a discussion between Jason Yust and
(I presume) you, which features i.a. the assumption of a cognitive
scale in one's musical perception. This is an issue I intend to
examine as best as I can, but no time for more words today.
(Even though it is implied in the following discussion.)

M. Schulter wrote:
> [first quuoting Paul Erlich]
> > 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?

Q: are there any other simple proportions approximate to 9:12:16 --
that is, any other way to fine-tune D-G-C? A: I cannot find any.
In the dualistic approach, the two alternative interpretations of
this combination are (a) undertones of D and (b) overtones of C.

I interpret the "pleasantness" of 9:12:16 -- also the fact
that we can fine tune it by ear as a chord -- as an indication
of a whole that is larger than its parts.

This effect cannot be unconditional, of course. It depends
on which overtones are musically significant in one's cognitive
setup (I will return to this in another message), on the harmonic
profiles of the tones, on whether the listener's perception is
moderated by a model scale or a system of scales (processing tones
"categorically") -- I bet the list expands easily.

With a tempered system of tuning, I see two plain options wrt 9:12:16.
Because there is no neigboring simple proportion, either one assumes
that the effect of this proportion remains, or that only intervals
and no sonorities are relevant. In other words, one can assess how
well does a tuning system serve the one or the other approach.

A third, more perplexing approach would be to assume both approaches
(whole-sonorities and interval-networks) as valid in this case,
therefore a transistion from tuning "9:12:16" to the tempered one
(which moves the outer ratio closer to 7/4) would effect a change
of function, therefore a more serious matter. I do not think that
this is the case (imho, should I add).

> 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.

This is what I, too, vaguely remember about gagaku -- it would not be
hard to find out. However, I have not found either a theoretical or an
experimental grace in (or about) 27:36:48:64, yet. I have played four
tones, varying continuously two of them; result: nothing special about
the above proportion. Actually the sonority got more "interesting",
albeit more Gesualdo-like, as I contracted the total interval.
(Might it be C-D-F = {1/8 : 1/7 : 1/6} (common overtone C) with a
drone G?)

Three writers, three views. Any more?
Good day!
- George Kahrimanis

"Paul H. Erlich" wrote:
> You do not think that both approaches can be valid? Well, I do.

I only had this particular case in mind, not postulating a general rule.
To test this matter in earnest, show me a short progression to try
out by ear, such that you think sounds better in that temperament.

> I was thinking of the cases where D-G-C is used as a D7sus4 (no 3rd) chord,
> i.e., D is clearly the root (say it's doubled an octave or two lower by the
> bass). Your theory may not allow this interpretation but believe me, as a
> musician I know it is real.

"D", you say? "My" theory (Partch would agree, I think) says that D is
one of the two alternate roots (an incomplete Gm with added 4th,
common overtone D). (Of course you do know this
sort of calculus, only you do not apply it. BTW, I am still
grateful for catching a silly mistake of mine once.)

> In this case, bringing the seventh down toward
> 7/4 is helpful both in terms of making the 16:9 more
> consonant, and in terms of chord function.

I repeat, it might be so in certain progressions but I remain
to be convinced with an example. In the static case my ears simply do
not perceive the advantage, not yet in any case. (The stack of
three fourths is another story.)

-------------------
My postings so far may well look like I joined the list ready
for polemical exchanges. The truth is, I joined the list to look for
suggestions on a certain subject but I got distracted by the
argument on the ideal tuning of stacked fourths. (You understand,
I hope.)

-------------------
To dispell any notion of chauvinism: my last name is not Greek,
but Turkish (meaning "popular killer", a term of endearment, really)
coming from the name of the Persian God of Evil (%ahriman).