Re: [MMM] Digest Number 102

> 1. 11-tet harmonies
> From: sethares@...

> 54-63-67 resolving to
> 56-63
>
> So the final resolution is to a dyad, not a triad.
> Now its time for me to scratch my head... what are these chords?
> What are the roots? What kind of harmony is this?

Hello, there, Bill, and let's see if at least I can identify the steps,
intervals, and rough ratio approximations.

54-63-67

with an 11-note octave would be a sonority, if I'm correct, of

0-9-13

or, in 11-tET, roughly

0-982-1418 cents

The interval between the lower notes is not too far from 4:7 (about 969
cents); between the two outer notes, fairly close to 15:34, I guess, an
octave plus a "tone" about midway between 8:9 (204 cents) and 7:8 (about
231 cents).

The interval between the two upper voices is a 4-step or "ditone" of about
436 cents, an excellent approximation of a just 7:9.

In 22-tET terms, also, this would be the best approximation of 4:7:9,
which has been mentioned in this thread by Paul.

What's apparently happening here is that the lower two voices with the
near-4:7 are progressing by oblique motion (one voice moving, one
stationary):

54-63-67 (0-9-13 = 0-982-1418) roughly 4:7:9
56-63 (0-7 = 0-764) near-just 9:14

This new sonority, a 7-step at around 764 cents, is an excellent
approximation of 9:14 (about 765 cents). In conventional terms, this might
be called a "septimal minor sixth," just its octave complement the 4-step
might be called a "septimal major third" at very close to 7:9

In conventional terms, this might be called a "minor seventh resolving to
a minor sixth," with the lower voice ascending by the 11-tET/22-tET
"tone" of around 15:17.

If the lower voice descended from the 9-step to the 7-step interval, then
this might be analogous to a kind of "7-6 suspension" where a seventh
resolves to a sixth, one of the most characteristic features of
Renaissance and some later European styles.

With a 4:7:9, although here I may be going a bit outside of the medieval
and Renaissance European theory with which I'm most familiar, I'd say that
the lowest note is arguably the "root," both because it's an even power of
two, making it an octave of the "1" or "fundamental" in this kind of
approach; and because under at least some theories, a 4:7 above the lowest
note has some "anchoring" power to make that note felt as the firm
"basis" of the sonority.

If we did 0-13-18-26 steps or 0-709-982-1418 cents in 22-tET, the
approximation of 4:6:7:9, then the "anchoring" power of the 2:3 above the
lowest note would reinforce this "rootedness."

"What kind of harmony is this" is a good question, and when I heard _Dabo
Girl_ on your CD, I didn't say to myself, "That's some 11-tET takeoff on a
16th-century suspension with a voice resolving upward rather than
downward."

It may be unique kind of harmony, because the "harmonic geometry" of
11-tET is different, and trying to describe it in familiar Renaissance or
other terms is maybe a bit like trying to project a 3-D object on a 2-D
surface -- we may get an idea of some interesting aspect, but distort
others, sometimes not recognizing the distortion.

Anyway, I hope that this is the start of an answer, and that I haven't
committed any obvious errors in trying to respond offhand.

With peace and love,

Margo