Voice-Leading and Counterpoint

From going to the SMT Conference, and from reading Music Theory
Spectrum, one finds that voice-leading and counterpoint are still
very much alive in music theory. (To state the obvious). It might be
interesting to start incorporating that into tuning-math discussions.
(Among other things, like modes of limited transposition, etc.)

Anyway, for starters, take the simple progression ii-half-dim-7, V7.
Let's use Bb Major. You obtain C-Eb-Gb-Bb -> C-Eb-F-A. Fix C-Eb and
use 6:7 in the 7-limit. For F7 you get 4:5:6:7. For Chd7 you get
1/4:1/5:1/6:1/7. These are the tetrads, (in the stellated hexany and
elsewhere) as we all know.

So 1/1, 5/4, 3/2, 7/4, 21/10, 21/8 for the whole thing. The parts
that move (between Gb->F, Bb->A,) move 21/20 ! Why is that
interesting?

Well, 21/20 equals 25/24 * 126/125, which is a "normal" chromatic
semitone in the 5-limit * the discrepency used in my C(2*2) X C3
shoebox.
(It's 10/7:(6/5)^2).

Now let's go from ultraminor to ultramajor. Consider 7/6, 32/27,
6/5, 5/4, 81/64, 9/7. If you take this like a palindrome, multiply
from the outsides and see that you always get 3/2. Now let's take
ratios:

64/63, 81/80, 25/24, 81/80, 64/63
64/63 * 81/80 = 36/35, which also appears in my shoebox
and is 7/5:(6/5)^2.

This is the sequence ultraminor, diatonic minor, just minor, just
major, diatonic major, ultramajor.

I know I am just boring the experts and confusing the newbies.
To work with my shoebox (C(2*2) X C3) you map 9/8 -> 8/7, that is,
3^2->7^-1. Then you can take 10/7:(6/5)^2-> 128/125 (And then temper
out with the diesis) Or forgetting about that mapping, just consider
the ratios above: Increment in minor thirds (6/5). Either go to 7/5
for the tritone, and consider 36/35 at that point, and then 126/125
at the octave, or go to 10/7 at the tritone, using 126/125 at that
point, and 36/35 at the octave! Multiplying 126/125 and 36/35 and
tempering out the diesis gives 81/80. I still need to work out my
shoebox idea, the main idea there is that you go to 3 dimensions and
consider the 7-limit, treat 5 in a fluid way, so that 128/125 is
always tempered out and so forth. So bringing the mapping back,
continue on from 10/7, multiply by (6/5)^2, remap, get 256/245
use the diesis (shuffle 5's), and end up with 50/49! Reversing the
mapping and shuffling 5's again brings us back to 81/80.

So it's not enough to just work with fractions, one has to put the
puzzle pieces together in a meaningful way. But going back to my
original subject, does anyone have any thoughts on the two
tetrads=tetrahedrons (which are displaced by a fifth, of course),
and voice-leading considerations??

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
>
> From going to the SMT Conference, and from reading Music Theory
> Spectrum, one finds that voice-leading and counterpoint are still
> very much alive in music theory. (To state the obvious). It might
be
> interesting to start incorporating that into tuning-math
discussions.
> (Among other things, like modes of limited transposition, etc.)
>
> Anyway, for starters, take the simple progression ii-half-dim-7,
V7.
> Let's use Bb Major. You obtain C-Eb-Gb-Bb -> C-Eb-F-A. Fix C-Eb
and
> use 6:7 in the 7-limit. For F7 you get 4:5:6:7. For Chd7 you get
> 1/4:1/5:1/6:1/7. These are the tetrads, (in the stellated hexany
and
> elsewhere) as we all know.
>
> So 1/1, 5/4, 3/2, 7/4, 21/10, 21/8 for the whole thing. The parts
> that move (between Gb->F, Bb->A,) move 21/20 ! Why is that
> interesting?
>
> Well, 21/20 equals 25/24 * 126/125, which is a "normal" chromatic
> semitone in the 5-limit * the discrepency used in my C(2*2) X C3
> shoebox.
> (It's 10/7:(6/5)^2).
>
> Now let's go from ultraminor to ultramajor. Consider 7/6, 32/27,
> 6/5, 5/4, 81/64, 9/7. If you take this like a palindrome, multiply
> from the outsides and see that you always get 3/2. Now let's take
> ratios:
>
> 64/63, 81/80, 25/24, 81/80, 64/63
> 64/63 * 81/80 = 36/35, which also appears in my shoebox
> and is 7/5:(6/5)^2.
>
> This is the sequence ultraminor, diatonic minor, just minor, just
> major, diatonic major, ultramajor.
>
> I know I am just boring the experts and confusing the newbies.
> To work with my shoebox (C(2*2) X C3) you map 9/8 -> 8/7, that is,
> 3^2->7^-1. Then you can take 10/7:(6/5)^2-> 128/125 (And then
temper
> out with the diesis) Or forgetting about that mapping, just
consider
> the ratios above: Increment in minor thirds (6/5). Either go to
7/5
> for the tritone, and consider 36/35 at that point, and then
126/125
> at the octave, or go to 10/7 at the tritone, using 126/125 at that
> point, and 36/35 at the octave! Multiplying 126/125 and 36/35 and
> tempering out the diesis gives 81/80. I still need to work out my
> shoebox idea, the main idea there is that you go to 3 dimensions
and
> consider the 7-limit, treat 5 in a fluid way, so that 128/125 is
> always tempered out and so forth. So bringing the mapping back,
> continue on from 10/7, multiply by (6/5)^2, remap, get 256/245
> use the diesis (shuffle 5's), and end up with 50/49! Reversing the
> mapping and shuffling 5's again brings us back to 81/80.
>
> So it's not enough to just work with fractions, one has to put the
> puzzle pieces together in a meaningful way. But going back to my
> original subject, does anyone have any thoughts on the two
> tetrads=tetrahedrons (which are displaced by a fifth, of course),
> and voice-leading considerations??

I should add, briefly, that Chd7 and F7 are adjacent tetrads in the
stellated hexany. Also, interesting that 21/20 completes an otonal
tetrad in Carl's semi-stellated hexany. So it's not just a ratio,
but actually part of one of the otonal tetrads. Just a couple
tidbits.
>