generalized diatonic scales

I've just done a search of 14 different scales, most of them discussed at
some point here in the last year, for "generalized-diatonic" properties.

That could mean a number of things, but the main thing is: given a pattern
of scale degrees, you get _different_, _consonant_ chords in different modes
of the scale. It turns out that this property is already strong enough to
disqualify most of the scales I looked at, so you don't need to agree with
me on the rest of what else constitutes "generalized diatonicity" to find
this useful.

Here are all the scales that even remotely meet this one criteria, when the
following chords are considered consonant...

Triads - 4:5:6, 1/(4:5:6), 6:7:9, 5:6:7, 7:9:11, 9:11:15, 9:11:13

Tetrads - 4:5:6:7, 5:6:7:9, 1/(4:5:6:7), 10:12:15:17, 8:10:12:15,
10:12:15:18, 12:14:18:21, 6:7:9:11, 8:10:11:14

[1.] Standard diatonic scale in 31tet.

0 5 10 13 18 23 28 31
5 5 3 5 5 5 3

Scale pattern 1-3-5 yields six 5-limit triads (3 otonal and 3 utonal).
Pattern 1-3-5-7 yields two 8:10:12:15 tetrads and three 10:12:15:18 tetrads.

[2.] Dave Keenan's 125-cent generator, single-chain scale in 29tet.

0 3 6 9 11 14 17 20 23 26 29
3 3 3 2 3 3 3 3 3 3

Scale pattern 1-4-7 yields six 5-limit triads (3 otonal and 3 utonal).

[3.] Easley Blackwood's decatonic scale in 15tet.

0 2 3 5 6 8 9 11 12 14 15
2 1 2 1 2 1 2 1 2 1

Scale pattern 1-4-7 yields ten 5-limit triads (5 otonal and 5 utonal).
Degrees 1-4-7-10 yield five 8:10:12:15 and five 10:12:15:18 tetrads.

[4.] A decatonic scale I cooked up from two pentatonic chains of 7:4's
rooted a 5:4 apart, tuned in 31tet.

0 4 6 10 12 16 18 22 25 28 31
4 2 4 2 4 2 4 3 3 3

Scale pattern 1-4-7 yields four 5-limit triads (2 otonal and 2 utonal).

[5.] A decatonic scale I cooked up from two pentatonic chains of 3:2's
rooted a 7:4 apart and tuned in 31tet.

0 4 5 10 12 17 18 23 25 30 31
4 1 5 2 5 1 5 2 5 1

Scale pattern 1-4-7 yields four 5-limit triads (2 otonal and 2 utonal).

[6.] Paul Erlich's pentachordal major decatonic in 22tet.

0 2 4 7 9 11 13 16 18 20 22
2 2 3 2 2 2 3 2 2 2

Scale pattern 1-4-7 yields seven 5-limit triads (4 otonal and 3 utonal).
Degrees 1-4-7-9 give three 4:5:6:7 and three 10:12:15:17 tetrads.

Scales that didn't make this list include two of Paul Hahn's nonatonic
scales in 31tet, two tunings of a decatonic MOS proposed by Dave Keenan, 8-
and 11-tone versions of the chain-of-minor-thirds scale in 19tet, three
tunings of an 11-tone MOS suggested by Dave Keenan, and a scale by Heinz
Bohlen based on the 5:7:9 triad. The complete results of the search are
downloadable at:

http://lumma.org/Gd.zip

-Carl

Carl Lumma wrote,

>[6.] Paul Erlich's pentachordal major decatonic in 22tet.

> 0 2 4 7 9 11 13 16 18 20 22
> 2 2 3 2 2 2 3 2 2 2

>Scale pattern 1-4-7 yields seven 5-limit triads (4 otonal and 3 utonal).

Count again -- there's eight (4 otonal and 4 utonal).

>Degrees 1-4-7-9 give three 4:5:6:7 and three 10:12:15:17 tetrads.

Why do you call them 10:12:15:17 tetrads and not 1/(4:5:6:7) tetrads?

What about the symmetrical decatonic scale, where 1-4-7-9 gives _four_
4:5:6:7s and _four_ 1/(4:5:6:7)s?

And how about the diminished scale in 12-tET (or 28-tET, etc.) where 1-4-7
gives four 3:4:5 triads and four 1/(3:4:5) triads?

And how about the 14-out-of-26 scale I've discussed?