Augmented[9] MODMOS's suggest a new class of pentachordal scales with some neat properties

Hi all,

One of the results that I found most interesting from the MODMOS
search in 12-equal is that there exists two "pentachordal" scales that
are MODMOS's to augmented[9], just as Paul's pentachordal scales are
MODMOS's to pajara[10]. Classifying the MODMOS's of fractional-period
scales is a bit over my head right now, so here they are in step form
without recourse to the alterations made to the periodicity blocks.

12-equal is nearly ideal to tune for this, but I'll put this out in
45-equal because it has 7-limit implications. 15-equal is no good,
because the scale becomes improper. Replace every 4 with 1 and every 7
with 2 to get the 12-equal version.

The .'s are just to break things up and make it easier to read.

Starting with augmented[9] - 4 7 4 . 4 7 4 . 4 7 4

"Major" 9-note pentachordal scale - Augmented[9]#7
Augmented[9] MODMOS view: 4 7 4 . 4 7 4 . 7 4 4
Tetrachordally symmetric mode: 4 7 4 4 . 7 . 4 7 4 4
"Major" mode: 7 4 4 4 . 7 . 4 4 7 4

The major mode has 4:5:6 chords lying 3/2 away from the root in both
directions. A major 7 chord over the root, if transposed
"albitonically" up and down the scale, will turn into a dominant 7
chord with a raised 7th approximating 9/5 when it's lying on top of
the "V" chord, which of course would no longer have the number V here.

"Minor" 9-note pentachordal scale - Augmented[9]b8
Augmented[9] MODMOS view: 4 7 4 . 4 7 4 . 4 4 7
Tetrachordally symmetric mode: 4 4 7 4 . 7 . 4 4 7 4
"Minor" mode: 7 4 4 4 . 7 . 4 7 4 4

The minor chord has 10:12:15 chords lying 3/2 away from the root in
both directions. Additionally, over the "V" chord there's a dominant 7
chord with a raised 7th again approximating 9/5.

These seem very useful in the sense that they might be used to set up
"xenharmonic" 5-limit tonal structures that extend on the diatonic one
and also have remained buried within 12-tet, as did 22-equal's. Has
anyone explored these before? Might there be some interesting subgroup
one could set up over this to bring higher-limit harmonies into the
fold?

-Mike