Common Scales

Actually, one very simple practical upshot of all the musical
set theory I have been doing is the "discovery" that you can ---

1. Start with one of these septad scales:

a. Diatonic
b. Melodic Minor
c. Harmonic Major
d. Harmonic Minor
e. Hungarian

2. Alter one pitch (or choose not to)
3. Apply the M5 transform (or choose not to)

M5 is when you hold even pitches fixed and transpose odd ones by a tritone.

This will cover all septads, (and subsequently hexads, pentads, etc.)

One could tighten this up a bit and find just enough to cover
hexad-partitions (a hexad and/or it's complement, which might be a
Z-related pair) and then by subset and complementation, pentad-septads,
tetrad-octads, triad-nonads, dyad-dodecad, monad-modecad, null-dodecad
(empty set/whole collection)

In my grid 110209.xls which I posted on tuning-math Files I found
that there is a pattern to where hexachord complements land.
In fact every "block" will contain (at least the first four columns)
a hexad, it's M5 complement, it's regular complement (also inverted)
and the M5 of this.

PGH