Myna thoughts

I'm back looking at old, unfinished composing projects to complete
again, and was looking at myna, which I started something with back in
the days when I still called it "nonkliesmic". Whatever you call it,
it's the 7-limit 27&31 temperament, with commas 126/125, 1728/1715 and
so (126/125)/(1728/1715) = 2401/2400 also. It seems to have a natural
tendency towards small scale steps in the melody line, which I think
is connected to the fact that <4 6 9 11| is a val for the temperament.

In defense of this perhaps unlikely sounding claim, consider what
happens when you have two successive utonal tetrads which share a note
and such that reduced to 4-equal by the above (standard) val are the
same chord. If [0, 0, 0] reprsents the first tetrad, then the second
tetrad is one of the verticies of the cuboctahdron surrounding it,
namely one of [+-1, +-1, 0], [+-1, 0, +-1], [0, +-1, +-1]. Of these,
the six in the x+y+z=0 plane, where the tetrad is [x, y, z], have
roots which are closer in terms of the generator steps of myna, a
minor third, than the six with the same signs on the 1s. We have 1
step a 6/5, 2 a 10/7 and 3 a 12/7, with -1, -2, -3 being inverses. The
otonal tetrads with these as roots differ from the [0,0,0] tetrad by
intervals of 1, 15/14, 21/20, 25/24, 36/35, 49/48 and 50/49 and their
inverses. The other six involve intervals of 9/8 and 35/32. These are
16 and 20 generator steps repsectively, and not as characteristic of
myna. Myna conflates 50/49, 49/48 and 36/35 to four generator steps,
25/24 and 21/20 to eight generator steps, and 15/14 to twelve
generator steps. The smallest interval is the closest to the unison in
myna, the next smallest the next closest, and so on through 15/14 at
12, 35/32 at 16 and 9/8 at 20. So if smaller intervals somehow seem to
want to happen in myna, this may explain it.

The otonal tetrads of myna can be placed along a line of generator
steps. For Myna[17], with roots from -3 to 3 generators, the otonal
tetrads are related exactly as the corresponding JI tetrads, with no
comma pumps. The are the six cuboctahdral verticies in the x+y+z=0
plane, plus [0,0,0]. We can add to this seven more utonal tetrads,
which are just the minor versions of the otonal ones (subtract
[1,0,0].) This is because the maximal interval is the fifth, so to
each otonal tetrad in a chain of generators scale there is a
corresponding minorized version. However, the JI version of the scale
has 24 notes rather than 17. If we move up to Myna[19], which is a
MOS, we now have comma-pump connections from 4 to -3 generators, and
-4 to 3 generators.

Aside from these tetrads, myna also has diminished seventh chords of
the 6/5-6/5-6/5-7/6 variety, since 126/125 is a comma. There are also
what might be called "morewell chords", or 7/6-7/6-7/6-5/4 magic
chords, a feature of anything having the orwell comma of 1728/1715,
but particularly of orwell itself since 7/6 is the generator. In myna
generators these are the 0-3-6-9 chords, and hence there are plenty of
them in Myna[17] or Myna[19]. Even more diminished seventh chords, of
course, these being 0-1-2-3 chords.