12EDO Subscale Qantity/Rotation/"Nested" Inversions:

This is in response to a post about the quantity of
12EDO subscales that appeared last winter.

I'll restrict this discussion to the subscales with
between 4 and 8 pitch classes, since I think the
minimum interesting collection is 4 pitch classes,
since 4 pitch classes create a tetrahedron or
minimum 3-dimensional musical object in terms of
pitch classes.

According to 12EDO pitch class set theory, there
are 29 structurally distinct 4-tone sets, 38
structurally distinct 5-tone sets, 50 structurally
distinct -tone sets, 38 structurally distinct 7-tone sets,
and 29 structurally distinct 8-tone sets, for a grand total
of 184 structurally distinct subscales.

For reference, Larry Solomon's online 12EDO pitch
class set table is over here:

http://solomonsmusic.net/pcsets.htm

But that total does NOT take into account the
possible operation of "rotation" of the scale step pattern
occuring within these subscales. Each subscale can be rotated the
same number of times that it has different pitch classes, although
the resulting rotations are not necessarily
structurally distinct. Some will be redundant.

But including the rotation operation brings the total
up to 912 possible subscales, although the actual figure
would be somewhere between 184 and 912, since there will
be some rotations that are structurally redundant.

This brings me to the topic of what I call "nested
inversions" of 12EDO subscales. That involves aligning
or sandwhiching a subscale and it's inversion next to each
other in the minimum pitch/registral space. Here's the
diagram for the nested inversion of the 1-2-3 scale step
pattern 12EDO blues tetrachord:

Prime Form__________C___C#______D#__________F#
Inversion Form______C___________D#______F___F#

This "nesting" of a subscale and it's inversion is very
important, because each different rotation of a particular subscale
potentially results in a different superscale formed
by the combination of the prime and the inversion forms.

I have a Yahoo Group with MIDI file examples of this
nesting process over here:

/BluesNestedPentachordFiveToneScal
es/

You have to join the group in order to access the MIDI files,
which are in the group's files section.

Bill.Flavell at GMail.com