Fun with Pentachords/Septachords

Here's a recent posting of mine on tuning-math. Thought I would post
it here, even though it's kind of borderline with respect to being
mathematical, I think it has more of an atonal theory core to it:

> In 12-Et there are 35 pentachord/septachords, after reducing for
> direction (reversibility of necklaces) and Z-relation (There are
three Z-related pentachord/septachords (Three pairs of pentachords
where each pair shares a common interval vector). The Z-related
pentachords are:
>
> (0,1,2,5,9)---Z---(0,1,3,4,8)
> (0,1,2,5,8)---Z---(0,1,4,5,7)
> (0,1,3,5,6)---Z---(0,1,2,4,7)
>
> The first two are also related by "M5" symmetry: D4XS3 Group
symmetry.
> The last one is an "impassible" weakly-related 7/5 set complex: The
only
> way to go from the septachord to the pentachord is through a Z-
related
> hexachord (All the Z-related hexachords are complements of each
other)
> (0,1,3,5,6) is also special as a WR7/5SC in that the pentachord
does not
> fit into the complementary septachord. The other two WR7/5SC
complexes
> are:
>
> (0,1,4,7,8) and
> (0,1,2,4,7)
>
> Now there are also 35 hexachords, after reducing for direction and
Z-
> relation (or just complementarity, since all the Z-related pairs
are
> complements: there are 15). But let's stick to pentachords for now.
>
> My thinking is there is some numerical relationship between the 35
> pentachord/septachord types (and also 35 hexachord types) and
tuning
> considerations: The diatonic MOS scale (white keys), the pentatonic
MOS
> scale (black keys) number 7 and 5. There are 7 half steps in a P5
and 5
> half steps in a P4. 36/35 is an important comma. And so forth.
There is
> a fairly complicated formula for deriving the 35 hexachord types,
(And
> 38 pentachords, short of the Z-relations) There is no real formula
for
> calculating which sets are Z-related. Of course there is a formula
for
> measuring primitive sets (using the moebius function). Of course
with
> pentachords all of the sets are primitive.
>
> I don't have the formulas with me, but for pentachords it boils
down to
> taking the 66 pentachords (full count, with inverses) and averaging
with
> the symmetric ones (10) and gettin 38. With hexachords, there are
80 to
> start with, reducing for complementability: 80+8/2 (8 is the number
> which map perfectly into their complement) This gives us 44,
averaging
> with 26 (core of symmetry of complements, these are sets which are
> either symmetrical (13) or where the complement is also the inverse
> (13)) brings the count down to 35. So I ended up talking about
> hexachords anyway. The formulas for calculating these are based on
> Polya's counting method. (A different one from the one hitherfore
> presented on this newsgroup -- I'll post a new message tomorrow.)
>
> Thoughts anyone, of how reversible complementable necklaces (which
use
> group theory) could tie into tuning theory? Or even further, how
> interval vector counts (which one obtains after reducing for non-
> complementable and complementable Z-relations) could tie into
tuning
> considerations?

Some added thoughts: Add 8 to (0,1,3,5,6) (That "strangely-Z-related
impassible weakly-related seven-five set complex") and you obtain
(0,1,3,5,6,8). It's composed of V7 (on 8)and I (on 1) superimposed.
It's complement is the blues scale! (0,1,2,5,7,10). If you take the
complement of (0,1,3,5,6) you obtain a more symmetrical form of the
blues scale (0,3,4,5,6,7,10) which is the blues scale with added
major third (which is often in the harmony of blues: C-(E)-G-Bb)

Now the complement of (0,1,2,4,7), the Z-relation of (0,1,2,3,5,6) is
nothing special. But its kind of interesting that (0,1,3,5,6) in 7-
limit just intonation is (15,16,18,20,21) The ratio of 18/15 and
21/18 is (6/5)/(7/6) is 36/35 -- could this have anything to do with
the 35 pentachord/septachord types? Probably not... Remember there
are also 35 hexachords.
>
> Paul Hj