Fun with Pentachords/Septachords

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
consierations?

Paul Hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@m...> wrote:

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.
* * * * * * * * * * * * * * * * * * * * * *

> 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
> consierations?
>
> Paul Hj