Isomeric Sets, Affine Group and 22-tET

Well,

Here is my paper so far. I will post the Appendices to my Files Section.

For now, just for fun,

Notice that in 22-tET the Multiplicative Modulo Group (which generates
Aff(22)), is based on (1,3,5,7,9,13,15,17,19,21) or those numbers
totient to 22.

Now, Gene showed that the powers of 7 generate this set, indeed,
it is the first 10-cycle. Excluding M1 and M21, which aren't too
interesting, here are the other multipliers. Also cool to note
that M7 is based on the 7th step, a pure prime (5), and M13 on the
13th step, another pure prime (3). Since 18th step is a double, (7),
doesn't make this list. This is based on diaschismic tempering,
but I think Porcupine and Magic apply here too.

10-cycles

Multiplier and Ratio

M7: 5
M13: 3 or 1/21
M17: 6/7
M19: 1/35

5-cycles

M3: 35
M5: 7/6
M9: 21
M15: 1/5

M1 is based on 1-step, which can be anything from 49/48 to 36/35 or
21/20 etc.

Hope somebody glances at my paper:)My goal is to find a formula for any
isomeric relation, (Z-relation) in any temperament, right now I am
focussing on 2N-temperaments, and not so concerned with sets related by
complementation (like complementary hexachords that are Z-related). I
am more concerned with strange Z-relations...I intuitively believe
there is a "dual" relation between the affine thing and isomeric thing,
because isomeric relations seem to be based on 11 here, which is not
one of the affine multipliers. (More on that later, if anyone is
interested)...

The fun part is tying all this back into the tuning theory, of course.

Best wishes,

PGH

Paul G Hjelmstad wrote:
> Well,
> > Here is my paper so far. I will post the Appendices to my Files Section.

I find I get lost very rapidly :(

Anyway, there's an "of" on line 6 of page 2 that should be an "or".

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Paul G Hjelmstad wrote:
> > Well,
> >
> > Here is my paper so far. I will post the Appendices to my Files
Section.
>
> I find I get lost very rapidly :(
>
> Anyway, there's an "of" on line 6 of page 2 that should be
> an "or".
>
> Graham

Thanks. I am finding more typos myself. The paper needs a lot
of work, but I am confident I have some good ideas. (The last paper I
wrote was on ISDN for a Data Communications class, about 10 years
ago). I need to add citations and footnotes, at the very least.

I guess my main finding is that Z-relations cannot be random, because
the affine action segments everything 5-fold. My hunch also is that
isomericity and affinity (okay, I made those up!) are a kind of dual
relation. Throwing in complementability maybe complicates things too
much, or makes it all too dense, but you always have this with 2N
temperaments, and many are also Z-related. It also augments strange
Z-relations, thus:

A<-complement->B
|
(strange)
|
C<-complement->D

Where A and B (and C and D) are Z-related complements and A,B and C,D
are strange Z-relations.

I also find that the tritone (in this case the 11-step) is responsible
for the isomeric relation almost everytime. If there are none of
these, then I find a kind of "hidden tritone" relationship.

PGH