--- In tuning-math@yahoogroups.com, "Paul H" <phjelmstad@...> wrote:
>
> p. 154: Projective Planes and Difference Sets
>
> This Chapter is exciting to me for a couple reasons. They discuss
> 7-tET, 13t-ET and 31-tET. 43-tET unfortunately is not a projective plane,
although 43 is k^2-k+1 when k=7.
>
> 1. Difference Sets in these projective planes have geometric duals.
> And these duals of difference sets end up being the inversion of the
> first difference set! So here we have a completely different application of
duals (where points and lines are interchanged) from
> the kind discussed here more often (between commas and vals).
>
> The theme of this chapter of the book here is finding what sets are fruitful
for composition, and which temperaments, etc.
>
> A FLID is a Difference Set, where every interval is used once.
> 12-tET only has one, a tetrachord and its Z-relation (111111)
> which is (0,1,4,7) and (0,1,3,6) in canonical form. This is
> the "first Z-relation" in 12-tET. Pentachords have only 3 of them,
> and Hexachords have 15, which are also complementary sets.
>
>
> 2. I was excited to see them use the 13-point plane (13 points and
> 13 lines in the 26-node diagram) for 13-tET and its difference sets.
> This is exciting because it is the same diagram used for "An Elementary
Approach to the Monster" in constructing the Bi-Monster (Y555 or "M666", the
Beast Group)which connects down to the Monster by
> means of M X M. (Subgroup of M | 2, Wreath group of the Monster (Bi-Monster))
This also relates to Conway's M13 game, etc.
>
> There is a composer who has composed a piece based on the Monster group.
>
> Definitions:
>
> FLID: Flat line Interval Distribution.
>
> Difference Set: When all the differences in a chord, are...different!
>
> Z-relation: When 2 or more sets of different Tn/TnI type have
> the same interval vector.
>
> * * *
>
> I have mapped the Z-relation for 31-tET chords, but stopped at
> hexachords because the numbers and patterns became overwhelming.
> But there are definite patterns. I am not yet certain of the
> relationship between Z-related sets, difference sets, and
> the affine relation. They might be 3 completely independent
> ideas? I do know that the Z-relation "carries through" in
> the affine relationship. And that (more obvious) the affine
> relationship also preserves difference sets, therefore I guess
> it is safe to say that:
>
> "A FLID or Difference Set operated upon by the Affine Group
> produces a Z-related Set (Isomeric Set) or is just trivial
> (Inversion, or Transposition of the Set or both (Same Tn/TnI Type).
>
> So in this case all three properties are related. Somehow this
> relates Affine and Projective geometry too.
>
> PGH

A better way to state this would be to say:

A Z-related difference set (pair, or triple, or whatever) operated
upon by the affine group action will produce another Z-related difference set
(along with its pair or triple, or whatever)

Any Z-related collection of sets operated upon by the affine
group will produce another Z-related collection, doesn't have to
be a difference set. If it is not, you will get a scrambled interval
vector.

Any difference set operated upon by the affine action will create
another difference set, which will be Z-related if the first set
(collection) is, or will not be if the first set is not.

PGH