Mathematics and Music - John Fauvel

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

Difference Set: Sorry, when all Differences are the Same. The other
case, such as the Diatonic Collection <1,4,3,2,5,0> is another matter.

--- 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
>

--- 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