Smallest Perfect Universe

The smallest perfect universe is the Projective Geometry (3,2). It
has 3 dimensions, and 2 indicates the incidence number. It has 15
"points" 35 "lines" and 15 "hyperplanes."

My latest project is mapping the 35 partitions of the Interzone (hexachords with one or two tritones) onto these lines. As it works
out, 15 hexachord partitions (Lower, with two tritones) map to
the 15 lines of "Picture A" in Cullinane's illustration (which is
the part analogous with 15 points, and 15 hyperplanes). The remaining
20 hexachords (Upper, with one tritone) map to Pictures B and C
with 10 and 10 lines from PG(3,2)

This involves using the Exceptional Outer Automorphism of S6,
directly for Picture A, and using three copies for each line
of Pictures B and C, (that is, C(6(2,2,2)3!) -> C(6(3)) by
way of mapping C(6,2) -> C(6,(2,2,2)3! in Picture A and C(6,3)->
C(6,3) in Pictures B and C.

In my construction, which is fairly involved, we consider
hexagram bars for tritones, called C and D, (for 0 6 and {} on a line)
A and B correspond to non-tritone lines, A being on the left
(0) and B on the right (6) for each bar.

This maps perfectly to the 15 partitions, so for example,
CCDDAA would be based on one of C(6(2,2,2)3!) But I would need
a few pages of a paper to explain it. Briefly, one takes distances between C's and D's as the first two pairs of this structure. There is "checkering" so CC, DD distance "small" corresponds to AA and "large" to AB.

The other hexachords, map CDABAB for example to something like
(0 1 2) based on 0 1 for C,D distance, and 2 representing A B
boundary (AAAA, BAAA, BBAA, BBBA, BBBB) location. As with lower hexachords, there is "checkering" for "large" types so we map
AAAA->ABAB, AAAA->BBAB etc. This is just a rough overview but I will address any questions, and hope to post a paper to files section.

PGH

Basically, what it is, I've found a generic way to map any S(5,6,12) Steiner system to upper and lower interzone hexachords. These hexachords (hexads) are not themselves a Steiner system, not at all,
but a way to equate the seeds of any Steiner system (the lines of
PG(3,2)) with a complete set of chords, which could have transformational value when (someone) composes with Steiner systems.
Also these 70 (or 35) hexachords equate to representations used
in the construction of M24 also...leading to a quartertone system.

PGH

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
>
> The smallest perfect universe is the Projective Geometry (3,2). It
> has 3 dimensions, and 2 indicates the incidence number. It has 15
> "points" 35 "lines" and 15 "hyperplanes."
>
> My latest project is mapping the 35 partitions of the Interzone (hexachords with one or two tritones) onto these lines. As it works
> out, 15 hexachord partitions (Lower, with two tritones) map to
> the 15 lines of "Picture A" in Cullinane's illustration (which is
> the part analogous with 15 points, and 15 hyperplanes). The remaining
> 20 hexachords (Upper, with one tritone) map to Pictures B and C
> with 10 and 10 lines from PG(3,2)
>
> This involves using the Exceptional Outer Automorphism of S6,
> directly for Picture A, and using three copies for each line
> of Pictures B and C, (that is, C(6(2,2,2)3!) -> C(6(3)) by
> way of mapping C(6,2) -> C(6,(2,2,2)3! in Picture A and C(6,3)->
> C(6,3) in Pictures B and C.
>
> In my construction, which is fairly involved, we consider
> hexagram bars for tritones, called C and D, (for 0 6 and {} on a line)
> A and B correspond to non-tritone lines, A being on the left
> (0) and B on the right (6) for each bar.
>
> This maps perfectly to the 15 partitions, so for example,
> CCDDAA would be based on one of C(6(2,2,2)3!) But I would need
> a few pages of a paper to explain it. Briefly, one takes distances between C's and D's as the first two pairs of this structure. There is "checkering" so CC, DD distance "small" corresponds to AA and "large" to AB.
>
> The other hexachords, map CDABAB for example to something like
> (0 1 2) based on 0 1 for C,D distance, and 2 representing A B
> boundary (AAAA, BAAA, BBAA, BBBA, BBBB) location. As with lower hexachords, there is "checkering" for "large" types so we map
> AAAA->ABAB, AAAA->BBAB etc. This is just a rough overview but I will address any questions, and hope to post a paper to files section.
>
> PGH
>