Projective Real Plane etc

For Paul E -

So I finally have a little time to study Projective Geometry - (The
links Gene posted). Here's what I was thinking -

For group theory, I use this grid and study various symmetries:

0 3 6 9
4 7 10 1
8 11 2 5

The edges wrap around normally, so it's a torus as you said.

Now one could define this grid based on 3^a by 5^b, (translated to
12-tET pitch classes)

0 7 2 9
4 11 6 1
8 3 10 5

The columns wrap around normally, but the rows have to stagger down
a row at the right end to wrap back to the first column.

I know a projective plane does the twist thing in both columns
and rows, so I don't even know how that would apply in this situation.
A Klein bottle does the twist thing in only one dimension, but still
not the same. (This reminds me of the Map of all Chords seminar. I
know that had the Moebius twist...)

Notice that the second column moves up one and the third column
down one. With a little adjustment could one apply group-theoretical
symmetries to this grid (and thus relate tuning considerations
into group theory?)

So feel free to fill me in on theory I am missing. Thanks!

Oh, and you said that wedge products = {something else} in terms
of abstract-algebraic theory.

Oh, and lastly, could you give me that link that explains
Grassmanian Algebra?

Regards,

Paul Hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
>
> For Paul E -
>
> So I finally have a little time to study Projective Geometry - (The
> links Gene posted). Here's what I was thinking -
>
> For group theory, I use this grid and study various symmetries:
>
> 0 3 6 9
> 4 7 10 1
> 8 11 2 5
>
> The edges wrap around normally, so it's a torus as you said.
>
> Now one could define this grid based on 3^a by 5^b, (translated to
> 12-tET pitch classes)
>
> 0 7 2 9
> 4 11 6 1
> 8 3 10 5
>
> The columns wrap around normally, but the rows have to stagger down
> a row at the right end to wrap back to the first column.
>
> I know a projective plane does the twist thing in both columns
> and rows, so I don't even know how that would apply in this
situation.
> A Klein bottle does the twist thing in only one dimension, but still
> not the same. (This reminds me of the Map of all Chords seminar. I
> know that had the Moebius twist...)
>
> Notice that the second column moves up one and the third column
> down one. With a little adjustment could one apply group-theoretical
> symmetries to this grid (and thus relate tuning considerations
> into group theory?)

Of course another way to think about it would be (5)^a by (3/5)^b
You can split the rows into 2 by 2 and define (3/5)^2 = (7/5),
octave-equivalence being assumed throughout. Then you could go to
three dimensions, but its a pretty weird structure.

>
> So feel free to fill me in on theory I am missing. Thanks!
>
> Oh, and you said that wedge products = {something else} in terms
> of abstract-algebraic theory.
>
> Oh, and lastly, could you give me that link that explains
> Grassmanian Algebra?
>
> Regards,
>
> Paul Hj
>