Looped distorted lattices

There has been some interest expressed recently in representing ET's and
other non-JI scales as distorted lattices.

Some years ago I mapped 31-tET onto the surface of a toroid as a 5-limit
lattice. You can print out the lattice below (in a monospaced font), rule
your own lines in 3 colours to triangulate it, cut out the rectangle
(cutting a half character width or height inside the lines), loop and tape
it first side to side and then top to bottom. Unfortunately you have to
flatten it after the first looping to get it to loop in the other
dimension, unless you can print it on rubber. :-)

It seems like magic that what is apparently a 4 x 8 grid ends up having
only 31 members.

-------------------------------
| Gx |
| Cx |
| Fx |
| B# |
| E# |
| A# |
| D# |
| G# |
| C# |
| F# |
| B |
| E |
| A |
|D |
| G |
| C |
| F |
|b B|
| Eb |
| Ab |
| Db |
| Gb|
| Cb |
| Fb |
| Bbb |
| Ebb|
| Abb |
| Dbb |
| Gbb |
| Ax |
| Ex |
-------------------------------

Now what I'd love to see is a perspective drawing of of this as if on a
transparent torus floating in space, with the edges (rungs) of the
triangular lattice having thickness that diminishes with distance. Anyone
volunteering? It could, with some difficulty, be engraved on a clear
plastic torus, but where do you get one of those? Or someone with access to
fancy CAD software may be able to do it.

Now how do we visualise the corresponding 3-dimensional 7-limit lattice (so
every note only appears once)? It seems like it has to be looped around in
a fourth dimension. What would be a good 3D projection of it that could
then be rendered in 2D-perspective?
-- Dave Keenan
http://dkeenan.com

> I feel that your lattices would be improved if
> <snip>...(b) you chose angles in accordance with
> Canright's procedure so that the pitch order would
> appear naturally on one axis.

The pitch order on my lattices appears naturally
on the axis you can't see yet: the front-to-back axis. :)
I'm working on being able to rotate the view to see that.

Also, I've just recently drawn some 'octave'-specific
lattices using my formula, and they came out a little
strange. Partch's 11-limit diamond looks really wacky.
I'm experimenting with lots of different lattice ideas
right now.

-monzo

Joseph L. Monzo....................monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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