4D lattices

Paul Hahn wrote,

> 10/9 --- 5/3 --- 5/4
> / \ / \10/7 / \
> / \ / \ / \
> /14/9 \ / 7/6 \ / 7/4 \
>16/9 --- 4/3 --- 1/1 --- 3/2 --- 9/8
> \ 8/7 / \12/7 / \ 9/7 /
> \ / \ / \ /
> \ / 7/5 \ / \ /
> 8/5 --- 6/5 --- 9/5
>
>This is a projection of part of the 3D 7-limit lattice into a plane.
>The triangles are in the 3-5 plane, and the ratios inside the triads are
>either one layer above or below depending on whether they're inside
>major or minor triads. This shows all pitches separated from the 1/1 by
>intervals I consider consonant or primary within the 9-limit; however,
>without drawing lines which intersect and become confusing I can't
>draw lines for all those intervals, even when I'm not restricted to
>ASCII art. Several of the 9-limit intervals look like secondaries
>instead of primaries when represented this way. In 4 dimensions it
>would look like this:
>
> 5/3 --- 5/4
> / \10/7 / \
> 10/9 / \ / \ 3/2 --- 9/8
> / \ / 7/6 \ / 7/4 \ \ 9/7 /
> / \ 4/3 --- 1/1 --- 3/2 \ /
> /14/9 \ \ 8/7 / \12/7 / \ /
>16/9 --- 4/3 \ / \ / 9/5
> \ / 7/5 \ /
> 8/5 --- 6/5

This pitch set is just the 9-limit Partch diamond, by the way. There are
better ways of visualizing 4- and higher dimensional lattices, though,
but you'll have to use your imagination since ASCII isn't up to the
task. John Chalmers sent me some lattice diagrams of pitch sets where
there are four axes representing the prime numbers 3, 5, 7, and 11. The
diagrams were developed by Erv Wilson. Essentially, these are
projections of 4-d space onto two dimensions along a very special
direction.

The 3-d diagrams above (which are also used by Wilson) show the basic
tetrad with three pitches around an equilateral triangle and the fourth
pitch in the center. The fact that its six intervals are represented by
lines in six different directions helps one construct fairly complex
structures without pitches overlapping in the two-dimensional
projection. However, three steps of 7/4 (343/256) will lead to a pitch
at the same 2-d location as one step of 3/2 and one step of 5/4 (15/8).
So certain structures, like Wilson's stellated hexany, are better
represented with slightly irregular triangles.

The 4-d diagrams show the basic pentad with the five pitches around an
equilateral pentagon. Its ten intervals are represented by lines in only
five different directions. Complex structures will have many overlapping
lines, and it will become difficult to see at a glance which pitches are
really connected and which just happen to coincide with a pre-existing
line. However, since the diagonal of an equilateral pentagon is the
golden ratio times the side, new pitches along a given line will always
divide the distance between existing pitches on that line in the golden
ratio. The golden ratio is of course the only ratio that has this
iterative property. So the magic of the pentagon ensures that one can
construct very complex lattices in 4-d without worrying about the
pitches overlapping in 2-d. If one doesn't want the lines to overlap,
one can use slightly irregular pentagons.

When John sent me the diagrams, I started drawing my own to determine
whether the 4-d CPS scales fit together to tile 4-d space. They do. I
had a very intersting e-mail exchange with John and/or the Tuning list
at that time. Unfortunately, my e-mail rcords from 3/96 to 6/96, which
includes that stuff, got deleted.

For 5-d, Wilson uses an equilateral pentagon with one pitch in the
center . . .