semi-regular tilings

In 1999, I thought Archimedean tilings might offer a nice
way to look at tonespace. I came up with:
http://lumma.org/stuff/4.8.8.png

There's enough room here for the 9-limit, even though it's
more spread out (squares may not be the roots of chords,
for instance) and/or I'm not showing all of it (can anyone
say whether this tiling is complete?).

Being a keyboardist, I always thought this would make a
nice keyboard. Perhaps, in the vein of the chord buttons
on an accordion, pressing an octahedron might cause its
squares to play automatically.

If you prefer a lattice, you can take the dual of this.
Now the short links (squares) represent consonances while
the long links (diagonals) represent root motions that
preserve a common dyad.

Recently I thought I'd take a look at other Archimedean
tilings. The choices are shown here (Wikipedia):
http://tinyurl.com/34hn3s

Considering only the tilings with two different polygons
(one for roots and another for non-roots), we rule out
3.4.6.4 and 4.6.12.

Next, to have something compact (a sphere of given radius
contain the greatest possible number of consonant n-ads),
we should make the modulation between consonances in JI
that involves the most common tones as short as possible.
ASSs aside, JI admits to at most common-dyad modulation,
which occurs between otonal and utonal chords. Therefore
we want a tiling whose dual contains two basic kinds of
line segment: one for the common-dyad-preserving modulations
and another for the consonances themselves. Furthermore,
there can be no cycles of the modulation segment with an
odd number of steps, since the quality of the chords must
alternate between otonal and utonal.

Given these constraints, I believe we are left with:
4.8.8 (see above)
3.6.3.6 or 3.12.12 ( http://lumma.org/stuff/3.6.3.6.png )

In the 4.8.8 mapping, the consonances (length 1) are shorter
than the modulations (length sqrt(2)). In 3.6.3.6 they are
apparently the same length.

Other ways of mapping tonespace to these are possible.
For instance, 4.8.8 can be done with primes:
http://lumma.org/stuff/4.8.8-primes.png

I suppose an open question is: do temperament or ASS-based
mappings allow some of the other tilings to be used? For
example, if hexagons (rather than triangles) were roots in
3.6.3.6, we'd have modulation cycles of length 3. I doubt
any temperament that could achieve this would be very
accurate. . . .

-Carl