Here is a web page on polyhexes, which are best understood by looking at them:

http://www.geocities.com/alclarke0/PolyPages/Polyhexes.html

Polyhexes are relevant to 5-limit connected scales, since if we surround each note-class on the lattice with a hexagon, we get a polyhex. Since tempered scales are often images under a mapping of

5-limit just scales, this is also relevant to large numbers of tempered scales, in particular those associated to a planar temperament.

Since a scale has an orientation (pointing along the 3/2 axis, the 5/3 axis, or the 8/5 axis being distinct), there are even more of them than there are polyhexes. On the other hand, if we add conditions, such as being epimorphic, we cut that down again. If we take hn for some n in the 5-limit, we can color the lattice with n different colors. An n-polyhex lying in such a way that it covers all of the colors corresponds to an epimorphic and connected scale. It might be interesting to see what these colored lattices look like, if Joe or Paul want to take a shot at it.

The number of polyhexes for each n for the first few n are

1: 1

2: 1

3: 3

4: 7

5: 22

6: 82

7: 333

8: 1448

9: 6572

10: 30490

11: 143552

This is according to information on the page