"Generator lattices" (or how should we call them?)

This is a very useful kind of diagram if you're not only interested in making music in a single regular temperament / tuning, but if you want to freely modulate between different temperaments / tunings within a desired EDO (or EDT etc.).

The main idea only works for prime-numbered equal divisions, like 11, 13, 17, 19, 23, 29, 31 and so on, but it could be generalized to other equal divisions as well.
In the following, octave equivalency is assumed, i.e. I won't distinguish between a major ninth (whole tone + octave) and a major second (a whole tone).

Ok, here's the basic idea: If we stack fifths, we get something like this: F C G D A E B
This sequence can be split into two sequences of stacked whole tones:
F G A B, and C D E

So there's a clear relationship between linear tunings whose generators differ by a factor of 2, and it's easy to see that modulations work well between those. In addition, if we start with a MOS with an odd number of notes, we can modulate simply by applying chromas to every other note in our generator chain:
F C# G D# A E# B -> F G A B C# D# E#

Of course, the same principle works the other way around, and we can use this to modulate between tunings that are generated by stacked neutral thirds and fifths, respectively. Also, tunings whose generators differ by a factor of 3 are somewhat similar, too, though the relationship is not as strong.

So if we're interested in those relationships, it makes some sense to organize our generators by grouping them s.th. generators that differ by a small factor are adjacent.

For example, if we're in 31-EDO, we can organize all 15 generator pairs as following:
(generators are doubled from left to right, and tripled from top to bottom; [] indicate how the block repeats)

(13,18) (26, 5) (21,10) (11,20) (22, 9) [13,18]
( 8,23) (16,15) ( 1,30) ( 2,29) ( 4,27) [ 8,23]
(24, 7) (17,14) ( 3,28) ( 6,25) (12,19) [24, 7]
[10,21] [20,11] [ 9,22] [18,13] [ 5,26]

...so this is what I'd call a "generator lattice". If you have a better name, or there's already one, please tell me. ;)

Note that if we don't have a prime-numbered EDO, some generator pairs can be halfed or tripled in multiple ways, while others can't at all, so you won't get a single complete homogenous lattice there.

I made a nice implementation for 31-EDO with two paper sheet (a data sheet and a context sheet), you can read more about it here:
http://xenguitarist.com/index.php?PHPSESSID=fGBU13rCHvSMAUJR7hziS0&topic=434.0

Best
- Gedankenwelt