The main idea I haven't seen you talk about is to then look at the group \(\text{Hom}(G, \Bbb Z)\) of homomorphisms from the JI group to the integers. If we think of \(G\) as being embedded in a real vector space \(\Bbb R^n\), then this group \(G*\) can be thought of as being embedded as a lattice in the dual space. This is called the lattice or group of "vals", short for "valuations" (this name is directly from Gene, FWIW). Each val represents a way of mapping the JI group onto a single rank-1 regular temperament. In general, given some basis for \(G\), we have a basis for \(G*\), and we can notate vals as bras. Equal temperaments are naturally vals. For instance, the so-called 5-limit "patent val" for 12-EDO, which we get by just independently rounding each prime to its nearest tempered equivalent, is \(\langle 12, 19, 28|\). Thus, we can write \(\langle 12, 19, 28|-4, 4, -1\rangle = 0\) which says that the patent val for 12-EDO tempers out the syntonic comma of 81/80 (and hence "supports" meantone temperament, in the sense that the kernel is a superset of meantone's kernel). This equation has basically powered virtually all of modern Western music theory for the last few hundred years, so you know, it's maybe slightly important. (3/N)