There is a LOT more and I don't have time to write it all now, but just for interest: - We can identify regular temperaments as subspaces of \(\Bbb R^n\) intersecting the lattice \(G\) - We can also identify them as subspaces \(\Bbb R^{n*}\) intersecting the lattice of vals \(G*\); the corresponding vals "support" the temperament by tempering out the comma in question - We can stack vals vertically to form "mapping matrices" and put them in various normal forms, corresponding to reduced bases for the temperament - We can also use the exterior algebra to represent subspaces/pure subgroups in both the primal and dual spaces - We can look for the best tuning of a regular temperament; if we use Tenney-Euclidean height this is a simple least-squares problem - And much more There's some good theory here, much of it from Gene: https://legacy.xen.wiki/w/Mathematical_theory There's also an archive of Gene's older work: http://lumma.org/tuning/gws/home.htm. Some of the terminology was changed later on, and certain ways of doing things were improved on, but there is still a lot of good stuff there. Paul Erlich's "A Middle Path" paper (https://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf) is one of the most accessible introductions, though he also does things slightly differently and uses slightly different terminology. (4/4)