Justifying restretched tunings

If you stretch to get some form of TOP, and then restretch to make octaves pure, the method certainly works well enough but the justification for it strikes me as murky. It seems to me that if you want to stretch, you should start out with an optimum which is defined projectively. The most obvious way to do that would be to minimize Graham's badness measure, | T/||T|| ^ J|, where T lives on some subspace of tunings as a regular temperament tuning. T/||T|| is a unit vector, and (identifying opposite points of the n-sphere in question) so represents a point in projective space. Having found an optimum tuning projectively, it now makes sense to stretch it.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

>The most obvious way to do that would be to minimize Graham's badness measure, | T/||T|| ^ J|, where T lives on some subspace of tunings as a regular temperament tuning. T/||T|| is a unit vector, and (identifying opposite points of the n-sphere in question) so represents a point in projective space. Having found an optimum tuning projectively, it now makes sense to stretch it.

I expounded my wisdom on this topic without ever calculating an example of the tuning, so I thought I'd do septimal meantone again. I get a fifth of size 696.4989638, an obviously crucial difference from Syphilitic Tuning Disorder, which comes in at 696.4936084.