Improved tunings and error measurements for subgroups

I mentioned a while back that I thought the definition I had been using for tunings and error on subgroups was ad hoc and not entirely satisfactory. I've introduced a corrected version here:

http://xenharmonic.wikispaces.com/Lp+tuning

The basic premise is that we should start from norms on smonzos and not svals. If we have the Lp norm on monzos, we get a norm on subgroup monzos by restriction; if we try the same for vals, the result may or may not make sense. Once we have a norm on the smonzos of a subgroup, we get the dual norm on 1he svals in the usual way, which requires that we look at unit balls of monzos not in the full p-limit but in the subspace spanned by the subgroup. For L2 (Eucliean) norms, instead of the slick method using pseudoinverses I was reduced to using Lagrange multipliers; however it all works.

This is wonderful, thanks.

It looks like the gencom page could be fleshed out some more.

I would also be interesting to see a comparison of this new
procedure to Graham's method of handling subgroups.

-Carl

Gene wrote:

>I mentioned a while back that I thought the definition I had been
>using for tunings and error on subgroups was ad hoc and not entirely
>satisfactory. I've introduced a corrected version here:
>
> http://xenharmonic.wikispaces.com/Lp+tuning
>
>The basic premise is that we should start from norms on smonzos and
>not svals. If we have the Lp norm on monzos, we get a norm on subgroup
>monzos by restriction; if we try the same for vals, the result may or
>may not make sense. Once we have a norm on the smonzos of a subgroup,
>we get the dual norm on 1he svals in the usual way, which requires
>that we look at unit balls of monzos not in the full p-limit but in
>the subspace spanned by the subgroup. For L2 (Eucliean) norms, instead
>of the slick method using pseudoinverses I was reduced to using
>Lagrange multipliers; however it all works.
>