Full Lp-TOP algorithm for subgroup temperaments

This was a joint collaboration between myself and Keenan on IRC.

Theorem: the Lp-TOP tuning for a subgroup temperament S is the
restriction of the full-limit Lp-TOP tuning of the full-limit
temperament T with the same kernel as S to the subgroup in question.

Proof:
A subgroup temperament will be represented by a subspace of the space
of svals. Each sval supporting this temperament has a coset of vals
mapping to it under what I've been calling the V-map for the subgroup,
so the union of all of these cosets is a higher-dimensional subspace
of val space. This higher-dimensional subspace is going to be a
higher-rank full-limit temperament, but its kernel will be the same as
the subgroup temperament's kernel. We'll call this the "completion
temperament" for the subgroup temperament in question.

Define an "error map" for a temperament as a linear functional that
can be expressed as the difference between a tuning map supporting
that temperament and the JIP. If the temperament exists in some
subgroup, then we will refer to the JIP for that subgroup as the sJIP.
To find the Lp-TOP tuning for any subgroup temperament in the space of
svals, we simply need to find the subgroup tuning map sT for which the
error map sE, defined as sT - sJIP, is such that ||sE|| is minimal.

Since sE itself is a linear functional, it will have a coset of
full-limit linear functionals that map to it under the V-map. ||sE||
will then by given by the minimum of the norms of the linear
functionals in this coset. Call the error map in this coset with
minimum norm E. However, note that E isn't just the error map in this
coset with minimum norm, but that E is the error map which has minimal
norm over the entire set of error maps for the completion temperament;
there cannot be another error map for the completion temperament with
a smaller norm than E. If there were such error map F so that ||F|| <
||E||, then the subgroup tuning map sF that F maps to under the V-map
would have ||sF|| < ||sE||, but ||sE|| has been defined to be minimal,
a contradiction.

To turn E back into a tuning map T, we can add anything in the
preimage of the sJIP to it. Then, to turn T back into sT, we simply
apply the V-map. Note that, no matter which vector in the preimage of
the sJIP you add to E to obtain your tuning map T, every such T
obtainable in this way will map to the same sT. Since one such vector
in the preimage of the sJIP is the actual full-limit JIP, we can
simply use the actual JIP for this purpose, and say that T = E + JIP.
T is therefore the tuning map in the completion temperament which is
closest to the full-limit JIP, and it is guaranteed to map to the
Lp-TOP subgroup tuning map sT, which proves our theorem.

-Mike

On Fri, Jul 20, 2012 at 9:21 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> Theorem: the Lp-TOP tuning for a subgroup temperament S is the
> restriction of the full-limit Lp-TOP tuning of the full-limit
> temperament T with the same kernel as S to the subgroup in question.

In general, here's the algorithm for computing Lp-TOP on arbitrary
subgroups for arbitrary choice of p:

1) pick subgroup temperament
2) get kernel
3) turn subgroup kernel into full-limit kernel
4) get set of full-limit vals and tuning maps tempering out full-limit kernel
5) find the full-limit tuning map with least Lp distance to the full-limit JIP
6) multiply this tuning map by the V-map to convert it into a subgroup
tuning map again

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> In general, here's the algorithm for computing Lp-TOP on arbitrary
> subgroups for arbitrary choice of p:

Looks like I may need to rewrite the Lp tuning article.

On Fri, Jul 20, 2012 at 11:15 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > In general, here's the algorithm for computing Lp-TOP on arbitrary
> > subgroups for arbitrary choice of p:
>
> Looks like I may need to rewrite the Lp tuning article.

There's a lot more coming too, especially if you can figure out a way
to prove the conjecture I just posted.

One thing I wish we could tighten up a bit is the terminology though;
I get confused sometimes when working on this stuff. I think these 4
things should have different names

1) unweighted Lp norm on monzos
2) weighted Lp norm on monzos
3) associated unweighted Lp optimal tuning for a temperament
4) associated weighted Lp optimal tuning for a temperament

The first thing is already just called the Lp norm, which is fine.
The second one is currently called nothing, but examples of it are
called Tenney Height and the Tenney-Euclidean norm.
The third thing has no name, but an example of it is the Frobenius
tuning for L2. I'd like to study these more.
I dunno what the fourth one is called; Gene's calling it the Lp tuning
and I'm calling it the Lp-TOP tuning.

Can we sort the naming out somehow? How about: the first is the Lp
norm, the second one is the Tenney-Lp norm, the third is... I dunno,
OP-Lp? And the fourth is TOP-Lp? Or if you don't like that, anything
sensible works.

-Mike