Wow! Idealized "perfect" higher-limit extensions of temperaments

There are two important seminorms which we can define for multivals:

TEerr(W) = ||W^J||/||W||
TEbad(W) = ||W^J||

where W is a multival. If you construct a V-map M sending multivals to
smultivals, there will thus be an induced quotient seminorm on
smultivals given by

TEerr(M(W)) = min TEerr(M^-1(M(W))) = ||M(W^J)||/||M(W)||
TEbad(M(W)) = min TEbad(M^-1(M(W))) = ||M(W^J)|| = ||M(W)^M(J)||

This becomes extremely meaningful if you work backwards from
smultivals to multivals, or from multivals to higher-limit multivals.
So for instance, say you have a rank-2 temperament T in the 2.3.5.7
group and you want to look at 2.3.5.7.11 rank-2 extensions T* of it.
Then there will be some extension T* which has the same TE error as T,
and some other extension T* which has the same TE badness as T (not
sure if these are the same).

Keep in mind I'm using the word "extension" to mean a temperament on a
higher-limit but of the same rank as T, e.g. not just a higher-rank
temperament with the same kernel. This is the "ideal extension" for T
given any larger subgroup. This paves the way for us looking at
different higher-limit extensions of temperaments and seeing how close
they are to ideal.

-Mike

On Sat, Jul 21, 2012 at 7:10 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> Keep in mind I'm using the word "extension" to mean a temperament on a
> higher-limit but of the same rank as T, e.g. not just a higher-rank
> temperament with the same kernel. This is the "ideal extension" for T
> given any larger subgroup. This paves the way for us looking at
> different higher-limit extensions of temperaments and seeing how close
> they are to ideal.

Another simple corollary to this is - we also just proved, in the
thread about TOP that the higher-rank expansion with the same kernel
also has the same error as T too.

So, for instance, there exists an 11-limit rank-2 extension of 3-limit
Pythagorean T* which has the same TE error as 11-limit JI, which is to
say it has 0 error. If it has 0 error, then ||T*^J||/||T*|| = 0, so
||T*^J|| must = 0, so T* must be able to be expressed as
JIP^something. In the case of Pythagorean, it's <1 0 0 0 0| ^ JIP.

The above approach gives us a way to obtain partly generalized
higher-dimensional analogues of the JIP, in that they're higher-rank
temperaments for which the optimal tuning map is the JIP (like the
rank-1 temperament specified by the JIP itself). The set of multivals
denoted by JIP^anything, which strangely form a Fokker group, are the
generalized JIP's in question. However, there's still a sense in which
they aren't exactly higher-dimensional analogues to the JIP, since
they're projective and the JIP isn't.

-Mike