Tp tuning

So is Mike suggesting "Lp tuning" should be changed to "Tp tuning" on the Xenwiki or what, exactly?

On Sat, Aug 4, 2012 at 1:28 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> So is Mike suggesting "Lp tuning" should be changed to "Tp tuning" on the
> Xenwiki or what, exactly?

I'm working on it. Sorry, I'm going nuts here, I have so much crap to
do. Pulled an all nighter, have deadlines for my fledgling business
I'm way past overdue for, weekly Thursday gig got cancelled, blah
blah. It's a shitty week. :( I'm hoping to finish the article today.

Any vector space V has an associated V-map M describing the subgroup
it's supposed to represent. M can become a weighted V-map by
left-multiplying it with a weighting matrix W. Then, for any vector v
in V, we can define a function f_p(v) = ||W*V*v||_p, where it should
be noted that || . ||_p is usual the Lp-norm applied to the result of
that matrix multiplication. The function f_p then itself has the
structure of a norm on V, so we can call f_p(v) the Tp norm on v and
denote it ||v||_p or ||v||_Tp or ||v||_p^a.b/c.d/e or whatever.

IOW, the Tp norm is what I'm calling the thing that actually gives you
the correct weighted full-limit Lp norm for any interval, no matter
what subgroup you're on. So the T1 norm in the 2.5/3.9/7 subgroup of
|0 1 -2> is log(245*243).

Given this definition, the Tp norm reduces to the usual Lp norm in cases
a) when the subgroup is generated by only primes and prime powers, and
b) when you cheat and weight all your coordinates from the getgo

A is a condition I want to make really explicit. In cases like
2.9.21.13/11 or whatever, the norm on the space is not an Lp norm in
any coherent sense of the word, but rather an Lp norm is just used
halfway through the process as an intermediate step in calculating the
actual thing you want.

Yes, I think that the T1, T2, Tinf tunings would be a good name. I'd
be happy with the compromise and drop the whole Lp-TOP thing if you're
alright with that. Tp-TOP would be stupid anyway.

-Mike