Lowest-badness transversals

I'm hopelessly behind on tuning-math stuff that I'm supposed to do,
but since this is a really quick, simple, and useful idea, I'll post
it up anyway. I made a post about this before in the context of
mavila, but I'll be more to the point here.

Some people hate to call the porcupine generator 10/9, because it's 20
cents flat. However, 11/10 also maps to the porcupine generator; the
benefit of viewing the porcupine generator this way is that 11/10 is
still fairly low in complexity, but now the tempered approximation is
much more accurate, containing only about 5 cents of optimal error on
it. The only reason to pick 10/9 is that it's the JI interval mapping
to the porcupine generator which is lowest in complexity, but taking
tempered error into account can be useful as well for this reason.

This suggests a new sort of transversal, where for any tempered
interval, we account for not only the JI complexity of each interval
in its preimage, but account also for tempered error. This implies the
construction of some sort of a "tempered interval badness" function
which takes both of those two parameters into account; we then pick
the JI interval which is optimal with respect to this function instead
of complexity.

I don't feel like this is an insurmountable task; in fact, it's
probably pretty simple. Has anyone thought about how to do this, or
have any ideas about how to proceed? Here are some thoughts I have
about it:

1) The tempered error of any interval is dependent on the specific
tuning map we're using. To use this technique to obtain transversals
for abstract temperaments in general, we can just go with any ol'
optimal tuning.
2) It's probably simplest to include a free parameter, a la Cangwu badness.
3) If you really want to get algebraic about it, there's probably some
nice way to do this by playing with the dual norm to the Cangwu
badness norm, though I have no idea what it is.

Thoughts?

-Mike

Mike Battaglia <battaglia01@gmail.com> wrote:

> 3) If you really want to get algebraic
> about it, there's probably some nice way to do this by
> playing with the dual norm to the Cangwu badness norm,
> though I have no idea what it is.

Wouldn't the dual norm give each interval a badness that
you can minimize?

Graham

On Fri, Jan 18, 2013 at 2:20 PM, Graham Breed <gbreed@gmail.com> wrote:
>
> Mike Battaglia battaglia01@gmail.com> wrote:
>
> > 3) If you really want to get algebraic
> > about it, there's probably some nice way to do this by
> > playing with the dual norm to the Cangwu badness norm,
> > though I have no idea what it is.
>
> Wouldn't the dual norm give each interval a badness that
> you can minimize?
>
> Graham

Yes. I'm suggesting a modification of it.

I haven't worked it out precisely, but I think the Cangwu dual norm
will end up doing some sort of geometric thing that implicitly factors
in the JI size of the interval, and the complexity of the interval,
and then spins a badness out from that information. If complexity is
held constant, larger comma size yields more badness and vice versa.
More complexity and a smaller comma size should be roughly neutral.

The only way to evaluate "comma size" from an algebraic standpoint is
to send the comma to the JIP, which returns a scalar for every comma.
What happens if we instead send the interval to some other tuning map,
specifically one which gives you tempered interval error for some
temperament? So if we use some arbitrary error map E to do this, then
<E|M> for some monzo M no longer represents comma size, but the amount
of tempered error on M.

For instance, if E is the difference between the JIP and quarter-comma
meantone, then <E|-2 0 1> = 0, and <E|-1 1 0> = -5.377.

I was thinking you could then just replace the JIP with this tuning
map and run Cangwu badness exactly as you otherwise would. Now instead
of taking the JI size of the comma into account when factoring
badness, it's taking tempered interval error into account instead. The
thing it spits out will be some combination of JI complexity and
tempered interval error.

-Mike