Some more tweaks to spectral complexity

On Mon, Jun 27, 2011 at 8:20 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> Or, since using the spectrum might pose a few problems, consider the
> non-baby version - let Mi represent some monzo in the n-limit, and let
> Vp and Vg represent the temperament's period and generator vals:
>
> sum_i(1/(<Vg|Mi>*||Mi||)) over all monzos i in the entire lattice, or
> the n-limit tonality diamond if the lattice scares you

For a fractional-octave scale with p periods per octave, we note that
every successive iteration of the generator will end up yielding p
additional octave-equivalent notes in the temperament. That is, while
an additional generator in meantone yields one extra note in the
temperament, in injera it yields two. Fractional-octave period scales
thus come with a price. So let Me be the monzo denoting your choice of
equivalence interval, and Vp be the period val in the reduced mapping.
<Vp|Me> yields the number of periods per equivalence interval, so we
can multiply the aforementioned expression by that:

<Vp|Me> * sum_i(1/(<Vg|Mi>*||Mi||))

As a final note, it may be wise to consider the log of temperamental
complexity, rather than temperamental complexity itself - it's
definitely more noticeable if a temperament moves from 7 to 17 notes
than if it moves from 50 to 57 notes. This gives us the current
expression

log(<Vp|Me>) * sum_i (1/(log(<Vg|Mi>)*||Mi||))

This suggests a few ways to extend to higher rank temperaments, but
I'll leave it alone for now. My biggest problem is actually solving
this for "all monzos." Does anyone have any suggestions on how to do
that? Should I run PSLQ at every iteration of this summation to find
all of the solutions for <Vg|Mi> = i?

-Mike

Mike wrote:
>
>log(<Vp|Me>) * sum_i (1/(log(<Vg|Mi>)*||Mi||))
>
>This suggests a few ways to extend to higher rank temperaments, but
>I'll leave it alone for now. My biggest problem is actually solving
>this for "all monzos." Does anyone have any suggestions on how to do
>that? Should I run PSLQ at every iteration of this summation to find
>all of the solutions for <Vg|Mi> = i?

I think a reasonable complexity should at least converge
as more monzos are considered. You might check the behavior
of yours with an expanding origin-centered ball on the lattice.

-Carl

On Fri, Jul 1, 2011 at 4:45 PM, Carl Lumma <carl@lumma.org> wrote:
>
> I think a reasonable complexity should at least converge
> as more monzos are considered. You might check the behavior
> of yours with an expanding origin-centered ball on the lattice.

I guess I'll start by just testing all of the primes, which by the way
might itself work well enough to end this whole discussion there. I
sure hope it does. I won't be able to work it out until later though.

-Mike

On Sat, Jul 2, 2011 at 2:13 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
> On Fri, Jul 1, 2011 at 4:45 PM, Carl Lumma <carl@lumma.org> wrote:
>>
>> I think a reasonable complexity should at least converge
>> as more monzos are considered. You might check the behavior
>> of yours with an expanding origin-centered ball on the lattice.
>
> I guess I'll start by just testing all of the primes, which by the way
> might itself work well enough to end this whole discussion there. I
> sure hope it does. I won't be able to work it out until later though.

A more robust approach might be to consider that a temperament is a
projective subspace of the original JI lattice. Hence a point in the
temperament de-projects to a subspace in the original JI lattice of
dimensionality equal to the codimension of the temperament. If we had
a way to work out the "collective complexity" of this hypervolume all
at once, then we could really solve this problem from the ground up
and tie it back into the usual multilinear algebra approach. I feel
like I'm stumbling onto exterior algebra here... is this part of what
wedge products are all about?

I would love to just get the hypervolume of the subspace as measured
by the L2 norm, which would make things incredibly easy, except the
hypervolume is infinite.

-Mike

>A more robust approach might be to consider that a temperament is a
>projective subspace of the original JI lattice. Hence a point in the
>temperament de-projects to a subspace in the original JI lattice of
>dimensionality equal to the codimension of the temperament. If we had
>a way to work out the "collective complexity" of this hypervolume all
>at once, then we could really solve this problem from the ground up
>and tie it back into the usual multilinear algebra approach.

This is the kind of thing TE complexity - and other complexities
discussed on the wiki - do. It doesn't solve the prime-limit
problem though because you map primes as you add them and this
changes the picture.

-Carl