Finite-dimensional subgroup temperaments as equivalence classes of no-limit temperaments

A random interesting thing I was thinking about earlier...

Subgroup mapping matrices are dual to what Graham's calling mapping
matrices: they're matrices in which the columns are monzos, and which
take in vals or mappings from the left and spit in svals or subgroup
mappings on the right. For example, here's a subgroup mapping matrix S
which converts 2.3.5.7 to 2.3.7/5:

[1 0 0]
[0 1 0]
[0 0 -1]
[0 0 1]

The monzos in this matrix, which are the columns are indeed 2/1, 3/1,
and 7/5 respectively. Note what happens if we multiply now with a val
on the left: <12 19 28 34| * S = <12 19 6|, which is the restriction
of <12 19 28 34| onto the 2.3.7/5 subgroup. We can also multiply
entire mapping matrices this way. For instance, here's Pajara

[2 3 5 6]
[0 1 -2 -2]

this multiples the above matrix to become

[2 3 1]
[0 1 0]

which is the restriction of the pajara mapping to the 2.3.7/5 subgroup.

Interestingly, the 2.3.7/5 subgroup matrix has a nontrivial left null
space, which is span{<0 0 1 1|}. Therefore, much like a tempered
interval can be thought of as defining an affine subspace of interval
space, a subgroup val can be thought of as defining an affine subspace
of val space (!). This means that we can add any multiple of this val
to any row of the original pajara matrix and get the same result. For
instance, this matrix

[2 3 8 9]
[0 1 -9 -9]

also maps to [<2 3 1|, <0 1 0|], giving us 2.3.7/5-limit pajara, even
though the original matrix wasn't pajara at all.

I'm not really sure what it all means yet, but one immediately
interesting consequence is that we can think of vals on a finite JI
group as being equivalence classes of no-limit vals, and wedgies on a
finite JI group as being equivalence classes of no-limit wedgies. Food
for thought for now.

-Mike

On Mon, Jun 18, 2012 at 5:16 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> I'm not really sure what it all means yet, but one immediately
> interesting consequence is that we can think of vals on a finite JI
> group as being equivalence classes of no-limit vals, and wedgies on a
> finite JI group as being equivalence classes of no-limit wedgies. Food
> for thought for now.

One really interesting and strange thing to look at is the interior
product of a multival and a multimonzo, where the multival represents
a temperament and the multimonzo represents a subgroup. For this to
make any sense at all, you need the grade of the multimonzo to be
greater than or equal to the grade of the multival, or else you're
trying to project a rank-n temperament onto a rank<n subgroup, which
would be bad. Applying a multimonzo of grade n to a multival of grade
<= means you get either a scalar representing how contorted the
temperament is on the subgroup in question, or a multimonzo left over.

For instance, keeping with our example where we project 2.3.5.7 pajara
onto the 2.3.7/5 subgroup, <<2 -4 -4 -11 -12 2|| \/ |||-1 1 0 0>>> =
|-23 8 2 2>

Interpreting the resultant multimonzo, which in this example was
simply a monzo, is really strange. You can get some intuition for what
it means by considering applying a monzo to a multival: for instance,
if you take <<1 4 4|| \/ |-3 -1 2>, you get <7 11 16| left over. This
is the result of adding |-3 -1 2> to the kernel of <<1 4 4||. Or, in
other words, if you start with meantone and additionally temper out
25/24, you get <7 11 16| as a result.

Likewise, in this case, we're adding <<2 -4 -4 -11 -12 2|| to the
kernel of the subgroup which is |||-1 1 0 0>>>, and as a result we get
the subgroup defined by span{|-23 8 2 2>}. Again, this all depends on
the notion that subgroups are linear transformations -on val space-
with nontrivial kernel.

Weird stuff.

-Mike

On Mon, Jun 18, 2012 at 6:02 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> Likewise, in this case, we're adding <<2 -4 -4 -11 -12 2|| to the
> kernel of the subgroup which is |||-1 1 0 0>>>, and as a result we get
> the subgroup defined by span{|-23 8 2 2>}. Again, this all depends on
> the notion that subgroups are linear transformations -on val space-
> with nontrivial kernel.
>
> Weird stuff.

It also just occurred to me that, in general, exterior algebra is a
spectacularly bad way to deal with this stuff. While the whole idea
behind subgroup mapping matrices is sound, but you can't make the leap
to representing subgroups with multimonzos like you can make the leap
to representing temperaments with multivals.

For instance, if you're in 5-limit JI, then the multimonzos for 4.3.5,
2.9.5, and 2.3.25 are all the same thing, which is the trimonzo
|||2>>>. Exterior algebra can't distinguish between these different
subgroups in the same sense that it also can't distinguish between the
different contorted versions of the same mapping matrix.

For temperaments, which are linear transformations on interval space,
you can mostly handle this problem by sweeping contorsion under the
rug and just not dealing with it. But for subgroups, which are linear
transformations on val space, you can't do the same thing as long as
you want 2.9.5 and 2.3.25 to remain distinct from one another.

-Mike