Subgroup matrices, tmonzos and tvals, tmappings and smappings

Some new terminology:
tmonzo - element of a tempered interval space, short for "tempered monzo"
tval - linear functional on a tempered interval space, short for "tempered val"
tmapping - mapping matrix taking in monzos and outputting tmonzos, the
usual mapping matrix
smapping - mapping matrix taking in vals and outputting svals, the new
sort of mapping matrix I'll define now

This will all be explained further below.

Using the convention that monzos are columns and vals are rows, then
mapping matrices, or what I called "tmapping"s above, take in monzos
from the right and spit out tempered monzos, which are what I called
tmonzos. Tmappings take in a set of monzos, defining an equivalence
relation on them such that monzos are equivalent if they differ by an
element in the null space of the matrix, and spitting out a tmonzo,
which denotes what the monzos map to under the mapping. For instance,

[1 1 0]
[0 1 4]

takes in |0 1 0> and spits out |-1 1>, the former being a monzo and
the latter being a tmonzo. The things I called "tvals" are entries

Another interesting sort of matrix one in which the columns are
monzos, and which take in vals from the left. Such a matrix would
define an equivalence relation on vals in the same sense that
tempering defines an equivalence relation on monzos. For instance,
consider this mapping matrix, which is what I called an "smapping":

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

The monzos in this matrix represent 2/1 and 5/3, respectively.
Multiplying this by a val on the left, such as <12 19 28|, yields the
new val <12 9|, which you'll note is also the 2.5/3 mapping for 12p.
These matrices take in one or more vals, evaluate them at the monzos
which are the columns of the matrix, and output a sval on the
appropriate subgroup. Any set of vals will map to the same sval if
their mappings on the subgroup defined by the columns of the smatrix
are identical.

Furthermore, any nonsquare smapping will have a non-trivial left null
space, which defines an equivalence relation on vals. In the case of
the above matrix, the null space is <0 1 1|, which tells you that (for
instance, <12 19 28| and <12 20 29| map to the same thing under the
smapping, and they are indeed identical on the 2.5/3 subgroup.

Intuitively, one can think of the result of "tempering out a val" to
be a subgroup, in the same sense that "tempering out a monzo" leads to
a temperament.

-Mike

On Tue, Jun 12, 2012 at 5:20 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> Intuitively, one can think of the result of "tempering out a val" to
> be a subgroup, in the same sense that "tempering out a monzo" leads to
> a temperament.

Interestingly, for any tmapping matrix of full row rank, there will be
at least one smapping matrix which maps to the identity under the
tmapping. For instance, given the tmapping

[1 1 0]
[0 1 4]

then the smapping

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

Indeed maps to the identity under the tmapping above. This means that
the above tmapping takes monzos of the form k*|1 0 0> + l*|-1 1 0> and
maps them to |k l> under the tmapping.

While one might be tempted to say that the above tmapping thus
"represents the 2.3/2 subgroup," or "sends 2.3.5 monzos to 2.3/2
monzos," 2.3/2 is only one of infinitely many smappings which map to
the identity under the tmapping. For instance, here's another one

[1 3]
[0 -3]
[0 1]

This smapping, representing 2.40/27, also maps to the identity under
the above tmapping. In general, all smappings of the form
[(2/1)*(81/80)^n, (3/2)*(81/80)^m] will also map to the identity under
the above tmapping.

This is more than just a fancy way of saying that there's an infinite
number of sets of generators for meantone. In general, it also means
that while smappings are nice for taking in vals and spitting out
svals, there is in general no analogous mapping matrix which will send
monzos to smonzos - tmappings don't quite fit this bill (unless the
tmapping is square and nonsingular). This is because nonsquare
matrices have a nontrivial nullspace, and as such have an infinite set
of "inverse" matrices which map to the identity under the matrix.

Thus, for any nonsquare tmapping matrix which takes in monzos of some
JI group and purports to spit out smonzos on some subgroup, I can find
an infinite other set of valid subgroups of the original JI group
which map to the same thing as your chosen subgroup under the
tmapping. In fact, you can even say that the tmapping represents the
equivalence class of all such smappings which map to the identity, if
you like. (You can, of course, always use the tmapping which is the
pseudoinverse of your smapping and claim it represents the subgroup
specified by the smapping.)

The reverse is generally not true for smappings: they really do map
from group to subgroup in a unique and canonical way. In other words,
for any smapping, vals will map to the same thing if and only if they
really do agree on the subgroup specified by the columns of the
matrix. If the smapping isn't of full column rank, there will be an
infinite set of tmappings which reduce to the same thing on the
subgroup.

-Mike

On Tue, Jun 12, 2012 at 5:42 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> The reverse is generally not true for smappings: they really do map
> from group to subgroup in a unique and canonical way. In other words,
> for any smapping, vals will map to the same thing if and only if they
> really do agree on the subgroup specified by the columns of the
> matrix. If the smapping isn't of full column rank, there will be an
> infinite set of tmappings which reduce to the same thing on the
> subgroup.

We can formulate a few regular temperament ideas this way. For
instance, say we want to find the best 7-limit extension for meantone.
We start with the meantone mapping matrix

[<1 1 0|, <0 1 4|]

and then we envision it as the product of a higher-limit extended
mapping matrix E, and a smapping S. So

E * [|1 0 0 0>, |0 1 0 0|, |0 0 1 0>] = [<1 1 0|, <0 1 4|]

There are an infinite set of E which fit the bill, which state that
there are an infinite set of mapping matrices which agree with
meantone on the 2.3.5 subgroup. This is just a fancy way of saying
that there's an infinite set of mapping matrices which are 7-limit
extensions of meantone. We can pick an element out of this set in the
usual way, e.g. by minimizing some sort of "badness" function of E.

More interestingly, we can consider any mapping matrix M to be the
product of an infinite-limit extended mapping matrix E and an infinite
smapping S, the former of which is like a "no-limit" extension of
meantone, and the latter of which maps from no-limit vals to svals on
a finite subgroup:

E*S = M

The thing I called "superbadness" would then be some sort of function
of S and M. One could also look for the matrix E which minimizes some
other function and fits the bill.

-Mike