Decomposition of contorted matrices as per Carl

A long time ago, Carl suggested that <14 22 32| can be viewed as a
"tuning system" that "implements" the <7 11 16| "temperament." I've
come back to this idea repeatedly in the past few months as a very
elegant way to look at things.

To formalize this idea, any nxm matrix M in Hermite form can be
uniquely decomposed into a factorization of the form M=CT, where C is
an nxn square matrix and T is an nxm non-contorted matrix that's also
in Hermite normal form.

In this factorization, let M be known as the "mapping matrix," C be
known as the "contorsion matrix, and T be known as the "temperament
matrix." Furthermore, let it be said that the mapping specified by M
"implements" the temperament specified by T.

The absolute value of the determinant of C can then be called the
"index" or the "degree" of contorsion of the mapping. If the degree is
d, then the mapping can be said to be d-contorted. Note that this
value will always be equal to the absolute value of the GCD of the
corresponding multival for the temperament.

Of note is the fact that, a finite number of possible mappings exist
for any temperament and degree of contorsion. For instance, if you
consider all 2-contorted mappings of meantone, the only 3 possible
options are as follows, given in factorized form:

[2 0][1 0 -4]
[0 1][0 1 4]

[1 0][1 0 -4]
[0 2][0 1 4]

[1 1][1 0 -4]
[0 2][0 1 4]

The first is the thing like injera, the second is the thing like semaphore, and
the third is Vicentino. The contorsion matrix distinguishes these
three from one another, and as such preserves the lost information
about contorsion that would otherwise occur when you wedge together
two vals (the degree of contorsion is preserved, but not the specific
way in which intervals are subdivided).

An immediate thing that jumps out at me is that the contorsion
matrices themselves in this example are themselves in Hermite form. I
conjecture that this will always be the case in such decompositions.

Anyone have a counterexample?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> To formalize this idea, any nxm matrix M in Hermite form can be
> uniquely decomposed into a factorization of the form M=CT, where C is
> an nxn square matrix and T is an nxm non-contorted matrix that's also
> in Hermite normal form.

Interesting! But why not just use the Hermite normal form?

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > To formalize this idea, any nxm matrix M in Hermite form can be
> > uniquely decomposed into a factorization of the form M=CT, where C is
> > an nxn square matrix and T is an nxm non-contorted matrix that's also
> > in Hermite normal form.
>
> Interesting! But why not just use the Hermite normal form?

You can use that too, but this way has the benefit of showing you what the underlying temperament is which is being contorted. This can be useful, because sometimes it's not clear from the Hermite form of a mapping what the underlying temperament is.

My real aim with this is to come up with some nice dual relationships for mappings in this form. I still have yet to work that out nicely in terms of matrices though, so more on that to come.

-Mike