Linear Temperaments

A few questions -

I know how to derive generators-to-primes using commas in matrices.
How is it done using values?

Second question - how do you go the other way? That is, derive commas
from generators. I know this is more difficult because of (con)torsion

Are Linear Temperaments always based on commas?

Any information, even partial is appreciated.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> A few questions -
>
> I know how to derive generators-to-primes using commas in matrices.
> How is it done using values?
>
> Second question - how do you go the other way? That is, derive
commas
> from generators.

you need the mapping; then it's straightforward.

> Are Linear Temperaments always based on commas?

sure, it always can be, as long as it has a mapping.

> Any information, even partial is appreciated.

the mapping represents the primes in terms of the generators.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > A few questions -
> >
> > I know how to derive generators-to-primes using commas in
matrices.
> > How is it done using values?
> >
> > Second question - how do you go the other way? That is, derive
> commas
> > from generators.
>
> you need the mapping; then it's straightforward.

I derive commas from the wedgie in my software.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > A few questions -
> >
> > I know how to derive generators-to-primes using commas in
matrices.
> > How is it done using values?
> >
> > Second question - how do you go the other way? That is, derive
> commas
> > from generators.
>
> you need the mapping; then it's straightforward.
>
> > Are Linear Temperaments always based on commas?
>
> sure, it always can be, as long as it has a mapping.
>
> > Any information, even partial is appreciated.
>
> the mapping represents the primes in terms of the generators.

So, can a mapping (of primes in terms of generators) be derived from
two temperaments (say 12 & 19) without using commas? or wedgies? How
would that be done? Thanks

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:

> So, can a mapping (of primes in terms of generators) be derived from
> two temperaments (say 12 & 19) without using commas? or wedgies?
How
> would that be done? Thanks

A method I often use is to reduce the matrix whose rows are the two
vals in quesition to integer Hermite normal form; this means reducing
the front square part, the rest of the matrix following along. The
reason I so often use it is that in Maple, if the two vals are
written in terms of lists, v1, and v2, and I make a lists of lists
[v1, v2], then ihermite([v1, v2]) does this for me immediately.

http://mathworld.wolfram.com/HermiteNormalForm.html