Starling temperament mapping

Here's an opportunity to describe a planar temperament mapping in the
canonical form I've just proposed in 'Standardising temperament mappings'.

Starling temperament
discovered and named by Herman Miller, 1999

Target
[lg(2) lg(3) lg(5) lg(7)]
Generators (cents)
[1200 498.0 107.1]
Mapping
[1 2 2 1]
[0 -1 1 5]
[0 0 -1 -3]

Which means for example that lg(7) ~= 1*1200 + 5*498.0 + -3*107.1

Can someone please check that that agrees with the fact that the 125:126
(2^1 * 3^2 * 5^-3 * 7^1) is distributed, and remind me how you do that?

Gene,

Can you please give your proposed canonical form if it differs from this,
and explain the conventions involved.

Regards,
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

> Gene,
>
> Can you please give your proposed canonical form if it differs from this,
> and explain the conventions involved.

If we are looking at a d-dimensional temperament (ie, d is one less than the number of generators) then we have d+1 columns, and by taking
d+1 primes and leaving one out, we can wedge the wedgie with d of the rest of the primes, producing a sans 2, sans 3, sans 5 etc. column. If we have a gcd > 1, we divide out by that, and we make the top number of the column positive by multiplying by -1 if necessary.

We now have a mapping matrix which starts out with a diagonal square matrix part on top (since the sans p part will give zeros for all of the primes other than p, leaving only one nonzero entry.) This is my proposed normal form, not to be confused with Smith Normal Form though in fact it looks kind of similar. :)

Here's the 126/125 example:

126/125^3^5 = [1 0 0 -1]
126/125^2^5 = [0 -1 0 2]
126/125^2^3 = [0 0 1 3]

We change [0 -1 0 2] to [0 1 0 -2], and use these three to get the columns of the mapping matrix:

[ 1 0 0]
[ 0 1 0]
[ 0 0 1]
[-1 -2 3]

In this case, we have 1s along the diagonal, so the generators are
appromiately equal to 2, 3, and 5.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
>
> > Gene,
> >
> > Can you please give your proposed canonical form if it differs from this,
> > and explain the conventions involved.

I've been looking at whether this really works in general, and it seems it doesn't--we don't always get the wedgie back by wedging the rows, because we can get a mulitple (ie, torsion problems again.) But it's so neat I'll try to see if I can save it somehow.