Another example

Let's start from the 441-et in the 11-limit. If I reduce the 5x6
matrix whose first row is [441,1,0,0,0,0] (telling me 2 is 441 steps
in the 441-et) and last is [1526,0,0,0,1] (telling me 11 is 1526
steps in the 441-et) I will get vectors which span the same lattice,
but which should be smaller and closer to being orthogonal. If the
right hand column has smaller entries, that means they will have only
a few steps in the 441-et, and hence will be small intervals.
However, since I started from the 11-limit, these small intervals
will still span the entire 11-limit, just as 2,3,5,7,11 do.

When I do this, I do in fact get small intervals: 440/441, -1 steps;
539/540, -1 steps; 176/175, 4 steps; 1372/1375, -2 steps, and
11979/12005 0 steps--a 441 comma. If I leave off the column on the
right, counting 441-et steps, I get a square matrix which must be
unimodular--have determinant +-1, since it spans the 11-limit. I
therefore may invert it, getting a matrix [-h49,h12,h72,-h58,-h31]
of vals, which together with my intervals gives me a notation.

I now may remove some of these vals, and see what lattice the
remaining ones generate. They will span the dual group to the
corresponding basis elements. If I eliminate h72 and -h31, I get
something dual to <440/441, 539/540, 1372/1375>. If I apply lattice
basis reduction to these, I don't get anywhere, since they are
already reduced. However, I can reduce the dual lattice of vals,
getting reduced vals. When I do this, I get

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

These span the same group, dual to <440/441, 539/540, 1372/1375> as
the 49, 12, and 58-et 11-limit vals. We now want to find the
elements, corresponding to <440/441, 539/540, 1372/1375>, dual to
these. They will give us a temperament with the same kernel elements,
176/175 and 12005/11979 in the 11-limit, and thus an 11-limit planar
temperament. Since the vals have been reduced, however, we don't have
so many generator steps to get to the primes, meaning these will be
good choices for generators--or at least, better certainly than
441/440 and so forth!

The top three rows of the above matrix gives us a 3x3 unimodular
matrix, inverting it gives us

[ 1 0 0]
[ 7 -1 -3]
[-2 0 1]

The rows of this correspond to 2, 128/375, and 5/4, which we can use
as approximations to the generators we seek. The full matrix above
tells us how 2,3,5,7 and 11 are to be approximated using these three
generators, each row representing a prime. We can replace 128/375
with 375/256 if we like, putting all the generators in an octave; if
c~5/4 and d~375/256 the 5x3 matrix, after adjustment, tells us that

3 ~ 2^2 c^(-3) d
5 ~ 2^2 c
7 ~ 2^2 c^(-1) d^2
11 ~ 2^2 c d

This defines the temperament; we can use least squares or linear
programming to optimize and get tuning values for c and d. A quick
check shows that this system is practical, though the 5 is a little
sharp at 5.6 cents or thereabouts.

This example was worked more or less at random, so there are plenty
more of the same out there. Of course we can regard this as basically
the 11-limit version of the 126/125 temperament.