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.