Generators for 4000/3993 planar

This is a good example, since it doesn't wimp out on us by letting us
use {2,3,5,7} as a set of generators. Here, following the discussion
on sci.math, I am not meaning a specific tuning by "generator".

We may first form a set of vals by taking all 4-monzos
4000/3993^p^q^r, where p, q, and r are distinct primes from the set
{2,3,5,7,11}. We now reduce this to Hermite normal form and toss the
six columns of zeros, giving us the matrix

M = [<1 2 0 0 1|, <0 3 0 0 -1|, <0 0 1 0 1|, <0 0 0 1 0|]

If now we take

N = [|1 0 0 0 0>, |1 0 1 0 -1>, |0 0 1 0 0>, |0 0 0 1 0>]

Then NM is the 4x4 identity matrix. The rows of N are monzos
corresponding to 2, 10/11, 5, and 7 respectively. We can renomalize by
taking 11/10 instead of 10/11, and changing the sign of the second
column of M. We now have, ignoring questions of tuning, a basis for
the generators of the temperament given by {2, 11/10, 5, 7}. Referring
to the rows of M, we see that

2 ~ 2
3 ~ 2^2 (11/10)^(-3) = 4000/1331
5 ~ 5
7 ~ 7
11 ~ 2 (11/10) 5 = 11

Hence, up to 4000/3993, we can express the 11-limit in terms of
2^a * (11/10)^b 5^c 7^d. If we now optimize the tuning of these, we
get our optimized tuning of the 4000/3993 spacial temperament. Or, of
course, we could simply tune fifths 4000/3993, or three cents, sharp,
and keep the rest pure if we liked that idea.