Another approach to 9-limit

Here is a quite different approach which might prove useful. The An
lattice can be represented in (n+1)-space as [a0,a1,...,an], where
a0+...+an=0. For an A3 lattice of 5,7,9, we can set 5 to be
[-1 1 0 0], 7 [-1 0 1 0], and 9 [-1 0 0 1]. For the 11 limit we would
have 11 as [-1 0 0 0 1] etc. etc. Dot products of one of these basis
vectors with itself gives 2, with another gives 1, so distances have
been scaled up by a factor of sqrt(2), as in the fcc version of A3.

Now suppose we remove the restriction that the coordinates must add to
zero, and set

Note([a3 a5 a7 a9]) = 3^(a3+a5+a7+a9) 5^a5 7^a7 9^a9.

We again have multiple versions of the same note, with
Note([3,0,0,-1]) = 3^2 9^(-1) = 1. Modding out by this is a
possibility. The 11-limit version would be

Note([a3 a5 a7 a9 a11]) = 3^(a3+a5+a7+a11) 5^a5 7^a7 9^a9 11^a11

Still another possibility is the old reliable obvious one, of gluing a
3-lattice half way to the 9.