Val lattices

If we take the vals and put a norm on the vector space they live in,
we get a lattice. A very useful one is what I've been calling the val
lattice, with the L_inf norm the dual of the Tenney norm. However,
many other norms are possible. One possibility is to take n
independent vals, where n is the dimension of the space, and give them
the same Euclidean length. In particular the vals in what I've called
a "notation", which together form a unimodular matrix, can be give the
same length.

For example if I take 19,27,31, and 72, then the bilinear form

52*x2^2+8*x2*x3-14*x2*x5-30*x2*x7+58*x3^2-56*x3*x5-
22*x3*x7+33*x5^2-18*x5*x7+19*x7^2

has the same value of 1 for 19,27,31 and 72, where [x2,x3,x5,x7] is a
val or point in the val space. We get a Euclidean norm by taking the
square root. Vals which are sums or differences of any two of the
above have a norm of sqrt(2), and +-v1+-v2+-v3, for three of the
above vals, a norm of sqrt(3). The 4D cross polytope, or
hyper-octahedron, with verticies + or - the above vals has edges
corresponding to linear temperaments and faces corresponding to planar
temperaments.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

The 4D cross polytope, or
> hyper-octahedron, with verticies + or - the above vals has edges
> corresponding to linear temperaments and faces corresponding to planar
> temperaments.

It has 16 cells, and we may as well just pick the 19-27-31-72 cell and
look at that. This is a tetrahedron, with faces planar temperaments,
or 7-limit commas, and edges linear temperaments. Other regular
tetrahedra can be found in the space which have similar properties,
but the norm is a bit arbitary so I don't see too much importance in that.