Unlocking the mysteries of the wedge invariant

This can be defined in multiple ways as cross-products involving
intervals and vals, leading to multiple interpretations. We have for
instance

Interval product:

[u1,u2,u3,u4]^[v1,v2,v3,v4] = [[u2,u3,u4]X[v2,v3,v4],
[v3,v2,v1]X[u3,u2,u1]]

These cross-products can be thought of either as involving interval
classes, or intervals for which one of the prime powers are zero; from
the latter point of view something perpendicular would be a val which
mapped one of the primes to zero. We can exclude one of the primes in
turn, and take the cross product, and compare it to the wedge invariant,
and see how it can be interpreted in terms of vals.

If the wedge invariant is [a,b,c,d,e,f], we find we get in this way four
vals:

[ 0] [ a] [ b] [ c]
[ a] [ 0] [ f] [-e]
[ b] [-f] [ 0] [ d]
[ c] [ e] [-a] [ 0]

Similarly, a cross-product of two vals represents an interval in the
kernel of both, and so from the wedge invariant we get four commas:

2^f 3^-b 5^a, 3^d 5^e 7^f, 2^d 5^-c 7^b, 2^e 3^c 7^-a

It seems therefore that going from the wedge invariant to the temperament
it signifies is not too difficult.

Here is paultone as an example:

64/63^50/49 = h12^h22 = [2,-4,-4,2,12,-11]

We get the four vals:

[ 0] [ 2] [ -4] [ -4]
[ 2] [ 0] [-11] [-12]
[-4] [11] [ 0] [ 2]
[-4] [12] [ -2] [ 0]

All these have 64/63 and 50/49 in the kernel.

We also get the four commas:

2^-11 3^4 5^2 = 2025/2048
3^2 5^12 7^-11 = 2197265625/1977326743
2^2 5^4 7^-4 = (50/49)^2
2^12 3^-4 7^-2 = (64/63)^2

All these are commas of the temperament.