Finding the wedgie from the 2-part

By the 2-part I mean the first n entries, where n=pi(p)-1 is the
number of odd primes. If the period is an octave, this is just the
number of generator steps to get to the odd primes; otherwise it is
that times the number of periods in an octave.

In the 7-limit case, if we have [a, b, c] and want to extend it to a
wedgie, we can use the fact that the values we want are the exponents
of two for triprime commas. If 3 takes a generator steps, for
instance, and 5 takes b, then 3^b/5^a takes no generator steps, and if
we find the appropriate expoentent we have a comma for the
temperament. The upshot is that

<<a b c ||log2(3^b/5^a)|| ||log2(3^c/7^a)|| ||log2(5^c/7^b)|| ||

is going to give the wedgie if there is a reasonable one to give. Here
||x|| means the integer closest to x.

When doing this, we should check to see if the result is actually a
wedge product, which we can test for by seeing if, for the wedgie
<<a b c d e f|| we have a*e-b*f+c*d = 0. We can also remove common
factors and make the reduction to the standard wedgie. For 11 limit
and higher, the method is the same, but the tests to see if the result
is a wedge product get out of hand after a while, since they amount to
taking all combinations of four primes {p1,p2,p3,p4}, and getting the
corresponding {p1,p2,p3,p4} possible wedgie by selecting the
appropriate values from the possible wedgie being tested, and using
the same method as the 7-limit. These relationships are known as the
Pluecker relationships, or Plueker-Grassmann syzygies, and we can
discuss the whole thing in mathematical overkill mode by saying that
wedgies are exactly points on the Pluecker embedding of the Grassmannian.

We can often get one temperament from another by multiplying by some
positive integer m and reducing mod n, where n>m is relatively prime
to m. For instance, starting from [1 4 10], and reducing mod 31, we
get semififths multiplying by 2, mothra/cynder by 3, tritonic by 5,
miracle by 6, orwell by 7, valentine by 9, myna by 10, slender by 13
and hemithirds by 15, to name the interesting results. Less often we
can find something by multiplying by n but not reducing; in this case
the period will be a fraction of an octave. When the result is a
wedgie at all, it is normally not an interesting one but I will try to
see if I can find interesting examples.