More on extending generator mappings to temperaments

In message

/tuning-math/message/12139

I discussed how to extend a generator mapping [a,b,c] to a wedgie, and
hence a 7-limit temperament. Here is a purely algebraic definition of
what that is, under the additional assumption that the resulting
temperament must be supported by the et val <v2 v3 v5 v7|.

First we form the bival (and it is one)

<<v2*a v2*b v2*c v3*b-v5*a v3*c-v7*a v5*c-v7*b||

Now we reduce this in the standard way to a wedgie: we divide out any
common factors, and make the first nonzero coefficient positive. The
result is a wedgie where the generator part of the mapping is what we
want. While the result is guaranteed to be a wedgie for a temperament,
it probably will not be any good. If we take [1 4 10], for example,
then using the above method with the standard vals for 12, 19, 31, 43,
etc will give meantone. If the val does not support meantone, the
result is normally complete rubbish, and typically with n periods to
the octave, where n is the division we are using. Even when that
doesn't happen, as for instance with 94, which gives a temperament
with a 1/47 octave period, the result is still normally rubbish.
Buried in the rubbish, however, there seems to be a good r2
temperament--while one comma of the TM reduction is garbage,
corresponding to the period, there is another which is good, and is
the common comma of the val in question and meantone.