Geometric complexity

The complexity measures we have been using, based on linear temperaments,
give rise to problems when we try to generalize beyond the linear case.
Moreover, as can be seen in the examples of commas associated to 7-limit
JI scales I recently posted, the commas which seem to be the most
important for tempering JI scales are, reasonably enough, the ones with
small Euclidean length in terms of the lattice of octave classes.

I propose we scrap the linear approach and define complexity in terms of
lattice geometry. We can define a Euclidean metric by requiring that for
odd primes p, q and Euclidean distance from the unison L, that we have
the following:

L(p) = ln(p)
if p<q then L(p/q) = L(q/p) = ln(q)

This uniquely determines a Euclidean metric, with quadratic form

L(n)^2 = sum_{i,j} ln(p)^2 x_p x_q,

where x_p is the exponent of p n the factorization of n, and x_q the
exponent of q.

Choosing an orthonormal basis in this space, we define the geometric
complexity as the length of the wedge product of a set of
octave-equivalent commas (commas stripped of 2, a la Graham). Because of
the wedge product, this is independent of the choice of comma basis, and
depends only on the temperament.

Here are some examples; showing first Keenan-style weighted complexity,
geometric complexity as defined above, and
geometric complexity using a symmetric metric, where 3, 5, and 7 all have
the same length.

[50/49, 64/63] [2, -4, -4, 2, 12, -11] pajara
3.93867776085766 11.9251094548197 10.3923048454133

[81/80, 126/125] [1, 4, 10, 12, -13, 4] meantone
5.32244723964455 15.1018056341299 15.5884572681199

[225/224, 1029/1024] [6, -7, -2, 15, 20, -25] miracle
7.60914796670902 24.9266291754661 18.6279360101972

[225/224, 1728/1715] [7, -3, 8, 27, 7, -21] orwell
7.42151179771811 25.4206296354264 18.5472369909914

[2401/2400, 4375/4374] [18, 27, 18, -34, 22, 1] ennealimmal
16.9575882920421 58.8407776477707 39.2300904918661