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