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 ------------------------ Yahoo! Groups Sponsor ---------------------~--> Save on REALTOR Fees http://us.click.yahoo.com/Xw80LD/h1ZEAA/Ey.GAA/wHYolB/TM ---------------------------------------------------------------------~-> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxxxxxx Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ ____________________________________________________________ To learn how to configure this list via e-mail (subscribe, unsubscribe, etc.), send a message to listar@xxxxxxxxxxxxx with the subject line "info tuning-math". Or visit the website: < //www.freelists.org/list/tuning-math > .