Wedgie norm for 12-equal:
Take the two unison vectors
|7 0 -3>
|-4 4 -1>
Now find the determinant, and the "area" it represents, in each of
the basis planes:
|7 0| = 28*(e23) -> 28/lg2(5) = 12.059
|-4 4|
|7 -3| = -19*(e25) -> 19/lg2(3) = 11.988
|-4 -1|
|0 -3| = 12*(e35) -> 12 = 12
|4 -1|
sum = 36.047
If I just use the maximum (L_inf = 12.059) as a measure of notes per
acoustical octave, then I "predict" tempered octaves of 1194.1 cents.
If I use the sum (L_1), dividing by the "mystery constant" 3,
I "predict" tempered octaves of 1198.4 cents. Neither one is the TOP
value . . . :( . . . but what sorts of error criteria, if any, *do*
they optimize?
So the cross-checking I found for the 3-limit case in "Attn: Gene 2"
http://groups.yahoo.com/group/tuning-math/message/8799
doesn't seem to work in the 5-limit ET case for either the L_1 or
L_inf norms.
However, if I just add the largest and smallest values above:
28/lg2(5)+19/lg2(3)
I do predict the correct tempered octave (aside from a factor of 2),
1197.67406985219 cents.
So what sort of norm, if any, did I use to calculate complexity this
time? It's related to how we temper for TOP . . .