"Best" divisions

A very different slant on the question of what the interesting
divisons of an octave are is afforded by considering the complexity of
the minimal temperament as defining the complexity. This might be
regarded as a little artificial, since it assumes "best" values are
used; I used the "standard" val for n for the calculations below. Yet
the results are interesting, I think.

Suppose E is the maximum p-limit error and C is the minimal Graham
complexity for the odd limit we are using. Then E*G^(n/(n-2)), where n
= pi(p) is the number of primes in the p-limit, gives a measure for
the goodness, in some sense of goodness, of the division in question.
Measuring E in octave units and searching for E*G^3 less than 1/3 up
to 200 gives the following: 7, 10, and 17 are interesting as 25/24
systems; 39 and 42 as augmented systems; 31, 50, and 81 as meantone
systems; 53, 87, 140, and 193 as kleismic systems; 118 and 171 as
schismatic systems. All that makes a good deal of sense, from a
different perspective than usual.

Similarly, if in the 7-limit we search for values of E*G^2 less than
0.4, we end up with 9 as a beep system, 99 and 130 as hemiwuerschmidt
systems, 175 as a miracle system, and 171 as both an ennealimmal and a
tertiaseptal system. In the 11-limit everyone will be amazed to learn
that 72 stands out as a miracle system, though if you go even farther
342 and especially 612 are really hot stuff as hemiennealimmal systems.