A generalized approach to units of interval measure

The idea behind most of units of interval measure is that of a "coarse
reference" EDO + "fine division" scheme. The coarse reference gets you
in the ballpark of where you want to go, and the fine division dials
you in beyond that. I will call such things instance of a "two-tiered"
scheme of interval measure.

Examples: 12 is the coarse reference, there are 100 subdivisions =
cents. 10 is the coarse reference, there are 100 subdivisions =
millioctaves.

An EDO serves as a better coarse reference if it's
a) small, so that it's manageable and just serves to get you in the
ballpark of where you want to go, and
b) accurate in approximating important JI intervals, so that these
ballpark estimates are somewhat accurate.

It's clear that the EDOs which meet these two criteria the best are
the various zeta EDOs, which approximate JI over all intervals better
than anything else smaller than them.

This all says that there's some mathematical justification for using
"cents," other than their historical usage, in that they're based
around a low-numbered zeta EDO - 12-EDO. As a coarse reference, it
puts you in the ballpark of important JI intervals better than
anything else smaller than it. (Also noteworthy is that 12 is a highly
composite number, so as far as ballparking goes, you have immediate
references for the quarter-, third-, half- octave, etc.)

The list of zeta peak EDOs is as follows: 1, 2, 3, 4, 5, 7, 10, 12,
19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171... . So if we want
to look at some more xen ways of conceptualizing interval sizes, we
could instead base our unit of interval measure around 19-EDO or
22-EDO, giving us a more accurate coarse grid, though at the cost of
more notes. Alternatively, could go in the other direction and use
10-EDO or 7-EDO as a reference, the former being millioctaves, and the
latter being something without a name.

Example: using 7, 10, 12, 19, and 22, respectively, 7/4 would be
565.148, 807.355, 968.826, 1533.974, and 1776.181 "cents,"
respectively (e.g. 100 subdivisions of the coarse reference).

While this gives some sort of idea for how to handle the coarse EDO,
how do we handle the fine divisions? Is it always best to divide each
coarse step into 100 subdivisions exactly? I'll write about that in
the next email.

Mike