A generalized approach to units of interval measure, part 2: fine divisions

OK, finally time for part two in the series of posts about generalized
units of interval measure.

I started thinking about these ideas when Gene posted something to the
tuning list asking if millioctaves should be used instead of cents,
which I replied to here extolling the mathematical virtue of using
cents: (/tuning/topicId_100020.html#100021). I
then posted part one of this series generalizing the concept to
tuning-math here
(/tuning-math/message/21323), and
this is part 2.

To briefly summarize part one, I outlined what I called a two-tiered
interval measuring scheme, my reasoning for which I gave in the
response to Gene on the tuning list. In such a scheme, there's a
"coarse reference" EDO whose purpose is to get you in the ballpark of
the interval you want to measure, and then a "fine division" of each
step in the EDO to dial you in beyond that. I proposed that an EDO
serves as a better coarse reference if it's small, so that it's
manageable in size, and if it's harmonically accurate, so that these
ballpark estimates are pretty good for the most important JI
intervals. A list of "the best" EDOs meeting this criteria is given by
Gene's various lists of zeta EDOs.

Last time I talked about coarse references. This time, I propose that
fine divisions are better if they
1) are actually fine, so that motion up or down by one fine division
is at least smaller than the JND for pitch discrimination, which is
roughly 5 cents
2) are composite, so that they enable you to subdivide the step evenly
in various ways
3) have low-numbered factors that, if treated as fine divisions
themselves, approximate important JI intervals well

The zeta function again makes it possible to systematically find these
sorts of things, luckily!

Note that it's often (always?) the case that some zeta EDO z1 will
have a multiple that's also a zeta EDO z2; we say that z2 is a "zeta
multiple" of z1 (according to some specific list of zeta
gap/integral/peak/etc EDOs). So if we say that r = z2/z1, and if we
choose z1 for our coarse reference, then choosing a multiple of r as
our fine division immediately satisfies criteria #2-3, and if the
multiple is high enough to make the division really "fine," it
satisfies #1 as well. This works best if r is sufficiently small.

Although I haven't defined the notation yet, here's a quick example to
sanity check yourself: 12 has a zeta multiple in 72. So if we choose
12-EDO to be our coarse reference, then we want to pick a fine
division which is a multiple of 6, which is small enough to be under 5
cents, and which enables us to subdivide the octave in various even
ways. A useful division satisfying all of these criteria is 60, which
is a highly composite number with more factors than anything smaller,
and gives us a fine division of about 1.667 cents.

Some other examples of zeta multiples can be given in the thread here:
/tuning-math/message/21322

A notation is needed for the measure of any interval. One option is to
write it as a mixed fraction: s x/d, where s is the integer number of
steps (rounded down), d is the natural number of fine divisions per
step, and x is the real number of fine divisions needed to get you to
the interval. So some examples, for select intervals and units of
interval measure:

Coarse reference = 12, fine division = 100 (cents)
3/2: 7 1.96/100
5/4: 3 86.3/100
6/5: 3 15.6/100
7/4: 9 68.8/100
9/7: 4 35.0/100
11/8: 5 51.3/100
13/8 8 40.5/100

Coarse reference = 12, fine division = 60, this time rounded to the nearest step
3/2: 7 1/60
5/4: 3 52/60
6/5: 3 9/60
7/4: 9 41/60
9/7: 4 21/60
11/8: 5 31/60
13/8: 8 24/60

Coarse reference = 19, fine division = 360, also rounded to the nearest step
3/2: 11 41/360
5/4: 6 42/360
6/5: 4 359/360
7/4: 15 122/360
9/7: 6 320/360
11/8: 8 263/360
13/8: 13 111/360

This is nice and generalized, but still kind of a pain to read Next
time I'll write a post about using highly composite numbers to come up
with a one-size-fits-all type scheme, and introduce a few nicer
looking notations.

Mike