Osmium generators

Paul has taken up the challenge of finding the exact value of the
Osmium Meantone, which is the limit of the sequence 4/7, 7/12, 7/12,
11/19, 14/24, 18/31, 25/43, 32/55 ..., and is about 1/5-comma
meantone. If he solves that, this should be easy:

3/8, 4/11, 7/19, 7/19, 11/30, 14/38, 18/49, 25/68, 32/87, 43/117,
57/155... . If you multiply the denominators by 9, you get 72, 99,
171, 171, 270, 342, 441, 612, 783, 1053, 1395 ..., so this is the
Osmium version of a generator which doesn't yet have a name, so far
as I know, but which has been discussed several times.

(Why Osmium? I'll explain that after Paul either solves the problem
or asks for my solution.)

I'd also like to know what to do with the sequence 9, 10, 12, 19, 22,
31, 41, 53, 72, 94, ... It doesn't seem like Miracle, or Orwell
either, but the ets make sense, at any rate, even if the resultant
commas don't. Apparently this Osmium business doesn't always give you
a generator that makes much sense.