171 and 441

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--- In MakeMicroMusic@yahoogroups.com, "Aaron Krister Johnson"
<aaron@...> wrote:

> Sounds interesting....right now I'm looking at the properties of 171
> and 441 as JI approximations in my 'micro_composer' (old alias:
> 'et_compose' (as you know those divisions are great for 7-limit, but
> they are good in general one would think)...care to paste .scl files
> of the above suggestions?

Most obviously, these are 7-limit tunings which support ennealimmal
(the 7-limit 171&441 temperament.) There are two accurate ways to use
171 in the 11-limit:

U = <171 271 397 480 592|

V = <171 271 397 480 591|

"U" tempers out what I've called the 243/242-441/440-540/539 complex.
The factorization of 243/242 involves only 2, 3, and 11, and defines
11/9 as the 11-limit neutral third--that is, (3/2)/(11/9)^2 =
243/242. We then have that (11/9)/(49/40) = 441/440, and (60/49)/
(11/9) = 540/539. So the 11-limit planar temperament {243/242,
2401/2400} is supported by this version of 171edo.

"V" tempers out 385/384 instead, which is a very useful thing indeed;
it equates 48/35, which is across a diagonal of the hexany from 1/1,
with 11/8: (11/8)/(48/35) = 385/384. This gives you an amazingly
efficent and accurate means of squeezing 11-limit harmony into the 7-
limit lattice, and if you want a tuning for it which puts most of the
error burden on the 11-limit but allows some to fall on the 7-limit,
it's self-recommending. I'm thinking about exploring such things as
the convex closure of the eikosany in 385/384-spacial.

For 441/440, the patent val is most accurate, and it tempers out
4000/3993. If you add 2401/2400 and 4375/4374 to that, you get an
extension of ennealimmal which slices the 36/35 ennealimmal generator
into thirds, which is not nearly as efficent as slicing the period in
half, which is what hemiennealimmal does. But that might not matter
to you, if what you are after is having three 11/10 small seconds
come to a fourth: (4/3)/(11/10)^3 = 4000/3993.