359-tone equal temperament

Don't know how useful this would be, but I just "discovered" this one --
twice. Just having fun here, really.

I arrived at this in two different directions. The most obvious is its
Pythagorean nature, as the interval 3^359/2^569 is 1.8453 cents (a little
less than a schisma). I also ended up with this temperament by generating a
MIRACLE temperament with a generator of (3/2)^(1/6) ~ 116.9916 cents. It
came out to be an almost-equal 359-tone scale.

The 3-limit is very precise, but not as much as 665-tone (we ARE dealing
with tiny discrepancies here though). 7-limit is good, and 11-limit is
better. However, 5-limit doesn't work so well, as 5/4 is close to halfway
between degrees degrees 115 and 116, though favoring the latter. 13-, 17-
limit isn't any better, but 19-limit is great, just 0.02 cents off of the
nearest 359-tone degree.

6.4340 steps equals one Didymus comma; 7.0185 steps a Pythagorean comma. A
diaschisma is 5.8495 steps, a minor diesis 12.2834, a septimal comma 8.1565
steps, and an undecimal comma 15.9375.

Finally, 359 is one off of 360, a superset of 72-equal.

On the theme of 117-cent MIRACLE, that works out to 400-tet by way of
41-near equal. 116-tone to 300-tet via 31-near equal. So I retract my
referring to 1200-tet (cents) as arbritrary....

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:
> Don't know how useful this would be, but I just "discovered" this
one --
> twice. Just having fun here, really.

you're not alone :)

> I arrived at this in two different directions. The most obvious is
its
> Pythagorean nature, as the interval 3^359/2^569 is 1.8453 cents (a
little
> less than a schisma).

yup -- 359 could have been predicted as a pretty good pythagorean by
adding 53 and 306, the two most prominent ETs here:

http://sonic-arts.org/dict/pythag.htm

> I also ended up with this temperament by generating a
> MIRACLE temperament with a generator of (3/2)^(1/6) ~ 116.9916
cents. It
> came out to be an almost-equal 359-tone scale.

that's not all that great as a miracle generator, being almost all
the way to 41-equal in the usual "allowed" 41-equal-to-31-equal
spectrum.

> The 3-limit is very precise, but not as much as 665-tone (we ARE
dealing
> with tiny discrepancies here though).

306 stands out more there.

> but 19-limit is great, just 0.02 cents off of the
> nearest 359-tone degree.

you must be looking at just the 19th harmonic or the interval 19/16,
not the 19-limit. 19-limit accuracy means the accuracy of *all* the
19-limit consonances, *including* 5/4, 13/10, 20/19, etc.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:

> > but 19-limit is great, just 0.02 cents off of the
> > nearest 359-tone degree.
>
> you must be looking at just the 19th harmonic or the interval
19/16,
> not the 19-limit. 19-limit accuracy means the accuracy of *all* the
> 19-limit consonances, *including* 5/4, 13/10, 20/19, etc.

and 359 is not even *consistent* in the 19-limit:
http://sonic-arts.org/dict/consiste.htm

having fun yet?

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:
> Don't know how useful this would be, but I just "discovered" this
one --
> twice. Just having fun here, really.
>
> I arrived at this in two different directions. The most obvious is its
> Pythagorean nature, as the interval 3^359/2^569 is 1.8453 cents (a
little
> less than a schisma). I also ended up with this temperament by
generating a
> MIRACLE temperament with a generator of (3/2)^(1/6) ~ 116.9916 cents. It
> came out to be an almost-equal 359-tone scale.

Rather than Miracle, 359 makes more sense as a Hemiwuerschmidt et.
Hemiwuerschmidt is a 7-limit linear temperament with
[16, 2, 5, -34, -37, 6] as wedgie and [[1,-1,2,2], [0,16,2,5]] as
mapping, with 58/359 being a good Hemiwuerschmidt generator.