The best 5-limit temperaments

I did a search for commas which lead to temperaments such that if g is the rms of generator steps, and e is rms error in cents, then
g < 50, e < 50, and e g^3 < 500; the last being the 5-limit logarithmically flat badness measure. I obtained the following 32 commas, sorted by size:

5^49/2^90/3^15
3^62/2^17/5^35
2^144/3^22/5^47
3^47*5^14/2^107
2^54*5^2/3^37
5^51/2^36/3^52
2^37*3^25/5^33
3^10*5^16/2^53
2^91/3^12/5^31
2*5^18/3^27
3^35/2^16/5^17
2^38/3^2/5^15
5^34/2^52/3^17
3^8*5/2^15
5^19/2^14/3^19
2^23*3^6/5^14
2^8*3^14/5^13
2^9*5^5/3^13
5^6/2^6/3^5
3^3*5^7/2^21
2^17*3/5^8
2^2*3^9/5^7
2^11/3^4/5^2
3^4/2^4/5
5^5/2^10/3
2^7/5^3
2*5^3/3^5
2^3*3^4/5^4
5^2/2^3/3
3^3*5/2^7
2^4/3/5
3^3/5^2

Though it was slightly outside my generator range (I searched quite a bit farther than the above limits), I will also mention
2^161 3^(-84) 5^(-12); as a temperament this consists of 12-ets stacked by fourths and fifths, a sort of Pythagorean/12-et system. It is, of course, absurdly well in tune, and can also be thought of as 12-ets separated by schismas.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> I did a search for commas which lead to temperaments such that if g
is the rms of generator steps, and e is rms error in cents, then
> g < 50, e < 50, and e g^3 < 500;

I though we decided we were _not_ going to have a constraint on e,
and simply use an upper bound on badness combined with both lower and
upper bounds on g. What gives?

> Though it was slightly outside my generator range (I searched quite
a bit farther than the above limits), I will also mention
> 2^161 3^(-84) 5^(-12);

This just got a lot of discussion over at the tuning list.

>as a temperament this consists of 12-ets stacked by fourths and
>fifths, a sort of Pythagorean/12-et system. It is, of course,
>absurdly well in tune, and can also be thought of as 12-ets
>separated by schismas.

aka 612-tET!

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> I though we decided we were _not_ going to have a constraint on e,
> and simply use an upper bound on badness combined with both lower and
> upper bounds on g. What gives?

I decided that pushing it farther was getting into garbage territory, but there aren't many more that we would add in this way, and they could be included. Do we really want to temper out the fifth?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > I though we decided we were _not_ going to have a constraint on
e,
> > and simply use an upper bound on badness combined with both lower
and
> > upper bounds on g. What gives?
>
> I decided that pushing it farther was getting into garbage
territory, but there aren't many more that we would add in this way,
and they could be included. Do we really want to temper out the fifth?

Wouldn't the squared error of the fifth then take us outside our
allowed bound on badness?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> Wouldn't the squared error of the fifth then take us outside our
> allowed bound on badness?

Tempering out a fifth makes the error of a fifth equal to a fifth; since a major and minor third makes a fifth, it also makes the errors of the thirds a (neutral) third. The result, of course, makes absolutely no sense, but since you also don't need much in the way of generator steps to accomplish it, passes my badness limit.

3/2

Map:

[ 0 1]
[ 0 1]
[ 1 2]

Generators: a = .02945, b = 1

badness: 270
rms: 496
g: .8165
errors: [-702, -351, 351]