The four before meantone

Here are the second four 5-limit temperaments, which bring us up to
meantone:

648/625

Map:

[ 0 4]
[ 1 5]
[ 1 8]

Generators: a = 21.0205/64; b = 1/4

badness: 385
rms: 11.06
g: 3.266
errors: [-7.82, 7.82, 15.64]

64-et, anyone? It could also be used to temper the 12-et.

250/243

Map:

[ 0 1]
[-3 2]
[-5 3]

Generators: a = 2.9883/22; b = 1

badness: 360
rms: 7.98
g: 3.559
errors: [9.06, -1.29, -10.35]

One way to cure those 22-et major thirds of what ails them.

128/125

Map:

[ 0 3]
[-1 6]
[ 0 7]

Generators: a = 11.052/27 (~4/3); b = 1/3

badness: 142
rms: 9.68
g: 2.449
errors: [6.84, 13.69, 6.84]

When extended to the 7-limit, this becomes the

[ 0 3]
[-1 6]
[ 0 7]
[ 2 6]

system I've already mentioned in several contexts, such as the
15+12 system of the 27-et. Both as a 5-limit and a 7-limit system, it is good enough to deserve a name of its own.

3125/3072

Map:

[ 0 1]
[ 5 0]
[ 1 2]

Generators: a = 12.9822/41 (=6.016/19); b = 1

badness: 239
rms: 4.57
g: 3.74
errors: [-2.115, -6.346, -4.231]

Graham has named this one: Magic.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Here are the second four 5-limit temperaments, which bring us up to
> meantone:
>
> 648/625
>
> Map:
>
> [ 0 4]
> [ 1 5]
> [ 1 8]
>
> Generators: a = 21.0205/64; b = 1/4
>
> badness: 385
> rms: 11.06
> g: 3.266
> errors: [-7.82, 7.82, 15.64]
>
> 64-et, anyone?

octo-diminished.

> 250/243
>
> Map:
>
> [ 0 1]
> [-3 2]
> [-5 3]
>
> Generators: a = 2.9883/22; b = 1
>
> badness: 360
> rms: 7.98
> g: 3.559
> errors: [9.06, -1.29, -10.35]
>
> One way to cure those 22-et major thirds of what ails them.

Huh? These major thirds are much worse than those of 22-tET. Oh wait -
- the major third is the second entry under "errors"? So what on
earth is the third entry? Oh, it's the minor third!

>
>
> 128/125
>
> Map:
>
> [ 0 3]
> [-1 6]
> [ 0 7]
>
> Generators: a = 11.052/27 (~4/3); b = 1/3
>
> badness: 142
> rms: 9.68
> g: 2.449
> errors: [6.84, 13.69, 6.84]
>
> When extended to the 7-limit, this becomes the
>
> [ 0 3]
> [-1 6]
> [ 0 7]
> [ 2 6]
>
> system I've already mentioned in several contexts, such as the
> 15+12 system of the 27-et. Both as a 5-limit and a 7-limit system,
it is good enough to deserve a name of its own.

It's the augmented system, since the 6-tone MOS is commonly known as
the augmented scale.