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.