<225/224, 4000/3969>

Minkowski reduction: <335/224, 3125/3087>

Ets: 12, 29, 41, 53, 94

Map:

[ 1 2]

[ 2 3]

[-1 6]

[-3 8]

Reduced map:

[ 0 1]

[-1 2]

[ 8 -1]

[14 -3]

Generators: a = .4148831986 (~4/3) = 38.99902067 / 94; b = 1

Septimal schismic system; effectively this is 53+41

Errors and 94-et:

3: .185 .173

5: -3.44 -3.33

7: 1.21 1.39

<4375/4374, 225/224> Minkowski reduced

Ets: 19, 53, 72

Map:

[ 3 1]

[ 6 0]

[ 8 1]

[13 -3]

Adjusted map:

[ 0 1]

[ 6 0]

[ 5 1]

[22 -3]

Generators: a = .2639365655 (~6/5) = 19.00343272 / 72; b = 1

This is pretty much the 53+19 system of the 72 et, but a bit better

in tune; in particular the long string of generators to 7 helps to

get it in better tune.

Errors and 72-et:

3: -1.61 -1.96

5: -2.69 -2.98

7: -0.90 -2.16

Because the 7's are so inacessible, it's tempting to treat this as a

purely 5-limit system, in which case it's closer to the 53 et, with a

generator of 14.00435/53, and triads which are very good; this

version is the kleismic temperament. Another possibility would be a

planar temperament.

<1728/1715, 81/80>

Minkowski reduction: <1029/1024, 81/80>

Ets: 5, 26, 31, 57

Map (no adjustment)

[ 0 1]

[ 3 1]

[ 12 0]

[-1 3]

Generators: a = .1934362896 (~8/7) = 5.996524978 / 31; b = 1

Errors and 31-et

3: -5.58 -5.18

5: -0.83 0.78

7: -0.95 -1.08

Pretty much the 26+5 system of the 31-et.

<64/63, 686/685> Minkowski reduced

Ets: 10, 27

Map:

[1 -1]

[1 -3]

[5 4]

[4 0]

Adjusted map:

[ 0 1]

[ 2 1]

[-9 5]

[-4 4]

Generators: a = .2969233588 (~11/9) = 8.016930688 / 27; b = 1

Errors and 27-et

3: 10.66 9.15

5: 6.91 13.68

7: 5.94 8.95

This is one version of a neutral third temperament, though Graham

does not even mention the 27-et in connection to neutral thirds, so

it's hardly canonical.

--- In tuning-math@y..., genewardsmith@j... wrote:

> <225/224, 4000/3969>

>

> Minkowski reduction: <335/224, 3125/3087>

That 335 should really be 225, right?

>

> <64/63, 686/685> Minkowski reduced

That 685 should be 675, right?

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

> --- In tuning-math@y..., genewardsmith@j... wrote:

> > <225/224, 4000/3969>

> >

> > Minkowski reduction: <335/224, 3125/3087>

>

> That 335 should really be 225, right?

> >

> > <64/63, 686/685> Minkowski reduced

>

> That 685 should be 675, right?

Yes to both. Bad fingers, bad!