Again, everything checks as Minkowski reduced.

<3136/3125, 245/243>

ets: 19, 49, 68, 87

Map:

[ 2 -1]

[ 0 6]

[ 2 4]

[-1 13]

Adjusted map:

[ 0 1]

[-12 6]

[-10 6]

[-25 12]

Generators a = .3677022284 (~9/7) = 25.00375153 / 68; b = 1

Related to 49+19 and 68+19

Errors:

3: 3.13

5: 1.26

7: 0.11

This improves on the 68-et:

3: 3.93

5: 1.92

7: 1.76

<126/125, 49/48>

ets: 15, 19, 34

Map:

[1 -1]

[0 6]

[1 4]

[2 1]

Adjusted map:

[0 1]

[6 0]

[5 1]

[3 2]

Generators: a = .263886711 (~6/5) = 5.013847509 / 19; b = 1

Related systems: 19+15, 15+4

Errors:

3: -1.97

5: -2.99

7: -18.83

Comparison to 19 and 34

3: -7.22 3.93

5: -7.37 1.92

7: -21.46 -15.88

<3645/3584, 50/49>

ets: 12, 48

Map:

[ 0 12]

[ 0 19]

[-1 -2]

[-1 4]

Adjusted map:

[ 0 12]

[ 0 19]

[-1 28]

[-1 34]

Generators a = .01950640863 = 23.40769036 cents; b = 1/12 = 100 cents

Errors:

3: -1.96

5: -9.72

7: 7.77

Two 12-et keyboards, a 23.4 cent comma apart, would allow this one to

be tried. Is this a known idea? If not, it should be.

<6144/6125, 81/80>

ets: 7, 24, 31, 55

Map:

[1 1]

[1 3]

[0 8]

[6 -5]

Adjusted map:

[ 0 1]

[ 2 1]

[ 8 0]

[-11 6]

Generators a = .290240768 (~11/9) = 8.999116381 / 31; b = 1

This is for all intents and purposes the 24+7 system of the 31-et

Errors, and a comparison to the 31 et:

3: -5.25 -5.18

5: .51 .78

7: -.71 -1.08

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

> <3645/3584, 50/49>

>

> ets: 12, 48

>

> Map:

>

> [ 0 12]

> [ 0 19]

> [-1 -2]

> [-1 4]

>

> Adjusted map:

>

> [ 0 12]

> [ 0 19]

> [-1 28]

> [-1 34]

>

> Generators a = .01950640863 = 23.40769036 cents; b = 1/12 = 100

cents

>

> Errors:

>

> 3: -1.96

> 5: -9.72

> 7: 7.77

>

> Two 12-et keyboards, a 23.4 cent comma apart, would allow this one

to

> be tried. Is this a known idea? If not, it should be.

I did work with this many years ago -- nice to see it again. The 10-

cent-flat major third, umm, takes some getting used to. I do talk

about the 5-limit system of two 12-tET keyboards 15 cents apart a

lot. What is the Minkowski-reduced basis for that latter system? It

would seem that only one unison vector would be involved, but I can't

see how.

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

> I did work with this many years ago -- nice to see it again. The 10-

> cent-flat major third, umm, takes some getting used to. I do talk

> about the 5-limit system of two 12-tET keyboards 15 cents apart a

> lot. What is the Minkowski-reduced basis for that latter system? It

> would seem that only one unison vector would be involved, but I

can't

> see how.

I think I can do this one without the computer's help; the 15-cents

tells me that this is the 72 and 84 systems in the 5-limit, and they

both have the 12-et fifth, so they both have 3^12/2^19 in the kernel.

The map matrix has to be:

[ 0 12]

[ 0 19]

[-1 28]

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

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

> >It

> > would seem that only one unison vector would be involved, but I

> can't

> > see how.

>

> I think I can do this one without the computer's help; the 15-cents

> tells me that this is the 72 and 84 systems in the 5-limit, and they

> both have the 12-et fifth, so they both have 3^12/2^19 in the

kernel.

The Pythagorean comma . . . somehow when I thought about it before,

it wasn't so obvious . . . I guess I tend to think in terms of an

octave period, even though I'm so intimately familiar with a half-

octave-period system . . . but now it's completely obvious . . .

> The map matrix has to be:

>

> [ 0 12]

> [ 0 19]

> [-1 28]

Naturally.

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

>

> <126/125, 49/48>

>

> ets: 15, 19, 34

>

> Map:

>

> [1 -1]

> [0 6]

> [1 4]

> [2 1]

>

>

> Adjusted map:

>

> [0 1]

> [6 0]

> [5 1]

> [3 2]

>

> Generators: a = .263886711 (~6/5) = 5.013847509 / 19; b = 1

>

> Related systems: 19+15, 15+4

>

> Errors:

>

> 3: -1.97

> 5: -2.99

> 7: -18.83

Complexity 6, max. err. 18.83¢

>

> <3645/3584, 50/49>

>

> ets: 12, 48

>

> Map:

>

> [ 0 12]

> [ 0 19]

> [-1 -2]

> [-1 4]

>

> Adjusted map:

>

> [ 0 12]

> [ 0 19]

> [-1 28]

> [-1 34]

>

> Generators a = .01950640863 = 23.40769036 cents; b = 1/12 = 100

cents

>

> Errors:

>

> 3: -1.96

> 5: -9.72

> 7: 7.77

Complexity 12, max. err. 17.49¢

> <6144/6125, 81/80>

>

> ets: 7, 24, 31, 55

>

> Map:

>

> [1 1]

> [1 3]

> [0 8]

> [6 -5]

>

> Adjusted map:

>

> [ 0 1]

> [ 2 1]

> [ 8 0]

> [-11 6]

>

> Generators a = .290240768 (~11/9) = 8.999116381 / 31; b = 1

>

> This is for all intents and purposes the 24+7 system of the 31-et

>

> Errors, and a comparison to the 31 et:

>

> 3: -5.25 -5.18

> 5: .51 .78

> 7: -.71 -1.08

Complexity 19, max. error 5.76¢