Survey III

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¢