Survey V

<2401/2400, 2048/2025> Minkowski reduced

ets: 10, 58, 68

Map (no adjustment needed)

[ 0 2]
[-4 4]
[ 8 3]
[ 3 5]

a = .1030320504 (~15/14) = 7.006179427 / 68; b = 1/2

Error compared to 68:

3: 3.39 3.93
5: 2.79 1.92
7: 2.09 1.76

This system is effectively 58+10

<1029/1024, 686/675> Minkowski reduced

Map (no adjustment)

[ 0 2]
[-3 5]
[ 6 1]
[ 1 5]

Generators a = .3040426304 (~100/81) = 13.985961 / 46; b = 1/2

<4375/4374, 6144/6125>

Minkowski reduction: <4375/4374, 5120/5103>

ets: 7, 39, 46, 53, 99

Map:

[ 1 -2]
[ 3 -1]
[ 6 1]
[-2 13]

Adjusted map:

[ 0 1]
[ -5 3]
[-13 6]
[ 17 -2]

Generators: a = .2828456082 (~17/14, ~~11/9) = 28.00171521 / 99; b=1

This is essentially the 53+46 system of the 99-et; it's also related
to 46+7

Errors compared to 99:

3: 0.9713 1.0753
5: 1.2948 1.5651
7: 1.2245 0.8711

<3136/3125, 49/48>

Minkowski reduction: <3125/3072, 49/48>

Map:

[1 -1]
[0 10]
[2 0]
[2 3]

Adjusted map:

[ 0 1]
[10 0]
[ 2 2]
[ 5 2]

Generator: a = .1584971341 (~10/9, exactly 3^(1/10))
= 3.011445548 / 19

Errors compared to 19:

3: 0 -7.22
5: -5.92 -7.37
7: -17.84 -21.46

This is not your father's 19-et! In fact, with its perfect fifths,
it's not much like either the 19 or 25 et it can be played in. This
one is a genuine original, a temperament best left as a temperament.
Definately worth a look as a way of tempering 19 notes.

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

> <1029/1024, 686/675> Minkowski reduced
>
> Map (no adjustment)
>
> [ 0 2]
> [-3 5]
> [ 6 1]
> [ 1 5]
>
> Generators a = .3040426304 (~100/81) = 13.985961 / 46; b = 1/2

What about the errors?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
>
> > <1029/1024, 686/675> Minkowski reduced
> >
> > Map (no adjustment)
> >
> > [ 0 2]
> > [-3 5]
> > [ 6 1]
> > [ 1 5]
> >
> > Generators a = .3040426304 (~100/81) = 13.985961 / 46; b = 1/2
>
> What about the errors?

Did I forget those? I was supposed to include them, with a comparison
to the 46 et:

3: 3.49 2.39
5: 2.79 4.99
7: -3.98 -3.61

There doesn't seem to be much gained by not using 46-et for this. I
like this 686/675 in the kernel--it gives it character. It's also a
good system, with lots of 7-limit stuff in it.