Survey VII

<1029/1024, 245/243> Minkowski reduced

Ets: 5, 41, 46, 87

Map:

[ 1 1]
[ 1 4]
[-1 16]
[ 3 2]

Adjusted map:

[ 0 1]
[ 3 1]
[ 17 -1]
[-1 3]

Generators: a = .1953420071 (~8/7) = 16.99475462 / 87; b = 1

Error and 87-et:

3: 1.28 1.49
5: -1.34 -0.11
7: -3.24 -3.31

This is pretty nearly the 46+41 system of the 87-et, which fails to
take much advantage of the excellent thirds, but does have a lot of
1--3/2--7/4 chords to play with.

<4375/4374, 2401/2400> Minkowski reduced

Ets: 27, 72, 99, 171, 270, 441, 612

Map:

[ 0 9]
[-2 -1]
[-3 -2]
[-2 10]

Adjusted map:

[ 0 9]
[-2 15]
[-3 22]
[-2 26]

Generators: a = .04083262537 (~36/35) = 24.98956673 / 612; b = 1/9
(~27/25) = 68/612

Alternative adjusted map:

[0 9]
[2 13]
[3 19]
[2 24]

a' = 43.01043329/612 (~21/20) as generator

Errors vs 441 and 612 ets:

3: .0467 .0858 .0058
5: .0222 .0808 -.0392
7: -.1575 -.1184 -.1985

In case anyone cares about errors that small, it can be seen that
both 441 and 612 do a fine job with this system, which is another
familiar face, the ennealimmal temperament. I'd remark that it's good
enough even for Harry Partch but for the fact that it would get me in
trouble, so I'll point out that it's good enough even for Harry
Potter. Good enough for what I leave to Hogwarts to figure out, but
tempering the 72 notes remains one possibility.

<875/864, 50/49> Minkowski reduced

Ets: 22, 26, 48

Map:

[2 2]
[1 -3]
[3 0]
[4 1]

Adjusted map:

[0 2]
[4 1]
[3 3]
[3 4]

Generators: a = .2713428065 (~6/5) = 13.0244547 / 48; b = 1/2

Errors and the 48-et:

3: 0.49 -1.96
5: -9.48 -11.31
7: 8.01 6.17

Here's another system with very flat major thirds, and Paul, I
suppose, would prefer it in the 18+4 version of the 22-et, or 22+4 of
the 26 et, over 26+22 in the 48-et.

<4000/3969, 2048/2025>

Minkowski reduction: <3136/3125, 2048/2025>

Map (no adjustment)

[ 0 4]
[-1 8]
[ 2 6]
[ 5 3]

Generators: a = .4118714296 (~4/3) = 28/00725721 / 68; b = 1/4

Errors compared to 68:

3: 3.80 3.93
5: 2.18 1.92
7: 2.40 1.76

Effectively the 56+12 system of the 68-et.

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

> <875/864, 50/49> Minkowski reduced
>
> Ets: 22, 26, 48
>
> Map:
>
> [2 2]
> [1 -3]
> [3 0]
> [4 1]
>
> Adjusted map:
>
> [0 2]
> [4 1]
> [3 3]
> [3 4]
>
> Generators: a = .2713428065 (~6/5) = 13.0244547 / 48; b = 1/2
>
> Errors and the 48-et:
>
> 3: 0.49 -1.96
> 5: -9.48 -11.31
> 7: 8.01 6.17
>
> Here's another system with very flat major thirds, and Paul, I
> suppose, would prefer it in the 18+4 version of the 22-et, or 22+4
of
> the 26 et, over 26+22 in the 48-et.

Why would you say that? I only liked 26-tET in contexts where 81:80
was vanishing (probably since my brain could find something familiar
in such music) -- otherwise, I might not. But none of these comments
should be construed to mean that I'd be opposed to something *in
principle* -- only that the result didn't appeal to me when I tried
to get short-term musical satisfaction out of it.