A 19 tone circulating temperament with exact brats

The most interesting brats in the neighborhood of the 19-equal
meantone fifth would seem to be these (which I give together with the
associated brats)

-1 1 -1
0 5/3 infinity
1/4 2 2
3/7 7/3 1
1/2 5/2 4/5
2/3 3 1/2

Below I give a Scala file for a circulating temperament which makes
use of these brats. The brats, around a circle of fifths, go

-1 -1 -1 -1 -1 -1 -1 -1 0 3/7 390808/591947 1/2 1/2 1/2 1/2 1/4 0 0 -1

The 390808/591947 brat is approxiately 2/3, so all the interesting
brats get used. Instead of having this brat be approximate, we could
pick another, or spread the error, but it seems unlikely this would
make any real difference.

! cbrat19.scl
circulating 19-tone temperament with exact brats
! brats = -1 -1 -1 -1 -1 -1 -1 -1 0 3/7 390808/591947 1/2 1/2 1/2 1/2
1/4 0 0 -1
! adapted from secor_19wt.scl
19
!
3688037/3546660
9545591/8866650
197729/177333
686317/591110
6815759/5674656
441637/354666
1149379/886665
395458/295555
2468497/1773330
1280918/886665
264304/177333
28241/18188
1141103/709332
493284/295555
1026089/591110
528608/295555
329881/177333
686317/354666
2

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> The most interesting brats in the neighborhood of the 19-equal
> meantone fifth would seem to be these (which I give together with
the
> associated brats)
>
> -1 1 -1
> 0 5/3 infinity
> 1/4 2 2
> 3/7 7/3 1
> 1/2 5/2 4/5
> 2/3 3 1/2
>
> Below I give a Scala file for a circulating temperament which makes
> use of these brats. The brats, around a circle of fifths, go
>
> -1 -1 -1 -1 -1 -1 -1 -1 0 3/7 390808/591947 1/2 1/2 1/2 1/2 1/4 0
0 -1
>
> The 390808/591947 brat is approxiately 2/3, so all the interesting
> brats get used. Instead of having this brat be approximate, we could
> pick another, or spread the error, but it seems unlikely this would
> make any real difference.
>
> ! cbrat19.scl
> circulating 19-tone temperament with exact brats
> ! brats = -1 -1 -1 -1 -1 -1 -1 -1 0 3/7 390808/591947 1/2 1/2 1/2
1/2
> 1/4 0 0 -1
> ! adapted from secor_19wt.scl
> 19
> !
> 3688037/3546660
> 9545591/8866650
> 197729/177333
> 686317/591110
> 6815759/5674656
> 441637/354666
> 1149379/886665
> 395458/295555
> 2468497/1773330
> 1280918/886665
> 264304/177333
> 28241/18188
> 1141103/709332
> 493284/295555
> 1026089/591110
> 528608/295555
> 329881/177333
> 686317/354666
> 2

Hmmm, interesting!

But did you really intend that C-G be tempered as much as 11 cents
(beat rate ~10 Hz)?

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> But did you really intend that C-G be tempered as much as 11 cents
> (beat rate ~10 Hz)?

Not really a good choice. Below I give a revised version in a
transposed mode. This gives a sweet C major in the center of the -1
brat range, and makes the cheesiest key Fb=E# major. Now if only
they'd gone in for five split black keys, how much better life could
have been!

! cbrat19.scl
circulating 19-tone temperament with exact brats
! brats = -1 -1 -1 -1 -1 0 3/7 390808/591947 1/2 1/2 1/2 1/2 1/4 0 0
-1 -1 -1 -1
! adapted from secor_19wt.scl
19
!
686317/659762
354666/329881
3688037/3298810
9545591/8247025
395458/329881
2058951/1649405
6815759/5278096
441637/329881
2298758/1649405
2372748/1649405
2468497/1649405
2561836/1649405
528608/329881
1101399/659762
1141103/659762
2959704/1649405
3078267/1649405
3171648/1649405
2