Re: [tuning-math] 114 temperaments in the big parade

Gene wrote:
>Here are the temperaments I spoke of. I give the scale degrees, the
>circle of fifths, the major thirds and the brats, in that order. Note
>that the brats are all exact!

Impressive. Do rational scales also exist with exact beat ratios P5/M3
of 0 and 1/5?

Manuel

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> If anyone can explain how to evaluate these, I'm interested. Saying
> brute force doesn't help, since I don't know what I should try to
>force.

how about finding the ones that best approximate werckmeister III,
kirnberger II, kirnberger III, and valotti, for a start. be sure to
try all rotation of each.

! werck3.scl
!
Andreas Werckmeister's temperament III (the most famous one, 1681)
12
!
256/243
192.18000
32/27
390.22500
4/3
1024/729
696.09000
128/81
888.26999
16/9
1092.18000
2/1

! kirnberger2.scl
!
Kirnberger 2: 1/2 synt. comma. "Die Kunst des reinen Satzes" (1774)
12
!
135/128
9/8
32/27
5/4
4/3
45/32
3/2
405/256
895.11186
16/9
15/8
2/1

! kirnberger3.scl
!
Kirnberger 3: 1/4 synt. comma (1744)
12
!
135/128
193.15686
32/27
5/4
4/3
45/32
696.57843
405/256
889.73529
16/9
15/8
2/1

Vallotti & Young scale (Vallotti version)
12
!
94.135
196.090
298.045
392.180
501.955
592.180
698.045
796.090
894.135
1000.000
1090.225
2

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

Thanks; I was planning to look at beat ratios for some well-known
temperaments (starting from meantones wasn't getting me anywhere, for
one thing.)