Ratwolf

This is not exactly another circulating temperament of grail type; instead,
it is a version of meantone which tries its hardest to circulate. It
does that by having good thirds and by attempting to tame the wolf by
making it exactly 20/13, which is a 13-limit consonance. The name
comes from "rational wolf".

The 20/13 together with a 9/7 third make up what might be called a (major) ratwolf triad, for systems in which 351/350 is a "comma" (an interval identified with unison, such as 81/80 in meantone systems.) The major ratwolf triad is 1--9/7--20/13, with 9/7--20/13 playing the role of a 6/5 due to the 351/350~1. The minor ratwolf triad is 1--6/5--20/13, with 6/5--20/13 now identified as a 9/7. A relative of ratwolf is "wilwolf", which uses the Wilson meantone fifth (the positive real root of f^4 - 2f - 2 = 0.) This of course no longer has a pure 20/13, but it makes sense to use it when the wolf fifth does not make an appearance but the supermajor thirds are important, as these will now be in better tune, and we have the advantage of the subtle shimmering effect of the Wilson major triad.

Here is the triad breakdown. We have eight 5/4s, 2.96 cents flat, and
four 9/7s, 1.76 cents flat. We likewise have eight 6/5s, 3.15 cents
flat, and four 7/6s, 4.33 cents flat. The thirds in this vicinity of
meantone rock. We also of course have eleven meantone fifths, 6.12
cents flat, and a pure 20/13.

1 [433.2989450, 262.5386859, 695.8376309]
2 [383.3505284, 312.4871041, 695.8376325]
3 [383.3505268, 312.4871042, 695.8376310]
4 [383.3505264, 312.4871053, 695.8376317]
5 [383.3505267, 312.4871045, 695.8376312]
6 [433.2989446, 262.5386878, 695.8376324]
7 [383.3505282, 312.4871030, 695.8376312]
8 [433.2989453, 312.4871050, 745.7860503]
9 [383.3505283, 312.4871045, 695.8376328]
10 [383.3505270, 312.4871058, 695.8376328]
11 [433.2989450, 262.5386864, 695.8376314]
12 [383.3505283, 312.4871035, 695.8376318]

! ratwolf.scl
!
Eleven fifths of (418/5)^(1/11) and one 20/13 wolf
12
!
70.863424
191.675263
312.487105
383.350528
504.162369
575.025791
695.837632
766.701055
887.512896
1008.324736
1079.188160
1200.000000

Listen to them--the children of the night. What music they make!

Music in ratwolf

Music in wilwolf