Grail

I spent some time trying to find a temperament of a type which I named to myself the "holy grail" type. This is "grail", the first such a one I found. For a grail-type temperament, I wanted the following attributes:

(1) Near the key center, it functions as an honest meantone, with major thirds not too far off pure

(2) All fifths must be usable; no wolf fifths are allowed

(3) All thirds in all keys must be functional

The key idea is to have two pure 14/11 major thirds and one pure 9/7 major third in the remote keys. By giving them a precise just value, even if that value is not 5/4, we get something which doesn't sound bad. For the minor thirds, I was happy to get two fine 7/6 subminors and settled for a couple of 13/11 intermediate minor thirds.

Here are the fifths, major thirds, and minor thirds above each scale degree, in cents:

0 695.623009 391.246018 304.376991

1 708.753982 417.507964 304.376991

2 700.000000 382.457951 308.753982

3 708.788065 391.246018 273.703969

4 695.623008 404.376991 304.376991

5 695.623009 391.246018 291.246018

6 708.788067 435.084096 317.542049

7 700.000000 391.246017 317.542047

8 708.753982 404.376991 291.246017

9 695.623009 391.246018 304.376991

10 691.211935 382.457953 273.703971

11 691.211934 417.507965 308.753983

The major thirds on degree 1 and 11 are exact 14/11's, and on degree 6 we have the 9/7. On degree 3 and 10 we get minor thirds of 273.7 cents, which are good subminor thirds (to my mind, this is very acceptable.) On degrees 5 and 8 we have 291 cent minor thirds, whichis pretty close to a 13/11 at 289 cents. All the other minor thirds improve on those of 12-equal. We have three sharp fifths a tad flatter than the 22-et fifth, which is quite acceptable, two of 12-et's excellent fifths, three 5/17-comma meantone fifths and two 1/2-comma meantone fifths; the last (over degrees 10 and 11) are the only fifths which might be kind of problem, being "flattone" fifths a cent flatter than the 26-equal fifth.

Here is the Scala scl file:

! grail.scl
Holy Grail circulating temperament with two 14/11 and one 9/7 major third
12
!
86.869027
195.623009
304.376991
391.246018
504.376991
578.080960
695.623009
795.623009
895.623009
1013.165056
1086.869026
1200.000000

Music in graiil