Topic: Equaliized Grail

1 scales

Thread (1 messages)

From: Gene Ward Smith (2003-06-07)
Subject: Equaliized Grail

I've been listening to a lot of circulating temperaments, and grail
continues to knock my socks off. However, we have two fifths 1/2 comma
flat in grail and two fifths 1/2 comma sharp in rms grail, so I
decided to take a linear combination of the two and equalize the
error. This is achieved by taking 4 parts of grail and five parts of
rms grail, or 44.1348% grail to be more exact than Jon would prefer.
This should probably be considered the canonical version of grail, the
"bad" fifths are nine cents sharp or flat, which is
tolerable--certainly a long way from wolfdom.


0 C
 [392.4894413, 305.4637648, 697.9532061] 
 [307.1962474, 390.7569587, 697.9532061]

Major third six cents sharp, minor third ten cents flat.

1 C sharp
 [420.4916312, 290.4598958, 710.9515270] 
 [307.1962474, 403.7552796, 710.9515270]

14/11 third three cents sharp, another ten cent minor third. The fifth
is nine cents sharp.

2 D
 [383.4183699, 314.8731450, 698.2915149] 
 [309.5401042, 388.7514107, 698.2915149]

Major third less than three cents flat, minor third 6.1 cents flat.

3 E flat

 [390.7569587, 314.8731439, 705.6301026] 
 [272.4668434, 433.1632592, 705.6301026]

Major third 4.44 cents sharp, subminor third 5.6 cents sharp

4 E
 [403.7552796, 290.4598951, 694.2151747] 
 [305.4637648, 388.7514099, 694.2151747]

The major third is even sharper than 12-et and the minor third is ten
cents flat,
so this isn't one of the good keys; the fifth is 7.74 cents flat.

5 F
 [388.7514107, 305.4637642, 694.2151749] 
 [290.4598958, 403.7552791, 694.2151749]

A lovely 2.44 cent sharp major third and a 13/11 minor third a mere
1.25 cents sharp; another 7.74 cent flat fifth.

6 F sharp
 [433.1632592, 272.4668439, 705.6301031] 
 [314.8731450, 390.7569581, 705.6301031]

The 9/7 third is 1.92 cents flat; the minor third is 3/4 cent flat.
Not a key to fear.

7 G
 [388.7514099, 309.5401049, 698.2915148] 
 [314.8731439, 383.4183709, 698.2915148]

Another 2.44 cent sharp major third with a better fifth than F or C,
the minor third
is another excellent 3/4 cent flat one. G is a very nice key.
 
8 A flat
 [403.7552791, 307.1962474, 710.9515265] 
 [290.4598951, 420.4916314, 710.9515265]

The same clinker of a major third as with E and an even worse fifth,
one of the
nine cent sharp ones, but the good 13/11 minor third appears once more.

9 A
 [390.7569581, 307.1962474, 697.9532055] 
 [305.4637642, 392.4894413, 697.9532055]

The same 4.44 cent sharp major third as with E flat, the same ten cent
flat minor third as with E,
and the fifth spot-on four cents flat.

10 B flat
 [383.4183709, 309.5401042, 692.9584751] 
 [272.4668439, 420.4916312, 692.9584751]

The same 2.9 cent flat major thirds as D, the same 5.6 cent sharp
subminor third as E flat, and a nine-cent-flat fifth.

11 B
 [420.4916314, 272.4668434, 692.9584748] 
 [309.5401049, 383.4183699, 692.9584748]

The same three cent sharp 14/11 as C sharp, the same 6.1 cent flat
minor third as D, and the same
nine cent flat fifth as B flat.

! graileq.scl
56% RMS grail + 44% JI grail 
12
!
85.293194
196.244721
307.196247
392.489441
505.784825
579.663091
697.953206
796.244721
894.536236
1012.826350
1086.704616
1200.000000