RE: JI approximation to 22ET

Ben Johnston used a 22-tone 7-limit just tuning in his 4th string
quartet that was fairly equally spaced:

# ratio cents
0: 1/1 0
1: 28/27 63
2: 16/15 112
3: 10/9 182
4: 8/7 231
5: 7/6 267
6: 6/5 316
7: 5/4 386
8: 9/7 435
9: 21/16 471
10: 27/20 520
11: 45/32 590
12: 81/56 639
13: 3/2 702
14: 14/9 765
15: 8/5 814
16: 5/3 884
17: 12/7 933
18: 7/4 969
19: 9/5 1018
20: 15/8 1088
21: 27/14 1137

Kraehenbuehl & Schmidt also proposed a 22-tone 7-limit just tuning, with
"inflections" for some of the tones:

0: 1/1 0
1: 49/48 36
36/35 49
25/24 71
2: 21/20 84
16/15 112
15/14 119
3: 35/32 155
4: 9/8 204
8/7 231
5: 7/6 267
25/21 302
6: 6/5 316
128/105 343
7: 49/40 351
5/4 386
8: 9/7 435
9: 21/16 471
4/3 498
75/56 506
10: 27/20 520
48/35 547
11: 7/5 583
45/32 590
10/7 617
12: 35/24 653
13: 3/2 702
32/21 729
14: 49/32 738
54/35* 751
25/16 773
15: 63/40 786
8/5 814
45/28 821
16: 5/3 884
17: 27/16 906
12/7 933
18: 7/4 969
25/14 1004
19: 9/5 1018
175/96 1039
64/35 1045
20: 28/15 1081
15/8 1088
40/21 1116
21: 27/14 1137

*The original text had 56/35, a typo.

The number of steps assigned to all these ratios in both tunings agrees
with 22TET when displayed on a 7-limit lattice. Right, Manuel?


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