59edo

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The 59 equal division divides the octave into 59 equal steps of 20.339 cents each. Its best fifth is stretched about 9.91 cents from the just interval, and yet its major third is nearly pure (stretched only 0.127 cents), as the denominator of a convergent to log25. It is a good porcupine tuning, giving in fact the optimal patent val for 11-limit porcupine. This patent val tempers out 250/243 in the 5-limit, 64/63 and 16875/16807 in the 7-limit, and 55/54, 100/99 and 176/175 in the 11-limit. 59edo is an excellent tuning for the 2.9.5.21.11 11-limit 2*59 subgroup, on which it takes the same tuning and tempers out the same commas as 118et. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the 50&59 temperament with a subminor third generator provides an interesting temperament.

Using the flat fifth instead of the sharp one allows for the 12&35 temperament, which is a kind of bizarre cousin to garibaldi temperament with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth.

59edo is the 17th prime edo.

Degrees Cents Value
1 20.339
2 40.678
3 61.017
4 81.356
5 101.695
6 122.034
7 142.373
8 162.712
9 183.051
10 203.390
11 223.729
12 244.068
13 264.407
14 284.746
15 305.085
16 325.424
17 345.763
18 366.102
19 386.441
20 406.780
21 427.119
22 447.458
23 467.797
24 488.136
25 508.475
26 528.814
27 549.153
28 569.492
29 589.831
30 610.169
31 630.508
32 650.847
33 671.186
34 691.525
35 711.864
36 732.203
37 752.542
38 772.881
39 793.220
40 813.559
41 833.898
42 854.237
43 874.576
44 894.915
45 915.254
46 935.593
47 955.932
48 976.271
49 996.610
50 1016.949
51 1037.288
52 1057.627
53 1077.966
54 1098.305
55 1118.644
56 1138.983
57 1159.322
58 1179.661