59edo

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The 59 equal division divides the octave into 59 equal steps of 20.339 cents each.

Theory

59edo's 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.

Intervals

Degrees Cents Approximate Ratios
2.9.5.21.11.17 Subgroup Full 11-limit in Patent Val
0 0.000 1/1 1/1
1 20.339 81/80 50/49, 99/98
2 40.678 128/125 49/48
3 61.017 648/625 25/24, 81/80, 36/35, 33/32
4 81.356 21/20, 22/21
5 101.695 17/16, 18/17 16/15
6 122.034 15/14
7 142.373
8 162.712 11/10 10/9, 11/10, 12/11
9 183.051 10/9
10 203.390 9/8
11 223.729 9/8, 8/7
12 244.068
13 264.407 7/6
14 284.746 20/17
15 305.085
16 325.424 6/5, 11/9
17 345.763 11/9
18 366.102 21/17
19 386.441 5/4 5/4
20 406.780 81/64
21 427.119 32/25 32/25, 14/11
22 447.458 22/17 9/7
23 467.797 21/16
24 488.136 4/3, 21/16
25 508.475
26 528.814
27 549.153 11/8 27/20, 11/8, 15/11
28 569.492 25/18
29 589.831 45/32 7/5
30 610.169 64/45 10/7
31 630.508 36/25
32 650.847 16/11 40/27, 16/11, 22/15
33 671.186
34 691.525
35 711.864 3/2, 32/21
36 732.203 32/21
37 752.542 17/11 14/9
38 772.881 25/16 25/16, 11/7
39 793.220 128/81
40 813.559 8/5 8/5
41 833.898 34/21
42 854.237 18/11
43 874.576 5/3, 18/11
44 894.915
45 915.254 17/10
46 935.593 12/7
47 955.932
48 976.271 16/9, 7/4
49 996.610 16/9
50 1016.949 9/5
51 1037.288 20/11 9/5, 11/6, 20/11
52 1057.627
53 1077.966 28/15
54 1098.305 17/9, 32/17 15/8
55 1118.644 40/21, 21/11
56 1138.983 625/324 48/25, 160/81, 35/18, 64/33
57 1159.322 125/64 96/49
58 1179.661 160/81 49/25, 196/99
59 1200.000 2/1 2/1