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 pions 7mus
1 20.339 21.559 26.034 (1A.08B16)
2 40.678 43.119 52.068 (34.11616)
3 61.017 64.678 78.102 (4E.1A116)
4 81.356 86.237 104.136 (68.22816)
5 101.695 107.797 130.1695 (82.2B616)
6 122.034 129.356 156.203 (9C.34116)
7 142.373 150.915 182.237 (B6.3CC16)
8 162.712 172.475 208.271 (D0.45716)
9 183.051 194.034 234.305 (EA.4E216)
10 203.39 215.593 260.339 (104.56C816)
11 223.729 237.1525 286.373 (11E.5F716)
12 244.068 258.712 312.407 (138.68416)
13 264.407 280.271 338.441 (152.70D16)
14 284.746 301.8305 364.475 (16C.79816)
15 305.085 323.39 390.5085 (186.82316)
16 325.424 344.949 416.542 (1A0.8AE16)
17 345.763 366.5085 442.576 (1BA.93816)
18 366.102 388.068 468.61 (1D4.9C316)
19 386.441 409.627 494.644 (1EE.A4E16)
20 406.78 431.186 520.678 (208.AD916)
21 427.119 452.746 546.712 (222.B6416)
22 447.458 474.305 572.746 (23C.BEF16)
23 467.797 495.864 598.778 (256.C7916)
24 488.136 517.424 624.814 (270.D0416)
25 508.475 538.983 650.8475 (28A.D8F16)
26 528.814 560.542 676.881 (2A4.E1A16)
27 549.1525 582.102 702.915 (2BE.EA516)
28 569.4915 603.661 728.949 (2D8.F316)
29 589.8305 625.22 754.983 (2F2.FBB16)
30 610.1695 646.78 781.017 (30D.04516)
31 630.5085 668.339 807.051 (327.0D16)
32 650.8475 689.898 833.085 (341.15B16)
33 671.186 711.458 859.119 (35B.1EA16)
34 691.525 733.017 885.1525 (375.27116)
35 711.864 754.576 911.186 (38F.1FC16)
36 732.203 776.136 937.222 (3A9.39716)
37 752.542 797.695 963.254 (3C3.41116)
38 772.881 819.234 989.288 (3DD.49C16)
39 793.22 840.814 1015.322 (3F7.51716)
40 813.559 862.313 1041.356 (411.5B216)
41 833.898 883.932 1067.39 (42B.63D16)
42 854.237 905.4915 1093.424 (445.6C716)
43 874.576 927.051 1119.458 (45F.75216)
44 894.915 948.61 1145.4915 (479.7DD16)
45 915.254 970.1695 1171.525 (493.86816)
46 935.593 991.729 1197.569 (4AD.8F316)
47 955.932 1013.288 1223.593 (4C7.97C16)
48 976.271 1034.8475 1249.627 (4E1.A0916)
49 996.61 1056.407 1275.661 (4FB.A63816)
50 1016.949 1077.966 1301.695 (515.B1E16)
51 1037.288 1099.525 1327.729 (52F.BA916)
52 1057.627 1121.085 1353.763 (549.C3416,
53 1077.966 1142.644 1379.797 (563.CBF16)
54 1098.305 1164.213 1405.8305 (57D.D4A16)
55 1118.644 1185.763 1431.864 (597.DD816)
56 1138.983 1207.322 1457.898 (5B1.E5F16)
57 1159.322 1228.881 1483.932 (5CB.EEA16)
58 1179.661 1250.441 1509.966 (5E5.F7416)