# 59edo

(Redirected from 59-edo)

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