59edo
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. As every other step of 118edo, 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. 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 |