58edo
The 58 equal temperament, often abbreviated 58-tET, 58-EDO, or 58-ET, is the scale derived by dividing the octave into 58 equally-sized steps. Each step represents a frequency ratio of 20.69 cents. It tempers out 2048/2025, 126/125, 1728/1715, 144/143, 176/175, 896/891, 243/242, 5120/5103, 351/350, 364/363, 441/440, and 540/539, and is a strong system in the 11, 13 and 17-limits. It is the smallest equal temperament which is consistent through the 17-limit, and is also the first et to map the entire 11-limit tonality diamond to distinct scale steps, and hence the first et which can define a version of the famous 43-note Genesis scale of Harry Partch. It supports hemififths, myna, diaschismic, harry, mystery, buzzard and thuja temperaments, and supplies the optimal patent val for 7-, 11- and 13-limit diaschismic, 11- and 13-limit hemififths, 11- and 13-limit thuja, and 13-limit myna. It also supplies the optimal patent val for the 13-limit rank three temperaments thrush, bluebird, aplonis and jofur.
While the 17th harmonic is a cent and a half flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system. 58 = 2*29, and 58 shares the same excellent fifth with 29edo.
Scales
Intervals
| degree of 58edo | cents value | pions | 7mus | ratios |
| 0 | 0.00 | 1/1 | ||
| 1 | 20.69 | 21.93 | 26.48 (1A.7C16) | 56/55, 64/63, 81/80, 128/125 |
| 2 | 41.38 | 43.86 | 52.97 (34.F716) | 36/35, 49/48, 50/49, 55/54 |
| 3 | 62.07 | 65.79 | 79.45 (4F.7216) | 25/24, 26/25, 27/26, 28/27, 33/32 |
| 4 | 82.76 | 87.72 | 105.93 (69.EE16) | 21/20, 22/21 |
| 5 | 103.45 | 109.655 | 132.41 (84.6916) | 16/15, 17/16, 18/17 |
| 6 | 124.14 | 131.59 | 158.9 (9E.E5816) | 14/13, 15/14, 27/25 |
| 7 | 144.83 | 153.52 | 185.38 (B9.6116) | 12/11, 13/12 |
| 8 | 165.52 | 175.45 | 211.86 (D3.DD16) | 11/10 |
| 9 | 186.21 | 197.38 | 238.345 (EE.5816) | 10/9 |
| 10 | 206.9 | 219.31 | 264.83 (108.D316) | 9/8, 17/15 |
| 11 | 227.59 | 241.24 | 291.31 (123.4F16) | 8/7 |
| 12 | 248.28 | 263.17 | 317.79 (13D.CB16) | 15/13 |
| 13 | 268.97 | 285.1 | 344.28 (158.4716) | 7/6 |
| 14 | 289.655 | 307.035 | 370.76 (172.C216) | 13/11, 20/17 |
| 15 | 310.345 | 328.97 | 397.24 (18D.3E16) | 6/5 |
| 16 | 331.035 | 350.9 | 423.72 (1A7.B916) | 17/14 |
| 17 | 351.72 | 372.83 | 450.21 (1C2.3516) | 11/9, 16/13 |
| 18 | 372.41 | 394.76 | 476.69 (1DC.B116) | 21/17 |
| 19 | 393.1 | 416.69 | 503.17 (1F7.2C16) | 5/4 |
| 20 | 413.79 | 438.62 | 529.655 (211.A716) | 14/11 |
| 21 | 434.48 | 460.55 | 556.14 (22C.2316) | 9/7 |
| 22 | 455.17 | 482.48 | 582.62 (246.9F16) | 13/10, 17/13, 22/17 |
| 23 | 475.86 | 504.41 | 609.1 (261.1A816) | 21/16 |
| 24 | 496.55 | 526.345 | 635.59 (27B.9616) | 4/3 |
| 25 | 517.24 | 548.28 | 662.07 (296.1216) | 27/20 |
| 26 | 537.93 | 570.21 | 688.55 (2B0.5D16) | 15/11 |
| 27 | 558.62 | 592.14 | 715.035 (2CB.0916) | 11/8, 18/13 |
| 28 | 579.31 | 614.07 | 741.52 (2F5.8416) | 7/5 |
| 29 | 600 | 636 | 768 (30016) | 17/12, 24/17 |
| 30 | 620.69 | 657.93 | 794.48 (31A.7C16) | 10/7 |
| 31 | 641.38 | 679.86 | 820.97 (334.F716) | 13/9, 16/11 |
| 32 | 662.07 | 701.79 | 847.45 (34F.7216) | 22/15 |
| 33 | 682.76 | 723.72 | 873.93 (369.EE16) | 40/27 |
| 34 | 703.45 | 745.655 | 900.41 (384.6916) | 3/2 |
| 35 | 724.14 | 767.59 | 926.9 (39E.E5816) | 32/21 |
| 36 | 744.83 | 789.52 | 953.38 (3B9.6116) | 20/13, 26/17, 17/11 |
| 37 | 765.52 | 811.45 | 979.86 (3D3.DD16) | 14/9 |
| 38 | 786.21 | 833.38 | 1006.345 (3EE.5816) | 11/7 |
| 39 | 806.9 | 855.31 | 1032.83 (408.D316) | 8/5 |
| 40 | 827.59 | 877.24 | 1059.31 (423.4F16) | 34/21 |
| 41 | 848.28 | 899.17 | 1085.79 (43D.CB16) | 13/8, 18/11 |
| 42 | 868.97 | 921.1 | 1112.28 (458.4716) | 28/17 |
| 43 | 889.655 | 943.035 | 1138.76 (472.C216) | 5/3 |
| 44 | 910.345 | 964.97 | 1165.24 (48D.3E16) | 22/13, 17/10 |
| 45 | 931.035 | 986.9 | 1191.72 (4A7.B916) | 12/7 |
| 46 | 951.72 | 1008.83 | 1218.21 (4C2.3516) | 26/15 |
| 47 | 972.41 | 1030.76 | 1244.69 (4DC.B116) | 7/4 |
| 48 | 993.1 | 1052.69 | 1271.17 (4F7.2C16) | 16/9 |
| 49 | 1013.79 | 1074.62 | 1297.655 (511.A716) | 9/5 |
| 50 | 1034.48 | 1096.55 | 1324.14 (52C.2316) | 20/11 |
| 51 | 1055.17 | 1118.48 | 1350.62 (546.9F16) | 11/6, 24/13 |
| 52 | 1075.86 | 1140.41 | 1377.1 (561.1A816) | 13/7, 28/15 |
| 53 | 1096.55 | 1162.345 | 1403.59 (57B.9616) | 15/8, 32/17, 17/9 |
| 54 | 1117.24 | 1184.28 | 1430.07 (596.1216) | 40/21, 21/11 |
| 55 | 1137.93 | 1206.21 | 1456.55 (5B0.5D16) | 48/25, 25/13, 52/27, 27/14, 64/33 |
| 56 | 1158.62 | 1228.14 | 1483.035 (5CB.0916) | 35/18, 96/49, 49/25, 108/55 |
| 57 | 1179.31 | 1250.07 | 1509.52 (5F5.8416) | 55/28, 63/32, 160/81, 125/64 |
Rank two temperaments
| Period | Generator | Name |
|---|---|---|
| 1\1 | 1\58 | |
| 3\58 | ||
| 5\58 | ||
| 7\58 | ||
| 9\58 | ||
| 11\58 | Gorgik | |
| 13\58 | ||
| 15\58 | Myna | |
| 17\58 | Hemififths | |
| 19\58 | ||
| 21\58 | ||
| 23\58 | Buzzard | |
| 25\58 | ||
| 27\58 | Thuja | |
| 1\2 | 1\58 | |
| 2\58 | ||
| 3\58 | ||
| 4\58 | Harry | |
| 5\58 | Srutal/Diaschismic | |
| 6\58 | ||
| 7\58 | ||
| 8\58 | Echidna, Supers | |
| 9\58 | Secant | |
| 10\58 | ||
| 11\58 | ||
| 12\58 | Sruti | |
| 13\58 | ||
| 14\58 | ||
| 1\29 | 1\58 | Mystery |