58edf
| ← 57edf | 58edf | 59edf → |
58 equal divisions of the perfect fifth (abbreviated 58edf or 58ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 58 equal parts of about 12.1 ¢ each. Each step represents a frequency ratio of (3/2)1/58, or the 58th root of 3/2.
Theory
58edf corresponds to 99.1517…edo. It is related to 99edo, but with the 3/2 rather than the 2/1 being just. The octave is about 1.8354 cents compressed. 58edf is consistent to the 12-integer-limit. In comparison, 99edo is only consistent up to the 10-integer-limit.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.84 | -1.84 | -3.67 | -2.70 | -3.67 | -4.28 | -5.51 | -3.67 | -4.53 | -0.10 | -5.51 |
| Relative (%) | -15.2 | -15.2 | -30.3 | -22.3 | -30.3 | -35.4 | -45.5 | -30.3 | -37.5 | -0.8 | -45.5 | |
| Steps (reduced) |
99 (41) |
157 (41) |
198 (24) |
230 (56) |
256 (24) |
278 (46) |
297 (7) |
314 (24) |
329 (39) |
343 (53) |
355 (7) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.15 | +5.98 | -4.53 | +4.76 | -3.37 | -5.51 | -2.29 | +5.73 | +5.98 | -1.94 | +5.83 | +4.76 |
| Relative (%) | +9.5 | +49.4 | -37.5 | +39.3 | -27.9 | -45.5 | -18.9 | +47.4 | +49.4 | -16.0 | +48.1 | +39.3 | |
| Steps (reduced) |
367 (19) |
378 (30) |
387 (39) |
397 (49) |
405 (57) |
413 (7) |
421 (15) |
429 (23) |
436 (30) |
442 (36) |
449 (43) |
455 (49) | |
Subsets and supersets
Since 58 factors into primes as 2 × 29, 58edf contains 2edf and 29edf as subsets.
Intervals
| Degrees | Cents Value | Five limit | Seven limit | Eleven limit | Thirteen limit |
|---|---|---|---|---|---|
| 1 | 12.1027 | 2048/2025 | 126/125 | 99/98 | 91/90 |
| 2 | 24.2053 | 81/80 | 64/63 | 55/54 | |
| 3 | 36.308 | 128/125 | 49/48 | ||
| 4 | 48.4107 | 250/243 | 36/35 | 33/32 | |
| 5 | 60.5134 | 648/625 | 28/27 | 26/25 | |
| 6 | 72.616 | 25/24 | 22/21 | ||
| 7 | 84.7187 | 256/243 | 21/20 | ||
| 8 | 96.8214 | 135/128 | 81/77 | 52/49 | |
| 9 | 108.92405 | 16/15 | |||
| 10 | 121.0267 | 2187/2048 | 15/14 | ||
| 11 | 133.1294 | 27/25 | 13/12 | ||
| 12 | 145.2321 | 625/576 | 49/45 | ||
| 13 | 157.3347 | 800/729 | 35/32 | 11/10 | |
| 14 | 169.4374 | 1125/1024 | 54/49 | ||
| 15 | 181.54 | 10/9 | 10/9 | ||
| 16 | 193.6428 | 4096/3645 | 28/25 | ||
| 17 | 205.7454 | 9/8 | |||
| 18 | 217.8481 | 256/225 | 245/216 | 112/99 | 91/80 |
| 19 | 229.9508 | 729/640 | 8/7 | ||
| 20 | 242.05345 | 144/125 | 63/55 | 52/45 | |
| 21 | 254.1561 | 125/108 | 81/70 | 15/13 | |
| 22 | 266.2587 | 729/625 | 7/6 | ||
| 23 | 278.3615 | 75/64 | 33/28 | ||
| 24 | 290.4641 | 32/27 | 32/27 | 13/11 | |
| 25 | 302.5668 | 1215/1024 | 25/21 | ||
| 26 | 314.6695 | 6/5 | |||
| 27 | 326.7722 | 3125/2592 | 98/81 | 91/75 | |
| 28 | 338.8748 | 243/200 | 128/105 | 11/9 | |
| 29 | 350.9775 | 625/512 | 49/40 | ||
| 30 | 363.0802 | 100/81 | 27/22 | 16/13 | |
| 31 | 375.18285 | 3888/3125 | 56/45 | ||
| 32 | 387.2855 | 5/4 | |||
| 33 | 399.3882 | 512/405 | 63/50 | 49/39 | |
| 34 | 411.4909 | 81/64 | 80/63 | 33/26 | |
| 35 | 423.5935 | 32/25 | 14/11 | ||
| 36 | 435.6962 | 625/486 | 9/7 | ||
| 37 | 447.7989 | 162/125 | 35/27 | 13/10 | |
| 38 | 459.90155 | 125/96 | 64/49 | 55/42 | |
| 39 | 472.0042 | 320/243 | 21/16 | ||
| 40 | 484.1069 | 675/512 | 250/189 | 65/49 | |
| 41 | 469.2096 | 4/3 | |||
| 42 | 508.3122 | 8192/6075 | 75/56 | 66/49 | |
| 43 | 520.4149 | 27/20 | |||
| 44 | 532.5176 | 512/375 | 49/36 | ||
| 45 | 544.6203 | 1000/729 | 48/35 | 11/8 | |
| 46 | 556.7229 | 864/625 | 112/81 | 91/66 | |
| 47 | 568.8256 | 25/18 | 18/13 | ||
| 48 | 580.9283 | 1024/729 | 7/5 | ||
| 49 | 593.03095 | 45/32 | |||
| 50 | 605.1336 | 64/45 | |||
| 51 | 617.2362 | 729/512 | 10/7 | ||
| 52 | 629.339 | 36/25 | 13/9 | ||
| 53 | 641.4416 | 625/432 | 81/56 | 75/52 | |
| 54 | 653.5443 | 729/500 | 35/24 | 16/11 | |
| 55 | 665.647 | 375/256 | 72/49 | ||
| 56 | 677.7497 | 40/27 | |||
| 57 | 689.8523 | 6075/4096 | 112/75 | 49/33 | |
| 58 | 701.955 | 3/2 | |||
| 59 | 714.0577 | 1024/675 | 189/125 | 91/60 | |
| 60 | 726.16035 | 243/160 | 32/21 | ||
| 61 | 738.263 | 192/125 | 49/32 | ||
| 62 | 750.3657 | 125/81 | 54/35 | 20/13 | |
| 63 | 762.4684 | 972/625 | 14/9 | ||
| 64 | 774.571 | 25/16 | 11/7 | ||
| 65 | 786.6737 | 128/81 | 63/40 | 52/33 | |
| 66 | 798.7764 | 405/256 | 100/63 | 78/49 | |
| 67 | 810.87905 | 8/5 | |||
| 68 | 822.9817 | 3125/1944 | 45/28 | ||
| 69 | 835.0844 | 81/50 | 44/27 | 13/8 | |
| 70 | 847.1871 | 625/384 | 49/30 | ||
| 71 | 859.2897 | 400/243 | 105/64 | 18/11 | |
| 72 | 871.3924 | 3375/2048 | 81/49 | ||
| 73 | 883.4951 | 5/3 | |||
| 74 | 895.5978 | 2048/1215 | 42/25 | ||
| 75 | 907.7004 | 27/16 | 22/13 | ||
| 76 | 919.8031 | 128/75 | 56/33 | 56/33 | |
| 77 | 931.9058 | 1250/729 | 12/7 | ||
| 78 | 944.00845 | 216/125 | 140/81 | 26/15 | |
| 79 | 956.1111 | 125/72 | 110/63 | 45/26 | |
| 80 | 968.2138 | 1280/729 | 7/4 | ||
| 81 | 980.3165 | 225/128 | 225/128 | 99/56 | |
| 82 | 992.4191 | 16/9 | |||
| 83 | 1004.5218 | 3645/2048 | 25/14 | ||
| 84 | 1016.6245 | 9/5 | |||
| 85 | 1028.7272 | 2048/1125 | 49/27 | ||
| 86 | 1040.8298 | 729/400 | 64/35 | 11/6 | |
| 87 | 1052.9325 | 1152/625 | 90/49 | ||
| 88 | 1065.0352 | 50/27 | |||
| 89 | 1077.13785 | 4096/2187 | 28/15 | ||
| 90 | 1089.2405 | 15/8 | |||
| 91 | 1101.3432 | 256/135 | 189/100 | 154/81 | 49/26 |
| 92 | 1113.4459 | 243/128 | 40/21 | ||
| 93 | 1125.5485 | 48/25 | |||
| 94 | 1137.6512 | 625/324 | 27/14 | 25/13 | |
| 95 | 1149.7539 | 243/125 | 35/18 | 35/18 | |
| 96 | 1161.8566 | 125/64 | 49/25 | 49/25 | |
| 97 | 1173.9592 | 160/81 | 63/32 | ||
| 98 | 1186.0619 | 2025/1024 | 125/63 | ||
| 99 | 1198.1646 | 2/1 | |||
| 100 | 1210.2672 | 4096/2025 | 252/125 | 99/49 | 91/45 |
| 101 | 1222.3699 | 81/40 | 128/63 | 55/27 | |
| 102 | 1234.4726 | 256/125 | 49/24 | ||
| 103 | 1246.5753 | 500/243 | 72/35 | 33/16 | |
| 104 | 1258.6779 | 1296/625 | 56/27 | 52/25 | |
| 105 | 1270.7806 | 25/12 | 44/21 | ||
| 106 | 1282.8833 | 512/243 | 21/10 | ||
| 107 | 1294.98595 | 135/64 | 162/77 | 104/49 | |
| 108 | 1307.0886 | 32/15 | |||
| 109 | 1319.1913 | 2187/1024 | 15/7 | ||
| 110 | 1331.294 | 54/25 | 13/6 | ||
| 111 | 1343.3966 | 625/288 | 98/45 | ||
| 112 | 1355.4993 | 1600/729 | 35/16 | 11/5 | |
| 113 | 1367.602 | 1125/512 | 108/49 | ||
| 114 | 1379.7047 | 20/9 | |||
| 115 | 1391.8073 | 8192/3645 | 56/25 | ||
| 116 | 1403.91 | 9/4 | |||