51edf
Division of the just perfect fifth into 51 equal parts (51EDF) is related to 87 edo, but with the 3/2 rather than the 2/1 being just. The octave is compressed by about 2.5474 cents and the step size is about 13.7638 cents (corresponding to 87.1851 edo).
| ← 50edf | 51edf | 52edf → |
Unlike 87edo, it is only consistent up to the 6-integer-limit, with discrepancy for the 7th harmonic.
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.55 | -2.55 | -6.02 | +3.31 | +5.36 | +5.19 | -5.03 | -4.90 | -5.33 | +6.28 | +0.94 |
| Relative (%) | -18.5 | -18.5 | -43.7 | +24.1 | +38.9 | +37.7 | -36.6 | -35.6 | -38.7 | +45.7 | +6.8 | |
| Steps (reduced) |
87 (36) |
138 (36) |
202 (49) |
245 (41) |
302 (47) |
323 (17) |
356 (50) |
370 (13) |
394 (37) |
424 (16) |
432 (24) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.57 | -1.36 | -1.23 | -3.82 | -5.36 | +1.67 | -0.99 | +1.76 | -2.29 | +4.68 | +5.57 |
| Relative (%) | -18.7 | -9.9 | -8.9 | -27.7 | -38.9 | +12.1 | -7.2 | +12.8 | -16.6 | +34.0 | +40.4 | |
| Steps (reduced) |
454 (46) |
467 (8) |
473 (14) |
484 (25) |
499 (40) |
513 (3) |
517 (7) |
529 (19) |
536 (26) |
540 (30) |
550 (40) | |
Intervals
| Degree | Cents |
|---|---|
| 0 | |
| 1 | 13.7638 |
| 2 | 27.52765 |
| 3 | 41.2915 |
| 4 | 55.0555 |
| 5 | 68.8191 |
| 6 | 82.5829 |
| 7· | 96.3468 |
| 8 | 110.1106 |
| 9 | 123.8744 |
| 10 | 137.6382 |
| 11 | 151.0206 |
| 12· | 165.1659 |
| 13 | 178.9297 |
| 14 | 192.6935 |
| 15 | 206.45735 |
| 16 | 220.2212 |
| 17· | 233.985 |
| 18 | 248.7488 |
| 19 | 261.51265 |
| 20 | 275.2765 |
| 21 | 289.0403 |
| 22· | 302.8041 |
| 23 | 316.5679 |
| 24 | 330.3318 |
| 25 | 344.0955 |
| 26 | 357.8594 |
| 27 | 371.6232 |
| 28 | 385.3871 |
| 29 | 399.1509 |
| 30 | 412.9147 |
| 31 | 426.6785 |
| 32 | 440.44253 |
| 33 | 455.2062 |
| 34 | 467.97 |
| 35 | 481.7338 |
| 36 | 495.49765 |
| 37 | 509.2615 |
| 38 | 523.0253 |
| 39 | 536.7891 |
| 40 | 550.5529 |
| 41 | 564.3168 |
| 42 | 578.0806 |
| 43 | 591.8444 |
| 44 | 605.6082 |
| 45 | 619.3721 |
| 46 | 633.1359 |
| 47 | 646.8997 |
| 48 | 660.6635 |
| 49 | 674.42735 |
| 50 | 688.1912 |
| 51 | 701.955 |
| 52 | 715.7188 |
| 53 | 729.48365 |
| 54 | 743.2465 |
| 55 | 757.0103 |
| 56 | 770.7741 |
| 57 | 784.5379 |
| 58 | 798.3018 |
| 59 | 812.0656 |
| 60 | 825.8294 |
| 61 | 839.5932 |
| 62 | 853.3571 |
| 63 | 867.1209 |
| 64 | 880.8847 |
| 65 | 894.6485 |
| 66 | 908.41235 |
| 67 | 922.1762 |
| 68 | 935.94 |
| 69 | 949.7038 |
| 70 | 963.46765 |
| 71 | 977.2315 |
| 72 | 990.9952 |
| 73 | 1004.7591 |
| 74 | 1018.5229 |
| 75 | 1032.32868 |
| 76 | 1046.0506 |
| 77 | 1059.8144 |
| 78 | 1073.5782 |
| 79 | 1087.3421 |
| 80 | 1101.1059 |
| 81 | 1114.8697 |
| 82 | 1128.6335 |
| 83 | 1142.39735 |
| 84 | 1156.1612 |
| 85 | 1169.925 |
| 86 | 1183.6888 |
| 87 | 1197.45265 |
| 88 | 1211.2165 |
| 89 | 1224.9803 |
| 90 | 1238.7441 |
| 91 | 1252.5079 |
| 92 | 1266.2718 |
| 93 | 1280.0356 |
| 94 | 1293.7994 |
| 95 | 1307.5632 |
| 96 | 1321.3271 |
| 97 | 1335.0909 |
| 98 | 1348.8547 |
| 99 | 1362.6185 |
| 100 | 1376.3824 |
| 101 | 1390.1462 |
| 102 | 1403.91 |
|
Todo: complete table
Add a third column that comments on the intervals, either what JI they approximate, what they are named, or how they can be used musically. |