40edf
| ← 39edf | 40edf | 41edf → |
40 equal divisions of the perfect fifth (abbreviated 40edf or 40ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 40 equal parts of about 17.5 ¢ each. Each step represents a frequency ratio of (3/2)1/40, or the 40th root of 3/2.
Theory
40edf corresponds to 68.3805edo. It is related to the regular temperament which tempers out 2401/2400, 9801/9800, and 9453125/9437184 in the 11-limit, which is supported by 68edo, 274edo, 342edo, 410edo, 616edo, and 752edo among others.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -6.68 | -6.68 | +4.20 | +3.96 | +4.20 | +0.56 | -2.48 | +4.20 | -2.72 | +7.77 | -2.48 |
| Relative (%) | -38.0 | -38.0 | +23.9 | +22.6 | +23.9 | +3.2 | -14.1 | +23.9 | -15.5 | +44.3 | -14.1 | |
| Steps (reduced) |
68 (28) |
108 (28) |
137 (17) |
159 (39) |
177 (17) |
192 (32) |
205 (5) |
217 (17) |
227 (27) |
237 (37) |
245 (5) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.66 | -6.12 | -2.72 | +8.39 | +8.73 | -2.48 | -8.34 | +8.15 | -6.12 | +1.09 | -5.67 |
| Relative (%) | -3.8 | -34.9 | -15.5 | +47.8 | +49.7 | -14.1 | -47.5 | +46.5 | -34.9 | +6.2 | -32.3 | |
| Steps (reduced) |
253 (13) |
260 (20) |
267 (27) |
274 (34) |
280 (0) |
285 (5) |
290 (10) |
296 (16) |
300 (20) |
305 (25) |
309 (29) | |
Intervals
| degree | cents value | corresponding JI intervals |
comments |
|---|---|---|---|
| 0 | exact 1/1 | ||
| 1 | 17.5489 | 100/99, 99/98, 81/80 | |
| 2 | 35.09775 | 50/49, 49/48 | |
| 3 | 52.6466 | 33/32 | |
| 4 | 70.1955 | 25/24 | |
| 5 | 87.7444 | 20/19 | |
| 6 | 105.29325 | 17/16 | |
| 7 | 122.8421 | 15/14 | |
| 8 | 140.391 | 13/12 | |
| 9 | 157.9399 | 23/21 | |
| 10 | 175.48875 | 10/9 | |
| 11 | 193.0376 | 19/17 | |
| 12 | 210.5865 | 17/15 | |
| 13 | 228.1354 | 8/7 | |
| 14 | 245.68425 | 15/13 | |
| 15 | 263.2331 | 7/6 | |
| 16 | 280.782 | 20/17 | |
| 17 | 298.3309 | 19/16 | |
| 18 | 315.87975 | 6/5 | |
| 19 | 333.4286 | 63/52 | |
| 20 | 350.9775 | 60/49, 49/40 | |
| 21 | 368.5264 | 26/21 | |
| 22 | 386.07525 | 5/4 | |
| 23 | 403.6241 | 24/19 | |
| 24 | 421.173 | 51/40 | |
| 25 | 438.7219 | 9/7 | |
| 26 | 456.27075 | 13/10 | |
| 27 | 473.8196 | 21/16 | |
| 28 | 491.3685 | 4/3 | |
| 29 | 508.9174 | 66/49 | |
| 30 | 526.46625 | 200/147, 49/36, 27/20 | |
| 31 | 544.0151 | 11/8 | |
| 32 | 561.564 | 25/18 | |
| 33 | 579.1129 | 7/5 | |
| 34 | 596.66175 | 24/17 | |
| 35 | 614.2106 | 10/7 | |
| 36 | 631.7595 | 36/25 | |
| 37 | 649.3084 | 16/11 | |
| 38 | 666.85725 | 147/100, 72/49 | |
| 39 | 684.4061 | 297/200, 49/33, 40/27 | |
| 40 | 701.955 | exact 3/2 | just perfect fifth |
| 41 | 719.5039 | 50/33, 297/196, 243/160, 32/21 | |
| 42 | 737.05275 | 75/49, 49/32 | |
| 43 | 754.6016 | 99/64 | |
| 44 | 772.1505 | 25/16 | |
| 45 | 789.6994 | 30/19 | |
| 46 | 807.24825 | 51/32 | |
| 47 | 824.7971 | 45/28 | |
| 48 | 842.346 | 13/8 | |
| 49 | 859.8949 | 32/14 | |
| 50 | 877.44375 | 5/3 | |
| 51 | 894.9926 | 57/34 | |
| 52 | 912.5415 | 17/10 | |
| 53 | 930.0904 | 12/7 | |
| 54 | 947.63925 | 45/26 | |
| 55 | 965.1981 | 7/4 | |
| 56 | 982.737 | 30/17 | |
| 57 | 1000.2859 | 57/32 | |
| 58 | 1017.83275 | 9/5 | |
| 59 | 1035.3836 | 189/104 | |
| 60 | 1052.9325 | 90/49, 147/80 | |
| 61 | 1070.4814 | 13/7 | |
| 62 | 1088.03025 | 15/8 | |
| 63 | 1105.4791 | 36/19 | |
| 64 | 1123.128 | 153/80 | |
| 65 | 1140.6769 | 27/14 | |
| 66 | 1158.22575 | 39/20 | |
| 67 | 1175.7746 | 63/32 | |
| 68 | 1193.3235 | 2/1 | |
| 69 | 1210.8724 | 99/49 | |
| 70 | 1228.42125 | 300/147, 49/24, 81/40 | |
| 71 | 1246.9701 | 33/16 | |
| 72 | 1263.519 | 25/12 | |
| 73 | 1281.0679 | 21/10 | |
| 74 | 1298.61675 | 36/17 | |
| 75 | 1316.1756 | 15/7 | |
| 76 | 1333.7145 | 54/25 | |
| 77 | 1351.2634 | 24/11 | |
| 78 | 1368.81375 | 441/200, 108/49 | |
| 79 | 1386.3611 | 891/400, 49/22, 20/9 | |
| 80 | 1403.91 | exact 9/8 | |
Related regular temperaments
Adding one half of the octave as a generator, 40EDF leads the regular temperament which tempers out 2401/2400, 9801/9800, and 9453125/9437184 in the 11-limit.
11-limit 68&342
Commas: 2401/2400, 9801/9800, 9453125/9437184
POTE generator: ~99/98 = 17.545
Mapping: [<2 2 4 5 8|, <0 40 22 21 -37|]
EDOs: 68, 274, 342, 410, 616, 752
2.3.5.7.11.17 subgroup 68&342
Commas: 1089/1088, 1225/1224, 2401/2400, 24576/24565
POTE generator: ~99/98 = 17.546
Mapping: [<2 2 4 5 8 8|, <0 40 22 21 -37 6|]
EDOs: 68, 274, 342, 410, 616, 752
2.3.5.7.11.17.19 subgroup 68&342
Commas: 1089/1088, 1225/1224, 1445/1444, 1617/1615, 2401/2400
POTE generator: ~96/95 = 17.547
Mapping: [<2 2 4 5 8 8 8|, <0 40 22 21 -37 6 17|]
EDOs: 68, 274, 342, 410, 616, 752h
2.3.5.7.11.17.19.23 subgroup 68&342
Commas: 875/874, 1089/1088, 1225/1224, 1445/1444, 1617/1615, 2024/2023
POTE generator: ~96/95 = 17.546
Mapping: [<2 2 4 5 8 8 8 7|, <0 40 22 21 -37 6 17 70|]
EDOs: 68, 274, 342, 410, 616i, 752h