75edo
75edo divides the octave into 75 equal parts of exactly 16 cents each. In the 5-limit, it tempers out the tetracot comma, 20000/19683 and the semicomma 2109375/2097152, and provides a good tuning for tetracot temperament. In the 7-limit, it tenpers 225/224 and 1728/1715. In the 11-limit, 75e val scores lower in badness than the patent val, tempers 100/99 and 243/242, whereas the patent val tempers 99/98 and 121/120. In the 13-limit, it tempers 325/324 and 512/507, 17-limit 120/119 and 256/255 and 19-limit 190/189 and 250/247.
It provides the optimal patent val for the 12&51 temperament in the 7-limit and the 31&44 temperament in the 19-limit.
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.04 | -2.31 | +7.17 | +4.09 | -7.32 | +7.47 | -0.27 | +7.04 | +6.49 | -6.78 | -4.27 |
| Relative (%) | +12.8 | -14.5 | +44.8 | +25.6 | -45.7 | +46.7 | -1.7 | +44.0 | +40.5 | -42.4 | -26.7 | |
| Steps (reduced) |
119 (44) |
174 (24) |
211 (61) |
238 (13) |
259 (34) |
278 (53) |
293 (68) |
307 (7) |
319 (19) |
329 (29) |
339 (39) | |
Since it's part of the Fibonacci sequence beginning with 5 and 12, it closely approximates peppermint temperament. The size of its fifth is exactly 704c, which is very close to the peppermint fifth of 704.096c. This makes it suitable for neo-Gothic tunings.
Intervals
| Step | Cents |
| 0 | 0 |
| 1 | 16 |
| 2 | 32 |
| 3 | 48 |
| 4 | 64 |
| 5 | 80 |
| 6 | 96 |
| 7 | 112 |
| 8 | 128 |
| 9 | 144 |
| 10 | 160 |
| 11 | 176 |
| 12 | 192 |
| 13 | 208 |
| 14 | 224 |
| 15 | 240 |
| 16 | 256 |
| 17 | 272 |
| 18 | 288 |
| 19 | 304 |
| 20 | 320 |
| 21 | 336 |
| 22 | 352 |
| 23 | 368 |
| 24 | 384 |
| 25 | 400 |
| 26 | 416 |
| 27 | 432 |
| 28 | 448 |
| 29 | 464 |
| 30 | 480 |
| 31 | 496 |
| 32 | 512 |
| 33 | 528 |
| 34 | 544 |
| 35 | 560 |
| 36 | 576 |
| 37 | 592 |
| 38 | 608 |
| 39 | 624 |
| 40 | 640 |
| 41 | 656 |
| 42 | 672 |
| 43 | 688 |
| 44 | 704 |
| 45 | 720 |
| 46 | 736 |
| 47 | 752 |
| 48 | 768 |
| 49 | 784 |
| 50 | 800 |
| 51 | 816 |
| 52 | 832 |
| 53 | 848 |
| 54 | 864 |
| 55 | 880 |
| 56 | 896 |
| 57 | 912 |
| 58 | 928 |
| 59 | 944 |
| 60 | 960 |
| 61 | 976 |
| 62 | 992 |
| 63 | 1008 |
| 64 | 1024 |
| 65 | 1040 |
| 66 | 1056 |
| 67 | 1072 |
| 68 | 1088 |
| 69 | 1104 |
| 70 | 1120 |
| 71 | 1136 |
| 72 | 1152 |
| 73 | 1168 |
| 74 | 1184 |
| 75 | 1200 |