75edo
The 75 equal divisions of the octave (75edo), or the 75-tone equal temperament (75tet), 75 equal temperament (75et) when viewed from a regular temperament perspective, divides the octave into 75 equal parts of exactly 16 cents each.
Theory
In the 5-limit, 75et tempers out 20000/19683 (tetracot comma) and 2109375/2097152 (semicomma), and provides a good tuning for the tetracot temperament. In the 7-limit, it tempers 225/224 and 1728/1715. In the 11-limit, 75e val ⟨75 119 174 211 260] scores lower in error, and tempers 100/99 and 243/242, whereas the patent val ⟨75 119 174 211 259] 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 fog temperament in the 7-limit and the 31 & 44 temperament in the 19-limit.
Since 75 is 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. It also approximates the Carlos Beta scale well (4\75 ≈ 1\[Carlos Beta]).
Odd harmonics
| 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) | |
Intervals
| # | 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 |
| … | … |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [119 -75⟩ | [⟨75 119]] | -0.645 | 0.645 | 4.03 |
| 2.3.5 | 20000/19683, 2109375/2097152 | [⟨75 119 174]] | -0.099 | 0.936 | 5.85 |
| 2.3.5.7 | 225/224, 1728/1715, 15625/15309 | [⟨75 119 174 211]] | -0.713 | 1.337 | 8.36 |