75edo
| ← 74edo | 75edo | 76edo → |
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
75et tempers out 20000/19683 (tetracot comma) and 2109375/2097152 (semicomma) in the 5-limit, and provides a good tuning for the tetracot temperament. It tempers out 225/224 and 1728/1715 in the 7-limit, supporting bunya and orwell, and providing the optimal patent val for fog.
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.
Since 75 is part of the Fibonacci sequence beginning with 5 and 12, it closely approximates the peppermint temperament. The size of its fifth is exactly 704 ¢, which is very close to the peppermint fifth of 704.096 ¢. 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 |