94edo
The 94 equal temperament, often abbreviated 94-tET, 94-EDO, or 94-ET, results from dividing the octave into 94 equally-sized steps, where each step is 12.766 cents.
Theory
94edo is a remarkable all-around utility temperament, good from low prime limit to very high prime limit situations. It is the first equal temperament to be consistent through the 23-limit, and no other equal temperament is so consistent until 282 and 311 make their appearance.
The list of 23-limit commas it tempers out is huge, but it's worth noting that it tempers out 32805/32768 and is thus a schismatic system, that it tempers out 225/224 and 385/384 and so is a marvel system, and that it also tempers out 3125/3087, 4000/3969, 5120/5103 and 540/539. It provides the optimal patent val for the rank five temperament tempering out 275/273, and for a number of other temperaments, such as isis.
- See also: Table of 94edo intervals
Just approximation
Selected just intervals
| prime 2 | prime 3 | prime 5 | prime 7 | prime 11 | prime 13 | prime 17 | prime 19 | prime 23 | prime 29 | prime 31 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | 0.00 | +0.17 | -3.33 | +1.39 | -2.38 | +2.03 | -2.83 | -3.90 | -2.74 | +4.47 | +3.90 |
| relative (%) | 0.0 | +1.4 | -26.1 | +10.9 | -18.7 | +15.9 | -22.2 | -30.5 | -21.5 | +35.0 | +30.6 | |
Temperament measures
The following table shows TE temperament measures (RMS normalized by the rank) of 94et.
| 3-limit | 5-limit | 7-limit | 11-limit | 13-limit | 17-limit | 19-limit | 23-limit | 29-limit | 31-limit | ||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Octave stretch (¢) | -0.054 | +0.442 | +0.208 | +0.304 | +0.162 | +0.238 | +0.323 | +0.354 | +0.227 | +0.134 | |
| Error | absolute (¢) | 0.054 | 0.704 | 0.732 | 0.683 | 0.699 | 0.674 | 0.669 | 0.637 | 0.715 | 0.741 |
| relative (%) | 0.43 | 5.52 | 5.74 | 5.35 | 5.48 | 5.28 | 5.24 | 4.99 | 5.60 | 5.81 | |
94et has a lower relative error than any previous ETs in the 23-limit. The next ET that does better in this subgroup is 193.
Rank two temperaments
| Periods per octave |
Generator | Cents | Associated ratio |
Temperament |
|---|---|---|---|---|
| 1 | 3\94 | 38.298 | 49/48 | Slender |
| 1 | 5\94 | 63.830 | 25/24 | Sycamore / betic |
| 1 | 11\94 | 140.426 | 243/224 13/12 |
Tsaharuk Quanic |
| 1 | 19\94 | 242.553 | 147/128 | Septiquarter |
| 1 | 39\94 | 497.872 | 4/3 | Schismatic / Garibaldi |
| 2 | 2\94 | 25.532 | 64/63 | Ketchup |
| 2 | 11\94 | 140.426 | 27/25 | Fifive |
| 2 | 30\94 | 382.979 | 5/4 | Wizard / gizzard |
| 2 | 34\94 | 434.043 | 9/7 | Pogo / supers |
| 2 | 43\94 | 548.936 | 11/8 | Kleischismic |
Below are some 23-limit temperaments supported by 94et. It might be noted that 94, a very good tuning for garibaldi temperament, shows us how to extend it to the 23-limit.
- 46&94 <<8 30 -18 -4 -28 8 -24 2 ... ||
- 68&94 <<20 28 2 -10 24 20 34 52 ... ||
- 53&94 <<1 -8 -14 23 20 -46 -3 -35 ... || (one garibaldi)
- 41&94 <<1 -8 -14 23 20 48 -3 -35 ... || (another garibaldi, only differing in the mappings of 17 and 23)
- 135&94 <<1 -8 -14 23 20 48 -3 59 ... || (another garibaldi)
- 130&94 <<6 -48 10 -50 26 6 -18 -22 ... || (a pogo extension)
- 58&94 <<6 46 10 44 26 6 -18 -22 ... || (a supers extension)
- 50&94 <<24 -4 40 -12 10 24 22 6 ... ||
- 72&94 <<12 -2 20 -6 52 12 -36 -44 ... || (a gizzard extension)
- 80&94 <<18 44 30 38 -16 18 40 28 ... ||
- 94 solo <<12 -2 20 -6 -42 12 -36 -44 ... || (a rank one temperament!)
Temperaments for which 94 is a MOS:
- 311&94 <<3 70 -42 69 -34 50 85 83...||
- 422&94 <<8 124 -18 90 -28 102 164 96 ... ||