94edo
The 94 equal temperament, often abbreviated 94-tET, 94-EDO, or 94-ET, is the scale derived by dividing the octave into 94 equally-sized steps, where each step represents a frequency ratio of 12.766 cents. It 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 in the 23-odd-limit, and no other equal temperament is so consistent until 282 and 311 make their appearance.
Out of the infinite list of 23-limit commas it tempers out, 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.
Also, see the Table of 94edo intervals.
Rank two temperaments[edit]
| 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 |
| 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 | 43\94 | 548.936 | 11/8 | Kleischismic |
| 2 | 11\94 | 140.426 | 27/25 | Fifive |
| 2 | 34\94 | 434.043 | 9/7 | Pogo, supers |
| 2 | 30\94 | 382.979 | 5/4 | Wizard/gizzard |
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 ... ||
Selected just intervals by error[edit]
The following table shows how some prominent just intervals are represented in 94edo (ordered by absolute error).
| Interval, complement | Error (abs., in cents) |
| 4/3, 3/2 | 0.173 |
| 9/8, 16/9 | 0.345 |
| 14/13, 13/7 | 0.639 |
| 15/11, 22/15 | 0.781 |
| 11/10, 20/11 | 0.953 |
| 9/7, 14/9 | 1.042 |
| 7/6, 12/7 | 1.214 |
| 8/7, 7/4 | 1.387 |
| 18/13, 13/9 | 1.68 |
| 13/12, 24/13 | 1.853 |
| 16/13, 13/8 | 2.026 |
| 11/8, 16/11 | 2.382 |
| 12/11, 11/6 | 2.554 |
| 11/9, 18/11 | 2.727 |
| 16/15, 15/8 | 3.162 |
| 5/4, 8/5 | 3.335 |
| 6/5, 5/3 | 3.508 |
| 10/9, 9/5 | 3.68 |
| 14/11, 11/7 | 3.769 |
| 13/11, 22/13 | 4.407 |
| 15/14, 28/15 | 4.549 |
| 7/5, 10/7 | 4.722 |
| 15/13, 26/15 | 5.188 |
| 13/10, 20/13 | 5.361 |