94edo
94edo is most notable for being the sum of 53 and 41, and thus serving as a tuning of garibaldi that lies in between the two. It is additionally one of the first EDOs where the distance between steps is below certain thresholds of intonational accuracy on free-pitch instruments; in other words, 94edo could be seen as approximately the most accurate resolution a free-pitch instrument player or vocalist can be expected to manage.
It is also the first edo other than 72edo to distinguish all of the locking intervals.
94edo is not a hemipythagorean edo; being even with an odd mapping of the fifth (51\94) it has neither neutrals nor interordinals. It does, however, have the 600-cent tritone, mapped as with many other accurate edos of its size to 17/12. Its step is one half of the garibaldi comma.
Tuning theory
Edostep interpretations
One step of 94edo corresponds to:
- The magisma (the distance between 3/1 and a stack of five 5/4s)
- 245/243 (the distance between 5/3 and a stack of two 9/7s)
- 99/98 (the distance between 9/7 and 14/11)
- 243/242 (the distance between 11/9 and 27/22)
- 512/507 (the distance between 16/13 and 39/32)
- 896/891 (the distance between 81/64 and 14/11)
- 100/99, or S10 (the distance between 11/10 and 10/9)
- 121/120, or S11 (the distance between 12/11 and 11/10)
- 144/143, or S12 (the distance between 13/12 and 12/11)
- 169/168, or S13 (the distance between 14/13 and 13/12)
- 196/195, or S14 (the distance between 15/14 and 14/13)
- 256/255, or S16 (the distance between 17/16 and 16/15)
Two steps of 94edo corresponds to:
- The garibaldi comma
- 81/80, or S9 (the distance between 10/9 and 9/8)
- 64/63, or S8 (the distance between 9/8 and 8/7)
- 96/95 (the distance between 19/16 and 6/5)
- 65/64 (the distance between 5/4 and 16/13)
- 55/54 (the distance between 6/5 and 11/9)
Tuning accuracy
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | +0.2 | -3.3 | +1.4 | -2.4 | +2.0 | -2.8 | -3.9 | -2.7 | +4.5 | +3.9 | +4.0 | +5.0 | -0.9 | -1.7 | -5.4 | +0.4 | -6.2 | -2.7 | -1.0 | +2.0 |
| Relative (%) | 0.0 | +1.4 | -26.1 | +10.9 | -18.7 | +15.9 | -22.2 | -30.5 | -21.5 | +35.0 | +30.6 | +31.1 | +39.0 | -6.9 | -13.1 | -42.5 | +3.2 | -48.9 | -21.2 | -7.6 | +15.6 | |
| Steps
(reduced) |
94
(0) |
149
(55) |
218
(30) |
264
(76) |
325
(43) |
348
(66) |
384
(8) |
399
(23) |
425
(49) |
457
(81) |
466
(90) |
490
(20) |
504
(34) |
510
(40) |
522
(52) |
538
(68) |
553
(83) |
557
(87) |
570
(6) |
578
(14) |
582
(18) | |
94edo approximates all of the prime harmonics up to 73 except 29, 41, 53, and 61 to within 4 cents. The maximum possible error on a prime harmonic in 94edo is about 6.2 cents, this is less than the "kleisma"-resolution of 6.5c as noted by MidnightBlue, as stated in the intro. While 5 and 19 are off by more than 25 relative cents (and therefore result in an inconsistent mapping of their squares), the step size is small enough as to render inconsistencies largely inconsequential.
| View • Talk • EditEqual temperaments | |
|---|---|
| EDOs | |
| Macrotonal | 5 • 7 • 8 • 9 • 10 • 11 |
| 12-23 | 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 |
| 24-35 | 24 • 25 • 26 • 27 • 29 • 31 • 32 • 34 • 35 |
| 36-47 | 36 • 37 • 39 • 40 • 41 • 43 • 44 • 45 • 46 • 47 |
| 48-59 | 48 • 50 • 51 • 53 • 54 • 56 • 57 • 58 |
| 60-71 | 60 • 63 • 64 • 65 • 67 • 68 • 70 |
| 72-83 | 72 • 77 • 80 • 81 |
| 84-95 | 84 • 87 • 89 • 90 • 93 • 94 |
| Large EDOs | 99 • 104 • 106 • 111 • 118 • 130 • 140 • 152 • 159 • 171 • 217 • 224 • 239 • 270 • 306 • 311 • 612 • 665 |
| Nonoctave equal temperaments | |
| Tritave | 4 • 9 • 13 • 17 • 26 • 39 |
| Fifth | 8 • 9 • 11 • 20 |
| Other | |
