193edo
| ← 192edo | 193edo | 194edo → |
The 193 equal divisions of the octave (193edo), or the 193(-tone) equal temperament (193tet, 193et) when viewed from a regular temperament perspective, is the equal division of the octave into 193 parts of about 6.22 cents each.
Theory
193edo provides the optimal patent val for the sqrtphi temperament in the 13-, 17- and 19-limits, and for the 13-limit minos and vish temperaments. It is the 44th prime edo.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.64 | -0.82 | +1.12 | +2.05 | -1.15 | +0.74 | +0.93 | -0.30 | +2.55 | -0.99 |
| Relative (%) | +0.0 | +10.2 | -13.2 | +18.1 | +33.0 | -18.5 | +12.0 | +15.0 | -4.7 | +41.0 | -16.0 | |
| Steps (reduced) |
193 (0) |
306 (113) |
448 (62) |
542 (156) |
668 (89) |
714 (135) |
789 (17) |
820 (48) |
873 (101) |
938 (166) |
956 (184) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [306 -193⟩ | [⟨193 306]] | -0.2005 | 0.2005 | 3.23 |
| 2.3.5 | 15625/15552, [50 -33 1⟩ | [⟨193 306 448]] | -0.0158 | 0.3084 | 4.96 |
| 2.3.5.7 | 5120/5103, 15625/15552, 16875/16807 | [⟨193 306 448 542]] | -0.1118 | 0.3146 | 5.06 |
| 2.3.5.7.11 | 540/539, 1375/1372, 4375/4356, 5120/5103 | [⟨193 306 448 542 668]] | -0.2080 | 0.3408 | 5.48 |
| 2.3.5.7.11.13 | 325/324, 364/363, 540/539, 625/624, 4096/4095 | [⟨193 306 448 542 668 714]] | -0.1216 | 0.3662 | 5.89 |
| 2.3.5.7.11.13.17 | 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095 | [⟨193 306 448 542 668 714 789]] | -0.1302 | 0.3397 | 5.46 |
| 2.3.5.7.11.13.17.19 | 325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215 | [⟨193 306 448 542 668 714 789 820]] | -0.1414 | 0.3191 | 5.13 |
Rank-2 temperaments
| Periods
per octave |
Generator
(reduced) |
Cents
(reduced) |
Associated
ratio |
Temperament |
|---|---|---|---|---|
| 1 | 16\193 | 99.48 | 18/17 | Quintakwai/Quintakwoid |
| 1 | 18\193 | 111.92 | 16/15 | Vavoom |
| 1 | 39\193 | 242.49 | 147/128 | Septiquarter |
| 1 | 51\193 | 317.10 | 6/5 | Countercata (7-limit) |
| 1 | 56\193 | 348.19 | 11/9 | Eris |
| 1 | 61\193 | 379.28 | 56/45 | Marthirds |
| 1 | 67\193 | 416.58 | 14/11 | Sqrtphi |
| 1 | 79\193 | 491.19 | 3645/2744 | Fifthplus |
| 1 | 80\193 | 497.41 | 4/3 | Kwai |
Scales
- Approximation of sqrt (π): 159\193 (988.60104 cents), and of φ: 134\193 (833.16062 cents), both inside in the superdiatonic scale: 25 25 25 9 25 25 25 25 9