193edo
| ← 192edo | 193edo | 194edo → |
193 equal divisions of the octave (abbreviated 193edo or 193ed2), also called 193-tone equal temperament (193tet) or 193 equal temperament (193et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 193 equal parts of about 6.218 ¢ each. Each step represents a frequency ratio of 21/193, or the 193rd root of 2.
Theory
193edo is consistent to the 11-odd-limit, and almost consistent to the 23-odd-limit, the only failure being 13/11 and its octave complement. This makes it a strong 23-limit system.
As an equal temperament, 193et tempers out the kleisma in the 5-limit; 5120/5103 and 16875/16807 in the 7-limit; 540/539, 1375/1372, 3025/3024, 4375/4356 in the 11-limit; 325/324, 364/363, 625/624, 676/675, 1575/1573, 1716/1715, 4096/4095 in the 13-limit; 375/374, 442/441, 595/594, 715/714, 936/935, 1156/1155, 1225/1224, 2058/2057, 2431/2430 in the 17-limit; 400/399, 969/968, 1216/1215, 1445/1444, 1521/1520, 1540/1539, 1729/1728 in the 19-limit; and 460/459, 507/506, 529/528 in the 23-limit.
It provides the optimal patent val for the sqrtphi temperament in the 13-, 17- and 19-limit, and for the 13-limit minos and vish temperaments.
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 | +10 | -13 | +18 | +33 | -18 | +12 | +15 | -5 | +41 | -16 | |
| Steps (reduced) |
193 (0) |
306 (113) |
448 (62) |
542 (156) |
668 (89) |
714 (135) |
789 (17) |
820 (48) |
873 (101) |
938 (166) |
956 (184) | |
Subsets and supersets
193edo is the 44th prime edo.
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 |
| 2.3.5.7.11.13.17.19.23 | 325/324, 364/363, 375/374, 400/399, 442/441, 460/459, 507/506, 529/528 | [⟨193 306 448 542 668 714 789 820 873]] | -0.1184 | 0.3078 | 4.95 |
- 193et has a lower relative error in the 23-limit than any previous equal temperaments, past 190g and followed by 217.
- 193et is also notable in the 19-limit, where it has a lower absolute error than any previous equal temperaments, past 190g and followed by 212gh.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | 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 |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
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