193edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|193}} | |||
== Theory == | == Theory == | ||
193edo provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19- | 193edo provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[Swetismic temperaments #Minos|minos]] and [[Mirkwai family #Indra|vish]] temperaments. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|193}} | {{Harmonics in equal|193|columns=11}} | ||
=== Miscellaneous properties === | |||
193edo is the 44th [[prime edo]]. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 82: | Line 85: | ||
| 99.48 | | 99.48 | ||
| 18/17 | | 18/17 | ||
| [[Quindromeda family#Quintakwai|Quintakwai]]/[[Quindromeda family#Quintakwoid| | | [[Quindromeda family#Quintakwai|Quintakwai]] / [[Quindromeda family#Quintakwoid|quintakwoid]] | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 133: | Line 136: | ||
|} | |} | ||
==Scales== | == Scales == | ||
*Approximation of sqrt (π): '''159\193''' (988.60104 cents), and of φ: '''134\193''' (833.16062 cents), both inside in the [[7L 2s|superdiatonic]] scale: 25 25 25 9 25 25 25 25 9 | *Approximation of sqrt (π): '''159\193''' (988.60104 cents), and of φ: '''134\193''' (833.16062 cents), both inside in the [[7L 2s|superdiatonic]] scale: 25 25 25 9 25 25 25 25 9 | ||
Revision as of 08:17, 22 October 2022
| ← 192edo | 193edo | 194edo → |
Theory
193edo 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 (%) | +0.0 | +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) | |
Miscellaneous properties
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 |
Rank-2 temperaments
| Periods per 8ve |
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