193edo: Difference between revisions
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== Theory == | == Theory == | ||
193edo provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[ | 193edo provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[minos]] and [[Mirkwai family #Indra|vish]] temperaments. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|193 | {{Harmonics in equal|193}} | ||
=== | === Subsets and supersets === | ||
193edo is the 44th [[prime edo]]. | 193edo is the 44th [[prime edo]]. | ||
| Line 24: | Line 24: | ||
| 2.3 | | 2.3 | ||
| {{monzo| 306 -193 }} | | {{monzo| 306 -193 }} | ||
| | | {{mapping| 193 306 }} | ||
| -0.2005 | | -0.2005 | ||
| 0.2005 | | 0.2005 | ||
| Line 30: | Line 30: | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| 15625/15552, {{monzo|50 -33 1}} | | 15625/15552, {{monzo| 50 -33 1 }} | ||
| | | {{mapping| 193 306 448 }} | ||
| -0.0158 | | -0.0158 | ||
| 0.3084 | | 0.3084 | ||
| Line 38: | Line 38: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 5120/5103, 15625/15552, 16875/16807 | | 5120/5103, 15625/15552, 16875/16807 | ||
| | | {{mapping| 193 306 448 542 }} | ||
| -0.1118 | | -0.1118 | ||
| 0.3146 | | 0.3146 | ||
| Line 45: | Line 45: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 540/539, 1375/1372, 4375/4356, 5120/5103 | | 540/539, 1375/1372, 4375/4356, 5120/5103 | ||
| | | {{mapping| 193 306 448 542 668 }} | ||
| -0.2080 | | -0.2080 | ||
| 0.3408 | | 0.3408 | ||
| Line 52: | Line 52: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 325/324, 364/363, 540/539, 625/624, 4096/4095 | | 325/324, 364/363, 540/539, 625/624, 4096/4095 | ||
| | | {{mapping| 193 306 448 542 668 714 }} | ||
| -0.1216 | | -0.1216 | ||
| 0.3662 | | 0.3662 | ||
| Line 59: | Line 59: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095 | | 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095 | ||
| | | {{mapping| 193 306 448 542 668 714 789 }} | ||
| -0.1302 | | -0.1302 | ||
| 0.3397 | | 0.3397 | ||
| Line 66: | Line 66: | ||
| 2.3.5.7.11.13.17.19 | | 2.3.5.7.11.13.17.19 | ||
| 325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215 | | 325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215 | ||
| | | {{mapping| 193 306 448 542 668 714 789 820 }} | ||
| -0.1414 | | -0.1414 | ||
| 0.3191 | | 0.3191 | ||
| Line 78: | Line 78: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperament | ! Temperament | ||
|- | |- | ||
| Line 137: | Line 137: | ||
| [[Kwai]] | | [[Kwai]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
== 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 | ||
[[Category:Sqrtphi]] | [[Category:Sqrtphi]] | ||