193edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== 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 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, it [[tempering out|tempers out]] the [[15625/15552|kleisma]] in the [[5-limit]]; [[5120/5103]] and [[16875/16807]] in the [[7-limit]]; [[540/539]], [[1375/1372]], [[3025/3024]], and 4375/4356 in the [[11-limit]]; [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1575/1573]], [[1716/1715]], and [[4096/4095]] in the [[13-limit]]; [[375/374]], [[442/441]], [[595/594]], [[715/714]], [[936/935]], [[1156/1155]], [[1225/1224]], [[2058/2057]], and [[2431/2430]] in the [[17-limit]]; [[400/399]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]], [[1540/1539]], and [[1729/1728]] in the [[19-limit]]; and [[460/459]], [[507/506]], and [[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 [[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]]. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 24: | Line 29: | ||
| 2.3 | | 2.3 | ||
| {{monzo| 306 -193 }} | | {{monzo| 306 -193 }} | ||
| | | {{mapping| 193 306 }} | ||
| | | −0.2005 | ||
| 0.2005 | | 0.2005 | ||
| 3.23 | | 3.23 | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| 15625/15552, {{monzo|50 -33 1}} | | 15625/15552, {{monzo| 50 -33 1 }} | ||
| | | {{mapping| 193 306 448 }} | ||
| | | −0.0158 | ||
| 0.3084 | | 0.3084 | ||
| 4.96 | | 4.96 | ||
| Line 38: | Line 43: | ||
| 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 | ||
| 0.3146 | | 0.3146 | ||
| 5.06 | | 5.06 | ||
| Line 45: | Line 50: | ||
| 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 | ||
| 0.3408 | | 0.3408 | ||
| 5.48 | | 5.48 | ||
| Line 52: | Line 57: | ||
| 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 | ||
| 0.3662 | | 0.3662 | ||
| 5.89 | | 5.89 | ||
| Line 59: | Line 64: | ||
| 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 | ||
| 0.3397 | | 0.3397 | ||
| 5.46 | | 5.46 | ||
| Line 66: | Line 71: | ||
| 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 | ||
| 0.3191 | | 0.3191 | ||
| 5.13 | | 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 | |||
| {{mapping| 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 [[190edo|190g]] and followed by [[217edo|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 [[212edo|212gh]]. | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br> | ! Cents* | ||
! | ! Associated<br />ratio* | ||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 135: | Line 150: | ||
| [[Kwai]] | | [[Kwai]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if 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]] | ||