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{{Infobox ET}}
{{Infobox ET}}
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 [[EDO|equal division of the octave]] into 193 parts of about 6.22 [[cent]]s each.
{{ED intro}}


== Theory ==
== Theory ==
193edo provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limits, and for the 13-limit [[Swetismic temperaments #Minos|minos]] and [[Mirkwai family #Indra|vish]] temperaments. It is the 44th [[prime edo]].
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]].


== 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 8ve <br>stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 21: Line 29:
| 2.3
| 2.3
| {{monzo| 306 -193 }}
| {{monzo| 306 -193 }}
| [{{val| 193 306 }}]
| {{mapping| 193 306 }}
| -0.2005
| −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 }}
| [{{val| 193 306 448 }}]
| {{mapping| 193 306 448 }}
| -0.0158
| −0.0158
| 0.3084
| 0.3084
| 4.96
| 4.96
Line 35: Line 43:
| 2.3.5.7
| 2.3.5.7
| 5120/5103, 15625/15552, 16875/16807
| 5120/5103, 15625/15552, 16875/16807
| [{{val| 193 306 448 542 }}]
| {{mapping| 193 306 448 542 }}
| -0.1118
| −0.1118
| 0.3146
| 0.3146
| 5.06
| 5.06
Line 42: 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
| [{{val| 193 306 448 542 668 }}]
| {{mapping| 193 306 448 542 668 }}
| -0.2080
| −0.2080
| 0.3408
| 0.3408
| 5.48
| 5.48
Line 49: 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
| [{{val| 193 306 448 542 668 714 }}]
| {{mapping| 193 306 448 542 668 714 }}
| -0.1216
| −0.1216
| 0.3662
| 0.3662
| 5.89
| 5.89
Line 56: 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
| [{{val| 193 306 448 542 668 714 789 }}]
| {{mapping| 193 306 448 542 668 714 789 }}
| -0.1302
| −0.1302
| 0.3397
| 0.3397
| 5.46
| 5.46
Line 63: 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
| [{{val| 193 306 448 542 668 714 789 820 }}]
| {{mapping| 193 306 448 542 668 714 789 820 }}
| -0.1414
| −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
|-
per octave
! Periods<br />per 8ve
!Generator
! Generator*
(reduced)
! Cents*
!Cents
! Associated<br />ratio*
(reduced)
! Temperaments
!Associated
ratio
!Temperament
|-
|-
|1
| 1
|16\193
| 16\193
|99.48
| 99.48
|18/17
| 18/17
|[[Quindromeda family#Quintakwai|Quintakwai]]/[[Quindromeda family#Quintakwoid|Quintakwoid]]
| [[Quindromeda family#Quintakwai|Quintakwai]] / [[Quindromeda family#Quintakwoid|quintakwoid]]
|-
|-
|1
| 1
|18\193
| 18\193
|111.92
| 111.92
|16/15
| 16/15
|[[Vavoom]]
| [[Vavoom]]
|-
|-
|1
| 1
|39\193
| 39\193
|242.49
| 242.49
|147/128
| 147/128
|[[Septiquarter]]
| [[Septiquarter]]
|-
|-
|1
| 1
|51\193
| 51\193
|317.10
| 317.10
|6/5
| 6/5
|[[Countercata]] (7-limit)
| [[Countercata]] (7-limit)
|-
|-
|1
| 1
|56\193
| 56\193
|348.19
| 348.19
|11/9
| 11/9
|[[Mirkwai clan#Eris|Eris]]
| [[Eris]]
|-
|-
|1
| 1
|61\193
| 61\193
|379.28
| 379.28
|56/45
| 56/45
|[[Marthirds]]
| [[Marthirds]]
|-
|-
|1
| 1
|67\193
| 67\193
|416.58
| 416.58
|14/11
| 14/11
|[[Sqrtphi]]
| [[Sqrtphi]]
|-
|-
|1
| 1
|79\193
| 79\193
|491.19
| 491.19
|3645/2744
| 3645/2744
|[[Fifthplus]]
| [[Fifthplus]]
|-
|-
|1
| 1
|80\193
| 80\193
|497.41
| 497.41
|4/3
| 4/3
|[[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:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]
[[Category:Sqrtphi]]
[[Category:Sqrtphi]]
Retrieved from "https://en.xen.wiki/w/193edo"