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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-24 23:33:38 UTC</tt>.<br>
: The original revision id was <tt>268204982</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=&lt;span style="color: #003838; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&gt;193 tone equal temperament&lt;/span&gt;=
The //193-EDO// divides the octave into 193 equal parts of 6.21762 cents each. It provides the [[optimal patent val]] for [[Kleismic family#Sqrtphi|sqrtphi temperament]] in the 13-, 17- and 19-limits, and for 13-limit [[Swetismic temperaments#Minos-13-limit|minos]] and [[Mirkwai family#Indra-Vish|visdh]] temperaments.


Approximation of the intervals:
== Theory ==
Square root of Pi: **159\193** (988.60104 cents), and
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.
Phi: **134\193** (833.16062 cents), both inside of the scale [[7L 2s]]: 25 25 25 9 25 25 25 25 9</pre></div>
 
<h4>Original HTML content:</h4>
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.
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;193edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x193 tone equal temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="color: #003838; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&gt;193 tone equal temperament&lt;/span&gt;&lt;/h1&gt;
 
The &lt;em&gt;193-EDO&lt;/em&gt; divides the octave into 193 equal parts of 6.21762 cents each. It provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for &lt;a class="wiki_link" href="/Kleismic%20family#Sqrtphi"&gt;sqrtphi temperament&lt;/a&gt; in the 13-, 17- and 19-limits, and for 13-limit &lt;a class="wiki_link" href="/Swetismic%20temperaments#Minos-13-limit"&gt;minos&lt;/a&gt; and &lt;a class="wiki_link" href="/Mirkwai%20family#Indra-Vish"&gt;visdh&lt;/a&gt; temperaments.&lt;br /&gt;
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.
&lt;br /&gt;
 
Approximation of the intervals:&lt;br /&gt;
=== Prime harmonics ===
Square root of Pi: &lt;strong&gt;159\193&lt;/strong&gt; (988.60104 cents), and&lt;br /&gt;
{{Harmonics in equal|193}}
Phi: &lt;strong&gt;134\193&lt;/strong&gt; (833.16062 cents), both inside of the scale &lt;a class="wiki_link" href="/7L%202s"&gt;7L 2s&lt;/a&gt;: 25 25 25 9 25 25 25 25 9&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== Subsets and supersets ===
193edo is the 44th [[prime edo]].
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 306 -193 }}
| {{mapping| 193 306 }}
| −0.2005
| 0.2005
| 3.23
|-
| 2.3.5
| 15625/15552, {{monzo| 50 -33 1 }}
| {{mapping| 193 306 448 }}
| −0.0158
| 0.3084
| 4.96
|-
| 2.3.5.7
| 5120/5103, 15625/15552, 16875/16807
| {{mapping| 193 306 448 542 }}
| −0.1118
| 0.3146
| 5.06
|-
| 2.3.5.7.11
| 540/539, 1375/1372, 4375/4356, 5120/5103
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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
| {{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 ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 1
| 16\193
| 99.48
| 18/17
| [[Quindromeda family#Quintakwai|Quintakwai]] / [[Quindromeda family#Quintakwoid|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]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== 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
 
[[Category:Sqrtphi]]
Retrieved from "https://en.xen.wiki/w/193edo"