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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>2011-11-11 15:58:40 UTC</tt>.<br>
 
: The original revision id was <tt>274490056</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
212edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]] with [[harmonic]]s of 3 through 13 all tuned flat. It shares the [[3/1|3rd]], [[5/1|5th]], and [[13/1|13th]] [[harmonic]]s with [[53edo]], but the mapping differs for [[7/1|7]] and [[11/1|11]].  
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>
It [[tempering out|tempers out]] the same commas ([[15625/15552]], [[32805/32768]], [[amity comma|1600000/1594323]], etc.) as 53edo in the [[5-limit]]. In the [[7-limit]], it tempers out 2401/2400 ([[breedsma]]), 390625/388962 ([[dimcomp comma]]), and 4802000/4782969 ([[canousma]]). In the [[11-limit]], [[385/384]], [[1375/1372]], [[6250/6237]], [[9801/9800]], and [[14641/14580]]; in the [[13-limit]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], and [[10648/10647]].  
<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">The 212 equal temperament divides the octave into 212 equal parts of 5.660 cents each. 212 = 4 * 53, and it tempers out the same commas (32805/32768, 15625/15552 and 1600000/1594323) as 53 in the [[5-limit]]. In the [[7-limit]] it tempers out 2401/2400; in the [[11-limit]] 385/384 and 1375/1372; and in the [[13-limit]] 325/324, 625/624, 676/675 and 1001/1000. It is the [[optimal patent val]] for 7 and 13 limit [[Kleismic family|quadritikleismic temperament]], and the 13-limit rank three [[Breed family#Agni|agni temperament]]. Also, the 212b val gives a tuning almost identical to the POTE tuning for 5-limit meantone.</pre></div>
 
<h4>Original HTML content:</h4>
It is the [[optimal patent val]] for 7- and 13-limit [[quadritikleismic]] temperament, the 7-limit [[Kleismic rank three family #Rank-3 kleismic|rank-3 kleismic]] temperament, and the 13-limit rank-3 [[agni]] temperament. It enables [[marveltwin chords]], [[keenanismic chords]], [[sinbadmic chords]], and [[lambeth chords]] in the 13-odd-limit in addition to [[island chords]] in the 15-odd-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;212edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 212 equal temperament divides the octave into 212 equal parts of 5.660 cents each. 212 = 4 * 53, and it tempers out the same commas (32805/32768, 15625/15552 and 1600000/1594323) as 53 in the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;. In the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; it tempers out 2401/2400; in the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; 385/384 and 1375/1372; and in the &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; 325/324, 625/624, 676/675 and 1001/1000. It is the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for 7 and 13 limit &lt;a class="wiki_link" href="/Kleismic%20family"&gt;quadritikleismic temperament&lt;/a&gt;, and the 13-limit rank three &lt;a class="wiki_link" href="/Breed%20family#Agni"&gt;agni temperament&lt;/a&gt;. Also, the 212b val gives a tuning almost identical to the POTE tuning for 5-limit meantone.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
To the 13-limit we may add the [[prime harmonic|prime]] [[23/1|23]] without introducing too much extra error, tempering out [[484/483]] and [[507/506]]. The 212gh val shows some potential if the full [[23-limit]] is desired, where it notably tempers out [[289/288]] and [[361/360]]. Also, using 212bb val (where fifth is flattened by single step, approximately 1/4 comma) gives a tuning almost identical to the POTE tuning for 5-limit meantone. This is related to the fact that 212edo splits steps of 53edo, which are mapped to a syntonic comma, in four.
 
=== Prime harmonics ===
{{Harmonics in equal|212}}
 
=== Octave stretch ===
212edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[124edf]] or [[336edt]]. This improves the approximated harmonics 5, 7, 11, 13 and brings the flat mappings of 17 and 19 closer; only the 23 becomes less accurate as it is tuned sharp already.
 
=== Subsets and supersets ===
Since 212 factors into primes as {{nowrap| 2<sup>2</sup> × 53 }}, 212edo has subset edos {{EDOs| 2, 4, 53, and 106 }}. As such, it can be used to tune the 53rd-octave [[cartography]] temperament and the 106th-octave [[boiler]] temperment.
 
A step of 212edo is exactly 50 [[türk sent]]s.
 
== 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]] (%)
|-
| 2.3.5.7
| 2401/2400, 15625/15552, 32805/32768
| {{Mapping| 212 336 492 595 }}
| +0.243
| 0.244
| 4.30
|-
| 2.3.5.7.11
| 385/384, 1375/1372, 6250/6237, 14641/14580
| {{Mapping| 212 336 492 595 733 }}
| +0.325
| 0.273
| 4.82
|-
| 2.3.5.7.11.13
| 325/324, 385/384, 625/624, 1375/1372, 10648/10647
| {{Mapping| 212 336 492 595 733 784 }}
| +0.396
| 0.296
| 5.23
|-
| 2.3.5.7.11.13.17
| 289/288, 325/324, 385/384, 442/441, 625/624, 10648/10647
| {{Mapping| 212 336 492 595 733 784 866 }} (212g)
| +0.447
| 0.301
| 5.32
|-
| 2.3.5.7.11.13.17.19
| 289/288, 325/324, 361/360, 385/384, 442/441, 513/512, 625/624
| {{Mapping| 212 336 492 595 733 784 866 900 }} (212gh)
| +0.485
| 0.299
| 5.27
|-
| 2.3.5.7.11.13.17.19.23
| 289/288, 323/322, 325/324, 361/360, 385/384, 442/441, 484/483, 507/506
| {{Mapping| 212 336 492 595 733 784 866 900 959 }} (212gh)
| +0.430
| 0.321
| 5.67
|}
* 212et (212gh val) has a lower absolute error in the 19-limit than any previous equal temperaments, past [[193edo|193]] and followed by [[217edo|217]].
 
=== Rank-2 temperaments ===
Note: temperaments supported by 53et are not included.
 
{| 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
| 15\212
| 84.91
| 21/20
| [[Amicable]] / [[amorous]] / [[pseudoamical]]
|-
| 1
| 31\212
| 175.47
| 448/405
| [[Sesquiquartififths]]
|-
| 1
| 41\212
| 232.08
| 8/7
| [[Quadrawell]]
|-
| 1
| 67\212
| 379.25
| 56/45
| [[Marthirds]]
|-
| 2
| 11\212
| 62.26
| 28/27
| [[Eagle]]
|-
| 2
| 15\212
| 84.91
| 21/20
| [[Floral]]
|-
| 2
| 31\212
| 175.47
| 448/405
| [[Bisesqui]]
|-
| 2
| 97\212<br />(9\212)
| 549.06<br />(50.94)
| 11/8<br />(36/35)
| [[Kleischismic]]
|-
| 4
| 56\212<br />(3\212)
| 316.98<br />(16.98)
| 6/5<br />(126/125)
| [[Quadritikleismic]]
|-
| 4
| 88\212<br />(18\212)
| 498.11<br />(101.89)
| 4/3<br />(35/33)
| [[Quadrant]]
|-
| 53
| 41\212<br />(1\212)
| 232.08<br />(5.66)
| 8/7<br />(225/224)
| [[Schismerc]] / [[cartography]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=bvCeNcxUDnA ''Etude in Amicable, Bisesqui, and 53edo'']
 
[[Category:53edo]]
[[Category:Agni]]
[[Category:Kleismic]]
[[Category:Listen]]
[[Category:Quadritikleismic]]
Retrieved from "https://en.xen.wiki/w/212edo"