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208edo: Difference between revisions

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**Imported revision 213997760 - Original comment: **
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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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>2011-03-25 12:19:45 UTC</tt>.<br>
: The original revision id was <tt>213997760</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">The //208 equal division// divides the octave into 208 equal parts of size 5.769 cents each. It tempers out 15625/15552, the kleisma, and is the [[optimal patent val]] for the kleismic temperament [[Kleismic family|metakleismic]]. It is also the optimal patent val for the rank four 11-limit temperament tempering out 896/891, the pentacircle temperament. Other commas it tempers out include 2200/2187 in the 11-limit and 325/324, 352/351, 364/363 and 625/624 in the 13-limit.


208 = 16 * 13, and has divisors 2, 4, 8, 16, 13, 26, 52, 104.</pre></div>
== Theory ==
<h4>Original HTML content:</h4>
208edo is closely related to [[104edo]], but the mappings for [[harmonic]] [[5/1|5]] differ. As an equal temperament, it [[tempering out|tempers out]] [[15625/15552]], the kleisma, and is the [[optimal patent val]] for the kleismic temperament [[metakleismic]], and 7-, 11- and 13-limit rank-3 [[tolerant]] temperament. It is also the optimal patent val for the rank-4 [[11-limit]] temperament tempering out [[896/891]], the [[pentacircle]] temperament. Other commas it tempers out include [[2200/2187]] in the 11-limit and [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the 13-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;208edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;208 equal division&lt;/em&gt; divides the octave into 208 equal parts of size 5.769 cents each. It tempers out 15625/15552, the kleisma, and is the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for the kleismic temperament &lt;a class="wiki_link" href="/Kleismic%20family"&gt;metakleismic&lt;/a&gt;. It is also the optimal patent val for the rank four 11-limit temperament tempering out 896/891, the pentacircle temperament. Other commas it tempers out include 2200/2187 in the 11-limit and 325/324, 352/351, 364/363 and 625/624 in the 13-limit.&lt;br /&gt;
 
&lt;br /&gt;
=== Odd harmonics ===
208 = 16 * 13, and has divisors 2, 4, 8, 16, 13, 26, 52, 104.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Harmonics in equal|208}}
 
=== Subsets and supersets ===
Since 208 factors into 2<sup>4</sup> × 13, 208edo has subset edos {{EDOs| 2, 4, 8, 16, 13, 26, 52, and 104 }}.
 
== 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
| 15625/15552, {{monzo| 57 -33 -2 }}
| {{mapping| 208 330 483 }}
| −0.4301
| 0.5409
| 9.38
|-
| 2.3.5.7
| 2401/2400, 15625/15552, 179200/177147
| {{mapping| 208 330 483 584 }}
| −0.3586
| 0.4845
| 8.40
|-
| 2.3.5.7.11
| 896/891, 2200/2187, 2401/2400, 3025/3024
| {{mapping| 208 330 483 584 720 }}
| −0.4330
| 0.4582
| 7.94
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 676/675, 2401/2400
| {{mapping| 208 330 483 584 720 770 }}
| −0.4410
| 0.4187
| 7.26
|}
 
=== 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
| 47\208
| 271.15
| 1024/875
| [[Quasiorwell]]
|-
| 1
| 55\208
| 317.31
| 6/5
| [[Metakleismic]]
|-
| 4
| 55\208<br>(3\208)
| 317.31<br>(17.31)
| 6/5<br>(126/125)
| [[Quadritikleismic]] (7-limit)
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
[[Category:Metakleismic]]
[[Category:Tolerant]]
[[Category:Pentacircle]]
Retrieved from "https://en.xen.wiki/w/208edo"