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638edo: Difference between revisions

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Wikispaces>genewardsmith
**Imported revision 338108548 - 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>2012-05-21 20:16:09 UTC</tt>.<br>
 
: The original revision id was <tt>338108548</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
The equal temperament [[tempering out|tempers out]] the minortone comma, {{monzo| -16 35 -17 }}, in the 5-limit, [[4375/4374]] in the 7-limit, [[3025/3024]], [[9801/9800]], and 43923/43904, in the 11-limit; and [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]] and [[4225/4224]] in the 13-limit. It supplies the [[optimal patent val]] for [[quatracot]], the {{nowrap|224 &amp; 414}} temperament.
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>
=== Odd harmonics ===
<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 638 equal temperaments divide the octave into 638 equal parts of 1.881 cents each. It tempers out the minortone comma, 50031545098999707/50000000000000000, in the 5-limit, 4375/4374 in the 7-limit, 43923/43904, 3025/3024, 9801/9800 in the 11-limit, and 625/624, 729/728, 1575/1573 2200/2197 and 4225/4224 in the 13-limit. It supplies the optimal patent val for [[Ragismic microtemperaments#Quatracot|quatracot temperament]].</pre></div>
{{Harmonics in equal|638}}
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;638edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 638 equal temperaments divide the octave into 638 equal parts of 1.881 cents each. It tempers out the minortone comma, 50031545098999707/50000000000000000, in the 5-limit, 4375/4374 in the 7-limit, 43923/43904, 3025/3024, 9801/9800 in the 11-limit, and 625/624, 729/728, 1575/1573 2200/2197 and 4225/4224 in the 13-limit. It supplies the optimal patent val for &lt;a class="wiki_link" href="/Ragismic%20microtemperaments#Quatracot"&gt;quatracot temperament&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Subsets and supersets ===
Since 638 factors as {{factorisation|638}}, 638edo has subset edos {{EDOs| 2, 11, 22, 29, 58, and 319 }}.
 
== 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| -1011 638 }}
| {{mapping| 638 1011 }}
| +0.1223
| 0.1223
| 6.50
|-
| 2.3.5
| {{monzo| -51 19 9 }}, {{monzo| -16 35 -17 }}
| {{mapping| 638 1011 1481 }}
| +0.1869
| 0.1353
| 7.19
|-
| 2.3.5.7
| 4375/4374, 2100875/2097152, {{monzo| -11 5 11 -8 }}
| {{mapping| 638 1011 1481 1791 }}
| +0.1556
| 0.1291
| 6.86
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 825000/823543, 1265625/1261568
| {{mapping| 638 1011 1481 1791 2207 }}
| +0.1373
| 0.1212
| 6.44
|-
| 2.3.5.7.11.13
| 625/624, 729/728, 1575/1573, 2200/2197, 823680/823543
| {{mapping| 638 1011 1481 1791 2207 2361 }}
| +0.1043
| 0.1330
| 7.07
|}
 
=== 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
| 97\638
| 182.45
| 10/9
| [[Mitonic]]
|-
| 1
| 313\638
| 588.71
| 7/5
| [[Untriton]]
|-
| 2
| 94\638
| 176.80
| 448/405
| [[Quatracot]]
|}
<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:Quatracot]]
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