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137edo: Difference between revisions

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**Imported revision 214121468 - Original comment: **
 
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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 21:13:55 UTC</tt>.<br>
 
: The original revision id was <tt>214121468</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
137edo is a fairly accurate 5-limit temperament and also a strong no-7 19-limit temperament. The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]), {{monzo| -13 17 -6 }} ([[graviton]]), {{monzo| 8 14 -13 }} ([[parakleisma]]), and {{monzo| -29 -11 20 }} (gammic comma) in the 5-limit. Using the [[patent val]], it tempers out [[225/224]], [[1728/1715]], 2430/2401 in the 7-limit; [[243/242]] in the 11-limit; [[351/350]] in the 13-limit; [[375/374]] and [[442/441]] in the 17-limit; and [[324/323]] and [[495/494]] in the 19-limit. It provides the [[optimal patent val]] for 7-limit [[orwell]] temperament and for the planar temperament [[tempering out]] [[2430/2401]].
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>
=== Prime 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 //137 equal division// divides the octave into 137 equal parts of 8.759 cents each. It is the [[optimal patent val]] for 7-limit [[Semicomma family|orwell temperament]] and for the planar temperament tempering out 2430/2401. It tempers out 2109375/2097152 (the semicomma) in the 5-limit; 225/224 and 1728/1715 in the 7-limit; 243/242 in the 11-limit; 351/350 in the 13-limit; 375/374 and 442/441 in the 17-limit; and 324/323 and 495/494 in the 19-limit. Since it is a prime number, 137 has no proper divisors aside from 1.</pre></div>
{{Harmonics in equal|137}}
<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;137edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;137 equal division&lt;/em&gt; divides the octave into 137 equal parts of 8.759 cents each. It is the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for 7-limit &lt;a class="wiki_link" href="/Semicomma%20family"&gt;orwell temperament&lt;/a&gt; and for the planar temperament tempering out 2430/2401. It tempers out 2109375/2097152 (the semicomma) in the 5-limit; 225/224 and 1728/1715 in the 7-limit; 243/242 in the 11-limit; 351/350 in the 13-limit; 375/374 and 442/441 in the 17-limit; and 324/323 and 495/494 in the 19-limit. Since it is a prime number, 137 has no proper divisors aside from 1.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Subsets and supersets ===
137edo is the 33rd [[prime edo]], following [[131edo]] and before [[139edo]]. [[274edo]], which doubles it, provides a correction for its approximation to harmonic 7.  
 
== 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| -217 137 }}
| {{mapping| 137 217 }}
| +0.3865
| 0.3866
| 4.41
|-
| 2.3.5
| {{monzo| -21 3 7 }}, {{monzo| -13 17 -6 }}
| {{mapping| 137 217 318 }}
| +0.3887
| 0.3157
| 3.60
|}
 
=== 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
| 3\137
| 26.28
| 1594323/1562500
| [[Sfourth]] (5-limit)
|-
| 1
| 4\137
| 35.04
| 1990656/1953125
| [[Gammic]] (137d) / [[gammy]] (137)
|-
| 1
| 31\137
| 271.53
| 75/64
| [[Orwell]] (137e) / [[sabric]] (137d)
|-
| 1
| 36\137
| 315.33
| 6/5
| [[Parakleismic]]
|-
| 1
| 53\137
| 464.23
| 72/55
| [[Borwell]]
|-
| 1
| 59\137
| 516.79
| 27/20
| [[Marvo]] (137)
|-
| 1
| 63\137
| 551.82
| 11/8
| [[Emka]] (137d) / [[emkay]] (137)
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Diagrams ==
A diagram of 7-limit orwell based on the 31\137edo generator:
 
[[File:137edo_MOS_031_demo_correction.png|alt=137edo_MOS_031_demo_correction.png|137edo_MOS_031_demo_correction.png]]
 
[[:File:137edo_MOS_031.svg|137edo_MOS_031.svg]]
 
[[Category:Nuwell]]
[[Category:Orwell]]
[[Category:Orson]]
Retrieved from "https://en.xen.wiki/w/137edo"