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231edo: Difference between revisions

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**Imported revision 240162837 - 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>2011-07-06 03:44:03 UTC</tt>.<br>
 
: The original revision id was <tt>240162837</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
In the 5-limit, 231et [[tempering out|tempers out]] the [[kleisma]], 15625/15552, and in the 7-limit [[1029/1024]], so that it [[support]]s the [[tritikleismic]] temperament, and in fact provides the [[optimal patent val]]. In the 11-limit it tempers out [[385/384]], [[441/440]] and [[4000/3993]], leading to 11-limit tritikleismic for which it also gives the optimal patent val.
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>
231 years is the number of years in a 41 out of 231 leap week cycle, which corresponds to a {{nowrap|41 &amp; 149}} temperament tempering out 132055/131072, 166375/165888, and 2460375/2458624. This type of solar calendar leap rule scale may actually be of more use to harmony, since a 41 note subset mimics [[41edo]], a rather useful edo harmonically, and it preserves the simple commas mentioned above.
<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 //231 equal temperament// divides the octave into 231 equal parts of 5.195 cents each. In the 5-limit it tempers out the kleisma, 15625/15552, and in the 7-limit 1029/1024, so that it supports [[Kleismic family#Tritikleismic|tritikleismic temperament]], and in fact provides the [[optimal patent val]]. In the 11-limit it tempers out 385/384, 441/440 and 4000/3993, leading to 11-limit tritikleismic for which it also gives the optimal patent val.</pre></div>
 
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;231edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;231 equal temperament&lt;/em&gt; divides the octave into 231 equal parts of 5.195 cents each. In the 5-limit it tempers out the kleisma, 15625/15552, and in the 7-limit 1029/1024, so that it supports &lt;a class="wiki_link" href="/Kleismic%20family#Tritikleismic"&gt;tritikleismic temperament&lt;/a&gt;, and in fact provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt;. In the 11-limit it tempers out 385/384, 441/440 and 4000/3993, leading to 11-limit tritikleismic for which it also gives the optimal patent val.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Harmonics in equal|231}}
 
=== Subsets and supersets ===
231 = 3 × 7 × 11, with subset edos {{EDOs| 3, 7, 11, 21, 33, and 77 }}. Since it contains [[77edo]], it can be used for playing such a tuning of the [[Carlos Alpha]] scale.
 
== 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| -64 36 3 }}
| {{mapping| 231 366 536 }}
| +0.410
| 0.334
| 6.43
|-
| 2.3.5.7
| 1029/1024, 15625/15552, 823543/820125
| {{mapping| 231 366 536 648 }}
| +0.539
| 0.365
| 7.01
|-
| 2.3.5.7.11
| 385/384, 441/440, 4000/3993, 823543/820125
| {{mapping| 231 366 536 648 799 }}
| +0.469
| 0.354
| 6.81
|}
 
=== 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
| 26\231
| 135.06
| 27/25
| [[Superlimmal]]
|-
| 1
| 27\231
| 140.26
| 243/224
| [[Septichrome]]
|-
| 1
| 45\231
| 233.77
| 8/7
| [[Slendric]]
|-
| 1
| 61\231
| 316.88
| 6/5
| [[Hanson]]
|-
| 1
| 62\231
| 322.08
| 135/112
| [[Dee leap week]]
|-
| 1
| 73\231
| 379.22
| 56/45
| [[Marthirds]]
|-
| 3
| 61\231<br />(16\231)
| 316.88<br />(83.12)
| 6/5<br />(21/20)
| [[Tritikleismic]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=-bgUQ5BYnqM ''Sins of Stoicism''] (Demo Version, March 2022)
 
[[Category:Listen]]
[[Category:Tritikleismic]]
Retrieved from "https://en.xen.wiki/w/231edo"