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289edo: Difference between revisions

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**Imported revision 239769421 - Original comment: **
m renamed sextilififths to sextilifourths
 
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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>2011-07-02 16:26:00 UTC</tt>.<br>
 
: The original revision id was <tt>239769421</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
289edo is a strong 5-limit system with decent [[11-limit|11-]] and [[13-limit]] interpretations despite in[[consistency]] in the [[13-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] the [[schisma]], 32805/32768 in the [[5-limit]]; [[4375/4374]] and [[65625/65536]] in the [[7-limit]]; [[441/440]] and [[4000/3993]] in the 11-limit; and [[364/363]], [[676/675]], [[1001/1000]], [[1575/1573]] and [[2080/2079]] in the 13-limit.
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 is the [[optimal patent val]] for the [[13-limit]] rank-5 temperament tempering out 364/363, and the 13-limit [[history (temperament)|history]] temperament, which tempers out 364/363, 441/440 and 676/675. It provides a good tuning for the 11-limit version also. It is also the optimal patent val for [[sextilifourths]], [[quintaschis]], and [[quincy]] in both the 11- and 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;white-space: pre-wrap ! important" class="old-revision-html">The //289 equal temperament// divides the octave into 289 equal parts of 4.152 cents each. It is the [[optimal patent val]] for [[13-limit]] [[Werckismic temperaments#History|history temperament]], which tempers out 364/363, 441/440 and 1001/1000, and provides a good tuning for the 11-limit version also, and is also the optimal patent val for [[Schismatic family|sextilififths]] in both the 11- and 13-limit. It is uniquely consist in the 9-limit, and tempers out the schisma, 32805/32768 in the 5-limit; 4375/4374 and 65625/65536 in the 7-limit; 441/440 and 4000/3993 in the 11-limit; and 364/363, 676/675, 1001/1000, 1575/1573 and 2080/2079 in the 13-limit.</pre></div>
 
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;289edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;289 equal temperament&lt;/em&gt; divides the octave into 289 equal parts of 4.152 cents each. It is the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; &lt;a class="wiki_link" href="/Werckismic%20temperaments#History"&gt;history temperament&lt;/a&gt;, which tempers out 364/363, 441/440 and 1001/1000, and provides a good tuning for the 11-limit version also, and is also the optimal patent val for &lt;a class="wiki_link" href="/Schismatic%20family"&gt;sextilififths&lt;/a&gt; in both the 11- and 13-limit. It is uniquely consist in the 9-limit, and tempers out the schisma, 32805/32768 in the 5-limit; 4375/4374 and 65625/65536 in the 7-limit; 441/440 and 4000/3993 in the 11-limit; and 364/363, 676/675, 1001/1000, 1575/1573 and 2080/2079 in the 13-limit.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Harmonics in equal|289}}
 
=== Subsets and supersets ===
289 is 17 squared. In light of containing [[17edo]] as a subset, 289edo [[support]]s the [[chlorine]] temperament, which tempers out the [[septendecima]] {{monzo| -52 -17 34 }} and the ragisma 4375/4374.
 
== 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| -458 289 }}
| {{mapping| 289 458 }}
| +0.0709
| 0.0710
| 1.71
|-
| 2.3.5
| 32805/32768, {{monzo| 7 41 -31 }}
| {{mapping| 289 458 671 }}
| +0.0695
| 0.0580
| 1.40
|-
| 2.3.5.7
| 4375/4374, 32805/32768, 235298/234375
| {{mapping| 289 458 671 811 }}
| +0.1725
| 0.1854
| 4.46
|-
| 2.3.5.7.11
| 441/440, 4000/3993, 4375/4374, 32805/32768
| {{mapping| 289 458 671 811 1000 }}
| +0.0841
| 0.2423
| 5.83
|-
| 2.3.5.7.11.13
| 364/363, 441/440, 676/675, 4375/4374, 19773/19712
| {{mapping| 289 458 671 811 1000 1069 }}
| +0.1500
| 0.2657
| 6.40
|}
* 289et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[171edo|171]] and followed by [[323edo|323]].  
 
=== 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
| 4\289
| 16.61
| 100/99
| [[Quincy]]
|-
| 1
| 13\289
| 53.98
| 33/32
| [[Tridecafifths]]
|-
| 1
| 20\289
| 83.04
| 21/20
| [[Sextilifourths]]
|-
| 1
| 24\289
| 99.65
| 18/17
| [[Quintaschis]]
|-
| 1
| 76\289
| 315.57
| 6/5
| [[Acrokleismic]]
|-
| 1
| 86\289
| 357.09
| 768/625
| [[Dodifo]]
|-
| 1
| 108\289
| 448.44
| 35/27
| [[Semidimfourth]]
|-
| 1
| 120\289
| 498.27
| 4/3
| [[Pontiac]]
|-
| 1
| 135\289
| 560.55
| 864/625
| [[Whoosh]]
|-
| 17
| 93\289<br />(8\289)
| 386.16<br />(33.22)
| {{monzo| -23 5 9 -2 }}<br />(100352/98415)
| [[Chlorine]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
[[Category:History (temperament)]]
[[Category:Minor minthmic]]
[[Category:Quincy]]
[[Category:Quintaschis]]
[[Category:Sextilifourths]]
Retrieved from "https://en.xen.wiki/w/289edo"