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282edo: Difference between revisions

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**Imported revision 514613994 - 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>2014-06-22 03:24:02 UTC</tt>.<br>
 
: The original revision id was <tt>514613994</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
282edo is the smallest edo [[consistency|distinctly consistent]] through to the [[23-odd-limit]], and also the smallest consistent to the [[29-odd-limit]]. It shares the same 3rd, 7th, and 13th harmonics with [[94edo]] ({{nowrap| 282 {{=}} 3 × 94 }}), as well as [[11/10]] and [[20/17]] ([[support]]ing the [[Stearnsmic clan #Garistearn|garistearn]] temperament). It has a distinct sharp tendency for odd harmonics up to 29.  
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>
The equal temperament [[tempering out|tempers out]] [[6144/6125]] (porwell comma), 118098/117649 (stearnsma), and [[250047/250000]] (landscape comma) in the 7-limit, and [[540/539]] and [[5632/5625]] in the 11-limit, so that it provides the [[optimal patent val]] for the [[jupiter]] temperament; it also tempers out [[4000/3993]] and 234375/234256, providing the optimal patent val for [[septisuperfourth]] temperament. In the 13-limit, it tempers out [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], and [[10648/10647]].  
<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 //282 equal division// divides the octave into 282 equal parts of 4.255 cents each. It tempers out 16875/16807, 19683/19600 and 65625/65536 in the 7-limit, and 540/539 and 5632/5625 in the 11-limit, so that it provides the [[optimal patent val]] for [[Porwell family|jupiter temperament]]; it also tempers out 4000/3993 and 234375/234256, providing the optimal patent val for [[Porwell temperaments#Septisuperfourth|septisuperfourth]] temperament. In the 13-limit it tempers out 729/728, 1575/1573, 1716/1715 and 2080/2079. It is the smallest equal temperament uniquely [[consistent]] through to the 23 limit, and also the smallest consistent to the 29 limit. 282 has proper divisors 1, 2, 3, 6, 47, 94, and 141. It therefore divides the steps of 94et into three, but is not contorted beyond the 3-limit. </pre></div>
 
<h4>Original HTML content:</h4>
It allows [[essentially tempered chord]]s including [[swetismic chords]], [[squbemic chords]], and [[petrmic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-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;282edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;282 equal division&lt;/em&gt; divides the octave into 282 equal parts of 4.255 cents each. It tempers out 16875/16807, 19683/19600 and 65625/65536 in the 7-limit, and 540/539 and 5632/5625 in the 11-limit, so that it provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for &lt;a class="wiki_link" href="/Porwell%20family"&gt;jupiter temperament&lt;/a&gt;; it also tempers out 4000/3993 and 234375/234256, providing the optimal patent val for &lt;a class="wiki_link" href="/Porwell%20temperaments#Septisuperfourth"&gt;septisuperfourth&lt;/a&gt; temperament. In the 13-limit it tempers out 729/728, 1575/1573, 1716/1715 and 2080/2079. It is the smallest equal temperament uniquely &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt; through to the 23 limit, and also the smallest consistent to the 29 limit. 282 has proper divisors 1, 2, 3, 6, 47, 94, and 141. It therefore divides the steps of 94et into three, but is not contorted beyond the 3-limit.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== Prime harmonics ===
{{Harmonics in equal|282|columns=11}}
 
=== Subsets and supersets ===
Since 282 factors into primes as {{nowrap| 2 × 3 × 47 }}, 282edo has subset edos {{EDOs| 2, 3, 47, 94, and 141 }}.  
 
== 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
| {{Monzo| 32 -7 -9 }}, {{monzo| -7 22 -12 }}
| {{Mapping| 282 447 655 }}
| −0.1684
| 0.1671
| 3.93
|-
| 2.3.5.7
| 6144/6125, 118098/117649, 250047/250000
| {{Mapping| 282 447 655 792 }}
| −0.2498
| 0.2020
| 4.75
|-
| 2.3.5.7.11
| 540/539, 4000/3993, 5632/5625, 137781/137500
| {{Mapping| 282 447 655 792 976 }}
| −0.3081
| 0.2151
| 5.06
|-
| 2.3.5.7.11.13
| 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575
| {{Mapping| 282 447 655 792 976 1044 }}
| −0.3480
| 0.2156
| 5.07
|-
| 2.3.5.7.11.13.17
| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197
| {{Mapping| 282 447 655 792 976 1044 1153 }}
| −0.3481
| 0.1996
| 4.69
|-
| 2.3.5.7.11.13.17.19
| 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573
| {{Mapping| 282 447 655 792 976 1044 1153 1198 }}
| −0.3152
| 0.2061
| 4.84
|-
| 2.3.5.7.11.13.17.19.23
| 456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287
| {{Mapping| 282 447 655 792 976 1044 1153 1198 1276 }}
| −0.3173
| 0.1944
| 4.57
|}
* 282et has a lower relative error than any previous equal temperaments in the 23-limit, past [[270edo|270]] and before [[311edo|311]].
 
=== 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
| 13\282
| 55.32
| 33/32
| [[Escapade]]
|-
| 1
| 133\282
| 565.96
| 4096/2835
| [[Alphatrident]] (7-limit)
|-
| 2
| 13\282
| 55.32
| 33/32
| [[Septisuperfourth]]
|-
| 2
| 43\282
| 182.98
| 10/9
| [[Unidecmic]]
|-
| 3
| 33\282
| 140.43
| 243/224
| [[Septichrome]]
|-
| 3
| 37\282
| 157.45
| 35/32
| [[Nessafof]] (7-limit)
|-
| 6
| 51\282<br>(4\282)
| 217.02<br>(17.02)
| 17/15<br>(105/104)
| [[Stearnscape]]
|-
| 6
| 80\282<br>(14\282)
| 340.43<br>(59.57)
| 162/133<br>(88/85)
| [[Semiseptichrome]]
|-
| 6
| 117\282<br>(23\282)
| 497.87<br>(97.87)
| 4/3<br>(128/121)
| [[Sextile]]
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
 
[[Category:Jupiter]]
[[Category:Septisuperfourth]]
Retrieved from "https://en.xen.wiki/w/282edo"