Xenharmonic Wiki

282edo: Difference between revisions

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**Imported revision 240474327 - 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>2011-07-08 01:21:43 UTC</tt>.<br>
 
: The original revision id was <tt>240474327</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.</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.&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"