282edo: Difference between revisions

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**Imported revision 299722708 - 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>2012-02-08 11:04:47 UTC</tt>.<br>
 
: The original revision id was <tt>299722708</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.</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.&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]]

Latest revision as of 15:31, 16 March 2025

← 281edo 282edo 283edo →
Prime factorization 2 × 3 × 47
Step size 4.25532 ¢ 
Fifth 165\282 (702.128 ¢) (→ 55\94)
Semitones (A1:m2) 27:21 (114.9 ¢ : 89.36 ¢)
Consistency limit 29
Distinct consistency limit 23

282 equal divisions of the octave (abbreviated 282edo or 282ed2), also called 282-tone equal temperament (282tet) or 282 equal temperament (282et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 282 equal parts of about 4.26 ¢ each. Each step represents a frequency ratio of 21/282, or the 282nd root of 2.

Theory

282edo is the smallest edo 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 (282 = 3 × 94), as well as 11/10 and 20/17 (supporting the garistearn temperament). It has a distinct sharp tendency for odd harmonics up to 29.

The equal temperament 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.

It allows essentially tempered chords including swetismic chords, squbemic chords, and petrmic chords in the 13-odd-limit, in addition to nicolic chords in the 15-odd-limit.

Prime harmonics

Approximation of prime harmonics in 282edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.17 +0.92 +1.39 +1.87 +2.03 +1.43 +0.36 +1.51 +0.21 -0.35
Relative (%) +0.0 +4.1 +21.6 +32.6 +44.0 +47.6 +33.5 +8.4 +35.6 +4.9 -8.3
Steps
(reduced) 		282
(0) 		447
(165) 		655
(91) 		792
(228) 		976
(130) 		1044
(198) 		1153
(25) 		1198
(70) 		1276
(148) 		1370
(242) 		1397
(269)

Subsets and supersets

Since 282 factors into primes as 2 × 3 × 47, 282edo has subset edos 2, 3, 47, 94, and 141.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [32 -7 -9⟩, [-7 22 -12⟩ [⟨282 447 655]] −0.1684 0.1671 3.93
2.3.5.7 6144/6125, 118098/117649, 250047/250000 [⟨282 447 655 792]] −0.2498 0.2020 4.75
2.3.5.7.11 540/539, 4000/3993, 5632/5625, 137781/137500 [⟨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 [⟨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 [⟨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 [⟨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 [⟨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 270 and before 311.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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
(4\282) 		217.02
(17.02) 		17/15
(105/104) 		Stearnscape
6 80\282
(14\282) 		340.43
(59.57) 		162/133
(88/85) 		Semiseptichrome
6 117\282
(23\282) 		497.87
(97.87) 		4/3
(128/121) 		Sextile

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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