296edo: Difference between revisions

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**Imported revision 240004881 - Original comment: **
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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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>2011-07-05 01:56:42 UTC</tt>.<br>
 
: The original revision id was <tt>240004881</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
In the 5-limit, 296et not only [[tempering out|tempers out]] the [[semicomma]] of 5-limit orwell (orson) temperament, 2109375/2097152, it also provides its [[optimal patent val]], and tempers out the minortone comma, {{monzo| -16 35 -17 }}. It is also an interesting temperament in higher limits, being [[consistency|distinctly consistent]] through to the [[15-odd-limit]]. In the 7-limit it tempers out 4375/4374 ([[ragisma]]), 16875/16807 (mirkwai), and 118098/117649 (stearnsma), [[support]]ing 7-limit [[octoid]] and [[sabric]]. In the 11-limit, [[540/539]], 1375/1372, [[3025/3024]], [[4000/3993]], [[6250/6237]] and [[9801/9800]]; in the 13-limit, [[625/624]], [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], and [[6656/6655]], so that it also supports the 11- and 13-limit versions of octoid. It allows [[swetismic chords]] and [[squbemic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-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>
=== 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;white-space: pre-wrap ! important" class="old-revision-html">The //296 equal temperament// divides the octave into 296 equal parts of 4.054 cents each. In the 5-limit, it not only tempers out the semicomma of 5-limit orwell (orson) temperament, 2109375/2097152, it also provides its [[optimal patent val]], and tempers out the minortone comma, |-16 35 -17&gt;. It is also an interesting temperament in higher limits, being distinctly consistent through to the 15-limit. In the 7-limit it tempers out 4375/4374 and 16875/16807, supporting 7-limit [[Ragismic microtemperaments#Octoid|octoid temperament]]. In the 11-limit, it tempers out 1375/1372, 6250/6237, 540/539, 4000/3993 and 3205/3024, and in the 13-limit 625/624, 729/728, 1575/1573, 1716/1715, 2080/2079, so that it also supports the 11- and 13-limit versions of octoid.</pre></div>
{{Harmonics in equal|296|columns=11}}
<h4>Original HTML content:</h4>
 
<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;296edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;296 equal temperament&lt;/em&gt; divides the octave into 296 equal parts of 4.054 cents each. In the 5-limit, it not only tempers out the semicomma of 5-limit orwell (orson) temperament, 2109375/2097152, it also provides its &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt;, and tempers out the minortone comma, |-16 35 -17&amp;gt;. It is also an interesting temperament in higher limits, being distinctly consistent through to the 15-limit. In the 7-limit it tempers out 4375/4374 and 16875/16807, supporting 7-limit &lt;a class="wiki_link" href="/Ragismic%20microtemperaments#Octoid"&gt;octoid temperament&lt;/a&gt;. In the 11-limit, it tempers out 1375/1372, 6250/6237, 540/539, 4000/3993 and 3205/3024, and in the 13-limit 625/624, 729/728, 1575/1573, 1716/1715, 2080/2079, so that it also supports the 11- and 13-limit versions of octoid.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Subsets and supersets ===
Since 296 factors into {{factorisation|296}}, 296edo has subset edos {{EDOs| 2, 4, 8, 37, 74 and 148 }}.
 
== 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| -469 296 }}
| {{mapping| 296 469 }}
| +0.1904
| 0.1905
| 4.70
|-
| 2.3.5
| 2109375/2097152, {{monzo| -16 35 -17 }}
| {{mapping| 296 469 687 }}
| +0.2962
| 0.2158
| 5.32
|-
| 2.3.5.7
| 4375/4374, 16875/16807, 2100875/2097152
| {{mapping| 296 469 687 831 }}
| +0.2138
| 0.2350
| 5.80
|-
| 2.3.5.7.11
| 540/539, 1375/1372, 4000/3993, 2100875/2097152
| {{mapping| 296 469 687 831 1024 }}
| +0.1691
| 0.2284
| 5.63
|-
| 2.3.5.7.11.13
| 540/539, 625/624, 729/728, 1375/1372, 15379/15360
| {{mapping| 296 469 687 831 1024 1095 }}
| +0.2012
| 0.2206
| 5.44
|}
 
=== 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
| 45\296
| 182.43
| 10/9
| [[Mitonic]]
|-
| 1
| 67\296
| 271.62
| 75/64
| [[Sabric]]
|-
| 1
| 105\296
| 425.68
| 2625/2048
| [[Rainwell]]
|-
| 2
| 57\296
| 231/08
| 8/7
| [[Orga]]
|-
| 8
| 144\296<br />(4\296)
| 583.78<br />(16.22)
| 7/5<br />(126/125)
| [[Octoid]]
|-
| 37
| 67\296<br />(3\296)
| 271.62<br />(12.16)
| 117/100<br />(?)
| [[Dzelic]]
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
[[Category:Sabric]]

Latest revision as of 13:32, 13 March 2026

← 295edo 296edo 297edo →
Prime factorization 23 × 37
Step size 4.05405 ¢ 
Fifth 173\296 (701.351 ¢)
Semitones (A1:m2) 27:23 (109.5 ¢ : 93.24 ¢)
Consistency limit 15
Distinct consistency limit 15

296 equal divisions of the octave (abbreviated 296edo or 296ed2), also called 296-tone equal temperament (296tet) or 296 equal temperament (296et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 296 equal parts of about 4.05 ¢ each. Each step represents a frequency ratio of 21/296, or the 296th root of 2.

Theory

In the 5-limit, 296et not only tempers out the semicomma of 5-limit orwell (orson) temperament, 2109375/2097152, it also provides its optimal patent val, and tempers out the minortone comma, [-16 35 -17⟩. It is also an interesting temperament in higher limits, being distinctly consistent through to the 15-odd-limit. In the 7-limit it tempers out 4375/4374 (ragisma), 16875/16807 (mirkwai), and 118098/117649 (stearnsma), supporting 7-limit octoid and sabric. In the 11-limit, 540/539, 1375/1372, 3025/3024, 4000/3993, 6250/6237 and 9801/9800; in the 13-limit, 625/624, 729/728, 1575/1573, 1716/1715, 2080/2079, and 6656/6655, so that it also supports the 11- and 13-limit versions of octoid. It allows swetismic chords and squbemic chords in the 13-odd-limit, in addition to nicolic chords in the 15-odd-limit.

Prime harmonics

Approximation of prime harmonics in 296edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.60 -1.18 +0.09 +0.03 -1.34 +0.45 -1.57 +0.10 +0.15 -1.79
Relative (%) +0.0 -14.9 -29.1 +2.3 +0.8 -33.0 +11.1 -38.7 +2.6 +3.8 -44.2
Steps
(reduced) 		296
(0) 		469
(173) 		687
(95) 		831
(239) 		1024
(136) 		1095
(207) 		1210
(26) 		1257
(73) 		1339
(155) 		1438
(254) 		1466
(282)

Subsets and supersets

Since 296 factors into 23 × 37, 296edo has subset edos 2, 4, 8, 37, 74 and 148.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-469 296⟩ [⟨296 469]] +0.1904 0.1905 4.70
2.3.5 2109375/2097152, [-16 35 -17⟩ [⟨296 469 687]] +0.2962 0.2158 5.32
2.3.5.7 4375/4374, 16875/16807, 2100875/2097152 [⟨296 469 687 831]] +0.2138 0.2350 5.80
2.3.5.7.11 540/539, 1375/1372, 4000/3993, 2100875/2097152 [⟨296 469 687 831 1024]] +0.1691 0.2284 5.63
2.3.5.7.11.13 540/539, 625/624, 729/728, 1375/1372, 15379/15360 [⟨296 469 687 831 1024 1095]] +0.2012 0.2206 5.44

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 45\296 182.43 10/9 Mitonic
1 67\296 271.62 75/64 Sabric
1 105\296 425.68 2625/2048 Rainwell
2 57\296 231/08 8/7 Orga
8 144\296
(4\296) 		583.78
(16.22) 		7/5
(126/125) 		Octoid
37 67\296
(3\296) 		271.62
(12.16) 		117/100
(?) 		Dzelic

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

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