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296edo: Difference between revisions

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**Imported revision 240005561 - 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 02:08:23 UTC</tt>.<br>
: The original revision id was <tt>240005561</tt>.<br>
: The revision comment was: <tt></tt><br>
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>
<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.


296 is divisible by 2, 4, 8, 37, 74 and 148.</pre></div>
== Theory ==
<h4>Original HTML content:</h4>
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.
<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;br /&gt;
 
&lt;br /&gt;
=== Prime harmonics ===
296 is divisible by 2, 4, 8, 37, 74 and 148.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Harmonics in equal|296|columns=11}}
 
=== 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]]
Retrieved from "https://en.xen.wiki/w/296edo"