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328edo: Difference between revisions

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**Imported revision 240557849 - 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 17:09:49 UTC</tt>.<br>
 
: The original revision id was <tt>240557849</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
328edo is [[enfactoring|enfactored]] in the [[5-limit]], with the same tuning as [[164edo]], but the approximation of higher [[harmonic]]s are much improved. It has a sharp tendency, with harmonics 3 through 17 all tuned sharp. The equal temperament [[tempering out|tempers out]] [[2401/2400]], [[3136/3125]], and [[6144/6125]] in the 7-limit, [[9801/9800]], [[16384/16335]] and [[19712/19683]] in the 11-limit, [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]] in the 13-limit, [[936/935]], [[1156/1155]] and [[2601/2600]] in the 17-limit, so that it [[support]]s [[würschmidt]] and [[hemiwürschmidt]], and provides the [[optimal patent val]] for 7-limit hemiwürschmidt, 11- and 13-limit [[semihemiwür]], and 13-limit [[semiporwell]].  
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 //328 equal division// divides the octave into 328 equal parts of 3.659 cents each. It tempers out the wuerschmidt comma, 393216/390625, in the 5-limit and 3136/3125, 6144/6125 and 2401/2400 in the 7-limit, so that it supports wuerschmidt and hemiwuerschmidt temperaments, and provides the [[optimal patent val]] for 7-limit hemiwuerschmidt. </pre></div>
{{Harmonics in equal|328|intervals=prime|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;328edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;328 equal division&lt;/em&gt; divides the octave into 328 equal parts of 3.659 cents each. It tempers out the wuerschmidt comma, 393216/390625, in the 5-limit and 3136/3125, 6144/6125 and 2401/2400 in the 7-limit, so that it supports wuerschmidt and hemiwuerschmidt temperaments, and provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for 7-limit hemiwuerschmidt.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Subsets and supersets ===
Since 328 factors into {{Factorisation|328}}, 328edo has subset edos {{EDOs| 2, 4, 8, 41, 82, and 164 }}.
 
== 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]] (%)
|-
| 2.3.5.7
| 2401/2400, 3136/3125, 589824/588245
| {{mapping| 328 520 762 921 }}
| −0.298
| 0.229
| 6.27
|-
| 2.3.5.7.11
| 2401/2400, 3136/3125, 9801/9800, 19712/19683
| {{mapping| 328 520 762 921 1135 }}
| −0.303
| 0.205
| 5.61
|-
| 2.3.5.7.11.13
| 676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647
| {{mapping| 328 520 762 921 1135 1214 }}
| −0.295
| 0.188
| 5.15
|-
| 2.3.5.7.11.13.17
| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125
| {{mapping| 328 520 762 921 1135 1214 1341 }}
| −0.293
| 0.174
| 4.77
|}
 
=== Rank-2 temperaments ===
Note: 5-limit temperaments supported by 164et are not listed.
 
{| 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
| 53\328
| 193.90
| 28/25
| [[Hemiwürschmidt]]
|-
| 1
| 117\328
| 428.05
| 2800/2187
| [[Osiris]]
|-
| 2
| 17\328
| 62.20
| 28/27
| [[Eagle]]
|-
| 2
| 111\328<br />(53\328)
| 406.10<br />(193.90)
| 495/392<br />(28/25)
| [[Semihemiwürschmidt]]
|-
| 8
| 136\328<br />(13\328)
| 497.56<br />(47.56)
| 4/3<br />(36/35)
| [[Twilight]]
|-
| 41
| 49\328<br />(1\328)
| 179.27<br />(3.66)
| 567/512<br />(352/351)
| [[Hemicountercomp]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
[[Category:Hemiwürschmidt]]
[[Category:Semiporwell]]
Retrieved from "https://en.xen.wiki/w/328edo"