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320edo: Difference between revisions

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**Imported revision 240539109 - 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>2011-07-08 14:22:33 UTC</tt>.<br>
 
: The original revision id was <tt>240539109</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
320edo is [[consistent]] in the [[19-odd-limit]] and a fairly good tuning for the [[19-limit]]. It has a flat tendency for most [[prime harmonic]]s from 3 to 19, with the sole exception of [[17/1|17]].  
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>
As an equal temperament, it [[tempering out|tempers out]] 65625/65536 ([[horwell comma]]) and 420175/419904 ([[wizma]]) in the 7-limit and [[441/440]], [[8019/8000]] and [[9801/9800]] in the 11-limit, and so [[support]]s the [[varuna]] temperament, the rank-3 temperament tempering out 441/440, 8019/8000 and 9801/9800, for which it provides the [[optimal patent val]]. It also provides the optimal patent val for the rank-4 werckismic temperament tempering out 441/440. It tempers out [[729/728]], [[1001/1000]], [[1575/1573]], [[4225/4224]] and [[6656/6655]] in the 13-limit, leading to further temperaments for which it provides the optimal patent val, such as tempering out 441/440 with 729/728, 1001/1000 or both, or with 8019/8000, leading to an extension of varuna.  
<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 //320 equal division// divides the octave into 320 equal parts of precisely 3.75 cents each. It tempers out 65625/65532 and 420175/419904 in the 7-limit and 441/440, 8019/8000 and 9801/9800 in the 11-limit, and so supports [[Werckismic temperaments#Varuna|varuna temperament]], the rank three temperament tempering out 441/440, 8019/8000 and 9801/9800, for which it provides the [[optimal patent val]]. It also provides the [[optimal patent val]] for the rank four werckismic temperament tempering out 441/440. It tempers out 729/728, 1001/1000, 1575/1573, 4225/4224 and 6656/6655 in the 13-limit, leading to further temperaments for which it provides the optimal patent val, such as tempering out 441/440 with 729/728, 1001/1000 or both, or with 8019/8000, leading to a rank three temperament.</pre></div>
 
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;320edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;320 equal division&lt;/em&gt; divides the octave into 320 equal parts of precisely 3.75 cents each. It tempers out 65625/65532 and 420175/419904 in the 7-limit and 441/440, 8019/8000 and 9801/9800 in the 11-limit, and so supports &lt;a class="wiki_link" href="/Werckismic%20temperaments#Varuna"&gt;varuna temperament&lt;/a&gt;, the rank three temperament tempering out 441/440, 8019/8000 and 9801/9800, for which it provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt;. It also provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for the rank four werckismic temperament tempering out 441/440. It tempers out 729/728, 1001/1000, 1575/1573, 4225/4224 and 6656/6655 in the 13-limit, leading to further temperaments for which it provides the optimal patent val, such as tempering out 441/440 with 729/728, 1001/1000 or both, or with 8019/8000, leading to a rank three temperament.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Harmonics in equal|320|intervals=prime|columns=11}}
 
=== Subsets and supersets ===
Since 320 factors into 2<sup>6</sup> × 5, 320edo has subset edos {{EDOs| 2, 4, 5, 10, 16, 20, 32, 40, 64, 80, and 160 }}.
 
== 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| -507 320 }}
| {{Mapping| 320 507 }}
| +0.2224
| 0.2224
| 5.93
|-
| 2.3.5
| {{Monzo| 23 6 -14 }}, {{monzo| -28 25 -5 }}
| {{Mapping| 320 507 743 }}
| +0.1574
| 0.2036
| 5.43
|-
| 2.3.5.7
| 65625/65536, 235298/234375, 321489/320000
| {{Mapping| 320 507 743 898 }}
| +0.2361
| 0.2229
| 5.94
|-
| 2.3.5.7.11
| 441/440, 8019/8000, 41503/41472, 65625/65536
| {{Mapping| 320 507 743 898 1107 }}
| +0.1928
| 0.2173
| 5.80
|-
| 2.3.5.7.11.13
| 441/440, 729/728, 1001/1000, 4225/4224, 6656/6655
| {{Mapping| 320 507 743 898 1107 1184 }}
| +0.1845
| 0.1993
| 5.31
|-
| 2.3.5.7.11.13.17
| 441/440, 729/728, 833/832, 1001/1000, 1089/1088, 4225/4224
| {{Mapping| 320 507 743 898 1107 1184 1308 }}
| +0.1565
| 0.1968
| 5.25
|-
| 2.3.5.7.11.13.17.19
| 441/440, 513/512, 729/728, 833/832, 969/968, 1001/1000, 1521/1520
| {{Mapping| 320 507 743 898 1107 1184 1308 1359 }}
| +0.1741
| 0.1899
| 5.06
|}
 
=== 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
| 7\320
| 26.25
| {{Monzo| -2 13 -8 }}
| [[Sfourth]] (5-limit)
|-
| 1
| 131\320
| 491.25
| 3645/2744
| [[Fifthplus]]
|-
| 1
| 157\320
| 588.75
| 45/32
| [[Untriton]] (5-limit)
|-
| 1
| 93\320
| 348.75
| 6144/3757
| [[Hectosaros leap week]]
|-
| 2
| 19\320
| 71.25
| 25/24
| [[Narayana]]
|-
| 5
| 133\320<br>(5\320)
| 498.75<br>(18.75)
| 4/3<br>(81/80)
| [[Quintile]]
|-
| 8
| 133\320<br>(9\320)
| 566.25<br>(33.75)
| 104/75<br>(55/54)
| [[Octowerck]]
|-
| 10
| 19\320<br>(13\320)
| 71.25<br>(48.75)
| 25/24<br>(36/35)
| [[Decavish]]
|-
| 10
| 133\320<br>(5\320)
| 498.75<br>(18.75)
| 4/3<br>(81/80)
| [[Decile]]
|-
| 20
| 151\320<br>(7\320)
| 566.25<br>(26.25)
| 165/119<br>(?)
| [[Soviet ferris wheel]]
|-
| 32
| 133\320<br>(3\320)
| 498.75<br>(11.25)
| 4/3<br>(?)
| [[Bezique]]
|-
| 80
| 99\320<br>(3\320)
| 371.25<br>(11.25)
| 2275/1836<br>(?)
| [[Mercury]]
|-
| 80
| 133\320<br>(1\320)
| 498.75<br>(3.75)
| 4/3<br>(245/243)
| [[Octogintic]]
|}
<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:Varuna]]
[[Category:Werckismic]]
Retrieved from "https://en.xen.wiki/w/320edo"