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814edo: Difference between revisions

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**Imported revision 244799071 - 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>2011-08-08 03:20:12 UTC</tt>.<br>
 
: The original revision id was <tt>244799071</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
814edo is [[consistency|distinctly consistent]] to the [[17-odd-limit]] and is a strong 17-limit system. The equal temperament is [[enfactoring|enfactored]] in the 5-limit, tempering out the [[schisma]] as does 407et. In the 7-limit it tempers out [[2401/2400]] so that it [[support]]s and gives a good tuning for [[sesquiquartififths]]. In the 11-limit it tempers out [[9801/9800]], in the 13-limit [[4225/4224]] and [[6656/6655]], and in the 17-limit [[1701/1700]], [[2058/2057]], [[2601/2600]], [[4914/4913]] and [[5832/5831]]. The {{nowrap|171 &amp; 643}} temperament gives an extension of sesquiquartififths to the 17-limit for which 814edo provides the [[optimal patent val]].
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 //814 equal division// divides the octave into 814 equal parts of 1.474 cents each.It is uniquely [[consistent]] to the 17-limit and is a strong 17-limit system. It tempers out 32805/32768 in the 5-limit and 2401/2400 in the 7-limit, so that it supports and gives a good tuning for [[Schismatic family#Sesquiquartififths|sesquiquartififths temperament]]. In the 11-limit it tempers out 9801/9800, in the 13-limit 4224/4224 and 6656/6655, and in the 17-limit 1701/1700, 2058/2057, 2601/2600, 4914/4913 and 5832/5831. The 171&amp;643 temperament gives an extension of sesquiquartififths to the 17-limit for which 814edo provides the [[optimal patent val]].</pre></div>
{{Harmonics in equal|814}}
<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;814edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;814 equal division&lt;/em&gt; divides the octave into 814 equal parts of 1.474 cents each.It is uniquely &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt; to the 17-limit and is a strong 17-limit system. It tempers out 32805/32768 in the 5-limit and 2401/2400 in the 7-limit, so that it supports and gives a good tuning for &lt;a class="wiki_link" href="/Schismatic%20family#Sesquiquartififths"&gt;sesquiquartififths temperament&lt;/a&gt;. In the 11-limit it tempers out 9801/9800, in the 13-limit 4224/4224 and 6656/6655, and in the 17-limit 1701/1700, 2058/2057, 2601/2600, 4914/4913 and 5832/5831. The 171&amp;amp;643 temperament gives an extension of sesquiquartififths to the 17-limit for which 814edo provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Subsets and supersets ===
Since 814 factors into {{factorization|814}}, 814edo has subset edos {{EDOs| 2, 11, 22, 37, 74, and 407 }}.
 
== 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, 32805/32768, {{monzo| 25 20 -22 -2 }}
| {{mapping| 814 1290 1890 2285 }}
| +0.0695
| 0.0577
| 3.91
|-
| 2.3.5.7.11
| 2401/2400, 9801/9800, 32805/32768, 20155392/20131375
| {{mapping| 814 1290 1890 2285 2816 }}
| +0.0536
| 0.0605
| 4.11
|-
| 2.3.5.7.11.13
| 2401/2400, 4225/4224, 6656/6655, 9801/9800, 32805/32768
| {{mapping| 814 1290 1890 2285 2816 3012 }}
| +0.0552
| 0.0554
| 3.76
|-
| 2.3.5.7.11.13.17
| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4225/4224, 6656/6655
| {{mapping| 814 1290 1890 2285 2816 3012 3327 }}
| +0.0573
| 0.0528
| 3.50
|-
| 2.3.5.7.11.13.17.19
| 1445/1444, 1521/1520, 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2601/2600
| {{mapping| 814 1290 1890 2285 2816 3012 3327 3458 }}
| +0.0421
| 0.0629
| 4.27
|-
| 2.3.5.7.11.13.17.19.23
| 1445/1444, 1521/1520, 1701/1700, 1863/1862, 2058/2057, 2376/2375, 2401/2400, 2601/2600
| {{mapping| 814 1290 1890 2285 2816 3012 3327 3682 }}
| +0.0439
| 0.0595
| 4.04
|}
* 814et is notable in the 17- and 23-limit with lower absolute errors than any previous equal temperaments, beating [[764edo|764]] in the 17-limit and [[742edo|742i]] in the 23-limit, and is only bettered by [[935edo|935]] in either subgroup.
 
=== Rank-2 temperaments ===
Note: 5-limit temperaments supported by 407edo are not included.
 
{| 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
| 119\814
| 175.43
| 448/405
| [[Sesquiquartififths]]
|}
<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:Sesquiquartififths]]
Retrieved from "https://en.xen.wiki/w/814edo"