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2684edo: Difference between revisions

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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>2015-08-18 00:44:35 UTC</tt>.<br>
 
: The original revision id was <tt>556855705</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
2684edo is an extremely strong 13-limit system, with a lower 13-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until we reach [[5585edo]]. It is [[consistency|distinctly consistent]] through the [[17-odd-limit]], and is both a [[zeta edo|zeta peak and zeta integral edo]]. It is [[enfactoring|enfactored]] in the 2.3.5.13 [[subgroup]], with the same tuning as [[1342edo]], [[tempering out]] kwazy, {{monzo| -53 10 16 }}, senior, {{monzo| -17 62 -35 }} and egads, {{monzo| -36 52 51 }}. A 13-limit [[comma basis]] is {[[9801/9800]], [[10648/10647]], 140625/140608, 196625/196608, 823680/823543}; it also tempers out [[123201/123200]]. It is less accurate, but still quite accurate in the 17-limit; a comma basis is {[[4914/4913]], [[5832/5831]], 9801/9800, 10648/10647, [[28561/28560]], 140625/140608}.
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>
In higher limits, 2684edo is a good no-19s 31-limit tuning, with errors of 25% or less on all harmonics (except 19).  
<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 2684 division divides the octave into 2684 equal parts of 0.4471 cents each. It is a very strong 13-limit tuning, with a lower 13-limit [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] than any division until we reach [[5585edo]]. It is distinctly consistent though the 17 limit, and is both a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak and zeta integral edo]]. A basis for its 13-limit commas is {9801/9800, 10648/10647, 196625/196608, 823680/823543, 1399680/1399489}; it also tempers out 123201/123200. It factors as 2684 = 2^2 * 11 * 61, so that [[22edo|22]] is a divisor.</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;2684edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 2684 division divides the octave into 2684 equal parts of 0.4471 cents each. It is a very strong 13-limit tuning, with a lower 13-limit &lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&gt;relative error&lt;/a&gt; than any division until we reach &lt;a class="wiki_link" href="/5585edo"&gt;5585edo&lt;/a&gt;. It is distinctly consistent though the 17 limit, and is both a &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&gt;zeta peak and zeta integral edo&lt;/a&gt;. A basis for its 13-limit commas is {9801/9800, 10648/10647, 196625/196608, 823680/823543, 1399680/1399489}; it also tempers out 123201/123200. It factors as 2684 = 2^2 * 11 * 61, so that &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt; is a divisor.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Harmonics in equal|2684|columns=11}}
 
=== Subsets and supersets ===
Since 2684 factors into {{factorization|2684}}, 2684edo has subset edos {{EDOs| 2, 4, 11, 22, 44, 61, 122, 244, 671, and 1342 }}.
 
2684edo tunes the septimal comma, 64/63, to an exact 1/44th of the octave (61 steps). As a corollary, it supports the period-44 [[ruthenium]] temperament.
 
== 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
| 78125000/78121827, {{monzo| -5 10 5 -8 }}, {{monzo| -48 0 11 8 }}
| {{mapping| 2684 4254 6232 7535 }}
| +0.0030
| 0.0085
| 1.90
|-
| 2.3.5.7.11
| 9801/9800, 1771561/1771470, 35156250/35153041, 67110351/67108864
| {{mapping| 2684 4254 6232 7535 9825 }}
| +0.0089
| 0.0089
| 1.99
|-
| 2.3.5.7.11.13
| 9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543
| {{mapping| 2684 4254 6232 7535 9825 9932 }}
| +0.0041
| 0.0086
| 1.93
|-
| 2.3.5.7.11.13.17
| 4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608
| {{mapping| 2684 4254 6232 7535 9825 9932 10971 }}
| −0.0004
| 0.0136
| 3.04
|-
| 2.3.5.7.11.13.17.23
| 4761/4760, 4914/4913, 5832/5831, 8625/8624, 9801/9800, 10648/10647, 28561/28560
| {{mapping| 2684 4254 6232 7535 9825 9932 10971 12141 }}
| +0.0026
| 0.0150
| 3.36
|}
* 2684et holds a record for the lowest relative error in the 13-limit, past [[2190edo|2190]] and is only bettered by [[5585edo|5585]], which is more than twice its size. In terms of absolute error, it is narrowly beaten by [[3395edo|3395]].
* 2684et is also notable in the 11-limit, where it has the lowest absolute error, past [[1848edo|1848]] and before 3395.  
 
=== Rank-2 temperaments ===
Note: 5-limit temperaments supported by [[1342edo]] 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
| 353\2684
| 157.824
| 36756909/33554432
| [[Hemiegads]]
|-
| 44
| 1114\2684<br />(16\2684)
| 498.063<br />(7.154)
| 4/3<br />(18375/18304)
| [[Ruthenium]]
|-
| 61
| 557\2684<br />(29\2684)
| 249.031<br />(12.965)
| 11907/6875<br />(?)
| [[Promethium]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
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