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31st-octave temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{Infobox fractional-octave|31}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-01 18:23:43 UTC</tt>.<br>
This page collects rank-2 temperaments with a period that is 1/31 of an octave.
: The original revision id was <tt>281304484</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">[[toc|flat]]


By the ''31-3-comma'' is meant 617673396283947/562949953421312 = {{monzo| -49 31 }}, the amount (160.605 [[cent]]s) by which 31 just perfect fifths ([[3/2]]) exceed 18 [[octave]]s. This may not seem like much of a comma, but since [[31edo]] is such a strong 7-limit system, 11- and 13-limit temperaments based on the cycle of 31 fifths actually make sense. This approach leads to the prajapati temperament and the gallium temperament.


=31-3-comma=
31edo is accurate for harmonics 5 and 7, the 31-5-comma ({{monzo| 72 0 -31 }}, the amount by which 31 just major thirds ([[5/4]]) fall short of 10 octaves) and the 31-7-comma ({{monzo| -87 0 0 31 }}, the amount by which 31 septimal whole tones ([[8/7]]) fall short of 6 octaves) is tempered out by the following ETs: {{Optimal ET sequence| 31, 62, 93, 124, 155, 186, 217, 248, 279, 310, 341, 372, 403, 434, 465, 496, and 527 }}. Tempering out these commas leads to the birds temperament.
By the //31-3-comma// is meant 617673396283947/562949953421312 = &lt;-49 31|, the amount (160.605 cents) by which 31 just (3/2) fifths exceed 18 octaves. This may not seem like much of a comma, but since 31et is such a strong 7-limit system, 11- and 13-limit temperaments based on the cycle of 31 fifths actually make sense.


=Prajapati=
== 31-commatic ==
The Hindu  god Pradjapati is said to have created the universe by speaking aloud the odd numbers from 1 to 31. People with an interest in 31 may want to try this method themselves.
Subgroup: 2.3.5


Commas: 81/80, 126/125, 1029/1024
Comma list: {{monzo| -49 31 }}


POTE generator: ~176/175 = 6.519
{{Mapping|legend=1| -31 -49 0 | 0 0 1 }}


Map: [[&lt;31 49 72 87 107|, &lt;0 0 0 0 1|]
: mapping generators: ~531441/524288 = 1\31, ~5
EDOs: 31, 93, 124b, 155b, 186b
Badness: 0.0430


==13-limit==
[[Optimal tuning]] ([[CTE]]): ~5/4 = 386.314
Commas: 81/80, 126/125, 105/104, 512/507
 
[[Support]]ing [[ET]]s: {{EDOs|31, 62, 93}}
 
== 31-5-commatic ==
Subgroup: 2.3.5
 
Comma list: {{monzo| 72 0 -31 }}
 
{{Mapping|legend=1| 31 31 72 | 0 1 0 }}
 
[[Optimal tuning]] ([[CWE]]): ~128/125 = 1\31, ~3/2 = 702.133
 
[[Support]]ing [[ET]]s: 31, 217, 186, 248, 155, 465, 403, 279, 124, 93c, 62c, 682, 310, 620
 
== 31-17/13-commatic ==
A circle of 31 [[17/13]]'s closes at the octave with an error of only 2.74 cents.
 
Subgroup: 2.13.17
 
Comma list: {{Monzo|12 0 0 0 0 31 -31}}
 
{{Mapping|31 0 12|0 1 1|legend=2}}
 
: sval mapping generators: ~2.13.17 {{monzo|-5 -13 13}} = 1\31, ~13
 
[[Optimal tuning]] ([[CTE]]): ~13/8 = 840.488
 
== Birds ==
The birds temperament tempers out the 31-5 comma, {{monzo| 72 0 -31 }}, and the 31-7 comma, ({{monzo| -87 0 0 31 }}. The name comes from Isaiah 31:5 "As birds flying, so wil the Lord of hostes defend Ierusalem, defending also hee will deliuer it, and passing ouer, he will preserue it."
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 3136/3125, 823543/819200
 
[[Mapping]]: [{{val| 31 49 72 87 }}, {{val| 0 1 0 0 }}]
 
[[POTE generator]]: ~1029/1024 = 5.1551
 
{{Optimal ET sequence|legend=1| 31, 124, 155, 186, 217, 248, 465 }}
 
[[Badness]]: 0.099928
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 3136/3125, 41503/41472
 
Mapping: [{{val| 31 49 72 87 107 }}, {{val| 0 1 0 0 2 }}]
 
POTE generator: ~385/384 = 4.9377
 
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248, 961cd }}
 
Badness: 0.039921
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 441/440, 1001/1000, 3136/3125, 13720/13689
 
Mapping: [{{val| 31 49 72 87 107 115 }}, {{val| 0 1 0 0 2 -2 }}]
 
POTE generator: ~385/384 = 5.1703
 
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248, 465 }}
 
Badness: 0.035680
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 441/440, 833/832, 1001/1000, 1225/1224, 3136/3125
 
Mapping: [{{val| 31 49 72 87 107 115 127 }}, {{val| 0 1 0 0 2 -2 -2 }}]
 
POTE generator: ~385/384 = 5.2248
 
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248, 465 }}
 
Badness: 0.025890
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 343/342, 441/440, 476/475, 833/832, 1001/1000, 1445/1444
 
Mapping: [{{val| 31 49 72 87 107 115 127 132 }}, {{val| 0 1 0 0 2 -2 -2 -2 }}]
 
POTE generator: ~385/384 = 5.3169
 
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248h, 465h }}
 
Badness: 0.021271
 
== 217 & 1178 ==
The 217 & 1178 temperament combines two multiples of 31, which are large equal divisions consistent in the 21-odd-limit. 1395edo, also consistent in 21-odd-limit, is also a tuning.
 
Subgroup: 2.3.5.7
 
Comma list: 4375/4374, {{monzo|-153 42 7 25}}
 
{{Mapping|legend=1| 31 2 -38 197 | 0 3 7 -7 }}
 
: mapping generators: ~562711519881/549755813888 = 1\31, ~67108864/47258883 = 608.167
 
[[Optimal tuning]] ([[CTE]]): ~14553/10240 = 608.167
 
[[Support]]ing [[ET]]s: {{EDOs|217, 744c, 961, 1178, 1395, 1612, 2573}}
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 820125/819896, {{monzo|-37 12 -1  6  1}}
 
{{Mapping|legend=1| 31 2 -38 197 -97 | 0 3 7 -7 13 }}
 
: mapping generators: ~45/44 = 1\31, ~14553/10240 = 608.167
 
[[Optimal tuning]] ([[CTE]]): ~14553/10240 = 608.167
 
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 4225/4224, 4375/4374, 225000/224939, 18753525/18743296
 
{{Mapping|legend=1| 31 2 -38 197 -97 99 | 0 3 7 -7 13 1 }}
 
: mapping generators: ~45/44 = 1\31, ~14553/10240 = 608.167
 
[[Optimal tuning]] ([[CTE]]): ~14553/10240 = 608.167
 
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
 
=== 17-limit ===
 
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 4225/4224, 4375/4374, 14400/14399, 14875/14872, 56595/56576
 
{{Mapping|legend=1| 31 2 -38 197 -97 99 111 | 0 3 7 -7 13 1 1 }}
 
: mapping generators: ~45/44 = 1\31, ~1989/1400 = 608.167
 
[[Optimal tuning]] ([[CTE]]): ~1989/1400 = 608.167
 
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
 
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 4200/4199, 4225/4224, 4375/4374, 5929/5928, 5985/5984, 14875/14872
 
{{Mapping|legend=1| 31 2 -38 197 -97 99 111 6 | 0 3 7 -7 13 1 1 8 }}
 
: mapping generators: ~112651/110160 = 1\31, ~665/468 = 608.166
 
[[Optimal tuning]] ([[CTE]]): ~665/468 = 608.166
 
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
 
; Music
 
* ''[https://www.youtube.com/watch?v=c9e7MTsIDc4 Listening]'' by [[Eliora]] (2023) - 217 & 1178 and enneadecal in 1178edo tuning
 
== Prajapati ==
The Hindu god Pradjapati is said to have created the universe by speaking aloud the odd numbers from 1 to 31.
 
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 81/80, 126/125, 1029/1024
 
[[Mapping]]: [{{val| 31 49 72 87 107 }}, {{val| 0 0 0 0 1 }}]
 
[[POTE generator]]: ~176/175 = 6.519
 
{{Optimal ET sequence|legend=1| 31, 93, 124b, 155b, 186b }}
 
[[Badness]]: 0.042959
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 126/125, 105/104, 512/507
 
Mapping: [{{val| 31 49 72 87 107 115 }}, {{val| 0 0 0 0 1 0 }}]


POTE generator: ~66/65 = 9.171
POTE generator: ~66/65 = 9.171


Map: [[&lt;31 49 72 87 107 115|, &lt;0 0 0 0 1 0|]
{{Optimal ET sequence|legend=1| 31, 93f, 124bf }}
EDOs: 31, 93f, 124bf
Badness: 0.0379


==Kumhar==
Badness: 0.037885
Commas: 81/80, 126/125, 1029/1024, 144/143
 
=== Kumhar ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 126/125, 144/143, 1029/1024
 
Mapping: [{{val| 31 49 72 87 107 115 }}, {{val| 0 0 0 0 1 -1 }}]


POTE generator: ~196/195 = 10.120
POTE generator: ~196/195 = 10.120


Map: [[&lt;31 49 72 87 107 115|, &lt;0 0 0 0 1 -1|]
{{Optimal ET sequence|legend=1| 31, 62e, 93, 124b, 341b }}
EDOs: 31, 62e, 93, 124b, 341b
 
Badness: 0.0486
Badness: 0.048582


=Gallium=
== Gallium ==
The name of gallium temperament comes from Gallium, the 31st element.
The name of gallium temperament comes from the 31st element. Gallium preserves the 11-limit mapping of 31et, while adding 13, 17, and 19 on an independent generator chain, and this considerably improves the qualities of 13-limit and beyond.  


Commas: 81/80, 126/125, 99/98, 121/120
[[Subgroup]]: 2.3.5.7.11.13


POTE generator: ~16807/16640 = 15.541
[[Comma list]]: 81/80, 99/98, 121/120, 126/125


Map: [&lt;31 49 72 87 107 0|, &lt;0 0 0 0 0 1|]
{{Mapping|legend=1| 31 49 72 87 107 115 | 0 0 0 0 0 -1 }}
EDOs: 31, 62, 93e, 155bef
Badness: 0.0255


=Birds=
[[Optimal tuning]] ([[CTE]]): ~45/44 = 1\31, ~13/8 = 840.5276 (~144/143 = 11.0853)
The birds temperament tempers out the //31-5 comma//, |72 0 -31&gt;, which is the amount (24.275 cents) by which 31 just (5/4) major thirds fall short of ten octaves. The name comes from Isaiah 31:5 "As birds flying, so wil the Lord of hostes defend Ierusalem, defending also hee will deliuer it, and passing ouer, he will preserue it." It also tempers out the 31-7 comma, but sadly, combining the two commas leads to torsion.


Commas: 3136/3125, 823543/819200
{{Optimal ET sequence|legend=1| 31, 62, 93e, 155bef }}


POTE generator: ~1029/1024 = 5.155
[[Badness]]: 0.025484


Map: [&lt;31 49 72 87|, &lt;0 1 0 0|]
=== 17-limit ===
Wedgie: &lt;&lt;31 0 0 -72 -87 0||
Subgroup: 2.3.5.7.11.13.17
Edos: 31, 124, 155, 186, 217, 248, 465
Badness: 0.1000


==11-limit==
Comma list: 81/80, 99/98, 121/120, 126/125, 273/272
Commas: 441/440, 3136/3125, 41503/41472


POTE generator: ~385/384 = 4.938
Mapping: {{mapping| 31 49 72 87 107 115 127 | 0 0 0 0 0 -1 -1 }}


Map: [&lt;31 49 72 87 107|, &lt;0 1 0 0 2|]
Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.4879 (~144/143 = 11.1250)
EDOs: 31, 186e, 217, 248, 961cd
Badness: 0.0399


==13-limit==
{{Optimal ET sequence|legend=1| 31, 62, 93e, 155befg }}
Commas: 441/440, 3136/3125, 1001/1000, 13720/13689


POTE generator: ~385/384 = 5.170
Badness: 0.023421


Map: [&lt;31 49 72 87 107 115|, &lt;0 1 0 0 2 -2|]
=== 19-limit ===
EDOs: 31, 186e, 217, 248, 465
Subgroup: 2.3.5.7.11.13.17.19
Badness: 0.0357


Comma list: 81/80, 99/98, 121/120, 126/125, 153/152, 273/272


Mapping: {{mapping| 31 49 72 87 107 115 127 132 | 0 0 0 0 0 -1 -1 -1 }}


Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.1820 (~144/143 = 11.4309)


{{Optimal ET sequence|legend=1| 31, 62, 155befg }}


Badness: 0.019963


</pre></div>
{{Navbox fractional-octave}}
<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;31 comma temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:16:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;a href="#x31-3-comma"&gt;31-3-comma&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Prajapati"&gt;Prajapati&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#Gallium"&gt;Gallium&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Birds"&gt;Birds&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;
&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x31-3-comma"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;31-3-comma&lt;/h1&gt;
By the &lt;em&gt;31-3-comma&lt;/em&gt; is meant 617673396283947/562949953421312 = &amp;lt;-49 31|, the amount (160.605 cents) by which 31 just (3/2) fifths exceed 18 octaves. This may not seem like much of a comma, but since 31et is such a strong 7-limit system, 11- and 13-limit temperaments based on the cycle of 31 fifths actually make sense.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Prajapati"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Prajapati&lt;/h1&gt;
The Hindu  god Pradjapati is said to have created the universe by speaking aloud the odd numbers from 1 to 31. People with an interest in 31 may want to try this method themselves.&lt;br /&gt;
&lt;br /&gt;
Commas: 81/80, 126/125, 1029/1024&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~176/175 = 6.519&lt;br /&gt;
&lt;br /&gt;
Map: [[&amp;lt;31 49 72 87 107|, &amp;lt;0 0 0 0 1|]&lt;br /&gt;
EDOs: 31, 93, 124b, 155b, 186b&lt;br /&gt;
Badness: 0.0430&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Prajapati-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
Commas: 81/80, 126/125, 105/104, 512/507&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~66/65 = 9.171&lt;br /&gt;
&lt;br /&gt;
Map: [[&amp;lt;31 49 72 87 107 115|, &amp;lt;0 0 0 0 1 0|]&lt;br /&gt;
EDOs: 31, 93f, 124bf&lt;br /&gt;
Badness: 0.0379&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Prajapati-Kumhar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Kumhar&lt;/h2&gt;
Commas: 81/80, 126/125, 1029/1024, 144/143&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~196/195 = 10.120&lt;br /&gt;
&lt;br /&gt;
Map: [[&amp;lt;31 49 72 87 107 115|, &amp;lt;0 0 0 0 1 -1|]&lt;br /&gt;
EDOs: 31, 62e, 93, 124b, 341b&lt;br /&gt;
Badness: 0.0486&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Gallium"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Gallium&lt;/h1&gt;
The name of gallium temperament comes from Gallium, the 31st element.&lt;br /&gt;
&lt;br /&gt;
Commas: 81/80, 126/125, 99/98, 121/120&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~16807/16640 = 15.541&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;31 49 72 87 107 0|, &amp;lt;0 0 0 0 0 1|]&lt;br /&gt;
EDOs: 31, 62, 93e, 155bef&lt;br /&gt;
Badness: 0.0255&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Birds"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Birds&lt;/h1&gt;
The birds temperament tempers out the &lt;em&gt;31-5 comma&lt;/em&gt;, |72 0 -31&amp;gt;, which is the amount (24.275 cents) by which 31 just (5/4) major thirds fall short of ten octaves. The name comes from Isaiah 31:5 &amp;quot;As birds flying, so wil the Lord of hostes defend Ierusalem, defending also hee will deliuer it, and passing ouer, he will preserue it.&amp;quot; It also tempers out the 31-7 comma, but sadly, combining the two commas leads to torsion.&lt;br /&gt;
&lt;br /&gt;
Commas: 3136/3125, 823543/819200&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~1029/1024 = 5.155&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;31 49 72 87|, &amp;lt;0 1 0 0|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;31 0 0 -72 -87 0||&lt;br /&gt;
Edos: 31, 124, 155, 186, 217, 248, 465&lt;br /&gt;
Badness: 0.1000&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Birds-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;11-limit&lt;/h2&gt;
Commas: 441/440, 3136/3125, 41503/41472&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~385/384 = 4.938&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;31 49 72 87 107|, &amp;lt;0 1 0 0 2|]&lt;br /&gt;
EDOs: 31, 186e, 217, 248, 961cd&lt;br /&gt;
Badness: 0.0399&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Birds-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit&lt;/h2&gt;
Commas: 441/440, 3136/3125, 1001/1000, 13720/13689&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~385/384 = 5.170&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;31 49 72 87 107 115|, &amp;lt;0 1 0 0 2 -2|]&lt;br /&gt;
EDOs: 31, 186e, 217, 248, 465&lt;br /&gt;
Badness: 0.0357&lt;/body&gt;&lt;/html&gt;</pre></div>
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