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Trienstonic clan: 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>
The '''trienstonic clan''' of [[rank-2 temperament|rank-2]] [[temperament]]s are low-complexity, high-error temperaments that [[tempering out|temper out]] [[28/27]], the septimal third-tone or trienstonic comma. This equates very different intervals with each other; in particular, [[9/8]] with [[7/6]], [[8/7]] with [[32/27]], and [[4/3]] with [[9/7]]. Trienstonian is close to the edge of what can be sensibly called a temperament at all; in other words, it is an [[exotemperament]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-06 15:19:12 UTC</tt>.<br>
: The original revision id was <tt>330741712</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]]


Adding 16/15 to the trienstonic comma of 28/27 leads to father temperament, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 126/125 gives opossum. Other members of the clan discussed elsewhere are sharptone, sharp, and blacksmith.
== Trienstonian ==
This low-accuracy temperament is generated by a fifth, tuned very sharp such that a stack of three reach a ~7/4. [[5edo]] is the tuning that conflates 7/6~9/8 (+2 generator steps) with ~8/7 (-3 generator steps). If you do not care about the intervals of 9 in this temperament, you can tune the fifth sharper for the [[7-odd-limit]], leading to an [[5L 3s|oneirotonic]] scale or otherwise a [[5L 2s|diatonic]] scale with negative small steps. A tuning that prioritizes the 7-odd-limit also tunes the fifth sharper than the [[pentic]] range, instead generating an [[antipentic]] scale. Trienstonian can be considered the [[2.3.7 subgroup|2.3.7]] analog of [[mavila]] temperament, with extremely sharp fifths rather than extremely flat ones, being on the other side of 3\5 from [[archy]] fifths, just like how mavila fifths are on the other side of 4\7 from [[meantone]] fifths.


=Father=
[[Subgroup]]: 2.3.7
Commas: 16/15, 28/27


7-limit minimax
[[Comma list]]: 28/27
[&lt;1 0 0 0|, &lt;3/2 0 -1/4 1/4|, &lt;5/2 0 1/4 -1/4|, &lt;5/2 0 -3/4 3/4|]
[[Eigenmonzo subgroup]]: 2.7/5


9-limit eigenmonzo subgroup: 2.9/5
{{Mapping|legend=2| 1 0 -2 | 0 1 3 }}


[[POTE tuning|POTE generator]]: 742.002
{{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }}
: mapping generators: ~2, ~3


Map: [&lt;1 0 4 -2|, &lt;0 1 -1 3|]
[[Optimal tuning]]s:  
Wedgie: &lt;&lt;1 -1 3 -4 2 10||
* [[WE]]: ~2 = 1196.254{{c}}, ~3/2 = 719.306{{c}}
EDOs: 5, 8d, 13cd, 21bcd
: [[error map]]: {{val| -3.746 +13.604 -14.655 }}
Badness: 0.0213
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 719.606{{c}}
: error map: {{val| 0.000 +17.651 -10.007 }}


=Uncle=
{{Optimal ET sequence|legend=1| 2d, 3d, 5 }}
Commas: 28/27, 256/245


7-limit eigenmonzo subgroup: 2.5/3
[[Badness]] (Sintel): 0.235
9-limit eigenmonzo subgroup: 2.9/5


Map: [&lt;1 0 12 -2|, &lt;0 1 -6 3|]
=== Overview to extensions ===
Wedgie: &lt;&lt;1 -6 3 -12 2 24||
Adding 16/15 to 28/27 leads to father, 21/20 gives sharptone, 256/245 gives uncle, and 35/32 gives wallaby. These all use the same generators as trienstonian.  
EDOs: 13d, 18, 23bc, 41bcd
Badness: 0.0727


=Octokaidecal=
50/49 gives octokaidecal with a semi-octave period. 25/24 gives sharpie; 27/25 gives mite. Those split the generator in two. 1029/1000 gives parakangaroo; 126/125 gives opossum. Those split the generator in three. 128/125 gives inflated with a 1/3-octave period. Finally, 49/48 gives blackwood, with a 1/5-octave period.
Commas: 28/27, 50/49


7-limit eigenmonzo subgroup: 2.5
Members of the clan discussed elsewhere are:  
9-limit eigenmonzo subgroup: 2.5


[[POTE tuning|POTE generator]]: 728.874
* [[Antonian]] (+10/9 or +15/14) → [[Very low accuracy temperaments #Septimal antonian|Very low accuracy temperaments]]
* [[Father]] (+16/15) → [[Father family #Septimal father|Father family]]
* ''[[Sharptone]]'' (+21/20) → [[Meantone family #Sharptone|Meantone family]]
* ''[[Sharpie]]'' (+25/24) → [[Dicot family #Sharpie|Dicot family]]
* ''[[Mite]]'' (+27/25) → [[Bug family #Mite|Bug family]]
* ''[[Wallaby]]'' (+35/32) → [[Very low accuracy temperaments #Wallaby|Very low accuracy temperaments]]
* [[Blackwood]] (+49/48) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* ''[[Opossum]]'' (+126/125) → [[Porcupine family #Opossum|Porcupine family]]
* ''[[Inflated]]'' (+128/125) → [[Augmented family #Inflated|Augmented family]]


Map: [&lt;2 0 -5 -4|, &lt;0 1 3 3|]
Considered below are uncle, octokaidecal, and parakangaroo.
Wedgie: &lt;&lt;2 6 6 5 4 -2||
EDOs: 10, 18, 28b
Badness: 0.0367


==11-limit==
== Uncle ==
Commas: 28/27, 50/49, 55/54
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Uncle (5-limit)]].''


POTE generator: ~3/2 = 732.330
Uncle tempers out 256/245, mapping the interval class of 5 to -6 generator steps, as a major 2-step in oneirotonic or a diminished fifth in diatonic.  


Map: [&lt;2 0 -5 -4 7|, &lt;0 1 3 3 0|]
[[Subgroup]]: 2.3.5.7
EDOs: 10, 18e
Badness: 0.0302


=Opossum=
[[Comma list]]: 28/27, 256/245
Commas: 28/27, 126/125


7-limit eigenmonzo subgroup: 2.7
{{Mapping|legend=1| 1 0 12 -2 | 0 1 -6 3 }}
9-limit eigenmonzo subgroup: 2.7


[[POTE tuning|POTE generator]]: ~10/9 = 159.691
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1190.224{{c}}, ~3/2 = 725.221{{c}}
: [[error map]]: {{val| -9.776 +13.490 +3.707 -2.939 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 731.394{{c}}
: error map: {{val| 0.000 +29.439 +25.324 +25.355 }}


Map: [&lt;1 2 3 4|, &lt;0 -3 -5 -9|]
[[Minimax tuning]]:  
Wedgie: &lt;&lt;3 5 9 1 6 7||
* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3
EDOs: 7d, 8d, 15
* [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5
Badness: 0.0407


==11-limit==
{{Optimal ET sequence|legend=1| 5, 13d, 18, 23bc, 41bbcd }}
Commas: 28/27, 55/54, 77/75


11-limit eigenmonzo subgroup: 2.7
[[Badness]] (Sintel): 1.84


[[POTE tuning|POTE generator]]: ~10/9 = 159.807
== Octokaidecal ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Supersharp]].''
Octokaidecal extends trienstonian by tempering out [[50/49]], thus splitting the octave in half. It generates the [[8L 2s]] (taric) mos scale, with tunings on the other side of [[10edo]] as [[2L 8s]] (jaric). Compared to [[pajara]], decatonic thirds ([[8/7]] and [[7/6]]) and fourths ([[6/5]] and [[5/4]]) have their mappings reversed, meaning octokaidecal can be considered what is to pajara as [[mavila]] is to [[meantone]].


Map: [&lt;1 2 3 4 4|, &lt;0 -3 -5 -9 -4|]
[[Subgroup]]: 2.3.5.7
EDOs: 7d, 8d, 15
Badness: 0.0223


==13-limit==
[[Comma list]]: 28/27, 50/49
Commas: 28/27, 40/39, 55/54, 66/65


13-limit eigenmonzo subgroup: 2.7
{{Mapping|legend=1| 2 0 -5 -4 | 0 1 3 3 }}
15-limit eigenmonzo subgroup: 2.7


[[POTE tuning|POTE generator]]: ~10/9 = 158.805
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 596.984{{c}}, ~3/2 = 725.210{{c}} (~15/14 = 128.226{{c}})
: [[error map]]: {{val| -6.031 +17.224 -13.699 +0.774 }}
* [[CWE]]: ~7/5 = 600.000{{c}}, ~3/2 = 726.319{{c}} (~15/14 = 126.319{{c}})
: error map: {{val| 0.000 +24.364 -7.358 +10.130 }}


Map: [&lt;1 2 3 4 4 4|, &lt;0 -3 -5 -9 -4 -2|]
[[Minimax tuning]]:  
EDOs: 7d, 8d, 15, 38bcef
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
Badness: 0.0194


=Wallaby=
{{Optimal ET sequence|legend=1| 8d, 10, 18, 28b }}
Commas: 28/27, 35/32


POTE generator: ~3/2 = 691.351
[[Badness]] (Sintel): 0.930


Map: [&lt;1 0 7 -2|, &lt;0 1 -3 3|]
=== 11-limit ===
Wedgie: &lt;&lt;1 -3 3 -7 2 15||
Subgroup: 2.3.5.7.11
EDOs: 2d, 5c, 19ccdd
Badness: 0.0585


=Kangaroo=
Comma list: 28/27, 50/49, 55/54
Comma: 64000/59049


POTE generator: ~27/20 = 561.195
Mapping: {{mapping| 2 0 -5 -4 7 | 0 1 3 3 0 }}


Map: [&lt;1 0 -3|, &lt;0 3 10|]
Optimal tunings:
EDOs: 15, 47b, 62b
* WE: ~7/5 = 595.139{{c}}, ~3/2 = 726.397{{c}} (~15/14 = 131.258{{c}})
Badness: 0.2958
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 729.485{{c}} (~15/14 = 129.485{{c}})


==7-limit==
{{Optimal ET sequence|legend=0| 8d, 10, 18e }}
Commas: 28/27, 1029/1000


POTE generator: ~7/5 = 560.328
Badness (Sintel): 1.00


Map: [&lt;1 0 -3 -2|, &lt;0 3 10 9|]
== Parakangaroo ==
Wedgie: &lt;&lt;3 10 9 9 6 -7||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''
EDOs: 15
Badness: 0.0779


==11-limit==
This temperament used to be known as ''kangaroo'', but was decanonicalized in 2024 in favor of a more accurate extension. It splits the perfect twelfth into three generators of ~10/7; its ploidacot is alpha-tricot. [[15edo]] shows us an obvious tuning.
Commas: 28/27, 77/75, 245/242


POTE generator: ~7/5 = 560.155
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 0 -3 -2 -4|, &lt;0 3 10 9 14|]
[[Comma list]]: 28/27, 1029/1000
EDOs: 15
Badness: 0.0432


==13-limit==
{{Mapping|legend=1| 1 0 -3 -2 | 0 3 10 9 }}
Commas: 28/27, 40/39, 77/75, 147/143


POTE generator: ~7/5 = 559.770
: mapping generators: ~2, ~10/7


Map: [&lt;1 0 -3 -2 -4 0|, &lt;0 3 10 9 14 7|]
[[Optimal tuning]]s:
EDOs: 15
* [[WE]]: ~2 = 596.984{{c}}, ~10/7 = 638.135{{c}}
Badness: 0.0327
: [[error map]]: {{val| -2.883 +12.450 +3.685 -19.845 }}
</pre></div>
* [[CWE]]: ~2 = 1200.000{{c}}, ~10/7 = 639.302{{c}}
<h4>Original HTML content:</h4>
: error map: {{val| 0.000 +15.952 +6.710 -15.104 }}
<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;Trienstonic clan&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:24:&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:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;&lt;a href="#Father"&gt;Father&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt; | &lt;a href="#Uncle"&gt;Uncle&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Octokaidecal"&gt;Octokaidecal&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt; | &lt;a href="#Opossum"&gt;Opossum&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt; | &lt;a href="#Wallaby"&gt;Wallaby&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt; | &lt;a href="#Kangaroo"&gt;Kangaroo&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;br /&gt;
{{Optimal ET sequence|legend=1| 2cd, …, 13cd, 15 }}
Adding 16/15 to the trienstonic comma of 28/27 leads to father temperament, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 126/125 gives opossum. Other members of the clan discussed elsewhere are sharptone, sharp, and blacksmith.&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 1.97
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Father"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Father&lt;/h1&gt;
 
Commas: 16/15, 28/27&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
7-limit minimax&lt;br /&gt;
 
[&amp;lt;1 0 0 0|, &amp;lt;3/2 0 -1/4 1/4|, &amp;lt;5/2 0 1/4 -1/4|, &amp;lt;5/2 0 -3/4 3/4|]&lt;br /&gt;
Comma list: 28/27, 77/75, 245/242
&lt;a class="wiki_link" href="/Eigenmonzo%20subgroup"&gt;Eigenmonzo subgroup&lt;/a&gt;: 2.7/5&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 -3 -2 -4 | 0 3 10 9 14 }}
9-limit eigenmonzo subgroup: 2.9/5&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 742.002&lt;br /&gt;
* WE: ~2 = 1196.971{{c}}, ~10/7 = 638.230{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.480{{c}}
Map: [&amp;lt;1 0 4 -2|, &amp;lt;0 1 -1 3|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;1 -1 3 -4 2 10||&lt;br /&gt;
{{Optimal ET sequence|legend=0| 15 }}
EDOs: 5, 8d, 13cd, 21bcd&lt;br /&gt;
 
Badness: 0.0213&lt;br /&gt;
Badness (Sintel): 1.43
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Uncle"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Uncle&lt;/h1&gt;
=== 13-limit ===
Commas: 28/27, 256/245&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
7-limit eigenmonzo subgroup: 2.5/3&lt;br /&gt;
Comma list: 28/27, 40/39, 66/65, 147/143
9-limit eigenmonzo subgroup: 2.9/5&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 -3 -2 -4 0 | 0 3 10 9 14 7 }}
Map: [&amp;lt;1 0 12 -2|, &amp;lt;0 1 -6 3|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;1 -6 3 -12 2 24||&lt;br /&gt;
Optimal tunings:  
EDOs: 13d, 18, 23bc, 41bcd&lt;br /&gt;
* WE: ~2 = 1194.720{{c}}, ~10/7 = 637.413{{c}}
Badness: 0.0727&lt;br /&gt;
* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.609{{c}}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Octokaidecal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Octokaidecal&lt;/h1&gt;
{{Optimal ET sequence|legend=0| 15 }}
Commas: 28/27, 50/49&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.35
7-limit eigenmonzo subgroup: 2.5&lt;br /&gt;
 
9-limit eigenmonzo subgroup: 2.5&lt;br /&gt;
[[Category:Temperament clans]]
&lt;br /&gt;
[[Category:Trienstonic clan| ]] <!-- main article -->
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 728.874&lt;br /&gt;
[[Category:Rank 2]]
&lt;br /&gt;
Map: [&amp;lt;2 0 -5 -4|, &amp;lt;0 1 3 3|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;2 6 6 5 4 -2||&lt;br /&gt;
EDOs: 10, 18, 28b&lt;br /&gt;
Badness: 0.0367&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="Octokaidecal-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;11-limit&lt;/h2&gt;
Commas: 28/27, 50/49, 55/54&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~3/2 = 732.330&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 0 -5 -4 7|, &amp;lt;0 1 3 3 0|]&lt;br /&gt;
EDOs: 10, 18e&lt;br /&gt;
Badness: 0.0302&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="Opossum"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Opossum&lt;/h1&gt;
Commas: 28/27, 126/125&lt;br /&gt;
&lt;br /&gt;
7-limit eigenmonzo subgroup: 2.7&lt;br /&gt;
9-limit eigenmonzo subgroup: 2.7&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~10/9 = 159.691&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 3 4|, &amp;lt;0 -3 -5 -9|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;3 5 9 1 6 7||&lt;br /&gt;
EDOs: 7d, 8d, 15&lt;br /&gt;
Badness: 0.0407&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Opossum-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;11-limit&lt;/h2&gt;
Commas: 28/27, 55/54, 77/75&lt;br /&gt;
&lt;br /&gt;
11-limit eigenmonzo subgroup: 2.7&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~10/9 = 159.807&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 3 4 4|, &amp;lt;0 -3 -5 -9 -4|]&lt;br /&gt;
EDOs: 7d, 8d, 15&lt;br /&gt;
Badness: 0.0223&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="Opossum-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit&lt;/h2&gt;
Commas: 28/27, 40/39, 55/54, 66/65&lt;br /&gt;
&lt;br /&gt;
13-limit eigenmonzo subgroup: 2.7&lt;br /&gt;
15-limit eigenmonzo subgroup: 2.7&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~10/9 = 158.805&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 3 4 4 4|, &amp;lt;0 -3 -5 -9 -4 -2|]&lt;br /&gt;
EDOs: 7d, 8d, 15, 38bcef&lt;br /&gt;
Badness: 0.0194&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Wallaby"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Wallaby&lt;/h1&gt;
Commas: 28/27, 35/32&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~3/2 = 691.351&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 7 -2|, &amp;lt;0 1 -3 3|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;1 -3 3 -7 2 15||&lt;br /&gt;
EDOs: 2d, 5c, 19ccdd&lt;br /&gt;
Badness: 0.0585&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Kangaroo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Kangaroo&lt;/h1&gt;
Comma: 64000/59049&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~27/20 = 561.195&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 -3|, &amp;lt;0 3 10|]&lt;br /&gt;
EDOs: 15, 47b, 62b&lt;br /&gt;
Badness: 0.2958&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Kangaroo-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;7-limit&lt;/h2&gt;
Commas: 28/27, 1029/1000&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~7/5 = 560.328&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 -3 -2|, &amp;lt;0 3 10 9|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;3 10 9 9 6 -7||&lt;br /&gt;
EDOs: 15&lt;br /&gt;
Badness: 0.0779&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Kangaroo-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;11-limit&lt;/h2&gt;
Commas: 28/27, 77/75, 245/242&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~7/5 = 560.155&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 -3 -2 -4|, &amp;lt;0 3 10 9 14|]&lt;br /&gt;
EDOs: 15&lt;br /&gt;
Badness: 0.0432&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Kangaroo-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;13-limit&lt;/h2&gt;
Commas: 28/27, 40/39, 77/75, 147/143&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~7/5 = 559.770&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 -3 -2 -4 0|, &amp;lt;0 3 10 9 14 7|]&lt;br /&gt;
EDOs: 15&lt;br /&gt;
Badness: 0.0327&lt;/body&gt;&lt;/html&gt;</pre></div>
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