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Augmented family: Difference between revisions

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**Imported revision 321164642 - Original comment: **
Intro to some of these temps
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| en = Augmented family
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-04-16 14:41:48 UTC</tt>.<br>
| de = Übermässige Temperaturen
: The original revision id was <tt>321164642</tt>.<br>
}}
: The revision comment was: <tt></tt><br>
{{Technical data page}}
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
The '''augmented family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diesis a.k.a. augmented comma, [[128/125]], the amount by which three [[5/4]] major thirds fall short of an [[2/1|octave]], and so identifies the major third with the 1/3-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]].
<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]]


The 5-limit parent comma for the **augmented family** is 128/125, the [[128_125|diesis]]. Its [[monzo]] is |7 0 -3&gt;, and flipping that yields &lt;&lt;3 0 -7|| for the [[wedgie]]. Hence the period is 1/3 octave, and this is what is used for a major third. The [[generator]] can be taken as a fifth or a semitone, and [[12edo]], with its excellent fifth, is an obvious tuning for [[5-limit]] augmented, though a sharper fifth might be preferred to go with the sharp third.  
== Augmented ==
The [[period]] is 1/3 octave, and this is what is used for 5/4, the classical major third. The [[generator]] can be taken as a fifth or a semitone, and [[12edo]], with its excellent fifth, is an obvious tuning for [[5-limit]] augmented, though a sharper fifth might be preferred to go with the sharp third. Its [[ploidacot]] is triploid monocot.  


=Seven limit children=
[[Subgroup]]: 2.3.5
The second comma of the [[Normal lists|normal comma list]] defines which [[7-limit]] family member we are looking at. August adds 36/35, augene 64/63, hexe 256/245, hemiaug 245/243, and hemis 49/48. Hexe splits the period to 1/6 octave, and hemiaug the generator, giving quarter-tones instead of semitones.


=August=
[[Comma list]]: 128/125
[[Comma]]s: 36/35, 128/125


[[POTE tuning|POTE generator]]: 696.011
{{Mapping|legend=1| 3 0 7 | 0 1 0 }}


[[Map]]: [&lt;3 0 7 -1|, &lt;0 1 0 2|]
: mapping generators: ~5/4, ~3
[[Wedgie]]: &lt;&lt;3 0 6 -7 1 14||
[[edo|EDOs]]: [[9edo|9]], [[12edo|12]], [[45edo|45cd]], [[57edo|57cd]], [[69edo|69cd]]
[[Badness]]: 0.0265


==11-limit==
[[Optimal tuning]]s:
Commas: 36/35, 45/44, 56/55
* [[WE]]: ~5/4 = 399.0128{{c}}, ~3/2 = 704.8937{{c}} (~16/15 = 93.1320{{c}})
: [[error map]]: {{val| -2.962 -0.023 +6.776 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 705.0691{{c}} (~16/15 = 94.9309{{c}})
: error map: {{val| 0.000 +3.114 +13.686 }}


POTE generator: ~3/2 = 692.514
{{Optimal ET sequence|legend=1| 3, 9, 12, 27, 39, 51c, 90cc }}


Map: [&lt;3 0 7 -1 1|, &lt;0 1 0 2 2|]
[[Badness]] (Sintel): 0.523
EDOs: 9, 12, [[21edo|21]], [[33edo|33e]], [[45edo|45cde]]
Badness: 0.0202


==13-limit==
=== Overview to extensions ===
Commas: 27/26, 36/35, 45/44, 56/55
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which [[7-limit]] family member we are looking at. Augene adds [[64/63]], august [[36/35]], hexe [[256/245]], hemiaug [[245/243]], and triforce [[49/48]]. Hexe splits the [[period]] to 1/6 octave, and hemiaug the [[generator]], giving quartertones instead of semitones.


POTE generator: ~3/2 = 688.783
== Augene ==
{{Main| Augene }}


Map: [&lt;3 0 7 -1 1 -3|, &lt;0 1 0 2 2 3|]
Augene tempers out 64/63 and 126/125. It may be described as the {{nowrap| 12 & 15 }} temperament. [[27edo]] and [[39edo]] in the 39d val make for good tunings.  
EDOs: 9, 12, [[21edo|21]], [[33edo|33ef]], [[54edo|54bcef]]
Badness: 0.0184


==Augustus==
[[Subgroup]]: 2.3.5.7
Commas: 26/25, 36/35, 45/44, 56/55


POTE generator: ~3/2 = 685.356
[[Comma list]]: 64/63, 126/125


Map: [&lt;3 0 7 -1 1 11|, &lt;0 1 0 2 2 0|]
{{Mapping|legend=1| 3 0 7 18 | 0 1 0 -2 }}
EDOs: 9, 12, 21f
Badness: 0.0222


=Augene=
[[Optimal tuning]]s:
Commas: 64/63, 126/125
* [[WE]]: ~5/4 = 398.7461{{c}}, ~3/2 = 707.0335{{c}} (~21/20 = 90.4587{{c}})
: [[error map]]: {{val| -3.762 +1.317 +4.909 +2.060 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 709.3249{{c}} (~21/20 = 90.6751{{c}})
: error map: {{val| 0.000 +7.370 +13.686 +12.524 }}


[[POTE tuning|POTE generator]]: 709.257
{{Optimal ET sequence|legend=1| 12, 27, 39d, 66cd }}


Map: [&lt;3 0 7 18|, &lt;0 1 0 -2|]
[[Badness]] (Sintel): 0.628
Wedgie: &lt;&lt;3 0 -6 -7 -18 -14||
EDOs: [[12edo|12]], [[27edo|27]], [[39edo|39d]], [[66edo|66cd]]
Badness: 0.0248


==11-limit==
=== 11-limit ===
Commas: 56/55, 64/63, 100/99
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 711.177
Comma list: 56/55, 64/63, 100/99


Map: [&lt;3 0 7 18 20|, &lt;0 1 0 -2 -2|]
Mapping: {{mapping| 3 0 7 18 20 | 0 1 0 -2 -2 }}
EDOs: 12, 15, 27e
Badness: 0.0196


==Eugene==
Optimal tunings:
Commas: 55/54, 64/63, 77/75
* WE: ~5/4 = 398.4962{{c}}, ~3/2 = 708.5030{{c}} (~21/20 = 88.4895{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 711.6031{{c}} (~21/20 = 88.3969{{c}})


POTE generator: ~3/2 = 714.150
{{Optimal ET sequence|legend=0| 12, 15, 27e }}


Map: [&lt;3 0 7 18 -4|, &lt;0 1 0 -2 3|]
Badness (Sintel): 0.648
EDOs: 15, 27, 42
Badness: 0.0356


[[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3|Sad Like Winter Leaves]] by [[http://soundcloud.com/cityoftheasleep/sad-like-winter-trees|Igliashon Jones]]
==== 13-limit ====
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3|Galticeran Sonatina]] by [[http://soundcloud.com/joelgranttaylor/galticeran_sonatina|Joel Taylor]]
Subgroup: 2.3.5.7.11.13


=Inflated=
Comma list: 40/39, 56/55, 64/63, 66/65
Commas: 28/27, 128/125


POTE generator: ~3/2 = 722.719
Mapping: {{mapping| 3 0 7 18 20 16 | 0 1 0 -2 -2 -1 }}


Map: [&lt;3 0 7 -6|, &lt;0 1 0 3|]
Optimal tunings:  
Wedgie: &lt;&lt;3 0 9 -7 6 21||
* WE: ~5/4 = 398.0488{{c}}, ~3/2 = 708.5402{{c}} (~21/20 = 87.5574{{c}})
EDOs: 15, 48bc, 63bc, 78bc
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 712.6704{{c}} (~21/20 = 87.3296{{c}})
Badness: 0.0547


==11-limit==
{{Optimal ET sequence|legend=0| 12f, 15, 27eff }}
Commas: 28/27, 55/54, 128/125


POTE generator: ~3/2 = 722.663
Badness (Sintel): 0.859


Map: [&lt;3 0 7 -6|, &lt;0 1 0 3|]
==== Ogene ====
EDOs: 15, 48bce, 63bce, 78bce
Subgroup: 2.3.5.7.11.13
Badness: 0.0312


=Deflated=
Comma list: 56/55, 64/63, 91/90, 100/99
Commas: 21/20, 128/125


POTE generator: ~3/2 = 681.629
Mapping: {{mapping| 3 0 7 18 20 -8 | 0 1 0 -2 -2 4 }}


Map: [&lt;3 0 7 13|, &lt;0 1 0 -1|]
Optimal tunings:  
Wedgie: &lt;&lt;3 0 -3 -7 -13 -7||
* WE: ~5/4 = 398.6473{{c}}, ~3/2 = 710.1987{{c}} (~21/20 = 87.0959{{c}})
EDOs: 3, 9, 18bd, 21d, 30bd
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 712.5057{{c}} (~21/20 = 87.4943{{c}})
Badness: 0.0591


=Hexe=
{{Optimal ET sequence|legend=0| 12, 15, 27e, 69bceef }}
Commas: 50/49, 128/125


[[POTE tuning|POTE generator]]: 710.963
Badness (Sintel): 0.946


Map: [&lt;6 0 14 17|, &lt;0 1 0 0|]
==== Agene ====
Wedgie: &lt;&lt;6 0 0 -14 -17 0||
Subgroup: 2.3.5.7.11.13
EDOs: [[6edo|6]], [[12edo|12]], [[30edo|30d]], [[37edo|37d]], [[42edo|42d]], [[54edo|54cd]]
Badness: 0.0577


=Hemiaug=
Comma list: 56/55, 64/63, 78/77, 100/99
Commas: 128/125, 245/243


[[POTE tuning|POTE generator]]: ~28/27 = 45.118
Mapping: {{mapping| 3 0 7 18 20 35 | 0 1 0 -2 -2 -5 }}


Map: [&lt;3 1 7 -1|, &lt;0 2 0 5|]
Optimal tunings:  
Wedgie: &lt;&lt;6 0 15 -14 7 35||
* WE: ~5/4 = 398.5229{{c}}, ~3/2 = 707.0562{{c}} (~21/20 = 89.9897{{c}})
EDOs: [[24edo|24]], [[27edo|27]], [[105edo|105bc]]
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 710.1903{{c}} (~21/20 = 89.8097{{c}})
Badness: 0.0705


==11-limit==
{{Optimal ET sequence|legend=0| 12f, 27e, 66cdeeef }}
Commas: 56/55, 128/125, 245/243


POTE generator: ~28/27 = 45.788
Badness (Sintel): 0.955


EDOs: 24, 27e, 51ce, 78ce
=== Eugene ===
Badness: 0.0382
Subgroup: 2.3.5.7.11


=Triforce=
Comma list: 55/54, 64/63, 77/75
Commas: 49/48, 128/125


[[POTE tuning|POTE generator]]: ~7/6 = 247.049
Mapping: {{mapping| 3 0 7 18 -4 | 0 1 0 -2 3 }}


Map: [&lt;3 0 7 6|, &lt;0 2 0 1|]
Optimal tunings:  
Wedgie: &lt;&lt;6 0 3 -14 -12 7||
* WE: ~5/4 = 399.1743{{c}}, ~3/2 = 712.6763{{c}} (~21/20 = 85.6723{{c}})
EDOs: 6, 9, [[15edo|15]], [[24edo|24]], [[39edo|39]], [[63edo|63cd]], [[102edo|102cd]]
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 713.9414{{c}} (~21/20 = 86.0586{{c}})
Badness: 0.0202


[[image:triforce9.jpg]]
{{Optimal ET sequence|legend=0| 12e, 15, 27, 42 }}


==11-limit==
Badness (Sintel): 1.18
Commas: 56/55, 77/75, 128/125


[[POTE tuning|POTE generator]]: ~7/6 = 247.068
== August ==
August tempers out 36/35 and 225/224. It may be described as the {{nowrap| 9 & 12 }} temperament. Unlike augene, august calls for a flat tuning of the fifth, and besides [[12edo]], [[21edo]] is among the possible tunings.  


Map: [&lt;3 0 7 6 8|, &lt;0 2 0 1 1|]
[[Subgroup]]: 2.3.5.7
EDOs: 6, 9, 15, 24, 39, [[63edo|63cd]], [[102edo|102cd]]
Badness: 0.0262


==13-limit==
[[Comma list]]: 36/35, 128/125
Commas: 49/48, 56/55, 66/65, 77/75


[[POTE tuning|POTE generator]]: ~7/6 = 248.313
{{Mapping|legend=1| 3 0 7 -1 | 0 1 0 2 }}


Map: [&lt;3 0 7 6 8 4|, &lt;0 2 0 1 1 3|]
[[Optimal tuning]]s:  
EDOs: 6, 9, 15, 24, [[63edo|63cdf]], [[87edo|87cdf]]
* [[WE]]: ~5/4 = 399.1036{{c}}, ~3/2 = 694.4509{{c}} (~16/15 = 103.7564{{c}})
Badness: 0.0202
: [[error map]]: {{val| -2.689 -10.193 +7.412 +15.594 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 694.6812{{c}} (~16/15 = 105.3188{{c}})
: error map: {{val| 0.000 -7.274 +13.686 +20.537 }}


[[triphi|Triforce(9) with L:s = phi]]
{{Optimal ET sequence|legend=1| 9, 12, 45cd }}


=Niner=
[[Badness]] (Sintel): 0.670
Niner gives 9 as the complexity of an otonal tetrad, tying it with augene as a temperament supported by 27edo. Niner[18], therefore, has nine such tetrads.


Commas: 128/125, 686/675
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 707.167
Comma list: 36/35, 45/44, 56/55


Map: [&lt;9 0 21 11|, &lt;0 1 0 1|]
Mapping: {{mapping| 3 0 7 -1 1 | 0 1 0 2 2 }}
Wedgie: &lt;&lt;9 0 9 -21 -11 21||
 
EDOs: 9, 18, 27, 63c, 90c
Optimal tunings:  
Badness: 0.0672</pre></div>
* WE: ~5/4 = 398.9225{{c}}, ~3/2 = 690.6486{{c}} (~16/15 = 107.1966{{c}})
<h4>Original HTML content:</h4>
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 690.8519{{c}} (~16/15 = 109.1481{{c}})
<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;Augmented family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:36:&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:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;&lt;a href="#Seven limit children"&gt;Seven limit children&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt; | &lt;a href="#August"&gt;August&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt; | &lt;a href="#Augene"&gt;Augene&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt;&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt; | &lt;a href="#Inflated"&gt;Inflated&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt;&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt; | &lt;a href="#Deflated"&gt;Deflated&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt; | &lt;a href="#Hexe"&gt;Hexe&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt; | &lt;a href="#Hemiaug"&gt;Hemiaug&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt;&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt; | &lt;a href="#Triforce"&gt;Triforce&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt;&lt;!-- ws:end:WikiTextTocRule:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt;&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt; | &lt;a href="#Niner"&gt;Niner&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;br /&gt;
{{Optimal ET sequence|legend=0| 9, 12, 21, 33e }}
The 5-limit parent comma for the &lt;strong&gt;augmented family&lt;/strong&gt; is 128/125, the &lt;a class="wiki_link" href="/128_125"&gt;diesis&lt;/a&gt;. Its &lt;a class="wiki_link" href="/monzo"&gt;monzo&lt;/a&gt; is |7 0 -3&amp;gt;, and flipping that yields &amp;lt;&amp;lt;3 0 -7|| for the &lt;a class="wiki_link" href="/wedgie"&gt;wedgie&lt;/a&gt;. Hence the period is 1/3 octave, and this is what is used for a major third. The &lt;a class="wiki_link" href="/generator"&gt;generator&lt;/a&gt; can be taken as a fifth or a semitone, and &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, with its excellent fifth, is an obvious tuning for &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; augmented, though a sharper fifth might be preferred to go with the sharp third. &lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.668
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Seven limit children&lt;/h1&gt;
 
The second comma of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; family member we are looking at. August adds 36/35, augene 64/63, hexe 256/245, hemiaug 245/243, and hemis 49/48. Hexe splits the period to 1/6 octave, and hemiaug the generator, giving quarter-tones instead of semitones.&lt;br /&gt;
==== 13-limit ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="August"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;August&lt;/h1&gt;
 
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;s: 36/35, 128/125&lt;br /&gt;
Comma list: 27/26, 36/35, 45/44, 56/55
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 696.011&lt;br /&gt;
Mapping: {{mapping| 3 0 7 -1 1 -3 | 0 1 0 2 2 3 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Map"&gt;Map&lt;/a&gt;: [&amp;lt;3 0 7 -1|, &amp;lt;0 1 0 2|]&lt;br /&gt;
Optimal tunings:  
&lt;a class="wiki_link" href="/Wedgie"&gt;Wedgie&lt;/a&gt;: &amp;lt;&amp;lt;3 0 6 -7 1 14||&lt;br /&gt;
* WE: ~5/4 = 399.0956{{c}}, ~3/2 = 687.2261{{c}} (~16/15 = 110.9651{{c}})
&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/9edo"&gt;9&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/45edo"&gt;45cd&lt;/a&gt;, &lt;a class="wiki_link" href="/57edo"&gt;57cd&lt;/a&gt;, &lt;a class="wiki_link" href="/69edo"&gt;69cd&lt;/a&gt;&lt;br /&gt;
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 687.5057{{c}} (~16/15 = 112.4943{{c}})
&lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0265&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 9, 12f, 21, 33ef }}
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="August-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
 
Commas: 36/35, 45/44, 56/55&lt;br /&gt;
Badness (Sintel): 0.762
&lt;br /&gt;
 
POTE generator: ~3/2 = 692.514&lt;br /&gt;
==== Augustus ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Map: [&amp;lt;3 0 7 -1 1|, &amp;lt;0 1 0 2 2|]&lt;br /&gt;
 
EDOs: 9, 12, &lt;a class="wiki_link" href="/21edo"&gt;21&lt;/a&gt;, &lt;a class="wiki_link" href="/33edo"&gt;33e&lt;/a&gt;, &lt;a class="wiki_link" href="/45edo"&gt;45cde&lt;/a&gt;&lt;br /&gt;
Comma list: 26/25, 36/35, 45/44, 56/55
Badness: 0.0202&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 3 0 7 -1 1 11 | 0 1 0 2 2 0 }}
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="August-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
 
Commas: 27/26, 36/35, 45/44, 56/55&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~5/4 = 400.4230{{c}}, ~3/2 = 686.0809{{c}} (~16/15 = 114.7650{{c}})
POTE generator: ~3/2 = 688.783&lt;br /&gt;
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 685.8446{{c}} (~16/15 = 114.1554{{c}})
&lt;br /&gt;
 
Map: [&amp;lt;3 0 7 -1 1 -3|, &amp;lt;0 1 0 2 2 3|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 9, 12 }}
EDOs: 9, 12, &lt;a class="wiki_link" href="/21edo"&gt;21&lt;/a&gt;, &lt;a class="wiki_link" href="/33edo"&gt;33ef&lt;/a&gt;, &lt;a class="wiki_link" href="/54edo"&gt;54bcef&lt;/a&gt;&lt;br /&gt;
 
Badness: 0.0184&lt;br /&gt;
Badness (Sintel): 0.919
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="August-Augustus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Augustus&lt;/h2&gt;
== Inflated ==
Commas: 26/25, 36/35, 45/44, 56/55&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
POTE generator: ~3/2 = 685.356&lt;br /&gt;
[[Comma list]]: 28/27, 128/125
&lt;br /&gt;
 
Map: [&amp;lt;3 0 7 -1 1 11|, &amp;lt;0 1 0 2 2 0|]&lt;br /&gt;
{{Mapping|legend=1| 3 0 7 -6 | 0 1 0 3 }}
EDOs: 9, 12, 21f&lt;br /&gt;
 
Badness: 0.0222&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;br /&gt;
* [[WE]]: ~5/4 = 398.4023{{c}}, ~3/2 = 719.8327{{c}} (~25/24 = 76.9719{{c}})
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Augene"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Augene&lt;/h1&gt;
: [[error map]]: {{val| -3.762 +1.317 +4.909 +2.060 }}
Commas: 64/63, 126/125&lt;br /&gt;
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 721.0196{{c}} (~25/24 = 78.9804{{c}})
&lt;br /&gt;
: error map: {{val| 0.000 +19.065 +13.686 -5.767 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 709.257&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 3d, 12d, 15 }}
Map: [&amp;lt;3 0 7 18|, &amp;lt;0 1 0 -2|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;3 0 -6 -7 -18 -14||&lt;br /&gt;
[[Badness]] (Sintel): 1.39
EDOs: &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/27edo"&gt;27&lt;/a&gt;, &lt;a class="wiki_link" href="/39edo"&gt;39d&lt;/a&gt;, &lt;a class="wiki_link" href="/66edo"&gt;66cd&lt;/a&gt;&lt;br /&gt;
 
Badness: 0.0248&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Augene-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;11-limit&lt;/h2&gt;
 
Commas: 56/55, 64/63, 100/99&lt;br /&gt;
Comma list: 28/27, 55/54, 128/125
&lt;br /&gt;
 
POTE generator: ~3/2 = 711.177&lt;br /&gt;
Mapping: {{mapping| 3 0 7 -6 -4 | 0 1 0 3 3 }}
&lt;br /&gt;
 
Map: [&amp;lt;3 0 7 18 20|, &amp;lt;0 1 0 -2 -2|]&lt;br /&gt;
Optimal tunings:  
EDOs: 12, 15, 27e&lt;br /&gt;
* WE: ~5/4 = 398.4016{{c}}, ~3/2 = 719.7758{{c}} (~25/24 = 77.0275{{c}})
Badness: 0.0196&lt;br /&gt;
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 720.9386{{c}} (~25/24 = 79.0614{{c}})
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Augene-Eugene"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Eugene&lt;/h2&gt;
{{Optimal ET sequence|legend=0| 3de, 12de, 15 }}
Commas: 55/54, 64/63, 77/75 &lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.03
POTE generator: ~3/2 = 714.150&lt;br /&gt;
 
&lt;br /&gt;
== Deflated ==
Map: [&amp;lt;3 0 7 18 -4|, &amp;lt;0 1 0 -2 3|]&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
EDOs: 15, 27, 42&lt;br /&gt;
 
Badness: 0.0356&lt;br /&gt;
[[Comma list]]: 21/20, 128/125
&lt;br /&gt;
 
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3" rel="nofollow"&gt;Sad Like Winter Leaves&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://soundcloud.com/cityoftheasleep/sad-like-winter-trees" rel="nofollow"&gt;Igliashon Jones&lt;/a&gt;&lt;br /&gt;
{{Mapping|legend=1| 3 0 7 13 | 0 1 0 -1 }}
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3" rel="nofollow"&gt;Galticeran Sonatina&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://soundcloud.com/joelgranttaylor/galticeran_sonatina" rel="nofollow"&gt;Joel Taylor&lt;/a&gt;&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Inflated"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Inflated&lt;/h1&gt;
* [[WE]]: ~5/4 = 401.9566{{c}}, ~3/2 = 684.9634{{c}} (~16/15 = 118.9497{{c}})
Commas: 28/27, 128/125&lt;br /&gt;
: [[error map]]: {{val| +5.870 -11.122 +27.382 -34.224 }}
&lt;br /&gt;
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 682.2587{{c}} (~16/15 = 117.7413{{c}})
POTE generator: ~3/2 = 722.719&lt;br /&gt;
: error map: {{val| 0.000 -19.696 +13.686 -51.085 }}
&lt;br /&gt;
 
Map: [&amp;lt;3 0 7 -6|, &amp;lt;0 1 0 3|]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 3, 6b, 9 }}
Wedgie: &amp;lt;&amp;lt;3 0 9 -7 6 21||&lt;br /&gt;
 
EDOs: 15, 48bc, 63bc, 78bc&lt;br /&gt;
[[Badness]] (Sintel): 1.50
Badness: 0.0547&lt;br /&gt;
 
&lt;br /&gt;
=== 11-limit ===
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Inflated-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;11-limit&lt;/h2&gt;
Subgroup: 2.3.5.7.11
Commas: 28/27, 55/54, 128/125&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 21/20, 33/32, 128/125
POTE generator: ~3/2 = 722.663&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 3 0 7 13 15 | 0 1 0 -1 -1 }}
Map: [&amp;lt;3 0 7 -6|, &amp;lt;0 1 0 3|]&lt;br /&gt;
 
EDOs: 15, 48bce, 63bce, 78bce&lt;br /&gt;
Optimal tunings:  
Badness: 0.0312&lt;br /&gt;
* WE: ~5/4 = 402.1799{{c}}, ~3/2 = 683.7477{{c}} (~16/15 = 120.6120{{c}})
&lt;br /&gt;
* CWE: ~5/4 = 400.0000{{c}}, ~3/2 = 680.0162{{c}} (~16/15 = 119.9838{{c}})
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Deflated"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Deflated&lt;/h1&gt;
 
Commas: 21/20, 128/125&lt;br /&gt;
{{Optimal ET sequence|legend=0| 3, 6b, 9 }}
&lt;br /&gt;
 
POTE generator: ~3/2 = 681.629&lt;br /&gt;
Badness (Sintel): 1.23
&lt;br /&gt;
 
Map: [&amp;lt;3 0 7 13|, &amp;lt;0 1 0 -1|]&lt;br /&gt;
== Hexe ==
Wedgie: &amp;lt;&amp;lt;3 0 -3 -7 -13 -7||&lt;br /&gt;
Hexe tempers out 50/49 and may be described as {{nowrap| 6 & 12 }}, viewed as [[6edo|6et]] with an independent generator for prime 3. Its ploidacot is hexaploid monocot.
EDOs: 3, 9, 18bd, 21d, 30bd&lt;br /&gt;
 
Badness: 0.0591&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Hexe"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Hexe&lt;/h1&gt;
[[Comma list]]: 50/49, 128/125
Commas: 50/49, 128/125&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 6 0 14 17 | 0 1 0 0 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 710.963&lt;br /&gt;
 
&lt;br /&gt;
: mapping generators: ~28/25, ~3
Map: [&amp;lt;6 0 14 17|, &amp;lt;0 1 0 0|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;6 0 0 -14 -17 0||&lt;br /&gt;
[[Optimal tuning]]s:  
EDOs: &lt;a class="wiki_link" href="/6edo"&gt;6&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/30edo"&gt;30d&lt;/a&gt;, &lt;a class="wiki_link" href="/37edo"&gt;37d&lt;/a&gt;, &lt;a class="wiki_link" href="/42edo"&gt;42d&lt;/a&gt;, &lt;a class="wiki_link" href="/54edo"&gt;54cd&lt;/a&gt;&lt;br /&gt;
* [[WE]]: ~28/25 = 199.0488{{c}}, ~3/2 = 707.5815{{c}} (~25/24 = 88.6137{{c}})
Badness: 0.0577&lt;br /&gt;
: [[error map]]: {{val| +5.870 -11.122 +27.382 -34.224 }}
&lt;br /&gt;
* [[CWE]]: ~28/25 = 200.0000{{c}}, ~3/2 = 708.6907{{c}} (~25/24 = 91.3093{{c}})
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="Hemiaug"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Hemiaug&lt;/h1&gt;
: error map: {{val| 0.000 +6.735 +13.686 +31.174 }}
Commas: 128/125, 245/243&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 6, 12, 30d, 42dd, 54cdd }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~28/27 = 45.118&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 1.46
Map: [&amp;lt;3 1 7 -1|, &amp;lt;0 2 0 5|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;6 0 15 -14 7 35||&lt;br /&gt;
=== 11-limit ===
EDOs: &lt;a class="wiki_link" href="/24edo"&gt;24&lt;/a&gt;, &lt;a class="wiki_link" href="/27edo"&gt;27&lt;/a&gt;, &lt;a class="wiki_link" href="/105edo"&gt;105bc&lt;/a&gt;&lt;br /&gt;
Subgroup: 2.3.5.7.11
Badness: 0.0705&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 50/49, 56/55, 125/121
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Hemiaug-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;11-limit&lt;/h2&gt;
 
Commas: 56/55, 128/125, 245/243&lt;br /&gt;
Mapping: {{mapping| 6 0 14 17 21 | 0 1 0 0 0 }}
&lt;br /&gt;
 
POTE generator: ~28/27 = 45.788&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~28/25 = 198.6942{{c}}, ~3/2 = 709.6404{{c}} (~25/24 = 85.1362{{c}})
EDOs: 24, 27e, 51ce, 78ce&lt;br /&gt;
* CWE: ~28/25 = 200.0000{{c}}, ~3/2 = 711.8043{{c}} (~25/24 = 88.1957{{c}})
Badness: 0.0382&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 6, 12, 30dee, 42ddeee }}
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="Triforce"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Triforce&lt;/h1&gt;
 
Commas: 49/48, 128/125&lt;br /&gt;
Badness (Sintel): 1.27
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~7/6 = 247.049&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Map: [&amp;lt;3 0 7 6|, &amp;lt;0 2 0 1|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;6 0 3 -14 -12 7||&lt;br /&gt;
Comma list: 50/49, 56/55, 66/65, 105/104
EDOs: 6, 9, &lt;a class="wiki_link" href="/15edo"&gt;15&lt;/a&gt;, &lt;a class="wiki_link" href="/24edo"&gt;24&lt;/a&gt;, &lt;a class="wiki_link" href="/39edo"&gt;39&lt;/a&gt;, &lt;a class="wiki_link" href="/63edo"&gt;63cd&lt;/a&gt;, &lt;a class="wiki_link" href="/102edo"&gt;102cd&lt;/a&gt;&lt;br /&gt;
 
Badness: 0.0202&lt;br /&gt;
Mapping: {{mapping| 6 0 14 17 21 13 | 0 1 0 0 0 1 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextLocalImageRule:56:&amp;lt;img src=&amp;quot;/file/view/triforce9.jpg/271381934/triforce9.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/triforce9.jpg/271381934/triforce9.jpg" alt="triforce9.jpg" title="triforce9.jpg" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:56 --&gt;&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~28/25 = 198.4492{{c}}, ~3/2 = 704.4994{{c}} (~25/24 = 89.2973{{c}})
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Triforce-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;11-limit&lt;/h2&gt;
* CWE: ~28/25 = 200.0000{{c}}, ~3/2 = 706.6050{{c}} (~16/15 = 93.3950{{c}})
Commas: 56/55, 77/75, 128/125&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 6f, 12f }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~7/6 = 247.068&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.49
Map: [&amp;lt;3 0 7 6 8|, &amp;lt;0 2 0 1 1|]&lt;br /&gt;
 
EDOs: 6, 9, 15, 24, 39, &lt;a class="wiki_link" href="/63edo"&gt;63cd&lt;/a&gt;, &lt;a class="wiki_link" href="/102edo"&gt;102cd&lt;/a&gt;&lt;br /&gt;
== Triforce ==
Badness: 0.0262&lt;br /&gt;
[[File:triforce9.jpg|thumb|alt=triforce9.jpg|Lattice of triforce]]
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Triforce-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;13-limit&lt;/h2&gt;
Triforce tempers out 49/48 and may be described as {{nowrap| 9 & 15 }}. Its ploidacot is triploid alpha-dicot. [[24edo]] and [[39edo]] are among the possible tunings.  
Commas: 49/48, 56/55, 66/65, 77/75&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~7/6 = 248.313&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 49/48, 128/125
Map: [&amp;lt;3 0 7 6 8 4|, &amp;lt;0 2 0 1 1 3|]&lt;br /&gt;
 
EDOs: 6, 9, 15, 24, &lt;a class="wiki_link" href="/63edo"&gt;63cdf&lt;/a&gt;, &lt;a class="wiki_link" href="/87edo"&gt;87cdf&lt;/a&gt;&lt;br /&gt;
{{Mapping|legend=1| 3 0 7 6 | 0 2 0 1 }}
Badness: 0.0202&lt;br /&gt;
 
&lt;br /&gt;
: mapping generators: ~5/4, ~7/4
&lt;a class="wiki_link" href="/triphi"&gt;Triforce(9) with L:s = phi&lt;/a&gt;&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc17"&gt;&lt;a name="Niner"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Niner&lt;/h1&gt;
* [[WE]]: ~5/4 = 399.7480{{c}}, ~7/4 = 952.3507{{c}} (~35/32 = 152.8547{{c}})
Niner gives 9 as the complexity of an otonal tetrad, tying it with augene as a temperament supported by 27edo. Niner[18], therefore, has nine such tetrads.&lt;br /&gt;
: [[error map]]: {{val| -0.756 +2.746 +11.922 -17.987 }}
&lt;br /&gt;
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~7/4 = 952.7463{{c}} (~35/32 = 152.7463{{c}})
Commas: 128/125, 686/675&lt;br /&gt;
: error map: {{val| 0.000 +3.538 +13.686 -16.080 }}
&lt;br /&gt;
 
POTE generator: ~3/2 = 707.167&lt;br /&gt;
{{Optimal ET sequence|legend=1| 6, 9, 15, 24, 39 }}
&lt;br /&gt;
 
Map: [&amp;lt;9 0 21 11|, &amp;lt;0 1 0 1|]&lt;br /&gt;
[[Badness]] (Sintel): 1.39
Wedgie: &amp;lt;&amp;lt;9 0 9 -21 -11 21||&lt;br /&gt;
 
EDOs: 9, 18, 27, 63c, 90c&lt;br /&gt;
=== 11-limit ===
Badness: 0.0672&lt;/body&gt;&lt;/html&gt;</pre></div>
Subgroup: 2.3.5.7.11
 
Comma list: 49/48, 56/55, 77/75
 
Mapping: {{mapping| 3 0 7 6 8 | 0 2 0 1 1 }}
 
Optimal tunings:
* WE: ~5/4 = 399.7654{{c}}, ~7/4 = 952.3730{{c}} (~12/11 = 152.8421{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 952.7447{{c}} (~12/11 = 152.7447{{c}})
 
{{Optimal ET sequence|legend=0| 6, 9, 15, 24, 39 }}
 
Badness (Sintel): 0.865
 
; Music
* [https://cityoftheasleep.bandcamp.com/track/the-triforce-of-courage-24edo ''The Triforce of Courage (24edo)'']{{dead link}} by [[Igliashon Jones]] (2018)
* [https://www.chrisvaisvil.com/2-2-1-2-2-1-2-2-1-mode-of-15-edo/ ''2-2-1-2-2-1-2-2-1 mode of 15 edo''] [https://web.archive.org/web/20201127015017/http://micro.soonlabel.com/15-ET/20130831_221of15.mp3 play] by [[Chris Vaisvil]] (2013)
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 56/55, 66/65, 77/75
 
Mapping: {{mapping| 3 0 7 6 8 4 | 0 2 0 1 1 3 }}
 
Optimal tunings:
* WE: ~5/4 = 399.7107{{c}}, ~7/4 = 950.9983{{c}} (~12/11 = 151.5768{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 951.4465{{c}} (~12/11 = 151.4465{{c}})
 
{{Optimal ET sequence|legend=0| 6f, 9, 15, 24 }}
 
Badness (Sintel): 0.837
 
; Scales
* [[triphi]], Triforce[9] with L:s = phi
 
==== Semitriforce ====
This extension splits the period into 1/6-octave for ~44/39. Its ploidacot is hexaploid dicot.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 56/55, 77/75, 507/500
 
Mapping: {{mapping| 6 0 14 12 16 27 | 0 2 0 1 1 -1 }}
 
: mapping generators: ~44/39, ~7/4
 
Optimal tunings:  
* WE: ~44/39 = 199.8321{{c}}, ~7/4 = 952.5580{{c}} (~40/39 = 46.6024{{c}})
* CWE: ~44/39 = 200.0000{{c}}, ~7/4 = 953.2005{{c}} (~40/39 = 46.7995{{c}})
 
{{Optimal ET sequence|legend=0| 6, 18bd, 24 }}
 
Badness (Sintel): 2.44
 
== Hemiaug ==
Hemiaug tempers out 245/243 and may be described as {{nowrap| 24 & 27 }}. The generator may be taken as ~14/9, but also a neutral third or a neutral second that stand in for 11/9~16/13 and 12/11~13/12 in the higher limits, respectively. Hemiaug's ploidacot is triploid dicot. [[27edo]] makes for a recommendable tuning in the 7-limit, but [[51edo]] serves better in the higher limits.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 128/125, 245/243
 
{{Mapping|legend=1| 3 1 7 -1 | 0 2 0 5 }}
 
: mapping generators: ~5/4, ~14/9
 
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 398.9278{{c}}, ~14/9 = 752.8583{{c}} (~36/35 = 44.9973{{c}})
: [[error map]]: {{val| -3.217 +2.689 +6.181 -3.462 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~14/9 = 754.2078{{c}} (~36/35 = 45.7922{{c}})
: error map: {{val| 0.000 +6.461 +13.686 +2.213 }}
 
{{Optimal ET sequence|legend=1| 24, 27 }}
 
[[Badness]] (Sintel): 1.78
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 128/125, 243/242
 
Mapping: {{mapping| 3 1 7 -1 1 | 0 2 0 5 5 }}
 
Optimal tunings:
* WE: ~5/4 = 398.8946{{c}}, ~14/9 = 752.1272{{c}} (~36/35 = 45.6619{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~14/9 = 753.5000{{c}} (~36/35 = 46.5000{{c}})
 
{{Optimal ET sequence|legend=0| 24, 27e, 51ce }}
 
Badness (Sintel): 1.26
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 91/90, 128/125, 243/242
 
Mapping: {{mapping| 3 1 7 -1 1 13 | 0 2 0 5 5 -1 }}
 
Optimal tunings:  
* WE: ~5/4 = 399.1053{{c}}, ~14/9 = 752.0643{{c}} (~36/35 = 46.1463{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~14/9 = 753.3806{{c}} (~36/35 = 46.6194{{c}})
 
{{Optimal ET sequence|legend=0| 24, 27e, 51ce }}
 
Badness (Sintel): 1.25
 
== Hemiug ==
Hemiug tempers out 1323/1250 and may be described as {{nowrap| 21 & 24 }}. The generator is a similar interval but for ~32/21 instead of ~14/9, and the ploidacot is triploid dicot, the same as hemiaug.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 128/125, 1323/1250
 
{{Mapping|legend=1| 3 1 7 14 | 0 2 0 -3 }}
 
: mapping generators: ~5/4, ~32/21
 
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 400.1805{{c}}, ~32/21 = 748.2436{{c}} (~21/20 = 52.1174{{c}})
: [[error map]]: {{val| +0.542 -5.287 +14.950 -11.030 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~32/21 = 747.9138{{c}} (~21/20 = 52.0862{{c}})
: error map: {{val| 0.000 -6.127 +13.686 -12.567 }}
 
{{Optimal ET sequence|legend=1| 21, 24, 45c }}
 
[[Badness]] (Sintel): 3.49
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 128/125, 1323/1250
 
Mapping: {{mapping| 3 1 7 14 16 | 0 2 0 -3 -3 }}
 
Optimal tunings:
* WE: ~5/4 = 400.0637{{c}}, ~32/21 = 748.4638{{c}} (~33/32 = 51.6637{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~32/21 = 748.3383{{c}} (~33/32 = 51.6617{{c}})
 
{{Optimal ET sequence|legend=0| 21, 24 }}
 
Badness (Sintel): 2.25
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 66/65, 105/104, 507/500
 
Mapping: {{mapping| 3 1 7 14 16 13 | 0 2 0 -3 -3 -1 }}
 
Optimal tunings:  
* WE: ~5/4 = 399.8855{{c}}, ~32/21 = 748.2378{{c}} (~33/32 = 51.5332{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~32/21 = 748.4655{{c}} (~33/32 = 51.5345{{c}})
 
{{Optimal ET sequence|legend=0| 21, 24 }}
 
Badness (Sintel): 1.75
 
== Oodako ==
Oodako tempers out 2401/2400 and may be described as {{nowrap| 21 & 27 }}. It is generated by a quarter of a fifth, which stands in for ~28/25. Its ploidacot is triploid tetracot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 128/125, 2401/2400
 
{{Mapping|legend=1| 3 3 7 8 | 0 4 0 1 }}
 
: mapping generators: ~5/4, ~28/25
 
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 399.0296{{c}}, ~28/25 = 176.2174{{c}} (~49/48 = 46.5949{{c}})
: [[error map]]: {{val| -2.911 +0.004 +6.894 -0.371 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~28/25 = 176.2984{{c}} (~49/48 = 47.4031{{c}})
: error map: {{val| 0.000 +3.239 +13.686 +7.473 }}
 
{{Optimal ET sequence|legend=1| 6, 21, 27, 75c, 102ccd, 129bccd }}
 
[[Badness]] (Sintel): 2.86
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 128/125, 2401/2400
 
Mapping: {{mapping| 3 3 7 8 10 | 0 4 0 1 1 }}
 
Optimal tunings:  
* WE: ~5/4 = 398.6615{{c}}, ~11/10 = 176.3886{{c}} (~49/48 = 45.8843{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~11/10 = 176.5471{{c}} (~49/48 = 46.9059{{c}})
 
{{Optimal ET sequence|legend=0| 6, 21, 27e }}
 
Badness (Sintel): 1.96
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 78/77, 128/125, 507/500
 
Mapping: {{mapping| 3 3 7 8 10 12 | 0 4 0 1 1 -2 }}
 
Optimal tunings:  
* WE: ~5/4 = 398.8612{{c}}, ~11/10 = 176.0486{{c}} (~49/48 = 46.7640{{c}})
* CWE: ~5/4 = 400.0000{{c}}, ~11/10 = 176.3326{{c}} (~49/48 = 47.3348{{c}})
 
{{Optimal ET sequence|legend=0| 6, 21, 27e }}
 
Badness (Sintel): 1.75
 
== Hemisemiaug ==
Hemisemiaug tempers out 12005/11664 and splits both the period and generator of augmented in two. Its ploidacot is hexaploid alpha-dicot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 128/125, 12005/11664
 
{{Mapping|legend=1| 6 1 14 4 | 0 2 0 3 }}
 
: mapping generators: ~54/49, ~45/28
 
[[Optimal tuning]]s:  
* [[WE]]: ~54/49 = 199.5469{{c}}, ~45/28 = 853.5468{{c}} (~36/35 = 55.3594{{c}})
: [[error map]]: {{val| -2.719 +4.686 +7.342 -9.998 }}
* [[CWE]]: ~54/49 = 200.0000{{c}}, ~45/28 = 854.7144{{c}} (~36/35 = 54.7144{{c}})
: error map: {{val| 0.000 +7.474 +13.686 -4.683 }}
 
{{Optimal ET sequence|legend=1| 18, 24, 42 }}
 
[[Badness]] (Sintel): 5.34
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 128/125, 3773/3645
 
Mapping: {{mapping| 6 1 14 4 8 | 0 2 0 3 3 }}
 
Optimal tunings:
* WE: ~54/49 = 199.5188{{c}}, ~18/11 = 853.1623{{c}} (~36/35 = 55.0872{{c}})
* CWE: ~54/49 = 200.0000{{c}}, ~18/11 = 854.3545{{c}} (~36/35 = 54.3545{{c}})
 
{{Optimal ET sequence|legend=0| 18e, 24, 42e, 66ce, 108bccee }}
 
Badness (Sintel): 2.67
 
== Niner ==
Niner tempers out 686/675 and may be described as the {{nowrap| 9 & 27 }} temperament. Its ploidacot is enneaploid monocot. It gives 9 as the complexity of a [[harmonic seventh chord]], tying it with augene as a temperament supported by 27edo. Niner[18], therefore, has nine such tetrads. 27edo, [[36edo]] and [[63edo]] in the 63c val are among the possible tunings.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 128/125, 686/675
 
{{Mapping|legend=1| 9 0 21 11 | 0 1 0 1 }}
 
: mapping generators: ~49/45, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~49/45 = 133.0272{{c}}, ~3/2 = 705.5438{{c}} (~36/35 = 40.4075{{c}})
: [[error map]]: {{val| -2.755 +0.834 +7.259 -2.737 }}
* [[CWE]]: ~49/45 = 133.3333{{c}}, ~3/2 = 705.5157{{c}} (~36/35 = 38.8490{{c}})
: error map: {{val| 0.000 +3.561 +13.686 +3.356 }}
 
{{Optimal ET sequence|legend=1| 9, 18, 27, 63c, 90cc }}
 
[[Badness]] (Sintel): 1.70
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 128/125, 540/539
 
Mapping: {{mapping| 9 0 21 11 17 | 0 1 0 1 1 }}
 
Optimal tunings:
* WE: ~12/11 = 132.9553{{c}}, ~3/2 = 704.7217{{c}} (~36/35 = 39.9453{{c}})
* CWE: ~12/11 = 133.3333{{c}}, ~3/2 = 704.5723{{c}} (~36/35 = 37.9056{{c}})
 
{{Optimal ET sequence|legend=0| 9, 18e, 27e, 63cee }}
 
Badness (Sintel): 1.15
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 78/77, 91/90, 128/125
 
Mapping: {{mapping| 9 0 21 11 17 19 | 0 1 0 1 1 1 }}
 
Optimal tunings:  
* WE: ~14/13 = 133.0143{{c}}, ~3/2 = 705.1969{{c}} (~36/35 = 40.1256{{c}})
* CWE: ~14/13 = 133.3333{{c}}, ~3/2 = 705.0176{{c}} (~36/35 = 38.3510{{c}})
 
{{Optimal ET sequence|legend=0| 9, 18e, 27e }}
 
Badness (Sintel): 0.998
 
== Trug ==
Trug tempers out 360/343. It is generated by an interval of ~48/35, tuned very close to a perfect fourth, but the perfect fourth is mapped to three generator steps and a period. Its ploidacot is triploid alpha-tricot. 12edo is about as accurate as it can be tuned.  
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 128/125, 360/343
 
{{Mapping|legend=1| 3 1 7 6 | 0 3 0 2 }}
 
: mapping generators: ~5/4, ~48/35
 
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 398.2337{{c}}, ~48/35 = 499.7635{{c}} (~15/14 = 101.5299{{c}})
: [[error map]]: {{val| -2.755 +0.834 +7.259 -2.737 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~48/35 = 500.9654{{c}} (~15/14 = 100.9654{{c}})
: error map: {{val| 0.000 +3.561 +13.686 +3.356 }}
 
{{Optimal ET sequence|legend=1| 3b, 9bd, 12 }}
 
[[Badness]] (Sintel): 3.50
 
== External links ==
* [https://web.archive.org/web/20211201070113/https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Augmented family| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Augmented_family"