Augmented family: Difference between revisions

Revision as of 18:33, 9 May 2012

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[[toc]]

The 5-limit parent comma for the **augmented family** is 128/125, the [[128_125|diesis]]. Its [[monzo]] is |7 0 -3>, and flipping that yields <<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. 

=Seven limit children=
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]]s: 36/35, 128/125

[[POTE tuning|POTE generator]]: 696.011

[[Map]]: [<3 0 7 -1|, <0 1 0 2|]
[[Wedgie]]: <<3 0 6 -7 1 14||
[[edo|EDOs]]: [[9edo|9]], [[12edo|12]], [[45edo|45cd]], [[57edo|57cd]], [[69edo|69cd]]
[[Badness]]: 0.0265

==11-limit==
Commas: 36/35, 45/44, 56/55

POTE generator: ~3/2 = 692.514

Map: [<3 0 7 -1 1|, <0 1 0 2 2|]
EDOs: 9, 12, [[21edo|21]], [[33edo|33e]], [[45edo|45cde]]
Badness: 0.0202

==13-limit==
Commas: 27/26, 36/35, 45/44, 56/55

POTE generator: ~3/2 = 688.783

Map: [<3 0 7 -1 1 -3|, <0 1 0 2 2 3|]
EDOs: 9, 12, [[21edo|21]], [[33edo|33ef]], [[54edo|54bcef]]
Badness: 0.0184

==Augustus==
Commas: 26/25, 36/35, 45/44, 56/55

POTE generator: ~3/2 = 685.356

Map: [<3 0 7 -1 1 11|, <0 1 0 2 2 0|]
EDOs: 9, 12, 21f
Badness: 0.0222

=Augene=
Commas: 64/63, 126/125

[[POTE tuning|POTE generator]]: 709.257

Map: [<3 0 7 18|, <0 1 0 -2|]
Wedgie: <<3 0 -6 -7 -18 -14||
EDOs: [[12edo|12]], [[27edo|27]], [[39edo|39d]], [[66edo|66cd]]
Badness: 0.0248

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

POTE generator: ~3/2 = 711.177

Map: [<3 0 7 18 20|, <0 1 0 -2 -2|]
EDOs: 12, 15, 27e
Badness: 0.0196

===13-limit===
Commas: 40/39, 56/55, 64/63, 66/65

POTE generator: ~3/2 = 712.013

Map: [<3 0 7 18 20 16|, <0 1 0 -2 -2 -1|]
EDOs: 12f, 15, 27ef
Badness: 0.0208

===Ogene===
Commas: 56/55, 64/63, 91/90, 100/99

POTE generator: ~3/2 = 712.609

Map: [<3 0 7 18 20 -8|, <0 1 0 -2 -2 4|]
EDOs: 12, 15, 27e, 69bcef
Badness: 0.0229

==Eugene==
Commas: 55/54, 64/63, 77/75 

POTE generator: ~3/2 = 714.150

Map: [<3 0 7 18 -4|, <0 1 0 -2 3|]
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]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3|Galticeran Sonatina]] by [[http://soundcloud.com/joelgranttaylor/galticeran_sonatina|Joel Taylor]]

=Inflated=
Commas: 28/27, 128/125

POTE generator: ~3/2 = 722.719

Map: [<3 0 7 -6|, <0 1 0 3|]
Wedgie: <<3 0 9 -7 6 21||
EDOs: 15, 48bc, 63bc, 78bc
Badness: 0.0547

==11-limit==
Commas: 28/27, 55/54, 128/125

POTE generator: ~3/2 = 722.663

Map: [<3 0 7 -6|, <0 1 0 3|]
EDOs: 15, 48bce, 63bce, 78bce
Badness: 0.0312

=Deflated=
Commas: 21/20, 128/125

POTE generator: ~3/2 = 681.629

Map: [<3 0 7 13|, <0 1 0 -1|]
Wedgie: <<3 0 -3 -7 -13 -7||
EDOs: 3, 9, 18bd, 21d, 30bd
Badness: 0.0591

=Hexe=
Commas: 50/49, 128/125

[[POTE tuning|POTE generator]]: 710.963

Map: [<6 0 14 17|, <0 1 0 0|]
Wedgie: <<6 0 0 -14 -17 0||
EDOs: [[6edo|6]], [[12edo|12]], [[30edo|30d]], [[37edo|37d]], [[42edo|42d]], [[54edo|54cd]]
Badness: 0.0577

=Hemiaug=
Commas: 128/125, 245/243

[[POTE tuning|POTE generator]]: ~28/27 = 45.118

Map: [<3 1 7 -1|, <0 2 0 5|]
Wedgie: <<6 0 15 -14 7 35||
EDOs: [[24edo|24]], [[27edo|27]], [[105edo|105bc]]
Badness: 0.0705

==11-limit==
Commas: 56/55, 128/125, 245/243

POTE generator: ~28/27 = 45.788

Map: [<3 1 7 -1 1|, <0 2 0 5 5|]
EDOs: 24, 27e, 51ce, 78ce
Badness: 0.0382

==13-limit==
Commas: 56/55, 91/90, 128/125, 245/243

POTE generator: ~28/27 = 46.250

Map: [<3 1 7 -1 1 13|, <0 2 0 5 5 -1|]
EDOs: 24, 27e, 51ce, 78ce
Badness: 0.0302

=Triforce=
Commas: 49/48, 128/125

[[POTE tuning|POTE generator]]: ~7/6 = 247.049

Map: [<3 0 7 6|, <0 2 0 1|]
Wedgie: <<6 0 3 -14 -12 7||
EDOs: 6, 9, [[15edo|15]], [[24edo|24]], [[39edo|39]], [[63edo|63cd]], [[102edo|102cd]]
Badness: 0.0202

[[image:triforce9.jpg]]

==11-limit==
Commas: 56/55, 77/75, 128/125

[[POTE tuning|POTE generator]]: ~7/6 = 247.068

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

==13-limit==
Commas: 49/48, 56/55, 66/65, 77/75

[[POTE tuning|POTE generator]]: ~7/6 = 248.313

Map: [<3 0 7 6 8 4|, <0 2 0 1 1 3|]
EDOs: 6, 9, 15, 24, [[63edo|63cdf]], [[87edo|87cdf]]
Badness: 0.0202

[[triphi|Triforce(9) with L:s = phi]]

==Semitriforce==
Commas: 49/48, 56/55, 77/75, 507/500 

POTE generator: ~7/6 = 246.642

Map: [<6 0 14 12 16 27|, <0 2 0 1 1 -1|]
EDOs: 6, 24, 54cd, 78cd, 102cdf
Badness: 0.0592

=Hemiug=
Commas: 128/125, 1323/1250

POTE generator: ~21/20 = 52.093

Map: [<3 1 7 14|, <0 2 0 -3|]
Wedgie: <<6 0 -9 -14 -31 -21||
EDOs: 21, 24, 45c, 69cd
Badness: 0.1378


==11-limit==
Commas: 56/55, 128/125, 1323/1250

POTE generator: ~21/20 = 51.655

Map: [<3 1 7 14 16|, <0 2 0 -3 -3|]
EDOs: 21, 24, 69cd, 93cd
Badness: 0.0681

==13-limit==
Commas: 56/55, 66/65, 105/104, 507/500

POTE generator: ~21/20 = 51.548

Map: [<3 1 7 14 16 13|, <0 2 0 -3 -3 -1|]
EDOs: 21, 24, 69cdf, 93cdf
Badness: 0.0423

=Oodako=
Commas: 128/125, 2401/2400

POTE generator: ~8/7 = 223.3540873874507317

Map: [<3 3 7 8|, <0 4 0 1|]
Wedgie: <<12 0 3 -28 -29 7||
EDOs: 6, 21, 27, 75c, 102cd, 129bcd
Badness: 0.1132

==11-limit==
Commas: 56/55, 128/125, 3773/3750

POTE generator: ~8/7 = 223.019

Map: [<3 3 7 8 10|, <0 4 0 1 1|]
EDOs: 6, 21, 27e
Badness: 0.0592

==13-limit==
Commas: 56/55, 78/77, 128/125, 507/500

POTE generator: ~8/7 = 223.449

Map: [<3 3 7 8 10 12|, <0 4 0 1 1 -2|]
EDOs: 6, 21, 27e
Badness: 0.0423

=Hemisemiaug=
Commas: 128/125, 12005/11664

POTE generator: ~15/14 = 144.515

Map: [<6 1 14 4|, <0 2 0 3|]
Wedgie: <<12 0 18 -28 -5 42||
Edos: 18, 24, 42, 66c, 108bc
Badness: 0.2110

==11-limit==
Commas: 56/55, 128/125, 3773/3645

POTE generator: ~15/14 = 144.780

Map: [<6 1 14 4 8|, <0 2 0 3 3|]
EDOs: 24, 42e, 66ce, 108bce
Badness: 0.0807

=Niner=
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

POTE generator: ~3/2 = 707.167

Map: [<9 0 21 11|, <0 1 0 1|]
Wedgie: <<9 0 9 -21 -11 21||
EDOs: 9, 18, 27, 63c, 90c
Badness: 0.0672

==11-limit==
Commas: 56/55, 128/125, 540/539

POTE generator: ~3/2 = 706.726

Map: [<9 0 21 11 17|, <0 1 0 1 1|]
EDOs: 9, 27e, 36, 63ce
Badness: 0.0349

==13-limit==
Commas: 56/55, 78/77, 91/90, 128/125

POTE generator: ~3/2 = 706.889

Map: [<9 0 21 11 17 19|, <0 1 0 1 1 1|]
EDOs: 9, 27e, 36, 63ce
Badness: 0.0241

Original HTML content:

<html><head><title>Augmented family</title></head><body><!-- ws:start:WikiTextTocRule:64:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><div style="margin-left: 1em;"><a href="#Seven limit children">Seven limit children</a></div>
<!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><div style="margin-left: 1em;"><a href="#August">August</a></div>
<!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><div style="margin-left: 2em;"><a href="#August-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><div style="margin-left: 2em;"><a href="#August-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><div style="margin-left: 2em;"><a href="#August-Augustus">Augustus</a></div>
<!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><div style="margin-left: 1em;"><a href="#Augene">Augene</a></div>
<!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><div style="margin-left: 2em;"><a href="#Augene-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><div style="margin-left: 3em;"><a href="#Augene-11-limit-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><div style="margin-left: 3em;"><a href="#Augene-11-limit-Ogene">Ogene</a></div>
<!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><div style="margin-left: 2em;"><a href="#Augene-Eugene">Eugene</a></div>
<!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><div style="margin-left: 1em;"><a href="#Inflated">Inflated</a></div>
<!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --><div style="margin-left: 2em;"><a href="#Inflated-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --><div style="margin-left: 1em;"><a href="#Deflated">Deflated</a></div>
<!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><div style="margin-left: 1em;"><a href="#Hexe">Hexe</a></div>
<!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --><div style="margin-left: 1em;"><a href="#Hemiaug">Hemiaug</a></div>
<!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --><div style="margin-left: 2em;"><a href="#Hemiaug-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><div style="margin-left: 2em;"><a href="#Hemiaug-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><div style="margin-left: 1em;"><a href="#Triforce">Triforce</a></div>
<!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><div style="margin-left: 2em;"><a href="#Triforce-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><div style="margin-left: 2em;"><a href="#Triforce-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><div style="margin-left: 2em;"><a href="#Triforce-Semitriforce">Semitriforce</a></div>
<!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><div style="margin-left: 1em;"><a href="#Hemiug">Hemiug</a></div>
<!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><div style="margin-left: 2em;"><a href="#Hemiug-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><div style="margin-left: 2em;"><a href="#Hemiug-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><div style="margin-left: 1em;"><a href="#Oodako">Oodako</a></div>
<!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><div style="margin-left: 2em;"><a href="#Oodako-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><div style="margin-left: 2em;"><a href="#Oodako-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><div style="margin-left: 1em;"><a href="#Hemisemiaug">Hemisemiaug</a></div>
<!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><div style="margin-left: 2em;"><a href="#Hemisemiaug-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><div style="margin-left: 1em;"><a href="#Niner">Niner</a></div>
<!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><div style="margin-left: 2em;"><a href="#Niner-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><div style="margin-left: 2em;"><a href="#Niner-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --></div>
<!-- ws:end:WikiTextTocRule:97 --><br />
The 5-limit parent comma for the <strong>augmented family</strong> is 128/125, the <a class="wiki_link" href="/128_125">diesis</a>. Its <a class="wiki_link" href="/monzo">monzo</a> is |7 0 -3&gt;, and flipping that yields &lt;&lt;3 0 -7|| for the <a class="wiki_link" href="/wedgie">wedgie</a>. Hence the period is 1/3 octave, and this is what is used for a major third. The <a class="wiki_link" href="/generator">generator</a> can be taken as a fifth or a semitone, and <a class="wiki_link" href="/12edo">12edo</a>, with its excellent fifth, is an obvious tuning for <a class="wiki_link" href="/5-limit">5-limit</a> augmented, though a sharper fifth might be preferred to go with the sharp third. <br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h1>
The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> defines which <a class="wiki_link" href="/7-limit">7-limit</a> 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="August"></a><!-- ws:end:WikiTextHeadingRule:2 -->August</h1>
<a class="wiki_link" href="/Comma">Comma</a>s: 36/35, 128/125<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.011<br />
<br />
<a class="wiki_link" href="/Map">Map</a>: [&lt;3 0 7 -1|, &lt;0 1 0 2|]<br />
<a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;3 0 6 -7 1 14||<br />
<a class="wiki_link" href="/edo">EDOs</a>: <a class="wiki_link" href="/9edo">9</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/45edo">45cd</a>, <a class="wiki_link" href="/57edo">57cd</a>, <a class="wiki_link" href="/69edo">69cd</a><br />
<a class="wiki_link" href="/Badness">Badness</a>: 0.0265<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="August-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h2>
Commas: 36/35, 45/44, 56/55<br />
<br />
POTE generator: ~3/2 = 692.514<br />
<br />
Map: [&lt;3 0 7 -1 1|, &lt;0 1 0 2 2|]<br />
EDOs: 9, 12, <a class="wiki_link" href="/21edo">21</a>, <a class="wiki_link" href="/33edo">33e</a>, <a class="wiki_link" href="/45edo">45cde</a><br />
Badness: 0.0202<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="August-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h2>
Commas: 27/26, 36/35, 45/44, 56/55<br />
<br />
POTE generator: ~3/2 = 688.783<br />
<br />
Map: [&lt;3 0 7 -1 1 -3|, &lt;0 1 0 2 2 3|]<br />
EDOs: 9, 12, <a class="wiki_link" href="/21edo">21</a>, <a class="wiki_link" href="/33edo">33ef</a>, <a class="wiki_link" href="/54edo">54bcef</a><br />
Badness: 0.0184<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="August-Augustus"></a><!-- ws:end:WikiTextHeadingRule:8 -->Augustus</h2>
Commas: 26/25, 36/35, 45/44, 56/55<br />
<br />
POTE generator: ~3/2 = 685.356<br />
<br />
Map: [&lt;3 0 7 -1 1 11|, &lt;0 1 0 2 2 0|]<br />
EDOs: 9, 12, 21f<br />
Badness: 0.0222<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Augene"></a><!-- ws:end:WikiTextHeadingRule:10 -->Augene</h1>
Commas: 64/63, 126/125<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 709.257<br />
<br />
Map: [&lt;3 0 7 18|, &lt;0 1 0 -2|]<br />
Wedgie: &lt;&lt;3 0 -6 -7 -18 -14||<br />
EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/27edo">27</a>, <a class="wiki_link" href="/39edo">39d</a>, <a class="wiki_link" href="/66edo">66cd</a><br />
Badness: 0.0248<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Augene-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->11-limit</h2>
Commas: 56/55, 64/63, 100/99<br />
<br />
POTE generator: ~3/2 = 711.177<br />
<br />
Map: [&lt;3 0 7 18 20|, &lt;0 1 0 -2 -2|]<br />
EDOs: 12, 15, 27e<br />
Badness: 0.0196<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Augene-11-limit-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h3>
Commas: 40/39, 56/55, 64/63, 66/65<br />
<br />
POTE generator: ~3/2 = 712.013<br />
<br />
Map: [&lt;3 0 7 18 20 16|, &lt;0 1 0 -2 -2 -1|]<br />
EDOs: 12f, 15, 27ef<br />
Badness: 0.0208<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Augene-11-limit-Ogene"></a><!-- ws:end:WikiTextHeadingRule:16 -->Ogene</h3>
Commas: 56/55, 64/63, 91/90, 100/99<br />
<br />
POTE generator: ~3/2 = 712.609<br />
<br />
Map: [&lt;3 0 7 18 20 -8|, &lt;0 1 0 -2 -2 4|]<br />
EDOs: 12, 15, 27e, 69bcef<br />
Badness: 0.0229<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Augene-Eugene"></a><!-- ws:end:WikiTextHeadingRule:18 -->Eugene</h2>
Commas: 55/54, 64/63, 77/75 <br />
<br />
POTE generator: ~3/2 = 714.150<br />
<br />
Map: [&lt;3 0 7 18 -4|, &lt;0 1 0 -2 3|]<br />
EDOs: 15, 27, 42<br />
Badness: 0.0356<br />
<br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3" rel="nofollow">Sad Like Winter Leaves</a> by <a class="wiki_link_ext" href="http://soundcloud.com/cityoftheasleep/sad-like-winter-trees" rel="nofollow">Igliashon Jones</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3" rel="nofollow">Galticeran Sonatina</a> by <a class="wiki_link_ext" href="http://soundcloud.com/joelgranttaylor/galticeran_sonatina" rel="nofollow">Joel Taylor</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Inflated"></a><!-- ws:end:WikiTextHeadingRule:20 -->Inflated</h1>
Commas: 28/27, 128/125<br />
<br />
POTE generator: ~3/2 = 722.719<br />
<br />
Map: [&lt;3 0 7 -6|, &lt;0 1 0 3|]<br />
Wedgie: &lt;&lt;3 0 9 -7 6 21||<br />
EDOs: 15, 48bc, 63bc, 78bc<br />
Badness: 0.0547<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Inflated-11-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->11-limit</h2>
Commas: 28/27, 55/54, 128/125<br />
<br />
POTE generator: ~3/2 = 722.663<br />
<br />
Map: [&lt;3 0 7 -6|, &lt;0 1 0 3|]<br />
EDOs: 15, 48bce, 63bce, 78bce<br />
Badness: 0.0312<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Deflated"></a><!-- ws:end:WikiTextHeadingRule:24 -->Deflated</h1>
Commas: 21/20, 128/125<br />
<br />
POTE generator: ~3/2 = 681.629<br />
<br />
Map: [&lt;3 0 7 13|, &lt;0 1 0 -1|]<br />
Wedgie: &lt;&lt;3 0 -3 -7 -13 -7||<br />
EDOs: 3, 9, 18bd, 21d, 30bd<br />
Badness: 0.0591<br />
<br />
<!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name="Hexe"></a><!-- ws:end:WikiTextHeadingRule:26 -->Hexe</h1>
Commas: 50/49, 128/125<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 710.963<br />
<br />
Map: [&lt;6 0 14 17|, &lt;0 1 0 0|]<br />
Wedgie: &lt;&lt;6 0 0 -14 -17 0||<br />
EDOs: <a class="wiki_link" href="/6edo">6</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/30edo">30d</a>, <a class="wiki_link" href="/37edo">37d</a>, <a class="wiki_link" href="/42edo">42d</a>, <a class="wiki_link" href="/54edo">54cd</a><br />
Badness: 0.0577<br />
<br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Hemiaug"></a><!-- ws:end:WikiTextHeadingRule:28 -->Hemiaug</h1>
Commas: 128/125, 245/243<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~28/27 = 45.118<br />
<br />
Map: [&lt;3 1 7 -1|, &lt;0 2 0 5|]<br />
Wedgie: &lt;&lt;6 0 15 -14 7 35||<br />
EDOs: <a class="wiki_link" href="/24edo">24</a>, <a class="wiki_link" href="/27edo">27</a>, <a class="wiki_link" href="/105edo">105bc</a><br />
Badness: 0.0705<br />
<br />
<!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Hemiaug-11-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->11-limit</h2>
Commas: 56/55, 128/125, 245/243<br />
<br />
POTE generator: ~28/27 = 45.788<br />
<br />
Map: [&lt;3 1 7 -1 1|, &lt;0 2 0 5 5|]<br />
EDOs: 24, 27e, 51ce, 78ce<br />
Badness: 0.0382<br />
<br />
<!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Hemiaug-13-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->13-limit</h2>
Commas: 56/55, 91/90, 128/125, 245/243<br />
<br />
POTE generator: ~28/27 = 46.250<br />
<br />
Map: [&lt;3 1 7 -1 1 13|, &lt;0 2 0 5 5 -1|]<br />
EDOs: 24, 27e, 51ce, 78ce<br />
Badness: 0.0302<br />
<br />
<!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Triforce"></a><!-- ws:end:WikiTextHeadingRule:34 -->Triforce</h1>
Commas: 49/48, 128/125<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~7/6 = 247.049<br />
<br />
Map: [&lt;3 0 7 6|, &lt;0 2 0 1|]<br />
Wedgie: &lt;&lt;6 0 3 -14 -12 7||<br />
EDOs: 6, 9, <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/24edo">24</a>, <a class="wiki_link" href="/39edo">39</a>, <a class="wiki_link" href="/63edo">63cd</a>, <a class="wiki_link" href="/102edo">102cd</a><br />
Badness: 0.0202<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:98:&lt;img src=&quot;/file/view/triforce9.jpg/271381934/triforce9.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/triforce9.jpg/271381934/triforce9.jpg" alt="triforce9.jpg" title="triforce9.jpg" /><!-- ws:end:WikiTextLocalImageRule:98 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Triforce-11-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->11-limit</h2>
Commas: 56/55, 77/75, 128/125<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~7/6 = 247.068<br />
<br />
Map: [&lt;3 0 7 6 8|, &lt;0 2 0 1 1|]<br />
EDOs: 6, 9, 15, 24, 39, <a class="wiki_link" href="/63edo">63cd</a>, <a class="wiki_link" href="/102edo">102cd</a><br />
Badness: 0.0262<br />
<br />
<!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Triforce-13-limit"></a><!-- ws:end:WikiTextHeadingRule:38 -->13-limit</h2>
Commas: 49/48, 56/55, 66/65, 77/75<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~7/6 = 248.313<br />
<br />
Map: [&lt;3 0 7 6 8 4|, &lt;0 2 0 1 1 3|]<br />
EDOs: 6, 9, 15, 24, <a class="wiki_link" href="/63edo">63cdf</a>, <a class="wiki_link" href="/87edo">87cdf</a><br />
Badness: 0.0202<br />
<br />
<a class="wiki_link" href="/triphi">Triforce(9) with L:s = phi</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Triforce-Semitriforce"></a><!-- ws:end:WikiTextHeadingRule:40 -->Semitriforce</h2>
Commas: 49/48, 56/55, 77/75, 507/500 <br />
<br />
POTE generator: ~7/6 = 246.642<br />
<br />
Map: [&lt;6 0 14 12 16 27|, &lt;0 2 0 1 1 -1|]<br />
EDOs: 6, 24, 54cd, 78cd, 102cdf<br />
Badness: 0.0592<br />
<br />
<!-- ws:start:WikiTextHeadingRule:42:&lt;h1&gt; --><h1 id="toc21"><a name="Hemiug"></a><!-- ws:end:WikiTextHeadingRule:42 -->Hemiug</h1>
Commas: 128/125, 1323/1250<br />
<br />
POTE generator: ~21/20 = 52.093<br />
<br />
Map: [&lt;3 1 7 14|, &lt;0 2 0 -3|]<br />
Wedgie: &lt;&lt;6 0 -9 -14 -31 -21||<br />
EDOs: 21, 24, 45c, 69cd<br />
Badness: 0.1378<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:44:&lt;h2&gt; --><h2 id="toc22"><a name="Hemiug-11-limit"></a><!-- ws:end:WikiTextHeadingRule:44 -->11-limit</h2>
Commas: 56/55, 128/125, 1323/1250<br />
<br />
POTE generator: ~21/20 = 51.655<br />
<br />
Map: [&lt;3 1 7 14 16|, &lt;0 2 0 -3 -3|]<br />
EDOs: 21, 24, 69cd, 93cd<br />
Badness: 0.0681<br />
<br />
<!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><h2 id="toc23"><a name="Hemiug-13-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->13-limit</h2>
Commas: 56/55, 66/65, 105/104, 507/500<br />
<br />
POTE generator: ~21/20 = 51.548<br />
<br />
Map: [&lt;3 1 7 14 16 13|, &lt;0 2 0 -3 -3 -1|]<br />
EDOs: 21, 24, 69cdf, 93cdf<br />
Badness: 0.0423<br />
<br />
<!-- ws:start:WikiTextHeadingRule:48:&lt;h1&gt; --><h1 id="toc24"><a name="Oodako"></a><!-- ws:end:WikiTextHeadingRule:48 -->Oodako</h1>
Commas: 128/125, 2401/2400<br />
<br />
POTE generator: ~8/7 = 223.3540873874507317<br />
<br />
Map: [&lt;3 3 7 8|, &lt;0 4 0 1|]<br />
Wedgie: &lt;&lt;12 0 3 -28 -29 7||<br />
EDOs: 6, 21, 27, 75c, 102cd, 129bcd<br />
Badness: 0.1132<br />
<br />
<!-- ws:start:WikiTextHeadingRule:50:&lt;h2&gt; --><h2 id="toc25"><a name="Oodako-11-limit"></a><!-- ws:end:WikiTextHeadingRule:50 -->11-limit</h2>
Commas: 56/55, 128/125, 3773/3750<br />
<br />
POTE generator: ~8/7 = 223.019<br />
<br />
Map: [&lt;3 3 7 8 10|, &lt;0 4 0 1 1|]<br />
EDOs: 6, 21, 27e<br />
Badness: 0.0592<br />
<br />
<!-- ws:start:WikiTextHeadingRule:52:&lt;h2&gt; --><h2 id="toc26"><a name="Oodako-13-limit"></a><!-- ws:end:WikiTextHeadingRule:52 -->13-limit</h2>
Commas: 56/55, 78/77, 128/125, 507/500<br />
<br />
POTE generator: ~8/7 = 223.449<br />
<br />
Map: [&lt;3 3 7 8 10 12|, &lt;0 4 0 1 1 -2|]<br />
EDOs: 6, 21, 27e<br />
Badness: 0.0423<br />
<br />
<!-- ws:start:WikiTextHeadingRule:54:&lt;h1&gt; --><h1 id="toc27"><a name="Hemisemiaug"></a><!-- ws:end:WikiTextHeadingRule:54 -->Hemisemiaug</h1>
Commas: 128/125, 12005/11664<br />
<br />
POTE generator: ~15/14 = 144.515<br />
<br />
Map: [&lt;6 1 14 4|, &lt;0 2 0 3|]<br />
Wedgie: &lt;&lt;12 0 18 -28 -5 42||<br />
Edos: 18, 24, 42, 66c, 108bc<br />
Badness: 0.2110<br />
<br />
<!-- ws:start:WikiTextHeadingRule:56:&lt;h2&gt; --><h2 id="toc28"><a name="Hemisemiaug-11-limit"></a><!-- ws:end:WikiTextHeadingRule:56 -->11-limit</h2>
Commas: 56/55, 128/125, 3773/3645<br />
<br />
POTE generator: ~15/14 = 144.780<br />
<br />
Map: [&lt;6 1 14 4 8|, &lt;0 2 0 3 3|]<br />
EDOs: 24, 42e, 66ce, 108bce<br />
Badness: 0.0807<br />
<br />
<!-- ws:start:WikiTextHeadingRule:58:&lt;h1&gt; --><h1 id="toc29"><a name="Niner"></a><!-- ws:end:WikiTextHeadingRule:58 -->Niner</h1>
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.<br />
<br />
Commas: 128/125, 686/675<br />
<br />
POTE generator: ~3/2 = 707.167<br />
<br />
Map: [&lt;9 0 21 11|, &lt;0 1 0 1|]<br />
Wedgie: &lt;&lt;9 0 9 -21 -11 21||<br />
EDOs: 9, 18, 27, 63c, 90c<br />
Badness: 0.0672<br />
<br />
<!-- ws:start:WikiTextHeadingRule:60:&lt;h2&gt; --><h2 id="toc30"><a name="Niner-11-limit"></a><!-- ws:end:WikiTextHeadingRule:60 -->11-limit</h2>
Commas: 56/55, 128/125, 540/539<br />
<br />
POTE generator: ~3/2 = 706.726<br />
<br />
Map: [&lt;9 0 21 11 17|, &lt;0 1 0 1 1|]<br />
EDOs: 9, 27e, 36, 63ce<br />
Badness: 0.0349<br />
<br />
<!-- ws:start:WikiTextHeadingRule:62:&lt;h2&gt; --><h2 id="toc31"><a name="Niner-13-limit"></a><!-- ws:end:WikiTextHeadingRule:62 -->13-limit</h2>
Commas: 56/55, 78/77, 91/90, 128/125<br />
<br />
POTE generator: ~3/2 = 706.889<br />
<br />
Map: [&lt;9 0 21 11 17 19|, &lt;0 1 0 1 1 1|]<br />
EDOs: 9, 27e, 36, 63ce<br />
Badness: 0.0241</body></html>

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