Augmented family
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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 =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
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<html><head><title>Augmented family</title></head><body><!-- ws:start:WikiTextTocRule:58:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><div style="margin-left: 1em;"><a href="#Seven limit children">Seven limit children</a></div> <!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><div style="margin-left: 1em;"><a href="#August">August</a></div> <!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><div style="margin-left: 2em;"><a href="#August-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --><div style="margin-left: 2em;"><a href="#August-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><div style="margin-left: 2em;"><a href="#August-Augustus">Augustus</a></div> <!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><div style="margin-left: 1em;"><a href="#Augene">Augene</a></div> <!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><div style="margin-left: 2em;"><a href="#Augene-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><div style="margin-left: 3em;"><a href="#Augene-11-limit-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><div style="margin-left: 3em;"><a href="#Augene-11-limit-Ogene">Ogene</a></div> <!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><div style="margin-left: 2em;"><a href="#Augene-Eugene">Eugene</a></div> <!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><div style="margin-left: 1em;"><a href="#Inflated">Inflated</a></div> <!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><div style="margin-left: 2em;"><a href="#Inflated-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><div style="margin-left: 1em;"><a href="#Deflated">Deflated</a></div> <!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><div style="margin-left: 1em;"><a href="#Hexe">Hexe</a></div> <!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><div style="margin-left: 1em;"><a href="#Hemiaug">Hemiaug</a></div> <!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><div style="margin-left: 2em;"><a href="#Hemiaug-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><div style="margin-left: 2em;"><a href="#Hemiaug-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --><div style="margin-left: 1em;"><a href="#Triforce">Triforce</a></div> <!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --><div style="margin-left: 2em;"><a href="#Triforce-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><div style="margin-left: 2em;"><a href="#Triforce-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --><div style="margin-left: 2em;"><a href="#Triforce-Semitriforce">Semitriforce</a></div> <!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --><div style="margin-left: 1em;"><a href="#Oodako">Oodako</a></div> <!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><div style="margin-left: 2em;"><a href="#Oodako-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><div style="margin-left: 2em;"><a href="#Oodako-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><div style="margin-left: 1em;"><a href="#Hemisemiaug">Hemisemiaug</a></div> <!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><div style="margin-left: 2em;"><a href="#Hemisemiaug-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><div style="margin-left: 1em;"><a href="#Niner">Niner</a></div> <!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><div style="margin-left: 2em;"><a href="#Niner-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><div style="margin-left: 2em;"><a href="#Niner-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --></div> <!-- ws:end:WikiTextTocRule:88 --><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>, and flipping that yields <<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:<h1> --><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:<h1> --><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>: [<3 0 7 -1|, <0 1 0 2|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<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:<h2> --><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: [<3 0 7 -1 1|, <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:<h2> --><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: [<3 0 7 -1 1 -3|, <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:<h2> --><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: [<3 0 7 -1 1 11|, <0 1 0 2 2 0|]<br /> EDOs: 9, 12, 21f<br /> Badness: 0.0222<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h1> --><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: [<3 0 7 18|, <0 1 0 -2|]<br /> Wedgie: <<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:<h2> --><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: [<3 0 7 18 20|, <0 1 0 -2 -2|]<br /> EDOs: 12, 15, 27e<br /> Badness: 0.0196<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h3> --><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: [<3 0 7 18 20 16|, <0 1 0 -2 -2 -1|]<br /> EDOs: 12f, 15, 27ef<br /> Badness: 0.0208<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h3> --><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: [<3 0 7 18 20 -8|, <0 1 0 -2 -2 4|]<br /> EDOs: 12, 15, 27e, 69bcef<br /> Badness: 0.0229<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><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: [<3 0 7 18 -4|, <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:<h1> --><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: [<3 0 7 -6|, <0 1 0 3|]<br /> Wedgie: <<3 0 9 -7 6 21||<br /> EDOs: 15, 48bc, 63bc, 78bc<br /> Badness: 0.0547<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h2> --><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: [<3 0 7 -6|, <0 1 0 3|]<br /> EDOs: 15, 48bce, 63bce, 78bce<br /> Badness: 0.0312<br /> <br /> <!-- ws:start:WikiTextHeadingRule:24:<h1> --><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: [<3 0 7 13|, <0 1 0 -1|]<br /> Wedgie: <<3 0 -3 -7 -13 -7||<br /> EDOs: 3, 9, 18bd, 21d, 30bd<br /> Badness: 0.0591<br /> <br /> <!-- ws:start:WikiTextHeadingRule:26:<h1> --><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: [<6 0 14 17|, <0 1 0 0|]<br /> Wedgie: <<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:<h1> --><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: [<3 1 7 -1|, <0 2 0 5|]<br /> Wedgie: <<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:<h2> --><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: [<3 1 7 -1 1|, <0 2 0 5 5|]<br /> EDOs: 24, 27e, 51ce, 78ce<br /> Badness: 0.0382<br /> <br /> <!-- ws:start:WikiTextHeadingRule:32:<h2> --><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: [<3 1 7 -1 1 13|, <0 2 0 5 5 -1|]<br /> EDOs: 24, 27e, 51ce, 78ce<br /> Badness: 0.0302<br /> <br /> <!-- ws:start:WikiTextHeadingRule:34:<h1> --><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: [<3 0 7 6|, <0 2 0 1|]<br /> Wedgie: <<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:89:<img src="/file/view/triforce9.jpg/271381934/triforce9.jpg" alt="" title="" /> --><img src="/file/view/triforce9.jpg/271381934/triforce9.jpg" alt="triforce9.jpg" title="triforce9.jpg" /><!-- ws:end:WikiTextLocalImageRule:89 --><br /> <br /> <!-- ws:start:WikiTextHeadingRule:36:<h2> --><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: [<3 0 7 6 8|, <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:<h2> --><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: [<3 0 7 6 8 4|, <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:<h2> --><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: [<6 0 14 12 16 27|, <0 2 0 1 1 -1|]<br /> EDOs: 6, 24, 54cd, 78cd, 102cdf<br /> Badness: 0.0592<br /> <br /> <!-- ws:start:WikiTextHeadingRule:42:<h1> --><h1 id="toc21"><a name="Oodako"></a><!-- ws:end:WikiTextHeadingRule:42 -->Oodako</h1> Commas: 128/125, 2401/2400<br /> <br /> POTE generator: ~8/7 = 223.3540873874507317<br /> <br /> Map: [<3 3 7 8|, <0 4 0 1|]<br /> Wedgie: <<12 0 3 -28 -29 7||<br /> EDOs: 6, 21, 27, 75c, 102cd, 129bcd<br /> Badness: 0.1132<br /> <br /> <!-- ws:start:WikiTextHeadingRule:44:<h2> --><h2 id="toc22"><a name="Oodako-11-limit"></a><!-- ws:end:WikiTextHeadingRule:44 -->11-limit</h2> Commas: 56/55, 128/125, 3773/3750<br /> <br /> POTE generator: ~8/7 = 223.019<br /> <br /> Map: [<3 3 7 8 10|, <0 4 0 1 1|]<br /> EDOs: 6, 21, 27e<br /> Badness: 0.0592<br /> <br /> <!-- ws:start:WikiTextHeadingRule:46:<h2> --><h2 id="toc23"><a name="Oodako-13-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->13-limit</h2> Commas: 56/55, 78/77, 128/125, 507/500<br /> <br /> POTE generator: ~8/7 = 223.449<br /> <br /> Map: [<3 3 7 8 10 12|, <0 4 0 1 1 -2|]<br /> EDOs: 6, 21, 27e<br /> Badness: 0.0423<br /> <br /> <!-- ws:start:WikiTextHeadingRule:48:<h1> --><h1 id="toc24"><a name="Hemisemiaug"></a><!-- ws:end:WikiTextHeadingRule:48 -->Hemisemiaug</h1> Commas: 128/125, 12005/11664<br /> <br /> POTE generator: ~15/14 = 144.515<br /> <br /> Map: [<6 1 14 4|, <0 2 0 3|]<br /> Wedgie: <<12 0 18 -28 -5 42||<br /> Edos: 18, 24, 42, 66c, 108bc<br /> Badness: 0.2110<br /> <br /> <!-- ws:start:WikiTextHeadingRule:50:<h2> --><h2 id="toc25"><a name="Hemisemiaug-11-limit"></a><!-- ws:end:WikiTextHeadingRule:50 -->11-limit</h2> Commas: 56/55, 128/125, 3773/3645<br /> <br /> POTE generator: ~15/14 = 144.780<br /> <br /> Map: [<6 1 14 4 8|, <0 2 0 3 3|]<br /> EDOs: 24, 42e, 66ce, 108bce<br /> Badness: 0.0807<br /> <br /> <!-- ws:start:WikiTextHeadingRule:52:<h1> --><h1 id="toc26"><a name="Niner"></a><!-- ws:end:WikiTextHeadingRule:52 -->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: [<9 0 21 11|, <0 1 0 1|]<br /> Wedgie: <<9 0 9 -21 -11 21||<br /> EDOs: 9, 18, 27, 63c, 90c<br /> Badness: 0.0672<br /> <br /> <!-- ws:start:WikiTextHeadingRule:54:<h2> --><h2 id="toc27"><a name="Niner-11-limit"></a><!-- ws:end:WikiTextHeadingRule:54 -->11-limit</h2> Commas: 56/55, 128/125, 540/539<br /> <br /> POTE generator: ~3/2 = 706.726<br /> <br /> Map: [<9 0 21 11 17|, <0 1 0 1 1|]<br /> EDOs: 9, 27e, 36, 63ce<br /> Badness: 0.0349<br /> <br /> <!-- ws:start:WikiTextHeadingRule:56:<h2> --><h2 id="toc28"><a name="Niner-13-limit"></a><!-- ws:end:WikiTextHeadingRule:56 -->13-limit</h2> Commas: 56/55, 78/77, 91/90, 128/125<br /> <br /> POTE generator: ~3/2 = 706.889<br /> <br /> Map: [<9 0 21 11 17 19|, <0 1 0 1 1 1|]<br /> EDOs: 9, 27e, 36, 63ce<br /> Badness: 0.0241</body></html>