Augmented family
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- This revision was by author genewardsmith and made on 2011-11-20 06:03:27 UTC.
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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 ==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 [[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]] ==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]]: 445.118 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 ==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]]
Original HTML content:
<html><head><title>Augmented family</title></head><body><!-- ws:start:WikiTextTocRule:22:<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:22 --><!-- ws:start:WikiTextTocRule:23: --><div style="margin-left: 2em;"><a href="#x-Seven limit children">Seven limit children</a></div> <!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><div style="margin-left: 2em;"><a href="#x-August">August</a></div> <!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><div style="margin-left: 3em;"><a href="#x-August-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><div style="margin-left: 3em;"><a href="#x-August-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><div style="margin-left: 2em;"><a href="#x-Augene">Augene</a></div> <!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><div style="margin-left: 3em;"><a href="#x-Augene-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><div style="margin-left: 2em;"><a href="#x-Hexe">Hexe</a></div> <!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 2em;"><a href="#x-Hemiaug">Hemiaug</a></div> <!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 2em;"><a href="#x-Triforce">Triforce</a></div> <!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 3em;"><a href="#x-Triforce-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 3em;"><a href="#x-Triforce-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --></div> <!-- ws:end:WikiTextTocRule:34 -->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:<h2> --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h2> 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:<h2> --><h2 id="toc1"><a name="x-August"></a><!-- ws:end:WikiTextHeadingRule:2 -->August</h2> <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:<h3> --><h3 id="toc2"><a name="x-August-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h3> 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:<h3> --><h3 id="toc3"><a name="x-August-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h3> 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="x-Augene"></a><!-- ws:end:WikiTextHeadingRule:8 -->Augene</h2> 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:10:<h3> --><h3 id="toc5"><a name="x-Augene-11-limit"></a><!-- ws:end:WikiTextHeadingRule:10 -->11-limit</h3> 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 /> <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 /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="x-Hexe"></a><!-- ws:end:WikiTextHeadingRule:12 -->Hexe</h2> 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:14:<h2> --><h2 id="toc7"><a name="x-Hemiaug"></a><!-- ws:end:WikiTextHeadingRule:14 -->Hemiaug</h2> Commas: 128/125, 245/243<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 445.118<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:16:<h2> --><h2 id="toc8"><a name="x-Triforce"></a><!-- ws:end:WikiTextHeadingRule:16 -->Triforce</h2> 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:35:<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:35 --><br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h3> --><h3 id="toc9"><a name="x-Triforce-11-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->11-limit</h3> 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:20:<h3> --><h3 id="toc10"><a name="x-Triforce-13-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit</h3> 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></body></html>