Augmented family
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- This revision was by author genewardsmith and made on 2011-03-05 12:52:17 UTC.
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Original Wikitext content:
The 5-limit parent comma for the augmented family is 128/125, the 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== Commas: 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|| EDOs: [[9edo|9]], [[12edo|12]], [[69edo|69]], [[81edo|81]] 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, 21, 33, 45, 78 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, 21, 54 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]], [[66edo|66]] Badness: 0.0248 ==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]], [[54edo|54]] 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]] 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, 15, 24, 39, 102 Badness: 0.0202 ===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, 63, 102 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, 87 Badness: 0.0202
Original HTML content:
<html><head><title>Augmented family</title></head><body>The 5-limit parent comma for the augmented family is 128/125, the diesis. 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 5-limit 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 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.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-August"></a><!-- ws:end:WikiTextHeadingRule:2 -->August</h2> Commas: 36/35, 128/125<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.011<br /> <br /> Map: [<3 0 7 -1|, <0 1 0 2|]<br /> Wedgie: <<3 0 6 -7 1 14||<br /> EDOs: <a class="wiki_link" href="/9edo">9</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/69edo">69</a>, <a class="wiki_link" href="/81edo">81</a><br /> Badness: 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, 21, 33, 45, 78<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, 21, 54<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="/66edo">66</a><br /> Badness: 0.0248<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="x-Hexe"></a><!-- ws:end:WikiTextHeadingRule:10 -->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="/54edo">54</a><br /> Badness: 0.0577<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="x-Hemiaug"></a><!-- ws:end:WikiTextHeadingRule:12 -->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><br /> Badness: 0.0705<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="x-Triforce"></a><!-- ws:end:WikiTextHeadingRule:14 -->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, 15, 24, 39, 102<br /> Badness: 0.0202<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h3> --><h3 id="toc8"><a name="x-Triforce-11-limit"></a><!-- ws:end:WikiTextHeadingRule:16 -->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, 63, 102<br /> Badness: 0.0262<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h3> --><h3 id="toc9"><a name="x-Triforce-13-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->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, 87<br /> Badness: 0.0202</body></html>