Hemimean family
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[[toc|flat]] The hemimean comma, 3136/3125, is the ratio between the diesis and the tritonic diesis, or jubilisma; that is, (128/125)/(50/49). This is a no-threes comma, with [[Spectrum of a temperament|spectrum]] 5/4, 7/5, 4/3, 6/5, 8/7, 7/6, 9/8, 10/9, 9/7. [[Comma]]: 3136/3125, hemimean 7 and 9 limit minimax [|1 0 0 0>, |0 1 0 0>, |6/5 0 0 2/5>, |0 0 0 1>] [[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3 Lattice basis: 28/25 length 0.5055 3/2 length 1.5849 Angle(28/25, 3/2) = 90 degrees Map to lattice: [<0 0 2 5|, <0 1 0 0|] Map: [<1 0 0 -3|, <0 1 0 0|, <0 0 2 5|] [[Generator|Generators]]: 2, 3, 56/25 [[edo|Edos]]: [[217edo|217]], [[229edo|229]], [[328edo|328]], [[359edo|359]], [[446edo|446]], [[545edo|545c]], [[675edo|675c]] Badness: 0.000160 [[Projection pair]]s: 5 3136/625 7 68841472/9765625 to 2.3.25/7 =Belobog= [[Comma|Commas]]: 3136/3125, 441/440 11 limit minimax [|1 0 0 0 0>, |27/22 6/11 -5/22 -3/11 5/22>, |24/11 -4/11 -2/11 2/11 2/11>, |27/11 -10/11 -5/11 5/11 5/11>, |24/11 -4/11 -13/11 2/11 13/11>] [[Eigenmonzo|Eigenmonzos]]: 2, 11/10, 9/7 Lattice basis: 28/25 0.3829 16/15 1.1705 Angle(28/25, 16/15) = 93.2696 Map to lattice: [<0 -2 2 5 4|, <0 -1 0 0 -2|] Map: [<1 0 0 -3 -9|, <0 1 0 0 2|, [0 0 2 5 8|] [[Generator|Generators]]: 2, 3, 56/25 [[edo|Edos]]: 12, 31, 87, 118, 130, 217, [[248edo|248]], [[378edo|378]], [[626edo|626]], [[961edo|961cd]] Badness: 0.000609 Projection pairs: 5 3136/625 7 68841472/9765625 11 1700108992512/152587890625 to 2.3.25/7 Scales: [[belobog31]] ==13-limit== Commas: 441/440, 1001/1000, 3136/3125 Map: [<1 0 0 -3 -9 15|, <0 1 0 0 2 -2|, <0 0 2 5 8 -7|] EDOs: 12, 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e Badness: 0.00111 ==Bellowblog== Commas: 196/195, 352/351, 625/624 Map: [<1 0 0 -3 -9 -4|, <0 1 0 0 2 -1|, <0 0 2 5 8 8|] EDOs: 31, 56, 87, 118, 205d, 263f, 304f, 391df, 509df Badness: 0.00126 =Siebog= [[Comma|Commas]]: 3136/3125, 540/539 11 limit minimax [|1 0 0 0 0>, |0 1 0 0 0>, |8/5 3/5 1/5 0 -1/5>, |1 3/2 1/2 0 -1/2>, |8/5 3/5 -4/5 0 4/5>] [[Eigenmonzo|Eigenmonzos]]: 2, 11/10, 4/3 Map: [<1 0 0 -3 8|, <0 1 0 0 3|, <0 0 2 5 -8|] [[Generator|Generators]]: 2, 3, 768/343 [[edo|Edos]]: 19, 31, 80, 99d, 111, [[130edo|130]], [[241edo|241]], 340ce, 371ce, 470cde, 711cde Badness: 0.000870 =Triglav= Commas: 3136/3125, 3025/3024 Map: [<1 0 2 2 1|, <0 1 2 5 2|, <0 0 -4 -10 -1|] EDOs: 31, 80, 87, 111, 118, 198, 316, 545c, 861ce Badness: 0.000819
Original HTML content:
<html><head><title>Hemimean family</title></head><body><!-- ws:start:WikiTextTocRule:10:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Belobog">Belobog</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Siebog">Siebog</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Triglav">Triglav</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> <!-- ws:end:WikiTextTocRule:16 --><br /> <br /> The hemimean comma, 3136/3125, is the ratio between the diesis and the tritonic diesis, or jubilisma; that is, (128/125)/(50/49). This is a no-threes comma, with <a class="wiki_link" href="/Spectrum%20of%20a%20temperament">spectrum</a> 5/4, 7/5, 4/3, 6/5, 8/7, 7/6, 9/8, 10/9, 9/7.<br /> <br /> <br /> <a class="wiki_link" href="/Comma">Comma</a>: 3136/3125, hemimean<br /> <br /> 7 and 9 limit minimax<br /> [|1 0 0 0>, |0 1 0 0>, |6/5 0 0 2/5>, |0 0 0 1>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 7/6, 4/3<br /> <br /> Lattice basis: 28/25 length 0.5055 3/2 length 1.5849<br /> Angle(28/25, 3/2) = 90 degrees<br /> Map to lattice: [<0 0 2 5|, <0 1 0 0|]<br /> <br /> Map: [<1 0 0 -3|, <0 1 0 0|, <0 0 2 5|] <br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 3, 56/25<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/217edo">217</a>, <a class="wiki_link" href="/229edo">229</a>, <a class="wiki_link" href="/328edo">328</a>, <a class="wiki_link" href="/359edo">359</a>, <a class="wiki_link" href="/446edo">446</a>, <a class="wiki_link" href="/545edo">545c</a>, <a class="wiki_link" href="/675edo">675c</a><br /> Badness: 0.000160<br /> <br /> <a class="wiki_link" href="/Projection%20pair">Projection pair</a>s: 5 3136/625 7 68841472/9765625 to 2.3.25/7<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Belobog"></a><!-- ws:end:WikiTextHeadingRule:0 -->Belobog</h1> <a class="wiki_link" href="/Comma">Commas</a>: 3136/3125, 441/440<br /> <br /> 11 limit minimax<br /> [|1 0 0 0 0>, |27/22 6/11 -5/22 -3/11 5/22>, <br /> |24/11 -4/11 -2/11 2/11 2/11>, <br /> |27/11 -10/11 -5/11 5/11 5/11>, <br /> |24/11 -4/11 -13/11 2/11 13/11>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 11/10, 9/7<br /> <br /> Lattice basis: 28/25 0.3829 16/15 1.1705<br /> Angle(28/25, 16/15) = 93.2696<br /> Map to lattice: [<0 -2 2 5 4|, <0 -1 0 0 -2|]<br /> <br /> Map: [<1 0 0 -3 -9|, <0 1 0 0 2|, [0 0 2 5 8|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 3, 56/25<br /> <a class="wiki_link" href="/edo">Edos</a>: 12, 31, 87, 118, 130, 217, <a class="wiki_link" href="/248edo">248</a>, <a class="wiki_link" href="/378edo">378</a>, <a class="wiki_link" href="/626edo">626</a>, <a class="wiki_link" href="/961edo">961cd</a><br /> Badness: 0.000609<br /> <br /> Projection pairs: 5 3136/625 7 68841472/9765625 11 1700108992512/152587890625 to 2.3.25/7<br /> <br /> Scales: <a class="wiki_link" href="/belobog31">belobog31</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Belobog-13-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->13-limit</h2> Commas: 441/440, 1001/1000, 3136/3125<br /> <br /> Map: [<1 0 0 -3 -9 15|, <0 1 0 0 2 -2|, <0 0 2 5 8 -7|]<br /> EDOs: 12, 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e<br /> Badness: 0.00111<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Belobog-Bellowblog"></a><!-- ws:end:WikiTextHeadingRule:4 -->Bellowblog</h2> Commas: 196/195, 352/351, 625/624<br /> <br /> Map: [<1 0 0 -3 -9 -4|, <0 1 0 0 2 -1|, <0 0 2 5 8 8|]<br /> EDOs: 31, 56, 87, 118, 205d, 263f, 304f, 391df, 509df<br /> Badness: 0.00126<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Siebog"></a><!-- ws:end:WikiTextHeadingRule:6 -->Siebog</h1> <a class="wiki_link" href="/Comma">Commas</a>: 3136/3125, 540/539<br /> <br /> 11 limit minimax<br /> [|1 0 0 0 0>, |0 1 0 0 0>, |8/5 3/5 1/5 0 -1/5>, <br /> |1 3/2 1/2 0 -1/2>, |8/5 3/5 -4/5 0 4/5>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 11/10, 4/3<br /> <br /> Map: [<1 0 0 -3 8|, <0 1 0 0 3|, <0 0 2 5 -8|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 3, 768/343<br /> <a class="wiki_link" href="/edo">Edos</a>: 19, 31, 80, 99d, 111, <a class="wiki_link" href="/130edo">130</a>, <a class="wiki_link" href="/241edo">241</a>, 340ce, 371ce, 470cde, 711cde<br /> Badness: 0.000870<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="Triglav"></a><!-- ws:end:WikiTextHeadingRule:8 -->Triglav</h1> Commas: 3136/3125, 3025/3024<br /> <br /> Map: [<1 0 2 2 1|, <0 1 2 5 2|, <0 0 -4 -10 -1|]<br /> EDOs: 31, 80, 87, 111, 118, 198, 316, 545c, 861ce<br /> Badness: 0.000819</body></html>