Schismatic family
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The 5-limit parent comma for the schismatic family is the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymus comma (81/80), or alternatively put, the difference between a just major third and a just diminished fourth. Its [[monzo]] is |-15 8 1>, and flipping that yields <<1 -8 -15|| for the [[Wedgies and Multivals|wedgie]]. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768. The 5-limit version of the temperament is a [[Microtempering|microtemperament]], sometimes called helmholtz or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schisma schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. [[POTE tuning|POTE generator]]: 701.736 Map: [<1 0 15|, <0 1 -8|] EDOs: 12, 29, 41, 53, 118, 171, 289, 460, 749 ==Seven limit children== The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding |25 -14 0 -1> gives garibaldi, |-44 26 0 1> grackle, |6 -2 0 -1> schism and |-59 39 0 -1> pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2> and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3>, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3> with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4> and slices the fifth in four. ==Garibaldi== Commas: 225/224, 3125/3087 7-limit minimax tuning: 7-limit: [|1 0 0 0>, |5/3 1/15 0 -1/15>, |5/3 -8/15 0 8/15>, |5/3 -14/15 0 14/15>] [[Eigenmonzo]]s: 2, 7/6 9-limit: [|1 0 0 0>, |25/16 1/8 0 -1/16>, |5/2 -1 0 1/2>, |25/8 -7/4 0 7/8>] Eigenmonzos: 2, 9/7 [[POTE tuning|POTE generator]]: 702.085 Map: [<1 0 15 25|, <0 1 -8 -14|] Wedgie: <<1 -8 -14 -15 -25 -10|| EDOs: 12, 29, 41, 53, 94 Badness: 0.0216 ===11-limit=== Commas: 225/224, 385/384, 2200/2187 Minimax tuning: [|1 0 0 0 0>, |25/16 1/8 0 -1/16 0>, |5/2 -1 0 1/2 0>, |25/8 -7/4 0 7/8 0>, |47/16 23/8 0 -23/16 0>] Eigenmonzos: 2, 9/7 [[POTE tuning|POTE generator]]: 702.157 Map: [<1 0 15 25 -33|, <0 1 -8 -14 23|] [[edo|Edos]]: [[41edo|41]], [[53edo|53]], [[94edo|94]], [[135edo|135]] Badness: 0.0274 ====13-limit==== Commas: 225/224, 275/273, 325/325, 385/384 [[POTE tuning|POTE generator]]: 702.113 Map: [<1 0 15 25 -33 -28|, <0 1 -8 -14 23 20|] EDOs: 12, 41, 53, 94 Badness: 0.0207 ===Cassandra=== Commas: 100/99, 225/224, 245/242 [[POTE tuning|POTE generator]]: 702.321 Map: [<1 0 15 25 32|, <0 1 -8 -14 -18|] EDOs: 12, 29, 41, 422 Badness: 0.0236 ====13-limit Cassandra==== Commas: 100/99, 105/104, 196/195, 245/242 [[POTE tuning|POTE generator]]: 702.559 Map: [<1 0 15 25 32 37|, <0 1 -8 -14 -18 -21|] EDOs: 12, 29, 41, 234 Badness: 0.0207 ==Guiron== [[Comma|Commas]]: 1029/1024, 10976/10935 [[Minimax tuning]]: 7+9 limit: [|1 0 0 0>, |15/8 0 -1/8 0>, |0 0 1 0>, |65/24 0 1/24 0>] [[Eigenmonzo|Eigenmonzos]]: 2, 5/4 [[POTE tuning|POTE generator]]: 233.930 Map: [<1 1 7 3|, <0 3 -24 -1|] Wedgie: <<3 -24 -1 -45 -10 65|| EDOs: 36, 41, 77, 118 Badness: 0.0475 ===11-limit=== [[Comma|Commas]]: 385/384, 441/440, 10976/10935 [[Minimax tuning]]: [|1 0 0 0 0>, |15/8 0 -1/8 0 0>, |0 0 1 0 0>, |65/24 0 1/24 0 0>, |37/6 0 -7/6 0 0>] [[Eigenmonzo|Eigenmonzos]]: 2, 5/4 [[POTE tuning|POTE generator]]: 233.931 Map: [<1 1 7 3 -2|, <0 3 -24 -1 28|] [[edo|Edos]]: [[41edo|41]], [[77edo|77]], [[118edo|118]], [[159edo|159]], [[200edo|200]] Badness: 0.0266 ==Pontiac== Commas: 32805/32768, 4375/4374 7-limit minimax: [|1 0 0 0>, |74/47 0 -1/47 1/47>, |113/47 0 8/47 -8/47>, |113/47 0 -39/47 39/47>] Eigenmonzos: 2, 7/5 9-limit minimax: [|1 0 0 0>, |3/2 1/5 -1/10 0>, |3 -8/5 4/5 0>, |-1/2 39/5 -39/10 0>] Eigenmonzos: 2, 10/9 [[POTE tuning|POTE generator]]: 701.757 Map: [<1 0 15 -59|, <0 1 -8 39|] Wedgie: <<1 -8 39 -15 59 113|| [[edo|Edos]]: [[53edo|53]], [[118edo|118]], [[171edo|171]], [[1079edo|1079]], [[1250edo|1250]], [[1421edo|1421]] Badness: 0.0141 ==Grackle== Commas: 126/125, 32805/32768 7-limit minimax Eigenmonzos: 2, 7/6 9-limit minimax Eigenmonzos: 2, 9/7 [[POTE tuning|POTE generator]]: 701.239 Map: [<1 0 15 -44|, <0 1 -8 -26|] Wedgie: <<1 -8 -26 -15 -44 -38|| [[edo|Edos]]: [[77edo|77]], [[89edo|89]], [[101edo|101]] Badness: 0.070 ==Bischismic== Commas: 3136/3125, 32805/32768 7-limit minimax Eigenmonzos: 2, 7/6 9-limit minimax Eigenmonzos: 2, 9/7 [[POTE tuning|POTE generator]]: 701.592 Map: [<2 0 30 69|, <0 1 -8 -20|] Wedgie: <<2 -16 -40 -30 -69 -48|| [[edo|Edos]]: [[118edo|118]], [[130edo|130]], [[248edo|248]], [[378edo|378]], [[508edo|508]] Badness: 0.0547 ==Term== Commas: 32805/32768, 250047/250000 7-limit minimax Eigenmonzos: 2, 6/5 9-limit minimax Eigenmonzos: 2, 9/7 [[POTE tuning|POTE generator]]: 701.742 Map: [<3 0 45 94|, <0 1 -8 -18|] Wedgie: <<3 -24 -54 -45 -94 -58|| [[edo|Edos]]: [[171edo|171]], [[1038edo|1038]], [[1209edo|1209]] Badness: 0.0200 ==Sesquiquartififths== Commas: 2401/2400, 32805/32768 7-limit minimax Eigenmonzos: 2, 7/6 9-limit minimax Eigenmonzos: 2, 9/7 [[POTE tuning|POTE generator]]: 175.434 Map: [<1 1 7 5|, <0 4 -32 -15|] Wedgie: <<4 -32 -15 -60 -35 55|| [[edo|Edos]]: [[171edo|171]], [[643edo|643]] Badness: 0.0112
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<html><head><title>Schismatic family</title></head><body>The 5-limit parent comma for the schismatic family is the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymus comma (81/80), or alternatively put, the difference between a just major third and a just diminished fourth. Its <a class="wiki_link" href="/monzo">monzo</a> is |-15 8 1>, and flipping that yields <<1 -8 -15|| for the <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768. <br /> <br /> The 5-limit version of the temperament is a <a class="wiki_link" href="/Microtempering">microtemperament</a>, sometimes called helmholtz or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. <a class="wiki_link" href="/53edo">53edo</a> is a possible tuning for schismatic, but you need <a class="wiki_link" href="/118edo">118edo</a> if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schisma schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.736<br /> <br /> Map: [<1 0 15|, <0 1 -8|]<br /> EDOs: 12, 29, 41, 53, 118, 171, 289, 460, 749<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. Adding |25 -14 0 -1> gives garibaldi, |-44 26 0 1> grackle, |6 -2 0 -1> schism and |-59 39 0 -1> pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2> and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3>, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3> with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4> and slices the fifth in four.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-Garibaldi"></a><!-- ws:end:WikiTextHeadingRule:2 -->Garibaldi</h2> Commas: 225/224, 3125/3087<br /> <br /> 7-limit minimax tuning:<br /> 7-limit: [|1 0 0 0>, |5/3 1/15 0 -1/15>,<br /> |5/3 -8/15 0 8/15>, |5/3 -14/15 0 14/15>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 7/6<br /> <br /> 9-limit: [|1 0 0 0>, |25/16 1/8 0 -1/16>, <br /> |5/2 -1 0 1/2>, |25/8 -7/4 0 7/8>]<br /> Eigenmonzos: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 702.085<br /> <br /> Map: [<1 0 15 25|, <0 1 -8 -14|]<br /> Wedgie: <<1 -8 -14 -15 -25 -10||<br /> EDOs: 12, 29, 41, 53, 94<br /> Badness: 0.0216<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="x-Garibaldi-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h3> Commas: 225/224, 385/384, 2200/2187<br /> <br /> Minimax tuning:<br /> [|1 0 0 0 0>, |25/16 1/8 0 -1/16 0>, |5/2 -1 0 1/2 0>,<br /> |25/8 -7/4 0 7/8 0>, |47/16 23/8 0 -23/16 0>]<br /> Eigenmonzos: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 702.157<br /> <br /> Map: [<1 0 15 25 -33|, <0 1 -8 -14 23|]<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/94edo">94</a>, <a class="wiki_link" href="/135edo">135</a><br /> Badness: 0.0274<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h4> --><h4 id="toc3"><a name="x-Garibaldi-11-limit-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h4> Commas: 225/224, 275/273, 325/325, 385/384<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 702.113<br /> <br /> Map: [<1 0 15 25 -33 -28|, <0 1 -8 -14 23 20|]<br /> EDOs: 12, 41, 53, 94<br /> Badness: 0.0207<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="x-Garibaldi-Cassandra"></a><!-- ws:end:WikiTextHeadingRule:8 -->Cassandra</h3> Commas: 100/99, 225/224, 245/242<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 702.321<br /> <br /> Map: [<1 0 15 25 32|, <0 1 -8 -14 -18|]<br /> EDOs: 12, 29, 41, 422<br /> Badness: 0.0236<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h4> --><h4 id="toc5"><a name="x-Garibaldi-Cassandra-13-limit Cassandra"></a><!-- ws:end:WikiTextHeadingRule:10 -->13-limit Cassandra</h4> Commas: 100/99, 105/104, 196/195, 245/242<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 702.559<br /> <br /> Map: [<1 0 15 25 32 37|, <0 1 -8 -14 -18 -21|]<br /> EDOs: 12, 29, 41, 234<br /> Badness: 0.0207<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="x-Guiron"></a><!-- ws:end:WikiTextHeadingRule:12 -->Guiron</h2> <a class="wiki_link" href="/Comma">Commas</a>: 1029/1024, 10976/10935<br /> <br /> <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br /> 7+9 limit: [|1 0 0 0>, |15/8 0 -1/8 0>, <br /> |0 0 1 0>, |65/24 0 1/24 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 5/4<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 233.930<br /> <br /> Map: [<1 1 7 3|, <0 3 -24 -1|]<br /> Wedgie: <<3 -24 -1 -45 -10 65||<br /> EDOs: 36, 41, 77, 118<br /> Badness: 0.0475<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h3> --><h3 id="toc7"><a name="x-Guiron-11-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->11-limit</h3> <a class="wiki_link" href="/Comma">Commas</a>: 385/384, 441/440, 10976/10935<br /> <br /> <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br /> [|1 0 0 0 0>, |15/8 0 -1/8 0 0>, <br /> |0 0 1 0 0>, |65/24 0 1/24 0 0>, <br /> |37/6 0 -7/6 0 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 5/4<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 233.931<br /> <br /> Map: [<1 1 7 3 -2|, <0 3 -24 -1 28|]<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/77edo">77</a>, <a class="wiki_link" href="/118edo">118</a>, <a class="wiki_link" href="/159edo">159</a>, <a class="wiki_link" href="/200edo">200</a><br /> Badness: 0.0266<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="x-Pontiac"></a><!-- ws:end:WikiTextHeadingRule:16 -->Pontiac</h2> Commas: 32805/32768, 4375/4374<br /> <br /> 7-limit minimax:<br /> [|1 0 0 0>, |74/47 0 -1/47 1/47>, |113/47 0 8/47 -8/47>, <br /> |113/47 0 -39/47 39/47>]<br /> Eigenmonzos: 2, 7/5<br /> <br /> 9-limit minimax:<br /> [|1 0 0 0>, |3/2 1/5 -1/10 0>, <br /> |3 -8/5 4/5 0>, |-1/2 39/5 -39/10 0>]<br /> Eigenmonzos: 2, 10/9<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.757<br /> <br /> Map: [<1 0 15 -59|, <0 1 -8 39|]<br /> Wedgie: <<1 -8 39 -15 59 113||<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/118edo">118</a>, <a class="wiki_link" href="/171edo">171</a>, <a class="wiki_link" href="/1079edo">1079</a>, <a class="wiki_link" href="/1250edo">1250</a>, <a class="wiki_link" href="/1421edo">1421</a><br /> Badness: 0.0141<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="x-Grackle"></a><!-- ws:end:WikiTextHeadingRule:18 -->Grackle</h2> Commas: 126/125, 32805/32768<br /> <br /> 7-limit minimax<br /> Eigenmonzos: 2, 7/6<br /> <br /> 9-limit minimax<br /> Eigenmonzos: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.239<br /> <br /> Map: [<1 0 15 -44|, <0 1 -8 -26|]<br /> Wedgie: <<1 -8 -26 -15 -44 -38||<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/77edo">77</a>, <a class="wiki_link" href="/89edo">89</a>, <a class="wiki_link" href="/101edo">101</a><br /> Badness: 0.070<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h2> --><h2 id="toc10"><a name="x-Bischismic"></a><!-- ws:end:WikiTextHeadingRule:20 -->Bischismic</h2> Commas: 3136/3125, 32805/32768<br /> <br /> 7-limit minimax<br /> Eigenmonzos: 2, 7/6<br /> <br /> 9-limit minimax<br /> Eigenmonzos: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.592<br /> <br /> Map: [<2 0 30 69|, <0 1 -8 -20|]<br /> Wedgie: <<2 -16 -40 -30 -69 -48||<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/118edo">118</a>, <a class="wiki_link" href="/130edo">130</a>, <a class="wiki_link" href="/248edo">248</a>, <a class="wiki_link" href="/378edo">378</a>, <a class="wiki_link" href="/508edo">508</a><br /> Badness: 0.0547<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h2> --><h2 id="toc11"><a name="x-Term"></a><!-- ws:end:WikiTextHeadingRule:22 -->Term</h2> Commas: 32805/32768, 250047/250000<br /> <br /> 7-limit minimax<br /> Eigenmonzos: 2, 6/5<br /> <br /> 9-limit minimax<br /> Eigenmonzos: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.742<br /> <br /> Map: [<3 0 45 94|, <0 1 -8 -18|]<br /> Wedgie: <<3 -24 -54 -45 -94 -58||<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/171edo">171</a>, <a class="wiki_link" href="/1038edo">1038</a>, <a class="wiki_link" href="/1209edo">1209</a><br /> Badness: 0.0200<br /> <br /> <!-- ws:start:WikiTextHeadingRule:24:<h2> --><h2 id="toc12"><a name="x-Sesquiquartififths"></a><!-- ws:end:WikiTextHeadingRule:24 -->Sesquiquartififths</h2> Commas: 2401/2400, 32805/32768<br /> <br /> 7-limit minimax<br /> Eigenmonzos: 2, 7/6<br /> <br /> 9-limit minimax<br /> Eigenmonzos: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.434<br /> <br /> Map: [<1 1 7 5|, <0 4 -32 -15|]<br /> Wedgie: <<4 -32 -15 -60 -35 55||<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/171edo">171</a>, <a class="wiki_link" href="/643edo">643</a><br /> Badness: 0.0112</body></html>