Kleismic family
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The 5-limit parent comma for the kleismic family is 15625/15552, the kleisma. Its monzo is |-6 -5 6>, and flipping that yields <<6 5 -6|| for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)^5 = 5/2 * 15625/15552. 14/53 is about perfect as a generator, though 9/34 also makes sense and using [[19edo]] is possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]]. [[POTE tuning|POTE generator]]: 317.007 Map: [<1 0 1|, <0 6 5|] ==Seven limit children== The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives keemun, 4375/4374, the ragisma, gives catakleismic, 5120/5103, hemifamity, gives countercata, 6144/6125, the porwell comma, gives hemikleismic, 245/243, sensamagic, gives clyde, 1029/1024, the gamelisma, gives tritikleismic, and 2401/2400, the breedsma, gives quadritikleismic. Keemun, catakleismic and countercata all have octave period and use the minor third as a generator; catakleismic and countercata define the 7/4 more complexly but more accurately than keemun. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3 octave period with minor third generator, and quadritikleismic a 1/4 octave period with the minor third generator. ==Keemun== [[Comma|Commas]]: 49/48, 126/125 [[POTE tuning|POTE generator]]: 316.473 Map: [<1 0 1 2|, <0 6 5 3|] Wedgie: <<6 5 3 -6 -12 -7|| EDOs: [[15edo|15]], [[19edo|19]], [[91edo|91]] Badness: 0.0274 ==Catakleismic== [[Comma|Commas]]: 225/224, 4375/4374 [[POTE tuning|POTE generator]]: 316.732 Map: [<1 0 1 -3|, <0 6 5 22|] Wedgie: <<6 5 22 -6 18 37|| EDOs: [[19edo|19]], [[53edo|15]], [[72edo|72]], [[197edo|197]], [[269edo|269]] Badness: 0.0215 ===11-limit=== [[Comma|Commas]]: 225/224, 385/384, 4375/4374 [[POTE tuning|POTE generator]]: 316.719 Map: [<1 0 1 -3 9|, <0 6 5 22 -21|] EDOs: [[15edo|15]], [[19edo|19]], [[53edo|53]], [[72edo|72]], [[269edo|269]], [[341edo|341]] Badness: 0.0218 ===13-limit=== [[Comma|Commas]]: 169/168, 225/224, 325/324, 540/539 [[POTE tuning|POTE generator]]: 316.738 Map: [<1 0 1 -3 9 0|, <0 6 5 22 -21 14|] EDOs: [[15edo|15]], [[19edo|19]], [[53edo|53]], [[72edo|72]], [[197edo|197]], [[269edo|269]], [[466edo|466]] Badness: 0.0169 ==Countercata== [[Comma|Commas]]: 15625/15552, 5120/5103 [[POTE tuning|POTE generator]]: 317.121 Map: [<1 0 1 11|, <0 6 5 -31|] Wedgie: <<6 5 -31 -6 -66 -86|| EDOs: 15, 19, 34, 53, 87, 140, 333, 473, 806 Badness: 0.0521 ===11-limit=== Commas: 385/384, 2200/2187, 3388/3375 POTE generaor: ~6/5 = 317.162 Map: [<1 0 1 11 -5|, <0 6 5 -31 32|] EDOs: 34, 53, 87, 140, 227 Badness: 0.0398 ==Hemikleismic== Commas: 4000/3969, 6144/6125 [[POTE tuning|POTE generator]]: 158.649 Map: [<1 0 1 4|, <0 12 10 -9|] EDOs: 53, 121 ==Clyde== [[Comma|Commas]]: 245/243, 3136/3125 7 and 9 limit minimax [|1 0 0 0>, |6/25 0 0 12/25>, |6/5 0 0 2/5>, |0 0 0 1>] [[Eigenmonzo|Eigenmonzos]]: 2, 7 [[POTE tuning|POTE generator]]: 441.335 Algebraic generator: real root of 5x^3-6x-3, the Poussami generator. Approximately 441.309 [[Cent|cents]]. Associated recurrence relationship quickly converges. Map: [<1 6 6 12|, <0 -12 -10 -25|] [[Generator|Generators]]: 2, 9/7 [[Edo|Edos]]: [[19edo|19]], [[49edo|49]], [[68edo|68]], [[87edo|87]], [[155edo|155]] ==Tritikleismic== Commas: 15625/15552, 1029/1024 [[POTE tuning|POTE generator]]: 316.872 Map: [<3 0 3 10|, <0 6 5 -2|] Wedgie: <<18 15 -6 -18 -60 -56|| EDOs: 12, 15, 57, 72, 159, 231 Badness: 0.0563 ===11-limit=== Commas: 385/384, 441/440, 4000/3993 [[POTE tuning|POTE generator]]: 316.881 Map: [<3 0 3 10 8|, <0 6 5 -2 3|] EDOs: 12, 15, 57, 72, 159, 231 Badness: 0.0193 ===13-limit=== Commas: 325/324, 364/363, 441/440, 625/624 [[POTE tuning|POTE generator]]: 316.959 Map: [<3 0 3 10 8 0|, <0 6 5 -2 3 14|] EDOs: 12, 15, 72, 87, 159, 867, 1026 Badness: 0.0157 ==Quadritikleismic== Commas: 15625/15552, 2401/2400 [[POTE tuning|POTE generator]]: 316.9999 Map: [<4 0 4 7|, <0 6 5 4|] Wedgie: <<24 20 16 -24 -42 -19|| EDOs: [[68edo|68]], [[72edo|72]], [[140edo|140]], [[212edo|212]], [[1200edo|1200]] Badness: 0.0392 ===11-limit=== Commas: 385/384, 1375/1372, 6250/6237 [[POTE tuning|POTE generator]]: 316.925 Map: [<4 0 4 7 17|, <0 6 5 4 -3|] EDOs: [[68edo|68]], [[72edo|72]], [[140edo|140]], [[212edo|212]], [[284edo|284]], [[496edo|496]], [[780edo|780]] Badness: 0.0234 ===13-limit=== Commas: 325/324, 385/384, 625/624, 1573/1568 [[POTE tuning|POTE generator]]: 316.989 Map: [<4 0 4 7 17 0|, <0 6 5 4 -3 14|] EDOs: 68, 72, 140, 212 Badness: 0.0187
Original HTML content:
<html><head><title>Kleismic family</title></head><body>The 5-limit parent comma for the kleismic family is 15625/15552, the kleisma. Its monzo is |-6 -5 6>, and flipping that yields <<6 5 -6|| for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)^5 = 5/2 * 15625/15552. 14/53 is about perfect as a generator, though 9/34 also makes sense and using <a class="wiki_link" href="/19edo">19edo</a> is possible. Other tunings include <a class="wiki_link" href="/72edo">72edo</a>, <a class="wiki_link" href="/87edo">87edo</a> and <a class="wiki_link" href="/140edo">140edo</a>.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 317.007<br /> <br /> Map: [<1 0 1|, <0 6 5|]<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. 875/864, the keemic comma, gives keemun, 4375/4374, the ragisma, gives catakleismic, 5120/5103, hemifamity, gives countercata, 6144/6125, the porwell comma, gives hemikleismic, 245/243, sensamagic, gives clyde, 1029/1024, the gamelisma, gives tritikleismic, and 2401/2400, the breedsma, gives quadritikleismic. Keemun, catakleismic and countercata all have octave period and use the minor third as a generator; catakleismic and countercata define the 7/4 more complexly but more accurately than keemun. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3 octave period with minor third generator, and quadritikleismic a 1/4 octave period with the minor third generator.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-Keemun"></a><!-- ws:end:WikiTextHeadingRule:2 -->Keemun</h2> <a class="wiki_link" href="/Comma">Commas</a>: 49/48, 126/125<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.473<br /> <br /> Map: [<1 0 1 2|, <0 6 5 3|]<br /> Wedgie: <<6 5 3 -6 -12 -7||<br /> EDOs: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/91edo">91</a><br /> Badness: 0.0274<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x-Catakleismic"></a><!-- ws:end:WikiTextHeadingRule:4 -->Catakleismic</h2> <a class="wiki_link" href="/Comma">Commas</a>: 225/224, 4375/4374<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.732<br /> <br /> Map: [<1 0 1 -3|, <0 6 5 22|]<br /> Wedgie: <<6 5 22 -6 18 37||<br /> EDOs: <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/53edo">15</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/197edo">197</a>, <a class="wiki_link" href="/269edo">269</a><br /> Badness: 0.0215<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="x-Catakleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h3> <a class="wiki_link" href="/Comma">Commas</a>: 225/224, 385/384, 4375/4374<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.719<br /> <br /> Map: [<1 0 1 -3 9|, <0 6 5 22 -21|]<br /> EDOs: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/269edo">269</a>, <a class="wiki_link" href="/341edo">341</a><br /> Badness: 0.0218<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="x-Catakleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h3> <a class="wiki_link" href="/Comma">Commas</a>: 169/168, 225/224, 325/324, 540/539<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.738<br /> <br /> Map: [<1 0 1 -3 9 0|, <0 6 5 22 -21 14|]<br /> EDOs: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/197edo">197</a>, <a class="wiki_link" href="/269edo">269</a>, <a class="wiki_link" href="/466edo">466</a><br /> Badness: 0.0169<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="x-Countercata"></a><!-- ws:end:WikiTextHeadingRule:10 -->Countercata</h2> <a class="wiki_link" href="/Comma">Commas</a>: 15625/15552, 5120/5103<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 317.121<br /> <br /> Map: [<1 0 1 11|, <0 6 5 -31|]<br /> Wedgie: <<6 5 -31 -6 -66 -86||<br /> EDOs: 15, 19, 34, 53, 87, 140, 333, 473, 806<br /> Badness: 0.0521<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h3> --><h3 id="toc6"><a name="x-Countercata-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->11-limit</h3> Commas: 385/384, 2200/2187, 3388/3375<br /> <br /> POTE generaor: ~6/5 = 317.162<br /> <br /> Map: [<1 0 1 11 -5|, <0 6 5 -31 32|]<br /> EDOs: 34, 53, 87, 140, 227<br /> Badness: 0.0398<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="x-Hemikleismic"></a><!-- ws:end:WikiTextHeadingRule:14 -->Hemikleismic</h2> Commas: 4000/3969, 6144/6125<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 158.649<br /> <br /> Map: [<1 0 1 4|, <0 12 10 -9|]<br /> EDOs: 53, 121<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="x-Clyde"></a><!-- ws:end:WikiTextHeadingRule:16 -->Clyde</h2> <a class="wiki_link" href="/Comma">Commas</a>: 245/243, 3136/3125<br /> <br /> 7 and 9 limit minimax<br /> [|1 0 0 0>, |6/25 0 0 12/25>, |6/5 0 0 2/5>, |0 0 0 1>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 441.335<br /> <br /> Algebraic generator: real root of 5x^3-6x-3, the Poussami generator. Approximately 441.309 <a class="wiki_link" href="/Cent">cents</a>. Associated recurrence relationship quickly converges.<br /> <br /> Map: [<1 6 6 12|, <0 -12 -10 -25|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 9/7<br /> <a class="wiki_link" href="/Edo">Edos</a>: <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/49edo">49</a>, <a class="wiki_link" href="/68edo">68</a>, <a class="wiki_link" href="/87edo">87</a>, <a class="wiki_link" href="/155edo">155</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="x-Tritikleismic"></a><!-- ws:end:WikiTextHeadingRule:18 -->Tritikleismic</h2> Commas: 15625/15552, 1029/1024<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.872<br /> <br /> Map: [<3 0 3 10|, <0 6 5 -2|]<br /> Wedgie: <<18 15 -6 -18 -60 -56||<br /> EDOs: 12, 15, 57, 72, 159, 231<br /> Badness: 0.0563<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h3> --><h3 id="toc10"><a name="x-Tritikleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->11-limit</h3> Commas: 385/384, 441/440, 4000/3993<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.881<br /> <br /> Map: [<3 0 3 10 8|, <0 6 5 -2 3|]<br /> EDOs: 12, 15, 57, 72, 159, 231<br /> Badness: 0.0193<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h3> --><h3 id="toc11"><a name="x-Tritikleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->13-limit</h3> Commas: 325/324, 364/363, 441/440, 625/624<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.959<br /> <br /> Map: [<3 0 3 10 8 0|, <0 6 5 -2 3 14|]<br /> EDOs: 12, 15, 72, 87, 159, 867, 1026<br /> Badness: 0.0157<br /> <br /> <!-- ws:start:WikiTextHeadingRule:24:<h2> --><h2 id="toc12"><a name="x-Quadritikleismic"></a><!-- ws:end:WikiTextHeadingRule:24 -->Quadritikleismic</h2> Commas: 15625/15552, 2401/2400<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.9999<br /> <br /> Map: [<4 0 4 7|, <0 6 5 4|]<br /> Wedgie: <<24 20 16 -24 -42 -19||<br /> EDOs: <a class="wiki_link" href="/68edo">68</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/140edo">140</a>, <a class="wiki_link" href="/212edo">212</a>, <a class="wiki_link" href="/1200edo">1200</a><br /> Badness: 0.0392<br /> <br /> <!-- ws:start:WikiTextHeadingRule:26:<h3> --><h3 id="toc13"><a name="x-Quadritikleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->11-limit</h3> Commas: 385/384, 1375/1372, 6250/6237<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.925<br /> <br /> Map: [<4 0 4 7 17|, <0 6 5 4 -3|]<br /> EDOs: <a class="wiki_link" href="/68edo">68</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/140edo">140</a>, <a class="wiki_link" href="/212edo">212</a>, <a class="wiki_link" href="/284edo">284</a>, <a class="wiki_link" href="/496edo">496</a>, <a class="wiki_link" href="/780edo">780</a><br /> Badness: 0.0234<br /> <br /> <!-- ws:start:WikiTextHeadingRule:28:<h3> --><h3 id="toc14"><a name="x-Quadritikleismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->13-limit</h3> Commas: 325/324, 385/384, 625/624, 1573/1568<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.989<br /> <br /> Map: [<4 0 4 7 17 0|, <0 6 5 4 -3 14|]<br /> EDOs: 68, 72, 140, 212<br /> Badness: 0.0187</body></html>