Semicomma family
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[[toc]] =Orson= The 5-limit parent comma for the **semicomma family** is the semicomma, 2109375/2097152 = |-21 3 7>. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor thirds. **Orson**, the [[5-limit]] temperament tempering it out, has a [[generator]] of 75/64, which is sharper than 7/6 by 225/224 when untempered, and less sharp than that in any good orson tempering, for example [[53edo]] or [[84edo]]. These give tunings to the generator which are sharp of 7/6 by less than five [[cent]]s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell. Comma: 2109375/2097152 [[Tuning Ranges of Regular Temperaments|valid range]]: [257.143, 276.923] (14b to 13) nice range: [271.229, 271.708] strict range: [271.229, 271.708] [[POTE tuning|POTE generator]]: ~75/64 = 271.627 Map: [<1 0 3|, <0 7 -3|] EDOs: 22, 31, 53, 190, 243, 296, 645c Badness: 0.0408 ==Seven limit children== The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding 65536/64625 leads to orwell, but we could also add 1029/1024, leading to the 31&159 temperament with wedgie <<21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&243 temperament with wedgie <<28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&243 temperament with wedgie <<7 -3 61 -21 77 150||. =Orwell= Main article: [[Orwell]] So called because 19\84 (as a [[fraction of the octave]]) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with [[22edo|22]], [[31edo|31]], [[53edo|53]] and [[84edo|84]] equal, and may be described as the 22&31 temperament, or <<7 -3 8 -21 -7 27||. It's a good system in the [[7-limit]] and naturally extends into the [[11-limit]]. [[84edo]], with the 19\84 generator, provides a good tuning for the 5, 7 and 11 limits, but it does use its second-best 11. However, the 19\84 generator is remarkably close to the 11-limit [[POTE tuning]], as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. [[53edo]] might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out 2430/2401, the nuwell comma, 1728/1715, the orwellisma, 225/224, the septimal kleisma, and 6144/6125, the porwell comma. The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell. Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has [[Retuning 12edo to Orwell9|considerable harmonic resources]] despite its absence of 5-limit triads. The 13 note MOS has those, and of course the 22 and 31 note MOS are very well supplied with everything. [[Comma|Commas]]: 225/224, 1728/1715 7-limit [|1 0 0 0>, |14/11 0 -7/11 7/11>, |27/11 0 3/11 -3/11>, |27/11 0 -8/11 8/11>] [[Fractional monzos|Eigenmonzos]]: 2, 7/5 9-limit [|1 0 0 0>, |21/17 14/17 -7/17 0>, |42/17 -6/17 3/17 0>, |41/17 16/17 -8/17 0>] [[Eigenmonzo|Eigenmonzos]]: 2, 10/9 valid range: [266.667, 272.727] (9 to 22) nice range: [266.871, 271.708] strict range: [266.871, 271.708] [[POTE tuning|POTE generator]]: ~7/6 = 271.509 Algebraic generators: Sabra3, the real root of 12x^3-7x-48. Map: [<1 0 3 1|, <0 7 -3 8|] Wedgie: <<7 -3 8 -21 -7 27|| EDOs: 22, 31, 53, 84, 137, 221d, 358d Badness: 0.0207 ==11-limit== [[Comma|Commas]]: 99/98, 121/120, 176/175 [[Minimax tuning]] [|1 0 0 0 0>, |14/11 0 -7/11 7/11 0>, |27/11 0 3/11 -3/11 0>, |27/11 0 -8/11 8/11 0>, |37/11 0 -2/11 2/11 0>] [[Eigenmonzo|Eigenmonzos]]: 2, 7/5 valid range: [270.968, 272.727] (31 to 22) nice range: [266.871, 275.659] strict range: [270.968, 272.727] [[POTE tuning|POTE generator]]: ~7/6 = 271.426 Map: [<1 0 3 1 3|, <0 7 -3 8 2|] [[edo|Edos]]: [[22edo|22]], [[31edo|31]], [[53edo|53]], [[84edo|84e]] Badness: 0.0152 ==13-limit== Commas: 99/98, 121/120, 176/175, 275/273 valid range: [270.968, 271.698] (31 to 53) nice range: [266.871, 275.659] strict range: [270.968, 271.698] [[POTE tuning|POTE generator]]: ~7/6 = 271.546 Map: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|] EDOs: 22, 31, 53, 84e, 137e Badness: 0.0197 [[Orwell#Music|Music in Orwell]] ==Blair== Commas: 65/64, 78/77, 91/90, 99/98 POTE generator: ~7/6 = 271.301 Map: [<1 0 3 1 3 3|, <0 7 -3 8 2 3|] EDOs: 9, 22, 31f Badness: 0.0231 ==Newspeak== Commas: 225/224, 441/440, 1728/1715 POTE tuning: ~7/6 = 271.288 Map: [<1 0 3 1 -4|, <0 7 -3 8 33|] EDOs: 31, 84, 115, 376b, 491bd, 606bde Badness: 0.0314 ==Winston== Commas: 66/65, 99/98, 105/104, 121/120 [[POTE tuning|POTE generator]]: ~7/6 = 271.088 Map: [<1 0 3 1 3 1|, <0 7 -3 8 2 12|] EDOs: 22f, 31 Badness: 0.0199 =Doublethink= Commas: 99/98, 121/120, 169/168, 176/175 POTE tuning: ~13/12 = 135.723 Map: [<1 0 3 1 3 2|, <0 14 -6 16 4 15|] EDOs: 9, 35, 44, 53, 62, 115ef, 168ef Badness: 0.0271 =Borwell= Commas: 225/224, 243/242, 1728/1715 POTE generator: ~55/36 = 735.752 Map: [<1 7 0 9 17|, <0 -14 6 -16 -35|] EDOs: 31, 106, 137, 442bd Badness: 0.0384 =Triwell= Commas: 1029/1024, 235298/234375 POTE generator: ~448/375 = 309.472 Map: [<1 7 0 1|, <0 -21 9 7]] Wedgie: <<21 -9 -7 -63 -70 9|| EDOs: 31, 97, 128, 159, 190 Badness: 0.0806 ==11-limit== Commas: 385/384, 441/440, 456533/455625 POTE generator: ~448/375 = 309.471 Map: [<1 7 0 1 13|, <0 -21 9 7 -37]] EDOs: 31, 97, 128, 159, 190 Badness: 0.0298
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<html><head><title>Semicomma family</title></head><body><!-- ws:start:WikiTextTocRule:24:<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:24 --><!-- ws:start:WikiTextTocRule:25: --><div style="margin-left: 1em;"><a href="#Orson">Orson</a></div> <!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><div style="margin-left: 2em;"><a href="#Orson-Seven limit children">Seven limit children</a></div> <!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><div style="margin-left: 1em;"><a href="#Orwell">Orwell</a></div> <!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><div style="margin-left: 2em;"><a href="#Orwell-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><div style="margin-left: 2em;"><a href="#Orwell-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 2em;"><a href="#Orwell-Blair">Blair</a></div> <!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 2em;"><a href="#Orwell-Newspeak">Newspeak</a></div> <!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 2em;"><a href="#Orwell-Winston">Winston</a></div> <!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 1em;"><a href="#Doublethink">Doublethink</a></div> <!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 1em;"><a href="#Borwell">Borwell</a></div> <!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 1em;"><a href="#Triwell">Triwell</a></div> <!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 2em;"><a href="#Triwell-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --></div> <!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Orson"></a><!-- ws:end:WikiTextHeadingRule:0 -->Orson</h1> The 5-limit parent comma for the <strong>semicomma family</strong> is the semicomma, 2109375/2097152 = |-21 3 7>. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor thirds. <strong>Orson</strong>, the <a class="wiki_link" href="/5-limit">5-limit</a> temperament tempering it out, has a <a class="wiki_link" href="/generator">generator</a> of 75/64, which is sharper than 7/6 by 225/224 when untempered, and less sharp than that in any good orson tempering, for example <a class="wiki_link" href="/53edo">53edo</a> or <a class="wiki_link" href="/84edo">84edo</a>. These give tunings to the generator which are sharp of 7/6 by less than five <a class="wiki_link" href="/cent">cent</a>s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.<br /> <br /> Comma: 2109375/2097152<br /> <br /> <a class="wiki_link" href="/Tuning%20Ranges%20of%20Regular%20Temperaments">valid range</a>: [257.143, 276.923] (14b to 13)<br /> nice range: [271.229, 271.708]<br /> strict range: [271.229, 271.708]<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~75/64 = 271.627<br /> <br /> Map: [<1 0 3|, <0 7 -3|]<br /> EDOs: 22, 31, 53, 190, 243, 296, 645c<br /> Badness: 0.0408<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Orson-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:2 -->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 65536/64625 leads to orwell, but we could also add 1029/1024, leading to the 31&159 temperament with wedgie <<21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&243 temperament with wedgie <<28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&243 temperament with wedgie <<7 -3 61 -21 77 150||.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Orwell"></a><!-- ws:end:WikiTextHeadingRule:4 -->Orwell</h1> Main article: <a class="wiki_link" href="/Orwell">Orwell</a><br /> So called because 19\84 (as a <a class="wiki_link" href="/fraction%20of%20the%20octave">fraction of the octave</a>) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/53edo">53</a> and <a class="wiki_link" href="/84edo">84</a> equal, and may be described as the 22&31 temperament, or <<7 -3 8 -21 -7 27||. It's a good system in the <a class="wiki_link" href="/7-limit">7-limit</a> and naturally extends into the <a class="wiki_link" href="/11-limit">11-limit</a>. <a class="wiki_link" href="/84edo">84edo</a>, with the 19\84 generator, provides a good tuning for the 5, 7 and 11 limits, but it does use its second-best 11. However, the 19\84 generator is remarkably close to the 11-limit <a class="wiki_link" href="/POTE%20tuning">POTE tuning</a>, as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. <a class="wiki_link" href="/53edo">53edo</a> might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out 2430/2401, the nuwell comma, 1728/1715, the orwellisma, 225/224, the septimal kleisma, and 6144/6125, the porwell comma.<br /> <br /> The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell.<br /> <br /> Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has <a class="wiki_link" href="/Retuning%2012edo%20to%20Orwell9">considerable harmonic resources</a> despite its absence of 5-limit triads. The 13 note MOS has those, and of course the 22 and 31 note MOS are very well supplied with everything.<br /> <br /> <a class="wiki_link" href="/Comma">Commas</a>: 225/224, 1728/1715<br /> <br /> 7-limit<br /> [|1 0 0 0>, |14/11 0 -7/11 7/11>,<br /> |27/11 0 3/11 -3/11>, |27/11 0 -8/11 8/11>]<br /> <a class="wiki_link" href="/Fractional%20monzos">Eigenmonzos</a>: 2, 7/5<br /> <br /> 9-limit<br /> [|1 0 0 0>, |21/17 14/17 -7/17 0>,<br /> |42/17 -6/17 3/17 0>, |41/17 16/17 -8/17 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 10/9<br /> <br /> valid range: [266.667, 272.727] (9 to 22)<br /> nice range: [266.871, 271.708]<br /> strict range: [266.871, 271.708]<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~7/6 = 271.509<br /> Algebraic generators: Sabra3, the real root of 12x^3-7x-48.<br /> <br /> Map: [<1 0 3 1|, <0 7 -3 8|]<br /> Wedgie: <<7 -3 8 -21 -7 27||<br /> EDOs: 22, 31, 53, 84, 137, 221d, 358d<br /> Badness: 0.0207<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Orwell-11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h2> <a class="wiki_link" href="/Comma">Commas</a>: 99/98, 121/120, 176/175<br /> <br /> <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a><br /> [|1 0 0 0 0>, |14/11 0 -7/11 7/11 0>, |27/11 0 3/11 -3/11 0>,<br /> |27/11 0 -8/11 8/11 0>, |37/11 0 -2/11 2/11 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 7/5<br /> <br /> valid range: [270.968, 272.727] (31 to 22)<br /> nice range: [266.871, 275.659]<br /> strict range: [270.968, 272.727]<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~7/6 = 271.426<br /> <br /> Map: [<1 0 3 1 3|, <0 7 -3 8 2|]<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/84edo">84e</a><br /> Badness: 0.0152<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Orwell-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h2> Commas: 99/98, 121/120, 176/175, 275/273<br /> <br /> valid range: [270.968, 271.698] (31 to 53)<br /> nice range: [266.871, 275.659]<br /> strict range: [270.968, 271.698] <br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~7/6 = 271.546<br /> <br /> Map: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|]<br /> EDOs: 22, 31, 53, 84e, 137e<br /> Badness: 0.0197<br /> <br /> <a class="wiki_link" href="/Orwell#Music">Music in Orwell</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="Orwell-Blair"></a><!-- ws:end:WikiTextHeadingRule:10 -->Blair</h2> Commas: 65/64, 78/77, 91/90, 99/98<br /> <br /> POTE generator: ~7/6 = 271.301<br /> <br /> Map: [<1 0 3 1 3 3|, <0 7 -3 8 2 3|]<br /> EDOs: 9, 22, 31f<br /> Badness: 0.0231<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="Orwell-Newspeak"></a><!-- ws:end:WikiTextHeadingRule:12 -->Newspeak</h2> Commas: 225/224, 441/440, 1728/1715<br /> <br /> POTE tuning: ~7/6 = 271.288<br /> <br /> Map: [<1 0 3 1 -4|, <0 7 -3 8 33|]<br /> EDOs: 31, 84, 115, 376b, 491bd, 606bde<br /> Badness: 0.0314<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="Orwell-Winston"></a><!-- ws:end:WikiTextHeadingRule:14 -->Winston</h2> Commas: 66/65, 99/98, 105/104, 121/120<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~7/6 = 271.088<br /> <br /> Map: [<1 0 3 1 3 1|, <0 7 -3 8 2 12|]<br /> EDOs: 22f, 31<br /> Badness: 0.0199<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h1> --><h1 id="toc8"><a name="Doublethink"></a><!-- ws:end:WikiTextHeadingRule:16 -->Doublethink</h1> Commas: 99/98, 121/120, 169/168, 176/175<br /> <br /> POTE tuning: ~13/12 = 135.723<br /> <br /> Map: [<1 0 3 1 3 2|, <0 14 -6 16 4 15|]<br /> EDOs: 9, 35, 44, 53, 62, 115ef, 168ef<br /> Badness: 0.0271<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h1> --><h1 id="toc9"><a name="Borwell"></a><!-- ws:end:WikiTextHeadingRule:18 -->Borwell</h1> Commas: 225/224, 243/242, 1728/1715<br /> <br /> POTE generator: ~55/36 = 735.752<br /> <br /> Map: [<1 7 0 9 17|, <0 -14 6 -16 -35|]<br /> EDOs: 31, 106, 137, 442bd<br /> Badness: 0.0384<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h1> --><h1 id="toc10"><a name="Triwell"></a><!-- ws:end:WikiTextHeadingRule:20 -->Triwell</h1> Commas: 1029/1024, 235298/234375<br /> <br /> POTE generator: ~448/375 = 309.472<br /> <br /> Map: [<1 7 0 1|, <0 -21 9 7]]<br /> Wedgie: <<21 -9 -7 -63 -70 9||<br /> EDOs: 31, 97, 128, 159, 190<br /> Badness: 0.0806<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h2> --><h2 id="toc11"><a name="Triwell-11-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->11-limit</h2> Commas: 385/384, 441/440, 456533/455625<br /> <br /> POTE generator: ~448/375 = 309.471<br /> <br /> Map: [<1 7 0 1 13|, <0 -21 9 7 -37]]<br /> EDOs: 31, 97, 128, 159, 190<br /> Badness: 0.0298</body></html>