Semicomma family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 303361900 - Original comment: ** |
Wikispaces>guest **Imported revision 306992522 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:guest|guest]] and made on <tt>2012-03-01 21:09:23 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>306992522</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc]] | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc]] | ||
=Orson= | |||
=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. | 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. | ||
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Badness: 0.0408 | Badness: 0.0408 | ||
==Seven limit children== | ==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||. | 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= | =Orwell= | ||
Main article: [[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. | 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. | ||
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Badness: 0.0207 | Badness: 0.0207 | ||
==11-limit== | ==11-limit== | ||
[[Comma|Commas]]: 99/98, 121/120, 176/175 | [[Comma|Commas]]: 99/98, 121/120, 176/175 | ||
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Badness: 0.0152 | Badness: 0.0152 | ||
==13-limit== | ==13-limit== | ||
Commas: 99/98, 121/120, 176/175, 275/273 | Commas: 99/98, 121/120, 176/175, 275/273 | ||
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Badness: 0.0197 | Badness: 0.0197 | ||
==Blair== | ==Blair== | ||
Commas: 65/64, 78/77, 91/90, 99/98 | Commas: 65/64, 78/77, 91/90, 99/98 | ||
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Badness: 0.0231 | Badness: 0.0231 | ||
==Newspeak== | ==Newspeak== | ||
Commas: 225/224, 441/440, 1728/1715 | Commas: 225/224, 441/440, 1728/1715 | ||
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Badness: 0.0314 | Badness: 0.0314 | ||
==Winston== | ==Winston== | ||
Commas: 66/65, 99/98, 105/104, 121/120 | Commas: 66/65, 99/98, 105/104, 121/120 | ||
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Badness: 0.0199 | Badness: 0.0199 | ||
=Doublethink= | =Doublethink= | ||
Commas: 99/98, 121/120, 169/168, 176/175 | Commas: 99/98, 121/120, 169/168, 176/175 | ||
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Map: [<1 0 3 1 3 2|, <0 14 -6 16 4 15|] | Map: [<1 0 3 1 3 2|, <0 14 -6 16 4 15|] | ||
EDOs: 9, 44, 53, 115ef, 168ef | EDOs: 9, 35, 44, 53, 61, 115ef, 168ef | ||
Badness: 0.0271 | Badness: 0.0271 | ||
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Badness: 0.0384 | Badness: 0.0384 | ||
=Triwell= | =Triwell= | ||
Commas: 1029/1024, 235298/234375 | Commas: 1029/1024, 235298/234375 | ||
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Badness: 0.0806 | Badness: 0.0806 | ||
==11-limit== | ==11-limit== | ||
Commas: 385/384, 441/440, 456533/455625 | Commas: 385/384, 441/440, 456533/455625 | ||
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<!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#Music">Music</a></div> | <!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#Music">Music</a></div> | ||
<!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --></div> | <!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --></div> | ||
<!-- ws:end:WikiTextTocRule:40 --> | <!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Orson"></a><!-- ws:end:WikiTextHeadingRule:0 -->Orson</h1> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><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&gt;. 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 /> | ||
The 5-limit parent comma for the <strong>semicomma family</strong> is the semicomma, 2109375/2097152 = |-21 3 7&gt;. 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 /> | <br /> | ||
Comma: 2109375/2097152<br /> | Comma: 2109375/2097152<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Orson-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:2 -->Seven limit children</h2> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><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&amp;159 temperament with wedgie &lt;&lt;21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&amp;243 temperament with wedgie &lt;&lt;28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&amp;243 temperament with wedgie &lt;&lt;7 -3 61 -21 77 150||.<br /> | 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&amp;159 temperament with wedgie &lt;&lt;21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&amp;243 temperament with wedgie &lt;&lt;28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&amp;243 temperament with wedgie &lt;&lt;7 -3 61 -21 77 150||.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Orwell"></a><!-- ws:end:WikiTextHeadingRule:4 -->Orwell</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Orwell"></a><!-- ws:end:WikiTextHeadingRule:4 -->Orwell</h1> | ||
Main article: <a class="wiki_link" href="/Orwell">Orwell</a><br /> | 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&amp;31 temperament, or &lt;&lt;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 /> | 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&amp;31 temperament, or &lt;&lt;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 /> | <br /> | ||
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<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Orwell-11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><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 /> | <a class="wiki_link" href="/Comma">Commas</a>: 99/98, 121/120, 176/175<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a><br /> | <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a><br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Orwell-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><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 /> | Commas: 99/98, 121/120, 176/175, 275/273<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~7/6 = 271.546<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~7/6 = 271.546<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Orwell-Blair"></a><!-- ws:end:WikiTextHeadingRule:10 -->Blair</h2> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Orwell-Blair"></a><!-- ws:end:WikiTextHeadingRule:10 -->Blair</h2> | ||
Commas: 65/64, 78/77, 91/90, 99/98<br /> | Commas: 65/64, 78/77, 91/90, 99/98<br /> | ||
<br /> | <br /> | ||
POTE generator: ~7/6 = 271.301<br /> | POTE generator: ~7/6 = 271.301<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Orwell-Newspeak"></a><!-- ws:end:WikiTextHeadingRule:12 -->Newspeak</h2> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Orwell-Newspeak"></a><!-- ws:end:WikiTextHeadingRule:12 -->Newspeak</h2> | ||
Commas: 225/224, 441/440, 1728/1715<br /> | Commas: 225/224, 441/440, 1728/1715<br /> | ||
<br /> | <br /> | ||
POTE tuning: ~7/6 = 271.288<br /> | POTE tuning: ~7/6 = 271.288<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Orwell-Winston"></a><!-- ws:end:WikiTextHeadingRule:14 -->Winston</h2> | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Orwell-Winston"></a><!-- ws:end:WikiTextHeadingRule:14 -->Winston</h2> | ||
Commas: 66/65, 99/98, 105/104, 121/120<br /> | Commas: 66/65, 99/98, 105/104, 121/120<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~7/6 = 271.088<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~7/6 = 271.088<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Doublethink"></a><!-- ws:end:WikiTextHeadingRule:16 -->Doublethink</h1> | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Doublethink"></a><!-- ws:end:WikiTextHeadingRule:16 -->Doublethink</h1> | ||
Commas: 99/98, 121/120, 169/168, 176/175<br /> | Commas: 99/98, 121/120, 169/168, 176/175<br /> | ||
<br /> | <br /> | ||
POTE tuning: ~13/12 = 135.723<br /> | POTE tuning: ~13/12 = 135.723<br /> | ||
<br /> | <br /> | ||
Map: [&lt;1 0 3 1 3 2|, &lt;0 14 -6 16 4 15|]<br /> | Map: [&lt;1 0 3 1 3 2|, &lt;0 14 -6 16 4 15|]<br /> | ||
EDOs: 9, 44, 53, 115ef, 168ef<br /> | EDOs: 9, 35, 44, 53, 61, 115ef, 168ef<br /> | ||
Badness: 0.0271<br /> | Badness: 0.0271<br /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Triwell"></a><!-- ws:end:WikiTextHeadingRule:20 -->Triwell</h1> | <!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Triwell"></a><!-- ws:end:WikiTextHeadingRule:20 -->Triwell</h1> | ||
Commas: 1029/1024, 235298/234375<br /> | Commas: 1029/1024, 235298/234375<br /> | ||
<br /> | <br /> | ||
POTE generator: ~448/375 = 309.472<br /> | POTE generator: ~448/375 = 309.472<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Triwell-11-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Triwell-11-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->11-limit</h2> | ||
Commas: 385/384, 441/440, 456533/455625<br /> | Commas: 385/384, 441/440, 456533/455625<br /> | ||
<br /> | <br /> | ||
POTE generator: ~448/375 = 309.471<br /> | POTE generator: ~448/375 = 309.471<br /> | ||