Breedsmic temperaments: Difference between revisions
Wikispaces>genewardsmith **Imported revision 295856398 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 300276506 - 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:genewardsmith|genewardsmith]] and made on <tt>2012- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-09 16:16:33 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>300276506</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> | ||
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EDOs: 7, 17, 41, 58, 99 | EDOs: 7, 17, 41, 58, 99 | ||
Badness: 0.0191 | Badness: 0.0191 | ||
=Semihemi= | |||
Commas: 2401/2400, 3388/3375, 9801/9800 | |||
POTE generator: ~49/40 = 351.505 | |||
Map: [<2 0 -35 -15 -47|, <0 2 25 13 34|] | |||
EDOs: 58, 140, 198, 734bc, 932bcd, 1130bcd | |||
Badness: 42.487 | |||
==13-limit== | |||
Commas: 352/351, 676/675, 847/845, 1716/1715 | |||
POTE generator: ~49/40 = 351.502 | |||
Map: [<2 0 -35 -15 -47 -37|, <0 2 25 13 34 28|] | |||
EDOs: 58, 140, 198, 536f, 734bcf, 932bcdf | |||
Badness: 0.0212 | |||
=Tertiaseptal= | =Tertiaseptal= | ||
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</pre></div> | </pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Breedsmic temperaments</title></head><body><!-- ws:start:WikiTextTocRule: | <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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Breedsmic temperaments</title></head><body><!-- ws:start:WikiTextTocRule:56:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><a href="#Hemififths">Hemififths</a><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --> | <a href="#Semihemi">Semihemi</a><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --> | <a href="#Tertiaseptal">Tertiaseptal</a><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --> | <a href="#Harry">Harry</a><!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --> | <a href="#Quasiorwell">Quasiorwell</a><!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --> | <a href="#Decoid">Decoid</a><!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --> | <a href="#Neominor">Neominor</a><!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --><!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --> | <a href="#Emmthird">Emmthird</a><!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --> | <a href="#Quinmite">Quinmite</a><!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --> | <a href="#Unthirds">Unthirds</a><!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --><!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --> | <a href="#Newt">Newt</a><!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:85 --><br /> | ||
<br /> | <br /> | ||
Breedsmic temperaments are rank two temperaments tempering out the breedsma, |-5 -1 -2 4&gt; = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.<br /> | Breedsmic temperaments are rank two temperaments tempering out the breedsma, |-5 -1 -2 4&gt; = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.<br /> | ||
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Badness: 0.0191<br /> | Badness: 0.0191<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Tertiaseptal"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Semihemi"></a><!-- ws:end:WikiTextHeadingRule:6 -->Semihemi</h1> | ||
Commas: 2401/2400, 3388/3375, 9801/9800<br /> | |||
<br /> | |||
POTE generator: ~49/40 = 351.505<br /> | |||
<br /> | |||
Map: [&lt;2 0 -35 -15 -47|, &lt;0 2 25 13 34|]<br /> | |||
EDOs: 58, 140, 198, 734bc, 932bcd, 1130bcd<br /> | |||
Badness: 42.487<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Semihemi-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h2> | |||
Commas: 352/351, 676/675, 847/845, 1716/1715<br /> | |||
<br /> | |||
POTE generator: ~49/40 = 351.502<br /> | |||
<br /> | |||
Map: [&lt;2 0 -35 -15 -47 -37|, &lt;0 2 25 13 34 28|]<br /> | |||
EDOs: 58, 140, 198, 536f, 734bcf, 932bcdf<br /> | |||
Badness: 0.0212<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Tertiaseptal"></a><!-- ws:end:WikiTextHeadingRule:10 -->Tertiaseptal</h1> | |||
Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. <a class="wiki_link" href="/171edo">171edo</a> makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.<br /> | Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. <a class="wiki_link" href="/171edo">171edo</a> makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0130<br /> | Badness: 0.0130<br /> | ||
<br /> | <br /> | ||
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Commas: 243/242, 441/440, 65625/65536<br /> | Commas: 243/242, 441/440, 65625/65536<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0356<br /> | Badness: 0.0356<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Tertiaseptal-Tertia"></a><!-- ws:end:WikiTextHeadingRule:14 -->Tertia</h2> | ||
Commas: 385/384, 1331/1323, 1375/1372<br /> | Commas: 385/384, 1331/1323, 1375/1372<br /> | ||
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Badness: 0.0302<br /> | Badness: 0.0302<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Harry"></a><!-- ws:end:WikiTextHeadingRule:16 -->Harry</h1> | ||
Commas: 2401/2400, 19683/19600<br /> | Commas: 2401/2400, 19683/19600<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0341<br /> | Badness: 0.0341<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Harry-11-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->11-limit</h2> | ||
Commas: 243/242, 441/440, 4000/3993<br /> | Commas: 243/242, 441/440, 4000/3993<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0159<br /> | Badness: 0.0159<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Harry-13-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit</h2> | ||
Commas: 243/242, 351/350, 441/440, 676/675<br /> | Commas: 243/242, 351/350, 441/440, 676/675<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0130<br /> | Badness: 0.0130<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Quasiorwell"></a><!-- ws:end:WikiTextHeadingRule:22 -->Quasiorwell</h1> | ||
In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1&gt;. It has a wedgie &lt;&lt;38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&amp;270 temperament, and as one might expect, 61/270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.<br /> | In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1&gt;. It has a wedgie &lt;&lt;38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&amp;270 temperament, and as one might expect, 61/270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.<br /> | ||
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Badness: 0.0358<br /> | Badness: 0.0358<br /> | ||
<br /> | <br /> | ||
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Commas: 2401/2400, 3025/3024, 5632/5625<br /> | Commas: 2401/2400, 3025/3024, 5632/5625<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0175<br /> | Badness: 0.0175<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Quasiorwell-13-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->13-limit</h2> | ||
Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095<br /> | Commas: 1001/1000, 1716/1715, 3025/3024, 4096/4095<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0179<br /> | Badness: 0.0179<br /> | ||
<br /> | <br /> | ||
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Commas: 2401/2400, 67108864/66976875<br /> | Commas: 2401/2400, 67108864/66976875<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0339<br /> | Badness: 0.0339<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Decoid-11-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->11-limit</h2> | ||
Commas: 2401/2400, 5832/5825, 9801/9800<br /> | Commas: 2401/2400, 5832/5825, 9801/9800<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0187<br /> | Badness: 0.0187<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Decoid-13-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->13-limit</h2> | ||
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224<br /> | Commas: 676/675, 1001/1000, 1716/1715, 4225/4224<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0135<br /> | Badness: 0.0135<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Neominor"></a><!-- ws:end:WikiTextHeadingRule:34 -->Neominor</h1> | ||
Commas: 2401/2400, 177147/175616<br /> | Commas: 2401/2400, 177147/175616<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0882<br /> | Badness: 0.0882<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Neominor-11-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->11-limit</h2> | ||
Commas: 243/242, 441/440, 35937/35840<br /> | Commas: 243/242, 441/440, 35937/35840<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0280<br /> | Badness: 0.0280<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Neominor-13-limit"></a><!-- ws:end:WikiTextHeadingRule:38 -->13-limit</h2> | ||
Commas: 169/168, 243/242, 364/363, 441/440<br /> | Commas: 169/168, 243/242, 364/363, 441/440<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0269<br /> | Badness: 0.0269<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:40:&lt;h1&gt; --><h1 id="toc20"><a name="Emmthird"></a><!-- ws:end:WikiTextHeadingRule:40 -->Emmthird</h1> | ||
The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.<br /> | The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0167<br /> | Badness: 0.0167<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:42:&lt;h1&gt; --><h1 id="toc21"><a name="Quinmite"></a><!-- ws:end:WikiTextHeadingRule:42 -->Quinmite</h1> | ||
Commas: 2401/2400, 1959552/1953125<br /> | Commas: 2401/2400, 1959552/1953125<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0373<br /> | Badness: 0.0373<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:44:&lt;h1&gt; --><h1 id="toc22"><a name="Unthirds"></a><!-- ws:end:WikiTextHeadingRule:44 -->Unthirds</h1> | ||
Commas: 2401/2400, 68359375/68024448<br /> | Commas: 2401/2400, 68359375/68024448<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0753<br /> | Badness: 0.0753<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><h2 id="toc23"><a name="Unthirds-11-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->11-limit</h2> | ||
Commas: 2401/2400, 3025/3024, 4000/3993<br /> | Commas: 2401/2400, 3025/3024, 4000/3993<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0229<br /> | Badness: 0.0229<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:48:&lt;h2&gt; --><h2 id="toc24"><a name="Unthirds-13-limit"></a><!-- ws:end:WikiTextHeadingRule:48 -->13-limit</h2> | ||
Commas: 625/624, 1575/1573, 2080/2079, 2401/2400<br /> | Commas: 625/624, 1575/1573, 2080/2079, 2401/2400<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0209<br /> | Badness: 0.0209<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:50:&lt;h1&gt; --><h1 id="toc25"><a name="Newt"></a><!-- ws:end:WikiTextHeadingRule:50 -->Newt</h1> | ||
Commas: 2401/2400, 33554432/33480783<br /> | Commas: 2401/2400, 33554432/33480783<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0419<br /> | Badness: 0.0419<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:52:&lt;h2&gt; --><h2 id="toc26"><a name="Newt-11-limit"></a><!-- ws:end:WikiTextHeadingRule:52 -->11-limit</h2> | ||
Commas: 2401/2400, 3025/3024, 19712/19683<br /> | Commas: 2401/2400, 3025/3024, 19712/19683<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0195<br /> | Badness: 0.0195<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:54:&lt;h2&gt; --><h2 id="toc27"><a name="Newt-13-limit"></a><!-- ws:end:WikiTextHeadingRule:54 -->13-limit</h2> | ||
Commas: 2080/2079, 2401/2400, 3025/3024, 4096/4095<br /> | Commas: 2080/2079, 2401/2400, 3025/3024, 4096/4095<br /> | ||
<br /> | <br /> | ||