Ragismic microtemperaments: Difference between revisions
Wikispaces>genewardsmith **Imported revision 238903247 - 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>2011-06- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-28 12:51:20 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>239160771</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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Map: [<9 1 1 2|, <0 2 3 2|] | Map: [<9 1 1 2|, <0 2 3 2|] | ||
Wedgie: <<18 27 18 1 -22 -34|| | Wedgie: <<18 27 18 1 -22 -34|| | ||
EDOs: 27, 45, 72, 99, 171, 270, 441, 612 | EDOs: [[27edo|27]], [[45edo|45]], [[72edo|72]], [[99edo|99]], [[171edo|171]], [[270edo|270]], [[441edo|441]], [[612edo|612]], [[3600edo|3600] | ||
Badness: 0.00361 | Badness: 0.00361 | ||
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<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>Ragismic microtemperaments</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>Ragismic microtemperaments</title></head><body><!-- ws:start:WikiTextTocRule:32:&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:32 --><!-- ws:start:WikiTextTocRule:33: --><a href="#Ennealimmal">Ennealimmal</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | <a href="#Abigail">Abigail</a><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --> | <a href="#Nearly Micro">Nearly Micro</a><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --> | <a href="#Octoid">Octoid</a><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --> | <a href="#Amity">Amity</a><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --> | <a href="#Parakleismic">Parakleismic</a><!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --> | ||
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The ragisma is 4375/4374, the smallest 7-limit superparticular ratio. Since (10/9)^4 = 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low complexity in temperaments tempering out the ragisma, though when looking at microtemperaments the word &quot;relatively&quot; should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low cmplexity, with the same caveat about &quot;relatively&quot;; however 27/25 is the period for ennealimmal.<br /> | The ragisma is 4375/4374, the smallest 7-limit superparticular ratio. Since (10/9)^4 = 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low complexity in temperaments tempering out the ragisma, though when looking at microtemperaments the word &quot;relatively&quot; should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low cmplexity, with the same caveat about &quot;relatively&quot;; however 27/25 is the period for ennealimmal.<br /> | ||
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Map: [&lt;9 1 1 2|, &lt;0 2 3 2|]<br /> | Map: [&lt;9 1 1 2|, &lt;0 2 3 2|]<br /> | ||
Wedgie: &lt;&lt;18 27 18 1 -22 -34||<br /> | Wedgie: &lt;&lt;18 27 18 1 -22 -34||<br /> | ||
EDOs: 27, | EDOs: <a class="wiki_link" href="/27edo">27</a>, <a class="wiki_link" href="/45edo">45</a>, <a class="wiki_link" href="/72edo">72</a>, <a class="wiki_link" href="/99edo">99</a>, <a class="wiki_link" href="/171edo">171</a>, <a class="wiki_link" href="/270edo">270</a>, <a class="wiki_link" href="/441edo">441</a>, <a class="wiki_link" href="/612edo">612</a>, <a class="wiki_link" href="/3600edo">3600]Badness: 0.00361==11 limit hemiennealimmal==Commas: 2401/2400, 4375/4374, 3025/3024POTE generator: 99/98: 17.6219 or 6/5: 315.7114Map: [&lt;18 0 -1 22 48|, &lt;0 2 3 2 1|]EDOs: 72, 198, 270, 342, 612, 954, 1566Badness: 0.00628==13 limit hemiennealimmal==Commas: 676/675, 1001/1000, 1716/1715, 3025/3024POTE generator ~99/98 = 17.7504Map: [&lt;18 0 -1 22 48 -19|, &lt;0 2 3 2 1 6|]EDOs: 72, 198, 270Badness: 0.0125=Gamera=Commas: 4375/4374, 589824/588245POTE generator ~8/7 = 230.336Map: [&lt;1 6 10 3|, &lt;0 -23 -40 -1|]EDOs: 26, 73, 99, 224, 323, 422, 735Badness: 0.0376=Supermajor=The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.0002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of &lt;&lt;37 46 75 -13 15 45||. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.Commas: 4375/4374, 52734375/52706752POTE generator: ~9/7 = 435.082Map: [&lt;1 15 19 30|, &lt;0 -37 -46 -75|]EDOs: 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214Badness: 0.0108=Enneadecal=Enndedecal temperament tempers out the enneadeca, |-14 -19 19&gt;, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo</a> up to just ones. <a class="wiki_link" href="/171edo">171edo</a> is a good tuning for either the 5 or 7 limits, and <a class="wiki_link" href="/494edo">494edo</a> shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use <a class="wiki_link" href="/665edo">665edo</a> for a tuning.<br /> | ||
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EDOs: 72, 198, 270, 342, 612, 954, | |||
Commas: 676/675, 1001/1000, 1716/1715, 3025/ | |||
EDOs: 72, 198, | |||
Commas: 4375/4374, 589824/ | |||
EDOs: 26, 73, 99, 224, 323, 422, | |||
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.0002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of &lt;&lt;37 46 75 -13 15 45||. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS. | |||
Commas: 4375/4374, 52734375/ | |||
EDOs: 11, 80, 171, 764, 1106, 1277, 3660, 4937, | |||
Enndedecal temperament tempers out the enneadeca, |-14 -19 19&gt;, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of | |||
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Commas: 4375/4374, 703125/702464<br /> | Commas: 4375/4374, 703125/702464<br /> | ||
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Badness: 0.0110<br /> | Badness: 0.0110<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Abigail"></a><!-- ws:end:WikiTextHeadingRule:2 -->Abigail</h1> | ||
Commas: 4375/4374, 2147483648/2144153025<br /> | Commas: 4375/4374, 2147483648/2144153025<br /> | ||
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Badness: 0.0370<br /> | Badness: 0.0370<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Abigail-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h2> | ||
Comma: 3025/3024, 4375/4374, 20614528/20588575<br /> | Comma: 3025/3024, 4375/4374, 20614528/20588575<br /> | ||
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Badness: 0.0129<br /> | Badness: 0.0129<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Abigail-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h2> | ||
Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095<br /> | Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095<br /> | ||
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Badness: 0.00886<br /> | Badness: 0.00886<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Nearly Micro"></a><!-- ws:end:WikiTextHeadingRule:8 -->Nearly Micro</h1> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Octoid"></a><!-- ws:end:WikiTextHeadingRule:10 -->Octoid</h1> | ||
Commas: 4375/4374, 16875/16807<br /> | Commas: 4375/4374, 16875/16807<br /> | ||
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Badness: 0.0427<br /> | Badness: 0.0427<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Octoid-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->11-limit</h2> | ||
Commas: 540/539, 1375/1372, 4000/3993<br /> | Commas: 540/539, 1375/1372, 4000/3993<br /> | ||
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Badness: 0.0141<br /> | Badness: 0.0141<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Octoid-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h2> | ||
Commas: 540/539, 1375/1372, 4000/3993, 625/624<br /> | Commas: 540/539, 1375/1372, 4000/3993, 625/624<br /> | ||
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Badness: 0.0153<br /> | Badness: 0.0153<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Amity"></a><!-- ws:end:WikiTextHeadingRule:16 -->Amity</h1> | ||
The generator for amity temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, 5120/5103 and 6144/6125. It can also be described as the 46&amp;53 temperament, or by its wedgie, &lt;&lt;5 13 -17 9 -41 -76||. <a class="wiki_link" href="/99edo">99edo</a> is a good tuning for amity, with generator 28/99, and MOS of 11, 18, 25, 32, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.<br /> | The generator for amity temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, 5120/5103 and 6144/6125. It can also be described as the 46&amp;53 temperament, or by its wedgie, &lt;&lt;5 13 -17 9 -41 -76||. <a class="wiki_link" href="/99edo">99edo</a> is a good tuning for amity, with generator 28/99, and MOS of 11, 18, 25, 32, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.<br /> | ||
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In the 5-limit amity is a genuine microtemperament, with 58/205 being a possible tuning. Another good choice is (64/5)^(1/13), which gives pure major thirds.<br /> | In the 5-limit amity is a genuine microtemperament, with 58/205 being a possible tuning. Another good choice is (64/5)^(1/13), which gives pure major thirds.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Amity-5-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->5-limit</h2> | ||
Comma: 1600000/1594323<br /> | Comma: 1600000/1594323<br /> | ||
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Badness: 0.0220<br /> | Badness: 0.0220<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Amity-7-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->7-limit</h2> | ||
Commas: 4375/4374, 5120/5103<br /> | Commas: 4375/4374, 5120/5103<br /> | ||
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Badness: 0.0236<br /> | Badness: 0.0236<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Amity-Hitchcock"></a><!-- ws:end:WikiTextHeadingRule:22 -->Hitchcock</h2> | ||
Commas: 121/120, 176/175, 2200/2187<br /> | Commas: 121/120, 176/175, 2200/2187<br /> | ||
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Badness: 0.0352<br /> | Badness: 0.0352<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Amity-Hemiamity"></a><!-- ws:end:WikiTextHeadingRule:24 -->Hemiamity</h2> | ||
Commas: 4375/4374, 5120/5103, 3025/3024<br /> | Commas: 4375/4374, 5120/5103, 3025/3024<br /> | ||
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EDOs: 14, 46, 106, 152, 350<br /> | EDOs: 14, 46, 106, 152, 350<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name="Parakleismic"></a><!-- ws:end:WikiTextHeadingRule:26 -->Parakleismic</h1> | ||
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |8 14 -13&gt;, with the <a class="wiki_link" href="/118edo">118edo</a> tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being &lt;&lt;13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie &lt;&lt;13 14 35 -36 ...|| adding 385/384. For the 7-limit <a class="wiki_link" href="/99edo">99edo</a> may be preferred, but in the 11-limit it is best to stick with 118. <br /> | In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |8 14 -13&gt;, with the <a class="wiki_link" href="/118edo">118edo</a> tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being &lt;&lt;13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie &lt;&lt;13 14 35 -36 ...|| adding 385/384. For the 7-limit <a class="wiki_link" href="/99edo">99edo</a> may be preferred, but in the 11-limit it is best to stick with 118. <br /> | ||
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Badness: 0.0433<br /> | Badness: 0.0433<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Parakleismic-7-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->7-limit</h2> | ||
Commas: 3136/3125, 5475/4374<br /> | Commas: 3136/3125, 5475/4374<br /> | ||
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Badness: 0.0274<br /> | Badness: 0.0274<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Parakleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->11-limit</h2> | ||
Commas: 385/384, 3136/3125, 4375/4374<br /> | Commas: 385/384, 3136/3125, 4375/4374<br /> | ||
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