Ragismic microtemperaments: Difference between revisions
Wikispaces>genewardsmith **Imported revision 240098653 - 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>2011-07-05 17: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-05 17:47:14 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>240104945</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: 19, 152, 171, 665, 836, 1007, 2185 | EDOs: 19, 152, 171, 665, 836, 1007, 2185 | ||
Badness: 0.0110 | Badness: 0.0110 | ||
=Deca= | |||
Commas: 4375/4374, 165288374272/164794921875 | |||
POTE generator: ~460992/390625 = 284.423 | |||
Map: [<10 4 2 9|, <0 5 6 11|] | |||
EDOs: 80, 190, 270, 1270, 1540, 1810, 2080 | |||
Badness: 0.0806 | |||
==11-limit== | |||
Commas: 3025/3024, 4375/4374, 422576/421875 | |||
POTE generator: ~33/28 = 284.418 | |||
Map: [<10 4 2 9 18|, <0 5 6 11 7|] | |||
EDOs: 80, 190, 270, 1000, 1270 | |||
Badness: 0.0243 | |||
==13-limit== | |||
Commas: 1001/1000, 3025/3024, 4225/4224, 4375/4374 | |||
POTE generator: ~33/28 = 284.398 | |||
Map: [<10 4 2 9 18 37|, <0 5 6 11 7 0|] | |||
EDOs: 80, 190, 270, 730, 1000 | |||
Badness: 0.0168 | |||
=Mitonic= | =Mitonic= | ||
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Badness: 0.0497</pre></div> | Badness: 0.0497</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>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:54:&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:54 --><!-- ws:start:WikiTextTocRule:55: --><a href="#Ennealimmal">Ennealimmal</a><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><!-- 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="#Gamera">Gamera</a><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --> | <a href="#Supermajor">Supermajor</a><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --> | <a href="#Enneadecal">Enneadecal</a><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --> | <a href="#Deca">Deca</a><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --> | <a href="#Mitonic">Mitonic</a><!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --> | <a href="#Abigail">Abigail</a><!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --> | <a href="#Nearly Micro">Nearly Micro</a><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --> | <a href="#Octoid">Octoid</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="#Amity">Amity</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: --><!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --> | <a href="#Parakleismic">Parakleismic</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: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:82 --><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 /> | 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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Badness: 0.0110<br /> | Badness: 0.0110<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Mitonic"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Deca"></a><!-- ws:end:WikiTextHeadingRule:16 -->Deca</h1> | ||
Commas: 4375/4374, 165288374272/164794921875<br /> | |||
<br /> | |||
POTE generator: ~460992/390625 = 284.423<br /> | |||
<br /> | |||
Map: [&lt;10 4 2 9|, &lt;0 5 6 11|]<br /> | |||
EDOs: 80, 190, 270, 1270, 1540, 1810, 2080<br /> | |||
Badness: 0.0806<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Deca-11-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->11-limit</h2> | |||
Commas: 3025/3024, 4375/4374, 422576/421875<br /> | |||
<br /> | |||
POTE generator: ~33/28 = 284.418<br /> | |||
<br /> | |||
Map: [&lt;10 4 2 9 18|, &lt;0 5 6 11 7|]<br /> | |||
EDOs: 80, 190, 270, 1000, 1270<br /> | |||
Badness: 0.0243<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Deca-13-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit</h2> | |||
Commas: 1001/1000, 3025/3024, 4225/4224, 4375/4374<br /> | |||
<br /> | |||
POTE generator: ~33/28 = 284.398<br /> | |||
<br /> | |||
Map: [&lt;10 4 2 9 18 37|, &lt;0 5 6 11 7 0|]<br /> | |||
EDOs: 80, 190, 270, 730, 1000<br /> | |||
Badness: 0.0168<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Mitonic"></a><!-- ws:end:WikiTextHeadingRule:22 -->Mitonic</h1> | |||
Commas: 4375/4374, 2100875/2097152<br /> | Commas: 4375/4374, 2100875/2097152<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0252<br /> | Badness: 0.0252<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Abigail"></a><!-- ws:end:WikiTextHeadingRule:24 -->Abigail</h1> | ||
Commas: 4375/4374, 2147483648/2144153025<br /> | Commas: 4375/4374, 2147483648/2144153025<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0370<br /> | Badness: 0.0370<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Abigail-11-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->11-limit</h2> | ||
Comma: 3025/3024, 4375/4374, 20614528/20588575<br /> | Comma: 3025/3024, 4375/4374, 20614528/20588575<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0129<br /> | Badness: 0.0129<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Abigail-13-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->13-limit</h2> | ||
Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095<br /> | Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095<br /> | ||
<br /> | <br /> | ||
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Badness: 0.00886<br /> | Badness: 0.00886<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h1&gt; --><h1 id="toc15"><a name="Nearly Micro"></a><!-- ws:end:WikiTextHeadingRule:30 -->Nearly Micro</h1> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Octoid"></a><!-- ws:end:WikiTextHeadingRule:32 -->Octoid</h1> | ||
Commas: 4375/4374, 16875/16807<br /> | Commas: 4375/4374, 16875/16807<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0427<br /> | Badness: 0.0427<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Octoid-11-limit"></a><!-- ws:end:WikiTextHeadingRule:34 -->11-limit</h2> | ||
Commas: 540/539, 1375/1372, 4000/3993<br /> | Commas: 540/539, 1375/1372, 4000/3993<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0141<br /> | Badness: 0.0141<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Octoid-13-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->13-limit</h2> | ||
Commas: 540/539, 1375/1372, 4000/3993, 625/624<br /> | Commas: 540/539, 1375/1372, 4000/3993, 625/624<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0153<br /> | Badness: 0.0153<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><h1 id="toc19"><a name="Amity"></a><!-- ws:end:WikiTextHeadingRule:38 -->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 /> | ||
<br /> | <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 /> | 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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Amity-5-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->5-limit</h2> | ||
Comma: 1600000/1594323<br /> | Comma: 1600000/1594323<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0220<br /> | Badness: 0.0220<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:42:&lt;h2&gt; --><h2 id="toc21"><a name="Amity-7-limit"></a><!-- ws:end:WikiTextHeadingRule:42 -->7-limit</h2> | ||
Commas: 4375/4374, 5120/5103<br /> | Commas: 4375/4374, 5120/5103<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0236<br /> | Badness: 0.0236<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:44:&lt;h2&gt; --><h2 id="toc22"><a name="Amity-Hitchcock"></a><!-- ws:end:WikiTextHeadingRule:44 -->Hitchcock</h2> | ||
Commas: 121/120, 176/175, 2200/2187<br /> | Commas: 121/120, 176/175, 2200/2187<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0352<br /> | Badness: 0.0352<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><h2 id="toc23"><a name="Amity-Hemiamity"></a><!-- ws:end:WikiTextHeadingRule:46 -->Hemiamity</h2> | ||
Commas: 4375/4374, 5120/5103, 3025/3024<br /> | Commas: 4375/4374, 5120/5103, 3025/3024<br /> | ||
<br /> | <br /> | ||
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EDOs: 14, 46, 106, 152, 350<br /> | EDOs: 14, 46, 106, 152, 350<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:48:&lt;h1&gt; --><h1 id="toc24"><a name="Parakleismic"></a><!-- ws:end:WikiTextHeadingRule:48 -->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 /> | ||
<br /> | <br /> | ||
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Badness: 0.0433<br /> | Badness: 0.0433<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:50:&lt;h2&gt; --><h2 id="toc25"><a name="Parakleismic-7-limit"></a><!-- ws:end:WikiTextHeadingRule:50 -->7-limit</h2> | ||
Commas: 3136/3125, 5475/4374<br /> | Commas: 3136/3125, 5475/4374<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0274<br /> | Badness: 0.0274<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:52:&lt;h2&gt; --><h2 id="toc26"><a name="Parakleismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:52 -->11-limit</h2> | ||
Commas: 385/384, 3136/3125, 4375/4374<br /> | Commas: 385/384, 3136/3125, 4375/4374<br /> | ||
<br /> | <br /> | ||