Diaschismic family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 322883116 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 322884492 - 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>2012-04-19 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-04-19 19:03:39 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>322884492</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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The 5-limit parent comma for the **diaschismic family** is 2048/2025, the [[diaschisma]]. Its monzo is |11 -4 -2>, and flipping that yields <<2 -4 -11|| for the wedgie for 5-limit diaschismic, or **srutal**, temperament. This tells us the period is half an octave, the GCD of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]] or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities. | The 5-limit parent comma for the **diaschismic family** is 2048/2025, the [[diaschisma]]. Its monzo is |11 -4 -2>, and flipping that yields <<2 -4 -11|| for the wedgie for 5-limit diaschismic, or **srutal**, temperament. This tells us the period is half an octave, the GCD of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]] or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities. | ||
[[POTE tuning|POTE generator]]: 704.898 | [[POTE tuning|POTE generator]]: ~3/2 = 704.898 | ||
Map: [<2 0 11|, <0 1 -2|] | Map: [<2 0 11|, <0 1 -2|] | ||
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==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. Pajara derives from 64/63 and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, echidna 1728/1715 and shrutar 245/243, the sensamagic comma. The pajara, diaschismic and keen keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36_35|36/35]], the septimal quarter-tone) and echidna has a generator of 9/7. | The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Pajara derives from 64/63 and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, echidna 1728/1715 and shrutar 245/243, the sensamagic comma. The pajara, diaschismic and keen keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36_35|36/35]], the septimal quarter-tone) and echidna has a generator of 9/7. Adding 4375/4374 does no significant tuning damage, so for that we keep the 5-limit label srutal. | ||
=Srutal= | |||
Commas: 2048/2025, 4375/4374 | |||
POTE generator: ~3/2 = 704.814 | |||
Map: [<2 0 11 -42|, <0 1 -2 15|] | |||
Wedgie: <<2 -4 30 -11 42 81|| | |||
EDOs: 46, 80, 126, 206cd, 332bcd | |||
Badness: 0.0915 | |||
==11-limit== | |||
Commas: 176/175, 896/891, 1331/1323 | |||
POTE generator: ~3/2 = 704.856 | |||
Map: [<2 0 11 -42 -28|, <0 1 -2 15 11|] | |||
EDOs: 46, 80, 126, 206cd | |||
Badness: 0.0353 | |||
==13-limit== | |||
Commas: 169/168, 176/175, 325/324, 364/363 | |||
POTE generator: ~3/2 = 704.881 | |||
Map: [<2 0 11 -42 -28 -18|, <0 1 -2 15 11 8|] | |||
EDOs: 34d, 46, 80, 206cd, 286bcde | |||
Badness: 0.0253 | |||
=Srutal= | =Srutal= | ||
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Map: [<2 0 11 -15 -1 9|, <0 2 -4 13 5 -1|] | Map: [<2 0 11 -15 -1 9|, <0 2 -4 13 5 -1|] | ||
EDOs: 24, 34d, 58, 150cdef, 208cdef | EDOs: 24, 34d, 58, 150cdef, 208cdef | ||
Badness: 0.0238</pre></div> | Badness: 0.0238 | ||
</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>Diaschismic family</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>Diaschismic family</title></head><body><!-- ws:start:WikiTextTocRule:70:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><div style="margin-left: 2em;"><a href="#x-Seven limit children">Seven limit children</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><div style="margin-left: 1em;"><a href="#Srutal">Srutal</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><div style="margin-left: 2em;"><a href="#Srutal-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><div style="margin-left: 2em;"><a href="#Srutal-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><div style="margin-left: 1em;"><a href="#Srutal">Srutal</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --><div style="margin-left: 2em;"><a href="#Srutal-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --><div style="margin-left: 1em;"><a href="#Pajara">Pajara</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><div style="margin-left: 2em;"><a href="#Pajara-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --><div style="margin-left: 2em;"><a href="#Pajara-Pajarous">Pajarous</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --><div style="margin-left: 2em;"><a href="#Pajara-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><div style="margin-left: 2em;"><a href="#Pajara-Pajaro">Pajaro</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><div style="margin-left: 1em;"><a href="#Diaschismic">Diaschismic</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><div style="margin-left: 2em;"><a href="#Diaschismic-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><div style="margin-left: 2em;"><a href="#Diaschismic-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><div style="margin-left: 2em;"><a href="#Diaschismic-17-limit">17-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><div style="margin-left: 1em;"><a href="#Keen">Keen</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><div style="margin-left: 2em;"><a href="#Keen-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><div style="margin-left: 1em;"><a href="#Bidia">Bidia</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><div style="margin-left: 2em;"><a href="#Bidia-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><div style="margin-left: 2em;"><a href="#Bidia-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><div style="margin-left: 1em;"><a href="#Echidna">Echidna</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><div style="margin-left: 2em;"><a href="#Echidna-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><div style="margin-left: 2em;"><a href="#Echidna-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><div style="margin-left: 2em;"><a href="#Echidna-17-limit">17-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><div style="margin-left: 1em;"><a href="#Echidnic">Echidnic</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><div style="margin-left: 2em;"><a href="#Echidnic-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><div style="margin-left: 2em;"><a href="#Echidnic-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><div style="margin-left: 1em;"><a href="#Shrutar">Shrutar</a></div> | |||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><div style="margin-left: 2em;"><a href="#Shrutar-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><div style="margin-left: 2em;"><a href="#Shrutar-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><div style="margin-left: 2em;"><a href="#Shrutar-17-limit">17-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><div style="margin-left: 2em;"><a href="#Shrutar-19-limit">19-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><div style="margin-left: 1em;"><a href="#Sruti">Sruti</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><div style="margin-left: 2em;"><a href="#Sruti-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><div style="margin-left: 2em;"><a href="#Sruti-13-limit">13-limit</a></div> | |||
<!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --></div> | |||
<!-- ws:end:WikiTextTocRule:106 -->The 5-limit parent comma for the <strong>diaschismic family</strong> is 2048/2025, the <a class="wiki_link" href="/diaschisma">diaschisma</a>. Its monzo is |11 -4 -2&gt;, and flipping that yields &lt;&lt;2 -4 -11|| for the wedgie for 5-limit diaschismic, or <strong>srutal</strong>, temperament. This tells us the period is half an octave, the GCD of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. <a class="wiki_link" href="/34edo">34edo</a> is a good tuning choice, with <a class="wiki_link" href="/46edo">46edo</a>, <a class="wiki_link" href="/56edo">56edo</a>, <a class="wiki_link" href="/58edo">58edo</a> or <a class="wiki_link" href="/80edo">80edo</a> being other possibilities. Both <a class="wiki_link" href="/12edo">12edo</a> and <a class="wiki_link" href="/22edo">22edo</a> support it, and retuning them to a MOS of diaschismic gives two scale possibilities.<br /> | |||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 704.898<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 704.898<br /> | ||
<br /> | <br /> | ||
Map: [&lt;2 0 11|, &lt;0 1 -2|]<br /> | Map: [&lt;2 0 11|, &lt;0 1 -2|]<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h2> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->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. Pajara derives from 64/63 and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, echidna 1728/1715 and shrutar 245/243, the sensamagic comma. The pajara, diaschismic and keen keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as <a class="wiki_link" href="/36_35">36/35</a>, the septimal quarter-tone) and echidna has a generator of 9/7.<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. Pajara derives from 64/63 and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, echidna 1728/1715 and shrutar 245/243, the sensamagic comma. The pajara, diaschismic and keen keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as <a class="wiki_link" href="/36_35">36/35</a>, the septimal quarter-tone) and echidna has a generator of 9/7. Adding 4375/4374 does no significant tuning damage, so for that we keep the 5-limit label srutal.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Srutal"></a><!-- ws:end:WikiTextHeadingRule:2 -->Srutal</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Srutal"></a><!-- ws:end:WikiTextHeadingRule:2 -->Srutal</h1> | ||
Commas: 2048/2025, 4375/4374<br /> | |||
<br /> | |||
POTE generator: ~3/2 = 704.814<br /> | |||
<br /> | |||
Map: [&lt;2 0 11 -42|, &lt;0 1 -2 15|]<br /> | |||
Wedgie: &lt;&lt;2 -4 30 -11 42 81||<br /> | |||
EDOs: 46, 80, 126, 206cd, 332bcd<br /> | |||
Badness: 0.0915<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Srutal-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Srutal-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h2> | ||
Commas: 176/175, 896/891, 1331/1323<br /> | |||
<br /> | |||
POTE generator: ~3/2 = 704.856<br /> | |||
<br /> | |||
Map: [&lt;2 0 11 -42 -28|, &lt;0 1 -2 15 11|]<br /> | |||
EDOs: 46, 80, 126, 206cd<br /> | |||
Badness: 0.0353<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Srutal-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h2> | |||
Commas: 169/168, 176/175, 325/324, 364/363<br /> | |||
<br /> | |||
POTE generator: ~3/2 = 704.881<br /> | |||
<br /> | |||
Map: [&lt;2 0 11 -42 -28 -18|, &lt;0 1 -2 15 11 8|]<br /> | |||
EDOs: 34d, 46, 80, 206cd, 286bcde<br /> | |||
Badness: 0.0253<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Srutal"></a><!-- ws:end:WikiTextHeadingRule:8 -->Srutal</h1> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Srutal-11-limit"></a><!-- ws:end:WikiTextHeadingRule:10 -->11-limit</h2> | |||
Commas: 176/175, 896/891, 1331/1323<br /> | Commas: 176/175, 896/891, 1331/1323<br /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Pajara"></a><!-- ws:end:WikiTextHeadingRule:12 -->Pajara</h1> | ||
Main article: <a class="wiki_link" href="/Pajara">Pajara</a><br /> | Main article: <a class="wiki_link" href="/Pajara">Pajara</a><br /> | ||
Pajara, with wedgie &lt;&lt;2 -4 -4 -11 -12 2|| is closely associated with 22et (not to mention <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a>) but other tunings are possible. The 1/2 octave period serves as both a <a class="wiki_link" href="/10_7">10/7</a> and a <a class="wiki_link" href="/7_5">7/5</a>. Aside from 22et, 34 with the val &lt;34 54 79 96| and 56 with the val &lt;56 89 130 158| are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12et and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.<br /> | Pajara, with wedgie &lt;&lt;2 -4 -4 -11 -12 2|| is closely associated with 22et (not to mention <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a>) but other tunings are possible. The 1/2 octave period serves as both a <a class="wiki_link" href="/10_7">10/7</a> and a <a class="wiki_link" href="/7_5">7/5</a>. Aside from 22et, 34 with the val &lt;34 54 79 96| and 56 with the val &lt;56 89 130 158| are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12et and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.<br /> | ||
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Badness: 0.0200<br /> | Badness: 0.0200<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Pajara-11-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->11-limit</h2> | ||
Commas: 50/49, 64/63, 99/98<br /> | Commas: 50/49, 64/63, 99/98<br /> | ||
<br /> | <br /> | ||
| Line 386: | Line 446: | ||
Badness: 0.0203<br /> | Badness: 0.0203<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Pajara-Pajarous"></a><!-- ws:end:WikiTextHeadingRule:16 -->Pajarous</h2> | ||
Commas: 50/49, 55/54, 64/63<br /> | Commas: 50/49, 55/54, 64/63<br /> | ||
<br /> | <br /> | ||
| Line 395: | Line 455: | ||
Badness: 0.0283<br /> | Badness: 0.0283<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Pajara-13-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->13-limit</h2> | ||
Commas: 50/49, 55/54, 64/63, 65/63<br /> | Commas: 50/49, 55/54, 64/63, 65/63<br /> | ||
<br /> | <br /> | ||
| Line 404: | Line 464: | ||
Badness: 0.0252<br /> | Badness: 0.0252<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Pajara-Pajaro"></a><!-- ws:end:WikiTextHeadingRule:20 -->Pajaro</h2> | ||
Commas: 40/39, 50/49, 55/54, 64/63<br /> | Commas: 40/39, 50/49, 55/54, 64/63<br /> | ||
<br /> | <br /> | ||
| Line 413: | Line 473: | ||
Badness: 0.0274<br /> | Badness: 0.0274<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Diaschismic"></a><!-- ws:end:WikiTextHeadingRule:22 -->Diaschismic</h1> | ||
A simpler characterization than the one given by the normal comma list is that diaschismic adds 126/125 or 5120/5103 to the set of commas, and it can also be called 46&amp;58. However described, diaschismic has wedgie &lt;&lt;2 -4 -16 -11 -31 -26||, with a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. <a class="wiki_link" href="/58edo">58et</a> provides an excellent tuning, but an alternative is to make <a class="wiki_link" href="/7_4">7/4</a> just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58et.<br /> | A simpler characterization than the one given by the normal comma list is that diaschismic adds 126/125 or 5120/5103 to the set of commas, and it can also be called 46&amp;58. However described, diaschismic has wedgie &lt;&lt;2 -4 -16 -11 -31 -26||, with a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. <a class="wiki_link" href="/58edo">58et</a> provides an excellent tuning, but an alternative is to make <a class="wiki_link" href="/7_4">7/4</a> just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58et.<br /> | ||
<br /> | <br /> | ||
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EDOs: 46, 58, 104c, 162c<br /> | EDOs: 46, 58, 104c, 162c<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Diaschismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:24 -->11-limit</h2> | ||
Commas: 126/125, 176/175, 896/891<br /> | Commas: 126/125, 176/175, 896/891<br /> | ||
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EDOs: 46, 58, 104c, 162ce<br /> | EDOs: 46, 58, 104c, 162ce<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Diaschismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->13-limit</h2> | ||
Commas: 126/125, 196/195, 364/363, 2048/2025<br /> | Commas: 126/125, 196/195, 364/363, 2048/2025<br /> | ||
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EDOs: <a class="wiki_link" href="/46edo">46</a>, <a class="wiki_link" href="/58edo">58</a>, <a class="wiki_link" href="/104edo">104c</a>, <a class="wiki_link" href="/162edo">162cef</a><br /> | EDOs: <a class="wiki_link" href="/46edo">46</a>, <a class="wiki_link" href="/58edo">58</a>, <a class="wiki_link" href="/104edo">104c</a>, <a class="wiki_link" href="/162edo">162cef</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Diaschismic-17-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->17-limit</h2> | ||
Commas: 126/125, 136/135, 176/175, 196/195, 256/255<br /> | Commas: 126/125, 136/135, 176/175, 196/195, 256/255<br /> | ||
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EDOs: 46, 58, 104c<br /> | EDOs: 46, 58, 104c<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h1&gt; --><h1 id="toc15"><a name="Keen"></a><!-- ws:end:WikiTextHeadingRule:30 -->Keen</h1> | ||
Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie &lt;&lt;2 -4 18 -11 23 53||. It may also be described as the 22&amp;56 temperament. <a class="wiki_link" href="/78edo">78et</a> is a good tuning choice, and remains a good one in the 11-limit, where keen, &lt;&lt;2 -4 18 -12 ...||, is really more interesting, adding 100/99 and 385/384 to the commas.<br /> | Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie &lt;&lt;2 -4 18 -11 23 53||. It may also be described as the 22&amp;56 temperament. <a class="wiki_link" href="/78edo">78et</a> is a good tuning choice, and remains a good one in the 11-limit, where keen, &lt;&lt;2 -4 18 -12 ...||, is really more interesting, adding 100/99 and 385/384 to the commas.<br /> | ||
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EDOs: 22, 56, 78, 134b, 212b, 290b<br /> | EDOs: 22, 56, 78, 134b, 212b, 290b<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Keen-11-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->11-limit</h2> | ||
Commas: 100/99, 385/384, 1232/1215<br /> | Commas: 100/99, 385/384, 1232/1215<br /> | ||
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EDOs: 22, 56, 78, 212bf, 290bf<br /> | EDOs: 22, 56, 78, 212bf, 290bf<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Bidia"></a><!-- ws:end:WikiTextHeadingRule:34 -->Bidia</h1> | ||
Bidia adds 3136/3125 to the commas, splitting the period into 1/4 octave. It may be called the 12&amp;56 temperament.<br /> | Bidia adds 3136/3125 to the commas, splitting the period into 1/4 octave. It may be called the 12&amp;56 temperament.<br /> | ||
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Badness: 0.0565<br /> | Badness: 0.0565<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Bidia-11-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->11-limit</h2> | ||
Commas: 176/175, 896/891, 1375/1372<br /> | Commas: 176/175, 896/891, 1375/1372<br /> | ||
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Badness: 0.0402<br /> | Badness: 0.0402<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Bidia-13-limit"></a><!-- ws:end:WikiTextHeadingRule:38 -->13-limit</h2> | ||
Commas: 176/175, 325/324, 640/637, 896/891<br /> | Commas: 176/175, 325/324, 640/637, 896/891<br /> | ||
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Badness: 0.0411<br /> | Badness: 0.0411<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:40:&lt;h1&gt; --><h1 id="toc20"><a name="Echidna"></a><!-- ws:end:WikiTextHeadingRule:40 -->Echidna</h1> | ||
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie &lt;&lt;6 -12 10 -33 -1 57|| and may be called the 22&amp;58 temperament. <a class="wiki_link" href="/58edo">58et</a> or <a class="wiki_link" href="/80edo">80et</a> make for good tunings, or their vals can be add to &lt;138 219 321 388|.<br /> | Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie &lt;&lt;6 -12 10 -33 -1 57|| and may be called the 22&amp;58 temperament. <a class="wiki_link" href="/58edo">58et</a> or <a class="wiki_link" href="/80edo">80et</a> make for good tunings, or their vals can be add to &lt;138 219 321 388|.<br /> | ||
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Badness: 0.0580<br /> | Badness: 0.0580<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:42:&lt;h2&gt; --><h2 id="toc21"><a name="Echidna-11-limit"></a><!-- ws:end:WikiTextHeadingRule:42 -->11-limit</h2> | ||
Commas: 176/175, 896/891, 540/539<br /> | Commas: 176/175, 896/891, 540/539<br /> | ||
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Badness: 0.0260<br /> | Badness: 0.0260<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:44:&lt;h2&gt; --><h2 id="toc22"><a name="Echidna-13-limit"></a><!-- ws:end:WikiTextHeadingRule:44 -->13-limit</h2> | ||
Commas: 176/175, 351/350, 364/363, 540/539<br /> | Commas: 176/175, 351/350, 364/363, 540/539<br /> | ||
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Badness: 0.0237<br /> | Badness: 0.0237<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><h2 id="toc23"><a name="Echidna-17-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->17-limit</h2> | ||
Commas: 136/135, 176/175, 221/220, 256/255, 540/539<br /> | Commas: 136/135, 176/175, 221/220, 256/255, 540/539<br /> | ||
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Badness: 0.0203<br /> | Badness: 0.0203<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:48:&lt;h1&gt; --><h1 id="toc24"><a name="Echidnic"></a><!-- ws:end:WikiTextHeadingRule:48 -->Echidnic</h1> | ||
Commas: 686/675, 1029/1024<br /> | Commas: 686/675, 1029/1024<br /> | ||
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Badness: 0.0722<br /> | Badness: 0.0722<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:50:&lt;h2&gt; --><h2 id="toc25"><a name="Echidnic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:50 -->11-limit</h2> | ||
Commas: 385/384, 441/440, 686/675<br /> | Commas: 385/384, 441/440, 686/675<br /> | ||
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Badness: 0.0451<br /> | Badness: 0.0451<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:52:&lt;h2&gt; --><h2 id="toc26"><a name="Echidnic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:52 -->13-limit</h2> | ||
Commas: 91/90, 169/168, 385/384, 441/440<br /> | Commas: 91/90, 169/168, 385/384, 441/440<br /> | ||
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Compositions:<br /> | Compositions:<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:445:http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 --><a class="wiki_link_ext" href="http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3" rel="nofollow">http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3</a><!-- ws:end:WikiTextUrlRule:445 --><br /> | ||
(the description says &quot;lemba&quot; which has a similar sale structure but different mapping for 5)<br /> | (the description says &quot;lemba&quot; which has a similar sale structure but different mapping for 5)<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:54:&lt;h1&gt; --><h1 id="toc27"><a name="Shrutar"></a><!-- ws:end:WikiTextHeadingRule:54 -->Shrutar</h1> | ||
Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. With wedgie &lt;&lt;4 -8 14 -22 11 55||, it can also be described as 22&amp;46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. <a class="wiki_link" href="/68edo">68edo</a> makes for a good tuning, but another and excellent choice is a generator of 14^(1/7), making 7s just.<br /> | Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. With wedgie &lt;&lt;4 -8 14 -22 11 55||, it can also be described as 22&amp;46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. <a class="wiki_link" href="/68edo">68edo</a> makes for a good tuning, but another and excellent choice is a generator of 14^(1/7), making 7s just.<br /> | ||
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EDOs: 22, 46, 68, 182b, 250bc<br /> | EDOs: 22, 46, 68, 182b, 250bc<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:56:&lt;h2&gt; --><h2 id="toc28"><a name="Shrutar-11-limit"></a><!-- ws:end:WikiTextHeadingRule:56 -->11-limit</h2> | ||
Commas: 2048/2025, 245/243, 121/120<br /> | Commas: 2048/2025, 245/243, 121/120<br /> | ||
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EDOs: 22, 46, 68, 114, 296bce, 410bce<br /> | EDOs: 22, 46, 68, 114, 296bce, 410bce<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:58:&lt;h2&gt; --><h2 id="toc29"><a name="Shrutar-13-limit"></a><!-- ws:end:WikiTextHeadingRule:58 -->13-limit</h2> | ||
Commas: 121/120, 176/175, 196/195, 245/243<br /> | Commas: 121/120, 176/175, 196/195, 245/243<br /> | ||
<br /> | <br /> | ||
| Line 604: | Line 664: | ||
Badness: 0.0281<br /> | Badness: 0.0281<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:60:&lt;h2&gt; --><h2 id="toc30"><a name="Shrutar-17-limit"></a><!-- ws:end:WikiTextHeadingRule:60 -->17-limit</h2> | ||
Commas: 121/120, 136/135, 154/153, 176/175, 196/195<br /> | Commas: 121/120, 136/135, 154/153, 176/175, 196/195<br /> | ||
<br /> | <br /> | ||
| Line 613: | Line 673: | ||
Badness: 0.0187<br /> | Badness: 0.0187<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:62:&lt;h2&gt; --><h2 id="toc31"><a name="Shrutar-19-limit"></a><!-- ws:end:WikiTextHeadingRule:62 -->19-limit</h2> | ||
Commas: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342<br /> | Commas: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342<br /> | ||
<br /> | <br /> | ||
| Line 622: | Line 682: | ||
Badness: 0.0175<br /> | Badness: 0.0175<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:64:&lt;h1&gt; --><h1 id="toc32"><a name="Sruti"></a><!-- ws:end:WikiTextHeadingRule:64 -->Sruti</h1> | ||
Commas: 2048/2025, 19683/19600<br /> | Commas: 2048/2025, 19683/19600<br /> | ||
<br /> | <br /> | ||
| Line 632: | Line 692: | ||
Badness: 0.1174<br /> | Badness: 0.1174<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:66:&lt;h2&gt; --><h2 id="toc33"><a name="Sruti-11-limit"></a><!-- ws:end:WikiTextHeadingRule:66 -->11-limit</h2> | ||
Commas: 176/175, 243/242, 896/891<br /> | Commas: 176/175, 243/242, 896/891<br /> | ||
<br /> | <br /> | ||
| Line 641: | Line 701: | ||
Badness: 0.0415<br /> | Badness: 0.0415<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:68:&lt;h2&gt; --><h2 id="toc34"><a name="Sruti-13-limit"></a><!-- ws:end:WikiTextHeadingRule:68 -->13-limit</h2> | ||
Commas: 144/143, 176/175, 351/350, 676/675<br /> | Commas: 144/143, 176/175, 351/350, 676/675<br /> | ||
<br /> | <br /> | ||