Schismatic family: Difference between revisions
Wikispaces>xenwolf **Imported revision 238466687 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 239340937 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-29 12:29:01 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>239340937</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The 5-limit parent comma for the schismatic family is the [[schisma]] of 32805/32768, which is the amount by which the Pythagorean comma exceeds the [[Didymus comma]] ([[81_80|81/80]]), or alternatively put, the difference between a just major third and a just diminished fourth. Its [[monzo]] is |-15 8 1>, and flipping that yields <<1 -8 -15|| for the [[Wedgies and Multivals|wedgie]]. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768. | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]] | ||
=Five limit= | |||
The 5-limit parent comma for the schismatic family is the [[schisma]] of 32805/32768, which is the amount by which the Pythagorean comma exceeds the [[Didymus comma]] ([[81_80|81/80]]), or alternatively put, the difference between a just major third and a just diminished fourth. Its [[monzo]] is |-15 8 1>, and flipping that yields <<1 -8 -15|| for the [[Wedgies and Multivals|wedgie]]. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768. | |||
The 5-limit version of the temperament is a [[Microtempering|microtemperament]], sometimes called helmholtz or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schisma schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. | The 5-limit version of the temperament is a [[Microtempering|microtemperament]], sometimes called helmholtz or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schisma schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. | ||
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EDOs: [[12edo|12]], [[29edo|29]], [[41edo|41]], [[53edo|53]], [[118edo|118]], [[171edo|171]], [[289edo|289]], 460, 749 | EDOs: [[12edo|12]], [[29edo|29]], [[41edo|41]], [[53edo|53]], [[118edo|118]], [[171edo|171]], [[289edo|289]], 460, 749 | ||
=Seven limit children= | |||
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding |25 -14 0 -1> gives garibaldi, |-44 26 0 1> grackle, |6 -2 0 -1> schism and |-59 39 0 -1> pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2> and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3>, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3> with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4> and slices the fifth in four. | The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding |25 -14 0 -1> gives garibaldi, |-44 26 0 1> grackle, |6 -2 0 -1> schism and |-59 39 0 -1> pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2> and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3>, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3> with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4> and slices the fifth in four. | ||
=Garibaldi= | =Garibaldi= | ||
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Badness: 0.0112</pre></div> | Badness: 0.0112</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>Schismatic family</title></head><body>The 5-limit parent comma for the schismatic family is the <a class="wiki_link" href="/schisma">schisma</a> of 32805/32768, which is the amount by which the Pythagorean comma exceeds the <a class="wiki_link" href="/Didymus%20comma">Didymus comma</a> (<a class="wiki_link" href="/81_80">81/80</a>), or alternatively put, the difference between a just major third and a just diminished fourth. Its <a class="wiki_link" href="/monzo">monzo</a> is |-15 8 1&gt;, and flipping that yields &lt;&lt;1 -8 -15|| for the <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768. <br /> | <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>Schismatic family</title></head><body><!-- ws:start:WikiTextTocRule:30:&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:30 --><!-- ws:start:WikiTextTocRule:31: --><a href="#Five limit">Five limit</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --> | <a href="#Seven limit children">Seven limit children</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#Garibaldi">Garibaldi</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --> | <a href="#Guiron">Guiron</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="#Pontiac">Pontiac</a><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --> | <a href="#Grackle">Grackle</a><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --> | <a href="#Bischismic">Bischismic</a><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --> | <a href="#Term">Term</a><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --> | <a href="#Sesquiquartififths">Sesquiquartififths</a><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --> | ||
<!-- ws:end:WikiTextTocRule:46 --><br /> | |||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Five limit"></a><!-- ws:end:WikiTextHeadingRule:0 -->Five limit</h1> | |||
The 5-limit parent comma for the schismatic family is the <a class="wiki_link" href="/schisma">schisma</a> of 32805/32768, which is the amount by which the Pythagorean comma exceeds the <a class="wiki_link" href="/Didymus%20comma">Didymus comma</a> (<a class="wiki_link" href="/81_80">81/80</a>), or alternatively put, the difference between a just major third and a just diminished fourth. Its <a class="wiki_link" href="/monzo">monzo</a> is |-15 8 1&gt;, and flipping that yields &lt;&lt;1 -8 -15|| for the <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768. <br /> | |||
<br /> | <br /> | ||
The 5-limit version of the temperament is a <a class="wiki_link" href="/Microtempering">microtemperament</a>, sometimes called helmholtz or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. <a class="wiki_link" href="/53edo">53edo</a> is a possible tuning for schismatic, but you need <a class="wiki_link" href="/118edo">118edo</a> if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schisma schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering.<br /> | The 5-limit version of the temperament is a <a class="wiki_link" href="/Microtempering">microtemperament</a>, sometimes called helmholtz or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. <a class="wiki_link" href="/53edo">53edo</a> is a possible tuning for schismatic, but you need <a class="wiki_link" href="/118edo">118edo</a> if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schisma schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering.<br /> | ||
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EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/29edo">29</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/118edo">118</a>, <a class="wiki_link" href="/171edo">171</a>, <a class="wiki_link" href="/289edo">289</a>, 460, 749<br /> | EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/29edo">29</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/118edo">118</a>, <a class="wiki_link" href="/171edo">171</a>, <a class="wiki_link" href="/289edo">289</a>, 460, 749<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:2 -->Seven limit children</h1> | ||
The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> defines which 7-limit family member we are looking at. Adding |25 -14 0 -1&gt; gives garibaldi, |-44 26 0 1&gt; grackle, |6 -2 0 -1&gt; schism and |-59 39 0 -1&gt; pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2&gt; and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3&gt;, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3&gt; with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4&gt; and slices the fifth in four.<br /> | The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> defines which 7-limit family member we are looking at. Adding |25 -14 0 -1&gt; gives garibaldi, |-44 26 0 1&gt; grackle, |6 -2 0 -1&gt; schism and |-59 39 0 -1&gt; pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2&gt; and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3&gt;, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3&gt; with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4&gt; and slices the fifth in four.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Garibaldi"></a><!-- ws:end:WikiTextHeadingRule:4 -->Garibaldi</h1> | |||
<!-- ws:start:WikiTextHeadingRule: | |||
Commas: 225/224, 3125/3087<br /> | Commas: 225/224, 3125/3087<br /> | ||
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Badness: 0.0216<br /> | Badness: 0.0216<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Garibaldi-11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h2> | ||
Commas: 225/224, 385/384, 2200/2187<br /> | Commas: 225/224, 385/384, 2200/2187<br /> | ||
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Badness: 0.0274<br /> | Badness: 0.0274<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Garibaldi-11-limit-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h3> | ||
Commas: 225/224, 275/273, 325/325, 385/384<br /> | Commas: 225/224, 275/273, 325/325, 385/384<br /> | ||
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Badness: 0.0207<br /> | Badness: 0.0207<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Garibaldi-Cassandra"></a><!-- ws:end:WikiTextHeadingRule:10 -->Cassandra</h2> | ||
Commas: 100/99, 225/224, 245/242<br /> | Commas: 100/99, 225/224, 245/242<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0236<br /> | Badness: 0.0236<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Garibaldi-Cassandra-13-limit Cassandra"></a><!-- ws:end:WikiTextHeadingRule:12 -->13-limit Cassandra</h3> | ||
Commas: 100/99, 105/104, 196/195, 245/242<br /> | Commas: 100/99, 105/104, 196/195, 245/242<br /> | ||
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Badness: 0.0207<br /> | Badness: 0.0207<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Guiron"></a><!-- ws:end:WikiTextHeadingRule:14 -->Guiron</h1> | ||
<a class="wiki_link" href="/Comma">Commas</a>: 1029/1024, 10976/10935<br /> | <a class="wiki_link" href="/Comma">Commas</a>: 1029/1024, 10976/10935<br /> | ||
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Badness: 0.0475<br /> | Badness: 0.0475<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Guiron-11-limit"></a><!-- ws:end:WikiTextHeadingRule:16 -->11-limit</h2> | ||
<a class="wiki_link" href="/Comma">Commas</a>: 385/384, 441/440, 10976/10935<br /> | <a class="wiki_link" href="/Comma">Commas</a>: 385/384, 441/440, 10976/10935<br /> | ||
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Badness: 0.0266<br /> | Badness: 0.0266<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Guiron-13-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->13-limit</h2> | ||
Commas: 196/195, 352/351, 385/384, 729/728<br /> | Commas: 196/195, 352/351, 385/384, 729/728<br /> | ||
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Badness: 0.0284<br /> | Badness: 0.0284<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Pontiac"></a><!-- ws:end:WikiTextHeadingRule:20 -->Pontiac</h1> | ||
Commas: 32805/32768, 4375/4374<br /> | Commas: 32805/32768, 4375/4374<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0141<br /> | Badness: 0.0141<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Grackle"></a><!-- ws:end:WikiTextHeadingRule:22 -->Grackle</h1> | ||
Commas: 126/125, 32805/32768<br /> | Commas: 126/125, 32805/32768<br /> | ||
<br /> | <br /> | ||
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Badness: 0.070<br /> | Badness: 0.070<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Bischismic"></a><!-- ws:end:WikiTextHeadingRule:24 -->Bischismic</h1> | ||
Commas: 3136/3125, 32805/32768<br /> | Commas: 3136/3125, 32805/32768<br /> | ||
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Badness: 0.0547<br /> | Badness: 0.0547<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name="Term"></a><!-- ws:end:WikiTextHeadingRule:26 -->Term</h1> | ||
Commas: 32805/32768, 250047/250000<br /> | Commas: 32805/32768, 250047/250000<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0200<br /> | Badness: 0.0200<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Sesquiquartififths"></a><!-- ws:end:WikiTextHeadingRule:28 -->Sesquiquartififths</h1> | ||
Commas: 2401/2400, 32805/32768<br /> | Commas: 2401/2400, 32805/32768<br /> | ||
<br /> | <br /> | ||