Superpyth: Difference between revisions
Wikispaces>genewardsmith **Imported revision 319886746 - Original comment: ** |
Wikispaces>Kosmorsky **Imported revision 320803970 - 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:Kosmorsky|Kosmorsky]] and made on <tt>2012-04-15 19:04:26 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>320803970</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">[[toc|flat]] | <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]] | ||
**Superpyth**, a member of the [[Archytas clan]], has 4/3 as a generator, and the Archytas comma 64/63 is [[tempering out|tempered out]], so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for [[meantone]] and [[12edo]], with the understanding that, for example, A# is sharper than Bb (in contrast to meantone where A# is flatter than Bb, or 12edo where they are identical). An interesting coincidence is that the [[@http://en.wikipedia.org/wiki/Plastic_number|plastic number]]has a value of ~486.822 cents, which, taken as a generator and assuming an octave period, constitutes a variety of superpyth. | |||
**Superpyth**, a member of the [[Archytas clan]], has 4/3 as a generator, and the Archytas comma 64/63 is [[tempering out|tempered out]], so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for [[meantone]] and [[12edo]], with the understanding that, for example, A# is sharper than Bb (in contrast to meantone where A# is flatter than Bb, or 12edo where they are identical). | |||
If the 5th harmonic is used at all, it is mapped to -9 generators, so C-D# is 5/4. So superpyth is "the opposite of" septimal meantone in several different ways: meantone has 4/3 tempered wide so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 4/3 tempered narrow so that intervals of 7 are simple while intervals of 5 are complex. | If the 5th harmonic is used at all, it is mapped to -9 generators, so C-D# is 5/4. So superpyth is "the opposite of" septimal meantone in several different ways: meantone has 4/3 tempered wide so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 4/3 tempered narrow so that intervals of 7 are simple while intervals of 5 are complex. | ||
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MOSes include 5, 7, 12, 17, and 22. | MOSes include 5, 7, 12, 17, and 22. | ||
=Superpyth= | =Superpyth= | ||
Commas: 64/63, 245/243 | Commas: 64/63, 245/243 | ||
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Badness: 0.0323 | Badness: 0.0323 | ||
==11-limit== | ==11-limit== | ||
Commas: 64/63, 100/99, 245/243 | Commas: 64/63, 100/99, 245/243 | ||
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Badness: 0.0250 | Badness: 0.0250 | ||
==13-limit== | ==13-limit== | ||
Commas: 64/63, 78/77, 91/90, 100/99 | Commas: 64/63, 78/77, 91/90, 100/99 | ||
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Badness: 0.0247 | Badness: 0.0247 | ||
=Suprapyth= | =Suprapyth= | ||
Commas: 55/54, 64/63, 99/98 | Commas: 55/54, 64/63, 99/98 | ||
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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>Superpyth</title></head><body><!-- ws:start:WikiTextTocRule:26:&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:26 --><!-- ws:start:WikiTextTocRule:27: --><a href="#Superpyth">Superpyth</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Suprapyth">Suprapyth</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- 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: --><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | <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>Superpyth</title></head><body><!-- ws:start:WikiTextTocRule:26:&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:26 --><!-- ws:start:WikiTextTocRule:27: --><a href="#Superpyth">Superpyth</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Suprapyth">Suprapyth</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- 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: --><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | ||
<!-- ws:end:WikiTextTocRule:40 --> | <!-- ws:end:WikiTextTocRule:40 --><strong>Superpyth</strong>, a member of the <a class="wiki_link" href="/Archytas%20clan">Archytas clan</a>, has 4/3 as a generator, and the Archytas comma 64/63 is <a class="wiki_link" href="/tempering%20out">tempered out</a>, so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for <a class="wiki_link" href="/meantone">meantone</a> and <a class="wiki_link" href="/12edo">12edo</a>, with the understanding that, for example, A# is sharper than Bb (in contrast to meantone where A# is flatter than Bb, or 12edo where they are identical). An interesting coincidence is that the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Plastic_number" rel="nofollow" target="_blank">plastic number</a>has a value of ~486.822 cents, which, taken as a generator and assuming an octave period, constitutes a variety of superpyth.<br /> | ||
<strong>Superpyth</strong>, a member of the <a class="wiki_link" href="/Archytas%20clan">Archytas clan</a>, has 4/3 as a generator, and the Archytas comma 64/63 is <a class="wiki_link" href="/tempering%20out">tempered out</a>, so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for <a class="wiki_link" href="/meantone">meantone</a> and <a class="wiki_link" href="/12edo">12edo</a>, with the understanding that, for example, A# is sharper than Bb (in contrast to meantone where A# is flatter than Bb, or 12edo where they are identical).<br /> | |||
<br /> | <br /> | ||
If the 5th harmonic is used at all, it is mapped to -9 generators, so C-D# is 5/4. So superpyth is &quot;the opposite of&quot; septimal meantone in several different ways: meantone has 4/3 tempered wide so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 4/3 tempered narrow so that intervals of 7 are simple while intervals of 5 are complex.<br /> | If the 5th harmonic is used at all, it is mapped to -9 generators, so C-D# is 5/4. So superpyth is &quot;the opposite of&quot; septimal meantone in several different ways: meantone has 4/3 tempered wide so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 4/3 tempered narrow so that intervals of 7 are simple while intervals of 5 are complex.<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Superpyth"></a><!-- ws:end:WikiTextHeadingRule:0 -->Superpyth</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Superpyth"></a><!-- ws:end:WikiTextHeadingRule:0 -->Superpyth</h1> | ||
Commas: 64/63, 245/243<br /> | Commas: 64/63, 245/243<br /> | ||
<br /> | <br /> | ||
POTE generator: ~3/2 = 710.291<br /> | POTE generator: ~3/2 = 710.291<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Superpyth-11-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Superpyth-11-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->11-limit</h2> | ||
Commas: 64/63, 100/99, 245/243<br /> | Commas: 64/63, 100/99, 245/243<br /> | ||
<br /> | <br /> | ||
POTE generator: ~3/2 = 710.175<br /> | POTE generator: ~3/2 = 710.175<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Superpyth-13-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Superpyth-13-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->13-limit</h2> | ||
Commas: 64/63, 78/77, 91/90, 100/99<br /> | Commas: 64/63, 78/77, 91/90, 100/99<br /> | ||
<br /> | <br /> | ||
POTE generator: ~3/2 = 710.479<br /> | POTE generator: ~3/2 = 710.479<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Suprapyth"></a><!-- ws:end:WikiTextHeadingRule:6 -->Suprapyth</h1> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Suprapyth"></a><!-- ws:end:WikiTextHeadingRule:6 -->Suprapyth</h1> | ||
Commas: 55/54, 64/63, 99/98<br /> | Commas: 55/54, 64/63, 99/98<br /> | ||
<br /> | <br /> | ||
POTE generator: ~3/2 = 709.495<br /> | POTE generator: ~3/2 = 709.495<br /> | ||