Superpyth: Difference between revisions

Revision as of 19:04, 15 April 2012

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This revision was by author Kosmorsky and made on 2012-04-15 19:04:26 UTC.

The original revision id was 320803970.

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Original Wikitext content:

[[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.

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 intervals of 11 are desired the simplest reasonable way is to map 11/8 to 6 generators (so 11/8 is a "diminished fifth"), by tempering out 99/98.
This temperament is called "**supra**", or "**suprapyth**" if you include 5 as well.

MOSes include 5, 7, 12, 17, and 22.

=Superpyth= 
Commas: 64/63, 245/243

POTE generator: ~3/2 = 710.291

Map: [<1 0 -12 6|, <0 1 9 -2|]
Wedgie: <<1 9 -2 12 -6 -30||
EDOs: 5, 17, 22, 27, 49
Badness: 0.0323

==11-limit== 
Commas: 64/63, 100/99, 245/243

POTE generator: ~3/2 = 710.175

Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|]
EDOs: 22, 27e, 49
Badness: 0.0250

==13-limit== 
Commas: 64/63, 78/77, 91/90, 100/99

POTE generator: ~3/2 = 710.479

Map: [<1 0 -12 6 -22 -17|, <0 1 9 -2 16 13|]
EDOs: 22, 27e, 49, 76bcde
Badness: 0.0247

=Suprapyth= 
Commas: 55/54, 64/63, 99/98

POTE generator: ~3/2 = 709.495

Map: [<1 0 -12 6 13|, <0 1 9 -2 -6|]
EDOs: 5, 17, 22
Badness: 0.0328

==Interval chains== 
===Basic superpyth (2.3.7)=== 
|| 1146.61 || 437.29 || 927.97 || 218.64 || 709.32 || 0 || 490.68 || 981.36 || 272.03 || 762.71 || 53.39 ||
|| 27/14 || 9/7 || 12/7 || 9/8~8/7 || 3/2 || 1/1 || 4/3 || 7/4~16/9 || 7/6 || 14/9 || 28/27 ||
===Full 7-limit superpyth=== 
|| 613.20 || 1102.91 || 392.62 || 882.33 || 172.04 || 661.75 || 1151.46 || 441.16 || 930.87 || 220.58 || 710.29 || 0 || 489.71 || 979.42 || 269.13 || 758.84 || 48.54 || 538.25 || 1027.96 || 317.67 || 807.38 || 97.09 || 586.80 ||
|| 10/7 || 15/8 || 5/4 || 5/3 || 10/9 ||   || 27/14 || 9/7 || 12/7 || 9/8~8/7 || 3/2 || 1/1 || 4/3 || 7/4~16/9 || 7/6 || 14/9 || 28/27 ||   || 9/5 || 6/5 || 8/5 || 16/15 || 7/5 ||
===Supra (2.3.7.11)=== 
|| 857.54 || 150.35 || 643.15 || 1135.96 || 428.77 || 921.58 || 214.38 || 707.19 || 0 || 492.81 || 985.62 || 278.42 || 771.23 || 64.04 || 556.85 || 1049.65 || 342.46 ||
|| 18/11 || 12/11 || 16/11 || 27/14 || 14/11~9/7 || 12/7 || 9/8~8/7 || 3/2 || 1/1 || 4/3 || 7/4~16/9 || 7/6 || 14/9~11/7 || 33/32~28/27 || 11/8 || 11/6 || 11/9 ||
===Full 11-limit suprapyth=== 
|| 604.44 || 1094.94 || 385.45 || 875.96 || 166.46 || 656.97 || 1147.47 || 437.98 || 928.48 || 218.99 || 709.49 || 0 || 490.51 || 981.01 || 271.52 || 762.02 || 52.53 || 543.03 || 1033.54 || 324.04 || 814.55 || 105.06 || 595.56 ||
|| 10/7 || 15/8 || 5/4 || 18/11~5/3 || 12/11~10/9 || 16/11 || 27/14 || 14/11~9/7 || 12/7 || 9/8~8/7 || 3/2 || 1/1 || 4/3 || 7/4~16/9 || 7/6 || 14/9~11/7 || 33/32~28/27 || 11/8 || 9/5~11/6 || 6/5~11/9 || 8/5 || 16/15 || 7/5 ||

==MOSes== 
===5-note (LsLss, proper)=== 
See [[2L 3s]].
===7-note (LLLsLLs, improper)=== 
See [[5L 2s]]. In contrast to the meantone diatonic scale, the superpyth diatonic is slightly improper.
===12-note (LsLsLssLsLss, borderline improper)=== 
See [[5L 7s]]. The boundary of propriety is [[17edo]].

Original HTML content:

<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 --><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 />
<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 />
<br />
If intervals of 11 are desired the simplest reasonable way is to map 11/8 to 6 generators (so 11/8 is a &quot;diminished fifth&quot;), by tempering out 99/98.<br />
This temperament is called &quot;<strong>supra</strong>&quot;, or &quot;<strong>suprapyth</strong>&quot; if you include 5 as well.<br />
<br />
MOSes include 5, 7, 12, 17, and 22.<br />
<br />
<!-- 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 />
<br />
POTE generator: ~3/2 = 710.291<br />
<br />
Map: [&lt;1 0 -12 6|, &lt;0 1 9 -2|]<br />
Wedgie: &lt;&lt;1 9 -2 12 -6 -30||<br />
EDOs: 5, 17, 22, 27, 49<br />
Badness: 0.0323<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>
 Commas: 64/63, 100/99, 245/243<br />
<br />
POTE generator: ~3/2 = 710.175<br />
<br />
Map: [&lt;1 0 -12 6 -22|, &lt;0 1 9 -2 16|]<br />
EDOs: 22, 27e, 49<br />
Badness: 0.0250<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>
 Commas: 64/63, 78/77, 91/90, 100/99<br />
<br />
POTE generator: ~3/2 = 710.479<br />
<br />
Map: [&lt;1 0 -12 6 -22 -17|, &lt;0 1 9 -2 16 13|]<br />
EDOs: 22, 27e, 49, 76bcde<br />
Badness: 0.0247<br />
<br />
<!-- 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 />
<br />
POTE generator: ~3/2 = 709.495<br />
<br />
Map: [&lt;1 0 -12 6 13|, &lt;0 1 9 -2 -6|]<br />
EDOs: 5, 17, 22<br />
Badness: 0.0328<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Suprapyth-Interval chains"></a><!-- ws:end:WikiTextHeadingRule:8 -->Interval chains</h2>
 <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Suprapyth-Interval chains-Basic superpyth (2.3.7)"></a><!-- ws:end:WikiTextHeadingRule:10 -->Basic superpyth (2.3.7)</h3>
 

<table class="wiki_table">
    <tr>
        <td>1146.61<br />
</td>
        <td>437.29<br />
</td>
        <td>927.97<br />
</td>
        <td>218.64<br />
</td>
        <td>709.32<br />
</td>
        <td>0<br />
</td>
        <td>490.68<br />
</td>
        <td>981.36<br />
</td>
        <td>272.03<br />
</td>
        <td>762.71<br />
</td>
        <td>53.39<br />
</td>
    </tr>
    <tr>
        <td>27/14<br />
</td>
        <td>9/7<br />
</td>
        <td>12/7<br />
</td>
        <td>9/8~8/7<br />
</td>
        <td>3/2<br />
</td>
        <td>1/1<br />
</td>
        <td>4/3<br />
</td>
        <td>7/4~16/9<br />
</td>
        <td>7/6<br />
</td>
        <td>14/9<br />
</td>
        <td>28/27<br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Suprapyth-Interval chains-Full 7-limit superpyth"></a><!-- ws:end:WikiTextHeadingRule:12 -->Full 7-limit superpyth</h3>
 

<table class="wiki_table">
    <tr>
        <td>613.20<br />
</td>
        <td>1102.91<br />
</td>
        <td>392.62<br />
</td>
        <td>882.33<br />
</td>
        <td>172.04<br />
</td>
        <td>661.75<br />
</td>
        <td>1151.46<br />
</td>
        <td>441.16<br />
</td>
        <td>930.87<br />
</td>
        <td>220.58<br />
</td>
        <td>710.29<br />
</td>
        <td>0<br />
</td>
        <td>489.71<br />
</td>
        <td>979.42<br />
</td>
        <td>269.13<br />
</td>
        <td>758.84<br />
</td>
        <td>48.54<br />
</td>
        <td>538.25<br />
</td>
        <td>1027.96<br />
</td>
        <td>317.67<br />
</td>
        <td>807.38<br />
</td>
        <td>97.09<br />
</td>
        <td>586.80<br />
</td>
    </tr>
    <tr>
        <td>10/7<br />
</td>
        <td>15/8<br />
</td>
        <td>5/4<br />
</td>
        <td>5/3<br />
</td>
        <td>10/9<br />
</td>
        <td><br />
</td>
        <td>27/14<br />
</td>
        <td>9/7<br />
</td>
        <td>12/7<br />
</td>
        <td>9/8~8/7<br />
</td>
        <td>3/2<br />
</td>
        <td>1/1<br />
</td>
        <td>4/3<br />
</td>
        <td>7/4~16/9<br />
</td>
        <td>7/6<br />
</td>
        <td>14/9<br />
</td>
        <td>28/27<br />
</td>
        <td><br />
</td>
        <td>9/5<br />
</td>
        <td>6/5<br />
</td>
        <td>8/5<br />
</td>
        <td>16/15<br />
</td>
        <td>7/5<br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Suprapyth-Interval chains-Supra (2.3.7.11)"></a><!-- ws:end:WikiTextHeadingRule:14 -->Supra (2.3.7.11)</h3>
 

<table class="wiki_table">
    <tr>
        <td>857.54<br />
</td>
        <td>150.35<br />
</td>
        <td>643.15<br />
</td>
        <td>1135.96<br />
</td>
        <td>428.77<br />
</td>
        <td>921.58<br />
</td>
        <td>214.38<br />
</td>
        <td>707.19<br />
</td>
        <td>0<br />
</td>
        <td>492.81<br />
</td>
        <td>985.62<br />
</td>
        <td>278.42<br />
</td>
        <td>771.23<br />
</td>
        <td>64.04<br />
</td>
        <td>556.85<br />
</td>
        <td>1049.65<br />
</td>
        <td>342.46<br />
</td>
    </tr>
    <tr>
        <td>18/11<br />
</td>
        <td>12/11<br />
</td>
        <td>16/11<br />
</td>
        <td>27/14<br />
</td>
        <td>14/11~9/7<br />
</td>
        <td>12/7<br />
</td>
        <td>9/8~8/7<br />
</td>
        <td>3/2<br />
</td>
        <td>1/1<br />
</td>
        <td>4/3<br />
</td>
        <td>7/4~16/9<br />
</td>
        <td>7/6<br />
</td>
        <td>14/9~11/7<br />
</td>
        <td>33/32~28/27<br />
</td>
        <td>11/8<br />
</td>
        <td>11/6<br />
</td>
        <td>11/9<br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Suprapyth-Interval chains-Full 11-limit suprapyth"></a><!-- ws:end:WikiTextHeadingRule:16 -->Full 11-limit suprapyth</h3>
 

<table class="wiki_table">
    <tr>
        <td>604.44<br />
</td>
        <td>1094.94<br />
</td>
        <td>385.45<br />
</td>
        <td>875.96<br />
</td>
        <td>166.46<br />
</td>
        <td>656.97<br />
</td>
        <td>1147.47<br />
</td>
        <td>437.98<br />
</td>
        <td>928.48<br />
</td>
        <td>218.99<br />
</td>
        <td>709.49<br />
</td>
        <td>0<br />
</td>
        <td>490.51<br />
</td>
        <td>981.01<br />
</td>
        <td>271.52<br />
</td>
        <td>762.02<br />
</td>
        <td>52.53<br />
</td>
        <td>543.03<br />
</td>
        <td>1033.54<br />
</td>
        <td>324.04<br />
</td>
        <td>814.55<br />
</td>
        <td>105.06<br />
</td>
        <td>595.56<br />
</td>
    </tr>
    <tr>
        <td>10/7<br />
</td>
        <td>15/8<br />
</td>
        <td>5/4<br />
</td>
        <td>18/11~5/3<br />
</td>
        <td>12/11~10/9<br />
</td>
        <td>16/11<br />
</td>
        <td>27/14<br />
</td>
        <td>14/11~9/7<br />
</td>
        <td>12/7<br />
</td>
        <td>9/8~8/7<br />
</td>
        <td>3/2<br />
</td>
        <td>1/1<br />
</td>
        <td>4/3<br />
</td>
        <td>7/4~16/9<br />
</td>
        <td>7/6<br />
</td>
        <td>14/9~11/7<br />
</td>
        <td>33/32~28/27<br />
</td>
        <td>11/8<br />
</td>
        <td>9/5~11/6<br />
</td>
        <td>6/5~11/9<br />
</td>
        <td>8/5<br />
</td>
        <td>16/15<br />
</td>
        <td>7/5<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Suprapyth-MOSes"></a><!-- ws:end:WikiTextHeadingRule:18 -->MOSes</h2>
 <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Suprapyth-MOSes-5-note (LsLss, proper)"></a><!-- ws:end:WikiTextHeadingRule:20 -->5-note (LsLss, proper)</h3>
 See <a class="wiki_link" href="/2L%203s">2L 3s</a>.<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Suprapyth-MOSes-7-note (LLLsLLs, improper)"></a><!-- ws:end:WikiTextHeadingRule:22 -->7-note (LLLsLLs, improper)</h3>
 See <a class="wiki_link" href="/5L%202s">5L 2s</a>. In contrast to the meantone diatonic scale, the superpyth diatonic is slightly improper.<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Suprapyth-MOSes-12-note (LsLsLssLsLss, borderline improper)"></a><!-- ws:end:WikiTextHeadingRule:24 -->12-note (LsLsLssLsLss, borderline improper)</h3>
 See <a class="wiki_link" href="/5L%207s">5L 7s</a>. The boundary of propriety is <a class="wiki_link" href="/17edo">17edo</a>.</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2012-04-12 13:13:17 UTC</tt>.<br>
+: 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>319886746</tt>.<br>
+: The original revision id was <tt>320803970</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>
 <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]]
+**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.
@@ Line 17: / Line 16: @@
 MOSes include 5, 7, 12, 17, and 22.
 =Superpyth=
 Commas: 64/63, 245/243
@@ Line 27: / Line 26: @@
 Badness: 0.0323
 ==11-limit==
 Commas: 64/63, 100/99, 245/243
@@ Line 36: / Line 35: @@
 Badness: 0.0250
 ==13-limit==
 Commas: 64/63, 78/77, 91/90, 100/99
@@ Line 45: / Line 44: @@
 Badness: 0.0247
 =Suprapyth=
 Commas: 55/54, 64/63, 99/98
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 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Superpyth&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:26:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;a href="#Superpyth"&gt;Superpyth&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt; | &lt;a href="#Suprapyth"&gt;Suprapyth&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;strong&gt;Superpyth&lt;/strong&gt;, a member of the &lt;a class="wiki_link" href="/Archytas%20clan"&gt;Archytas clan&lt;/a&gt;, has 4/3 as a generator, and the Archytas comma 64/63 is &lt;a class="wiki_link" href="/tempering%20out"&gt;tempered out&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; and &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Plastic_number" rel="nofollow" target="_blank"&gt;plastic number&lt;/a&gt;has a value of ~486.822 cents, which, taken as a generator and assuming an octave period, constitutes a variety of superpyth.&lt;br /&gt;
-&lt;strong&gt;Superpyth&lt;/strong&gt;, a member of the &lt;a class="wiki_link" href="/Archytas%20clan"&gt;Archytas clan&lt;/a&gt;, has 4/3 as a generator, and the Archytas comma 64/63 is &lt;a class="wiki_link" href="/tempering%20out"&gt;tempered out&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; and &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, 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).&lt;br /&gt;
 &lt;br /&gt;
 If the 5th harmonic is used at all, it is mapped to -9 generators, so C-D# is 5/4. So superpyth is &amp;quot;the opposite of&amp;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.&lt;br /&gt;
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Superpyth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Superpyth&lt;/h1&gt;
-Commas: 64/63, 245/243&lt;br /&gt;
+ Commas: 64/63, 245/243&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~3/2 = 710.291&lt;br /&gt;
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Superpyth-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
-Commas: 64/63, 100/99, 245/243&lt;br /&gt;
+ Commas: 64/63, 100/99, 245/243&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~3/2 = 710.175&lt;br /&gt;
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Superpyth-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
-Commas: 64/63, 78/77, 91/90, 100/99&lt;br /&gt;
+ Commas: 64/63, 78/77, 91/90, 100/99&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~3/2 = 710.479&lt;br /&gt;
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Suprapyth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Suprapyth&lt;/h1&gt;
-Commas: 55/54, 64/63, 99/98&lt;br /&gt;
+ Commas: 55/54, 64/63, 99/98&lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~3/2 = 709.495&lt;br /&gt;

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