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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<span style="display: block; text-align: right;">[[:de:Superpyth Deutsch]]</span>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
__FORCETOC__
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2016-07-05 04:39:47 UTC</tt>.<br>
-----
: The original revision id was <tt>586573097</tt>.<br>
'''Superpyth''', a member of the [[Archytas_clan|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|meantone]] and [[12edo|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.
: 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">&lt;span style="display: block; text-align: right;"&gt;[[xenharmonie/Superpyth|Deutsch]]
&lt;/span&gt;
[[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 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.
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.
 
This temperament is called "'''supra'''", or "'''suprapyth'''" if you include 5 as well.


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


Line 26: Line 18:


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


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


Line 36: Line 31:


Map: [&lt;1 0 -12 6 -22|, &lt;0 1 9 -2 16|]
Map: [&lt;1 0 -12 6 -22|, &lt;0 1 9 -2 16|]
EDOs: 22, 27e, 49
EDOs: 22, 27e, 49
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


Line 45: Line 42:


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


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


Line 54: Line 53:


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


==Interval chains==  
==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 ||
===Basic superpyth (2.3.7)===
|| 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===  
{| class="wikitable"
|| 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 ||
| | 1146.61
===Supra (2.3.7.11)===  
| | 437.29
|| 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 ||
| | 927.97
|| 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 ||
| | 218.64
===Full 11-limit suprapyth===  
| | 709.32
|| 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 ||
| | 0
|| 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 ||
| | 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===
 
{| class="wikitable"
|-
| | 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)===
 
{| class="wikitable"
|-
| | 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===


==MOSes==
{| class="wikitable"
===5-note (LsLss, proper)===
|-
See [[2L 3s]].
| | 604.44
===7-note (LLLsLLs, improper)===
| | 1094.94
See [[5L 2s]]. In contrast to the meantone diatonic scale, the superpyth diatonic is slightly improper.
| | 385.45
===12-note (LsLsLssLsLss, borderline improper)===
| | 875.96
See [[5L 7s]]. The boundary of propriety is [[17edo]].
| | 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
|}


=Music=  
==MOSes==
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3|12of22studyPentUp4thsMstr]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3|12of22gamelan1b]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3|12of22study3 (children's story)]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3|12of22study7]]
By [[Joel Grant Taylor]], all in Superpyth[12] in 22edo tuning.</pre></div>
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Superpyth&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="display: block; text-align: right;"&gt;&lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/Superpyth"&gt;Deutsch&lt;/a&gt;&lt;br /&gt;
&lt;/span&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextTocRule:28:&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:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;a href="#Superpyth"&gt;Superpyth&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&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;a href="#Suprapyth"&gt;Suprapyth&lt;/a&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;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt; | &lt;a href="#Music"&gt;Music&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;
&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;br /&gt;
&lt;hr /&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;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;
&lt;br /&gt;
If intervals of 11 are desired the simplest reasonable way is to map 11/8 to 6 generators (so 11/8 is a &amp;quot;diminished fifth&amp;quot;), by tempering out 99/98.&lt;br /&gt;
This temperament is called &amp;quot;&lt;strong&gt;supra&lt;/strong&gt;&amp;quot;, or &amp;quot;&lt;strong&gt;suprapyth&lt;/strong&gt;&amp;quot; if you include 5 as well.&lt;br /&gt;
&lt;br /&gt;
MOSes include 5, 7, 12, 17, and 22.&lt;br /&gt;
&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;
&lt;br /&gt;
POTE generator: ~3/2 = 710.291&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 -12 6|, &amp;lt;0 1 9 -2|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;1 9 -2 12 -6 -30||&lt;br /&gt;
EDOs: 5, 17, 22, 27, 49&lt;br /&gt;
Badness: 0.0323&lt;br /&gt;
&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;
&lt;br /&gt;
POTE generator: ~3/2 = 710.175&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 -12 6 -22|, &amp;lt;0 1 9 -2 16|]&lt;br /&gt;
EDOs: 22, 27e, 49&lt;br /&gt;
Badness: 0.0250&lt;br /&gt;
&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;
&lt;br /&gt;
POTE generator: ~3/2 = 710.479&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 -12 6 -22 -17|, &amp;lt;0 1 9 -2 16 13|]&lt;br /&gt;
EDOs: 22, 27e, 49, 76bcde&lt;br /&gt;
Badness: 0.0247&lt;br /&gt;
&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;
&lt;br /&gt;
POTE generator: ~3/2 = 709.495&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 -12 6 13|, &amp;lt;0 1 9 -2 -6|]&lt;br /&gt;
EDOs: 5, 17, 22&lt;br /&gt;
Badness: 0.0328&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Suprapyth-Interval chains"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Interval chains&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Suprapyth-Interval chains-Basic superpyth (2.3.7)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Basic superpyth (2.3.7)&lt;/h3&gt;


&lt;table class="wiki_table"&gt;
===5-note (LsLss, proper)===
    &lt;tr&gt;
See [[2L_3s|2L 3s]].
        &lt;td&gt;1146.61&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;437.29&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;927.97&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;218.64&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;709.32&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;490.68&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;981.36&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;272.03&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;762.71&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;53.39&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;27/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8~8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4~16/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;28/27&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="Suprapyth-Interval chains-Full 7-limit superpyth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Full 7-limit superpyth&lt;/h3&gt;
===7-note (LLLsLLs, improper)===
See [[5L_2s|5L 2s]]. In contrast to the meantone diatonic scale, the superpyth diatonic is slightly improper.


&lt;table class="wiki_table"&gt;
===12-note (LsLsLssLsLss, borderline improper)===
    &lt;tr&gt;
See [[5L_7s|5L 7s]]. The boundary of propriety is [[17edo|17edo]].
        &lt;td&gt;613.20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1102.91&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;392.62&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;882.33&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;172.04&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;661.75&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1151.46&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;441.16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;930.87&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;220.58&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;710.29&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;489.71&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;979.42&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;269.13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;758.84&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;48.54&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;538.25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1027.96&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;317.67&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;807.38&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;97.09&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;586.80&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8~8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4~16/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;28/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="Suprapyth-Interval chains-Supra (2.3.7.11)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Supra (2.3.7.11)&lt;/h3&gt;
=Music=
[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3 12of22studyPentUp4thsMstr]


&lt;table class="wiki_table"&gt;
[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3 12of22gamelan1b]
    &lt;tr&gt;
        &lt;td&gt;857.54&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;150.35&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;643.15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1135.96&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;428.77&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;921.58&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;214.38&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;707.19&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;492.81&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;985.62&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;278.42&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;771.23&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;64.04&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;556.85&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1049.65&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;342.46&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/11~9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8~8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4~16/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/9~11/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;33/32~28/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="Suprapyth-Interval chains-Full 11-limit suprapyth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Full 11-limit suprapyth&lt;/h3&gt;
[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 12of22study3 (children's story)]


&lt;table class="wiki_table"&gt;
[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 12of22study7]
    &lt;tr&gt;
        &lt;td&gt;604.44&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1094.94&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;385.45&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;875.96&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;166.46&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;656.97&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1147.47&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;437.98&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;928.48&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;218.99&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;709.49&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;490.51&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;981.01&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.52&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;762.02&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;52.53&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;543.03&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1033.54&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;324.04&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;814.55&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;105.06&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;595.56&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;18/11~5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/11~10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/11~9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8~8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4~16/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/9~11/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;33/32~28/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/5~11/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5~11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
By [[Joel_Grant_Taylor|Joel Grant Taylor]], all in Superpyth[12] in 22edo tuning.
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Suprapyth-MOSes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;MOSes&lt;/h2&gt;
[[Category:archytas]]
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&gt;&lt;a name="Suprapyth-MOSes-5-note (LsLss, proper)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;5-note (LsLss, proper)&lt;/h3&gt;
[[Category:todo:add_definition]]
See &lt;a class="wiki_link" href="/2L%203s"&gt;2L 3s&lt;/a&gt;.&lt;br /&gt;
[[Category:todo:intro]]
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&gt;&lt;a name="Suprapyth-MOSes-7-note (LLLsLLs, improper)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;7-note (LLLsLLs, improper)&lt;/h3&gt;
See &lt;a class="wiki_link" href="/5L%202s"&gt;5L 2s&lt;/a&gt;. In contrast to the meantone diatonic scale, the superpyth diatonic is slightly improper.&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="Suprapyth-MOSes-12-note (LsLsLssLsLss, borderline improper)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;12-note (LsLsLssLsLss, borderline improper)&lt;/h3&gt;
See &lt;a class="wiki_link" href="/5L%207s"&gt;5L 7s&lt;/a&gt;. The boundary of propriety is &lt;a class="wiki_link" href="/17edo"&gt;17edo&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc13"&gt;&lt;a name="Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Music&lt;/h1&gt;
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3" rel="nofollow"&gt;12of22studyPentUp4thsMstr&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3" rel="nofollow"&gt;12of22gamelan1b&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3" rel="nofollow"&gt;12of22study3 (children's story)&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3" rel="nofollow"&gt;12of22study7&lt;/a&gt;&lt;br /&gt;
By &lt;a class="wiki_link" href="/Joel%20Grant%20Taylor"&gt;Joel Grant Taylor&lt;/a&gt;, all in Superpyth[12] in 22edo tuning.&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

de:Superpyth Deutsch

Superpyth, a member of the Archytas clan, has 4/3 as a generator, and the Archytas comma 64/63 is 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 plastic numberhas 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.

Music

12of22studyPentUp4thsMstr

12of22gamelan1b

12of22study3 (children's story)

12of22study7

By Joel Grant Taylor, all in Superpyth[12] in 22edo tuning.

Retrieved from "https://en.xen.wiki/index.php?title=Superpyth&oldid=3599"