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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<span style="display: block; text-align: right;">Other languages: [[:de:Pajara Deutsch]]</span>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-28 14:04:59 UTC</tt>.<br>
: The original revision id was <tt>597406232</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">&lt;span style="display: block; text-align: right;"&gt;Other languages: [[xenharmonie/Pajara|Deutsch]]
&lt;/span&gt;
Pajara (pronounced /p&lt;span class="IPA"&gt;əˈd͡ʒɑːr&lt;/span&gt;ə/, with the J as in "jar") is a temperament with a half-octave period that represents both 7/5 and 10/7, so 50/49 is tempered out and it is in the [[jubilismic clan]]. The generator is in the neighborhood of 105-110 cents, so that period + generator represents 3/2. Period minus 2 generators is 5/4, which, if you work it out, implies that 2048/2025 is tempered out, so pajara is also in the [[diaschismic family]]. Finally, two 4/3s (or a 2/1 minus two generators) represents 7/4 as well as 16/9, so 64/63 is tempered out and pajara is in the [[Archytas clan]]. Tempering out any two of these commas (among others) produces the unique temperament, pajara.


The 10-note MOS and LsssLsssss almost-MOS are called the symmetric and pentachordal decatonic scales and were independently invented/discovered by [[Paul Erlich]] and [[Gene Ward Smith]]. They are often thought of as subsets of [[22edo]], without much loss of generality and accuracy.
Pajara (pronounced /p<span style="">əˈd͡ʒɑːr</span>ə/, with the J as in "jar") is a temperament with a half-octave period that represents both 7/5 and 10/7, so 50/49 is tempered out and it is in the [[Jubilismic_clan|jubilismic clan]]. The generator is in the neighborhood of 105-110 cents, so that period + generator represents 3/2. Period minus 2 generators is 5/4, which, if you work it out, implies that 2048/2025 is tempered out, so pajara is also in the [[Diaschismic_family|diaschismic family]]. Finally, two 4/3s (or a 2/1 minus two generators) represents 7/4 as well as 16/9, so 64/63 is tempered out and pajara is in the [[Archytas_clan|Archytas clan]]. Tempering out any two of these commas (among others) produces the unique temperament, pajara.


==Interval chains==
The 10-note MOS and LsssLsssss almost-MOS are called the symmetric and pentachordal decatonic scales and were independently invented/discovered by [[Paul_Erlich|Paul Erlich]] and [[Gene_Ward_Smith|Gene Ward Smith]]. They are often thought of as subsets of [[22edo|22edo]], without much loss of generality and accuracy.
There are two different mappings of the 11 limit. One is just called "pajara" and is slightly more complex but suffers almost no loss of accuracy compared to the 7 limit. The other, called "pajarous" to avoid confusion, loses some accuracy and there's little reason to use it unless you're using [[22edo]], which is the intersection of both systems.
===Basic 7-limit pajara===
|| 771.81 || 878.86 || 985.90 || 1092.95 || 0. || 107.05 || 214.10 || 321.14 || 428.19 ||
|| 14/9 || 5/3 || 7/4~16/9 ||  || 1/1 ||  || 9/8~8/7 || 6/5 || 9/7 ||
|| 171.81 || 278.86 || 385.90 || 492.95 || 600. || 707.05 || 814.10 || 921.14 || 1028.19 ||
|| 10/9 || 7/6 || 5/4 || 4/3 || 7/5~10/7 || 3/2 || 8/5 || 12/7 || 9/5 ||
===11-limit pajara===
|| 344.92 || 451.80 || 558.69 || 665.57 || 772.46 || 879.34 || 986.23 || 1093.11 || 0. || 106.89 || 213.77 || 320.66 || 427.54 || 534.43 || 641.31 || 748.20 || 855.08 ||
|| 11/9 ||  || 11/8 ||  || 14/9~11/7 || 5/3 || 7/4~16/9 ||  || 1/1 ||  || 9/8~8/7 || 6/5 || 14/9~9/7 ||  || 16/11 ||  || 18/11 ||
|| 944.92 || 1051.80 || 1158.69 || 65.57 || 172.46 || 279.34 || 386.23 || 493.11 || 600. || 706.89 || 813.77 || 920.66 || 1027.54 || 1134.43 || 41.31 || 148.20 || 255.08 ||
||  || 11/6 ||  ||  || 11/10~10/9 || 7/6 || 5/4 || 4/3 || 7/5~10/7 || 3/2 || 8/5 || 12/7 || 9/5 ||  ||  || 12/11 ||  ||
===Pajarous===
|| 432.96 || 542.54 || 652.11 || 761.69 || 871.27 || 980.85 || 1090.42 || 0. || 109.58 || 219.15 || 328.73 || 438.31 || 547.89 || 657.46 || 767.04 ||
|| 14/11 ||  || 16/11 || 14/9 || 18/11~5/3 || 7/4~16/9 ||  || 1/1 ||  || 9/8~8/7 || 6/5~11/9 || 9/7 || 11/8 ||  || 11/7 ||
|| 1032.96 || 1142.54 || 52.11 || 161.69 || 271.27 || 380.85 || 490.42 || 600. || 709.58 || 819.15 || 928.73 || 1038.31 || 1147.89 || 57.46 || 167.04 ||
|| 20/11 ||  ||  || 12/11~10/9 || 7/6 || 5/4 || 4/3 || 7/5~10/7 || 3/2 || 8/5 || 12/7 || 9/5~11/6 ||  ||  || 11/10 ||


==MOSes==  
==Interval chains==
===10-note (proper)===  
There are two different mappings of the 11 limit. One is just called "pajara" and is slightly more complex but suffers almost no loss of accuracy compared to the 7 limit. The other, called "pajarous" to avoid confusion, loses some accuracy and there's little reason to use it unless you're using [[22edo|22edo]], which is the intersection of both systems.
See [[2L 8s]].
 
The true MOS is called the "symmetric" decatonic scale, because it repeats exactly at the half-octave, so the symmetric scale starting from 7/5~10/7 is the same as the symmetric scale starting from 1/1. The near-MOS, LsssLsssss, in which only the 5-step interval violates the "no more than 2 intervals per class" rule, is called the "pentachordal" decatonic, because it consists of two identical "pentachords" plus a split 9/8~8/7 whole tone to complete the octave.
===Basic 7-limit pajara===
 
{| class="wikitable"
|-
| | 771.81
| | 878.86
| | 985.90
| | 1092.95
| | 0.
| | 107.05
| | 214.10
| | 321.14
| | 428.19
|-
| | 14/9
| | 5/3
| | 7/4~16/9
| |
| | 1/1
| |
| | 9/8~8/7
| | 6/5
| | 9/7
|-
| | 171.81
| | 278.86
| | 385.90
| | 492.95
| | 600.
| | 707.05
| | 814.10
| | 921.14
| | 1028.19
|-
| | 10/9
| | 7/6
| | 5/4
| | 4/3
| | 7/5~10/7
| | 3/2
| | 8/5
| | 12/7
| | 9/5
|}
 
===11-limit pajara===
 
{| class="wikitable"
|-
| | 344.92
| | 451.80
| | 558.69
| | 665.57
| | 772.46
| | 879.34
| | 986.23
| | 1093.11
| | 0.
| | 106.89
| | 213.77
| | 320.66
| | 427.54
| | 534.43
| | 641.31
| | 748.20
| | 855.08
|-
| | 11/9
| |
| | 11/8
| |
| | 14/9~11/7
| | 5/3
| | 7/4~16/9
| |
| | 1/1
| |
| | 9/8~8/7
| | 6/5
| | 14/9~9/7
| |
| | 16/11
| |
| | 18/11
|-
| | 944.92
| | 1051.80
| | 1158.69
| | 65.57
| | 172.46
| | 279.34
| | 386.23
| | 493.11
| | 600.
| | 706.89
| | 813.77
| | 920.66
| | 1027.54
| | 1134.43
| | 41.31
| | 148.20
| | 255.08
|-
| |
| | 11/6
| |
| |
| | 11/10~10/9
| | 7/6
| | 5/4
| | 4/3
| | 7/5~10/7
| | 3/2
| | 8/5
| | 12/7
| | 9/5
| |
| |
| | 12/11
| |
|}
 
===Pajarous===


===12-note (proper)===
{| class="wikitable"
See [[10L 2s]].
|-
| | 432.96
| | 542.54
| | 652.11
| | 761.69
| | 871.27
| | 980.85
| | 1090.42
| | 0.
| | 109.58
| | 219.15
| | 328.73
| | 438.31
| | 547.89
| | 657.46
| | 767.04
|-
| | 14/11
| |
| | 16/11
| | 14/9
| | 18/11~5/3
| | 7/4~16/9
| |
| | 1/1
| |
| | 9/8~8/7
| | 6/5~11/9
| | 9/7
| | 11/8
| |
| | 11/7
|-
| | 1032.96
| | 1142.54
| | 52.11
| | 161.69
| | 271.27
| | 380.85
| | 490.42
| | 600.
| | 709.58
| | 819.15
| | 928.73
| | 1038.31
| | 1147.89
| | 57.46
| | 167.04
|-
| | 20/11
| |
| |
| | 12/11~10/9
| | 7/6
| | 5/4
| | 4/3
| | 7/5~10/7
| | 3/2
| | 8/5
| | 12/7
| | 9/5~11/6
| |
| |
| | 11/10
|}


==Spectrum of Pajara Tunings by Eigenmonzos==  
==MOSes==
||~ EDO degree ||~ Eigenmonzo ||~ Decatonic seventh ||
|| 7\12 ||  || 700.000 ||
||  || 3/2 || 701.955 ||
|| 41\70 ||  || 702.857 ||
|| 34\58 ||  || 703.448 ||
|| 61\104 ||  || 703.846 ||
|| 27\46 ||  || 704.348 ||
||  || 14/11 || 704.377 ||
||  || 10/9 || 704.399 ||
|| 74\126 ||  || 704.762 ||
|| 47\80 ||  || 705.000 ||
|| 114\194 ||  || 705.155 ||
||  || 6/5 || 705.214 (5 and 15 limit minimax) ||
|| 67\114 ||  || 705.263 ||
|| 87\148 ||  || 705.405 ||
|| 20\34 ||  || 705.882 ||
|| 93\158 ||  || 706.329 ||
|| 73\124 ||  || 706.452 ||
|| 126\214 ||  || 706.542 ||
||  || 11/9 || 706.574 ||
|| 53\90 ||  || 706.667 ||
|| 139\236 ||  || 706.780 ||
||  || 5/4 || 706.843 (7 and 11 limit POTT) ||
|| 86\146 ||  || 706.849 ||
|| 119\202 ||  || 706.931 ||
|| 33\56 ||  || 707.143 ||
||  || 12/11 || 707.234 ||
|| 112\190 ||  || 707.368 ||
||  || 15/11 || 707.390 ||
|| 79\134 ||  || 707.463 ||
|| 125\212 ||  || 707.547 ||
|| 46\78 ||  || 707.692 ||
|| 105\178 ||  || 707.865 ||
|| 59\100 ||  || 708.000 ||
||  || 11/8 || 708.114 ||
|| 72\122 ||  || 708.196 ||
||  || 11/10 || 708.749 (11 limit minimax) ||
||  || 9/7 || 708.771 ||
|| 13\22 ||  || 709.091 ||
|| 58\98 ||  || 710.204 ||
|| 45\76 ||  || 710.526 ||
|| 122\206 ||  || 710.680 ||
|| 77\130 ||  || 710.769 ||
|| 109\184 ||  || 710.870 ||
||  || 7/6 || 711.043 (7 limit minimax) ||
|| 32\54 ||  || 711.111 ||
||  || 13/11 || 711.151 (13 limit minimax) ||
|| 83\140 ||  || 711.429 ||
|| 51\86 ||  || 711.628 ||
||  || 16/15 || 711.731 ||
|| 70\118 ||  || 711.864 ||
|| 19\32 ||  || 712.500 ||
|| 44\74 ||  || 713.5135 ||
||  || 13/10 || 713.553 ||
|| 25\42 ||  || 714.286 ||
|| 31\52 ||  || 715.385 ||
||  || 8/7 || 715.587 ||
|| 6\10 ||  || 720.000 ||


==References==  
===10-note (proper)===
* Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [[http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]]
See [[2L_8s|2L 8s]].


=Music=
The true MOS is called the "symmetric" decatonic scale, because it repeats exactly at the half-octave, so the symmetric scale starting from 7/5~10/7 is the same as the symmetric scale starting from 1/1. The near-MOS, LsssLsssss, in which only the 5-step interval violates the "no more than 2 intervals per class" rule, is called the "pentachordal" decatonic, because it consists of two identical "pentachords" plus a split 9/8~8/7 whole tone to complete the octave.
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3|12-22hexachordal Dirge]] and
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3|12-22hexachordal Sonatina]] both by [[Joel Grant Taylor]], in the hexachordal dodecatonic MODMOS.
[[http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3|Smoke Filled Bar]] by [[http://chrisvaisvil.com/?p=2403|Chris Vaisvil]], also in 12-22h.
[[https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs|Chord Sequence in Paul Erlich's Decatonic Major]] by Jake Freivald</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;pajara&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="display: block; text-align: right;"&gt;Other languages: &lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/Pajara"&gt;Deutsch&lt;/a&gt;&lt;br /&gt;
&lt;/span&gt;&lt;br /&gt;
Pajara (pronounced /p&lt;span class="IPA"&gt;əˈd͡ʒɑːr&lt;/span&gt;ə/, with the J as in &amp;quot;jar&amp;quot;) is a temperament with a half-octave period that represents both 7/5 and 10/7, so 50/49 is tempered out and it is in the &lt;a class="wiki_link" href="/jubilismic%20clan"&gt;jubilismic clan&lt;/a&gt;. The generator is in the neighborhood of 105-110 cents, so that period + generator represents 3/2. Period minus 2 generators is 5/4, which, if you work it out, implies that 2048/2025 is tempered out, so pajara is also in the &lt;a class="wiki_link" href="/diaschismic%20family"&gt;diaschismic family&lt;/a&gt;. Finally, two 4/3s (or a 2/1 minus two generators) represents 7/4 as well as 16/9, so 64/63 is tempered out and pajara is in the &lt;a class="wiki_link" href="/Archytas%20clan"&gt;Archytas clan&lt;/a&gt;. Tempering out any two of these commas (among others) produces the unique temperament, pajara.&lt;br /&gt;
&lt;br /&gt;
The 10-note MOS and LsssLsssss almost-MOS are called the symmetric and pentachordal decatonic scales and were independently invented/discovered by &lt;a class="wiki_link" href="/Paul%20Erlich"&gt;Paul Erlich&lt;/a&gt; and &lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&gt;Gene Ward Smith&lt;/a&gt;. They are often thought of as subsets of &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;, without much loss of generality and accuracy.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Interval chains"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Interval chains&lt;/h2&gt;
There are two different mappings of the 11 limit. One is just called &amp;quot;pajara&amp;quot; and is slightly more complex but suffers almost no loss of accuracy compared to the 7 limit. The other, called &amp;quot;pajarous&amp;quot; to avoid confusion, loses some accuracy and there's little reason to use it unless you're using &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;, which is the intersection of both systems.&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Interval chains-Basic 7-limit pajara"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Basic 7-limit pajara&lt;/h3&gt;


&lt;table class="wiki_table"&gt;
===12-note (proper)===
    &lt;tr&gt;
See [[10L_2s|10L 2s]].
        &lt;td&gt;771.81&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;878.86&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;985.90&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1092.95&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;107.05&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;214.10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;321.14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;428.19&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4~16/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8~8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;171.81&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;278.86&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;385.90&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;492.95&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;600.&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;707.05&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;814.10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;921.14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1028.19&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/5~10/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Interval chains-11-limit pajara"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit pajara&lt;/h3&gt;
==Spectrum of Pajara Tunings by Eigenmonzos==


&lt;table class="wiki_table"&gt;
{| class="wikitable"
    &lt;tr&gt;
|-
        &lt;td&gt;344.92&lt;br /&gt;
! | EDO degree
&lt;/td&gt;
! | Eigenmonzo
        &lt;td&gt;451.80&lt;br /&gt;
! | Decatonic seventh
&lt;/td&gt;
|-
        &lt;td&gt;558.69&lt;br /&gt;
| | 7\12
&lt;/td&gt;
| |
        &lt;td&gt;665.57&lt;br /&gt;
| | 700.000
&lt;/td&gt;
|-
        &lt;td&gt;772.46&lt;br /&gt;
| |
&lt;/td&gt;
| | 3/2
        &lt;td&gt;879.34&lt;br /&gt;
| | 701.955
&lt;/td&gt;
|-
        &lt;td&gt;986.23&lt;br /&gt;
| | 41\70
&lt;/td&gt;
| |
        &lt;td&gt;1093.11&lt;br /&gt;
| | 702.857
&lt;/td&gt;
|-
        &lt;td&gt;0.&lt;br /&gt;
| | 34\58
&lt;/td&gt;
| |
        &lt;td&gt;106.89&lt;br /&gt;
| | 703.448
&lt;/td&gt;
|-
        &lt;td&gt;213.77&lt;br /&gt;
| | 61\104
&lt;/td&gt;
| |
        &lt;td&gt;320.66&lt;br /&gt;
| | 703.846
&lt;/td&gt;
|-
        &lt;td&gt;427.54&lt;br /&gt;
| | 27\46
&lt;/td&gt;
| |
        &lt;td&gt;534.43&lt;br /&gt;
| | 704.348
&lt;/td&gt;
|-
        &lt;td&gt;641.31&lt;br /&gt;
| |
&lt;/td&gt;
| | 14/11
        &lt;td&gt;748.20&lt;br /&gt;
| | 704.377
&lt;/td&gt;
|-
        &lt;td&gt;855.08&lt;br /&gt;
| |
&lt;/td&gt;
| | 10/9
    &lt;/tr&gt;
| | 704.399
    &lt;tr&gt;
|-
        &lt;td&gt;11/9&lt;br /&gt;
| | 74\126
&lt;/td&gt;
| |
        &lt;td&gt;&lt;br /&gt;
| | 704.762
&lt;/td&gt;
|-
        &lt;td&gt;11/8&lt;br /&gt;
| | 47\80
&lt;/td&gt;
| |
        &lt;td&gt;&lt;br /&gt;
| | 705.000
&lt;/td&gt;
|-
        &lt;td&gt;14/9~11/7&lt;br /&gt;
| | 114\194
&lt;/td&gt;
| |
        &lt;td&gt;5/3&lt;br /&gt;
| | 705.155
&lt;/td&gt;
|-
        &lt;td&gt;7/4~16/9&lt;br /&gt;
| |
&lt;/td&gt;
| | 6/5
        &lt;td&gt;&lt;br /&gt;
| | 705.214 (5 and 15 limit minimax)
&lt;/td&gt;
|-
        &lt;td&gt;1/1&lt;br /&gt;
| | 67\114
&lt;/td&gt;
| |
        &lt;td&gt;&lt;br /&gt;
| | 705.263
&lt;/td&gt;
|-
        &lt;td&gt;9/8~8/7&lt;br /&gt;
| | 87\148
&lt;/td&gt;
| |
        &lt;td&gt;6/5&lt;br /&gt;
| | 705.405
&lt;/td&gt;
|-
        &lt;td&gt;14/9~9/7&lt;br /&gt;
| | 20\34
&lt;/td&gt;
| |
        &lt;td&gt;&lt;br /&gt;
| | 705.882
&lt;/td&gt;
|-
        &lt;td&gt;16/11&lt;br /&gt;
| | 93\158
&lt;/td&gt;
| |
        &lt;td&gt;&lt;br /&gt;
| | 706.329
&lt;/td&gt;
|-
        &lt;td&gt;18/11&lt;br /&gt;
| | 73\124
&lt;/td&gt;
| |
    &lt;/tr&gt;
| | 706.452
    &lt;tr&gt;
|-
        &lt;td&gt;944.92&lt;br /&gt;
| | 126\214
&lt;/td&gt;
| |
        &lt;td&gt;1051.80&lt;br /&gt;
| | 706.542
&lt;/td&gt;
|-
        &lt;td&gt;1158.69&lt;br /&gt;
| |
&lt;/td&gt;
| | 11/9
        &lt;td&gt;65.57&lt;br /&gt;
| | 706.574
&lt;/td&gt;
|-
        &lt;td&gt;172.46&lt;br /&gt;
| | 53\90
&lt;/td&gt;
| |
        &lt;td&gt;279.34&lt;br /&gt;
| | 706.667
&lt;/td&gt;
|-
        &lt;td&gt;386.23&lt;br /&gt;
| | 139\236
&lt;/td&gt;
| |
        &lt;td&gt;493.11&lt;br /&gt;
| | 706.780
&lt;/td&gt;
|-
        &lt;td&gt;600.&lt;br /&gt;
| |
&lt;/td&gt;
| | 5/4
        &lt;td&gt;706.89&lt;br /&gt;
| | 706.843 (7 and 11 limit POTT)
&lt;/td&gt;
|-
        &lt;td&gt;813.77&lt;br /&gt;
| | 86\146
&lt;/td&gt;
| |
        &lt;td&gt;920.66&lt;br /&gt;
| | 706.849
&lt;/td&gt;
|-
        &lt;td&gt;1027.54&lt;br /&gt;
| | 119\202
&lt;/td&gt;
| |
        &lt;td&gt;1134.43&lt;br /&gt;
| | 706.931
&lt;/td&gt;
|-
        &lt;td&gt;41.31&lt;br /&gt;
| | 33\56
&lt;/td&gt;
| |
        &lt;td&gt;148.20&lt;br /&gt;
| | 707.143
&lt;/td&gt;
|-
        &lt;td&gt;255.08&lt;br /&gt;
| |
&lt;/td&gt;
| | 12/11
    &lt;/tr&gt;
| | 707.234
    &lt;tr&gt;
|-
        &lt;td&gt;&lt;br /&gt;
| | 112\190
&lt;/td&gt;
| |
        &lt;td&gt;11/6&lt;br /&gt;
| | 707.368
&lt;/td&gt;
|-
        &lt;td&gt;&lt;br /&gt;
| |
&lt;/td&gt;
| | 15/11
        &lt;td&gt;&lt;br /&gt;
| | 707.390
&lt;/td&gt;
|-
        &lt;td&gt;11/10~10/9&lt;br /&gt;
| | 79\134
&lt;/td&gt;
| |
        &lt;td&gt;7/6&lt;br /&gt;
| | 707.463
&lt;/td&gt;
|-
        &lt;td&gt;5/4&lt;br /&gt;
| | 125\212
&lt;/td&gt;
| |
        &lt;td&gt;4/3&lt;br /&gt;
| | 707.547
&lt;/td&gt;
|-
        &lt;td&gt;7/5~10/7&lt;br /&gt;
| | 46\78
&lt;/td&gt;
| |
        &lt;td&gt;3/2&lt;br /&gt;
| | 707.692
&lt;/td&gt;
|-
        &lt;td&gt;8/5&lt;br /&gt;
| | 105\178
&lt;/td&gt;
| |
        &lt;td&gt;12/7&lt;br /&gt;
| | 707.865
&lt;/td&gt;
|-
        &lt;td&gt;9/5&lt;br /&gt;
| | 59\100
&lt;/td&gt;
| |
        &lt;td&gt;&lt;br /&gt;
| | 708.000
&lt;/td&gt;
|-
        &lt;td&gt;&lt;br /&gt;
| |
&lt;/td&gt;
| | 11/8
        &lt;td&gt;12/11&lt;br /&gt;
| | 708.114
&lt;/td&gt;
|-
        &lt;td&gt;&lt;br /&gt;
| | 72\122
&lt;/td&gt;
| |
    &lt;/tr&gt;
| | 708.196
&lt;/table&gt;
|-
| |
| | 11/10
| | 708.749 (11 limit minimax)
|-
| |
| | 9/7
| | 708.771
|-
| | 13\22
| |
| | 709.091
|-
| | 58\98
| |
| | 710.204
|-
| | 45\76
| |
| | 710.526
|-
| | 122\206
| |
| | 710.680
|-
| | 77\130
| |
| | 710.769
|-
| | 109\184
| |
| | 710.870
|-
| |
| | 7/6
| | 711.043 (7 limit minimax)
|-
| | 32\54
| |
| | 711.111
|-
| |
| | 13/11
| | 711.151 (13 limit minimax)
|-
| | 83\140
| |
| | 711.429
|-
| | 51\86
| |
| | 711.628
|-
| |
| | 16/15
| | 711.731
|-
| | 70\118
| |
| | 711.864
|-
| | 19\32
| |
| | 712.500
|-
| | 44\74
| |
| | 713.5135
|-
| |
| | 13/10
| | 713.553
|-
| | 25\42
| |
| | 714.286
|-
| | 31\52
| |
| | 715.385
|-
| |
| | 8/7
| | 715.587
|-
| | 6\10
| |
| | 720.000
|}


&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Interval chains-Pajarous"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Pajarous&lt;/h3&gt;
==References==
<ul><li>Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]</li></ul>


&lt;table class="wiki_table"&gt;
=Music=
    &lt;tr&gt;
[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge] and
        &lt;td&gt;432.96&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;542.54&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;652.11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;761.69&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;871.27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;980.85&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1090.42&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;109.58&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;219.15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;328.73&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;438.31&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;547.89&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;657.46&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;767.04&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;18/11~5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4~16/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8~8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5~11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1032.96&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1142.54&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;52.11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;161.69&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;271.27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;380.85&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;490.42&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;600.&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;709.58&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;819.15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;928.73&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1038.31&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1147.89&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;57.46&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;167.04&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;20/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/11~10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/5~10/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/5~11/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/10&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 12-22hexachordal Sonatina] both by [[Joel_Grant_Taylor|Joel Grant Taylor]], in the hexachordal dodecatonic MODMOS.
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-MOSes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;MOSes&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-MOSes-10-note (proper)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;10-note (proper)&lt;/h3&gt;
See &lt;a class="wiki_link" href="/2L%208s"&gt;2L 8s&lt;/a&gt;.&lt;br /&gt;
The true MOS is called the &amp;quot;symmetric&amp;quot; decatonic scale, because it repeats exactly at the half-octave, so the symmetric scale starting from 7/5~10/7 is the same as the symmetric scale starting from 1/1. The near-MOS, LsssLsssss, in which only the 5-step interval violates the &amp;quot;no more than 2 intervals per class&amp;quot; rule, is called the &amp;quot;pentachordal&amp;quot; decatonic, because it consists of two identical &amp;quot;pentachords&amp;quot; plus a split 9/8~8/7 whole tone to complete the octave.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x-MOSes-12-note (proper)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;12-note (proper)&lt;/h3&gt;
See &lt;a class="wiki_link" href="/10L%202s"&gt;10L 2s&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="x-Spectrum of Pajara Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Spectrum of Pajara Tunings by Eigenmonzos&lt;/h2&gt;


&lt;table class="wiki_table"&gt;
[http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 Smoke Filled Bar] by [http://chrisvaisvil.com/?p=2403 Chris Vaisvil], also in 12-22h.
    &lt;tr&gt;
        &lt;th&gt;EDO degree&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Decatonic seventh&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7\12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;700.000&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;701.955&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;41\70&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;702.857&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;34\58&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;703.448&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;61\104&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;703.846&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;27\46&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;704.348&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;704.377&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;704.399&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;74\126&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;704.762&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;47\80&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;705.000&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;114\194&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;705.155&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;705.214 (5 and 15 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;67\114&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;705.263&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;87\148&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;705.405&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;20\34&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;705.882&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;93\158&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;706.329&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;73\124&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;706.452&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;126\214&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;706.542&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;706.574&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;53\90&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;706.667&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;139\236&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;706.780&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;706.843 (7 and 11 limit POTT)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;86\146&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;706.849&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;119\202&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;706.931&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;33\56&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;707.143&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;707.234&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;112\190&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;707.368&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;707.390&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;79\134&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;707.463&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;125\212&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;707.547&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;46\78&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;707.692&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;105\178&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;707.865&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;59\100&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;708.000&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;708.114&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;72\122&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;708.196&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;708.749 (11 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;708.771&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13\22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;709.091&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;58\98&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;710.204&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;45\76&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;710.526&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;122\206&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;710.680&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;77\130&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;710.769&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;109\184&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;710.870&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;711.043 (7 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;32\54&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;711.111&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;711.151 (13 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;83\140&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;711.429&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;51\86&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;711.628&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;711.731&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;70\118&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;711.864&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;19\32&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;712.500&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;44\74&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;713.5135&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;713.553&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;25\42&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;714.286&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;31\52&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;715.385&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;715.587&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6\10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;720.000&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
[https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs Chord Sequence in Paul Erlich's Decatonic Major] by Jake Freivald
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="x-References"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;References&lt;/h2&gt;
[[Category:erlich]]
&lt;ul&gt;&lt;li&gt;Erlich, Paul. &amp;quot;Tuning, Tonality and 22-Tone Temperament.&amp;quot; Xenharmonicon 17, 1998. &lt;a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf" rel="nofollow"&gt;http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
[[Category:pajara]]
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Music&lt;/h1&gt;
[[Category:temperament]]
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3" rel="nofollow"&gt;12-22hexachordal Dirge&lt;/a&gt; and&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3" rel="nofollow"&gt;12-22hexachordal Sonatina&lt;/a&gt; both by &lt;a class="wiki_link" href="/Joel%20Grant%20Taylor"&gt;Joel Grant Taylor&lt;/a&gt;, in the hexachordal dodecatonic MODMOS.&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3" rel="nofollow"&gt;Smoke Filled Bar&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2403" rel="nofollow"&gt;Chris Vaisvil&lt;/a&gt;, also in 12-22h.&lt;br /&gt;
&lt;a class="wiki_link_ext" href="https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs" rel="nofollow"&gt;Chord Sequence in Paul Erlich's Decatonic Major&lt;/a&gt; by Jake Freivald&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

Other languages: de:Pajara Deutsch

Pajara (pronounced /pəˈd͡ʒɑːrə/, with the J as in "jar") is a temperament with a half-octave period that represents both 7/5 and 10/7, so 50/49 is tempered out and it is in the jubilismic clan. The generator is in the neighborhood of 105-110 cents, so that period + generator represents 3/2. Period minus 2 generators is 5/4, which, if you work it out, implies that 2048/2025 is tempered out, so pajara is also in the diaschismic family. Finally, two 4/3s (or a 2/1 minus two generators) represents 7/4 as well as 16/9, so 64/63 is tempered out and pajara is in the Archytas clan. Tempering out any two of these commas (among others) produces the unique temperament, pajara.

The 10-note MOS and LsssLsssss almost-MOS are called the symmetric and pentachordal decatonic scales and were independently invented/discovered by Paul Erlich and Gene Ward Smith. They are often thought of as subsets of 22edo, without much loss of generality and accuracy.

Interval chains

There are two different mappings of the 11 limit. One is just called "pajara" and is slightly more complex but suffers almost no loss of accuracy compared to the 7 limit. The other, called "pajarous" to avoid confusion, loses some accuracy and there's little reason to use it unless you're using 22edo, which is the intersection of both systems.

Basic 7-limit pajara

771.81 878.86 985.90 1092.95 0. 107.05 214.10 321.14 428.19
14/9 5/3 7/4~16/9 1/1 9/8~8/7 6/5 9/7
171.81 278.86 385.90 492.95 600. 707.05 814.10 921.14 1028.19
10/9 7/6 5/4 4/3 7/5~10/7 3/2 8/5 12/7 9/5

11-limit pajara

344.92 451.80 558.69 665.57 772.46 879.34 986.23 1093.11 0. 106.89 213.77 320.66 427.54 534.43 641.31 748.20 855.08
11/9 11/8 14/9~11/7 5/3 7/4~16/9 1/1 9/8~8/7 6/5 14/9~9/7 16/11 18/11
944.92 1051.80 1158.69 65.57 172.46 279.34 386.23 493.11 600. 706.89 813.77 920.66 1027.54 1134.43 41.31 148.20 255.08
11/6 11/10~10/9 7/6 5/4 4/3 7/5~10/7 3/2 8/5 12/7 9/5 12/11

Pajarous

432.96 542.54 652.11 761.69 871.27 980.85 1090.42 0. 109.58 219.15 328.73 438.31 547.89 657.46 767.04
14/11 16/11 14/9 18/11~5/3 7/4~16/9 1/1 9/8~8/7 6/5~11/9 9/7 11/8 11/7
1032.96 1142.54 52.11 161.69 271.27 380.85 490.42 600. 709.58 819.15 928.73 1038.31 1147.89 57.46 167.04
20/11 12/11~10/9 7/6 5/4 4/3 7/5~10/7 3/2 8/5 12/7 9/5~11/6 11/10

MOSes

10-note (proper)

See 2L 8s.

The true MOS is called the "symmetric" decatonic scale, because it repeats exactly at the half-octave, so the symmetric scale starting from 7/5~10/7 is the same as the symmetric scale starting from 1/1. The near-MOS, LsssLsssss, in which only the 5-step interval violates the "no more than 2 intervals per class" rule, is called the "pentachordal" decatonic, because it consists of two identical "pentachords" plus a split 9/8~8/7 whole tone to complete the octave.

12-note (proper)

See 10L 2s.

Spectrum of Pajara Tunings by Eigenmonzos

EDO degree Eigenmonzo Decatonic seventh
7\12 700.000
3/2 701.955
41\70 702.857
34\58 703.448
61\104 703.846
27\46 704.348
14/11 704.377
10/9 704.399
74\126 704.762
47\80 705.000
114\194 705.155
6/5 705.214 (5 and 15 limit minimax)
67\114 705.263
87\148 705.405
20\34 705.882
93\158 706.329
73\124 706.452
126\214 706.542
11/9 706.574
53\90 706.667
139\236 706.780
5/4 706.843 (7 and 11 limit POTT)
86\146 706.849
119\202 706.931
33\56 707.143
12/11 707.234
112\190 707.368
15/11 707.390
79\134 707.463
125\212 707.547
46\78 707.692
105\178 707.865
59\100 708.000
11/8 708.114
72\122 708.196
11/10 708.749 (11 limit minimax)
9/7 708.771
13\22 709.091
58\98 710.204
45\76 710.526
122\206 710.680
77\130 710.769
109\184 710.870
7/6 711.043 (7 limit minimax)
32\54 711.111
13/11 711.151 (13 limit minimax)
83\140 711.429
51\86 711.628
16/15 711.731
70\118 711.864
19\32 712.500
44\74 713.5135
13/10 713.553
25\42 714.286
31\52 715.385
8/7 715.587
6\10 720.000

References

Music

12-22hexachordal Dirge and

12-22hexachordal Sonatina both by Joel Grant Taylor, in the hexachordal dodecatonic MODMOS.

Smoke Filled Bar by Chris Vaisvil, also in 12-22h.

Chord Sequence in Paul Erlich's Decatonic Major by Jake Freivald

Retrieved from "https://en.xen.wiki/index.php?title=Pajara&oldid=3006"