Pajara: Difference between revisions

Revision as of 19:18, 17 August 2011

IMPORTED REVISION FROM WIKISPACES

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This revision was by author keenanpepper and made on 2011-08-17 19:18:51 UTC.

The original revision id was 246585785.

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Original Wikitext content:

Pajara (pronounced 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 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]].

==References== 
* Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [[http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]]

Original HTML content:

<html><head><title>pajara</title></head><body>Pajara (pronounced with the J as in &quot;jar&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 <a class="wiki_link" href="/jubilismic%20clan">jubilismic clan</a>. 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 <a class="wiki_link" href="/diaschismic%20family">diaschismic family</a>. 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 <a class="wiki_link" href="/Archytas%20clan">Archytas clan</a>. Tempering out any two of these commas (among others) produces the unique temperament, pajara.<br />
<br />
The 10-note MOS and LsssLsssss almost-MOS are called the symmetric and pentachordal decatonic scales and were invented/discovered by <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a> and <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a>. They are often thought of as subsets of <a class="wiki_link" href="/22edo">22edo</a>, without much loss of generality and accuracy.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Interval chains"></a><!-- ws:end:WikiTextHeadingRule:0 -->Interval chains</h2>
 There are two different mappings of the 11 limit. One is just called &quot;pajara&quot; and is slightly more complex but suffers almost no loss of accuracy compared to the 7 limit. The other, called &quot;pajarous&quot; to avoid confusion, loses some accuracy and there's little reason to use it unless you're using <a class="wiki_link" href="/22edo">22edo</a>, which is the intersection of both systems.<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Interval chains-Basic 7-limit pajara"></a><!-- ws:end:WikiTextHeadingRule:2 -->Basic 7-limit pajara</h3>
 

<table class="wiki_table">
    <tr>
        <td>771.81<br />
</td>
        <td>878.86<br />
</td>
        <td>985.90<br />
</td>
        <td>1092.95<br />
</td>
        <td>0.<br />
</td>
        <td>107.05<br />
</td>
        <td>214.10<br />
</td>
        <td>321.14<br />
</td>
        <td>428.19<br />
</td>
    </tr>
    <tr>
        <td>14/9<br />
</td>
        <td>5/3<br />
</td>
        <td>7/4~16/9<br />
</td>
        <td><br />
</td>
        <td>1/1<br />
</td>
        <td><br />
</td>
        <td>9/8~8/7<br />
</td>
        <td>6/5<br />
</td>
        <td>9/7<br />
</td>
    </tr>
    <tr>
        <td>171.81<br />
</td>
        <td>278.86<br />
</td>
        <td>385.90<br />
</td>
        <td>492.95<br />
</td>
        <td>600.<br />
</td>
        <td>707.05<br />
</td>
        <td>814.10<br />
</td>
        <td>921.14<br />
</td>
        <td>1028.19<br />
</td>
    </tr>
    <tr>
        <td>10/9<br />
</td>
        <td>7/6<br />
</td>
        <td>5/4<br />
</td>
        <td>4/3<br />
</td>
        <td>7/5~10/7<br />
</td>
        <td>3/2<br />
</td>
        <td>8/5<br />
</td>
        <td>12/7<br />
</td>
        <td>9/5<br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-Interval chains-11-limit pajara"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit pajara</h3>
 

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

<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x-Interval chains-Pajarous"></a><!-- ws:end:WikiTextHeadingRule:6 -->Pajarous</h3>
 

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

<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x-MOSes"></a><!-- ws:end:WikiTextHeadingRule:8 -->MOSes</h2>
 <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-MOSes-10-note (proper)"></a><!-- ws:end:WikiTextHeadingRule:10 -->10-note (proper)</h3>
 See <a class="wiki_link" href="/2L%208s">2L 8s</a>.<br />
The true MOS is called the &quot;symmetric&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.<br />
The near-MOS, LsssLsssss, in which only the 5-step interval violates the &quot;no more than 2 intervals per class&quot; rule, is called the &quot;pentachordal&quot; decatonic, because it consists of two identical &quot;pentachords&quot; plus a split 9/8~8/7 whole tone to complete the octave.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x-MOSes-12-note (proper)"></a><!-- ws:end:WikiTextHeadingRule:12 -->12-note (proper)</h3>
 See <a class="wiki_link" href="/10L%202s">10L 2s</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="x-References"></a><!-- ws:end:WikiTextHeadingRule:14 -->References</h2>
 <ul><li>Erlich, Paul. &quot;Tuning, Tonality and 22-Tone Temperament.&quot; Xenharmonicon 17, 1998. <a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf" rel="nofollow">http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf</a></li></ul></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>2011-08-17 17:52:29 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-08-17 19:18:51 UTC</tt>.<br>
-: The original revision id was <tt>246567081</tt>.<br>
+: The original revision id was <tt>246585785</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>
@@ Line 31: / Line 31: @@
 ===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]].</pre></div>
+See [[10L 2s]].
+==References==
+* Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [[http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]]</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;Pajara (pronounced 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;
@@ Line 414: / Line 420: @@
   &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.&lt;br /&gt;
+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;/body&gt;&lt;/html&gt;</pre></div>
+  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-References"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;References&lt;/h2&gt;
+ &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;/body&gt;&lt;/html&gt;</pre></div>

Pajara: Difference between revisions

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IMPORTED REVISION FROM WIKISPACES

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