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Wikispaces>JosephRuhf **Imported revision 597388978 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-28 10:57:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>597388978</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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|| 20\34 || 705.882 || | || 20\34 || 705.882 || | ||
|| 11/9 || 706.574 || | || 11/9 || 706.574 || | ||
|| 5/4 || 706.843 | || 5/4 || 706.843 (7 and 11 limit POTT) || | ||
|| 33\56 || 707.143 || | |||
|| 12/11 || 707.234 || | || 12/11 || 707.234 || | ||
|| 15/11 || 707.390 || | || 15/11 || 707.390 || | ||
| Line 497: | Line 498: | ||
<td>5/4<br /> | <td>5/4<br /> | ||
</td> | </td> | ||
<td>706.843 | <td>706.843 (7 and 11 limit POTT)<br /> | ||
</td> | </td> | ||
</tr> | |||
<tr> | |||
<td>33\56<br /> | <td>33\56<br /> | ||
</td> | </td> | ||
Revision as of 10:57, 28 October 2016
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author JosephRuhf and made on 2016-10-28 10:57:24 UTC.
- The original revision id was 597388978.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
<span style="display: block; text-align: right;">Other languages: [[xenharmonie/Pajara|Deutsch]] </span> Pajara (pronounced /p<span class="IPA">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]]. 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== ||~ Eigenmonzo ||~ Fifth || || 14/11 || 704.377 || || 10/9 || 704.399 || || 6/5 || 705.214 (5 and 15 limit minimax) || || 20\34 || 705.882 || || 11/9 || 706.574 || || 5/4 || 706.843 (7 and 11 limit POTT) || || 33\56 || 707.143 || || 12/11 || 707.234 || || 15/11 || 707.390 || || 11/8 || 708.114 || || 11/10 || 708.749 (11 limit minimax) || || 9/7 || 708.771 || || 13\22 || 709.091 || || 7/6 || 711.043 (7 limit minimax) || || 32\54 || 711.111 || || 13/11 || 711.151 (13 limit minimax) || || 16/15 || 711.731 || || 13/10 || 713.553 || ==References== * Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [[http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]] =Music= [[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
Original HTML content:
<html><head><title>pajara</title></head><body><span style="display: block; text-align: right;">Other languages: <a class="wiki_link" href="http://xenharmonie.wikispaces.com/Pajara">Deutsch</a><br />
</span><br />
Pajara (pronounced /p<span class="IPA">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 <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 independently 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:<h2> --><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 "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 <a class="wiki_link" href="/22edo">22edo</a>, which is the intersection of both systems.<br />
<!-- ws:start:WikiTextHeadingRule:2:<h3> --><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:<h3> --><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:<h3> --><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:<h2> --><h2 id="toc4"><a name="x-MOSes"></a><!-- ws:end:WikiTextHeadingRule:8 -->MOSes</h2>
<!-- ws:start:WikiTextHeadingRule:10:<h3> --><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 "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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:<h3> --><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:<h2> --><h2 id="toc7"><a name="x-Spectrum of Pajara Tunings by Eigenmonzos"></a><!-- ws:end:WikiTextHeadingRule:14 -->Spectrum of Pajara Tunings by Eigenmonzos</h2>
<table class="wiki_table">
<tr>
<th>Eigenmonzo<br />
</th>
<th>Fifth<br />
</th>
</tr>
<tr>
<td>14/11<br />
</td>
<td>704.377<br />
</td>
</tr>
<tr>
<td>10/9<br />
</td>
<td>704.399<br />
</td>
</tr>
<tr>
<td>6/5<br />
</td>
<td>705.214 (5 and 15 limit minimax)<br />
</td>
</tr>
<tr>
<td>20\34<br />
</td>
<td>705.882<br />
</td>
</tr>
<tr>
<td>11/9<br />
</td>
<td>706.574<br />
</td>
</tr>
<tr>
<td>5/4<br />
</td>
<td>706.843 (7 and 11 limit POTT)<br />
</td>
</tr>
<tr>
<td>33\56<br />
</td>
<td>707.143<br />
</td>
</tr>
<tr>
<td>12/11<br />
</td>
<td>707.234<br />
</td>
</tr>
<tr>
<td>15/11<br />
</td>
<td>707.390<br />
</td>
</tr>
<tr>
<td>11/8<br />
</td>
<td>708.114<br />
</td>
</tr>
<tr>
<td>11/10<br />
</td>
<td>708.749 (11 limit minimax)<br />
</td>
</tr>
<tr>
<td>9/7<br />
</td>
<td>708.771<br />
</td>
</tr>
<tr>
<td>13\22<br />
</td>
<td>709.091<br />
</td>
</tr>
<tr>
<td>7/6<br />
</td>
<td>711.043 (7 limit minimax)<br />
</td>
</tr>
<tr>
<td>32\54<br />
</td>
<td>711.111<br />
</td>
</tr>
<tr>
<td>13/11<br />
</td>
<td>711.151 (13 limit minimax)<br />
</td>
</tr>
<tr>
<td>16/15<br />
</td>
<td>711.731<br />
</td>
</tr>
<tr>
<td>13/10<br />
</td>
<td>713.553<br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="x-References"></a><!-- ws:end:WikiTextHeadingRule:16 -->References</h2>
<ul><li>Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." 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><br />
<!-- ws:start:WikiTextHeadingRule:18:<h1> --><h1 id="toc9"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:18 -->Music</h1>
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3" rel="nofollow">12-22hexachordal Dirge</a> and<br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3" rel="nofollow">12-22hexachordal Sonatina</a> both by <a class="wiki_link" href="/Joel%20Grant%20Taylor">Joel Grant Taylor</a>, in the hexachordal dodecatonic MODMOS.<br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3" rel="nofollow">Smoke Filled Bar</a> by <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2403" rel="nofollow">Chris Vaisvil</a>, also in 12-22h.<br />
<a class="wiki_link_ext" href="https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs" rel="nofollow">Chord Sequence in Paul Erlich's Decatonic Major</a> by Jake Freivald</body></html>