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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{interwiki |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | | en = Pajara |
| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-06-04 20:26:54 UTC</tt>.<br>
| | | de = Pajara |
| : The original revision id was <tt>342627210</tt>.<br>
| | | es = |
| : The revision comment was: <tt></tt><br>
| | | ja = |
| 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>
| | {{Infobox regtemp |
| <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">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.
| | | Title = Pajara |
| | | Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.17 |
| | | Comma basis = [[50/49]], [[64/63]] (7-limit);<br>[[50/49]], [[64/63]], [[99/98]] (11-limit);<br>[[50/49]], [[64/63]], [[85/84]], [[99/98]]<br>(2.3.5.7.11.17) |
| | | Edo join 1 = 12 | Edo join 2 = 22 |
| | | Mapping = 2; 1 -2 -2 -6 1 |
| | | Generators = 3/2 | Generators tuning = 707.4 | Optimization method = CWE |
| | | MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]] |
| | | Pergen = (P8/2, P5) |
| | | Odd limit 1 = 9 | Mistuning 1 = 17.5 | Complexity 1 = 10 |
| | | Odd limit 2 = 2.3.5.7.11.17 21 | Mistuning 2 = 22.4 | Complexity 2 = 22 |
| | }} |
| | '''Pajara''' (pronounced /pəˈd͡ʒɑːrə/, with the J as in "jar") is a [[regular temperament|temperament]] with a half-octave [[period]] that represents both [[7/5]] and [[10/7]], so [[50/49]] is [[tempering out|tempered out]] and it is in the [[jubilismic clan]]. The [[generator]] is a [[3/2|perfect fifth]] in the neighborhood of 707–711 [[cent]]s, or that minus a half-octave period, which is a semitone representing [[15/14]] and [[16/15]]. One period minus 2 such semitones is [[~]][[5/4]], which, if you work it out, implies that [[2048/2025]] is tempered out, so pajara is also in the [[diaschismic family]]. In fact, it shares the same structure as 5-limit [[diaschismic]]. Finally, two 4/3's (or an octave minus two semitones) 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.
| | Pajara has fairly low accuracy overall, due to the ~5/4 and ~7/4 necessarily being separated by 600 cents via vanishing of [[50/49]]. However, if one accepts the accuracy of [[12edo]] in the 5-limit, they would probably accept the accuracy of pajara as well. The vanishing of [[50/49]] means that [[49/48]] and [[25/24]] are tempered to the same interval, and allows for a simple alteration to produce the subharmonic sixth chord [[70:84:105:120|1/(12:10:8:7)]] with 6/5 and 12/7 by flattening the third and seventh the same amount from the harmonic seventh chord, [[4:5:6:7]]. |
|
| |
|
| ==Interval chains==
| | Pajara has [[mos scale]]s of 10, 12, and 22 notes. The 10-note mos, Pajara[10], is notable for sharing a number of desirable properties with [[5L 2s|diatonic]], while having fundamentally different categories; for example, the ~7/4 is a now major 8-step, rather than a minor 6-step. This mos and the LsssLsssss [[modmos]] are called the ''symmetric'' and ''pentachordal'' decatonic scales and were independently invented/discovered by [[Paul Erlich]]<ref>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]</ref> and [[Gene Ward Smith]]. They are often thought of as subsets of [[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==
| | As does all diaschismic temperaments, pajara has a natural extension to prime [[17/1|17]], obtained by tempering out [[136/135]], [[256/255]], and [[289/288]]. This extension notably also tempers out [[120/119]], which equates the 1/(12:10:8:7) utonal tetrad with the otonal [[10:12:15:17]]. |
| ===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 [[Diaschismic family #Pajara]] for technical data. See [[Pajara extensions]] for a discussion on the 11-limit extensions. |
| See [[10L 2s]]. | |
|
| |
|
| ==References== | | == Interval chains == |
| * Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [[http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]]
| | 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. It is best tuned flat of 22edo, with the optimum at around 707-708 cents. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but it equates [[12/11]] with [[10/9]], and the only tuning equating [[11/10]] with both is 22edo. |
|
| |
|
| =Music=
| | In the following tables, odd harmonics 1–21 and their inverses are in '''bold'''. |
| [[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3|12-22hexachordal Dirge]] by [[Joel Grant Taylor]], in the hexachordal dodecatonic MODMOS.</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"><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">
| | {| class="wikitable center-1 right-2 right-4" |
| <tr>
| | |+ style="font-size: 105%;" | Pajara ({{nowrap| 12 & 22 }}) |
| <td>771.81<br />
| | |- |
| </td>
| | ! rowspan="2" | # |
| <td>878.86<br />
| | ! colspan="2" | Period 0 |
| </td>
| | ! colspan="2" | Period 1 |
| <td>985.90<br />
| | |- |
| </td>
| | ! Cents* |
| <td>1092.95<br />
| | ! Approximate ratios |
| </td>
| | ! Cents* |
| <td>0.<br />
| | ! Approximate ratios |
| </td>
| | |- |
| <td>107.05<br />
| | | 0 |
| </td>
| | | 0.0 |
| <td>214.10<br />
| | | '''1/1''' |
| </td>
| | | 600.0 |
| <td>321.14<br />
| | | 7/5, 10/7, 17/12, 24/17 |
| </td>
| | |- |
| <td>428.19<br />
| | | 1 |
| </td>
| | | 707.4 |
| </tr>
| | | '''3/2''', '''32/21''' |
| <tr>
| | | 107.4 |
| <td>14/9<br />
| | | 15/14, '''16/15''', '''17/16''',<br>18/17, 21/20 |
| </td>
| | |- |
| <td>5/3<br />
| | | 2 |
| </td>
| | | 214.7 |
| <td>7/4~16/9<br />
| | | '''8/7''', '''9/8''', 17/15 |
| </td>
| | | 814.7 |
| <td><br />
| | | '''8/5''', 34/21 |
| </td>
| | |- |
| <td>1/1<br />
| | | 3 |
| </td>
| | | 922.1 |
| <td><br />
| | | 12/7, 17/10 |
| </td>
| | | 322.1 |
| <td>9/8~8/7<br />
| | | 6/5, 17/14 |
| </td>
| | |- |
| <td>6/5<br />
| | | 4 |
| </td>
| | | 429.5 |
| <td>9/7<br />
| | | 9/7, 14/11 |
| </td>
| | | 1029.5 |
| </tr>
| | | 9/5, 20/11 |
| <tr>
| | |- |
| <td>171.81<br />
| | | 5 |
| </td>
| | | 1136.9 |
| <td>278.86<br />
| | | 21/11, 27/14, 48/25, <br>64/33, 96/49 |
| </td>
| | | 536.9 |
| <td>385.90<br />
| | | 15/11, 27/20 |
| </td>
| | |- |
| <td>492.95<br />
| | | 6 |
| </td>
| | | 644.2 |
| <td>600.<br />
| | | '''16/11''', 36/25, 72/49 |
| </td>
| | | 44.2 |
| <td>707.05<br />
| | | 45/44, 56/55, 81/80 |
| </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>
| | {| class="wikitable center-1 right-2 right-4" |
|
| | |+ style="font-size: 105%;" | Pajarous ({{nowrap| 10 & 22 }}) |
| | |- |
| | ! rowspan="2" | # |
| | ! colspan="2" | Period 0 |
| | ! colspan="2" | Period 1 |
| | |- |
| | ! Cents* |
| | ! Approximate ratios |
| | ! Cents* |
| | ! Approximate ratios |
| | |- |
| | | 0 |
| | | 0.0 |
| | | '''1/1''' |
| | | 600.0 |
| | | 7/5, 10/7, 17/12, 24/17 |
| | |- |
| | | 1 |
| | | 709.5 |
| | | '''3/2''', '''32/21''' |
| | | 109.5 |
| | | 15/14, '''16/15''', '''17/16''',<br>18/17, 21/20 |
| | |- |
| | | 2 |
| | | 219.1 |
| | | '''8/7''', '''9/8''', 17/15 |
| | | 819.1 |
| | | '''8/5''', 34/21 |
| | |- |
| | | 3 |
| | | 928.6 |
| | | 12/7, 17/10 |
| | | 328.6 |
| | | 6/5, 11/9, 17/14 |
| | |- |
| | | 4 |
| | | 438.2 |
| | | 9/7, 22/17 |
| | | 1038.2 |
| | | 9/5, 11/6 |
| | |- |
| | | 5 |
| | | 1147.7 |
| | | 27/14, 48/25, 55/28, <br>88/45, 96/49 |
| | | 547.7 |
| | | '''11/8''', 27/20 |
| | |- |
| | | 6 |
| | | 657.3 |
| | | 22/15 |
| | | 57.3 |
| | | 22/21, 33/32, 81/80 |
| | |} |
| | <nowiki/>* In 2.3.5.7.11.17-subgroup CWE tuning, octave-reduced |
|
| |
|
| <table class="wiki_table">
| | == Chords and harmony == |
| <tr>
| | {{See also| Chords of pajara }} |
| <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>
| | In pajara, a decatonic system of interval classification based on the [[2L 8s]] (jaric) [[mos scale]] is preferred over the [[diatonic]] interval classification system traditionally used in western music, which is used in [[meantone]]. If we count scale degrees similarly to diatonic, then [[2/1]] is a "hendecave" (11ve), as there are 10 scale degrees, and we repeat at 2/1 at the 11th. In this system, [[3/2]] is a perfect 7th, and [[4/3]] is a perfect 5th. The intervals [[5/4]] and [[6/5]] are major and minor decatonic 4ths respectively, rather than being major and minor 3rds by diatonic interval classification in meantone. Importantly, [[7/4]] is now a major decatonic 9th, with [[12/7]] being its minor counterpart. This is in contrast to diatonic, where 7/4 is considered a subminor 7th, and 12/7 a supermajor 6th. |
|
| |
|
| |
|
| <table class="wiki_table">
| | By decatonic interval classification, the [[4:5:6:7]] tetrad is written as P1–M4–P7–M9. It can be considered the ''major tetrad'', since the non-perfect intervals, those being the decatonic 4th and 9th, are both major intervals. If we instead use a minor interval for the 4th and 9th; that is, a P1–m4–P7–m9 chord, then we get a tetrad approximating [[70:84:105:120|1/(12:10:8:7)]], which can be considered the ''minor tetrad''. |
| <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 />
| | {{Todo|complete section}} |
| <!-- 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>
| | == Scales == |
| See <a class="wiki_link" href="/2L%208s">2L 8s</a>.<br />
| | === 10-note (proper) === |
| 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. 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 /> | | {{Main| 2L 8s }} |
| <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>
| | The true mos is called the ''symmetric'' decatonic scale, because it repeats exactly at the half-octave, so the symmetric scale starting from {{nowrap|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 rule of no more than 2 intervals per class, is called the ''pentachordal'' decatonic, because it consists of two identical [[pentachord]]s plus a split {{nowrap|9/8~8/7}} whole tone to complete the octave. |
| See <a class="wiki_link" href="/10L%202s">10L 2s</a>.<br />
| | |
| <br />
| | === 12-note (proper) === |
| <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="x-References"></a><!-- ws:end:WikiTextHeadingRule:14 -->References</h2>
| | {{Main| 10L 2s }} |
| <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><br />
| | |
| <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:16 -->Music</h1>
| | === Scala files === |
| <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> by <a class="wiki_link" href="/Joel%20Grant%20Taylor">Joel Grant Taylor</a>, in the hexachordal dodecatonic MODMOS.</body></html></pre></div>
| | * [[Pajara12]] |
| | * [[12-22h]] |
| | |
| | == Tunings == |
| | As with [[archy]], there is a tradeoff in pajara between accuracy of 3 and accuracy of 7. Unlike tunings of archy which the fifth is around 710–712{{c}}, however, pajara is conventionally tuned flat of 22edo, since tunings sharp of about 710{{c}} lose a large degree of accuracy in 5/4 and especially 6/5. |
| | |
| | === Norm-based tunings === |
| | {| class="wikitable mw-collapsible mw-collapsed" |
| | |+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings |
| | |- |
| | ! rowspan="2" | |
| | ! colspan="3" | Euclidean |
| | |- |
| | ! Constrained |
| | ! Constrained & skewed |
| | ! Destretched |
| | |- |
| | ! Tenney |
| | | CTE: ~3/2 = 708.3557{{c}} |
| | | CWE: ~3/2 = 707.3438{{c}} |
| | | POTE: ~3/2 = 707.0477{{c}} |
| | |} |
| | {| class="wikitable mw-collapsible mw-collapsed" |
| | |+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings |
| | |- |
| | ! rowspan="2" | |
| | ! colspan="3" | Euclidean |
| | |- |
| | ! Constrained |
| | ! Constrained & skewed |
| | ! Destretched |
| | |- |
| | ! Tenney |
| | | CTE: ~3/2 = 708.1993{{c}} |
| | | CWE: ~3/2 = 707.1826{{c}} |
| | | POTE: ~3/2 = 706.8851{{c}} |
| | |} |
| | |
| | === Target tunings === |
| | {| class="wikitable center-all mw-collapsible mw-collapsed" |
| | |+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings |
| | |- |
| | ! rowspan="2" | Target |
| | ! colspan="2" | Minimax |
| | |- |
| | ! Generator |
| | ! Eigenmonzo* |
| | |- |
| | | 7-odd-limit |
| | | ~3/2 = 709.363{{c}} |
| | | 35/24 |
| | |- |
| | | 9-odd-limit |
| | | ~3/2 = 708.128{{c}} |
| | | 35/18 |
| | |- |
| | | 11-odd-limit |
| | | ~3/2 = 708.128{{c}} |
| | | 35/18 |
| | |} |
| | |
| | === Tuning spectrum === |
| | {| class="wikitable center-all left-4" |
| | |- |
| | ! Edo<br>generator |
| | ! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval]]) |
| | ! Generator (¢) |
| | ! Comments |
| | |- |
| | | 7\12 |
| | | |
| | | 700.000 |
| | | Lower bound of 9- and 11-odd-limit diamond monotone |
| | |- |
| | | |
| | | 3/2 |
| | | 701.955 |
| | | |
| | |- |
| | | 34\58 |
| | | |
| | | 703.448 |
| | | 58ddee val |
| | |- |
| | | 27\46 |
| | | |
| | | 704.348 |
| | | 46de val |
| | |- |
| | | |
| | | 11/7 |
| | | 704.377 |
| | | |
| | |- |
| | | |
| | | 9/5 |
| | | 704.399 |
| | | |
| | |- |
| | | 47\80 |
| | | |
| | | 705.000 |
| | | 80ddee val |
| | |- |
| | | |
| | | 5/3 |
| | | 705.214 |
| | | 5-odd-limit minimax |
| | |- |
| | | 20\34 |
| | | |
| | | 705.882 |
| | | 34d val |
| | |- |
| | | |
| | | 11/9 |
| | | 706.574 |
| | | |
| | |- |
| | | 53\90 |
| | | |
| | | 706.667 |
| | | 90dde val |
| | |- |
| | | |
| | | 5/4 |
| | | 706.843 |
| | | 7- and 11-limit POTT |
| | |- |
| | | 33\56 |
| | | |
| | | 707.143 |
| | | 56d val |
| | |- |
| | | |
| | | 11/6 |
| | | 707.234 |
| | | |
| | |- |
| | | |
| | | 15/11 |
| | | 707.390 |
| | | |
| | |- |
| | | 46\78 |
| | | |
| | | 707.692 |
| | | 78dd val |
| | |- |
| | | |
| | | 11/8 |
| | | 708.114 |
| | | 11- and 15-odd-limit minimax |
| | |- |
| | | |
| | |36/35 |
| | |708.128 |
| | |9-odd-limit minimax |
| | |- |
| | | |
| | | 11/10 |
| | | 708.749 |
| | | |
| | |- |
| | | |
| | | 9/7 |
| | | 708.771 |
| | | |
| | |- |
| | | 13\22 |
| | | |
| | | 709.091 |
| | | Upper bound of 11-odd-limit diamond monotone |
| | |- |
| | | |
| | |48/35 |
| | |709.363 |
| | |7-odd-limit minimax |
| | |- |
| | | |
| | | 7/6 |
| | | 711.043 |
| | | |
| | |- |
| | | 32\54 |
| | | |
| | | 711.111 |
| | | 54e val |
| | |- |
| | | |
| | | 15/8 |
| | | 711.731 |
| | | |
| | |- |
| | | 19\32 |
| | | |
| | | 712.500 |
| | | 32e val |
| | |- |
| | | 25\42 |
| | | |
| | | 714.286 |
| | | 42cee val |
| | |- |
| | | |
| | | 7/4 |
| | | 715.587 |
| | | |
| | |- |
| | | 6\10 |
| | | |
| | | 720.000 |
| | | 10e val, upper bound of 9-odd-limit diamond monotone |
| | |} |
| | |
| | == Music == |
| | ; [[Jake Freivald]] |
| | * [https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs ''Chord Sequence in Paul Erlich's Decatonic Major''] (2014) – in Pajara[10], 22edo tuning |
| | |
| | ; [[Joel Grant Taylor]] |
| | * [https://web.archive.org/web/20201127012345/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 ''Dirge''] – in the hexachordal dodecatonic modmos, [[12-22h]] |
| | * [https://web.archive.org/web/20201127012408/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 ''Sonatina''] – ditto |
| | |
| | ; [[Chris Vaisvil]] |
| | * ''Smoke Filled Bar'' (2012) – [https://www.chrisvaisvil.com/smoke-filled-bar/ blog] | [https://web.archive.org/web/20230530093324/http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 play] – in 12-22h. |
| | |
| | == References == |
| | <references/> |
| | |
| | [[Category:Pajara| ]] <!-- main article --> |
| | [[Category:Rank-2 temperaments]] |
| | [[Category:Archytas clan]] |
| | [[Category:Diaschismic family]] |
| | [[Category:Jubilismic clan]] |