22edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 242771043 - Original comment: ** |
Wikispaces>keenanpepper **Imported revision 245875573 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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:keenanpepper|keenanpepper]] and made on <tt>2011-08-14 12:44:57 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>245875573</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> | ||
| Line 54: | Line 54: | ||
In the 7-limit 22-et tempers out certain commas also tempered out by 12-et; this relates 12 equal to 22 in a way different from the way in which meantone systems are akin to it. Both [[50_49|50/49]], (the [[jubilee comma]]), and [[64_63|64/63]], (the [[septimal comma]]), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritons of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the [[septimal kleisma]], so that the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. A septimal comma not tempered out by 12-et which 22-et does temper out is 1728/1715, the [[orwell comma]]; and the [[orwell tetrad]] is also a chord of 22-et. | In the 7-limit 22-et tempers out certain commas also tempered out by 12-et; this relates 12 equal to 22 in a way different from the way in which meantone systems are akin to it. Both [[50_49|50/49]], (the [[jubilee comma]]), and [[64_63|64/63]], (the [[septimal comma]]), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritons of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the [[septimal kleisma]], so that the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. A septimal comma not tempered out by 12-et which 22-et does temper out is 1728/1715, the [[orwell comma]]; and the [[orwell tetrad]] is also a chord of 22-et. | ||
===Linear Temperaments=== | |||
||~ Periods | |||
per octave ||~ Generator ||~ Temperaments || | |||
|| 1 || 1\22 || || | |||
|| 1 || 3\22 || [[Porcupine]] || | |||
|| 1 || 5\22 || [[Orwell]] || | |||
|| 1 || 7\22 || [[Magic]] || | |||
|| 1 || 9\22 || [[Superpyth]] || | |||
|| 2 || 1\22 || [[Shrutar]] || | |||
|| 2 || 2\22 || [[Pajara]] || | |||
|| 2 || 3\22 || [[Hedgehog]]/[[echidna]] || | |||
|| 2 || 4\22 || [[Astrology]]/[[wizard]] || | |||
|| 2 || 5\22 || [[Doublewide]] || | |||
|| 11 || 1\22 || (unnamed) || | |||
===Commas=== | ===Commas=== | ||
22 EDO tempers out the following commas. (Note: This assumes the val < 22 35 51 62 76 81 |.) | 22 EDO tempers out the following commas. (Note: This assumes the val < 22 35 51 62 76 81 |.) | ||
| Line 121: | Line 135: | ||
|| < 22 35 51 62 76 81 | ||</pre></div> | || < 22 35 51 62 76 81 | ||</pre></div> | ||
<h4>Original HTML content:</h4> | <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>22edo</title></head><body><!-- ws:start:WikiTextTocRule: | <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>22edo</title></head><body><!-- ws:start:WikiTextTocRule:18:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><a href="#Theory">Theory</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Compositions">Compositions</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:28 --><hr /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Theory"></a><!-- ws:end:WikiTextHeadingRule:0 -->Theory</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Theory"></a><!-- ws:end:WikiTextHeadingRule:0 -->Theory</h1> | ||
<br /> | <br /> | ||
| Line 342: | Line 356: | ||
In the 7-limit 22-et tempers out certain commas also tempered out by 12-et; this relates 12 equal to 22 in a way different from the way in which meantone systems are akin to it. Both <a class="wiki_link" href="/50_49">50/49</a>, (the <a class="wiki_link" href="/jubilee%20comma">jubilee comma</a>), and <a class="wiki_link" href="/64_63">64/63</a>, (the <a class="wiki_link" href="/septimal%20comma">septimal comma</a>), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritons of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the <a class="wiki_link" href="/septimal%20kleisma">septimal kleisma</a>, so that the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. A septimal comma not tempered out by 12-et which 22-et does temper out is 1728/1715, the <a class="wiki_link" href="/orwell%20comma">orwell comma</a>; and the <a class="wiki_link" href="/orwell%20tetrad">orwell tetrad</a> is also a chord of 22-et.<br /> | In the 7-limit 22-et tempers out certain commas also tempered out by 12-et; this relates 12 equal to 22 in a way different from the way in which meantone systems are akin to it. Both <a class="wiki_link" href="/50_49">50/49</a>, (the <a class="wiki_link" href="/jubilee%20comma">jubilee comma</a>), and <a class="wiki_link" href="/64_63">64/63</a>, (the <a class="wiki_link" href="/septimal%20comma">septimal comma</a>), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritons of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the <a class="wiki_link" href="/septimal%20kleisma">septimal kleisma</a>, so that the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. A septimal comma not tempered out by 12-et which 22-et does temper out is 1728/1715, the <a class="wiki_link" href="/orwell%20comma">orwell comma</a>; and the <a class="wiki_link" href="/orwell%20tetrad">orwell tetrad</a> is also a chord of 22-et.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="Theory-Properties of 22 equal temperament-Commas"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="Theory-Properties of 22 equal temperament-Linear Temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->Linear Temperaments</h3> | ||
<table class="wiki_table"> | |||
<tr> | |||
<th>Periods<br /> | |||
per octave<br /> | |||
</th> | |||
<th>Generator<br /> | |||
</th> | |||
<th>Temperaments<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td>1<br /> | |||
</td> | |||
<td>1\22<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>1<br /> | |||
</td> | |||
<td>3\22<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Porcupine">Porcupine</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>1<br /> | |||
</td> | |||
<td>5\22<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Orwell">Orwell</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>1<br /> | |||
</td> | |||
<td>7\22<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Magic">Magic</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>1<br /> | |||
</td> | |||
<td>9\22<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Superpyth">Superpyth</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>2<br /> | |||
</td> | |||
<td>1\22<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Shrutar">Shrutar</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>2<br /> | |||
</td> | |||
<td>2\22<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Pajara">Pajara</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>2<br /> | |||
</td> | |||
<td>3\22<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Hedgehog">Hedgehog</a>/<a class="wiki_link" href="/echidna">echidna</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>2<br /> | |||
</td> | |||
<td>4\22<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Astrology">Astrology</a>/<a class="wiki_link" href="/wizard">wizard</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>2<br /> | |||
</td> | |||
<td>5\22<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/Doublewide">Doublewide</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11<br /> | |||
</td> | |||
<td>1\22<br /> | |||
</td> | |||
<td>(unnamed)<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Theory-Properties of 22 equal temperament-Commas"></a><!-- ws:end:WikiTextHeadingRule:6 -->Commas</h3> | |||
22 EDO tempers out the following commas. (Note: This assumes the val &lt; 22 35 51 62 76 81 |.)<br /> | 22 EDO tempers out the following commas. (Note: This assumes the val &lt; 22 35 51 62 76 81 |.)<br /> | ||
| Line 755: | Line 872: | ||
</table> | </table> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Theory-Properties of 22 equal temperament-A Superpythagorean System"></a><!-- ws:end:WikiTextHeadingRule:8 -->A Superpythagorean System</h3> | ||
<br /> | <br /> | ||
The 22edo fifth, measuring approximately 709.1 cents, is wider than the 702-cent <a class="wiki_link" href="/3-limit">3-limit</a> fifth, thus making 22edo a &quot;super-pythagorean&quot; system. As with any superpyth, a chain of fifths produces relatively wide major thirds and narrow minor thirds. In the case of 22edo, the thirds are stretched out to the <a class="wiki_link" href="/7-limit">7-limit</a> ; the <a class="wiki_link" href="/subminor%20third">subminor third</a> comes close to <a class="wiki_link" href="/7_6">7/6</a> and the <a class="wiki_link" href="/supermajor%20third">supermajor third</a> to <a class="wiki_link" href="/9_7">9/7</a>. Thus, the resulting diatonic scale, which no longer approximates 5-limit thirds, sounds oddly consonant. The ratio of major 2nd to minor 2nd in this diatonic scale is stretched out to 4:1, with the M2 falling between 9/8 and <a class="wiki_link" href="/8_7">8/7</a>, and the m2 falling close to a quarter-tone.<br /> | The 22edo fifth, measuring approximately 709.1 cents, is wider than the 702-cent <a class="wiki_link" href="/3-limit">3-limit</a> fifth, thus making 22edo a &quot;super-pythagorean&quot; system. As with any superpyth, a chain of fifths produces relatively wide major thirds and narrow minor thirds. In the case of 22edo, the thirds are stretched out to the <a class="wiki_link" href="/7-limit">7-limit</a> ; the <a class="wiki_link" href="/subminor%20third">subminor third</a> comes close to <a class="wiki_link" href="/7_6">7/6</a> and the <a class="wiki_link" href="/supermajor%20third">supermajor third</a> to <a class="wiki_link" href="/9_7">9/7</a>. Thus, the resulting diatonic scale, which no longer approximates 5-limit thirds, sounds oddly consonant. The ratio of major 2nd to minor 2nd in this diatonic scale is stretched out to 4:1, with the M2 falling between 9/8 and <a class="wiki_link" href="/8_7">8/7</a>, and the m2 falling close to a quarter-tone.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Theory-Properties of 22 equal temperament-11edo"></a><!-- ws:end:WikiTextHeadingRule:10 -->11edo</h3> | ||
<br /> | <br /> | ||
As 22 is divisible by 11, a 22edo instrument can play any music in <a class="wiki_link" href="/11edo">11edo</a>, in the same way that 12edo can play 6edo (the whole tone scale).<br /> | As 22 is divisible by 11, a 22edo instrument can play any music in <a class="wiki_link" href="/11edo">11edo</a>, in the same way that 12edo can play 6edo (the whole tone scale).<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Theory-External links"></a><!-- ws:end:WikiTextHeadingRule:12 -->External links</h2> | ||
<br /> | <br /> | ||
<a class="wiki_link_ext" href="http://lumma.org/tuning/erlich/erlich-decatonic.pdf" rel="nofollow">Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament''</a><br /> | <a class="wiki_link_ext" href="http://lumma.org/tuning/erlich/erlich-decatonic.pdf" rel="nofollow">Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament''</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Theory-References"></a><!-- ws:end:WikiTextHeadingRule:14 -->References</h2> | ||
<br /> | <br /> | ||
Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]<br /> | Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]<br /> | ||
| Line 773: | Line 890: | ||
<br /> | <br /> | ||
<hr /> | <hr /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:16 -->Compositions</h1> | ||
<br /> | <br /> | ||
<a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/TIBIA.mp3" rel="nofollow">Tibia</a> by <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a><br /> | <a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/TIBIA.mp3" rel="nofollow">Tibia</a> by <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a><br /> | ||