Semicomma family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 247968755 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 255683340 - 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:genewardsmith|genewardsmith]] and made on <tt>2011- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-19 12:07:54 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>255683340</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 18: | Line 18: | ||
==Orwell== | ==Orwell== | ||
So called because 19\84 (as a [[fraction of the octave]]) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with [[22edo|22]], [[31edo|31]], [[53edo|53 | So called because 19\84 (as a [[fraction of the octave]]) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with [[22edo|22]], [[31edo|31]], [[53edo|53]] and [[84edo|84]] equal, and may be described as the 22&31 temperament, or <<7 -3 8 -21 -7 27||. It's a good system in the [[7-limit]] and naturally extends into the [[11-limit]]. [[84edo]], with the 19\84 generator, provides a good tuning for the 5, 7 and 11 limits, but it does use its second-best 11. However, the 19\84 generator is remarkably close to the 11-limit [[POTE tuning]], as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. [[53edo]] might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out 2430/2401, the nuwell comma, 1728/1715, the orwellisma, 225/224, the septimal kleisma, and 6144/6125, the porwell comma. | ||
The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell. | The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell. | ||
| Line 74: | Line 74: | ||
EDOs: 9, 22, 31, 53, 137 | EDOs: 9, 22, 31, 53, 137 | ||
Badness: 0.0197 | Badness: 0.0197 | ||
=Borwell= | |||
Commas: 225/224, 243/242, 1728/1715 | |||
POTE generator: | |||
Map: [<1 7 0 9 17|, <0 -14 6 -16 -35|] | |||
EDOs: 31, 106, 137 | |||
Badness: 0.0384 | |||
=Music= | =Music= | ||
| Line 80: | Line 89: | ||
[[http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3|Orwellian Cameras]] by [[Chris Vaisvil]]</pre></div> | [[http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3|Orwellian Cameras]] by [[Chris Vaisvil]]</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>Semicomma family</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>Semicomma family</title></head><body><!-- ws:start:WikiTextTocRule:16:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><div style="margin-left: 1em;"><a href="#Seven limit children">Seven limit children</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 2em;"><a href="#Seven limit children-Orwell">Orwell</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><div style="margin-left: 2em;"><a href="#Seven limit children-Vital statistics">Vital statistics</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><div style="margin-left: 1em;"><a href="#x11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><div style="margin-left: 1em;"><a href="#Winston">Winston</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><div style="margin-left: 1em;"><a href="#Julia">Julia</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><div style="margin-left: 1em;"><a href="#Borwell">Borwell</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><div style="margin-left: 1em;"><a href="#Music">Music</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --></div> | ||
<!-- ws:end:WikiTextTocRule:25 -->The 5-limit parent comma for the <strong>semicomma family</strong> is the semicomma, 2109375/2097152 = |-21 3 7&gt;. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor thirds. <strong>Orson</strong>, the <a class="wiki_link" href="/5-limit">5-limit</a> temperament tempering it out, has a <a class="wiki_link" href="/generator">generator</a> of 75/64, which is sharper than 7/6 by 225/224 when untempered, and less sharp than that in any good orson tempering, for example <a class="wiki_link" href="/53edo">53edo</a> or <a class="wiki_link" href="/84edo">84edo</a>. These give tunings to the generator which are sharp of 7/6 by less than five <a class="wiki_link" href="/cent">cent</a>s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.<br /> | |||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 271.627<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 271.627<br /> | ||
| Line 99: | Line 109: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Seven limit children-Orwell"></a><!-- ws:end:WikiTextHeadingRule:2 -->Orwell</h2> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Seven limit children-Orwell"></a><!-- ws:end:WikiTextHeadingRule:2 -->Orwell</h2> | ||
So called because 19\84 (as a <a class="wiki_link" href="/fraction%20of%20the%20octave">fraction of the octave</a>) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/53edo">53 | So called because 19\84 (as a <a class="wiki_link" href="/fraction%20of%20the%20octave">fraction of the octave</a>) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/53edo">53</a> and <a class="wiki_link" href="/84edo">84</a> equal, and may be described as the 22&amp;31 temperament, or &lt;&lt;7 -3 8 -21 -7 27||. It's a good system in the <a class="wiki_link" href="/7-limit">7-limit</a> and naturally extends into the <a class="wiki_link" href="/11-limit">11-limit</a>. <a class="wiki_link" href="/84edo">84edo</a>, with the 19\84 generator, provides a good tuning for the 5, 7 and 11 limits, but it does use its second-best 11. However, the 19\84 generator is remarkably close to the 11-limit <a class="wiki_link" href="/POTE%20tuning">POTE tuning</a>, as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. <a class="wiki_link" href="/53edo">53edo</a> might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out 2430/2401, the nuwell comma, 1728/1715, the orwellisma, 225/224, the septimal kleisma, and 6144/6125, the porwell comma.<br /> | ||
<br /> | <br /> | ||
The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell.<br /> | The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell.<br /> | ||
| Line 156: | Line 166: | ||
Badness: 0.0197<br /> | Badness: 0.0197<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Borwell"></a><!-- ws:end:WikiTextHeadingRule:12 -->Borwell</h1> | ||
Commas: 225/224, 243/242, 1728/1715<br /> | |||
<br /> | |||
POTE generator: <br /> | |||
<br /> | |||
Map: [&lt;1 7 0 9 17|, &lt;0 -14 6 -16 -35|]<br /> | |||
EDOs: 31, 106, 137<br /> | |||
Badness: 0.0384<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:14 -->Music</h1> | |||
<a class="wiki_link_ext" href="http://www.archive.org/details/TrioInOrwell" rel="nofollow">Trio in Orwell</a> <a class="wiki_link_ext" href="http://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a><br /> | <a class="wiki_link_ext" href="http://www.archive.org/details/TrioInOrwell" rel="nofollow">Trio in Orwell</a> <a class="wiki_link_ext" href="http://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a><br /> | ||
<a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101705" rel="nofollow">one drop of rain</a>, <a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101704" rel="nofollow">i've come with a bucket of roses</a>, and <a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839071" rel="nofollow">my own house</a> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a><br /> | <a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101705" rel="nofollow">one drop of rain</a>, <a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101704" rel="nofollow">i've come with a bucket of roses</a>, and <a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839071" rel="nofollow">my own house</a> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a><br /> | ||
<a class="wiki_link_ext" href="http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3" rel="nofollow">Orwellian Cameras</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a></body></html></pre></div> | <a class="wiki_link_ext" href="http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3" rel="nofollow">Orwellian Cameras</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a></body></html></pre></div> | ||