Semicomma family: Difference between revisions
Wikispaces>xenwolf **Imported revision 247384505 - 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:xenwolf|xenwolf]] and made on <tt>2011-08-21 11: | : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-08-21 11:19:41 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>247384829</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">The 5-limit parent comma for the semicomma family is the semicomma, 2109375/2097152 = |-21 3 7>. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor thirds. **Orson**, the [[5-limit]] temperament tempering it out, has a [[generator]] 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 [[53edo]] or [[84edo]]. These give tunings to the generator which are sharp of 7/6 by less than five [[cent]]s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell. | <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">[[toc]] | ||
The 5-limit parent comma for the semicomma family is the semicomma, 2109375/2097152 = |-21 3 7>. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor thirds. **Orson**, the [[5-limit]] temperament tempering it out, has a [[generator]] 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 [[53edo]] or [[84edo]]. These give tunings to the generator which are sharp of 7/6 by less than five [[cent]]s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell. | |||
[[POTE tuning|POTE generator]]: 271.627 | [[POTE tuning|POTE generator]]: 271.627 | ||
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EDOs: 22, 31, 53, 190, 253, 296 | EDOs: 22, 31, 53, 190, 253, 296 | ||
=Seven limit children= | |||
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding 65536/64625 leads to orwell, but we could also add 1029/1024, leading to the 31&159 temperament with wedgie <<21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&243 temperament with wedgie <<28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&243 temperament with wedgie <<7 -3 61 -21 77 150||. | The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding 65536/64625 leads to orwell, but we could also add 1029/1024, leading to the 31&159 temperament with wedgie <<21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&243 temperament with wedgie <<28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&243 temperament with wedgie <<7 -3 61 -21 77 150||. | ||
==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-EDO]] and [[84edo]], 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. | 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-EDO]] and [[84edo]], 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. | ||
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Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has considerable harmonic resources despite its absence of 5-limit triads. The 13 note MOS has those, and of course the 22 and 31 note MOS are very well supplied with everything. | Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has considerable harmonic resources despite its absence of 5-limit triads. The 13 note MOS has those, and of course the 22 and 31 note MOS are very well supplied with everything. | ||
==Vital statistics== | |||
[[Comma|Commas]]: 225/224, 1728/1715 | [[Comma|Commas]]: 225/224, 1728/1715 | ||
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EDOs: 22, 31, 53, 84, 137 | EDOs: 22, 31, 53, 84, 137 | ||
=11-limit= | |||
[[Comma|Commas]]: 99/98, 121/120, 176/175 | [[Comma|Commas]]: 99/98, 121/120, 176/175 | ||
| Line 56: | Line 57: | ||
Badness: 99/98, 121/120, 176/175 | Badness: 99/98, 121/120, 176/175 | ||
=Winston= | |||
Commas: 66/65, 99/98, 105/104, 121/120 | Commas: 66/65, 99/98, 105/104, 121/120 | ||
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Badness: 0.0199 | Badness: 0.0199 | ||
=Julia= | |||
Commas: 99/98, 121/120, 176/175, 275/273 | Commas: 99/98, 121/120, 176/175, 275/273 | ||
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Badness: 0.0197 | Badness: 0.0197 | ||
=Music= | |||
http://www.archive.org/details/TrioInOrwell by [[Gene Ward Smith]] | http://www.archive.org/details/TrioInOrwell by [[Gene Ward Smith]] | ||
[[http://soundclick.com/share?songid=9101705|one drop of rain]], [[http://soundclick.com/share?songid=9101704|i've come with a bucket of roses]], and [[http://soundclick.com/share?songid=8839071|my own house]] by [[Andrew Heathwaite]] | [[http://soundclick.com/share?songid=9101705|one drop of rain]], [[http://soundclick.com/share?songid=9101704|i've come with a bucket of roses]], and [[http://soundclick.com/share?songid=8839071|my own house]] by [[Andrew Heathwaite]] | ||
http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3 by [[Chris Vaisvil]]</pre></div> | http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3 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>The 5-limit parent comma for the semicomma family 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 /> | <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:14:&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:14 --><!-- ws:start:WikiTextTocRule:15: --><div style="margin-left: 1em;"><a href="#Seven limit children">Seven limit children</a></div> | ||
<!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><div style="margin-left: 2em;"><a href="#Seven limit children-Orwell">Orwell</a></div> | |||
<!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><div style="margin-left: 2em;"><a href="#Seven limit children-Vital statistics">Vital statistics</a></div> | |||
<!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 1em;"><a href="#x11-limit">11-limit</a></div> | |||
<!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><div style="margin-left: 1em;"><a href="#Winston">Winston</a></div> | |||
<!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><div style="margin-left: 1em;"><a href="#Julia">Julia</a></div> | |||
<!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><div style="margin-left: 1em;"><a href="#Music">Music</a></div> | |||
<!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --></div> | |||
<!-- ws:end:WikiTextTocRule:22 -->The 5-limit parent comma for the semicomma family 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 86: | Line 95: | ||
EDOs: 22, 31, 53, 190, 253, 296<br /> | EDOs: 22, 31, 53, 190, 253, 296<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt; | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h1> | ||
The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> defines which 7-limit family member we are looking at. Adding 65536/64625 leads to orwell, but we could also add 1029/1024, leading to the 31&amp;159 temperament with wedgie &lt;&lt;21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&amp;243 temperament with wedgie &lt;&lt;28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&amp;243 temperament with wedgie &lt;&lt;7 -3 61 -21 77 150||.<br /> | The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> defines which 7-limit family member we are looking at. Adding 65536/64625 leads to orwell, but we could also add 1029/1024, leading to the 31&amp;159 temperament with wedgie &lt;&lt;21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&amp;243 temperament with wedgie &lt;&lt;28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&amp;243 temperament with wedgie &lt;&lt;7 -3 61 -21 77 150||.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt; | <!-- 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-EDO</a> and <a class="wiki_link" href="/84edo">84edo</a>, 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 /> | 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-EDO</a> and <a class="wiki_link" href="/84edo">84edo</a>, 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 /> | ||
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Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has considerable harmonic resources despite its absence of 5-limit triads. The 13 note MOS has those, and of course the 22 and 31 note MOS are very well supplied with everything.<br /> | Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has considerable harmonic resources despite its absence of 5-limit triads. The 13 note MOS has those, and of course the 22 and 31 note MOS are very well supplied with everything.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt; | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Seven limit children-Vital statistics"></a><!-- ws:end:WikiTextHeadingRule:4 -->Vital statistics</h2> | ||
<a class="wiki_link" href="/Comma">Commas</a>: 225/224, 1728/1715<br /> | <a class="wiki_link" href="/Comma">Commas</a>: 225/224, 1728/1715<br /> | ||
<br /> | <br /> | ||
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EDOs: 22, 31, 53, 84, 137<br /> | EDOs: 22, 31, 53, 84, 137<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt; | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="x11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h1> | ||
<a class="wiki_link" href="/Comma">Commas</a>: 99/98, 121/120, 176/175<br /> | <a class="wiki_link" href="/Comma">Commas</a>: 99/98, 121/120, 176/175<br /> | ||
<br /> | <br /> | ||
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Badness: 99/98, 121/120, 176/175<br /> | Badness: 99/98, 121/120, 176/175<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt; | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Winston"></a><!-- ws:end:WikiTextHeadingRule:8 -->Winston</h1> | ||
Commas: 66/65, 99/98, 105/104, 121/120<br /> | Commas: 66/65, 99/98, 105/104, 121/120<br /> | ||
<br /> | <br /> | ||
| Line 138: | Line 147: | ||
Badness: 0.0199<br /> | Badness: 0.0199<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt; | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Julia"></a><!-- ws:end:WikiTextHeadingRule:10 -->Julia</h1> | ||
Commas: 99/98, 121/120, 176/175, 275/273<br /> | Commas: 99/98, 121/120, 176/175, 275/273<br /> | ||
<br /> | <br /> | ||
| Line 147: | Line 156: | ||
Badness: 0.0197<br /> | Badness: 0.0197<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt; | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:12 -->Music</h1> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:125:http://www.archive.org/details/TrioInOrwell --><a class="wiki_link_ext" href="http://www.archive.org/details/TrioInOrwell" rel="nofollow">http://www.archive.org/details/TrioInOrwell</a><!-- ws:end:WikiTextUrlRule:125 --> 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 /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:126:http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3 --><a class="wiki_link_ext" href="http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3" rel="nofollow">http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3</a><!-- ws:end:WikiTextUrlRule:126 --> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a></body></html></pre></div> | ||