Porcupine
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**Porcupine** is a [[Regular Temperaments|linear temperament]] in the [[porcupine family]] that tempers out 250/243, the porcupine [[Comma|comma]], and whose generator is somewhere around 160-165 cents. It can be thought of as a 5-[[Harmonic Limit|limit]], 7-limit, or 11-limit temperament, or a 2.3.5.11 [[Subgroup temperaments|subgroup temperament]]. It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy. The basic 5-limit harmonic structure of porcupine can be understood simply by noting that tempering out 250/243 makes (4/3)^2 equivalent to (6/5)^3. In perhaps more familiar musical terms, this means two "perfect fourths" equals three "minor thirds". As a consequence of this, 4/3 is divided into 3 equal parts, and 6/5 is divided into 2 of those same equal parts. This is obviously in stark contrast to [[12edo]], and to meantone, in which neither 4/3 nor 6/5 can be divided into any number of equal parts. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many porcupine scales. ==Interval chain== Main article: [[Porcupine intervals]] ||~ Generators ||~ Cents ||~ Ratios || || 0 ||> 0.00 ||= 1/1 || || 1 ||> 162.75 ||= 12/11~11/10~10/9 || || 2 ||> 325.50 ||= 6/5~11/9 || || 3 ||> 488.25 ||= 4/3 || || 4 ||> 651.00 ||= 16/11~22/15 || || 5 ||> 813.75 ||= 8/5 || || 6 ||> 976.50 ||= 7/4~16/9 || || 7 ||> 1139.25 ||= 48/25~160/81 || || 8 ||> 102.00 ||= 16/15~21/20 || || 9 ||> 264.75 ||= 7/6 || || 10 ||> 427.50 ||= 14/11 || || 11 ||> 590.25 ||= 7/5 || || 12 ||> 753.00 ||= 14/9 || The specific tuning shown is the full 11-limit [[POTE tuning]], but of course there is a range of acceptible porcupine tunings that includes generators as small as 160 cents ([[15edo]]) and as large as 165.5 cents ([[29edo]]). (However, the 29edo patent val does not support 11-limit porcupine proper, not annihilating 64/63.) 12/11, 11/10, and 10/9 are all represented by the same interval, the generator. This makes chords such as 8:9:10:11:12 exceptionally common and easy to find. The 11/9 interval, usually considered a "neutral third", is in porcupine identical to the 6/5 "minor third". This means that the 27/20 "acute fourth" of the JI diatonic scale is equivalent to 11/8 (rather than becoming 4/3 as in meantone). The characteristic small interval of porcupine, which is 60.75 cents in this tuning but can range from <50 to 80 cents in general, represents both 25/24 and 81/80. [[media type="custom" key="11980245"]] ==History== <span class="commentBody">Porcupine temperament/scales were discovered by [[Dave Keenan]], but didn't have a name until [[Herman Miller]] mentioned that his Mizarian Porcupine Overture in 15-tET had a section that pumps the 250:243 comma. Although this music did not use a Porcupine MOS or MODMOS (which would have 7 or 8 notes), the name was adopted for such scales as well, once the essentially one-to-one relationship between vanishing commas and sequences of DE scales was fully evident. It was clear that even though Herman's piece was in 15, 22 was a porcupine tuning par excellence, and that was an interesting development in itself.</span> ==See also== [[Chords of porcupine]] ==Musical examples== * "[[http://sites.google.com/site/teamouse/home#TOC-Mizarian-music|Mizarian Porcupine Overture]]", Herman Miller, 1999. (15edo, namesake of the temperament) * "[[http://www.myspace.com/paulerlich/music/songs/glassic-in-22-tone-equal-temperament-45202095|Glassic]]", Paul Erlich, [[22edo]] (at least the beginning part is in porcupine). * "<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3|Night on Porcupine Mountain]]////</span></span>", Gene Ward Smith and Modest Mussorgsky, [[22edo]]. * "[[http://soundclick.com/share.cfm?id=8839060|being a]]", Andrew Heathwaite, 2010, 22edo, mode 3 1 3 3 3 3 3 3 of Porcupine[8]. * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://micro.soonlabel.com/15-ET/daily20110619_millers_porcupine_7a.mp3|Playing Gently with Miller's Porcupine]]////</span></span>, [[Chris Vaisvil]] * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-indian.mp3|15 Porcupines in India]]////</span></span>, Sarangi, Tambura and Sitar improvisation by [[Chris Vaisvil]] * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-piano.mp3|15 Quills]]////</span></span> piano solo by Chris Vaisvil * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-prickly-side-of-love.mp3|Prickly Side of Love]]////</span></span> - rock band in Porcupine Temperament with vocals by Chris Vaisvil * <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://micro.soonlabel.com/15-ET/daily20120102-porcupine-organ.mp3|Porcupine Organ Composition]]////</span> by [[Chris Vaisvil]] * ////[[file:xenharmonic/AmongOtherThings2.mp3|Among Other Things 2]]//// by Petr Pařízek ==Images== [[image:porcupine8.jpg]]
Original HTML content:
<html><head><title>Porcupine</title></head><body><strong>Porcupine</strong> is a <a class="wiki_link" href="/Regular%20Temperaments">linear temperament</a> in the <a class="wiki_link" href="/porcupine%20family">porcupine family</a> that tempers out 250/243, the porcupine <a class="wiki_link" href="/Comma">comma</a>, and whose generator is somewhere around 160-165 cents. It can be thought of as a 5-<a class="wiki_link" href="/Harmonic%20Limit">limit</a>, 7-limit, or 11-limit temperament, or a 2.3.5.11 <a class="wiki_link" href="/Subgroup%20temperaments">subgroup temperament</a>. It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.<br />
<br />
The basic 5-limit harmonic structure of porcupine can be understood simply by noting that tempering out 250/243 makes (4/3)^2 equivalent to (6/5)^3. In perhaps more familiar musical terms, this means two "perfect fourths" equals three "minor thirds". As a consequence of this, 4/3 is divided into 3 equal parts, and 6/5 is divided into 2 of those same equal parts. This is obviously in stark contrast to <a class="wiki_link" href="/12edo">12edo</a>, and to meantone, in which neither 4/3 nor 6/5 can be divided into any number of equal parts. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many porcupine scales.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:1:<h2> --><h2 id="toc0"><a name="x-Interval chain"></a><!-- ws:end:WikiTextHeadingRule:1 -->Interval chain</h2>
Main article: <a class="wiki_link" href="/Porcupine%20intervals">Porcupine intervals</a><br />
<table class="wiki_table">
<tr>
<th>Generators<br />
</th>
<th>Cents<br />
</th>
<th>Ratios<br />
</th>
</tr>
<tr>
<td>0<br />
</td>
<td style="text-align: right;">0.00<br />
</td>
<td style="text-align: center;">1/1<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td style="text-align: right;">162.75<br />
</td>
<td style="text-align: center;">12/11~11/10~10/9<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td style="text-align: right;">325.50<br />
</td>
<td style="text-align: center;">6/5~11/9<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td style="text-align: right;">488.25<br />
</td>
<td style="text-align: center;">4/3<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td style="text-align: right;">651.00<br />
</td>
<td style="text-align: center;">16/11~22/15<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td style="text-align: right;">813.75<br />
</td>
<td style="text-align: center;">8/5<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td style="text-align: right;">976.50<br />
</td>
<td style="text-align: center;">7/4~16/9<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td style="text-align: right;">1139.25<br />
</td>
<td style="text-align: center;">48/25~160/81<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td style="text-align: right;">102.00<br />
</td>
<td style="text-align: center;">16/15~21/20<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td style="text-align: right;">264.75<br />
</td>
<td style="text-align: center;">7/6<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td style="text-align: right;">427.50<br />
</td>
<td style="text-align: center;">14/11<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td style="text-align: right;">590.25<br />
</td>
<td style="text-align: center;">7/5<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td style="text-align: right;">753.00<br />
</td>
<td style="text-align: center;">14/9<br />
</td>
</tr>
</table>
The specific tuning shown is the full 11-limit <a class="wiki_link" href="/POTE%20tuning">POTE tuning</a>, but of course there is a range of acceptible porcupine tunings that includes generators as small as 160 cents (<a class="wiki_link" href="/15edo">15edo</a>) and as large as 165.5 cents (<a class="wiki_link" href="/29edo">29edo</a>). (However, the 29edo patent val does not support 11-limit porcupine proper, not annihilating 64/63.)<br />
12/11, 11/10, and 10/9 are all represented by the same interval, the generator. This makes chords such as 8:9:10:11:12 exceptionally common and easy to find.<br />
The 11/9 interval, usually considered a "neutral third", is in porcupine identical to the 6/5 "minor third". This means that the 27/20 "acute fourth" of the JI diatonic scale is equivalent to 11/8 (rather than becoming 4/3 as in meantone).<br />
The characteristic small interval of porcupine, which is 60.75 cents in this tuning but can range from <50 to 80 cents in general, represents both 25/24 and 81/80.<br />
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<!-- ws:start:WikiTextHeadingRule:3:<h2> --><h2 id="toc1"><a name="x-History"></a><!-- ws:end:WikiTextHeadingRule:3 -->History</h2>
<span class="commentBody">Porcupine temperament/scales were discovered by <a class="wiki_link" href="/Dave%20Keenan">Dave Keenan</a>, but didn't have a name until <a class="wiki_link" href="/Herman%20Miller">Herman Miller</a> mentioned that his Mizarian Porcupine Overture in 15-tET had a section that pumps the 250:243 comma. Although this music did not use a Porcupine MOS or MODMOS (which would have 7 or 8 notes), the name was adopted for such scales as well, once the essentially one-to-one relationship between vanishing commas and sequences of DE scales was fully evident. It was clear that even though Herman's piece was in 15, 22 was a porcupine tuning par excellence, and that was an interesting development in itself.</span><br />
<br />
<!-- ws:start:WikiTextHeadingRule:5:<h2> --><h2 id="toc2"><a name="x-See also"></a><!-- ws:end:WikiTextHeadingRule:5 -->See also</h2>
<a class="wiki_link" href="/Chords%20of%20porcupine">Chords of porcupine</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:7:<h2> --><h2 id="toc3"><a name="x-Musical examples"></a><!-- ws:end:WikiTextHeadingRule:7 -->Musical examples</h2>
<ul><li>"<a class="wiki_link_ext" href="http://sites.google.com/site/teamouse/home#TOC-Mizarian-music" rel="nofollow">Mizarian Porcupine Overture</a>", Herman Miller, 1999. (15edo, namesake of the temperament)</li><li>"<a class="wiki_link_ext" href="http://www.myspace.com/paulerlich/music/songs/glassic-in-22-tone-equal-temperament-45202095" rel="nofollow">Glassic</a>", Paul Erlich, <a class="wiki_link" href="/22edo">22edo</a> (at least the beginning part is in porcupine).</li><li>"<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3" rel="nofollow">Night on Porcupine Mountain</a></span></span>", Gene Ward Smith and Modest Mussorgsky, <a class="wiki_link" href="/22edo">22edo</a>.</li><li>"<a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=8839060" rel="nofollow">being a</a>", Andrew Heathwaite, 2010, 22edo, mode 3 1 3 3 3 3 3 3 of Porcupine[8].</li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/daily20110619_millers_porcupine_7a.mp3" rel="nofollow">Playing Gently with Miller's Porcupine</a></span></span>, <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a></li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-indian.mp3" rel="nofollow">15 Porcupines in India</a></span></span>, Sarangi, Tambura and Sitar improvisation by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a></li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-piano.mp3" rel="nofollow">15 Quills</a></span></span> piano solo by Chris Vaisvil</li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-prickly-side-of-love.mp3" rel="nofollow">Prickly Side of Love</a></span></span> - rock band in Porcupine Temperament with vocals by Chris Vaisvil</li><li><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/daily20120102-porcupine-organ.mp3" rel="nofollow">Porcupine Organ Composition</a></span> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a></li><li><a href="http://xenharmonic.wikispaces.com/file/view/AmongOtherThings2.mp3/319978024/AmongOtherThings2.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/AmongOtherThings2.mp3/319978024/AmongOtherThings2.mp3');">Among Other Things 2</a> by Petr Pařízek</li></ul><!-- ws:start:WikiTextHeadingRule:9:<h2> --><h2 id="toc4"><a name="x-Images"></a><!-- ws:end:WikiTextHeadingRule:9 -->Images</h2>
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