Porcupine
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- This revision was by author keenanpepper and made on 2011-10-31 17:36:22 UTC.
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Original Wikitext content:
**Porcupine** is a [[Regular Temperaments|linear temperament]] 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]] temperament, a 7-limit one, an 11-limit one, or a 2.3.5.11 [[Subgroup temperaments|subgroup temperament]]. In the 2.3.5.11 subgroup it is one of the best temperaments, 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== || 0 || 162.75 || 325.50 || 488.25 || 651.00 || 813.75 || 976.50 || 1139.25 || 102.00 || 264.75 || 427.50 || 590.25 || 753.00 || || 1/1 || 12/11~11/10~10/9 || 6/5~11/9 || 4/3 || 16/11 || 8/5 || 16/9~7/4 || 48/25~160/81 || 16/15 || 7/6 || 14/11 || 7/5 || 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 cents. 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 "wolf fifth" 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.
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> 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> temperament, a 7-limit one, an 11-limit one, or a 2.3.5.11 <a class="wiki_link" href="/Subgroup%20temperaments">subgroup temperament</a>. In the 2.3.5.11 subgroup it is one of the best temperaments, 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:0:<h2> --><h2 id="toc0"><a name="x-Interval chain"></a><!-- ws:end:WikiTextHeadingRule:0 -->Interval chain</h2>
<table class="wiki_table">
<tr>
<td>0<br />
</td>
<td>162.75<br />
</td>
<td>325.50<br />
</td>
<td>488.25<br />
</td>
<td>651.00<br />
</td>
<td>813.75<br />
</td>
<td>976.50<br />
</td>
<td>1139.25<br />
</td>
<td>102.00<br />
</td>
<td>264.75<br />
</td>
<td>427.50<br />
</td>
<td>590.25<br />
</td>
<td>753.00<br />
</td>
</tr>
<tr>
<td>1/1<br />
</td>
<td>12/11~11/10~10/9<br />
</td>
<td>6/5~11/9<br />
</td>
<td>4/3<br />
</td>
<td>16/11<br />
</td>
<td>8/5<br />
</td>
<td>16/9~7/4<br />
</td>
<td>48/25~160/81<br />
</td>
<td>16/15<br />
</td>
<td>7/6<br />
</td>
<td>14/11<br />
</td>
<td>7/5<br />
</td>
<td>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 cents.<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 "wolf fifth" 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.</body></html>