Porcupine: Difference between revisions

Revision as of 17:36, 31 October 2011

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author keenanpepper and made on 2011-10-31 17:36:22 UTC.

The original revision id was 270426250.

The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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 &quot;perfect fourths&quot; equals three &quot;minor thirds&quot;. 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 &quot;equal tetrachord&quot; formed by dividing 4/3 into 3 equal parts is a characteristic feature of many porcupine scales.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><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 &quot;neutral third&quot;, is in porcupine identical to the 6/5 &quot;minor third&quot;. This means that the 27/20 &quot;wolf fifth&quot; 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 &lt;50 to 80 cents in general, represents both 25/24 and 81/80.</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:guest|guest]] and made on <tt>2011-10-31 17:01:16 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-10-31 17:36:22 UTC</tt>.<br>
-: The original revision id was <tt>270416010</tt>.<br>
+: The original revision id was <tt>270426250</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>
@@ Line 8: / Line 8: @@
 <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">**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.</pre></div>
+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.
+/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 &lt;50 to 80 cents in general, represents both 25/24 and 81/80.</pre></div>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Porcupine&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;strong&gt;Porcupine&lt;/strong&gt; is a &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;linear temperament&lt;/a&gt; that tempers out 250/243, the porcupine &lt;a class="wiki_link" href="/Comma"&gt;comma&lt;/a&gt;, and whose generator is somewhere around 160-165 cents. It can be thought of as a 5-&lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;limit&lt;/a&gt; temperament, a 7-limit one, an 11-limit one, or a 2.3.5.11 &lt;a class="wiki_link" href="/Subgroup%20temperaments"&gt;subgroup temperament&lt;/a&gt;. In the 2.3.5.11 subgroup it is one of the best temperaments, with a unique combination of efficiency and accuracy.&lt;br /&gt;
 &lt;br /&gt;
-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 &amp;quot;perfect fourths&amp;quot; equals three &amp;quot;minor thirds&amp;quot;. 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 &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, and to meantone in which neither 4/3 nor 6/5 can be divided into any number of equal parts. The &amp;quot;equal tetrachord&amp;quot; formed by dividing 4/3 into 3 equal parts is a characteristic feature of many porcupine scales.&lt;/body&gt;&lt;/html&gt;</pre></div>
+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 &amp;quot;perfect fourths&amp;quot; equals three &amp;quot;minor thirds&amp;quot;. 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 &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, and to meantone in which neither 4/3 nor 6/5 can be divided into any number of equal parts. The &amp;quot;equal tetrachord&amp;quot; formed by dividing 4/3 into 3 equal parts is a characteristic feature of many porcupine scales.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Interval chain"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Interval chain&lt;/h2&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;162.75&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;325.50&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;488.25&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;651.00&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;813.75&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;976.50&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1139.25&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;102.00&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;264.75&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;427.50&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;590.25&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;753.00&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;1/1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;12/11~11/10~10/9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;6/5~11/9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;4/3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;16/11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;8/5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;16/9~7/4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;48/25~160/81&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;16/15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7/6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;14/11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7/5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;14/9&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+The specific tuning shown is the full 11-limit &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE tuning&lt;/a&gt;, but of course there is a range of acceptible porcupine tunings that includes generators as small as 160 cents (&lt;a class="wiki_link" href="/15edo"&gt;15edo&lt;/a&gt;) and as large as 165 cents.&lt;br /&gt;
+/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.&lt;br /&gt;
+The 11/9 interval, usually considered a &amp;quot;neutral third&amp;quot;, is in porcupine identical to the 6/5 &amp;quot;minor third&amp;quot;. This means that the 27/20 &amp;quot;wolf fifth&amp;quot; of the JI diatonic scale is equivalent to 11/8 (rather than becoming 4/3 as in meantone).&lt;br /&gt;
+The characteristic small interval of porcupine, which is 60.75 cents in this tuning but can range from &amp;lt;50 to 80 cents in general, represents both 25/24 and 81/80.&lt;/body&gt;&lt;/html&gt;</pre></div>

Porcupine: Difference between revisions

Revision as of 17:36, 31 October 2011

IMPORTED REVISION FROM WIKISPACES

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Original HTML content:

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