Cluster MOS

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This revision was by author keenanpepper and made on 2012-05-11 04:08:38 UTC.
The original revision id was 333417886.
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Original Wikitext content:

A cluster temperament (named by [[Keenan Pepper]]) is a very particular kind of rank-2 temperament whose generator is quite near a rational fraction of an octave. Therefore some MOS of the temperament is quasi-equal (which should be reasonably sized for it to be a good cluster temperament, usually between 5 and 10 notes per octave). But not only that; in a cluster temperament, the different versions of each interval, differing by a chroma ("diminished", "minor", "major", "augmented"...) include many nearby JI intervals that are individually recognizable, yet conceptually grouped into the same category because they're so close.

An example of something that is **not** a cluster temperament is [[amity]], because although the amity generator is within 4 cents of 2\7, making amity[7] near equal, amity is too complex of a temperament and most of the intervals differing by a chroma do not represent simple JI intervals at all. For example, the list of amity "thirds" includes ...6/5 (339.5) (363.2) 5/4... where the intervals given in cents are not representable as simple JI intervals (243/200 and 100/81 are about as simple as you can get).

Another way to describe this property is that the chroma of the near-equal MOS is a kind of "super-comma", a set of many useful commas that are tempered to become the same, non-vanishing, interval. It should be obvious that "cluster temperament" is a vague, qualitative phrase and not mathematically well-defined.

=Examples= 
==Slendric== 
Chroma: 49/48~64/63
|| Steps || "Diminished" || "Minor" || "Major" || "Augmented" ||
|| 1 || 9/8 || 8/7 || 7/6 || 32/27 ||
|| 2 || 9/7 || 21/16 || 4/3 ||   ||
|| 3 ||   || 3/2 || 32/21 || 14/9 ||
|| 4 || 27/16 || 12/7 || 7/4 || 16/9 ||
Slendric has two quite different extensions that are both also cluster scales:
===Mothra=== 
Chroma: 33/32~36/35~49/48~55/54~56/55~64/63
|| Steps ||   || "Diminished" || "Minor" || "Major" || "Augmented" ||   ||
|| 1 || 12/11 || 10/9~9/8 || 8/7 || 7/6 || 6/5 || 11/9 ||
|| 2 || 5/4 || 14/11~9/7 || 21/16 || 4/3 || 11/8 || 7/5 ||
|| 3 || 10/7 || 16/11 || 3/2 || 32/21 || 14/9~11/7 || 8/5 ||
|| 4 || 18/11 || 5/3 || 12/7 || 7/4 || 16/9~9/5 || 11/6 ||
===Rodan=== 
Chroma: 49/48~55/54~56/55~64/63~81/80~99/98
|| Steps ||   ||   || "Diminished" || "Minor" || "Major" || "Augmented" ||   ||   ||
|| 1 || 12/11 || 10/9 || 9/8 || 8/7 || 7/6 || 32/27 || 6/5 || 11/9 ||
|| 2 || 5/4 || 14/11 || 9/7 || 21/16 || 4/3 || 27/20 || 11/8 || 7/5 ||
|| 3 || 10/7 || 16/11 || 40/27 || 3/2 || 32/21 || 14/9 || 11/7 || 8/5 ||
|| 4 || 18/11 || 5/3 || 27/16 || 12/7 || 7/4 || 16/9 || 9/5 || 11/6 ||
==Modus== 
Chroma: 40/39~45/44~55/54~66/65~81/80~121/120
|| Steps || "Diminished" || "Minor" || "Major" || "Augmented" ||
|| 1 || 16/15 || 13/12~12/11 || 11/10~10/9 || 9/8 ||
|| 2 || 13/11 || 6/5 || 11/9 || 5/4 ||
|| 3 || 13/10 || 4/3 || 27/20 || 11/8 ||
|| 4 || 16/11 || 40/27 || 3/2 || 20/13 ||
|| 5 || 8/5 || 18/11 || 5/3 || 22/13~27/16 ||
|| 6 || 16/9 || 9/5 || 11/6 || 15/8 ||
==Miracle== 
Chroma: 45/44~49/48~50/49~55/54~56/55~64/63
|| Steps ||   || "Diminished" || "Minor" || "Major" || "Augmented" ||   ||
|| 1 ||   || 22/21~21/20 || 16/15~15/14 || 12/11 || 10/9 ||   ||
|| 2 || 11/10 || 9/8 || 8/7 || 7/6 || 32/27 ||   ||
|| 3 ||   || 6/5 || 11/9 || 5/4 || 14/11 ||   ||
|| 4 ||   || 9/7 || 21/16 || 4/3 ||   ||   ||
|| 5 ||   || 11/8 || 7/5 || 10/7 || 16/11 ||   ||
|| 6 ||   ||   || 3/2 || 32/21 || 14/9 ||   ||
|| 7 ||   || 11/7 || 8/5 || 18/11 || 5/3 ||   ||
|| 8 ||   || 27/16 || 12/7 || 7/4 || 16/9 || 20/11 ||
|| 9 ||   || 9/5 || 11/6 || 15/8 || 21/11 ||   ||

Original HTML content:

<html><head><title>Cluster temperament</title></head><body>A cluster temperament (named by <a class="wiki_link" href="/Keenan%20Pepper">Keenan Pepper</a>) is a very particular kind of rank-2 temperament whose generator is quite near a rational fraction of an octave. Therefore some MOS of the temperament is quasi-equal (which should be reasonably sized for it to be a good cluster temperament, usually between 5 and 10 notes per octave). But not only that; in a cluster temperament, the different versions of each interval, differing by a chroma (&quot;diminished&quot;, &quot;minor&quot;, &quot;major&quot;, &quot;augmented&quot;...) include many nearby JI intervals that are individually recognizable, yet conceptually grouped into the same category because they're so close.<br />
<br />
An example of something that is <strong>not</strong> a cluster temperament is <a class="wiki_link" href="/amity">amity</a>, because although the amity generator is within 4 cents of 2\7, making amity[7] near equal, amity is too complex of a temperament and most of the intervals differing by a chroma do not represent simple JI intervals at all. For example, the list of amity &quot;thirds&quot; includes ...6/5 (339.5) (363.2) 5/4... where the intervals given in cents are not representable as simple JI intervals (243/200 and 100/81 are about as simple as you can get).<br />
<br />
Another way to describe this property is that the chroma of the near-equal MOS is a kind of &quot;super-comma&quot;, a set of many useful commas that are tempered to become the same, non-vanishing, interval. It should be obvious that &quot;cluster temperament&quot; is a vague, qualitative phrase and not mathematically well-defined.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:0 -->Examples</h1>
 <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Examples-Slendric"></a><!-- ws:end:WikiTextHeadingRule:2 -->Slendric</h2>
 Chroma: 49/48~64/63<br />


<table class="wiki_table">
    <tr>
        <td>Steps<br />
</td>
        <td>&quot;Diminished&quot;<br />
</td>
        <td>&quot;Minor&quot;<br />
</td>
        <td>&quot;Major&quot;<br />
</td>
        <td>&quot;Augmented&quot;<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>9/8<br />
</td>
        <td>8/7<br />
</td>
        <td>7/6<br />
</td>
        <td>32/27<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>9/7<br />
</td>
        <td>21/16<br />
</td>
        <td>4/3<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td><br />
</td>
        <td>3/2<br />
</td>
        <td>32/21<br />
</td>
        <td>14/9<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>27/16<br />
</td>
        <td>12/7<br />
</td>
        <td>7/4<br />
</td>
        <td>16/9<br />
</td>
    </tr>
</table>

Slendric has two quite different extensions that are both also cluster scales:<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="Examples-Slendric-Mothra"></a><!-- ws:end:WikiTextHeadingRule:4 -->Mothra</h3>
 Chroma: 33/32~36/35~49/48~55/54~56/55~64/63<br />


<table class="wiki_table">
    <tr>
        <td>Steps<br />
</td>
        <td><br />
</td>
        <td>&quot;Diminished&quot;<br />
</td>
        <td>&quot;Minor&quot;<br />
</td>
        <td>&quot;Major&quot;<br />
</td>
        <td>&quot;Augmented&quot;<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>12/11<br />
</td>
        <td>10/9~9/8<br />
</td>
        <td>8/7<br />
</td>
        <td>7/6<br />
</td>
        <td>6/5<br />
</td>
        <td>11/9<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>5/4<br />
</td>
        <td>14/11~9/7<br />
</td>
        <td>21/16<br />
</td>
        <td>4/3<br />
</td>
        <td>11/8<br />
</td>
        <td>7/5<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>10/7<br />
</td>
        <td>16/11<br />
</td>
        <td>3/2<br />
</td>
        <td>32/21<br />
</td>
        <td>14/9~11/7<br />
</td>
        <td>8/5<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>18/11<br />
</td>
        <td>5/3<br />
</td>
        <td>12/7<br />
</td>
        <td>7/4<br />
</td>
        <td>16/9~9/5<br />
</td>
        <td>11/6<br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Examples-Slendric-Rodan"></a><!-- ws:end:WikiTextHeadingRule:6 -->Rodan</h3>
 Chroma: 49/48~55/54~56/55~64/63~81/80~99/98<br />


<table class="wiki_table">
    <tr>
        <td>Steps<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>&quot;Diminished&quot;<br />
</td>
        <td>&quot;Minor&quot;<br />
</td>
        <td>&quot;Major&quot;<br />
</td>
        <td>&quot;Augmented&quot;<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>12/11<br />
</td>
        <td>10/9<br />
</td>
        <td>9/8<br />
</td>
        <td>8/7<br />
</td>
        <td>7/6<br />
</td>
        <td>32/27<br />
</td>
        <td>6/5<br />
</td>
        <td>11/9<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>5/4<br />
</td>
        <td>14/11<br />
</td>
        <td>9/7<br />
</td>
        <td>21/16<br />
</td>
        <td>4/3<br />
</td>
        <td>27/20<br />
</td>
        <td>11/8<br />
</td>
        <td>7/5<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>10/7<br />
</td>
        <td>16/11<br />
</td>
        <td>40/27<br />
</td>
        <td>3/2<br />
</td>
        <td>32/21<br />
</td>
        <td>14/9<br />
</td>
        <td>11/7<br />
</td>
        <td>8/5<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>18/11<br />
</td>
        <td>5/3<br />
</td>
        <td>27/16<br />
</td>
        <td>12/7<br />
</td>
        <td>7/4<br />
</td>
        <td>16/9<br />
</td>
        <td>9/5<br />
</td>
        <td>11/6<br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Examples-Modus"></a><!-- ws:end:WikiTextHeadingRule:8 -->Modus</h2>
 Chroma: 40/39~45/44~55/54~66/65~81/80~121/120<br />


<table class="wiki_table">
    <tr>
        <td>Steps<br />
</td>
        <td>&quot;Diminished&quot;<br />
</td>
        <td>&quot;Minor&quot;<br />
</td>
        <td>&quot;Major&quot;<br />
</td>
        <td>&quot;Augmented&quot;<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>16/15<br />
</td>
        <td>13/12~12/11<br />
</td>
        <td>11/10~10/9<br />
</td>
        <td>9/8<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>13/11<br />
</td>
        <td>6/5<br />
</td>
        <td>11/9<br />
</td>
        <td>5/4<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>13/10<br />
</td>
        <td>4/3<br />
</td>
        <td>27/20<br />
</td>
        <td>11/8<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>16/11<br />
</td>
        <td>40/27<br />
</td>
        <td>3/2<br />
</td>
        <td>20/13<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>8/5<br />
</td>
        <td>18/11<br />
</td>
        <td>5/3<br />
</td>
        <td>22/13~27/16<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>16/9<br />
</td>
        <td>9/5<br />
</td>
        <td>11/6<br />
</td>
        <td>15/8<br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Examples-Miracle"></a><!-- ws:end:WikiTextHeadingRule:10 -->Miracle</h2>
 Chroma: 45/44~49/48~50/49~55/54~56/55~64/63<br />


<table class="wiki_table">
    <tr>
        <td>Steps<br />
</td>
        <td><br />
</td>
        <td>&quot;Diminished&quot;<br />
</td>
        <td>&quot;Minor&quot;<br />
</td>
        <td>&quot;Major&quot;<br />
</td>
        <td>&quot;Augmented&quot;<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td><br />
</td>
        <td>22/21~21/20<br />
</td>
        <td>16/15~15/14<br />
</td>
        <td>12/11<br />
</td>
        <td>10/9<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>11/10<br />
</td>
        <td>9/8<br />
</td>
        <td>8/7<br />
</td>
        <td>7/6<br />
</td>
        <td>32/27<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td><br />
</td>
        <td>6/5<br />
</td>
        <td>11/9<br />
</td>
        <td>5/4<br />
</td>
        <td>14/11<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td><br />
</td>
        <td>9/7<br />
</td>
        <td>21/16<br />
</td>
        <td>4/3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td><br />
</td>
        <td>11/8<br />
</td>
        <td>7/5<br />
</td>
        <td>10/7<br />
</td>
        <td>16/11<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3/2<br />
</td>
        <td>32/21<br />
</td>
        <td>14/9<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td><br />
</td>
        <td>11/7<br />
</td>
        <td>8/5<br />
</td>
        <td>18/11<br />
</td>
        <td>5/3<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td><br />
</td>
        <td>27/16<br />
</td>
        <td>12/7<br />
</td>
        <td>7/4<br />
</td>
        <td>16/9<br />
</td>
        <td>20/11<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td><br />
</td>
        <td>9/5<br />
</td>
        <td>11/6<br />
</td>
        <td>15/8<br />
</td>
        <td>21/11<br />
</td>
        <td><br />
</td>
    </tr>
</table>

</body></html>
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