Cluster MOS: Difference between revisions

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: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2012-05-11 04:12:38 UTC</tt>.<br>

: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2012-05-11 04:42:32 UTC</tt>.<br>

: The original revision id was <tt>~~333419106~~</tt>.<br>

: The original revision id was <tt>333428046</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>

<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">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.

<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">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 (or "cluster") 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.

Rather than simply denoting one of a list of rank-2 temperaments, the phrase "cluster scale" is also associated with a compositional philosophy. It is often said that temperaments such as slendric are melodically "bad" or have "bad MOS structure", because some MOS (in this case slendric[5]) is "too equal" and the next higher MOSes are "too unequal". But this can be thought of as a feature rather than a bug. Rather than forcing them into the MOS framework, one can think of cluster scales as having two hierarchical levels of melodic structure: the "step" (a step of the quasi-equal MOS), and the "chroma", and a chroma is so much smaller than a step that the steps seldom go out of order no matter how many chromas are involved.

=Examples=

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|| 9 || || 9/5 || 11/6 || 15/8 || 21/11 || ||</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"><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 />

<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>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 (or &quot;cluster&quot;) 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 />

Rather than simply denoting one of a list of rank-2 temperaments, the phrase &quot;cluster scale&quot; is also associated with a compositional philosophy. It is often said that temperaments such as slendric are melodically &quot;bad&quot; or have &quot;bad MOS structure&quot;, because some MOS (in this case slendric[5]) is &quot;too equal&quot; and the next higher MOSes are &quot;too unequal&quot;. But this can be thought of as a feature rather than a bug. Rather than forcing them into the MOS framework, one can think of cluster scales as having two hierarchical levels of melodic structure: the &quot;step&quot; (a step of the quasi-equal MOS), and the &quot;chroma&quot;, and a chroma is so much smaller than a step that the steps seldom go out of order no matter how many chromas are involved.<br />

<br />

<h1 id="toc0"><a name="Examples"></a>Examples</h1>

@@ 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:keenanpepper|keenanpepper]] and made on <tt>2012-05-11 04:12:38 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2012-05-11 04:42:32 UTC</tt>.<br>
-: The original revision id was <tt>333419106</tt>.<br>
+: The original revision id was <tt>333428046</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>
 <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">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.
+<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">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 (or "cluster") 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.
+Rather than simply denoting one of a list of rank-2 temperaments, the phrase "cluster scale" is also associated with a compositional philosophy. It is often said that temperaments such as slendric are melodically "bad" or have "bad MOS structure", because some MOS (in this case slendric[5]) is "too equal" and the next higher MOSes are "too unequal". But this can be thought of as a feature rather than a bug. Rather than forcing them into the MOS framework, one can think of cluster scales as having two hierarchical levels of melodic structure: the "step" (a step of the quasi-equal MOS), and the "chroma", and a chroma is so much smaller than a step that the steps seldom go out of order no matter how many chromas are involved.
 =Examples=
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 || 9 ||   || 9/5 || 11/6 || 15/8 || 21/11 ||   ||</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;Cluster temperament&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A cluster temperament (named by &lt;a class="wiki_link" href="/Keenan%20Pepper"&gt;Keenan Pepper&lt;/a&gt;) 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 (&amp;quot;diminished&amp;quot;, &amp;quot;minor&amp;quot;, &amp;quot;major&amp;quot;, &amp;quot;augmented&amp;quot;...) include many nearby JI intervals that are individually recognizable, yet conceptually grouped into the same category because they're so close.&lt;br /&gt;
+<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;Cluster temperament&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A cluster temperament (named by &lt;a class="wiki_link" href="/Keenan%20Pepper"&gt;Keenan Pepper&lt;/a&gt;) 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 (&amp;quot;diminished&amp;quot;, &amp;quot;minor&amp;quot;, &amp;quot;major&amp;quot;, &amp;quot;augmented&amp;quot;...) include many nearby JI intervals that are individually recognizable, yet conceptually grouped into the same category (or &amp;quot;cluster&amp;quot;) because they're so close.&lt;br /&gt;
 &lt;br /&gt;
 An example of something that is &lt;strong&gt;not&lt;/strong&gt; a cluster temperament is &lt;a class="wiki_link" href="/amity"&gt;amity&lt;/a&gt;, 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 &amp;quot;thirds&amp;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).&lt;br /&gt;
 &lt;br /&gt;
 Another way to describe this property is that the chroma of the near-equal MOS is a kind of &amp;quot;super-comma&amp;quot;, a set of many useful commas that are tempered to become the same, non-vanishing, interval. It should be obvious that &amp;quot;cluster temperament&amp;quot; is a vague, qualitative phrase and not mathematically well-defined.&lt;br /&gt;
+&lt;br /&gt;
+Rather than simply denoting one of a list of rank-2 temperaments, the phrase &amp;quot;cluster scale&amp;quot; is also associated with a compositional philosophy. It is often said that temperaments such as slendric are melodically &amp;quot;bad&amp;quot; or have &amp;quot;bad MOS structure&amp;quot;, because some MOS (in this case slendric[5]) is &amp;quot;too equal&amp;quot; and the next higher MOSes are &amp;quot;too unequal&amp;quot;. But this can be thought of as a feature rather than a bug. Rather than forcing them into the MOS framework, one can think of cluster scales as having two hierarchical levels of melodic structure: the &amp;quot;step&amp;quot; (a step of the quasi-equal MOS), and the &amp;quot;chroma&amp;quot;, and a chroma is so much smaller than a step that the steps seldom go out of order no matter how many chromas are involved.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Examples&lt;/h1&gt;