Tour of regular temperaments: Difference between revisions

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<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:~~jake.huryn~~|~~jake.huryn~~]] and made on <tt>~~2017~~-05~~-15 15~~:59:23 UTC</tt>.<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-25 05:36:36 UTC</tt>.<br>

: The original revision id was <tt>~~612914477~~</tt>.<br>

: The original revision id was <tt>627987009</tt>.<br>

: The revision comment was: <tt></tt><br>

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=Rank-2 (including linear) temperaments[[#lineartemperaments]]=

P-limit rank-2 ~~temperaments map~~ all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 ~~temperaments are~~ said to have two generators, though ~~they~~ may have any number of step-sizes. This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a linear temperament. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-ET by tempering out the Pythagorean comma.

A p-limit rank-2 temperament maps all intervals of p-limit JI using a set of 2-dimensional coordinates, thus a rank-2 temperament is said to have two generators, though it may have any number of step-sizes. This means that a rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a linear temperament. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-ET by tempering out the Pythagorean comma.

Regular temperaments of ranks two and three are cataloged [[Optimal patent val|here]]. Rank-2 temperaments are also listed [[Proposed names for rank 2 temperaments|here]] by their generator mappings and [[Map of rank-2 temperaments|here]] by their generator size. There is also [[Graham Breed]]'s [[http://x31eq.com/catalog2.html|giant list of regular temperaments]].

Regular temperaments of ranks two and three are cataloged [[Optimal patent val|here]]. Rank-2 temperaments are also listed [[Proposed names for rank 2 temperaments|here]] by their generator mappings and [[Map of rank-2 temperaments|here]] by their generator size. See also the [[pergen|pergens]] page. There is also [[Graham Breed]]'s [[http://x31eq.com/catalog2.html|giant list of regular temperaments]].

==Families==

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<h1 id="toc4"><a name="Rank-2 (including linear) temperaments"></a>Rank-2 (including linear) temperaments<a name="lineartemperaments"></a></h1>

<br />

P-limit rank-2 ~~temperaments map~~ all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 ~~temperaments are~~ said to have two generators, though ~~they~~ may have any number of step-sizes. This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a linear temperament. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-ET by tempering out the Pythagorean comma.<br />

A p-limit rank-2 temperament maps all intervals of p-limit JI using a set of 2-dimensional coordinates, thus a rank-2 temperament is said to have two generators, though it may have any number of step-sizes. This means that a rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a linear temperament. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-ET by tempering out the Pythagorean comma.<br />

<br />

Regular temperaments of ranks two and three are cataloged <a class="wiki_link" href="/Optimal%20patent%20val">here</a>. Rank-2 temperaments are also listed <a class="wiki_link" href="/Proposed%20names%20for%20rank%202%20temperaments">here</a> by their generator mappings and <a class="wiki_link" href="/Map%20of%20rank-2%20temperaments">here</a> by their generator size. There is also <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>'s <a class="wiki_link_ext" href="http://x31eq.com/catalog2.html" rel="nofollow">giant list of regular temperaments</a>.<br />

Regular temperaments of ranks two and three are cataloged <a class="wiki_link" href="/Optimal%20patent%20val">here</a>. Rank-2 temperaments are also listed <a class="wiki_link" href="/Proposed%20names%20for%20rank%202%20temperaments">here</a> by their generator mappings and <a class="wiki_link" href="/Map%20of%20rank-2%20temperaments">here</a> by their generator size. See also the <a class="wiki_link" href="/pergen">pergens</a> page. There is also <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>'s <a class="wiki_link_ext" href="http://x31eq.com/catalog2.html" rel="nofollow">giant list of regular temperaments</a>.<br />

<br />

<h2 id="toc5"><a name="Rank-2 (including linear) temperaments-Families"></a>Families</h2>

@@ 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:jake.huryn|jake.huryn]] and made on <tt>2017-05-15 15:59:23 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-25 05:36:36 UTC</tt>.<br>
-: The original revision id was <tt>612914477</tt>.<br>
+: The original revision id was <tt>627987009</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 32: / Line 32: @@
 =Rank-2 (including linear) temperaments[[#lineartemperaments]]=
-P-limit rank-2 temperaments map all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 temperaments are said to have two generators, though they may have any number of step-sizes. This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a linear temperament. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-ET by tempering out the Pythagorean comma.
+A p-limit rank-2 temperament maps all intervals of p-limit JI using a set of 2-dimensional coordinates, thus a rank-2 temperament is said to have two generators, though it may have any number of step-sizes. This means that a rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a linear temperament. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-ET by tempering out the Pythagorean comma.
-Regular temperaments of ranks two and three are cataloged [[Optimal patent val|here]]. Rank-2 temperaments are also listed [[Proposed names for rank 2 temperaments|here]] by their generator mappings and [[Map of rank-2 temperaments|here]] by their generator size. There is also [[Graham Breed]]'s [[http://x31eq.com/catalog2.html|giant list of regular temperaments]].
+Regular temperaments of ranks two and three are cataloged [[Optimal patent val|here]]. Rank-2 temperaments are also listed [[Proposed names for rank 2 temperaments|here]] by their generator mappings and [[Map of rank-2 temperaments|here]] by their generator size. See also the [[pergen|pergens]] page. There is also [[Graham Breed]]'s [[http://x31eq.com/catalog2.html|giant list of regular temperaments]].
 ==Families==
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 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Rank-2 (including linear) temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Rank-2 (including linear) temperaments&lt;!-- ws:start:WikiTextAnchorRule:368:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@lineartemperaments&amp;quot; title=&amp;quot;Anchor: lineartemperaments&amp;quot;/&amp;gt; --&gt;&lt;a name="lineartemperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:368 --&gt;&lt;/h1&gt;
   &lt;br /&gt;
-P-limit rank-2 temperaments map all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 temperaments are said to have two generators, though they may have any number of step-sizes. This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a linear temperament. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-ET by tempering out the Pythagorean comma.&lt;br /&gt;
+A p-limit rank-2 temperament maps all intervals of p-limit JI using a set of 2-dimensional coordinates, thus a rank-2 temperament is said to have two generators, though it may have any number of step-sizes. This means that a rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a linear temperament. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-ET by tempering out the Pythagorean comma.&lt;br /&gt;
 &lt;br /&gt;
-Regular temperaments of ranks two and three are cataloged &lt;a class="wiki_link" href="/Optimal%20patent%20val"&gt;here&lt;/a&gt;. Rank-2 temperaments are also listed &lt;a class="wiki_link" href="/Proposed%20names%20for%20rank%202%20temperaments"&gt;here&lt;/a&gt; by their generator mappings and &lt;a class="wiki_link" href="/Map%20of%20rank-2%20temperaments"&gt;here&lt;/a&gt; by their generator size. There is also &lt;a class="wiki_link" href="/Graham%20Breed"&gt;Graham Breed&lt;/a&gt;'s &lt;a class="wiki_link_ext" href="http://x31eq.com/catalog2.html" rel="nofollow"&gt;giant list of regular temperaments&lt;/a&gt;.&lt;br /&gt;
+Regular temperaments of ranks two and three are cataloged &lt;a class="wiki_link" href="/Optimal%20patent%20val"&gt;here&lt;/a&gt;. Rank-2 temperaments are also listed &lt;a class="wiki_link" href="/Proposed%20names%20for%20rank%202%20temperaments"&gt;here&lt;/a&gt; by their generator mappings and &lt;a class="wiki_link" href="/Map%20of%20rank-2%20temperaments"&gt;here&lt;/a&gt; by their generator size. See also the &lt;a class="wiki_link" href="/pergen"&gt;pergens&lt;/a&gt; page. There is also &lt;a class="wiki_link" href="/Graham%20Breed"&gt;Graham Breed&lt;/a&gt;'s &lt;a class="wiki_link_ext" href="http://x31eq.com/catalog2.html" rel="nofollow"&gt;giant list of regular temperaments&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Rank-2 (including linear) temperaments-Families"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Families&lt;/h2&gt;