Tour of regular temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-05 23:56:28 UTC</tt>.<br>

: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-05 23:58:54 UTC</tt>.<br>

: The original revision id was <tt>~~240147963~~</tt>.<br>

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Regular temperaments are non-Just tunings wherein the infinite number of intervals in p-limit Just intonation (or any subgroup thereof) are mapped to a smaller (though possibly still infinite) set of tempered intervals, by "tempering" (deliberately mistuning) some of the ratios such that a comma (or set of commas) "vanishes" by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas a tempered out. Temperaments effectively reduce the "dimensionality" of JI, thus simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals would be located by a four-dimensional set of coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which commas (and how many) are tempered out, and intervals can be located with a set of one-, two-, or three-dimensional coordinates (depending on the number of commas that have been tempered out, or in other words the "rank" of the temperament).

A rank r [[http://en.wikipedia.org/wiki/Regular_temperament|regular temperament]] in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of [[comma|commas]] tempered out by the temperament, or a set of r independent [[Vals and Tuning Space|vals]] defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the [[comma pump examples|comma pumps]] of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.

A rank r [[http://en.wikipedia.org/wiki/Regular_temperament|regular temperament]] in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r vals. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of [[comma|commas]] tempered out by the temperament, or a set of r independent [[Vals and Tuning Space|vals]] defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the [[comma pump examples|comma pumps]] of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.

=[[edo|Equal temperaments]]=

[[Equal Temperaments|Equal temperaments]] (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. ~~An n~~-limit ET is simply ~~an n~~-limit temperament that uses a single generator (a "Rank 1") temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve "fun" results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.

[[Equal Temperaments|Equal temperaments]] (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator (making it a "Rank 1") temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve "fun" results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.

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

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Regular temperaments of ranks two and three are cataloged [[Optimal patent val|here]]. Other pages listing them are [[Paul Erlich]]'s [[Catalog of five-limit rank two temperaments|catalog of 5-limit rank two temperaments]], and a [[Proposed names for rank 2 temperaments|page]] listing higher limit rank two temperaments.

N-limit Rank 2 temperaments map all intervals of n-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. Rank-2 temperaments can be reduced to a related rank-1 temperaments 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-TET by tempering out the Pythagorean comma.

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. Rank-2 temperaments can be reduced to a related rank-1 temperaments 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-TET by tempering out the Pythagorean comma.

As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the [[Normal lists|normal comma list]] of the various temperaments, where a **comma** is a small interval, not a square or cube or other power, which is tempered out by the temperament.

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Regular temperaments are non-Just tunings wherein the infinite number of intervals in p-limit Just intonation (or any subgroup thereof) are mapped to a smaller (though possibly still infinite) set of tempered intervals, by &quot;tempering&quot; (deliberately mistuning) some of the ratios such that a comma (or set of commas) &quot;vanishes&quot; by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful &quot;puns&quot; as commas a tempered out. Temperaments effectively reduce the &quot;dimensionality&quot; of JI, thus simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals would be located by a four-dimensional set of coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which commas (and how many) are tempered out, and intervals can be located with a set of one-, two-, or three-dimensional coordinates (depending on the number of commas that have been tempered out, or in other words the &quot;rank&quot; of the temperament).<br />

<br />

A rank r <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_temperament" rel="nofollow">regular temperament</a> in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. An <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a> can be defined in various ways, for instance by giving a set of <a class="wiki_link" href="/comma">commas</a> tempered out by the temperament, or a set of r independent <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">vals</a> defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the <a class="wiki_link" href="/comma%20pump%20examples">comma pumps</a> of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.<br />

A rank r <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_temperament" rel="nofollow">regular temperament</a> in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r vals. An <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a> can be defined in various ways, for instance by giving a set of <a class="wiki_link" href="/comma">commas</a> tempered out by the temperament, or a set of r independent <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">vals</a> defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the <a class="wiki_link" href="/comma%20pump%20examples">comma pumps</a> of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.<br />

<br />

<h1 id="toc1"><a name="Equal temperaments"></a><a class="wiki_link" href="/edo">Equal temperaments</a></h1>

<br />

<a class="wiki_link" href="/Equal%20Temperaments">Equal temperaments</a> (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. ~~An n~~-limit ET is simply ~~an n~~-limit temperament that uses a single generator (a &quot;Rank 1&quot;) temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an &quot;equal division&quot; of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve &quot;fun&quot; results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.<br />

<a class="wiki_link" href="/Equal%20Temperaments">Equal temperaments</a> (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator (making it a &quot;Rank 1&quot;) temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an &quot;equal division&quot; of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve &quot;fun&quot; results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.<br />

<br />

<h1 id="toc2"><a name="Rank 2 (including &quot;linear&quot;) temperaments"></a>Rank 2 (including &quot;linear&quot;) temperaments<a name="lineartemperaments"></a></h1>

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Regular temperaments of ranks two and three are cataloged <a class="wiki_link" href="/Optimal%20patent%20val">here</a>. Other pages listing them are <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a>'s <a class="wiki_link" href="/Catalog%20of%20five-limit%20rank%20two%20temperaments">catalog of 5-limit rank two temperaments</a>, and a <a class="wiki_link" href="/Proposed%20names%20for%20rank%202%20temperaments">page</a> listing higher limit rank two temperaments.<br />

<br />

N-limit Rank 2 temperaments map all intervals of n-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. Rank-2 temperaments can be reduced to a related rank-1 temperaments 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-TET by tempering out the Pythagorean comma.<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. Rank-2 temperaments can be reduced to a related rank-1 temperaments 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-TET by tempering out the Pythagorean comma.<br />

<br />

As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> of the various temperaments, where a <strong>comma</strong> is a small interval, not a square or cube or other power, which is tempered out by the temperament.<br />

@@ 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:igliashon|igliashon]] and made on <tt>2011-07-05 23:56:28 UTC</tt>.<br>
+: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-05 23:58:54 UTC</tt>.<br>
-: The original revision id was <tt>240147963</tt>.<br>
+: The original revision id was <tt>240148149</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 12: / Line 12: @@
 Regular temperaments are non-Just tunings wherein the infinite number of intervals in p-limit Just intonation (or any subgroup thereof) are mapped to a smaller (though possibly still infinite) set of tempered intervals, by "tempering" (deliberately mistuning) some of the ratios such that a comma (or set of commas) "vanishes" by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas a tempered out. Temperaments effectively reduce the "dimensionality" of JI, thus simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals would be located by a four-dimensional set of coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which commas (and how many) are tempered out, and intervals can be located with a set of one-, two-, or three-dimensional coordinates (depending on the number of commas that have been tempered out, or in other words the "rank" of the temperament).
-A rank r [[http://en.wikipedia.org/wiki/Regular_temperament|regular temperament]] in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of [[comma|commas]] tempered out by the temperament, or a set of r independent [[Vals and Tuning Space|vals]] defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the [[comma pump examples|comma pumps]] of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.
+A rank r [[http://en.wikipedia.org/wiki/Regular_temperament|regular temperament]] in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r vals. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of [[comma|commas]] tempered out by the temperament, or a set of r independent [[Vals and Tuning Space|vals]] defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the [[comma pump examples|comma pumps]] of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.
 =[[edo|Equal temperaments]]=
-[[Equal Temperaments|Equal temperaments]] (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. An n-limit ET is simply an n-limit temperament that uses a single generator (a "Rank 1") temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve "fun" results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.
+[[Equal Temperaments|Equal temperaments]] (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator (making it a "Rank 1") temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve "fun" results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.
 =Rank 2 (including "linear") temperaments[[#lineartemperaments]]=
@@ Line 22: / Line 22: @@
 Regular temperaments of ranks two and three are cataloged [[Optimal patent val|here]]. Other pages listing them are [[Paul Erlich]]'s [[Catalog of five-limit rank two temperaments|catalog of 5-limit rank two temperaments]], and a [[Proposed names for rank 2 temperaments|page]] listing higher limit rank two temperaments.
-N-limit Rank 2 temperaments map all intervals of n-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. Rank-2 temperaments can be reduced to a related rank-1 temperaments 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-TET by tempering out the Pythagorean comma.
+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. Rank-2 temperaments can be reduced to a related rank-1 temperaments 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-TET by tempering out the Pythagorean comma.
 As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the [[Normal lists|normal comma list]] of the various temperaments, where a **comma** is a small interval, not a square or cube or other power, which is tempered out by the temperament.
@@ Line 274: / Line 274: @@
 Regular temperaments are non-Just tunings wherein the infinite number of intervals in p-limit Just intonation (or any subgroup thereof) are mapped to a smaller (though possibly still infinite) set of tempered intervals, by &amp;quot;tempering&amp;quot; (deliberately mistuning) some of the ratios such that a comma (or set of commas) &amp;quot;vanishes&amp;quot; by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful &amp;quot;puns&amp;quot; as commas a tempered out. Temperaments effectively reduce the &amp;quot;dimensionality&amp;quot; of JI, thus simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals would be located by a four-dimensional set of coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which commas (and how many) are tempered out, and intervals can be located with a set of one-, two-, or three-dimensional coordinates (depending on the number of commas that have been tempered out, or in other words the &amp;quot;rank&amp;quot; of the temperament).&lt;br /&gt;
 &lt;br /&gt;
-A rank r &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_temperament" rel="nofollow"&gt;regular temperament&lt;/a&gt; in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. An &lt;a class="wiki_link" href="/abstract%20regular%20temperament"&gt;abstract regular temperament&lt;/a&gt; can be defined in various ways, for instance by giving a set of &lt;a class="wiki_link" href="/comma"&gt;commas&lt;/a&gt; tempered out by the temperament, or a set of r independent &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;vals&lt;/a&gt; defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the &lt;a class="wiki_link" href="/comma%20pump%20examples"&gt;comma pumps&lt;/a&gt; of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.&lt;br /&gt;
+A rank r &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_temperament" rel="nofollow"&gt;regular temperament&lt;/a&gt; in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r vals. An &lt;a class="wiki_link" href="/abstract%20regular%20temperament"&gt;abstract regular temperament&lt;/a&gt; can be defined in various ways, for instance by giving a set of &lt;a class="wiki_link" href="/comma"&gt;commas&lt;/a&gt; tempered out by the temperament, or a set of r independent &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;vals&lt;/a&gt; defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the &lt;a class="wiki_link" href="/comma%20pump%20examples"&gt;comma pumps&lt;/a&gt; of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Equal temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;a class="wiki_link" href="/edo"&gt;Equal temperaments&lt;/a&gt;&lt;/h1&gt;
   &lt;br /&gt;
-&lt;a class="wiki_link" href="/Equal%20Temperaments"&gt;Equal temperaments&lt;/a&gt; (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. An n-limit ET is simply an n-limit temperament that uses a single generator (a &amp;quot;Rank 1&amp;quot;) temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an &amp;quot;equal division&amp;quot; of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve &amp;quot;fun&amp;quot; results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.&lt;br /&gt;
+&lt;a class="wiki_link" href="/Equal%20Temperaments"&gt;Equal temperaments&lt;/a&gt; (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator (making it a &amp;quot;Rank 1&amp;quot;) temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an &amp;quot;equal division&amp;quot; of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve &amp;quot;fun&amp;quot; results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.&lt;br /&gt;
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 Regular temperaments of ranks two and three are cataloged &lt;a class="wiki_link" href="/Optimal%20patent%20val"&gt;here&lt;/a&gt;. Other pages listing them are &lt;a class="wiki_link" href="/Paul%20Erlich"&gt;Paul Erlich&lt;/a&gt;'s &lt;a class="wiki_link" href="/Catalog%20of%20five-limit%20rank%20two%20temperaments"&gt;catalog of 5-limit rank two temperaments&lt;/a&gt;, and a &lt;a class="wiki_link" href="/Proposed%20names%20for%20rank%202%20temperaments"&gt;page&lt;/a&gt; listing higher limit rank two temperaments.&lt;br /&gt;
 &lt;br /&gt;
-N-limit Rank 2 temperaments map all intervals of n-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. Rank-2 temperaments can be reduced to a related rank-1 temperaments 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-TET by tempering out the Pythagorean comma.&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. Rank-2 temperaments can be reduced to a related rank-1 temperaments 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-TET by tempering out the Pythagorean comma.&lt;br /&gt;
 &lt;br /&gt;
 As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; of the various temperaments, where a &lt;strong&gt;comma&lt;/strong&gt; is a small interval, not a square or cube or other power, which is tempered out by the temperament.&lt;br /&gt;