Regular temperament: Difference between revisions
Wikispaces>clumma **Imported revision 578814015 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:clumma|clumma]] and made on <tt>2016- | : This revision was by author [[User:clumma|clumma]] and made on <tt>2016-07-27 14:26:54 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>588208191</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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=Characterizing a regular temperament= | |||
=Characterizing | |||
===The [[Wedgies and Multivals|wedgie]]=== | ===The [[Wedgies and Multivals|wedgie]]=== | ||
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<!-- ws:start:WikiTextTocRule:26:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><a href="#Characterizing | <!-- ws:start:WikiTextTocRule:26:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><a href="#Characterizing a regular temperament">Characterizing a regular temperament</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | <a href="#Translation between methods of specifying temperaments">Translation between methods of specifying temperaments</a><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --> | <a href="#The Geometry of Regular Temperaments">The Geometry of Regular Temperaments</a><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Characterizing a regular temperament"></a><!-- ws:end:WikiTextHeadingRule:0 -->Characterizing a regular temperament</h1> | |||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Characterizing | |||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="Characterizing a regular temperament--The wedgie"></a><!-- ws:end:WikiTextHeadingRule:2 -->The <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a></h3> | |||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="Characterizing | |||
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This uses <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">multilinear algebra</a> to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the <a class="wiki_link" href="/Interior%20product">interior product</a> of a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for a p-limit temperament with the p-limit monzos.<br /> | This uses <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">multilinear algebra</a> to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the <a class="wiki_link" href="/Interior%20product">interior product</a> of a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for a p-limit temperament with the p-limit monzos.<br /> | ||
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As explained on the <a class="wiki_link" href="/Interior%20product#Applications">interior product</a> page, if W is the r-wedgie defining the rank r temperament, then the tuning of map for the temperament can be defined via an (r-1)-multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q) • V.<br /> | As explained on the <a class="wiki_link" href="/Interior%20product#Applications">interior product</a> page, if W is the r-wedgie defining the rank r temperament, then the tuning of map for the temperament can be defined via an (r-1)-multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q) • V.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="Characterizing | <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="Characterizing a regular temperament--Normal val lists"></a><!-- ws:end:WikiTextHeadingRule:4 --><a class="wiki_link" href="/Normal%20lists">Normal val lists</a></h3> | ||
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Given a list of vals, we may <a class="wiki_link" href="/Saturation">saturate</a> it and reduce it using the <a class="wiki_link" href="/Normal%20lists">Hermite normal form</a> to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [&lt;1 1 3 3|, &lt;0 6 -7 -2|] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].<br /> | Given a list of vals, we may <a class="wiki_link" href="/Saturation">saturate</a> it and reduce it using the <a class="wiki_link" href="/Normal%20lists">Hermite normal form</a> to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [&lt;1 1 3 3|, &lt;0 6 -7 -2|] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Characterizing | <!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Characterizing a regular temperament--The Frobenius projection map"></a><!-- ws:end:WikiTextHeadingRule:6 -->The <a class="wiki_link" href="/Tenney-Euclidean%20Tuning">Frobenius projection map</a></h3> | ||
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Given any list of monzos, or any list of vals, we may compute the associated Frobenius projection map. This corresponds uniquely with an abstract regular temperament. The intervals of the abstract temperament may be defined via multiplication by the projection map, leading to <a class="wiki_link" href="/fractional%20monzos">fractional monzos</a> which are actually the tunings of these intervals in <a class="wiki_link" href="/Fractional%20monzos">Frobenius tuning</a>. However, using the Frobenius projection map to define the abstract temperament by no means commits us to Frobenius tuning.<br /> | Given any list of monzos, or any list of vals, we may compute the associated Frobenius projection map. This corresponds uniquely with an abstract regular temperament. The intervals of the abstract temperament may be defined via multiplication by the projection map, leading to <a class="wiki_link" href="/fractional%20monzos">fractional monzos</a> which are actually the tunings of these intervals in <a class="wiki_link" href="/Fractional%20monzos">Frobenius tuning</a>. However, using the Frobenius projection map to define the abstract temperament by no means commits us to Frobenius tuning.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Characterizing | <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Characterizing a regular temperament--transversals"></a><!-- ws:end:WikiTextHeadingRule:8 --><a class="wiki_link" href="/Just%20intonation%20subgroups">Just intonation subgroups</a> and <a class="wiki_link" href="/Transversal">transversals</a></h3> | ||
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A relatively concrete approach, but one which is not canonically defined, is to define a <a class="wiki_link" href="/transversal">transversal</a> for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament.<br /> | A relatively concrete approach, but one which is not canonically defined, is to define a <a class="wiki_link" href="/transversal">transversal</a> for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament.<br /> | ||
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For example, for <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a> [2, 15/14] defines a rank two 7-limit subgroup whose <a class="wiki_link" href="/Normal%20lists">normal interval list</a> is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.<br /> | For example, for <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a> [2, 15/14] defines a rank two 7-limit subgroup whose <a class="wiki_link" href="/Normal%20lists">normal interval list</a> is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Characterizing | <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Characterizing a regular temperament--Normal comma lists"></a><!-- ws:end:WikiTextHeadingRule:10 --><a class="wiki_link" href="/Normal%20lists">Normal comma lists</a></h3> | ||
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The normal comma list uniquely defines the abstract temperament, and has the advantage of showing family relationships even more clearly than the normal val list. Intervals of the temperament may be defined after computing another means of representing the temperament such as the normal val list.<br /> | The normal comma list uniquely defines the abstract temperament, and has the advantage of showing family relationships even more clearly than the normal val list. Intervals of the temperament may be defined after computing another means of representing the temperament such as the normal val list.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Characterizing | <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Characterizing a regular temperament--Reduced row echelon form"></a><!-- ws:end:WikiTextHeadingRule:12 --><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Row_echelon_form" rel="nofollow">Reduced row echelon form</a></h3> | ||
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If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.<br /> | If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.<br /> | ||