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Wikispaces>genewardsmith **Imported revision 181293029 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 197087796 - 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:genewardsmith|genewardsmith]] and made on <tt> | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-29 18:10:09 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>197087796</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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An important example is provided by [[Regular Temperaments|regular temperaments]], where if a particular tuning for the temperament is written additively, the generators can be taken as intervals expressed in terms of cents, and if multiplicatively, as intervals given as frequency ratios. Another example is provided by [[just intonation subgroups]], where the generators are a finite set of positive rational numbers. | An important example is provided by [[Regular Temperaments|regular temperaments]], where if a particular tuning for the temperament is written additively, the generators can be taken as intervals expressed in terms of cents, and if multiplicatively, as intervals given as frequency ratios. Another example is provided by [[just intonation subgroups]], where the generators are a finite set of positive rational numbers. | ||
These two example converge when we seek generators for the abstract temperament rather than any particular tuning of it. One way to obtain these is to use the [[http://mathworld.wolfram.com/InteriorProduct.html|interior product]] of a [[Wedgies and Multivals|wedgie]] for a p-limit temperament with the p-limit monzos. A less abstract approach is to define a [[transversal]] for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament. | These two example converge when we seek generators for the [[Abstract regular temperament|abstract temperament]] rather than any particular tuning of it. One way to obtain these is to use the [[http://mathworld.wolfram.com/InteriorProduct.html|interior product]] of a [[Wedgies and Multivals|wedgie]] for a p-limit temperament with the p-limit monzos. A less abstract approach is to define a [[transversal]] for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament. | ||
For example, for [[Gamelismic clan|miracle temperament]] [2, 15/14] defines a rank two 7-limit subgroup whose [[Normal lists|normal interval list]] 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. | For example, for [[Gamelismic clan|miracle temperament]] [2, 15/14] defines a rank two 7-limit subgroup whose [[Normal lists|normal interval list]] 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. | ||
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An important example is provided by <a class="wiki_link" href="/Regular%20Temperaments">regular temperaments</a>, where if a particular tuning for the temperament is written additively, the generators can be taken as intervals expressed in terms of cents, and if multiplicatively, as intervals given as frequency ratios. Another example is provided by <a class="wiki_link" href="/just%20intonation%20subgroups">just intonation subgroups</a>, where the generators are a finite set of positive rational numbers.<br /> | An important example is provided by <a class="wiki_link" href="/Regular%20Temperaments">regular temperaments</a>, where if a particular tuning for the temperament is written additively, the generators can be taken as intervals expressed in terms of cents, and if multiplicatively, as intervals given as frequency ratios. Another example is provided by <a class="wiki_link" href="/just%20intonation%20subgroups">just intonation subgroups</a>, where the generators are a finite set of positive rational numbers.<br /> | ||
<br /> | <br /> | ||
These two example converge when we seek generators for the abstract temperament rather than any particular tuning of it. One way to obtain these is to use the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/InteriorProduct.html" rel="nofollow">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. A less abstract approach 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 /> | These two example converge when we seek generators for the <a class="wiki_link" href="/Abstract%20regular%20temperament">abstract temperament</a> rather than any particular tuning of it. One way to obtain these is to use the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/InteriorProduct.html" rel="nofollow">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. A less abstract approach 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 /> | ||
<br /> | <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 /> | 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 /> | ||
<br /> | <br /> | ||
Alternatively, using &quot;v&quot; to represent the interior product, we have mir = &lt;&lt;6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0&gt; is &lt;0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1&gt; is &lt;1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0&gt; is also &lt;1 1 3 3|. Unlike the case for just intonation subgroups, the rank two group for the abstract temperament in terms of interior products is already exactly the same.</body></html></pre></div> | Alternatively, using &quot;v&quot; to represent the interior product, we have mir = &lt;&lt;6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0&gt; is &lt;0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1&gt; is &lt;1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0&gt; is also &lt;1 1 3 3|. Unlike the case for just intonation subgroups, the rank two group for the abstract temperament in terms of interior products is already exactly the same.</body></html></pre></div> | ||