Tuning ranges of regular temperaments: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 518800726 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 518800762 - 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>2014-08-18 02: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-08-18 02:32:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>518800762</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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Given a rank r p-limit regular temperament, we may define a tuning range by finding the [[http://en.wikipedia.org/wiki/Convex_hull|convex hull]] in [[Vals and Tuning Space|tuning space]] of the tunings with one [[Eigenmonzo subgroup|eigenmonzo]] 2 (pure octaves tunings) and the rest a set of r-1 members of the p-limit [[tonality diamond]], when this tuning is defined. This is the //nice// tuning range. We may define another tuning range by requiring that the tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also [[http://en.wikipedia.org/wiki/Monotonic_function|monotone]] weakly increasing. This we may call the //monotone// tuning range. A tuning which is both nice and monotone is a //strict// tuning and this defines the strict tuning range. A tuning which is monotone but not nice is a //lax// tuning. | Given a rank r p-limit regular temperament, we may define a tuning range by finding the [[http://en.wikipedia.org/wiki/Convex_hull|convex hull]] in [[Vals and Tuning Space|tuning space]] of the tunings with one [[Eigenmonzo subgroup|eigenmonzo]] 2 (pure octaves tunings) and the rest a set of r-1 members of the p-limit [[tonality diamond]], when this tuning is defined. This is the //nice// tuning range. We may define another tuning range by requiring that the tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also [[http://en.wikipedia.org/wiki/Monotonic_function|monotone]] weakly increasing. This we may call the //monotone// tuning range. A tuning which is both nice and monotone is a //strict// tuning and this defines the strict tuning range. A tuning which is monotone but not nice is a //lax// tuning. | ||
While nice tunings are alway guaranteed to occur, strict or lax tunings are not. For instance, from the tuning map [<1 0 5|, <0 1 -2|] for the temperament tempering out 45/32 we find that all tunings are of the form <1 0 5| + a<0 1 -2| = <1 a 5-2a|. Applying this to the list of steps between the 5-limit tonality diamond [6/5, 25/24, 16/15, 9/8] we obtain [3a-4, 7-5a, a-1, 2a-3] from which it follows that a≥4/3, a≤7/5, a≥1, a≥3/2, the solution set of which is empty. Hence there are no monotone tunings and | While nice tunings are alway guaranteed to occur, strict or lax tunings are not. For instance, from the tuning map [<1 0 5|, <0 1 -2|] for the temperament tempering out 45/32 we find that all tunings are of the form <1 0 5| + a<0 1 -2| = <1 a 5-2a|. Applying this to the list of steps between the 5-limit tonality diamond [6/5, 25/24, 16/15, 9/8] we obtain [3a-4, 7-5a, a-1, 2a-3] from which it follows that a≥4/3, a≤7/5, a≥1, a≥3/2, the solution set of which is empty. Hence there are no monotone tunings and therefore no strict or lax tunings. | ||
</pre></div> | </pre></div> | ||
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Given a rank r p-limit regular temperament, we may define a tuning range by finding the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_hull" rel="nofollow">convex hull</a> in <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">tuning space</a> of the tunings with one <a class="wiki_link" href="/Eigenmonzo%20subgroup">eigenmonzo</a> 2 (pure octaves tunings) and the rest a set of r-1 members of the p-limit <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a>, when this tuning is defined. This is the <em>nice</em> tuning range. We may define another tuning range by requiring that the tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Monotonic_function" rel="nofollow">monotone</a> weakly increasing. This we may call the <em>monotone</em> tuning range. A tuning which is both nice and monotone is a <em>strict</em> tuning and this defines the strict tuning range. A tuning which is monotone but not nice is a <em>lax</em> tuning.<br /> | Given a rank r p-limit regular temperament, we may define a tuning range by finding the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_hull" rel="nofollow">convex hull</a> in <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">tuning space</a> of the tunings with one <a class="wiki_link" href="/Eigenmonzo%20subgroup">eigenmonzo</a> 2 (pure octaves tunings) and the rest a set of r-1 members of the p-limit <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a>, when this tuning is defined. This is the <em>nice</em> tuning range. We may define another tuning range by requiring that the tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Monotonic_function" rel="nofollow">monotone</a> weakly increasing. This we may call the <em>monotone</em> tuning range. A tuning which is both nice and monotone is a <em>strict</em> tuning and this defines the strict tuning range. A tuning which is monotone but not nice is a <em>lax</em> tuning.<br /> | ||
<br /> | <br /> | ||
While nice tunings are alway guaranteed to occur, strict or lax tunings are not. For instance, from the tuning map [&lt;1 0 5|, &lt;0 1 -2|] for the temperament tempering out 45/32 we find that all tunings are of the form &lt;1 0 5| + a&lt;0 1 -2| = &lt;1 a 5-2a|. Applying this to the list of steps between the 5-limit tonality diamond [6/5, 25/24, 16/15, 9/8] we obtain [3a-4, 7-5a, a-1, 2a-3] from which it follows that a≥4/3, a≤7/5, a≥1, a≥3/2, the solution set of which is empty. Hence there are no monotone tunings and | While nice tunings are alway guaranteed to occur, strict or lax tunings are not. For instance, from the tuning map [&lt;1 0 5|, &lt;0 1 -2|] for the temperament tempering out 45/32 we find that all tunings are of the form &lt;1 0 5| + a&lt;0 1 -2| = &lt;1 a 5-2a|. Applying this to the list of steps between the 5-limit tonality diamond [6/5, 25/24, 16/15, 9/8] we obtain [3a-4, 7-5a, a-1, 2a-3] from which it follows that a≥4/3, a≤7/5, a≥1, a≥3/2, the solution set of which is empty. Hence there are no monotone tunings and therefore no strict or lax tunings.</body></html></pre></div> | ||
Revision as of 02:32, 18 August 2014
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2014-08-18 02:32:24 UTC.
- The original revision id was 518800762.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
There are various methods which have been suggested for defining tuning ranges appropriate to a given regular temperament. Given a rank r p-limit regular temperament, we may define a tuning range by finding the [[http://en.wikipedia.org/wiki/Convex_hull|convex hull]] in [[Vals and Tuning Space|tuning space]] of the tunings with one [[Eigenmonzo subgroup|eigenmonzo]] 2 (pure octaves tunings) and the rest a set of r-1 members of the p-limit [[tonality diamond]], when this tuning is defined. This is the //nice// tuning range. We may define another tuning range by requiring that the tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also [[http://en.wikipedia.org/wiki/Monotonic_function|monotone]] weakly increasing. This we may call the //monotone// tuning range. A tuning which is both nice and monotone is a //strict// tuning and this defines the strict tuning range. A tuning which is monotone but not nice is a //lax// tuning. While nice tunings are alway guaranteed to occur, strict or lax tunings are not. For instance, from the tuning map [<1 0 5|, <0 1 -2|] for the temperament tempering out 45/32 we find that all tunings are of the form <1 0 5| + a<0 1 -2| = <1 a 5-2a|. Applying this to the list of steps between the 5-limit tonality diamond [6/5, 25/24, 16/15, 9/8] we obtain [3a-4, 7-5a, a-1, 2a-3] from which it follows that a≥4/3, a≤7/5, a≥1, a≥3/2, the solution set of which is empty. Hence there are no monotone tunings and therefore no strict or lax tunings.
Original HTML content:
<html><head><title>Tuning Ranges of Regular Temperaments</title></head><body>There are various methods which have been suggested for defining tuning ranges appropriate to a given regular temperament.<br /> <br /> Given a rank r p-limit regular temperament, we may define a tuning range by finding the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_hull" rel="nofollow">convex hull</a> in <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">tuning space</a> of the tunings with one <a class="wiki_link" href="/Eigenmonzo%20subgroup">eigenmonzo</a> 2 (pure octaves tunings) and the rest a set of r-1 members of the p-limit <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a>, when this tuning is defined. This is the <em>nice</em> tuning range. We may define another tuning range by requiring that the tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Monotonic_function" rel="nofollow">monotone</a> weakly increasing. This we may call the <em>monotone</em> tuning range. A tuning which is both nice and monotone is a <em>strict</em> tuning and this defines the strict tuning range. A tuning which is monotone but not nice is a <em>lax</em> tuning.<br /> <br /> While nice tunings are alway guaranteed to occur, strict or lax tunings are not. For instance, from the tuning map [<1 0 5|, <0 1 -2|] for the temperament tempering out 45/32 we find that all tunings are of the form <1 0 5| + a<0 1 -2| = <1 a 5-2a|. Applying this to the list of steps between the 5-limit tonality diamond [6/5, 25/24, 16/15, 9/8] we obtain [3a-4, 7-5a, a-1, 2a-3] from which it follows that a≥4/3, a≤7/5, a≥1, a≥3/2, the solution set of which is empty. Hence there are no monotone tunings and therefore no strict or lax tunings.</body></html>