Tuning ranges 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:genewardsmith|genewardsmith]] and made on <tt>2014-08-18 11:19:50 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-08-18 13:22:48 UTC</tt>.<br>

: The original revision id was <tt>~~518830312~~</tt>.<br>

: The original revision id was <tt>518840966</tt>.<br>

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

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While strict tunings are alway guaranteed to occur, 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 lax tunings of this temperament.

However, for the kind of temperaments of the most interest, the lax tuning range does exist and contains the strict tuning range, and hence the names. For a more typical example, consider marvel temperament. Using the Hermite normal form tuning map again, we find that all marvel tunings are of the form <1 a b 2a+ab-5 12-a-3b|. Applying this to the steps of the 11-limit tonality diamond, we obtain eight inequalities, the solution set of which is the union of {30/19 ≤ a ≤ 49/31, 2+a/5 ≤ b ≤ 4a-4} with {49/31 ≤ a ≤ 35/22, 2+a/5 ≤ b ≤ 3-3a/7}, which is the triangular region bounded by the tunings for 19, 22, and 31.

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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>strict</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>lax</em> tuning range. <br />

<br />

While strict tunings are alway guaranteed to occur, 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 lax tunings of this temperament.</body></html></pre></div>

While strict tunings are alway guaranteed to occur, 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 lax tunings of this temperament.<br />

<br />

However, for the kind of temperaments of the most interest, the lax tuning range does exist and contains the strict tuning range, and hence the names. For a more typical example, consider marvel temperament. Using the Hermite normal form tuning map again, we find that all marvel tunings are of the form &lt;1 a b 2a+ab-5 12-a-3b|. Applying this to the steps of the 11-limit tonality diamond, we obtain eight inequalities, the solution set of which is the union of {30/19 ≤ a ≤ 49/31, 2+a/5 ≤ b ≤ 4a-4} with {49/31 ≤ a ≤ 35/22, 2+a/5 ≤ b ≤ 3-3a/7}, which is the triangular region bounded by the tunings for 19, 22, and 31.</body></html></pre></div>

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2014-08-18 11:19:50 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-08-18 13:22:48 UTC</tt>.<br>
-: The original revision id was <tt>518830312</tt>.<br>
+: The original revision id was <tt>518840966</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: @@
 While strict tunings are alway guaranteed to occur, 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 lax tunings of this temperament.
+However, for the kind of temperaments of the most interest, the lax tuning range does exist and contains the strict tuning range, and hence the names. For a more typical example, consider marvel temperament. Using the Hermite normal form tuning map again, we find that all marvel tunings are of the form &lt;1 a b 2a+ab-5 12-a-3b|. Applying this to the steps of the 11-limit tonality diamond, we obtain eight inequalities, the solution set of which is the union of {30/19 ≤ a ≤ 49/31, 2+a/5 ≤ b ≤ 4a-4} with {49/31 ≤ a ≤ 35/22, 2+a/5 ≤ b ≤ 3-3a/7}, which is the triangular region bounded by the tunings for 19, 22, and 31.
 </pre></div>
 <h4>Original HTML content:</h4>
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 Given a rank r p-limit regular temperament, we may define a tuning range by finding the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_hull" rel="nofollow"&gt;convex hull&lt;/a&gt; in &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;tuning space&lt;/a&gt; of the tunings with one &lt;a class="wiki_link" href="/Eigenmonzo%20subgroup"&gt;eigenmonzo&lt;/a&gt; 2 (pure octaves tunings) and the rest a set of r-1 members of the p-limit &lt;a class="wiki_link" href="/tonality%20diamond"&gt;tonality diamond&lt;/a&gt;, when this tuning is defined. This is the &lt;em&gt;strict&lt;/em&gt; 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Monotonic_function" rel="nofollow"&gt;monotone&lt;/a&gt; weakly increasing. This we may call the &lt;em&gt;lax&lt;/em&gt; tuning range. &lt;br /&gt;
 &lt;br /&gt;
-While strict tunings are alway guaranteed to occur, lax tunings are not. For instance, from the tuning map [&amp;lt;1 0 5|, &amp;lt;0 1 -2|] for the temperament tempering out 45/32 we find that all tunings are of the form &amp;lt;1 0 5| + a&amp;lt;0 1 -2| = &amp;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 lax tunings of this temperament.&lt;/body&gt;&lt;/html&gt;</pre></div>
+While strict tunings are alway guaranteed to occur, lax tunings are not. For instance, from the tuning map [&amp;lt;1 0 5|, &amp;lt;0 1 -2|] for the temperament tempering out 45/32 we find that all tunings are of the form &amp;lt;1 0 5| + a&amp;lt;0 1 -2| = &amp;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 lax tunings of this temperament.&lt;br /&gt;
+&lt;br /&gt;
+However, for the kind of temperaments of the most interest, the lax tuning range does exist and contains the strict tuning range, and hence the names. For a more typical example, consider marvel temperament. Using the Hermite normal form tuning map again, we find that all marvel tunings are of the form &amp;lt;1 a b 2a+ab-5 12-a-3b|. Applying this to the steps of the 11-limit tonality diamond, we obtain eight inequalities, the solution set of which is the union of {30/19 ≤ a ≤ 49/31, 2+a/5 ≤ b ≤ 4a-4} with {49/31 ≤ a ≤ 35/22, 2+a/5 ≤ b ≤ 3-3a/7}, which is the triangular region bounded by the tunings for 19, 22, and 31.&lt;/body&gt;&lt;/html&gt;</pre></div>