Tuning ranges of regular temperaments: Difference between revisions

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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 //valid// tuning range. A tuning which is both nice and valid is a //strict// tuning and this defines the strict tuning range. 

While nice tunings are always guaranteed to occur, valid 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 notes of 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 valid tunings of this temperament.

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. This is the valid range. The nice tuning range is a quadralateral, with vertices, given in terms of frequency ratios rather than log base 2 or cents, [[2, 4096/1375, 5, 524288/75625, 11], [2, 3, 224/45, 1568/225, 30375/2744], [2, 1620/539, 4/3*sqrt(14), 291600/41503, 44/15*sqrt(14)], [2, 3, 5, 225/32, 4096/375]]. The three verticies with all rational number values for the approximate 3 and 5 are not in the valid range, so that only the  [2, 1620/539, 4/3*sqrt(14), 291600/41503, 44/15*sqrt(14)] tuning is valid and hence strict. Other examples of strict tunings are 41p/41, 53p/53, 72p/72 etc.; however 19p/19, 22p/22 and 31p/31 are not in the nice range.

In **Tuning Continua and Keyboard Layouts** in the premiere issue of //Journal of Mathematics and Music//, Milne, Sethares and Plamodon define valid tunings; they discuss nice tunings in **X_System** in the Open University’s repository.

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 <a class="wiki_link" href="/regular%20temperament">regular temperament</a>.<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>valid</em> tuning range. A tuning which is both nice and valid is a <em>strict</em> tuning and this defines the strict tuning range. <br />
<br />
While nice tunings are always guaranteed to occur, valid 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 notes of 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 valid tunings of this temperament.<br />
<br />
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. This is the valid range. The nice tuning range is a quadralateral, with vertices, given in terms of frequency ratios rather than log base 2 or cents, [[2, 4096/1375, 5, 524288/75625, 11], [2, 3, 224/45, 1568/225, 30375/2744], [2, 1620/539, 4/3*sqrt(14), 291600/41503, 44/15*sqrt(14)], [2, 3, 5, 225/32, 4096/375]]. The three verticies with all rational number values for the approximate 3 and 5 are not in the valid range, so that only the  [2, 1620/539, 4/3*sqrt(14), 291600/41503, 44/15*sqrt(14)] tuning is valid and hence strict. Other examples of strict tunings are 41p/41, 53p/53, 72p/72 etc.; however 19p/19, 22p/22 and 31p/31 are not in the nice range.<br />
<br />
In <strong>Tuning Continua and Keyboard Layouts</strong> in the premiere issue of <em>Journal of Mathematics and Music</em>, Milne, Sethares and Plamodon define valid tunings; they discuss nice tunings in <strong>X_System</strong> in the Open University’s repository.</body></html>

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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>2014-08-20 22:37:42 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-08-20 22:58:38 UTC</tt>.<br>
-: The original revision id was <tt>519136530</tt>.<br>
+: The original revision id was <tt>519138058</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>
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 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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 //normal// tuning range. A tuning which is both nice and normal is a //strict// tuning and this defines the strict tuning range.
+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 //valid// tuning range. A tuning which is both nice and valid is a //strict// tuning and this defines the strict tuning range.
-While nice tunings are always guaranteed to occur, normal 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 notes of 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 normal tunings of this temperament.
+While nice tunings are always guaranteed to occur, valid 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 notes of 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 valid tunings of this temperament.
-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. This is the normal range. The nice tuning range is a quadralateral, with vertices, given in terms of frequency ratios rather than log base 2 or cents, [[2, 4096/1375, 5, 524288/75625, 11], [2, 3, 224/45, 1568/225, 30375/2744], [2, 1620/539, 4/3*sqrt(14), 291600/41503, 44/15*sqrt(14)], [2, 3, 5, 225/32, 4096/375]]. The three verticies with all rational number values for the approximate 3 and 5 are not in the normal range, so that only the  [2, 1620/539, 4/3*sqrt(14), 291600/41503, 44/15*sqrt(14)] tuning is normal and hence strict. Other examples of strict tunings are 41p/41, 53p/53, 72p/72 etc.; however 19p/19, 22p/22 and 31p/31 are not in the nice range.</pre></div>
+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. This is the valid range. The nice tuning range is a quadralateral, with vertices, given in terms of frequency ratios rather than log base 2 or cents, [[2, 4096/1375, 5, 524288/75625, 11], [2, 3, 224/45, 1568/225, 30375/2744], [2, 1620/539, 4/3*sqrt(14), 291600/41503, 44/15*sqrt(14)], [2, 3, 5, 225/32, 4096/375]]. The three verticies with all rational number values for the approximate 3 and 5 are not in the valid range, so that only the  [2, 1620/539, 4/3*sqrt(14), 291600/41503, 44/15*sqrt(14)] tuning is valid and hence strict. Other examples of strict tunings are 41p/41, 53p/53, 72p/72 etc.; however 19p/19, 22p/22 and 31p/31 are not in the nice range.
+In **Tuning Continua and Keyboard Layouts** in the premiere issue of //Journal of Mathematics and Music//, Milne, Sethares and Plamodon define valid tunings; they discuss nice tunings in **X_System** in the Open University’s repository.</pre></div>
 <h4>Original HTML content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Tuning Ranges of Regular Temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;There are various methods which have been suggested for defining tuning ranges appropriate to a given &lt;a class="wiki_link" href="/regular%20temperament"&gt;regular temperament&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-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;nice&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;normal&lt;/em&gt; tuning range. A tuning which is both nice and normal is a &lt;em&gt;strict&lt;/em&gt; tuning and this defines the strict tuning range. &lt;br /&gt;
+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;nice&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;valid&lt;/em&gt; tuning range. A tuning which is both nice and valid is a &lt;em&gt;strict&lt;/em&gt; tuning and this defines the strict tuning range. &lt;br /&gt;
+&lt;br /&gt;
+While nice tunings are always guaranteed to occur, valid 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 notes of 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 valid tunings of this temperament.&lt;br /&gt;
 &lt;br /&gt;
-While nice tunings are always guaranteed to occur, normal 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 notes of 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 normal tunings of this temperament.&lt;br /&gt;
+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. This is the valid range. The nice tuning range is a quadralateral, with vertices, given in terms of frequency ratios rather than log base 2 or cents, [[2, 4096/1375, 5, 524288/75625, 11], [2, 3, 224/45, 1568/225, 30375/2744], [2, 1620/539, 4/3*sqrt(14), 291600/41503, 44/15*sqrt(14)], [2, 3, 5, 225/32, 4096/375]]. The three verticies with all rational number values for the approximate 3 and 5 are not in the valid range, so that only the  [2, 1620/539, 4/3*sqrt(14), 291600/41503, 44/15*sqrt(14)] tuning is valid and hence strict. Other examples of strict tunings are 41p/41, 53p/53, 72p/72 etc.; however 19p/19, 22p/22 and 31p/31 are not in the nice range.&lt;br /&gt;
 &lt;br /&gt;
-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. This is the normal range. The nice tuning range is a quadralateral, with vertices, given in terms of frequency ratios rather than log base 2 or cents, [[2, 4096/1375, 5, 524288/75625, 11], [2, 3, 224/45, 1568/225, 30375/2744], [2, 1620/539, 4/3*sqrt(14), 291600/41503, 44/15*sqrt(14)], [2, 3, 5, 225/32, 4096/375]]. The three verticies with all rational number values for the approximate 3 and 5 are not in the normal range, so that only the  [2, 1620/539, 4/3*sqrt(14), 291600/41503, 44/15*sqrt(14)] tuning is normal and hence strict. Other examples of strict tunings are 41p/41, 53p/53, 72p/72 etc.; however 19p/19, 22p/22 and 31p/31 are not in the nice range.&lt;/body&gt;&lt;/html&gt;</pre></div>
+In &lt;strong&gt;Tuning Continua and Keyboard Layouts&lt;/strong&gt; in the premiere issue of &lt;em&gt;Journal of Mathematics and Music&lt;/em&gt;, Milne, Sethares and Plamodon define valid tunings; they discuss nice tunings in &lt;strong&gt;X_System&lt;/strong&gt; in the Open University’s repository.&lt;/body&gt;&lt;/html&gt;</pre></div>

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