TOP tuning: 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>2015-04-~~26 23~~:16:14 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-04-27 00:17:19 UTC</tt>.<br>

: The original revision id was <tt>~~548764630~~</tt>.<br>

: The original revision id was <tt>548768580</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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=Finding the tuning=

For a temperament with both intrinsic and extrinsic primes, we may find the set of TOP tunings by first computing the tuning of the intrinsic primes. Then if r is an extrinsic prime, the tuning may be anything in the range where APE(r) ≤ E. The limit of the [[Lp tuning]] as p tends to 1 and the centroid of the region of TOP tunings both lead to choosing the JI tuning for r. This produces the canonical TOP tuning, called TIPTOP. To find the TIPTOP tuning one method is to solve for all the potential TOP tunings of the intrinsic primes, find the tuning with least error, and then tune all the extrinsic primes purely. An alternative method is to first set up a linear programming problem; if T is a val with indeterminate coefficients T = <t₁ t₂ ... tₖ| then minimize E subject to nonnegativity and the linear constraints {tₙ/log₂(pₙ) - 1 ≤ E, 1 - tₙ/log₂(pₙ) ≤ E, <T|cₖ> = 0} where the pₙ are the primes of the temperament, and the cₖ are the commas. We then may replace the tuning of all of the extrinsic primes with the pure JI tuning to get TIPTOP.</pre></div>

For a temperament with both intrinsic and extrinsic primes, we may find the set of TOP tunings by first computing the tuning of the intrinsic primes. Then if r is an extrinsic prime, the tuning may be anything in the range where APE(r) ≤ E. The limit of the [[Lp tuning]] as p tends to 1 and the centroid of the region of TOP tunings both lead to choosing the JI tuning for r. This produces the canonical TOP tuning, called TIPTOP. To find the TIPTOP tuning one method is to solve for all the potential TOP tunings of the intrinsic primes, find the tuning with least error, and then tune all the extrinsic primes purely. An alternative method is to first set up a linear programming problem; if T is a val with indeterminate coefficients T = <t₁ t₂ ... tₖ| then minimize E subject to nonnegativity and the linear constraints {tₙ/log₂(pₙ) - 1 ≤ E, 1 - tₙ/log₂(pₙ) ≤ E, <T|cₖ> = 0} where the pₙ are the primes of the temperament, and the cₖ are the commas. We then may replace the tuning of all of the extrinsic primes with the pure JI tuning to get TIPTOP.

We may solve the sharp semigroup equations exactly to obtain solutions in the transcendental extension Q(log₂(q₁), log₂(t₂), ..., log₂(qₖ)) where the qₙ are the intrinsic primes other than 2. For example, take 5-limit meantone. Since 2 and 5 divide 80 and 3 divides 81. this is an intrinsic temperament. Solving for the TOP tuning either by linear programing or checking all the potential TOP tunings, we find the sharp semigroup is generated by {2, 1/3, 5}. Solving the sharp semigroup equations gives us a TOP tuning T = <log₂(6561)/log₂(6480) log₂(2025)/log₂(6480) log₂</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"><html><head><title>TOP tuning</title></head><body><h1 id="toc0"><a name="Proportional error"></a>Proportional error</h1>

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<br />

<h1 id="toc3"><a name="Finding the tuning"></a>Finding the tuning</h1>

For a temperament with both intrinsic and extrinsic primes, we may find the set of TOP tunings by first computing the tuning of the intrinsic primes. Then if r is an extrinsic prime, the tuning may be anything in the range where APE(r) ≤ E. The limit of the <a class="wiki_link" href="/Lp%20tuning">Lp tuning</a> as p tends to 1 and the centroid of the region of TOP tunings both lead to choosing the JI tuning for r. This produces the canonical TOP tuning, called TIPTOP. To find the TIPTOP tuning one method is to solve for all the potential TOP tunings of the intrinsic primes, find the tuning with least error, and then tune all the extrinsic primes purely. An alternative method is to first set up a linear programming problem; if T is a val with indeterminate coefficients T = &lt;t₁ t₂ ... tₖ| then minimize E subject to nonnegativity and the linear constraints {tₙ/log₂(pₙ) - 1 ≤ E, 1 - tₙ/log₂(pₙ) ≤ E, &lt;T|cₖ&gt; = 0} where the pₙ are the primes of the temperament, and the cₖ are the commas. We then may replace the tuning of all of the extrinsic primes with the pure JI tuning to get TIPTOP.</body></html></pre></div>

For a temperament with both intrinsic and extrinsic primes, we may find the set of TOP tunings by first computing the tuning of the intrinsic primes. Then if r is an extrinsic prime, the tuning may be anything in the range where APE(r) ≤ E. The limit of the <a class="wiki_link" href="/Lp%20tuning">Lp tuning</a> as p tends to 1 and the centroid of the region of TOP tunings both lead to choosing the JI tuning for r. This produces the canonical TOP tuning, called TIPTOP. To find the TIPTOP tuning one method is to solve for all the potential TOP tunings of the intrinsic primes, find the tuning with least error, and then tune all the extrinsic primes purely. An alternative method is to first set up a linear programming problem; if T is a val with indeterminate coefficients T = &lt;t₁ t₂ ... tₖ| then minimize E subject to nonnegativity and the linear constraints {tₙ/log₂(pₙ) - 1 ≤ E, 1 - tₙ/log₂(pₙ) ≤ E, &lt;T|cₖ&gt; = 0} where the pₙ are the primes of the temperament, and the cₖ are the commas. We then may replace the tuning of all of the extrinsic primes with the pure JI tuning to get TIPTOP.<br />

<br />

We may solve the sharp semigroup equations exactly to obtain solutions in the transcendental extension Q(log₂(q₁), log₂(t₂), ..., log₂(qₖ)) where the qₙ are the intrinsic primes other than 2. For example, take 5-limit meantone. Since 2 and 5 divide 80 and 3 divides 81. this is an intrinsic temperament. Solving for the TOP tuning either by linear programing or checking all the potential TOP tunings, we find the sharp semigroup is generated by {2, 1/3, 5}. Solving the sharp semigroup equations gives us a TOP tuning T = &lt;log₂(6561)/log₂(6480) log₂(2025)/log₂(6480) log₂</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>2015-04-26 23:16:14 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-04-27 00:17:19 UTC</tt>.<br>
-: The original revision id was <tt>548764630</tt>.<br>
+: The original revision id was <tt>548768580</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 22: / Line 22: @@
 =Finding the tuning=
-For a temperament with both intrinsic and extrinsic primes, we may find the set of TOP tunings by first computing the tuning of the intrinsic primes. Then if r is an extrinsic prime, the tuning may be anything in the range where APE(r)  ≤  E. The  limit of the [[Lp tuning]] as p tends to 1 and the centroid of the region of TOP tunings both lead to choosing the JI tuning for r. This produces the canonical TOP tuning, called TIPTOP. To find the TIPTOP tuning one method is to solve for all the potential TOP tunings of the intrinsic primes, find the tuning with least error, and then tune all the extrinsic primes purely. An alternative method is to first set up a linear programming problem; if T is a val with indeterminate coefficients T = &lt;t₁ t₂  ... tₖ| then minimize E subject to nonnegativity and the linear constraints {tₙ/log₂(pₙ) - 1 ≤ E, 1 - tₙ/log₂(pₙ) ≤ E, &lt;T|cₖ&gt; = 0} where the pₙ are the primes of the temperament, and the cₖ are the commas. We then may replace the tuning of all of the extrinsic primes with the pure JI tuning to get TIPTOP.</pre></div>
+For a temperament with both intrinsic and extrinsic primes, we may find the set of TOP tunings by first computing the tuning of the intrinsic primes. Then if r is an extrinsic prime, the tuning may be anything in the range where APE(r)  ≤  E. The  limit of the [[Lp tuning]] as p tends to 1 and the centroid of the region of TOP tunings both lead to choosing the JI tuning for r. This produces the canonical TOP tuning, called TIPTOP. To find the TIPTOP tuning one method is to solve for all the potential TOP tunings of the intrinsic primes, find the tuning with least error, and then tune all the extrinsic primes purely. An alternative method is to first set up a linear programming problem; if T is a val with indeterminate coefficients T = &lt;t₁ t₂  ... tₖ| then minimize E subject to nonnegativity and the linear constraints {tₙ/log₂(pₙ) - 1 ≤ E, 1 - tₙ/log₂(pₙ) ≤ E, &lt;T|cₖ&gt; = 0} where the pₙ are the primes of the temperament, and the cₖ are the commas. We then may replace the tuning of all of the extrinsic primes with the pure JI tuning to get TIPTOP.
+We may solve the sharp semigroup equations exactly to obtain solutions in the transcendental extension Q(log₂(q₁), log₂(t₂), ..., log₂(qₖ)) where the qₙ are the intrinsic primes other than 2. For example, take 5-limit meantone. Since 2 and 5 divide 80 and 3 divides 81. this is an intrinsic temperament. Solving for the TOP tuning either by linear programing or checking all the potential TOP tunings, we find the sharp semigroup is generated by {2, 1/3, 5}. Solving the sharp semigroup equations gives us a TOP tuning T = &lt;log₂(6561)/log₂(6480) log₂(2025)/log₂(6480) log₂</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;TOP tuning&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Proportional error"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Proportional error&lt;/h1&gt;
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Finding the tuning"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Finding the tuning&lt;/h1&gt;
-For a temperament with both intrinsic and extrinsic primes, we may find the set of TOP tunings by first computing the tuning of the intrinsic primes. Then if r is an extrinsic prime, the tuning may be anything in the range where APE(r)  ≤  E. The  limit of the &lt;a class="wiki_link" href="/Lp%20tuning"&gt;Lp tuning&lt;/a&gt; as p tends to 1 and the centroid of the region of TOP tunings both lead to choosing the JI tuning for r. This produces the canonical TOP tuning, called TIPTOP. To find the TIPTOP tuning one method is to solve for all the potential TOP tunings of the intrinsic primes, find the tuning with least error, and then tune all the extrinsic primes purely. An alternative method is to first set up a linear programming problem; if T is a val with indeterminate coefficients T = &amp;lt;t₁ t₂  ... tₖ| then minimize E subject to nonnegativity and the linear constraints {tₙ/log₂(pₙ) - 1 ≤ E, 1 - tₙ/log₂(pₙ) ≤ E, &amp;lt;T|cₖ&amp;gt; = 0} where the pₙ are the primes of the temperament, and the cₖ are the commas. We then may replace the tuning of all of the extrinsic primes with the pure JI tuning to get TIPTOP.&lt;/body&gt;&lt;/html&gt;</pre></div>
+For a temperament with both intrinsic and extrinsic primes, we may find the set of TOP tunings by first computing the tuning of the intrinsic primes. Then if r is an extrinsic prime, the tuning may be anything in the range where APE(r)  ≤  E. The  limit of the &lt;a class="wiki_link" href="/Lp%20tuning"&gt;Lp tuning&lt;/a&gt; as p tends to 1 and the centroid of the region of TOP tunings both lead to choosing the JI tuning for r. This produces the canonical TOP tuning, called TIPTOP. To find the TIPTOP tuning one method is to solve for all the potential TOP tunings of the intrinsic primes, find the tuning with least error, and then tune all the extrinsic primes purely. An alternative method is to first set up a linear programming problem; if T is a val with indeterminate coefficients T = &amp;lt;t₁ t₂  ... tₖ| then minimize E subject to nonnegativity and the linear constraints {tₙ/log₂(pₙ) - 1 ≤ E, 1 - tₙ/log₂(pₙ) ≤ E, &amp;lt;T|cₖ&amp;gt; = 0} where the pₙ are the primes of the temperament, and the cₖ are the commas. We then may replace the tuning of all of the extrinsic primes with the pure JI tuning to get TIPTOP.&lt;br /&gt;
+&lt;br /&gt;
+We may solve the sharp semigroup equations exactly to obtain solutions in the transcendental extension Q(log₂(q₁), log₂(t₂), ..., log₂(qₖ)) where the qₙ are the intrinsic primes other than 2. For example, take 5-limit meantone. Since 2 and 5 divide 80 and 3 divides 81. this is an intrinsic temperament. Solving for the TOP tuning either by linear programing or checking all the potential TOP tunings, we find the sharp semigroup is generated by {2, 1/3, 5}. Solving the sharp semigroup equations gives us a TOP tuning T = &amp;lt;log₂(6561)/log₂(6480) log₂(2025)/log₂(6480) log₂&lt;/body&gt;&lt;/html&gt;</pre></div>