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-~~30 14~~:44:03 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-05-01 15:51:41 UTC</tt>.<br>

: The original revision id was <tt>~~549338662~~</tt>.<br>

: The original revision id was <tt>549464692</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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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 = <3q₃/log₂(6480) (8q₃ + 2q₃q₅)/log₂(6480) 8q₃q₅/log₂(6480)|. Here q₃ = log₂(3), q₅ = log₂(5), and the denominator can also be written 4 + 4q₃ + q₅. A more complex example including an extrinsic prime is 13-limit [[Rastmic temperaments#Parahemif|parahemif temperament]]. Setting D = 22 + q₁₁ + 5q₁₃, we have T = <(2q₁₁ +10q₁₃)/D (18q₁₁ + 2q₁₃)/D q₅ (102q₁₁ - 62q₁₃)/D 44q₁₁/D 44q₁₃/D|. Note that all the prime tunings except for that of 5 lie in the field Q(q₁₁, q₁₃), where 1/2, 11 and 13 generate the sharp semigroup; 5 is of course the extrinsic prime. The tuning of the other primes is the same as the tuning for hemif temperament, which has the same commas, generated by {144/143, 243/242, 364/363}, and the same sharp semigroup, but which tempers the 2.3.7.11.13 subgroup.

If we want a pure-octaves tuning, we may divide the TIPTOP tuning by the tuning of 2, giving what may be called the Pure Octaves TIPTOP tuning, or POTT. The POTT tuning is sometimes rather simple in form; for 5- and 7-limit meantone and 11-limit meanpop, we get the 1/4-comma tuning which is also the eigenmonzo 5 minimax tuning. For 7- and 11-limit pajara, we also get the eigenmonzo 5 tuning, with pure 5/4s, and for 13-limit POTT, we get the eigenmonzo 13 tuning. For 5-, 7-, 11-, and 13-limit myna, the POTT tuning is pure 3s. And so forth, for many other examples. </pre></div>

If we want a pure-octaves tuning, we may divide the TIPTOP tuning by the tuning of 2, giving what may be called the Pure Octaves TIPTOP tuning, or POTT. The POTT tuning is sometimes rather simple in form; for 5- and 7-limit meantone and 11-limit meanpop, we get the 1/4-comma tuning which is also the eigenmonzo 5 minimax tuning. For 7- and 11-limit pajara, we also get the eigenmonzo 5 tuning, with pure 5/4s, and for 13-limit POTT, we get the eigenmonzo 13 tuning. For 5-, 7-, 11-, and 13-limit myna, the POTT tuning is pure 3s. And so forth, for many other examples. The same thing can happen in higher ranks: 7-limit starling has the 3 and 7 eigenvalue tuning, and 11 and 13 limit thrush the 3 and 11 eighenvalue tuning, etc.</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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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;3q₃/log₂(6480) (8q₃ + 2q₃q₅)/log₂(6480) 8q₃q₅/log₂(6480)|. Here q₃ = log₂(3), q₅ = log₂(5), and the denominator can also be written 4 + 4q₃ + q₅. A more complex example including an extrinsic prime is 13-limit <a class="wiki_link" href="/Rastmic%20temperaments#Parahemif">parahemif temperament</a>. Setting D = 22 + q₁₁ + 5q₁₃, we have T = &lt;(2q₁₁ +10q₁₃)/D (18q₁₁ + 2q₁₃)/D q₅ (102q₁₁ - 62q₁₃)/D 44q₁₁/D 44q₁₃/D|. Note that all the prime tunings except for that of 5 lie in the field Q(q₁₁, q₁₃), where 1/2, 11 and 13 generate the sharp semigroup; 5 is of course the extrinsic prime. The tuning of the other primes is the same as the tuning for hemif temperament, which has the same commas, generated by {144/143, 243/242, 364/363}, and the same sharp semigroup, but which tempers the 2.3.7.11.13 subgroup.<br />

<br />

If we want a pure-octaves tuning, we may divide the TIPTOP tuning by the tuning of 2, giving what may be called the Pure Octaves TIPTOP tuning, or POTT. The POTT tuning is sometimes rather simple in form; for 5- and 7-limit meantone and 11-limit meanpop, we get the 1/4-comma tuning which is also the eigenmonzo 5 minimax tuning. For 7- and 11-limit pajara, we also get the eigenmonzo 5 tuning, with pure 5/4s, and for 13-limit POTT, we get the eigenmonzo 13 tuning. For 5-, 7-, 11-, and 13-limit myna, the POTT tuning is pure 3s. And so forth, for many other examples.</body></html></pre></div>

If we want a pure-octaves tuning, we may divide the TIPTOP tuning by the tuning of 2, giving what may be called the Pure Octaves TIPTOP tuning, or POTT. The POTT tuning is sometimes rather simple in form; for 5- and 7-limit meantone and 11-limit meanpop, we get the 1/4-comma tuning which is also the eigenmonzo 5 minimax tuning. For 7- and 11-limit pajara, we also get the eigenmonzo 5 tuning, with pure 5/4s, and for 13-limit POTT, we get the eigenmonzo 13 tuning. For 5-, 7-, 11-, and 13-limit myna, the POTT tuning is pure 3s. And so forth, for many other examples. The same thing can happen in higher ranks: 7-limit starling has the 3 and 7 eigenvalue tuning, and 11 and 13 limit thrush the 3 and 11 eighenvalue tuning, etc.</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>2015-04-30 14:44:03 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-05-01 15:51:41 UTC</tt>.<br>
-: The original revision id was <tt>549338662</tt>.<br>
+: The original revision id was <tt>549464692</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 26: / Line 26: @@
 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;3q₃/log₂(6480)  (8q₃ + 2q₃q₅)/log₂(6480)  8q₃q₅/log₂(6480)|. Here q₃ = log₂(3), q₅ =  log₂(5), and the denominator can also be written 4 + 4q₃ + q₅. A more complex example including an extrinsic prime is 13-limit [[Rastmic temperaments#Parahemif|parahemif temperament]]. Setting D = 22 + q₁₁ + 5q₁₃, we have T = &lt;(2q₁₁ +10q₁₃)/D (18q₁₁ + 2q₁₃)/D q₅ (102q₁₁ - 62q₁₃)/D 44q₁₁/D 44q₁₃/D|. Note that all the prime tunings except for that of 5 lie in the field Q(q₁₁, q₁₃), where 1/2, 11 and 13 generate the sharp semigroup; 5 is of course the extrinsic prime. The tuning of the other primes is the same as the tuning for hemif temperament, which has the same commas, generated by {144/143, 243/242, 364/363}, and the same sharp semigroup, but which tempers the 2.3.7.11.13 subgroup.
-If we want a pure-octaves tuning, we may divide the TIPTOP tuning by the tuning of 2, giving what may be called the Pure Octaves TIPTOP tuning, or POTT. The POTT tuning is sometimes rather simple in form; for 5- and 7-limit meantone and 11-limit meanpop, we get the 1/4-comma tuning which is also the eigenmonzo 5 minimax tuning. For 7- and 11-limit pajara, we also get the eigenmonzo 5 tuning, with pure 5/4s, and for 13-limit POTT, we get the eigenmonzo 13 tuning. For 5-, 7-, 11-, and 13-limit myna, the POTT tuning is pure 3s. And so forth, for many other examples. </pre></div>
+If we want a pure-octaves tuning, we may divide the TIPTOP tuning by the tuning of 2, giving what may be called the Pure Octaves TIPTOP tuning, or POTT. The POTT tuning is sometimes rather simple in form; for 5- and 7-limit meantone and 11-limit meanpop, we get the 1/4-comma tuning which is also the eigenmonzo 5 minimax tuning. For 7- and 11-limit pajara, we also get the eigenmonzo 5 tuning, with pure 5/4s, and for 13-limit POTT, we get the eigenmonzo 13 tuning. For 5-, 7-, 11-, and 13-limit myna, the POTT tuning is pure 3s. And so forth, for many other examples. The same thing can happen in higher ranks: 7-limit starling has the 3 and 7 eigenvalue tuning, and 11 and 13 limit thrush the 3 and 11 eighenvalue tuning, etc.</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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 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;3q₃/log₂(6480)  (8q₃ + 2q₃q₅)/log₂(6480)  8q₃q₅/log₂(6480)|. Here q₃ = log₂(3), q₅ =  log₂(5), and the denominator can also be written 4 + 4q₃ + q₅. A more complex example including an extrinsic prime is 13-limit &lt;a class="wiki_link" href="/Rastmic%20temperaments#Parahemif"&gt;parahemif temperament&lt;/a&gt;. Setting D = 22 + q₁₁ + 5q₁₃, we have T = &amp;lt;(2q₁₁ +10q₁₃)/D (18q₁₁ + 2q₁₃)/D q₅ (102q₁₁ - 62q₁₃)/D 44q₁₁/D 44q₁₃/D|. Note that all the prime tunings except for that of 5 lie in the field Q(q₁₁, q₁₃), where 1/2, 11 and 13 generate the sharp semigroup; 5 is of course the extrinsic prime. The tuning of the other primes is the same as the tuning for hemif temperament, which has the same commas, generated by {144/143, 243/242, 364/363}, and the same sharp semigroup, but which tempers the 2.3.7.11.13 subgroup.&lt;br /&gt;
 &lt;br /&gt;
-If we want a pure-octaves tuning, we may divide the TIPTOP tuning by the tuning of 2, giving what may be called the Pure Octaves TIPTOP tuning, or POTT. The POTT tuning is sometimes rather simple in form; for 5- and 7-limit meantone and 11-limit meanpop, we get the 1/4-comma tuning which is also the eigenmonzo 5 minimax tuning. For 7- and 11-limit pajara, we also get the eigenmonzo 5 tuning, with pure 5/4s, and for 13-limit POTT, we get the eigenmonzo 13 tuning. For 5-, 7-, 11-, and 13-limit myna, the POTT tuning is pure 3s. And so forth, for many other examples.&lt;/body&gt;&lt;/html&gt;</pre></div>
+If we want a pure-octaves tuning, we may divide the TIPTOP tuning by the tuning of 2, giving what may be called the Pure Octaves TIPTOP tuning, or POTT. The POTT tuning is sometimes rather simple in form; for 5- and 7-limit meantone and 11-limit meanpop, we get the 1/4-comma tuning which is also the eigenmonzo 5 minimax tuning. For 7- and 11-limit pajara, we also get the eigenmonzo 5 tuning, with pure 5/4s, and for 13-limit POTT, we get the eigenmonzo 13 tuning. For 5-, 7-, 11-, and 13-limit myna, the POTT tuning is pure 3s. And so forth, for many other examples. The same thing can happen in higher ranks: 7-limit starling has the 3 and 7 eigenvalue tuning, and 11 and 13 limit thrush the 3 and 11 eighenvalue tuning, etc.&lt;/body&gt;&lt;/html&gt;</pre></div>