TOP tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 549004500 - 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>2015-04- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-04-30 00:57:23 UTC</tt>.<br> | ||
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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. | 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 = <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.</pre></div> | 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, 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.</pre></div> | |||
<h4>Original HTML content:</h4> | <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><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Proportional error"></a><!-- ws:end:WikiTextHeadingRule:0 -->Proportional error</h1> | <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><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Proportional error"></a><!-- ws:end:WikiTextHeadingRule:0 -->Proportional error</h1> | ||
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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 /> | 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 /> | ||
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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.</body></html></pre></div> | 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 /> | ||
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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, 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.</body></html></pre></div> | |||