Optimal patent val: Difference between revisions
Wikispaces>genewardsmith **Imported revision 203937790 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 203938302 - 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>2011-02-22 07: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-22 07:37:04 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>203938302</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q <= p, if d is the absolute value in cents of the difference between the tuning of q given by the [[POTE tuning]] and the POTE tuning rounded to the nearest N-edo value, then d < 600/N, from which it follows that N < 600/d. Likewise, if e is the absolute value of the error of q in the patent val tuning, then e < 600/N and so N < 600/e. If N-edo defines an optimal patent val, then the patent val will be identical to the val obtained by rounding the POTE tuning to the nearest N-edo value. We have two distances from the patent val, one to the POTE tuning and one to the the JI tuning, both bounded by 600/N, and so by the triangle inequality the distance from the JI tuning to the POTE tuning, which is the error of the prime q in the POTE tuning, is bounded by 1200/N. Hence, N < 1200/error(q). If now we take the minimum value for 1200/error(prime) for all the odd primes up to p, we obtain an upper bound for N. | To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q <= p, if d is the absolute value in cents of the difference between the tuning of q given by the [[POTE tuning]] and the POTE tuning rounded to the nearest N-edo value, then d < 600/N, from which it follows that N < 600/d. Likewise, if e is the absolute value of the error of q in the patent val tuning, then e < 600/N and so N < 600/e. If N-edo defines an optimal patent val, then the patent val will be identical to the val obtained by rounding the POTE tuning to the nearest N-edo value. We have two distances from the patent val, one to the POTE tuning and one to the the JI tuning, both bounded by 600/N, and so by the triangle inequality the distance from the JI tuning to the POTE tuning, which is the error of the prime q in the POTE tuning, is bounded by 1200/N. Hence, N < 1200/error(q). If now we take the minimum value for 1200/error(prime) for all the odd primes up to p, we obtain an upper bound for N. | ||
Below are tabulated some values. In each case an identifier which uniquely identifies the temperament in question is given. In the codimension one case, where the temperament is defined by a single comma, the comma is given and used as a name. In other cases, for a temperament of rank n, n independent [[patent vals]] are given. Normally this is by way of integers conjoined by ampersands, such as 2&10 for 7-limit pajara. This tells us we can use the 7-limit patent vals for 2 and 10 to define the temperament. In case n independent patent vals cannot be found, the [[Normal lists|normal val list]] is given. Not that this differs from the method of [[Graham Breed]]'s [[http://x31eq.com/temper/|regular temperament finder]], which does not assume the numbers refer to patent vals. | Below are tabulated some values. In each case an identifier which uniquely identifies the temperament in question is given. In the codimension one case, where the temperament is defined by a single comma, the comma is given and used as a name. In other cases, for a temperament of rank n, n independent [[Patent val|patent vals]] are given. Normally this is by way of integers conjoined by ampersands, such as 2&10 for 7-limit pajara. This tells us we can use the 7-limit patent vals for 2 and 10 to define the temperament. In case n independent patent vals cannot be found, the [[Normal lists|normal val list]] is given. Not that this differs from the method of [[Graham Breed]]'s [[http://x31eq.com/temper/|regular temperament finder]], which does not assume the numbers refer to patent vals. | ||
==5-limit rank two== | ==5-limit rank two== | ||
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[[Augmented family|August]]: [[21edo]] 9&21 18.448 | [[Augmented family|August]]: [[21edo]] 9&21 18.448 | ||
[[Meantone family|Huygens]]: [[31edo]] [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|] 18.048 | [[Meantone family|Huygens]]: [[31edo]] [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|] 18.048 | ||
[[Semicomma family|Winston]]: [[31edo]] | [[Semicomma family|Winston]]: [[31edo]] 9&31 19.931 | ||
[[Gamelismic clan|Miracle]]: [[41edo]] 10&31 18.669</pre></div> | [[Gamelismic clan|Miracle]]: [[41edo]] 10&31 18.669</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
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To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q &lt;= p, if d is the absolute value in cents of the difference between the tuning of q given by the <a class="wiki_link" href="/POTE%20tuning">POTE tuning</a> and the POTE tuning rounded to the nearest N-edo value, then d &lt; 600/N, from which it follows that N &lt; 600/d. Likewise, if e is the absolute value of the error of q in the patent val tuning, then e &lt; 600/N and so N &lt; 600/e. If N-edo defines an optimal patent val, then the patent val will be identical to the val obtained by rounding the POTE tuning to the nearest N-edo value. We have two distances from the patent val, one to the POTE tuning and one to the the JI tuning, both bounded by 600/N, and so by the triangle inequality the distance from the JI tuning to the POTE tuning, which is the error of the prime q in the POTE tuning, is bounded by 1200/N. Hence, N &lt; 1200/error(q). If now we take the minimum value for 1200/error(prime) for all the odd primes up to p, we obtain an upper bound for N.<br /> | To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q &lt;= p, if d is the absolute value in cents of the difference between the tuning of q given by the <a class="wiki_link" href="/POTE%20tuning">POTE tuning</a> and the POTE tuning rounded to the nearest N-edo value, then d &lt; 600/N, from which it follows that N &lt; 600/d. Likewise, if e is the absolute value of the error of q in the patent val tuning, then e &lt; 600/N and so N &lt; 600/e. If N-edo defines an optimal patent val, then the patent val will be identical to the val obtained by rounding the POTE tuning to the nearest N-edo value. We have two distances from the patent val, one to the POTE tuning and one to the the JI tuning, both bounded by 600/N, and so by the triangle inequality the distance from the JI tuning to the POTE tuning, which is the error of the prime q in the POTE tuning, is bounded by 1200/N. Hence, N &lt; 1200/error(q). If now we take the minimum value for 1200/error(prime) for all the odd primes up to p, we obtain an upper bound for N.<br /> | ||
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Below are tabulated some values. In each case an identifier which uniquely identifies the temperament in question is given. In the codimension one case, where the temperament is defined by a single comma, the comma is given and used as a name. In other cases, for a temperament of rank n, n independent <a class="wiki_link" href="/ | Below are tabulated some values. In each case an identifier which uniquely identifies the temperament in question is given. In the codimension one case, where the temperament is defined by a single comma, the comma is given and used as a name. In other cases, for a temperament of rank n, n independent <a class="wiki_link" href="/Patent%20val">patent vals</a> are given. Normally this is by way of integers conjoined by ampersands, such as 2&amp;10 for 7-limit pajara. This tells us we can use the 7-limit patent vals for 2 and 10 to define the temperament. In case n independent patent vals cannot be found, the <a class="wiki_link" href="/Normal%20lists">normal val list</a> is given. Not that this differs from the method of <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>'s <a class="wiki_link_ext" href="http://x31eq.com/temper/" rel="nofollow">regular temperament finder</a>, which does not assume the numbers refer to patent vals.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-5-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:0 -->5-limit rank two</h2> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-5-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:0 -->5-limit rank two</h2> | ||
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<a class="wiki_link" href="/Augmented%20family">August</a>: <a class="wiki_link" href="/21edo">21edo</a> 9&amp;21 18.448<br /> | <a class="wiki_link" href="/Augmented%20family">August</a>: <a class="wiki_link" href="/21edo">21edo</a> 9&amp;21 18.448<br /> | ||
<a class="wiki_link" href="/Meantone%20family">Huygens</a>: <a class="wiki_link" href="/31edo">31edo</a> [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|] 18.048<br /> | <a class="wiki_link" href="/Meantone%20family">Huygens</a>: <a class="wiki_link" href="/31edo">31edo</a> [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|] 18.048<br /> | ||
<a class="wiki_link" href="/Semicomma%20family">Winston</a>: <a class="wiki_link" href="/31edo">31edo</a><br /> | <a class="wiki_link" href="/Semicomma%20family">Winston</a>: <a class="wiki_link" href="/31edo">31edo</a> 9&amp;31 19.931<br /> | ||
<a class="wiki_link" href="/Gamelismic%20clan">Miracle</a>: <a class="wiki_link" href="/41edo">41edo</a> 10&amp;31 18.669</body></html></pre></div> | <a class="wiki_link" href="/Gamelismic%20clan">Miracle</a>: <a class="wiki_link" href="/41edo">41edo</a> 10&amp;31 18.669</body></html></pre></div> | ||