Optimal patent val: 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:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-15 04:19:11 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-15 04:32:09 UTC</tt>.

: The original revision id was <tt>~~201854482~~</tt>.

: The original revision id was <tt>201856064</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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By tempering a JI scale using N-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in Scala using the Quantize command: either type in "Quantize N" on the bottom, or use the pull-down menu under "Modify" and type N (without a decimal point) into "Resolution".

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 ~~a distance~~ from the patent val to ~~both~~ the POTE tuning and the JI tuning, and so by the triangle inequality ~~f < 1200/N~~, ~~where f~~ is the error of the POTE tuning ~~for q~~. Hence, N < 1200/f. If now we take the minimum value for 1200/error(q) for all the odd primes up to p~~, where error(q) is the POTE tuning error of q~~, 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.

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By tempering a JI scale using N-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in Scala using the Quantize command: either type in &quot;Quantize N&quot; on the bottom, or use the pull-down menu under &quot;Modify&quot; and type N (without a decimal point) into &quot;Resolution&quot;.

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 ~~a distance~~ from the patent val to ~~both~~ the POTE tuning and the JI tuning, and so by the triangle inequality ~~f &lt; 1200/N~~, ~~where f~~ is the error of the POTE tuning ~~for q~~. Hence, N &lt; 1200/f. If now we take the minimum value for 1200/error(q) for all the odd primes up to p~~, where error(q) is the POTE tuning error of q~~, 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 &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.

Below are tabulated some values.

@@ 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>2011-02-15 04:19:11 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-15 04:32:09 UTC</tt>.<br>
-: The original revision id was <tt>201854482</tt>.<br>
+: The original revision id was <tt>201856064</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 10: / Line 10: @@
 By tempering a JI scale using N-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in Scala using the Quantize command: either type in "Quantize N" on the bottom, or use the pull-down menu under "Modify" and type N (without a decimal point) into "Resolution".
-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 [[POTE tuning]] 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 a distance from the patent val to both the POTE tuning and the JI tuning, and so by the triangle inequality f &lt; 1200/N, where f is the error of the POTE tuning for q. Hence, N &lt; 1200/f. If now we take the minimum value for 1200/error(q) for all the odd primes up to p, where error(q) is the POTE tuning error of q, 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 &lt;= 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 &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.
 Below are tabulated some values.
@@ Line 139: / Line 139: @@
 By tempering a JI scale using N-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in Scala using the Quantize command: either type in &amp;quot;Quantize N&amp;quot; on the bottom, or use the pull-down menu under &amp;quot;Modify&amp;quot; and type N (without a decimal point) into &amp;quot;Resolution&amp;quot;.&lt;br /&gt;
 &lt;br /&gt;
-To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q &amp;lt;= p, if d is the absolute value in cents of the difference between the tuning of q given by the &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE tuning&lt;/a&gt; and the POTE tuning rounded to the nearest N-edo value, then d &amp;lt; 600/N, from which it follows that N &amp;lt; 600/d. Likewise, if e is the absolute value of the error of q in the patent val tuning, then e &amp;lt; 600/N and so N &amp;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 a distance from the patent val to both the POTE tuning and the JI tuning, and so by the triangle inequality f &amp;lt; 1200/N, where f is the error of the POTE tuning for q. Hence, N &amp;lt; 1200/f. If now we take the minimum value for 1200/error(q) for all the odd primes up to p, where error(q) is the POTE tuning error of q, we obtain an upper bound for N.&lt;br /&gt;
+To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q &amp;lt;= p, if d is the absolute value in cents of the difference between the tuning of q given by the &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE tuning&lt;/a&gt; and the POTE tuning rounded to the nearest N-edo value, then d &amp;lt; 600/N, from which it follows that N &amp;lt; 600/d. Likewise, if e is the absolute value of the error of q in the patent val tuning, then e &amp;lt; 600/N and so N &amp;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 &amp;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.&lt;br /&gt;
 &lt;br /&gt;
 Below are tabulated some values.&lt;br /&gt;