The wedgie: 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>2012-01-08 17:52:22 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-08 18:22:21 UTC</tt>.<br>

: The original revision id was <tt>~~290417595~~</tt>.<br>

: The original revision id was <tt>290423187</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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Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if E ≤ 1/(C(n, 3)√lb(q)√lb(p)) then wedging K = <1 lb(3) lb(5) ... lb(p)| with the val consisting of 0 followed by the first n-1 coefficients of the wedgie and rounding will give the wedgie. Here p and q are the largest and second largest primes in the prime limit, lb(x) is log base two, and C(n, 3) is n choose three, n(n-1)(n-2)/6.

In general, we can reconstruct W by rounding (W∨2)∧K to the nearest integer coefficients, where K is the JI point <1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. ~~Since (W-(W∨2)∧K)∧K = W∧K,~~ we have ~~that relative error, ||W∧K||, equals ||Y∧K|| where Y = (W∨2)∧K. Hence relative error for W can be bounded in terms of the coefficients of W∨2. Moreover,~~ ||W|| = ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we can be conducting a search.

In general, we can reconstruct W by rounding Y = (W∨2)∧K to the nearest integer coefficients, where K is the JI point <1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. Then we have ||W|| = ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we can be conducting a search. Moreover, we have from Y∧K = ((W∨2)∧K)∧K = 0 that relative error, which is ||W∧K||, is ||((W-Y) + Y)∧K|| = ||(W-Y)∧K||, hence relative error is also bounded by the fact that ||W-Y|| is bounded.

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Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if E ≤ 1/(C(n, 3)√lb(q)√lb(p)) then wedging K = &lt;1 lb(3) lb(5) ... lb(p)| with the val consisting of 0 followed by the first n-1 coefficients of the wedgie and rounding will give the wedgie. Here p and q are the largest and second largest primes in the prime limit, lb(x) is log base two, and C(n, 3) is n choose three, n(n-1)(n-2)/6.<br />

<br />

In general, we can reconstruct W by rounding (W∨2)∧K to the nearest integer coefficients, where K is the JI point &lt;1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. ~~Since~~ (W-~~(W∨2~~)~~∧K)∧K = W∧K~~, ~~we have~~ that ~~relative error~~, ||~~W∧K~~||, ~~equals~~ ||~~Y∧K~~|| ~~where Y = (W∨2)∧K. Hence relative error for W~~ can be bounded ~~in terms~~ of ~~the~~ coefficients of W∨2. Moreover, ~~||W|| =~~ ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we can be conducting a search.</body></html></pre></div>

In general, we can reconstruct W by rounding Y = (W∨2)∧K to the nearest integer coefficients, where K is the JI point &lt;1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. Then we have ||W|| <h1 id="toc3"><a name="x||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we can be conducting a search. Moreover, we have from Y∧K"></a> ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we can be conducting a search. Moreover, we have from Y∧K </h1>

((W∨2)∧K)∧K = 0 that relative error, which is ||W∧K||, is ||((W-Y) + Y)∧K|| = ||(W-Y)∧K||, hence relative error is also bounded by the fact that ||W-Y|| is bounded.</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>2012-01-08 17:52:22 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-08 18:22:21 UTC</tt>.<br>
-: The original revision id was <tt>290417595</tt>.<br>
+: The original revision id was <tt>290423187</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 33: / Line 33: @@
 Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if E ≤ 1/(C(n, 3)√lb(q)√lb(p)) then wedging K = &lt;1 lb(3) lb(5) ... lb(p)| with the val consisting of 0 followed by the first n-1 coefficients of the wedgie and rounding will give the wedgie. Here p and q are the largest and second largest primes in the prime limit, lb(x) is log base two, and C(n, 3) is n choose three, n(n-1)(n-2)/6.
-In general, we can reconstruct W by rounding (W∨2)∧K to the nearest integer coefficients, where K is the JI point &lt;1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. Since (W-(W∨2)∧K)∧K = W∧K, we have that relative error, ||W∧K||, equals ||Y∧K|| where Y = (W∨2)∧K. Hence relative error for W can be bounded in terms of the coefficients of W∨2. Moreover, ||W|| = ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we can be conducting a search.
+In general, we can reconstruct W by rounding Y = (W∨2)∧K to the nearest integer coefficients, where K is the JI point &lt;1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. Then we have ||W|| = ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we can be conducting a search. Moreover, we have from Y∧K = ((W∨2)∧K)∧K = 0 that relative error, which is ||W∧K||, is ||((W-Y) + Y)∧K|| = ||(W-Y)∧K||, hence relative error is also bounded by the fact that ||W-Y|| is bounded.
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 Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if E ≤ 1/(C(n, 3)√lb(q)√lb(p)) then wedging K = &amp;lt;1 lb(3) lb(5) ... lb(p)| with the val consisting of 0 followed by the first n-1 coefficients of the wedgie and rounding will give the wedgie. Here p and q are the largest and second largest primes in the prime limit, lb(x) is log base two, and C(n, 3) is n choose three, n(n-1)(n-2)/6.&lt;br /&gt;
 &lt;br /&gt;
-In general, we can reconstruct W by rounding (W∨2)∧K to the nearest integer coefficients, where K is the JI point &amp;lt;1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. Since (W-(W∨2)∧K)∧K = W∧K, we have that relative error, ||W∧K||, equals ||Y∧K|| where Y = (W∨2)∧K. Hence relative error for W can be bounded in terms of the coefficients of W∨2. Moreover, ||W|| = ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we can be conducting a search.&lt;/body&gt;&lt;/html&gt;</pre></div>
+In general, we can reconstruct W by rounding Y = (W∨2)∧K to the nearest integer coefficients, where K is the JI point &amp;lt;1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. Then we have ||W||  &lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="x||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we can be conducting a search. Moreover, we have from Y∧K"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt; ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we can be conducting a search. Moreover, we have from Y∧K &lt;/h1&gt;
+ ((W∨2)∧K)∧K = 0 that relative error, which is ||W∧K||, is ||((W-Y) + Y)∧K|| = ||(W-Y)∧K||, hence relative error is also bounded by the fact that ||W-Y|| is bounded.&lt;/body&gt;&lt;/html&gt;</pre></div>