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>2012-01-05 13:10:16 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-05 14:12:11 UTC</tt>.<br>

: The original revision id was <tt>~~289829061~~</tt>.<br>

: The original revision id was <tt>289850121</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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+(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 B^2

[[math]]

For this we can conclude that d, e and f satisfy |d - q3b + q5a| ≤ 2B√q3q5, |e - q3c + q7a| ≤ 2B√q3q7 and |f - q5c + q7d| ≤ 2B√q5q7. This has an interesting interpretation: since <1 q3 q5 q7|∧<0 a b c| = <<a b c q3b-q5a q3c-q7a q5c-q7b||, if B ≤ 1/(4√q5q7), then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with <1 q3 q5 q7| and rounding to the nearest integer. This is not a very serious constraint to place on simple badness; it seems unlikely anyone would be interested in a temperament which did not fall well under this low standard. Hence we may compile lists of reasonable temperaments by presuming "reasonable" requires this bound to be met, searching through triples <<a b d ...|| up to some complexity bound, wedging with <1 q3 q5 q7| and rounding, then checking if the GCD is one and the Pfaffian af-be+cd is zero. Then we may toss everthing which does not meet the bound on simple badness; however, for a reasonable list we will want a tighter bound.

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+(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 B^2&lt;br/&gt;[[math]]

--><script type="math/tex">\displaystyle (\frac{d}{q_3q_5}-\frac{b}{q_5}+\frac{a}{q_3})^2+(\frac{e}{q_3q_7}-\frac{c}{q_7}+\frac{a}{q_3})^2+(\frac{f}{q_5q_7}-\frac{c}{q_7}+\frac{b}{q_5})^2 \\

+(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 B^2</script></body></html></pre></div>

+(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 B^2</script><br />

<br />

For this we can conclude that d, e and f satisfy |d - q3b + q5a| ≤ 2B√q3q5, |e - q3c + q7a| ≤ 2B√q3q7 and |f - q5c + q7d| ≤ 2B√q5q7. This has an interesting interpretation: since &lt;1 q3 q5 q7|∧&lt;0 a b c| = &lt;&lt;a b c q3b-q5a q3c-q7a q5c-q7b||, if B ≤ 1/(4√q5q7), then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with &lt;1 q3 q5 q7| and rounding to the nearest integer. This is not a very serious constraint to place on simple badness; it seems unlikely anyone would be interested in a temperament which did not fall well under this low standard. Hence we may compile lists of reasonable temperaments by presuming &quot;reasonable&quot; requires this bound to be met, searching through triples &lt;&lt;a b d ...|| up to some complexity bound, wedging with &lt;1 q3 q5 q7| and rounding, then checking if the GCD is one and the Pfaffian af-be+cd is zero. Then we may toss everthing which does not meet the bound on simple badness; however, for a reasonable list we will want a tighter bound.</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>2012-01-05 13:10:16 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-05 14:12:11 UTC</tt>.<br>
-: The original revision id was <tt>289829061</tt>.<br>
+: The original revision id was <tt>289850121</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: @@
 +(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 B^2
 [[math]]
+For this we can conclude that d, e and f satisfy |d - q3b + q5a| ≤ 2B√q3q5, |e - q3c + q7a| ≤ 2B√q3q7 and |f - q5c + q7d| ≤ 2B√q5q7. This has an interesting interpretation: since &lt;1 q3 q5 q7|∧&lt;0 a b c| = &lt;&lt;a  b  c  q3b-q5a  q3c-q7a  q5c-q7b||, if B ≤ 1/(4√q5q7), then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with &lt;1 q3 q5 q7| and rounding to the nearest integer. This is not a very serious constraint to place on simple badness; it seems unlikely anyone would be interested in a temperament which did not fall well under this low standard. Hence we may compile lists of reasonable temperaments by presuming "reasonable" requires this bound to be met, searching through triples &lt;&lt;a b d ...|| up to some complexity bound, wedging with &lt;1 q3 q5 q7| and rounding, then checking if the GCD is one and the Pfaffian af-be+cd is zero. Then we may toss everthing which does not meet the bound on simple badness; however, for a reasonable list we will want a tighter bound.
@@ Line 52: / Line 54: @@
 +(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 B^2&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\displaystyle (\frac{d}{q_3q_5}-\frac{b}{q_5}+\frac{a}{q_3})^2+(\frac{e}{q_3q_7}-\frac{c}{q_7}+\frac{a}{q_3})^2+(\frac{f}{q_5q_7}-\frac{c}{q_7}+\frac{b}{q_5})^2 \\
-+(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 B^2&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
++(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 B^2&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
+&lt;br /&gt;
+For this we can conclude that d, e and f satisfy |d - q3b + q5a| ≤ 2B√q3q5, |e - q3c + q7a| ≤ 2B√q3q7 and |f - q5c + q7d| ≤ 2B√q5q7. This has an interesting interpretation: since &amp;lt;1 q3 q5 q7|∧&amp;lt;0 a b c| = &amp;lt;&amp;lt;a  b  c  q3b-q5a  q3c-q7a  q5c-q7b||, if B ≤ 1/(4√q5q7), then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with &amp;lt;1 q3 q5 q7| and rounding to the nearest integer. This is not a very serious constraint to place on simple badness; it seems unlikely anyone would be interested in a temperament which did not fall well under this low standard. Hence we may compile lists of reasonable temperaments by presuming &amp;quot;reasonable&amp;quot; requires this bound to be met, searching through triples &amp;lt;&amp;lt;a b d ...|| up to some complexity bound, wedging with &amp;lt;1 q3 q5 q7| and rounding, then checking if the GCD is one and the Pfaffian af-be+cd is zero. Then we may toss everthing which does not meet the bound on simple badness; however, for a reasonable list we will want a tighter bound.&lt;/body&gt;&lt;/html&gt;</pre></div>