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 14:21:03 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-05 20:20:03 UTC</tt>.<br>

: The original revision id was <tt>~~289853361~~</tt>.<br>

: The original revision id was <tt>289930517</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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=Constrained wedgies=

Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] B, aka simple badness, constrains a 7-limit rank two wedgie W = <<a b c d e f||.

Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] E, aka simple badness, constrains a 7-limit rank two wedgie W = <<a b c d e f||.

By definition, B = ||J∧Z||, where Z is the weighted version of W; if q3, q5 and q7 are the logarithms base two of 3, 5, and 7, then Z = <<a/q3 b/q5 c/q7 d/(q3q5) e/(q3q7) f/(q5q7)||. From this we may conclude that

By definition, E = ||J∧Z||, where Z is the weighted version of W; if q3, q5 and q7 are the logarithms base two of 3, 5, and 7, then Z = <<a/q3 b/q5 c/q7 d/(q3q5) e/(q3q7) f/(q5q7)||. From this we may conclude that

[[math]]

\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

+(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 E^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 relative error; 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 relative error; however, for a reasonable list we will want a tighter bound.

For this we can conclude that d, e and f satisfy |d - q3b + q5a| ≤ 2E√q3q5, |e - q3c + q7a| ≤ 2E√q3q7 and |f - q5c + q7d| ≤ 2E√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 E ≤ 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 relative error; 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 ...|| (note that if all of these are zero, 2 is being tempered out) 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 relative error; however, for a reasonable list we will want a tighter bound.

If C = ||W|| is the TE complexity, then the formula for the [[Tenney-Euclidean+metrics#Logflat TE badness|logflat badness]] B in the 7-limit rank-two case is particularly simple: B = CE.

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<br />

<h1 id="toc2"><a name="Constrained wedgies"></a>Constrained wedgies</h1>

Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness">relative error</a> B, aka simple badness, constrains a 7-limit rank two wedgie W = &lt;&lt;a b c d e f||.<br />

Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness">relative error</a> E, aka simple badness, constrains a 7-limit rank two wedgie W = &lt;&lt;a b c d e f||.<br />

<br />

By definition, B = ||J∧Z||, where Z is the weighted version of W; if q3, q5 and q7 are the logarithms base two of 3, 5, and 7, then Z = &lt;&lt;a/q3 b/q5 c/q7 d/(q3q5) e/(q3q7) f/(q5q7)||. From this we may conclude that<br />

By definition, E = ||J∧Z||, where Z is the weighted version of W; if q3, q5 and q7 are the logarithms base two of 3, 5, and 7, then Z = &lt;&lt;a/q3 b/q5 c/q7 d/(q3q5) e/(q3q7) f/(q5q7)||. From this we may conclude that<br />

<br />

<!-- ws:start:WikiTextMathRule:0:

[[math]]&lt;br/&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 \\&lt;br /&gt;

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

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

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

<br />

For this we can conclude that d, e and f satisfy |d - q3b + q5a| ≤ 2E√q3q5, |e - q3c + q7a| ≤ 2E√q3q7 and |f - q5c + q7d| ≤ 2E√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 E ≤ 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 relative error; 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 ...|| (note that if all of these are zero, 2 is being tempered out) 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 relative error; however, for a reasonable list we will want a tighter bound. <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 relative error; 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 relative error; however, for a reasonable list we will want a tighter bound~~.</body></html></pre></div>

If C = ||W|| is the TE complexity, then the formula for the [[Tenney-Euclidean+metrics#Logflat TE badness|logflat badness]] B in the 7-limit rank-two case is particularly simple: B = CE.</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 14:21:03 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-05 20:20:03 UTC</tt>.<br>
-: The original revision id was <tt>289853361</tt>.<br>
+: The original revision id was <tt>289930517</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 18: / Line 18: @@
 =Constrained wedgies=
-Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] B, aka simple badness, constrains a 7-limit rank two wedgie W = &lt;&lt;a b c d e f||.
+Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] E, aka simple badness, constrains a 7-limit rank two wedgie W = &lt;&lt;a b c d e f||.
-By definition, B = ||J∧Z||, where Z is the weighted version of W; if q3, q5 and q7 are the logarithms base two of 3, 5, and 7, then Z = &lt;&lt;a/q3 b/q5 c/q7 d/(q3q5) e/(q3q7) f/(q5q7)||. From this we may conclude that
+By definition, E = ||J∧Z||, where Z is the weighted version of W; if q3, q5 and q7 are the logarithms base two of 3, 5, and 7, then Z = &lt;&lt;a/q3 b/q5 c/q7 d/(q3q5) e/(q3q7) f/(q5q7)||. From this we may conclude that
 [[math]]
 \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
++(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 E^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 relative error; 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 relative error; however, for a reasonable list we will want a tighter bound.
+For this we can conclude that d, e and f satisfy |d - q3b + q5a| ≤ 2E√q3q5, |e - q3c + q7a| ≤ 2E√q3q7 and |f - q5c + q7d| ≤ 2E√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 E ≤ 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 relative error; 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 ...|| (note that if all of these are zero, 2 is being tempered out) 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 relative error; however, for a reasonable list we will want a tighter bound.
+If C = ||W|| is the TE complexity, then the formula for the [[Tenney-Euclidean+metrics#Logflat TE badness|logflat badness]] B in the 7-limit rank-two case is particularly simple: B = CE.
@@ Line 45: / Line 47: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Constrained wedgies"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;Constrained wedgies&lt;/h1&gt;
-Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding &lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&gt;relative error&lt;/a&gt; B, aka simple badness, constrains a 7-limit rank two wedgie W = &amp;lt;&amp;lt;a b c d e f||.&lt;br /&gt;
+Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding &lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&gt;relative error&lt;/a&gt; E, aka simple badness, constrains a 7-limit rank two wedgie W = &amp;lt;&amp;lt;a b c d e f||.&lt;br /&gt;
 &lt;br /&gt;
-By definition, B = ||J∧Z||, where Z is the weighted version of W; if q3, q5 and q7 are the logarithms base two of 3, 5, and 7, then Z = &amp;lt;&amp;lt;a/q3 b/q5 c/q7 d/(q3q5) e/(q3q7) f/(q5q7)||. From this we may conclude that&lt;br /&gt;
+By definition, E = ||J∧Z||, where Z is the weighted version of W; if q3, q5 and q7 are the logarithms base two of 3, 5, and 7, then Z = &amp;lt;&amp;lt;a/q3 b/q5 c/q7 d/(q3q5) e/(q3q7) f/(q5q7)||. From this we may conclude that&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:0:
 [[math]]&amp;lt;br/&amp;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 \\&amp;lt;br /&amp;gt;
-+(\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]]
++(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 E^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;br /&gt;
++(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 E^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| ≤ 2E√q3q5, |e - q3c + q7a| ≤ 2E√q3q7 and |f - q5c + q7d| ≤ 2E√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 E ≤ 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 relative error; 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 ...|| (note that if all of these are zero, 2 is being tempered out) 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 relative error; however, for a reasonable list we will want a tighter bound. &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 relative error; 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 relative error; however, for a reasonable list we will want a tighter bound.&lt;/body&gt;&lt;/html&gt;</pre></div>
+If C = ||W|| is the TE complexity, then the formula for the [[Tenney-Euclidean+metrics#Logflat TE badness|logflat badness]] B in the 7-limit rank-two case is particularly simple: B = CE.&lt;/body&gt;&lt;/html&gt;</pre></div>