Recoverability: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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=The rank one case=

In the case of rank one, if W is an N-edo val, ie whose first coefficient is N. then W∨2 = N and (W∨2)∧J = NJ, so that rounding it gives the patent val for N. Hence, W is recoverable if and only if it is a patent val. However, ]W∧J[ < 600 cents is now a stringent condition for p>5, especially in higher prime limits. In the 5-limit we have 4, 5, 7, 8, 12, 15, 19, 22, 23, 26, 27, 31, 34 ... . The 7-limit grows somewhat restrictive: 19, 27, 31, 41, 49, 60, 68, 72, 80, 91, 99... . In the 11-limit we have 49, 72, 103, 239, 270, 342, 391, 414, 445, 494, 552, 612... . The 13-limit is already quite restrictive: 552, 954, 1133, 1236, 1506..., and the 17-limit starts off 4452, 5527, 7033, 7315, 9896... .

If W is the patent val for N-edo and ]W∧J[ > 600 cents, it can happen that the minimum value for recoverability relative error/simple badness, defined as ]V∧J[ for any N-edo val V, does not occur for V = W. Examples are 13-limit 12f, where ]12f∧J[ = 1217.949 is much smaller than ]12∧J[ = 3844.172, 5-limit 17c, ]17c∧J[ = 847.730 compared to ]17∧J[ = 1054.225, and 11-limit 27e, where ]27e∧J[ = 1169.472 is less than ]27∧J[ = 2405.855. Of course two possibilities can be close enough that both are plausible, as with 7-limit 34d at 1169.472 cents versus 34 at 1437.444 cents.

=Complete searches for temperaments=

By a complete search for regular temperaments is meant a search which is guaranteed to find all temperaments meeting certain specified conditions. Recoverability conditions provide one approach to these. The first segment of W∨2 consists of C(n-1, r-1) zeros, and the second segment of C(n-1, r) integers identical to the initial, 2 containing, segment of W. By beginning with such a multivector of integer coefficients, wedging with J, and rounding, we obtain a multivector which is a candidate for a p-limit rank r wedgie, defining a regular temperament. It will not in general be a wedgie, but all recoverable wedgies can be obtained in this way. Hence all that remains to do, as discussed in [[The wedgie]], is to test if the multivector in question is actually a wedgie, and also if it passes any further conditions on complexity, error of badness we wish to place on our list of wedgies.

By a complete search for regular temperaments is meant a search which is guaranteed to find all temperaments meeting certain specified conditions. Recoverability conditions provide one approach to these. The first segment of W∨2 consists of C(n-1, r-1) zeros, and the second segment of C(n-1, r) integers identical to the initial, 2 containing, segment of W. By beginning with such a multivector of integer coefficients, wedging with J, and rounding, we obtain a multivector which is a candidate for a p-limit rank r wedgie, defining a regular temperament. It will not in general be a wedgie, but all recoverable wedgies can be obtained in this way. Hence all that remains to do, as discussed in [[The wedgie]], is to test if the multivector in question is actually a wedgie, and also if it passes any further conditions on complexity, error of badness we wish to place on our list of wedgies.</pre></div>

</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Recoverability</title></head><body><a href="#Definition">Definition</a> | <a href="#Segments">Segments</a> | <a href="#The recoverability norm">The recoverability norm</a> | <a href="#The rank one case">The rank one case</a> | <a href="#Complete searches for temperaments">Complete searches for temperaments</a>

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<h1 id="toc3"><a name="The rank one case"></a>The rank one case</h1>

In the case of rank one, if W is an N-edo val, ie whose first coefficient is N. then W∨2 = N and (W∨2)∧J = NJ, so that rounding it gives the patent val for N. Hence, W is recoverable if and only if it is a patent val. However, ]W∧J[ &lt; 600 cents is now a stringent condition for p&gt;5, especially in higher prime limits. In the 5-limit we have 4, 5, 7, 8, 12, 15, 19, 22, 23, 26, 27, 31, 34 ... . The 7-limit grows somewhat restrictive: 19, 27, 31, 41, 49, 60, 68, 72, 80, 91, 99... . In the 11-limit we have 49, 72, 103, 239, 270, 342, 391, 414, 445, 494, 552, 612... . The 13-limit is already quite restrictive: 552, 954, 1133, 1236, 1506..., and the 17-limit starts off 4452, 5527, 7033, 7315, 9896... .<br />

<br />

If W is the patent val for N-edo and ]W∧J[ &gt; 600 cents, it can happen that the minimum value for recoverability relative error/simple badness, defined as ]V∧J[ for any N-edo val V, does not occur for V = W. Examples are 13-limit 12f, where ]12f∧J[ = 1217.949 is much smaller than ]12∧J[ = 3844.172, 5-limit 17c, ]17c∧J[ = 847.730 compared to ]17∧J[ = 1054.225, and 11-limit 27e, where ]27e∧J[ = 1169.472 is less than ]27∧J[ = 2405.855. Of course two possibilities can be close enough that both are plausible, as with 7-limit 34d at 1169.472 cents versus 34 at 1437.444 cents.<br />

<br />

<h1 id="toc4"><a name="Complete searches for temperaments"></a>Complete searches for temperaments</h1>

By a complete search for regular temperaments is meant a search which is guaranteed to find all temperaments meeting certain specified conditions. Recoverability conditions provide one approach to these. The first segment of W∨2 consists of C(n-1, r-1) zeros, and the second segment of C(n-1, r) integers identical to the initial, 2 containing, segment of W. By beginning with such a multivector of integer coefficients, wedging with J, and rounding, we obtain a multivector which is a candidate for a p-limit rank r wedgie, defining a regular temperament. It will not in general be a wedgie, but all recoverable wedgies can be obtained in this way. Hence all that remains to do, as discussed in <a class="wiki_link" href="/The%20wedgie">The wedgie</a>, is to test if the multivector in question is actually a wedgie, and also if it passes any further conditions on complexity, error of badness we wish to place on our list of wedgies.</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>2015-02-07 12:30:05 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-02-07 12:58:38 UTC</tt>.<br>
-: The original revision id was <tt>540092618</tt>.<br>
+: The original revision id was <tt>540094842</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 19: / Line 19: @@
 =The rank one case=
 In the case of rank one, if W is an N-edo val, ie whose first coefficient is N. then W∨2 = N and (W∨2)∧J = NJ, so that rounding it gives the patent val for N. Hence, W is recoverable if and only if it is a patent val. However, ]W∧J[ &lt; 600 cents is now a stringent condition for p&gt;5, especially in higher prime limits. In the 5-limit we have 4, 5, 7, 8, 12, 15, 19, 22, 23, 26, 27, 31, 34 ... . The 7-limit grows somewhat restrictive: 19, 27, 31, 41, 49, 60, 68, 72, 80, 91, 99... . In the 11-limit we have 49, 72, 103, 239, 270, 342, 391, 414, 445, 494, 552, 612... . The 13-limit is already quite restrictive: 552, 954, 1133, 1236, 1506..., and the 17-limit starts off 4452, 5527, 7033, 7315, 9896... .
+If W is the patent val for N-edo and ]W∧J[  &gt; 600 cents, it can happen that the minimum value for recoverability relative error/simple badness, defined as ]V∧J[ for any N-edo val V, does not occur for V = W. Examples are 13-limit 12f, where ]12f∧J[ = 1217.949 is much smaller than ]12∧J[ = 3844.172, 5-limit 17c, ]17c∧J[ = 847.730 compared to ]17∧J[ = 1054.225, and 11-limit 27e, where ]27e∧J[ = 1169.472 is less than ]27∧J[ = 2405.855. Of course two possibilities can be close enough that both are plausible, as with 7-limit 34d at 1169.472 cents versus 34 at 1437.444 cents.
 =Complete searches for temperaments=
-By a complete search for regular temperaments is meant a search which is guaranteed to find all temperaments meeting certain specified conditions. Recoverability conditions provide one approach to these. The first segment of W∨2 consists of C(n-1, r-1) zeros, and the second segment of C(n-1, r) integers identical to the initial, 2 containing, segment of W. By beginning with such a multivector of integer coefficients, wedging with J, and rounding, we obtain a multivector which is a candidate for a p-limit rank r wedgie, defining a regular temperament. It will not in general be a wedgie, but all recoverable wedgies can be obtained in this way. Hence all that remains to do, as discussed in [[The wedgie]], is to test if the multivector in question is actually a wedgie, and also if it passes any further conditions on complexity, error of badness we wish to place on our list of wedgies.
+By a complete search for regular temperaments is meant a search which is guaranteed to find all temperaments meeting certain specified conditions. Recoverability conditions provide one approach to these. The first segment of W∨2 consists of C(n-1, r-1) zeros, and the second segment of C(n-1, r) integers identical to the initial, 2 containing, segment of W. By beginning with such a multivector of integer coefficients, wedging with J, and rounding, we obtain a multivector which is a candidate for a p-limit rank r wedgie, defining a regular temperament. It will not in general be a wedgie, but all recoverable wedgies can be obtained in this way. Hence all that remains to do, as discussed in [[The wedgie]], is to test if the multivector in question is actually a wedgie, and also if it passes any further conditions on complexity, error of badness we wish to place on our list of wedgies.</pre></div>
-</pre></div>
 <h4>Original HTML content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Recoverability&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:10:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#Segments"&gt;Segments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt; | &lt;a href="#The recoverability norm"&gt;The recoverability norm&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#The rank one case"&gt;The rank one case&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt; | &lt;a href="#Complete searches for temperaments"&gt;Complete searches for temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;
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 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="The rank one case"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;The rank one case&lt;/h1&gt;
 In the case of rank one, if W is an N-edo val, ie whose first coefficient is N. then W∨2 = N and (W∨2)∧J = NJ, so that rounding it gives the patent val for N. Hence, W is recoverable if and only if it is a patent val. However, ]W∧J[ &amp;lt; 600 cents is now a stringent condition for p&amp;gt;5, especially in higher prime limits. In the 5-limit we have 4, 5, 7, 8, 12, 15, 19, 22, 23, 26, 27, 31, 34 ... . The 7-limit grows somewhat restrictive: 19, 27, 31, 41, 49, 60, 68, 72, 80, 91, 99... . In the 11-limit we have 49, 72, 103, 239, 270, 342, 391, 414, 445, 494, 552, 612... . The 13-limit is already quite restrictive: 552, 954, 1133, 1236, 1506..., and the 17-limit starts off 4452, 5527, 7033, 7315, 9896... .&lt;br /&gt;
+&lt;br /&gt;
+If W is the patent val for N-edo and ]W∧J[  &amp;gt; 600 cents, it can happen that the minimum value for recoverability relative error/simple badness, defined as ]V∧J[ for any N-edo val V, does not occur for V = W. Examples are 13-limit 12f, where ]12f∧J[ = 1217.949 is much smaller than ]12∧J[ = 3844.172, 5-limit 17c, ]17c∧J[ = 847.730 compared to ]17∧J[ = 1054.225, and 11-limit 27e, where ]27e∧J[ = 1169.472 is less than ]27∧J[ = 2405.855. Of course two possibilities can be close enough that both are plausible, as with 7-limit 34d at 1169.472 cents versus 34 at 1437.444 cents.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Complete searches for temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Complete searches for temperaments&lt;/h1&gt;
 By a complete search for regular temperaments is meant a search which is guaranteed to find all temperaments meeting certain specified conditions. Recoverability conditions provide one approach to these. The first segment of W∨2 consists of C(n-1, r-1) zeros, and the second segment of C(n-1, r) integers identical to the initial, 2 containing, segment of W. By beginning with such a multivector of integer coefficients, wedging with J, and rounding, we obtain a multivector which is a candidate for a p-limit rank r wedgie, defining a regular temperament. It will not in general be a wedgie, but all recoverable wedgies can be obtained in this way. Hence all that remains to do, as discussed in &lt;a class="wiki_link" href="/The%20wedgie"&gt;The wedgie&lt;/a&gt;, is to test if the multivector in question is actually a wedgie, and also if it passes any further conditions on complexity, error of badness we wish to place on our list of wedgies.&lt;/body&gt;&lt;/html&gt;</pre></div>