Harmonic entropy: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 515669366 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 515669392 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-06 13: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-06 13:43:26 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>515669392</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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<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;white-space: pre-wrap ! important" class="old-revision-html">[[math]] | <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;white-space: pre-wrap ! important" class="old-revision-html">[[math]] | ||
\newcommand{\cent}{{\mathrm{c}\mkern-6.5mu{\mid}}} | \newcommand{\cent}{{\mathrm{c}\mkern-6.5mu{\mid}}} | ||
[[math]] | [[math]] | ||
=Introduction= | =Introduction= | ||
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[[math]] | [[math]] | ||
p_d(b) = \int_{\ | p_d(b) = \int_{\cent(b_l)}^{\cent(b_u)} s_d(x) dx | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
q_d(b) = \frac{s_d(\ | q_d(b) = \frac{s_d(\cent(b))}{\sqrt{n_b \cdot d_b}} | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
q_d(b) = \frac{s_d(\ | q_d(b) = \frac{s_d(\cent(b))}{\max(n_b,d_b)} | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
q_d(b) = \frac{s_d(\ | q_d(b) = \frac{s_d(\cent(b))}{\|b\|} | ||
[[math]] | [[math]] | ||
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<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>Harmonic Entropy</title></head><body><!-- ws:start:WikiTextMathRule:0: | <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>Harmonic Entropy</title></head><body><!-- ws:start:WikiTextMathRule:0: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\newcommand{\cent}{{\mathrm{c}\mkern-6.5mu{\mid}}} | \newcommand{\cent}{{\mathrm{c}\mkern-6.5mu{\mid}}}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\newcommand{\cent}{{\mathrm{c}\mkern-6.5mu{\mid}}}</script><!-- ws:end:WikiTextMathRule:0 --><br /> | |||
--><script type="math/tex">\newcommand{\cent}{{\mathrm{c}\mkern-6.5mu{\mid}}} | |||
<!-- ws:start:WikiTextHeadingRule:13:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:13 -->Introduction</h1> | <!-- ws:start:WikiTextHeadingRule:13:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:13 -->Introduction</h1> | ||
<!-- ws:start:WikiTextTocRule:35:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 1em;"><a href="#Introduction">Introduction</a></div> | <!-- ws:start:WikiTextTocRule:35:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 1em;"><a href="#Introduction">Introduction</a></div> | ||
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<!-- ws:start:WikiTextMathRule:3: | <!-- ws:start:WikiTextMathRule:3: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
p_d(b) = \int_{\ | p_d(b) = \int_{\cent(b_l)}^{\cent(b_u)} s_d(x) dx&lt;br/&gt;[[math]] | ||
--><script type="math/tex">p_d(b) = \int_{\ | --><script type="math/tex">p_d(b) = \int_{\cent(b_l)}^{\cent(b_u)} s_d(x) dx</script><!-- ws:end:WikiTextMathRule:3 --><br /> | ||
<br /> | <br /> | ||
where s<span style="vertical-align: sub;">d</span>(x) is the spreading function associated with d<span style="line-height: 1.5;">, b</span><span style="line-height: 1.5; vertical-align: sub;">l</span><span style="line-height: 1.5;"> and b</span><span style="line-height: 1.5; vertical-align: sub;">u</span><span style="line-height: 1.5;"> are the domain lower and upper bounds associated with basis rational b, and ¢(f) = 1200·log2(f), or the &quot;cents&quot; function converting frequency ratios to cents. Normally, b</span><span style="line-height: 1.5; vertical-align: sub;">l </span><span style="line-height: 1.5;">is set equal to the mediant of b and its nearest lower neighbor (if it exists), or -∞ if not; likewise with b</span><span style="line-height: 1.5; vertical-align: sub;">u</span><span style="line-height: 1.5;">.</span><br /> | where s<span style="vertical-align: sub;">d</span>(x) is the spreading function associated with d<span style="line-height: 1.5;">, b</span><span style="line-height: 1.5; vertical-align: sub;">l</span><span style="line-height: 1.5;"> and b</span><span style="line-height: 1.5; vertical-align: sub;">u</span><span style="line-height: 1.5;"> are the domain lower and upper bounds associated with basis rational b, and ¢(f) = 1200·log2(f), or the &quot;cents&quot; function converting frequency ratios to cents. Normally, b</span><span style="line-height: 1.5; vertical-align: sub;">l </span><span style="line-height: 1.5;">is set equal to the mediant of b and its nearest lower neighbor (if it exists), or -∞ if not; likewise with b</span><span style="line-height: 1.5; vertical-align: sub;">u</span><span style="line-height: 1.5;">.</span><br /> | ||
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<!-- ws:start:WikiTextMathRule:4: | <!-- ws:start:WikiTextMathRule:4: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
q_d(b) = \frac{s_d(\ | q_d(b) = \frac{s_d(\cent(b))}{\sqrt{n_b \cdot d_b}}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">q_d(b) = \frac{s_d(\ | --><script type="math/tex">q_d(b) = \frac{s_d(\cent(b))}{\sqrt{n_b \cdot d_b}}</script><!-- ws:end:WikiTextMathRule:4 --><br /> | ||
<br /> | <br /> | ||
where the q<span style="font-size: 12px; vertical-align: sub;">d</span>(b) now represent the unnormalized &quot;probabilities&quot;, and n<span style="vertical-align: sub;">b</span> and d<span style="vertical-align: sub;">b</span> are the numerator and denominator, respectively, of basis rational b. Again, the set of basis rationals here is assumed to be all of those rationals of Tenney Height ≤ N for some N.<br /> | where the q<span style="font-size: 12px; vertical-align: sub;">d</span>(b) now represent the unnormalized &quot;probabilities&quot;, and n<span style="vertical-align: sub;">b</span> and d<span style="vertical-align: sub;">b</span> are the numerator and denominator, respectively, of basis rational b. Again, the set of basis rationals here is assumed to be all of those rationals of Tenney Height ≤ N for some N.<br /> | ||
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<!-- ws:start:WikiTextMathRule:5: | <!-- ws:start:WikiTextMathRule:5: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
q_d(b) = \frac{s_d(\ | q_d(b) = \frac{s_d(\cent(b))}{\max(n_b,d_b)}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">q_d(b) = \frac{s_d(\ | --><script type="math/tex">q_d(b) = \frac{s_d(\cent(b))}{\max(n_b,d_b)}</script><!-- ws:end:WikiTextMathRule:5 --><br /> | ||
<br /> | <br /> | ||
where this time the set of basis rationals is assumed to be all of those of Weil Height ≤ N for some N.<br /> | where this time the set of basis rationals is assumed to be all of those of Weil Height ≤ N for some N.<br /> | ||
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<!-- ws:start:WikiTextMathRule:6: | <!-- ws:start:WikiTextMathRule:6: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
q_d(b) = \frac{s_d(\ | q_d(b) = \frac{s_d(\cent(b))}{\|b\|}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">q_d(b) = \frac{s_d(\ | --><script type="math/tex">q_d(b) = \frac{s_d(\cent(b))}{\|b\|}</script><!-- ws:end:WikiTextMathRule:6 --><br /> | ||
<br /> | <br /> | ||
where ||b|| denotes a complexity function mapping from rational numbers to reals.<br /> | where ||b|| denotes a complexity function mapping from rational numbers to reals.<br /> | ||