Harmonic entropy: Difference between revisions

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:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-06 13:51:32 UTC</tt>.<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-06 13:51:35 UTC</tt>.<br>

: The original revision id was <tt>~~515669696~~</tt>.<br>

: The original revision id was <tt>515669702</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>

<h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">[[math]]

\newcommand{\cent}{\text{¢}}

[[math]]

=Introduction=

Line 38:

[[math]]

H(d) = -\sum_{b} p_d(b) \log_β p_d(b)

[[math]]

Line 52:

[[math]]

s_d(x) = \frac{1}{s\sqrt{2\pi}} e^{-\frac{(x-d)^2}{2s^2}}

[[math]]

Line 65:

[[math]]

p_d(b) = \int_{\cent(b_l)}^{\cent(b_u)} s_d(x) dx

[[math]]

Line 81:

[[math]]

q_d(b) = \frac{s_d(\cent(b))}{\sqrt{n_b \cdot d_b}}

[[math]]

Line 89:

[[math]]

q_d(b) = \frac{s_d(\cent(b))}{\max(n_b,d_b)}

[[math]]

Line 97:

[[math]]

q_d(b) = \frac{s_d(\cent(b))}{\|b\|}

[[math]]

Line 105:

[[math]]

p_d(b) = \frac{q_d(b)}{\sum_b q_d(b)}

[[math]]

Line 128:

[[math]]

H(d) = \frac{1}{1-a} \log_β \sum_b p_d(b)^a

[[math]]

Line 143:

===a=0: Harmonic Hartley Entropy===

[[math]]

H(d) = \log |R|

[[math]]

where |R| is the cardinality of the set of basis rationals. This assumes, in essence, an "infinitely dumb" auditory system which can do no better than picking a rational number from a uniform distribution completely at random. All dyads have the same Harmonic Hartley Entropy.

Line 149:

===a=1: Harmonic Shannon Entropy (Harmonic Entropy)===

[[math]]

H(d) = -\sum_{b} p_d(b) \log p_d(b)

[[math]]

This is Paul's original Harmonic Entropy. This can be thought of as an auditory system which simply selects a rational at random from the incoming distribution, weighted via the distribution itself.

Line 155:

===a=2: Harmonic Collision Entropy===

[[math]]

H(d) = -\log \sum_b p_d(b)^2 = -log (P_d = Q_d)

[[math]]

where P<span style="font-size: 90%; vertical-align: sub;">d</span> and Q<span style="font-size: 90%; vertical-align: sub;">d</span> are independent and identically distributed random variables corresponding to the same dyad, and the collision entropy is the same as the negative log of the probability that the two variables produce the same outcome.

Line 161:

===a=∞: Harmonic Min-Entropy===

[[math]]

-\log \max_b p_d(b)

[[math]]

This is the min-entropy, which simply takes the negative log of the largest probability in the distribution. This can be thought of as representing the "strength" of the incoming dyad from being "deciphered" by a "best-case" auditory system.

Line 174:

<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;

\newcommand{\cent}{\text{¢}}&lt;br/&gt;[[math]]

--><script type="math/tex">\newcommand{\cent}{\text{¢}}</script><br />

--><script type="math/tex"> \newcommand{\cent}{\text{¢}}</script><br />

<h1 id="toc0"><a name="Introduction"></a>Introduction</h1>

<div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#Introduction">Introduction</a></div>

Line 211:

<!-- ws:start:WikiTextMathRule:1:

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

H(d) = -\sum_{b} p_d(b) \log_β p_d(b)&lt;br/&gt;[[math]]

--><script type="math/tex">H(d) = -\sum_{b} p_d(b) \log_β p_d(b)</script><br />

--><script type="math/tex"> H(d) = -\sum_{b} p_d(b) \log_β p_d(b)</script><br />

<br />

where H(d) is the Shannon entropy of the dyad d, the b are all of the basis rationals in the set, the p<span style="font-size: 90%; vertical-align: sub;">d</span>(b) is the probability assigned to basis rational b given an input dyad of d, and the logarithm β reflects the units of information being used (by convention, we set β=2, corresponding to the use of bits). This is the Harmonic Entropy of the dyad d.<br />

Line 224:

<!-- ws:start:WikiTextMathRule:2:

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

s_d(x) = \frac{1}{s\sqrt{2\pi}} e^{-\frac{(x-d)^2}{2s^2}}&lt;br/&gt;[[math]]

--><script type="math/tex">s_d(x) = \frac{1}{s\sqrt{2\pi}} e^{-\frac{(x-d)^2}{2s^2}}</script><br />

--><script type="math/tex"> s_d(x) = \frac{1}{s\sqrt{2\pi}} e^{-\frac{(x-d)^2}{2s^2}}</script><br />

<br />

where <em>s</em> becomes the standard deviation of the Gaussian, being an ASCII-friendly version of the more familiar symbol σ for representing the standard deviation.<br />

Line 238:

<!-- ws:start:WikiTextMathRule:3:

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

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_{\cent(b_l)}^{\cent(b_u)} s_d(x) dx</script><br />

--><script type="math/tex"> p_d(b) = \int_{\cent(b_l)}^{\cent(b_u)} s_d(x) dx</script><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 />

Line 255:

<!-- ws:start:WikiTextMathRule:4:

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

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(\cent(b))}{\sqrt{n_b \cdot d_b}}</script><br />

--><script type="math/tex"> q_d(b) = \frac{s_d(\cent(b))}{\sqrt{n_b \cdot d_b}}</script><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 />

Line 264:

<!-- ws:start:WikiTextMathRule:5:

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

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(\cent(b))}{\max(n_b,d_b)}</script><br />

--><script type="math/tex"> q_d(b) = \frac{s_d(\cent(b))}{\max(n_b,d_b)}</script><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 />

Line 273:

<!-- ws:start:WikiTextMathRule:6:

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

q_d(b) = \frac{s_d(\cent(b))}{\|b\|}&lt;br/&gt;[[math]]

--><script type="math/tex">q_d(b) = \frac{s_d(\cent(b))}{\|b\|}</script><br />

--><script type="math/tex"> q_d(b) = \frac{s_d(\cent(b))}{\|b\|}</script><br />

<br />

where ||b|| denotes a complexity function mapping from rational numbers to reals.<br />

Line 282:

<!-- ws:start:WikiTextMathRule:7:

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

p_d(b) = \frac{q_d(b)}{\sum_b q_d(b)}&lt;br/&gt;[[math]]

--><script type="math/tex">p_d(b) = \frac{q_d(b)}{\sum_b q_d(b)}</script><br />

--><script type="math/tex"> p_d(b) = \frac{q_d(b)}{\sum_b q_d(b)}</script><br />

<br />

The p<span style="vertical-align: sub;">d</span>(b) are then used directly to compute the entropy.<br />

Line 306:

<!-- ws:start:WikiTextMathRule:8:

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

H(d) = \frac{1}{1-a} \log_β \sum_b p_d(b)^a&lt;br/&gt;[[math]]

--><script type="math/tex">H(d) = \frac{1}{1-a} \log_β \sum_b p_d(b)^a</script><br />

--><script type="math/tex"> H(d) = \frac{1}{1-a} \log_β \sum_b p_d(b)^a</script><br />

<br />

where the p_d(b) are the probabilities assigned by dyad d to each basis rational b. Being a q-analog, it is noteworthy that Rényi entropy converges to Shannon entropy in the limit as a→1, a fact which can be verified using L'Hôpital's rule as found <a class="wiki_link_ext" href="http://www.sonycsl.co.jp/person/nielsen/Note-HopitalRuleShannonRenyiTsallis.pdf" rel="nofollow" target="_blank">here</a>.<br />

Line 322:

<!-- ws:start:WikiTextMathRule:9:

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

H(d) = \log |R|&lt;br/&gt;[[math]]

--><script type="math/tex">H(d) = \log |R|</script><br />

--><script type="math/tex"> H(d) = \log |R|</script><br />

where |R| is the cardinality of the set of basis rationals. This assumes, in essence, an &quot;infinitely dumb&quot; auditory system which can do no better than picking a rational number from a uniform distribution completely at random. All dyads have the same Harmonic Hartley Entropy.<br />

<br />

Line 329:

<!-- ws:start:WikiTextMathRule:10:

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

H(d) = -\sum_{b} p_d(b) \log p_d(b)&lt;br/&gt;[[math]]

--><script type="math/tex">H(d) = -\sum_{b} p_d(b) \log p_d(b)</script><br />

--><script type="math/tex"> H(d) = -\sum_{b} p_d(b) \log p_d(b)</script><br />

This is Paul's original Harmonic Entropy. This can be thought of as an auditory system which simply selects a rational at random from the incoming distribution, weighted via the distribution itself.<br />

<br />

Line 336:

<!-- ws:start:WikiTextMathRule:11:

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

H(d) = -\log \sum_b p_d(b)^2 = -log (P_d = Q_d)&lt;br/&gt;[[math]]

--><script type="math/tex">H(d) = -\log \sum_b p_d(b)^2 = -log (P_d = Q_d)</script><br />

--><script type="math/tex"> H(d) = -\log \sum_b p_d(b)^2 = -log (P_d = Q_d)</script><br />

where P<span style="font-size: 90%; vertical-align: sub;">d</span> and Q<span style="font-size: 90%; vertical-align: sub;">d</span> are independent and identically distributed random variables corresponding to the same dyad, and the collision entropy is the same as the negative log of the probability that the two variables produce the same outcome.<br />

<br />

Line 343:

<!-- ws:start:WikiTextMathRule:12:

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

-\log \max_b p_d(b)&lt;br/&gt;[[math]]

--><script type="math/tex">-\log \max_b p_d(b)</script><br />

--><script type="math/tex"> -\log \max_b p_d(b)</script><br />

This is the min-entropy, which simply takes the negative log of the largest probability in the distribution. This can be thought of as representing the &quot;strength&quot; of the incoming dyad from being &quot;deciphered&quot; by a &quot;best-case&quot; auditory system.<br />

<br />

@@ 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:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-06 13:51:32 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-06 13:51:35 UTC</tt>.<br>
-: The original revision id was <tt>515669696</tt>.<br>
+: The original revision id was <tt>515669702</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>
 <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">[[math]]
-\newcommand{\cent}{\text{¢}}
+ \newcommand{\cent}{\text{¢}}
 [[math]]
 =Introduction=
@@ Line 38: / Line 38: @@
 [[math]]
-H(d) = -\sum_{b} p_d(b) \log_β p_d(b)
+ H(d) = -\sum_{b} p_d(b) \log_β p_d(b)
 [[math]]
@@ Line 52: / Line 52: @@
 [[math]]
-s_d(x) = \frac{1}{s\sqrt{2\pi}} e^{-\frac{(x-d)^2}{2s^2}}
+ s_d(x) = \frac{1}{s\sqrt{2\pi}} e^{-\frac{(x-d)^2}{2s^2}}
 [[math]]
@@ Line 65: / Line 65: @@
 [[math]]
-p_d(b) = \int_{\cent(b_l)}^{\cent(b_u)} s_d(x) dx
+ p_d(b) = \int_{\cent(b_l)}^{\cent(b_u)} s_d(x) dx
 [[math]]
@@ Line 81: / Line 81: @@
 [[math]]
-q_d(b) = \frac{s_d(\cent(b))}{\sqrt{n_b \cdot d_b}}
+ q_d(b) = \frac{s_d(\cent(b))}{\sqrt{n_b \cdot d_b}}
 [[math]]
@@ Line 89: / Line 89: @@
 [[math]]
-q_d(b) = \frac{s_d(\cent(b))}{\max(n_b,d_b)}
+ q_d(b) = \frac{s_d(\cent(b))}{\max(n_b,d_b)}
 [[math]]
@@ Line 97: / Line 97: @@
 [[math]]
-q_d(b) = \frac{s_d(\cent(b))}{\|b\|}
+ q_d(b) = \frac{s_d(\cent(b))}{\|b\|}
 [[math]]
@@ Line 105: / Line 105: @@
 [[math]]
-p_d(b) = \frac{q_d(b)}{\sum_b q_d(b)}
+ p_d(b) = \frac{q_d(b)}{\sum_b q_d(b)}
 [[math]]
@@ Line 128: / Line 128: @@
 [[math]]
-H(d) = \frac{1}{1-a} \log_β \sum_b p_d(b)^a
+ H(d) = \frac{1}{1-a} \log_β \sum_b p_d(b)^a
 [[math]]
@@ Line 143: / Line 143: @@
 ===a=0: Harmonic Hartley Entropy===
 [[math]]
-H(d) = \log |R|
+ H(d) = \log |R|
 [[math]]
 where |R| is the cardinality of the set of basis rationals. This assumes, in essence, an "infinitely dumb" auditory system which can do no better than picking a rational number from a uniform distribution completely at random. All dyads have the same Harmonic Hartley Entropy.
@@ Line 149: / Line 149: @@
 ===a=1: Harmonic Shannon Entropy (Harmonic Entropy)===
 [[math]]
-H(d) = -\sum_{b} p_d(b) \log p_d(b)
+ H(d) = -\sum_{b} p_d(b) \log p_d(b)
 [[math]]
 This is Paul's original Harmonic Entropy. This can be thought of as an auditory system which simply selects a rational at random from the incoming distribution, weighted via the distribution itself.
@@ Line 155: / Line 155: @@
 ===a=2: Harmonic Collision Entropy===
 [[math]]
-H(d) = -\log \sum_b p_d(b)^2 = -log (P_d = Q_d)
+ H(d) = -\log \sum_b p_d(b)^2 = -log (P_d = Q_d)
 [[math]]
 where P&lt;span style="font-size: 90%; vertical-align: sub;"&gt;d&lt;/span&gt; and Q&lt;span style="font-size: 90%; vertical-align: sub;"&gt;d&lt;/span&gt; are independent and identically distributed random variables corresponding to the same dyad, and the collision entropy is the same as the negative log of the probability that the two variables produce the same outcome.
@@ Line 161: / Line 161: @@
 ===a=∞: Harmonic Min-Entropy===
 [[math]]
--\log \max_b p_d(b)
+ -\log \max_b p_d(b)
 [[math]]
 This is the min-entropy, which simply takes the negative log of the largest probability in the distribution. This can be thought of as representing the "strength" of the incoming dyad from being "deciphered" by a "best-case" auditory system.
@@ Line 174: / Line 174: @@
 <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;Harmonic Entropy&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextMathRule:0:
 [[math]]&amp;lt;br/&amp;gt;
-\newcommand{\cent}{\text{¢}}&amp;lt;br/&amp;gt;[[math]]
+ \newcommand{\cent}{\text{¢}}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\newcommand{\cent}{\text{¢}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt; \newcommand{\cent}{\text{¢}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:13:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&gt;Introduction&lt;/h1&gt;
   &lt;!-- ws:start:WikiTextTocRule:35:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Introduction"&gt;Introduction&lt;/a&gt;&lt;/div&gt;
@@ Line 211: / Line 211: @@
 &lt;!-- ws:start:WikiTextMathRule:1:
 [[math]]&amp;lt;br/&amp;gt;
-H(d) = -\sum_{b} p_d(b) \log_β p_d(b)&amp;lt;br/&amp;gt;[[math]]
+ H(d) = -\sum_{b} p_d(b) \log_β p_d(b)&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;H(d) = -\sum_{b} p_d(b) \log_β p_d(b)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:1 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt; H(d) = -\sum_{b} p_d(b) \log_β p_d(b)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:1 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 where H(d) is the Shannon entropy of the dyad d, the b are all of the basis rationals in the set, the p&lt;span style="font-size: 90%; vertical-align: sub;"&gt;d&lt;/span&gt;(b) is the probability assigned to basis rational b given an input dyad of d, and the logarithm β reflects the units of information being used (by convention, we set β=2, corresponding to the use of bits). This is the Harmonic Entropy of the dyad d.&lt;br /&gt;
@@ Line 224: / Line 224: @@
 &lt;!-- ws:start:WikiTextMathRule:2:
 [[math]]&amp;lt;br/&amp;gt;
-s_d(x) = \frac{1}{s\sqrt{2\pi}} e^{-\frac{(x-d)^2}{2s^2}}&amp;lt;br/&amp;gt;[[math]]
+ s_d(x) = \frac{1}{s\sqrt{2\pi}} e^{-\frac{(x-d)^2}{2s^2}}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;s_d(x) = \frac{1}{s\sqrt{2\pi}} e^{-\frac{(x-d)^2}{2s^2}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:2 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt; s_d(x) = \frac{1}{s\sqrt{2\pi}} e^{-\frac{(x-d)^2}{2s^2}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:2 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 where &lt;em&gt;s&lt;/em&gt; becomes the standard deviation of the Gaussian, being an ASCII-friendly version of the more familiar symbol σ for representing the standard deviation.&lt;br /&gt;
@@ Line 238: / Line 238: @@
 &lt;!-- ws:start:WikiTextMathRule:3:
 [[math]]&amp;lt;br/&amp;gt;
-p_d(b) = \int_{\cent(b_l)}^{\cent(b_u)} s_d(x) dx&amp;lt;br/&amp;gt;[[math]]
+ p_d(b) = \int_{\cent(b_l)}^{\cent(b_u)} s_d(x) dx&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;p_d(b) = \int_{\cent(b_l)}^{\cent(b_u)} s_d(x) dx&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:3 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt; p_d(b) = \int_{\cent(b_l)}^{\cent(b_u)} s_d(x) dx&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:3 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 where s&lt;span style="vertical-align: sub;"&gt;d&lt;/span&gt;(x) is the spreading function associated with d&lt;span style="line-height: 1.5;"&gt;, b&lt;/span&gt;&lt;span style="line-height: 1.5; vertical-align: sub;"&gt;l&lt;/span&gt;&lt;span style="line-height: 1.5;"&gt; and b&lt;/span&gt;&lt;span style="line-height: 1.5; vertical-align: sub;"&gt;u&lt;/span&gt;&lt;span style="line-height: 1.5;"&gt; are the domain lower and upper bounds associated with basis rational b, and ¢(f) = 1200·log2(f), or the &amp;quot;cents&amp;quot; function converting frequency ratios to cents. Normally, b&lt;/span&gt;&lt;span style="line-height: 1.5; vertical-align: sub;"&gt;l &lt;/span&gt;&lt;span style="line-height: 1.5;"&gt;is set equal to the mediant of b and its nearest lower neighbor (if it exists), or -∞ if not; likewise with b&lt;/span&gt;&lt;span style="line-height: 1.5; vertical-align: sub;"&gt;u&lt;/span&gt;&lt;span style="line-height: 1.5;"&gt;.&lt;/span&gt;&lt;br /&gt;
@@ Line 255: / Line 255: @@
 &lt;!-- ws:start:WikiTextMathRule:4:
 [[math]]&amp;lt;br/&amp;gt;
-q_d(b) = \frac{s_d(\cent(b))}{\sqrt{n_b \cdot d_b}}&amp;lt;br/&amp;gt;[[math]]
+ q_d(b) = \frac{s_d(\cent(b))}{\sqrt{n_b \cdot d_b}}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;q_d(b) = \frac{s_d(\cent(b))}{\sqrt{n_b \cdot d_b}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:4 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt; q_d(b) = \frac{s_d(\cent(b))}{\sqrt{n_b \cdot d_b}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:4 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 where the q&lt;span style="font-size: 12px; vertical-align: sub;"&gt;d&lt;/span&gt;(b) now represent the unnormalized &amp;quot;probabilities&amp;quot;, and n&lt;span style="vertical-align: sub;"&gt;b&lt;/span&gt; and d&lt;span style="vertical-align: sub;"&gt;b&lt;/span&gt; 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.&lt;br /&gt;
@@ Line 264: / Line 264: @@
 &lt;!-- ws:start:WikiTextMathRule:5:
 [[math]]&amp;lt;br/&amp;gt;
-q_d(b) = \frac{s_d(\cent(b))}{\max(n_b,d_b)}&amp;lt;br/&amp;gt;[[math]]
+ q_d(b) = \frac{s_d(\cent(b))}{\max(n_b,d_b)}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;q_d(b) = \frac{s_d(\cent(b))}{\max(n_b,d_b)}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:5 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt; q_d(b) = \frac{s_d(\cent(b))}{\max(n_b,d_b)}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:5 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 where this time the set of basis rationals is assumed to be all of those of Weil Height ≤ N for some N.&lt;br /&gt;
@@ Line 273: / Line 273: @@
 &lt;!-- ws:start:WikiTextMathRule:6:
 [[math]]&amp;lt;br/&amp;gt;
-q_d(b) = \frac{s_d(\cent(b))}{\|b\|}&amp;lt;br/&amp;gt;[[math]]
+ q_d(b) = \frac{s_d(\cent(b))}{\|b\|}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;q_d(b) = \frac{s_d(\cent(b))}{\|b\|}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:6 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt; q_d(b) = \frac{s_d(\cent(b))}{\|b\|}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:6 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 where ||b|| denotes a complexity function mapping from rational numbers to reals.&lt;br /&gt;
@@ Line 282: / Line 282: @@
 &lt;!-- ws:start:WikiTextMathRule:7:
 [[math]]&amp;lt;br/&amp;gt;
-p_d(b) = \frac{q_d(b)}{\sum_b q_d(b)}&amp;lt;br/&amp;gt;[[math]]
+ p_d(b) = \frac{q_d(b)}{\sum_b q_d(b)}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;p_d(b) = \frac{q_d(b)}{\sum_b q_d(b)}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:7 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt; p_d(b) = \frac{q_d(b)}{\sum_b q_d(b)}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:7 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 The p&lt;span style="vertical-align: sub;"&gt;d&lt;/span&gt;(b) are then used directly to compute the entropy.&lt;br /&gt;
@@ Line 306: / Line 306: @@
 &lt;!-- ws:start:WikiTextMathRule:8:
 [[math]]&amp;lt;br/&amp;gt;
-H(d) = \frac{1}{1-a} \log_β \sum_b p_d(b)^a&amp;lt;br/&amp;gt;[[math]]
+ H(d) = \frac{1}{1-a} \log_β \sum_b p_d(b)^a&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;H(d) = \frac{1}{1-a} \log_β \sum_b p_d(b)^a&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:8 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt; H(d) = \frac{1}{1-a} \log_β \sum_b p_d(b)^a&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:8 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 where the p_d(b) are the probabilities assigned by dyad d to each basis rational b. Being a q-analog, it is noteworthy that Rényi entropy converges to Shannon entropy in the limit as a→1, a fact which can be verified using L'Hôpital's rule as found &lt;a class="wiki_link_ext" href="http://www.sonycsl.co.jp/person/nielsen/Note-HopitalRuleShannonRenyiTsallis.pdf" rel="nofollow" target="_blank"&gt;here&lt;/a&gt;.&lt;br /&gt;
@@ Line 322: / Line 322: @@
   &lt;!-- ws:start:WikiTextMathRule:9:
 [[math]]&amp;lt;br/&amp;gt;
-H(d) = \log |R|&amp;lt;br/&amp;gt;[[math]]
+ H(d) = \log |R|&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;H(d) = \log |R|&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:9 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt; H(d) = \log |R|&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:9 --&gt;&lt;br /&gt;
 where |R| is the cardinality of the set of basis rationals. This assumes, in essence, an &amp;quot;infinitely dumb&amp;quot; auditory system which can do no better than picking a rational number from a uniform distribution completely at random. All dyads have the same Harmonic Hartley Entropy.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 329: / Line 329: @@
   &lt;!-- ws:start:WikiTextMathRule:10:
 [[math]]&amp;lt;br/&amp;gt;
-H(d) = -\sum_{b} p_d(b) \log p_d(b)&amp;lt;br/&amp;gt;[[math]]
+ H(d) = -\sum_{b} p_d(b) \log p_d(b)&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;H(d) = -\sum_{b} p_d(b) \log p_d(b)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:10 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt; H(d) = -\sum_{b} p_d(b) \log p_d(b)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:10 --&gt;&lt;br /&gt;
 This is Paul's original Harmonic Entropy. This can be thought of as an auditory system which simply selects a rational at random from the incoming distribution, weighted via the distribution itself.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 336: / Line 336: @@
   &lt;!-- ws:start:WikiTextMathRule:11:
 [[math]]&amp;lt;br/&amp;gt;
-H(d) = -\log \sum_b p_d(b)^2 = -log (P_d = Q_d)&amp;lt;br/&amp;gt;[[math]]
+ H(d) = -\log \sum_b p_d(b)^2 = -log (P_d = Q_d)&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;H(d) = -\log \sum_b p_d(b)^2 = -log (P_d = Q_d)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:11 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt; H(d) = -\log \sum_b p_d(b)^2 = -log (P_d = Q_d)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:11 --&gt;&lt;br /&gt;
 where P&lt;span style="font-size: 90%; vertical-align: sub;"&gt;d&lt;/span&gt; and Q&lt;span style="font-size: 90%; vertical-align: sub;"&gt;d&lt;/span&gt; are independent and identically distributed random variables corresponding to the same dyad, and the collision entropy is the same as the negative log of the probability that the two variables produce the same outcome.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 343: / Line 343: @@
   &lt;!-- ws:start:WikiTextMathRule:12:
 [[math]]&amp;lt;br/&amp;gt;
--\log \max_b p_d(b)&amp;lt;br/&amp;gt;[[math]]
+ -\log \max_b p_d(b)&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;-\log \max_b p_d(b)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:12 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt; -\log \max_b p_d(b)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:12 --&gt;&lt;br /&gt;
 This is the min-entropy, which simply takes the negative log of the largest probability in the distribution. This can be thought of as representing the &amp;quot;strength&amp;quot; of the incoming dyad from being &amp;quot;deciphered&amp;quot; by a &amp;quot;best-case&amp;quot; auditory system.&lt;br /&gt;
 &lt;br /&gt;