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:

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-05 03:27:02 UTC</tt>.

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-05 03:27:31 UTC</tt>.

: The original revision id was <tt>~~515617078~~</tt>.

: The original revision id was <tt>515617088</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Line 50:

Setting N=10000 and s=1% yields the following curve, where the x-axis represents the width in cents of the dyad, and the y-axis represents //discordance//, measured in bits of Shannon entropy:

[[image:http://i.imgur.com/tNg7z1P.png width="800" height="404"]]

Other ways of generating the set of basis rationals have also been explored, most notably by bounding by [[@height|Weil height]] rather than Tenney height. The following is the curve that results for a Weil height bound 100 and s=1%, which has the primary effect of altering the slope of the curve:

Other ways of generating the set of basis rationals have also been explored, most notably by bounding by [[@height|Weil height]] rather than Tenney height. The following is the curve that results for a Weil height bound of N=100 and s=1%, which has the primary effect of altering the slope of the curve:

[[image:http://i.imgur.com/TZdU6eD.png width="800" height="410"]]

Line 146:

Setting N=10000 and s=1% yields the following curve, where the x-axis represents the width in cents of the dyad, and the y-axis represents discordance, measured in bits of Shannon entropy:

<img src="http://i.imgur.com/tNg7z1P.png" alt="external image tNg7z1P.png" title="external image tNg7z1P.png" style="height: 404px; width: 800px;" />

Other ways of generating the set of basis rationals have also been explored, most notably by bounding by <a class="wiki_link" href="/height" target="_blank">Weil height</a> rather than Tenney height. The following is the curve that results for a Weil height bound 100 and s=1%, which has the primary effect of altering the slope of the curve:

Other ways of generating the set of basis rationals have also been explored, most notably by bounding by <a class="wiki_link" href="/height" target="_blank">Weil height</a> rather than Tenney height. The following is the curve that results for a Weil height bound of N=100 and s=1%, which has the primary effect of altering the slope of the curve:

<img src="http://i.imgur.com/TZdU6eD.png" alt="external image TZdU6eD.png" title="external image TZdU6eD.png" style="height: 410px; width: 800px;" />

@@ 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>2014-07-05 03:27:02 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-05 03:27:31 UTC</tt>.<br>
-: The original revision id was <tt>515617078</tt>.<br>
+: The original revision id was <tt>515617088</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 50: / Line 50: @@
 Setting N=10000 and s=1% yields the following curve, where the x-axis represents the width in cents of the dyad, and the y-axis represents //discordance//, measured in bits of Shannon entropy:
 [[image:http://i.imgur.com/tNg7z1P.png width="800" height="404"]]
-Other ways of generating the set of basis rationals have also been explored, most notably by bounding by [[@height|Weil height]] rather than Tenney height. The following is the curve that results for a Weil height bound 100 and s=1%, which has the primary effect of altering the slope of the curve:
+Other ways of generating the set of basis rationals have also been explored, most notably by bounding by [[@height|Weil height]] rather than Tenney height. The following is the curve that results for a Weil height bound of N=100 and s=1%, which has the primary effect of altering the slope of the curve:
 [[image:http://i.imgur.com/TZdU6eD.png width="800" height="410"]]
@@ Line 146: / Line 146: @@
 Setting N=10000 and s=1% yields the following curve, where the x-axis represents the width in cents of the dyad, and the y-axis represents &lt;em&gt;discordance&lt;/em&gt;, measured in bits of Shannon entropy:&lt;br /&gt;
 &lt;!-- ws:start:WikiTextRemoteImageRule:47:&amp;lt;img src=&amp;quot;http://i.imgur.com/tNg7z1P.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 404px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://i.imgur.com/tNg7z1P.png" alt="external image tNg7z1P.png" title="external image tNg7z1P.png" style="height: 404px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:47 --&gt;&lt;br /&gt;
-Other ways of generating the set of basis rationals have also been explored, most notably by bounding by &lt;a class="wiki_link" href="/height" target="_blank"&gt;Weil height&lt;/a&gt; rather than Tenney height. The following is the curve that results for a Weil height bound 100 and s=1%, which has the primary effect of altering the slope of the curve:&lt;br /&gt;
+Other ways of generating the set of basis rationals have also been explored, most notably by bounding by &lt;a class="wiki_link" href="/height" target="_blank"&gt;Weil height&lt;/a&gt; rather than Tenney height. The following is the curve that results for a Weil height bound of N=100 and s=1%, which has the primary effect of altering the slope of the curve:&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextRemoteImageRule:48:&amp;lt;img src=&amp;quot;http://i.imgur.com/TZdU6eD.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 410px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://i.imgur.com/TZdU6eD.png" alt="external image TZdU6eD.png" title="external image TZdU6eD.png" style="height: 410px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:48 --&gt;&lt;br /&gt;