Logarithmic approximants: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-22 11:41:16 UTC</tt>.

: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-22 13:00:11 UTC</tt>.

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######Figure 3. Geometrical representation of argent temperament

By the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem|Gelfond-Schneider theorem ]] the frequency ratios of all argent intervals (//r// = 2√2//a//+//b//, where// a// and //b// are integers) are transcendental, with the exception of octave multiples (//a// = 0).

By the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem|Gelfond-Schneider theorem ]] the frequency ratios of all argent intervals (//r// = 2√2//a//+//b//, where// a// and //b// are integers) are transcendental, with the exception of octave multiples (//a// = 0). The frequency ratio of the tempered perfect eleventh (__8/3__ = __2.6666...__) is the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant|Gelfond-Schneider constant ]]or Hilbert number, 2√2 = 2.665144...

==Golden temperaments==

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######Figure 3. Geometrical representation of argent temperament

By the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem" rel="nofollow">Gelfond-Schneider theorem </a> the frequency ratios of all argent intervals (r = 2√2a+b, where a and b are integers) are transcendental, with the exception of octave multiples (a = 0).

By the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem" rel="nofollow">Gelfond-Schneider theorem </a> the frequency ratios of all argent intervals (r = 2√2a+b, where a and b are integers) are transcendental, with the exception of octave multiples (a = 0). The frequency ratio of the tempered perfect eleventh (8/3 = 2.6666...) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant" rel="nofollow">Gelfond-Schneider constant </a>or Hilbert number, 2√2 = 2.665144...

<h2 id="toc19"><a name="x4. Quadratic approximants-Golden temperaments"></a>Golden temperaments</h2>

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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:MartinGough|MartinGough]] and made on <tt>2015-02-22 11:41:16 UTC</tt>.<br>
+: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-22 13:00:11 UTC</tt>.<br>
-: The original revision id was <tt>541704146</tt>.<br>
+: The original revision id was <tt>541710524</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 379: / Line 379: @@
 &lt;span style="color: #ffffff;"&gt;######&lt;/span&gt;Figure 3. Geometrical representation of argent temperament
-By the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem|Gelfond-Schneider theorem ]] the frequency ratios of all argent intervals (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// = 2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;√2//a//+//b//&lt;/span&gt;, where//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; a&lt;/span&gt;// and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;// are integers) are transcendental, with the exception of octave multiples (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//a// = 0&lt;/span&gt;).
+By the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem|Gelfond-Schneider theorem ]] the frequency ratios of all argent intervals (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// = 2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;√2//a//+//b//&lt;/span&gt;, where//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; a&lt;/span&gt;// and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;// are integers) are transcendental, with the exception of octave multiples (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//a// = 0&lt;/span&gt;). The frequency ratio of the tempered perfect eleventh (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;__8/3__ = __2.6666...__&lt;/span&gt;) is the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant|Gelfond-Schneider constant ]]or Hilbert number, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;√2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 2.665144&lt;/span&gt;...
 ==Golden temperaments==
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 &lt;span style="color: #ffffff;"&gt;######&lt;/span&gt;Figure 3. Geometrical representation of argent temperament&lt;br /&gt;
 &lt;br /&gt;
-By the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem" rel="nofollow"&gt;Gelfond-Schneider theorem &lt;/a&gt; the frequency ratios of all argent intervals (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; = 2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;√2&lt;em&gt;a&lt;/em&gt;+&lt;em&gt;b&lt;/em&gt;&lt;/span&gt;, where&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; a&lt;/span&gt;&lt;/em&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;&lt;/em&gt; are integers) are transcendental, with the exception of octave multiples (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;a&lt;/em&gt; = 0&lt;/span&gt;).&lt;br /&gt;
+By the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem" rel="nofollow"&gt;Gelfond-Schneider theorem &lt;/a&gt; the frequency ratios of all argent intervals (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; = 2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;√2&lt;em&gt;a&lt;/em&gt;+&lt;em&gt;b&lt;/em&gt;&lt;/span&gt;, where&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; a&lt;/span&gt;&lt;/em&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;&lt;/em&gt; are integers) are transcendental, with the exception of octave multiples (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;a&lt;/em&gt; = 0&lt;/span&gt;). The frequency ratio of the tempered perfect eleventh (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;u&gt;8/3&lt;/u&gt; = &lt;u&gt;2.6666...&lt;/u&gt;&lt;/span&gt;) is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant" rel="nofollow"&gt;Gelfond-Schneider constant &lt;/a&gt;or Hilbert number, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;√2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 2.665144&lt;/span&gt;...&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:87:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="x4. Quadratic approximants-Golden temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:87 --&gt;Golden temperaments&lt;/h2&gt;