Logarithmic approximants: Difference between revisions

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which after squaring both sides leads to √2 ≈ 99/70, a well-known approximation which can be confirmed by noting that 99/70 = √(2 + 1/4900).

As a consequence of these relationships the tempered diatonic semitone (85.281 cents) is close to __21/20__ (84.467 cents), the tempered chromatic semitone (120.606 cents) is close to __15/14__ (119.443 cents), and the tempered Pythagorean comma (35.325 cents) is close to __50/49__ (34.976 cents).

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;...

=Sources and acknowledgements=  

This article is based on original research by [[Martin Gough]]. See [[file:Bimod Approx 2014-6-8.pdf|this paper]] for a fuller account of bimodular approximants.

Thanks to [[Gene Ward Smith]] for the Gelfond-Schneider result.</pre></div>

<h4>Original HTML content:</h4>

which after squaring both sides leads to √2 ≈ 99/70, a well-known approximation which can be confirmed by noting that 99/70 = √(2 + 1/4900).&lt;br /&gt;

As a consequence of these relationships the tempered diatonic semitone (85.281 cents) is close to &lt;u&gt;21/20&lt;/u&gt; (84.467 cents), the tempered chromatic semitone (120.606 cents) is close to &lt;u&gt;15/14&lt;/u&gt; (119.443 cents), and the tempered Pythagorean comma (35.325 cents) is close to &lt;u&gt;50/49&lt;/u&gt; (34.976 cents).&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:95:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc22"&gt;&lt;a name="Sources and acknowledgements"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:95 --&gt;Sources and acknowledgements&lt;/h1&gt;

  This article is based on original research by &lt;a class="wiki_link" href="/Martin%20Gough"&gt;Martin Gough&lt;/a&gt;. See &lt;a href="/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf" onclick="ws.common.trackFileLink('/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf');"&gt;this paper&lt;/a&gt; for a fuller account of bimodular approximants.&lt;br /&gt;

Thanks to &lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&gt;Gene Ward Smith&lt;/a&gt; for the Gelfond-Schneider result.&lt;/body&gt;&lt;/html&gt;</pre></div>