Goldonic series: Difference between revisions

Revision as of 23:31, 30 November 2015

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This revision was by author MasonGreen1 and made on 2015-11-30 23:31:54 UTC.

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A **goldonic series** or **golden series** is a series of frequencies that form a [[https://en.wikipedia.org/wiki/Geometric_progression|geometric progression]] whose generating interval is the golden ratio (1.61803....).

==Unique properties== 

The goldonic series is unique among geometric sequencies because only //<span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;">φ</span>// satisfies the equation //x//<span style="vertical-align: super;">n-1</span> //+ x//<span style="vertical-align: super;">n</span> //= x//<span style="vertical-align: super;">n+1</span>.

From an acoustic standpoint, the goldonic series contains some "harmonic-like" characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors).

Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in //both// directions and contains no fundamental.

Original HTML content:

<html><head><title>Goldonic series</title></head><body>A <strong>goldonic series</strong> or <strong>golden series</strong> is a series of frequencies that form a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Geometric_progression" rel="nofollow">geometric progression</a> whose generating interval is the golden ratio (1.61803....).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Unique properties"></a><!-- ws:end:WikiTextHeadingRule:0 -->Unique properties</h2>
 <br />
The goldonic series is unique among geometric sequencies because only <em><span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;">φ</span></em> satisfies the equation <em>x</em><span style="vertical-align: super;">n-1</span> <em>+ x</em><span style="vertical-align: super;">n</span> <em>= x</em><span style="vertical-align: super;">n+1</span>.<br />
<br />
From an acoustic standpoint, the goldonic series contains some &quot;harmonic-like&quot; characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors).<br />
<br />
Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in <em>both</em> directions and contains no fundamental.</body></html>

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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:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-30 23:31:11 UTC</tt>.<br>
+: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-30 23:31:54 UTC</tt>.<br>
-: The original revision id was <tt>568375357</tt>.<br>
+: The original revision id was <tt>568375387</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>
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 The goldonic series is unique among geometric sequencies because only //&lt;span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;"&gt;φ&lt;/span&gt;// satisfies the equation //x//&lt;span style="vertical-align: super;"&gt;n-1&lt;/span&gt; //+ x//&lt;span style="vertical-align: super;"&gt;n&lt;/span&gt; //= x//&lt;span style="vertical-align: super;"&gt;n+1&lt;/span&gt;.
-From an acoustic standpoint, the goldonic series contains some "harmonic-like" characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors)..</pre></div>
+From an acoustic standpoint, the goldonic series contains some "harmonic-like" characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors).
+Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in //both// directions and contains no fundamental.</pre></div>
 <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Goldonic series&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;strong&gt;goldonic series&lt;/strong&gt; or &lt;strong&gt;golden series&lt;/strong&gt; is a series of frequencies that form a &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Geometric_progression" rel="nofollow"&gt;geometric progression&lt;/a&gt; whose generating interval is the golden ratio (1.61803....).&lt;br /&gt;
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 The goldonic series is unique among geometric sequencies because only &lt;em&gt;&lt;span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;"&gt;φ&lt;/span&gt;&lt;/em&gt; satisfies the equation &lt;em&gt;x&lt;/em&gt;&lt;span style="vertical-align: super;"&gt;n-1&lt;/span&gt; &lt;em&gt;+ x&lt;/em&gt;&lt;span style="vertical-align: super;"&gt;n&lt;/span&gt; &lt;em&gt;= x&lt;/em&gt;&lt;span style="vertical-align: super;"&gt;n+1&lt;/span&gt;.&lt;br /&gt;
 &lt;br /&gt;
-From an acoustic standpoint, the goldonic series contains some &amp;quot;harmonic-like&amp;quot; characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors)..&lt;/body&gt;&lt;/html&gt;</pre></div>
+From an acoustic standpoint, the goldonic series contains some &amp;quot;harmonic-like&amp;quot; characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors).&lt;br /&gt;
+&lt;br /&gt;
+Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in &lt;em&gt;both&lt;/em&gt; directions and contains no fundamental.&lt;/body&gt;&lt;/html&gt;</pre></div>

Goldonic series: Difference between revisions

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