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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
==Introduction==
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2015-07-27 23:29:05 UTC</tt>.<br>
: The original revision id was <tt>555828349</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">==Introduction==  


The "golden ratio" or "phi" (Greek letter Φ / φ / &lt;span class="Unicode"&gt;ϕ ) may be defined by a/b such that a/b = (a+b)/a. It follows that ϕ&lt;/span&gt;-1 = 1/&lt;span class="Unicode"&gt;ϕ, and also that ϕ = (1+sqrt(5))/2, or approximately &lt;/span&gt;1.6180339887... &lt;span class="Unicode"&gt;ϕ is an irrational number that appears in many branches of mathematics.&lt;/span&gt;
The "golden ratio" or "phi" (Greek letter Φ / φ / <span style="">ϕ ) may be defined by a/b such that a/b = (a+b)/a. It follows that ϕ</span>-1 = 1/<span style="">ϕ, and also that ϕ = (1+sqrt(5))/2, or approximately </span>1.6180339887... <span style="">ϕ is an irrational number that appears in many branches of mathematics.</span>


[[@http://en.wikipedia.org/wiki/Golden_ratio|Wikipedia article on phi]]
[http://en.wikipedia.org/wiki/Golden_ratio Wikipedia article on phi]


==Musical applications==


==Musical applications==  
<span style="">Phi taken as a musical ratio (ϕ</span>*f where f=1/1) <span style="">is about 833.1 cents. This is sometimes called "acoustical phi".</span>


&lt;span class="Unicode"&gt;Phi taken as a musical ratio (ϕ&lt;/span&gt;*f where f=1/1) &lt;span class="Unicode"&gt;is about 833.1 cents. This is sometimes called "acoustical phi".&lt;/span&gt;
<span style="">As the ratios of successive terms of the Fibonacci sequence converge on phi, the just intonation intervals 3/2, 5/3, 8/5, 13/8, 21/13, ... converge on ~833.1 cents.</span>
&lt;span class="Unicode"&gt;As the ratios of successive terms of the Fibonacci sequence converge on phi, the just intonation intervals 3/2, 5/3, 8/5, 13/8, 21/13, ... converge on ~833.1 cents.&lt;/span&gt;


"Logarithmic phi", or 1200*&lt;span class="Unicode"&gt;ϕ cents = 1941.6 cents (or, octave-reduced, 741.6 cents) is also useful as a generator, for example in [[Erv Wilson]]'s "Golden Horagrams".&lt;/span&gt;
"Logarithmic phi", or 1200*<span style="">ϕ cents = 1941.6 cents (or, octave-reduced, 741.6 cents) is also useful as a generator, for example in [[Erv_Wilson|Erv Wilson]]'s "Golden Horagrams".</span>


==Additional reading==


==Additional reading==
[[Generating_a_scale_through_successive_divisions_of_the_octave_by_the_Golden_Ratio|Generating a scale through successive divisions of the octave by the Golden Ratio]]


[[Generating a scale through successive divisions of the octave by the Golden Ratio]]
<span style="">[[Phi_as_a_Generator|Phi]]</span>[[Phi_as_a_Generator| as a Generator]]


&lt;span class="w_hl"&gt;[[xenharmonic/Phi as a Generator|Phi]]&lt;/span&gt;[[xenharmonic/Phi as a Generator| as a Generator]]
[[Sqrtphi|sqrtphi]], a temperament based on the square root of phi (~416.5 cents) as a generator


[[sqrtphi]], a temperament based on the square root of phi (~416.5 cents) as a generator
<span style="">[[Golden_Meantone|Golden]]</span>[[Golden_Meantone| Meantone ]]


&lt;span class="w_hl"&gt;[[xenharmonic/Golden Meantone|Golden]]&lt;/span&gt;[[xenharmonic/Golden Meantone| Meantone ]]
[[833_Cent_Golden_Scale_(Bohlen)|833 Cent ]]<span style="">[[833_Cent_Golden_Scale_(Bohlen)|Golden]]</span>[[833_Cent_Golden_Scale_(Bohlen)| Scale (Bohlen) ]]


[[xenharmonic/833 Cent Golden Scale (Bohlen)|833 Cent ]]&lt;span class="w_hl"&gt;[[xenharmonic/833 Cent Golden Scale (Bohlen)|Golden]]&lt;/span&gt;[[xenharmonic/833 Cent Golden Scale (Bohlen)| Scale (Bohlen) ]]
[http://dkeenan.com/Music/NobleMediant.txt The Noble Mediant: Complex ratios and metastable musical intervals], by [[Margo_Schulter|Margo Schulter]] and [[Dave_Keenan|David Keenan]]


[[@http://dkeenan.com/Music/NobleMediant.txt|The Noble Mediant: Complex ratios and metastable musical intervals]], by [[Margo Schulter]] and [[Dave Keenan|David Keenan]]
[http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm 5- to 9-tone, octave-repeating scales from Wilson's Golden Horagrams of the Scale Tree], by David Finnamore
 
[[@http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm|5- to 9-tone, octave-repeating scales from Wilson's Golden Horagrams of the Scale Tree]], by David Finnamore</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;Golden Ratio&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Introduction&lt;/h2&gt;
&lt;br /&gt;
The &amp;quot;golden ratio&amp;quot; or &amp;quot;phi&amp;quot; (Greek letter Φ / φ / &lt;span class="Unicode"&gt;ϕ ) may be defined by a/b such that a/b = (a+b)/a. It follows that ϕ&lt;/span&gt;-1 = 1/&lt;span class="Unicode"&gt;ϕ, and also that ϕ = (1+sqrt(5))/2, or approximately &lt;/span&gt;1.6180339887... &lt;span class="Unicode"&gt;ϕ is an irrational number that appears in many branches of mathematics.&lt;/span&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Golden_ratio" rel="nofollow" target="_blank"&gt;Wikipedia article on phi&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Musical applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Musical applications&lt;/h2&gt;
&lt;br /&gt;
&lt;span class="Unicode"&gt;Phi taken as a musical ratio (ϕ&lt;/span&gt;*f where f=1/1) &lt;span class="Unicode"&gt;is about 833.1 cents. This is sometimes called &amp;quot;acoustical phi&amp;quot;.&lt;/span&gt;&lt;br /&gt;
&lt;span class="Unicode"&gt;As the ratios of successive terms of the Fibonacci sequence converge on phi, the just intonation intervals 3/2, 5/3, 8/5, 13/8, 21/13, ... converge on ~833.1 cents.&lt;/span&gt;&lt;br /&gt;
&lt;br /&gt;
&amp;quot;Logarithmic phi&amp;quot;, or 1200*&lt;span class="Unicode"&gt;ϕ cents = 1941.6 cents (or, octave-reduced, 741.6 cents) is also useful as a generator, for example in &lt;a class="wiki_link" href="/Erv%20Wilson"&gt;Erv Wilson&lt;/a&gt;'s &amp;quot;Golden Horagrams&amp;quot;.&lt;/span&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Additional reading"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Additional reading&lt;/h2&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Generating%20a%20scale%20through%20successive%20divisions%20of%20the%20octave%20by%20the%20Golden%20Ratio"&gt;Generating a scale through successive divisions of the octave by the Golden Ratio&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;span class="w_hl"&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Phi%20as%20a%20Generator"&gt;Phi&lt;/a&gt;&lt;/span&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Phi%20as%20a%20Generator"&gt; as a Generator&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/sqrtphi"&gt;sqrtphi&lt;/a&gt;, a temperament based on the square root of phi (~416.5 cents) as a generator&lt;br /&gt;
&lt;br /&gt;
&lt;span class="w_hl"&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Golden%20Meantone"&gt;Golden&lt;/a&gt;&lt;/span&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Golden%20Meantone"&gt; Meantone &lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/833%20Cent%20Golden%20Scale%20%28Bohlen%29"&gt;833 Cent &lt;/a&gt;&lt;span class="w_hl"&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/833%20Cent%20Golden%20Scale%20%28Bohlen%29"&gt;Golden&lt;/a&gt;&lt;/span&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/833%20Cent%20Golden%20Scale%20%28Bohlen%29"&gt; Scale (Bohlen) &lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://dkeenan.com/Music/NobleMediant.txt" rel="nofollow" target="_blank"&gt;The Noble Mediant: Complex ratios and metastable musical intervals&lt;/a&gt;, by &lt;a class="wiki_link" href="/Margo%20Schulter"&gt;Margo Schulter&lt;/a&gt; and &lt;a class="wiki_link" href="/Dave%20Keenan"&gt;David Keenan&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm" rel="nofollow" target="_blank"&gt;5- to 9-tone, octave-repeating scales from Wilson's Golden Horagrams of the Scale Tree&lt;/a&gt;, by David Finnamore&lt;/body&gt;&lt;/html&gt;</pre></div>
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