Golden ratio
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==Introduction== The "golden ratio" or "phi" (Greek letter Φ / φ / <span class="Unicode">ϕ ) may be defined by a/b such that a/b = (a+b)/a. It follows that ϕ</span>-1 = 1/<span class="Unicode">ϕ, and also that ϕ = (1+sqrt(5))/2, or approximately </span>1.6180339887... <span class="Unicode">ϕ is an irrational number that appears in many branches of mathematics.</span> [[@http://en.wikipedia.org/wiki/Golden_ratio|Wikipedia article on phi]] ==Musical applications== <span class="Unicode">Phi taken as a musical ratio (ϕ</span>*f where f=1/1) <span class="Unicode">is about 833.1 cents. This is sometimes called "acoustical phi".</span> <span class="Unicode">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> "Logarithmic phi", or 1200*<span class="Unicode">ϕ cents = 1941.6 cents (or, octave-reduced, 741.6 cents) is also useful.</span> ==Additional reading== <span class="w_hl">[[xenharmonic/Phi as a Generator|Phi]]</span>[[xenharmonic/Phi as a Generator| as a Generator ]] [[sqrtphi]], a temperament based on the square root of phi (~416.5 cents) as a generator <span class="w_hl">[[xenharmonic/Golden Meantone|Golden]]</span>[[xenharmonic/Golden Meantone| Meantone ]] [[xenharmonic/833 Cent Golden Scale (Bohlen)|833 Cent ]]<span class="w_hl">[[xenharmonic/833 Cent Golden Scale (Bohlen)|Golden]]</span>[[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]] and [[David Keenan]]
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<html><head><title>Golden Ratio</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Introduction"></a><!-- ws:end:WikiTextHeadingRule:0 -->Introduction</h2> <br /> The "golden ratio" or "phi" (Greek letter Φ / φ / <span class="Unicode">ϕ ) may be defined by a/b such that a/b = (a+b)/a. It follows that ϕ</span>-1 = 1/<span class="Unicode">ϕ, and also that ϕ = (1+sqrt(5))/2, or approximately </span>1.6180339887... <span class="Unicode">ϕ is an irrational number that appears in many branches of mathematics.</span><br /> <br /> <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Golden_ratio" rel="nofollow" target="_blank">Wikipedia article on phi</a><br /> <br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-Musical applications"></a><!-- ws:end:WikiTextHeadingRule:2 -->Musical applications</h2> <br /> <span class="Unicode">Phi taken as a musical ratio (ϕ</span>*f where f=1/1) <span class="Unicode">is about 833.1 cents. This is sometimes called "acoustical phi".</span><br /> <span class="Unicode">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><br /> <br /> "Logarithmic phi", or 1200*<span class="Unicode">ϕ cents = 1941.6 cents (or, octave-reduced, 741.6 cents) is also useful.</span><br /> <br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x-Additional reading"></a><!-- ws:end:WikiTextHeadingRule:4 -->Additional reading</h2> <br /> <span class="w_hl"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Phi%20as%20a%20Generator">Phi</a></span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Phi%20as%20a%20Generator"> as a Generator </a><br /> <br /> <a class="wiki_link" href="/sqrtphi">sqrtphi</a>, a temperament based on the square root of phi (~416.5 cents) as a generator<br /> <br /> <span class="w_hl"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Golden%20Meantone">Golden</a></span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Golden%20Meantone"> Meantone </a><br /> <br /> <a class="wiki_link" href="http://xenharmonic.wikispaces.com/833%20Cent%20Golden%20Scale%20%28Bohlen%29">833 Cent </a><span class="w_hl"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/833%20Cent%20Golden%20Scale%20%28Bohlen%29">Golden</a></span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/833%20Cent%20Golden%20Scale%20%28Bohlen%29"> Scale (Bohlen) </a><br /> <br /> <a class="wiki_link_ext" href="http://dkeenan.com/Music/NobleMediant.txt" rel="nofollow" target="_blank">The Noble Mediant: Complex ratios and metastable musical intervals</a>, by <a class="wiki_link" href="/Margo%20Schulter">Margo Schulter</a> and <a class="wiki_link" href="/David%20Keenan">David Keenan</a></body></html>