Golden ratio: Difference between revisions
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Wikispaces>spt3125 **Imported revision 479296358 - Original comment: ** |
Wikispaces>spt3125 **Imported revision 479297806 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2013-12-24 | : This revision was by author [[User:spt3125|spt3125]] and made on <tt>2013-12-24 19:03:47 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>479297806</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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<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> | <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> | "Logarithmic phi", or 1200*<span class="Unicode">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".</span> | ||
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[[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) ]] | [[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]]</pre></div> | [[@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</pre></div> | |||
<h4>Original HTML content:</h4> | <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"><html><head><title>Golden Ratio</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Introduction"></a><!-- ws:end:WikiTextHeadingRule:0 -->Introduction</h2> | <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"><html><head><title>Golden Ratio</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Introduction"></a><!-- ws:end:WikiTextHeadingRule:0 -->Introduction</h2> | ||
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<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 /> | <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 /> | <br /> | ||
&quot;Logarithmic phi&quot;, or 1200*<span class="Unicode">ϕ cents = 1941.6 cents (or, octave-reduced, 741.6 cents) is also useful.</span><br /> | &quot;Logarithmic phi&quot;, or 1200*<span class="Unicode">ϕ cents = 1941.6 cents (or, octave-reduced, 741.6 cents) is also useful as a generator, for example in <a class="wiki_link" href="/Erv%20Wilson">Erv Wilson</a>'s &quot;Golden Horagrams&quot;.</span><br /> | ||
<br /> | <br /> | ||
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<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 /> | <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 /> | <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="/ | <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="/Dave%20Keenan">David Keenan</a><br /> | ||
<br /> | |||
<a class="wiki_link_ext" href="http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm" rel="nofollow" target="_blank">5- to 9-tone, octave-repeating scales from Wilson's Golden Horagrams of the Scale Tree</a>, by David Finnamore</body></html></pre></div> | |||
Revision as of 19:03, 24 December 2013
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author spt3125 and made on 2013-12-24 19:03:47 UTC.
- The original revision id was 479297806.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
==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 as a generator, for example in [[Erv Wilson]]'s "Golden Horagrams".</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 [[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
Original HTML content:
<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 as a generator, for example in <a class="wiki_link" href="/Erv%20Wilson">Erv Wilson</a>'s "Golden Horagrams".</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="/Dave%20Keenan">David Keenan</a><br /> <br /> <a class="wiki_link_ext" href="http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm" rel="nofollow" target="_blank">5- to 9-tone, octave-repeating scales from Wilson's Golden Horagrams of the Scale Tree</a>, by David Finnamore</body></html>