Golden ratio: Difference between revisions

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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>

The '''golden ratio''' or '''phi''' (Greek letter <math>\varphi</math> or <math>\phi</math>) is an irrational number that appears in many branches of mathematics, defined as the <math>\frac{a}{b}</math> such that <math>\frac{a}{b} = \frac{a+b}{a}</math>. It follows that <math>\varphi - 1 = \frac1{\varphi}</math>, and also that <math>\varphi = \frac{1+\sqrt{5}}{2}</math>, or approximately 1.6180339887...

~~: 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 / <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>

== Musical applications ==

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

The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. These two versions of phi have completely different musical applications which can be read about in detail on their separate pages. [[Lemba]] is a notable [[regular temperament]] for approximating both versions of phi simultaneously, requiring only two of its [[generators]] for logarithmic phi, and only one each of its generator and [[period]] for acoustic phi.

== Compositions based on the golden ratio ==

* ''[[Star Nursery]]'' - [[Sean Archibald]] (2021)

* ''[[Abyss]]'' - [[T.C. Edwards]] (2024)

==~~Musical applications~~==

== External links ==

* [http://tonalsoft.com/enc/p/phi.aspx Phi Φ / phi φ] on [[Tonalsoft Encyclopedia]]

~~<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>~~

[[Category:Golden ratio]]

<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>

[[Category:Irrational intervals]]

~~"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==~~

~~[[Generating a scale through successive divisions of the octave by the Golden Ratio~~]]

~~<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</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"><html><head><title>Golden Ratio</title></head><body><h2 id="toc0"><a name="x-Introduction"></a>Introduction</h2>

~~<br />~~

The &quot;golden ratio&quot; or &quot;phi&quot; (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 />~~

<h2 id="toc1"><a name="x-Musical applications"></a>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 &quot;acoustical phi&quot;.</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 />~~

&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 />~~

<h2 id="toc2"><a name="x-Additional reading"></a>Additional reading</h2>

~~<br />~~

<a class="wiki_link" href="/Generating%20a%20scale%20through%20successive%20divisions%20of%20the%20octave%20by%20the%20Golden%20Ratio">Generating a scale through successive divisions of the octave by the Golden Ratio</a><br />

~~<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></pre></div>

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Wikipedia}}
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+The '''golden ratio''' or '''phi''' (Greek letter <math>\varphi</math> or <math>\phi</math>) is an irrational number that appears in many branches of mathematics, defined as the <math>\frac{a}{b}</math> such that <math>\frac{a}{b} = \frac{a+b}{a}</math>. It follows that <math>\varphi - 1 = \frac1{\varphi}</math>, and also that <math>\varphi = \frac{1+\sqrt{5}}{2}</math>, or approximately 1.6180339887...
-: 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;
+== Musical applications ==
-[[@http://en.wikipedia.org/wiki/Golden_ratio|Wikipedia article on phi]]
+The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. These two versions of phi have completely different musical applications which can be read about in detail on their separate pages. [[Lemba]] is a notable [[regular temperament]] for approximating both versions of phi simultaneously, requiring only two of its [[generators]] for logarithmic phi, and only one each of its generator and [[period]] for acoustic phi.
+== Compositions based on the golden ratio ==
+* ''[[Star Nursery]]'' - [[Sean Archibald]] (2021)
+* ''[[Abyss]]'' - [[T.C. Edwards]] (2024)
-==Musical applications==
+== External links ==
+* [http://tonalsoft.com/enc/p/phi.aspx Phi Φ / phi φ] on [[Tonalsoft Encyclopedia]]
-&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;
+[[Category:Golden ratio]]
-&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;
+[[Category:Irrational intervals]]
-"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;
-==Additional reading==
-[[Generating a scale through successive divisions of the octave by the Golden Ratio]]
-&lt;span class="w_hl"&gt;[[xenharmonic/Phi as a Generator|Phi]]&lt;/span&gt;[[xenharmonic/Phi as a Generator| as a Generator]]
-[[sqrtphi]], a temperament based on the square root of phi (~416.5 cents) as a generator
-&lt;span class="w_hl"&gt;[[xenharmonic/Golden Meantone|Golden]]&lt;/span&gt;[[xenharmonic/Golden Meantone| Meantone ]]
-[[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]] 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>
-<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>