Golden ratio: Difference between revisions

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<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:spt3125|spt3125]] and made on <tt>2013-12-24 18:33:32 UTC</tt>.<br>
: The original revision id was <tt>479296358</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.&lt;/span&gt;
 
 
==Additional reading==
 
&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 [[David Keenan]]</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.&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;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="/David%20Keenan"&gt;David Keenan&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 00:31, 2 December 2024

English Wikipedia has an article on:

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

Musical applications

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

External links

Retrieved from "https://en.xen.wiki/index.php?title=Golden_ratio&oldid=168107"