Golden ratio: Difference between revisions
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{{Wikipedia}} | {{Wikipedia}} | ||
The '''golden ratio''' or '''phi''' (Greek letter <math>\ | 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... | ||
== Musical applications == | == Musical applications == | ||
The | 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. | ||
== External links == | == External links == | ||
* [http://tonalsoft.com/enc/p/phi.aspx Phi Φ / phi φ] on [[Tonalsoft Encyclopedia]] | * [http://tonalsoft.com/enc/p/phi.aspx Phi Φ / phi φ] on [[Tonalsoft Encyclopedia]] | ||
[[Category:Golden ratio | [[Category:Golden ratio]] | ||
[[Category:Irrational intervals]] | [[Category:Irrational intervals]] | ||
Revision as of 08:51, 1 July 2024
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.