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The '''golden ratio''' or '''phi''' (Greek letter | {{Wikipedia}} | ||
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 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. A third interval, the [[phith root of phi]] ([math]\displaystyle{ \sqrt[\varphi]{\varphi} }[/math]), acts as a bridge between the two: it divides acoustic phi logarithmically by phi, enabling golden MOS scales with acoustic phi as the equave. | |||
[[Lemba]] is a notable [[regular temperament]] for approximating both acoustic and logarithmic 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) | |||
== | == External links == | ||
* [https://sevish.com/2017/golden-ratio-music-interval/ The Golden Ratio as a musical interval] by [[Sevish]] | |||
* [http://tonalsoft.com/enc/p/phi.aspx Phi Φ / phi φ] on [[Tonalsoft Encyclopedia]] | |||
[[Category:Golden ratio]] | |||
[[Category:Irrational intervals]] | |||
Latest revision as of 09:08, 26 November 2025
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. A third interval, the phith root of phi ([math]\displaystyle{ \sqrt[\varphi]{\varphi} }[/math]), acts as a bridge between the two: it divides acoustic phi logarithmically by phi, enabling golden MOS scales with acoustic phi as the equave.
Lemba is a notable regular temperament for approximating both acoustic and logarithmic 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)