Golden ratio: Difference between revisions

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

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 ==

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]]. [[Lemba]] is particularly notable for approximating both simply and accurately simultaneously, at a generator + a period for acoustic and 2 generators for logarithmic, making it an excellent choice for experimenting with phi based composition. [[Triforce]] is also essentially based on dividing the 1/3 octave period into logarithmic phi sized fractions.

[[Category:Golden ratio| ]]

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.

[[Category:~~Theory~~]]

[[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]]

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 {{Wikipedia}}
-The '''golden ratio''' or '''phi''' (Greek letter <math>\phi</math> / <math>\varphi</math>) may be defined by <math>\frac{a}{b}</math> such that <math>\frac{a}{b} = \frac{a+b}{a}</math>. It follows that <math>\varphi - 1 = 1 / \varphi</math>, and also that <math>\varphi = \frac{1+\sqrt{5}}{2}</math>, or approximately 1.6180339887... <math>\varphi</math> is an irrational number that appears in many branches of mathematics.
+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 ==
-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]]. [[Lemba]] is particularly notable for approximating both simply and accurately simultaneously, at a generator + a period for acoustic and 2 generators for logarithmic, making it an excellent choice for experimenting with phi based composition. [[Triforce]] is also essentially based on dividing the 1/3 octave period into logarithmic phi sized fractions.
-[[Category:Golden ratio| ]]
+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.
-[[Category:Theory]]
+[[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]]