Golden ratio: Difference between revisions

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The "golden ratio" or "phi" (Greek letter Φ / φ / ~~ϕ ) ~~may be~~ defined by a/b such that a/b = (a+b)/a. It follows that ϕ<~~/span~~>-1 = 1/~~ϕ, and also that ϕ = (1+sqrt(5))/2, or approximately ~~~~1.6180339887... ~~ϕ 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...

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

== Musical applications ==

~~Phi taken~~ as a ~~musical ratio (ϕ*f where f=1/1) ~~is ~~about 833.1 cents. This~~ is ~~sometimes called "acoustical~~ 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.

~~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~~.~~~~

~~"Logarithmic phi", or 1200*ϕ cents~~ = ~~1941.6 cents~~ (~~or, octave-reduced, 741.6 cents~~) ~~is also useful as a generator, for example in~~ [[~~Erv_Wilson|Erv Wilson~~]]'~~s "Golden Horagrams"~~.~~~~

== Compositions based on the golden ratio ==

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

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

== ~~Additional reading~~ ==

== External links ==

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

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

[[Category:Golden ratio]]

* [[Phi as a Generator]]

[[Category:Irrational intervals]]

* [[sqrtphi]], a temperament based on the square root of phi (~416.5 cents) as a generator

* [[Golden ~~meantone~~]]

* [[~~833 Cent Golden Scale (Bohlen)]]~~

* [http:~~//dkeenan.com/Music/NobleMediant.txt The Noble Mediant: Complex ratios and metastable musical~~ intervals]~~, by [[Margo Schulter]] and [[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~~

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-The "golden ratio" or "phi" (Greek letter Φ / φ / <span style="">ϕ ) may be defined by a/b such that a/b = (a+b)/a. It follows that ϕ</span>-1 = 1/<span style="">ϕ, and also that ϕ = (1+sqrt(5))/2, or approximately </span>1.6180339887... <span style="">ϕ is an irrational number that appears in many branches of mathematics.</span>
+{{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...
-[http://en.wikipedia.org/wiki/Golden_ratio Wikipedia article on phi]
 == Musical applications ==
-<span style="">Phi taken as a musical ratio (ϕ</span>*f where f=1/1) <span style="">is about 833.1 cents. This is sometimes called "acoustical phi".</span>
+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.
-<span style="">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>
-"Logarithmic phi", or 1200*<span style="">ϕ cents = 1941.6 cents (or, octave-reduced, 741.6 cents) is also useful as a generator, for example in [[Erv_Wilson|Erv Wilson]]'s "Golden Horagrams".</span>
+== Compositions based on the golden ratio ==
+* ''[[Star Nursery]]'' - [[Sean Archibald]] (2021)
+* ''[[Abyss]]'' - [[T.C. Edwards]] (2024)
-== Additional reading ==
+== External links ==
+* [http://tonalsoft.com/enc/p/phi.aspx Phi Φ / phi φ] on [[Tonalsoft Encyclopedia]]
-* [[Generating a scale through successive divisions of the octave by the Golden Ratio]]
+[[Category:Golden ratio]]
-* [[Phi as a Generator]]
+[[Category:Irrational intervals]]
-* [[sqrtphi]], a temperament based on the square root of phi (~416.5 cents) as a generator
-* [[Golden meantone]]
-* [[833 Cent Golden Scale (Bohlen)]]
-* [http://dkeenan.com/Music/NobleMediant.txt The Noble Mediant: Complex ratios and metastable musical intervals], by [[Margo Schulter]] and [[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