Golden ratio: Difference between revisions

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~~==Introduction==~~

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-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.

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

==Musical applications==

== 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".

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"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".

==Additional reading==

== Additional reading ==

~~[[Generating_a_scale_through_successive_divisions_of_the_octave_by_the_Golden_Ratio|Generating a scale through successive divisions of the octave by the Golden Ratio]]~~

~~[[Phi_as_a_Generator|Phi]][[Phi_as_a_Generator| as a Generator]]~~

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

~~[[Golden_Meantone|Golden]][[Golden_Meantone| Meantone ]]~~

~~[[833_Cent_Golden_Scale_(Bohlen)|833 Cent ]][[833_Cent_Golden_Scale_(Bohlen)|Golden]][[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|Margo Schulter]] and [[Dave_Keenan|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

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

* [[Phi as a Generator]]

* [[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

@@ Line 1: / Line 1: @@
-==Introduction==
 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>
 [http://en.wikipedia.org/wiki/Golden_ratio Wikipedia article on phi]
-==Musical applications==
+== 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>
@@ Line 13: / Line 11: @@
 "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>
-==Additional reading==
+== Additional reading ==
-[[Generating_a_scale_through_successive_divisions_of_the_octave_by_the_Golden_Ratio|Generating a scale through successive divisions of the octave by the Golden Ratio]]
-<span style="">[[Phi_as_a_Generator|Phi]]</span>[[Phi_as_a_Generator| as a Generator]]
-[[Sqrtphi|sqrtphi]], a temperament based on the square root of phi (~416.5 cents) as a generator
-<span style="">[[Golden_Meantone|Golden]]</span>[[Golden_Meantone| Meantone ]]
-[[833_Cent_Golden_Scale_(Bohlen)|833 Cent ]]<span style="">[[833_Cent_Golden_Scale_(Bohlen)|Golden]]</span>[[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|Margo Schulter]] and [[Dave_Keenan|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
+* [[Generating a scale through successive divisions of the octave by the Golden Ratio]]
+* [[Phi as a Generator]]
+* [[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