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| ==Introduction== | | {{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... |
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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>
| | == Musical applications == |
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| [http://en.wikipedia.org/wiki/Golden_ratio Wikipedia article on 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. |
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| ==Musical applications== | | == Compositions based on the golden ratio == |
| | * ''[[Star Nursery]]'' - [[Sean Archibald]] (2021) |
| | * ''[[Abyss]]'' - [[T.C. Edwards]] (2024) |
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| <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>
| | == External links == |
| | * [http://tonalsoft.com/enc/p/phi.aspx Phi Φ / phi φ] on [[Tonalsoft Encyclopedia]] |
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| <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>
| | [[Category:Golden ratio]] |
| | | [[Category:Irrational intervals]] |
| "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>
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| ==Additional reading==
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| [[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]]
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| <span style="">[[Phi_as_a_Generator|Phi]]</span>[[Phi_as_a_Generator| as a Generator]]
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| [[Sqrtphi|sqrtphi]], a temperament based on the square root of phi (~416.5 cents) as a generator
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| <span style="">[[Golden_Meantone|Golden]]</span>[[Golden_Meantone| Meantone ]]
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| [[833_Cent_Golden_Scale_(Bohlen)|833 Cent ]]<span style="">[[833_Cent_Golden_Scale_(Bohlen)|Golden]]</span>[[833_Cent_Golden_Scale_(Bohlen)| Scale (Bohlen) ]]
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| [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]]
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| [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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