Golden ratio: Difference between revisions
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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> | 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] | [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> | <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> | ||
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"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> | "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 == | ||
[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 | |||
Revision as of 14:39, 25 September 2020
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.
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".
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's "Golden Horagrams".
Additional reading
- 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)
- The Noble Mediant: Complex ratios and metastable musical intervals, by Margo Schulter and David Keenan
- 5- to 9-tone, octave-repeating scales from Wilson's Golden Horagrams of the Scale Tree, by David Finnamore