Golden ratio: Difference between revisions
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The | 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] | [http://en.wikipedia.org/wiki/Golden_ratio Wikipedia article on phi] | ||
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== 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". | |||
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* | "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 == | == Additional reading == | ||
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* [[833 Cent Golden Scale (Bohlen)]] | * [[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://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 | * [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 10:59, 26 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