Golden ratio: Difference between revisions

Line 1:

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.~~~~

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]

Line 5:

== 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".~~~~

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.~~~~

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".~~~~

"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 ==

Line 19:

* [[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

* [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: @@
-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 / 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]
@@ Line 5: / Line 5: @@
 == 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>
+Phi taken as a musical ratio (ϕ*f where f=1/1) is about 833.1 cents. This is sometimes called "acoustical 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>
+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*<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*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 ==
@@ Line 19: / Line 19: @@
 * [[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
+* [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]]