Golden ratio: Difference between revisions

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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.

==Phith root of Phi (<math>\sqrt[\varphi]{\varphi}</math>) ==

The '''phith root of phi''' (<math>\varphi^{1/\varphi}</math>, <math>\sqrt[\varphi]{\varphi}</math>, or approximately 514.878 cents) is another useful interval for generating golden scales. Similarly to [[logarithmic phi]] (<math>2^{\varphi}</math>), <math>\sqrt[\varphi]{\varphi}</math> can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the acoustic phi instead of the octave.

<math>\sqrt[\varphi]{\varphi}</math> divides [[acoustic phi]] logarithmically by phi, just as [[logarithmic phi]] divides the octave logarithmically by phi:

: 833.09¢ ÷ φ = 514.878¢

When using acoustic phi (~833.09¢) as the [[equave]] instead of the octave, <math>\varphi^{1/\varphi}</math> generates MOS scales at Fibonacci cardinalities (3, 5, 8, 13...). This creates fractal, self-similar scales where every interval relationship exhibits the golden ratio, which is useful for compositions that maximize golden properties while avoiding octave equivalence entirely.

[[49edo]] provides an exceptionally accurate approximation. Because 34 steps ≈ 832.65¢ (acoustic phi) *and* because 34 is a fibonacci number, it naturally follows that 21 steps, its previous fibonacci number, ≈ 514.29¢ (<math>\varphi^{1/\varphi}</math>). This pattern continues, creating a highly accurate golden scale of self-similar frequency relationships at step sizes 2, 3, 5, 8, 13, 21, 34.

[[7edo]] approximates this interval extremely well (0.59c off), as does all edos that are multiples of 7.

{{Interval Edo Approximation | interval = 134636/100000 | interval_name = <math>\varphi^{1/\varphi}</math>}}

== Compositions based on the golden ratio ==

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== External links ==

* [https://sevish.com/2017/golden-ratio-music-interval/ The Golden Ratio as a musical interval] by [[Sevish]]

* [http://tonalsoft.com/enc/p/phi.aspx Phi Φ / phi φ] on [[Tonalsoft Encyclopedia]]

[[Category:Golden ratio]]

[[Category:Irrational intervals]]

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 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.
+==Phith root of Phi (<math>\sqrt[\varphi]{\varphi}</math>) ==
+The '''phith root of phi''' (<math>\varphi^{1/\varphi}</math>, <math>\sqrt[\varphi]{\varphi}</math>,  or approximately 514.878 cents) is another useful interval for generating golden scales. Similarly to [[logarithmic phi]] (<math>2^{\varphi}</math>), <math>\sqrt[\varphi]{\varphi}</math> can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the acoustic phi instead of the octave.
+<math>\sqrt[\varphi]{\varphi}</math> divides [[acoustic phi]] logarithmically by phi, just as [[logarithmic phi]] divides the octave logarithmically by phi:
+: 833.09¢ ÷ φ = 514.878¢
+When using acoustic phi (~833.09¢) as the [[equave]] instead of the octave, <math>\varphi^{1/\varphi}</math> generates MOS scales at Fibonacci cardinalities (3, 5, 8, 13...). This creates fractal, self-similar scales where every interval relationship exhibits the golden ratio, which is useful for compositions that maximize golden properties while avoiding octave equivalence entirely.
+[[49edo]] provides an exceptionally accurate approximation. Because 34 steps ≈ 832.65¢ (acoustic phi) *and* because 34 is a fibonacci number, it naturally follows that 21 steps, its previous fibonacci number, ≈ 514.29¢  (<math>\varphi^{1/\varphi}</math>). This pattern continues, creating a highly accurate golden scale of self-similar frequency relationships at step sizes 2, 3, 5, 8, 13, 21, 34.
+[[7edo]] approximates this interval extremely well (0.59c off), as does all edos that are multiples of 7.
+{{Interval Edo Approximation | interval = 134636/100000 | interval_name = <math>\varphi^{1/\varphi}</math>}}
 == Compositions based on the golden ratio ==
@@ Line 11: / Line 26: @@
 == External links ==
+* [https://sevish.com/2017/golden-ratio-music-interval/ The Golden Ratio as a musical interval] by [[Sevish]]
 * [http://tonalsoft.com/enc/p/phi.aspx Phi Φ / phi φ] on [[Tonalsoft Encyclopedia]]
 [[Category:Golden ratio]]
 [[Category:Irrational intervals]]