Golden ratio: Difference between revisions

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== Musical applications ==
== Musical applications ==


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.  
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. A third interval, the [[phith root of phi]] ([math]\displaystyle{ \sqrt[\varphi]{\varphi} }[/math]), acts as a bridge between the two: it divides acoustic phi logarithmically by phi, enabling golden MOS scales with acoustic phi as the equave.


==Phith root of Phi (<math>\sqrt[\varphi]{\varphi}</math>) ==
[[Lemba]] is a notable [[regular temperament]] for approximating both acoustic and logarithmic phi simultaneously, requiring only two of its [[generators]] for logarithmic phi, and only one each of its generator and [[period]] for acoustic phi.  
 
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 ==
== Compositions based on the golden ratio ==

Latest revision as of 09:08, 26 November 2025

English Wikipedia has an article on:

The golden ratio or phi (Greek letter [math]\displaystyle{ \varphi }[/math] or [math]\displaystyle{ \phi }[/math]) is an irrational number that appears in many branches of mathematics, defined as the [math]\displaystyle{ \frac{a}{b} }[/math] such that [math]\displaystyle{ \frac{a}{b} = \frac{a+b}{a} }[/math]. It follows that [math]\displaystyle{ \varphi - 1 = \frac1{\varphi} }[/math], and also that [math]\displaystyle{ \varphi = \frac{1+\sqrt{5}}{2} }[/math], or approximately 1.6180339887...

Musical applications

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. A third interval, the phith root of phi ([math]\displaystyle{ \sqrt[\varphi]{\varphi} }[/math]), acts as a bridge between the two: it divides acoustic phi logarithmically by phi, enabling golden MOS scales with acoustic phi as the equave.

Lemba is a notable regular temperament for approximating both acoustic and logarithmic phi simultaneously, requiring only two of its generators for logarithmic phi, and only one each of its generator and period for acoustic phi.

Compositions based on the golden ratio

External links

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