|
|
| Line 4: |
Line 4: |
| == 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 == |