Golden ratio
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. 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]\displaystyle{ \sqrt[\varphi]{\varphi} }[/math])
The phith root of phi ([math]\displaystyle{ \varphi^{1/\varphi} }[/math], [math]\displaystyle{ \sqrt[\varphi]{\varphi} }[/math], or approximately 514.878 cents) is another useful interval for generating golden scales. Similarly to logarithmic phi ([math]\displaystyle{ 2^{\varphi} }[/math]), [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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.
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 7 | 3\7 | 514.29 | -0.59 | -0.34 |
| 14 | 6\14 | 514.29 | -0.59 | -0.69 |
| 21 | 9\21 | 514.29 | -0.59 | -1.03 |
| 28 | 12\28 | 514.29 | -0.59 | -1.38 |
| 35 | 15\35 | 514.29 | -0.59 | -1.72 |
| 42 | 18\42 | 514.29 | -0.59 | -2.07 |
| 49 | 21\49 | 514.29 | -0.59 | -2.41 |
| 56 | 24\56 | 514.29 | -0.59 | -2.76 |
| 63 | 27\63 | 514.29 | -0.59 | -3.10 |
| 70 | 30\70 | 514.29 | -0.59 | -3.45 |
| 77 | 33\77 | 514.29 | -0.59 | -3.79 |
Compositions based on the golden ratio
- Star Nursery - Sean Archibald (2021)
- Abyss - T.C. Edwards (2024)