Phith root of 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 a 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 approximations for [math]\displaystyle{ \sqrt[\varphi]{\varphi} }[/math] (514.88 ¢)
≤ 80edo, relative error ≤ 10%

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)

• Durationplex - Sean Archibald (2025)

External links

• The Golden Ratio as a musical interval by Sevish