The 5th is approximately 1.49 higher than the root in frequency which is close-ish to 1.618~ but it fails to accomplish the golden ratio of a+b is to a as a is to b. If you subtract the root frequency from the 5th frequency, and then subtract the octave frequency from the 5th frequency, then you now have a and b distances.

Distance A+B frequencies / A will not equal A/B as a golden ratio does.

For example:

C=16.35

G=24.5

C8=32.7

G-C=8.15

C8-G=8.2

(G-C)+(C8-G)=16.35

(G-C)/(C8-G)=0.99~

((G-C)+(C8-G))/(G-C)=2.006~

So the frequency range itself in distance isn't phi ratio properly.

I've seen some try interval count to equal phi, but those attempts tend to just use the raw interval value rather than the distance. For example using 12/7 instead of 12 - 7 to get b and then say

a: 7

b: 5

a+b: 12

a/b:1.4~

(a+b)/a:1.7~

So the interval idea doesn't produce phi either.

The self-similar nature of phi is fascinating but with sound, that would produce too similar and symmetrical of sound to be interesting.

When you start looking at chords and their propagations you start to spot that nice strong chords have a slight asymmetry to them; something phi is incapable of.

Cheers!