The claim about the golden ratio in music actually refers to form, not to frequency (though that doesn't stop people from making music with tunings related to the golden ratio, but anyway). The claim is that if you have some work with an AB form, the A and B sections will ideally have durations in the golden ratio, etc., because the golden ratio provides the best balance between durations or something. I think it's claimed that this proportion can be specifically found in the music of Mozart.
While we're at it, 2^(1/12) isn't particularly special either; it's mostly a coincidence and a compromise. "Ideal" frequency ratios are in small whole numbers, but these come with some mathematical challenges (like the fact that (3/2)^4 ≠ 5) and it was generally decided to settle on a compromise system with 12 equal steps to the octave rather than unequal steps with "nicer" numbers or, say, a different number of equal steps (like 19 or 31).