Forget About Phi
First, what is the golden ratio? It’s the ratio that you get if you divide up a line into two sections such that the ratio of the larger to the smaller is the same as the whole to the larger. This comes to about 1.618, and is often called phi. You can also approximate it by dividing two consecutive Fibonacci numbers.
What’s so important about this ratio? That’s the thing, there’s nothing special about it at all. Many claims are made about it (see the first link), but none are true. It’s not in the Parthenon, it’s not in any Renaissance paintings, it’s really not anywhere. It’s not even the most pleasing ratio to the eye, studies asking people to pick out the most pleasing rectangle (whatever that means) do not find ones involving phi to be the most pleasing.
Don’t believe me? Pick up The Golden Ratio: The Story of Phi, the World's Most Astonishing Number by Mario Livio. He discusses where phi does and does not show up, and it’s almost never anywhere sensational. It is found in certain places, such as plants and crystals. But phi was first defined because it appears in the pentagram, so it shouldn’t be entirely surprising that a geometric ratio appears in things constructed geometrically, like crystals.
As for plants, it most frequently appears in the ratios of the number of degrees between two leaves on a stalk. Livio hypothesizes that this is an evolutionary adaptation, that because phi is irrational it allows the most leaves to be exposed to the sun at once.
I can’t say so much about the second link, but since phi being pleasing optically is bunk, I’m skeptical that it’s pleasing musically as well.
Another thing to note is that nothing is ever “equal” to phi, just “close”. Often 1.6 is close enough for people to claim that phi is involved. When you think about how much wiggle room there is in measuring lengths (or times) and how many different places you can possibly measure, something “close” to phi is going to come up frequently. Don’t buy into it, it’s nothing special, just someone grasping for straws.
Besides, phi isn’t even fundamental. It’s not in any physics equation, or any equation at all (as far as I know). Marvel at pi or c or h or alpha or e or the other e, but not phi. Phi just doesn’t deserve it.