Let's say I had a song last for 300 seconds (5 minutes). How would I use the golden ratio (1.61803399) to find what point in the song the golden ratio is at (in seconds).
An example: Uptown Funk is 270 seconds long, the "break down" of the song is at 167 seconds. If you did 270 divided by 167, you get 1.61676647. I'm trying to reverse it, so you can find the time of ANY song where the golden ratio point would be.
If the song has length $T$ (with $T>0$), and you cut it at time $t$ then you want $$ \frac{t}{T} = \frac{T-t}{t} $$ as ratio equation.
This is equivalent to $$ (t/T)^2 = 1 - (t/T) $$ or $$ (t/T)^2 + (t/T) - 1 = 0 \iff \\ x^2 + x - 1 = 0 $$ for $x=t/T$. It has the solutions $$ x^2 + 2 (1/2) x + (1/2)^4 - 1 = (1/2)^4 \iff \\ (x + 1/2)^2 = 1 + 1/4 = 5/4 \iff \\ x = (\pm\sqrt{5}-1)/2 $$ where we use the positive solution $$ x = (\sqrt{5}-1)/2 = 0.618033\cdots $$ So your sought time is $$ t = x T $$