Responding to this part in the article:

> I’m not entirely sure about the intuitive meaning of “taking the derivative and dividing by 2π”. Is there some sort of fundamental connection to periodic functions?

If you have a function f(x) where x is measured in radians, and there are 2pi radians per turn, then you can change variables.

Let t represent turns. One turn is 2*pi rad, and you want t = 1 when you've gone all the way around in x, so t = x/2pi.

By the chain rule,

df(x)/dx = df(t)/dt dt/dx = 1/2pi * df(t)/dt

So I think this might be the meaning you're looking for when you do the rescaling of the derivative.

You're using turns as units instead of radians. cos(x=2pi)=cos(t=1)=1, and so on.