Soft to Harsh transition of wave shape

Question

Let h is variable with range [0,1]. If 1 represents harshness (noise osc),0 represents soft (sine), and 0.4 represents between harsh and soft (square).

Here is the order of common wave shape:

 0.0: sine
 0.1: triangle
 0.2: sawtooth
 0.4: square
 0.4 > noise (uncertainty)

Then, there must be transition between sine, triangle, sawtooth, and square.

And let x(t, h) is a signal over time that depends on defined harshness parameter.

How do you make a relationship of x, t and h?

Note: It doesn't need really exact to what I define. e.g. square at 0.7.

Here is harshness of wave shape:

Answer 1

You can make an argument that "harshness" as defined here is simply the amount of high frequency content. Every periodic function can be represented as a Fourier Series. For the triangle the amplitude of the harmonics drops with $1/n^2$ and for a square wave only with $1/n$, so you could use that as a harshness dial.

You'll probably need more than one parameter though, because it's not only the amplitude but also the phase that matters here.

For example, the graph below shows a square wave spectrum with the "correct" phase and one with simply a 0 phase. The time domain wave forms look completely different despite having the same magnitude spectrum. Interestingly enough, they do sound mostly identical, so you could argue that they have the same "harshness".