I think I have an idea about a 'solution' for this derived from the parameterization of the "normal" cylindrical helix.
x = a cos(t), y = a sin(t), and z = bt
where "a" equal the radius, and "b" equals the "pitch".
If "b" were a non-linear function of "t" then the "pitch" would vary non-linearly with "t". The challenge would be to come up with the correct non-linear function of "t" that produced the exact pitch values of "1" and "10" given in my original post.
Please comment if you have a more elegant solution, as I am not into reinventing the wheel.