Mathieu function rescale problem

The Mathieu functions are the solutions for the equation

$$y''+(a-2q\cos(2z))y=0$$

If we require the solution has the form

$$y(z) = e^{i r z}f(z)$$

where $f(z)$ is a periodic function with period of $2\pi$, then the parameter $a$ should satisfy Mathieu characteristic function

$$a=\text{MatheiuCharacteristicA}(r,q)$$

My question is what's the condition of $a$ should satisfy, if the solution can be written as

$$y(z)=e^{irz}f(z)$$

where $f(z)$ is a periodic function with period 1, i.e., $f(z+1)=f(z)$?

Update

It seems that the period of the original Mathieu function should be $\pi$ instead of $2\pi$, according to Wikipedia, MathWorld, and DLMF, and the documentation of Mathematica may be incorrect on that. That means the ordinary scaling of the equation works fine.

I recommend you to use Scilab Mathieu Functions Toolbox, not Mathematica.

The parameter a (or b) is a function of order r and value of q - $a_r(q)$ (or $b_r(q)$), the dependency may be illustrated as follows (from aforementioned toolbox).

3. Update ATOMS database with atomsSystemUpdate
4. Install Mathieu Functions Toolbox with atomsInstall('Mathieu')