As the name suggests natural frequency is natural. It depends on the mass and stiffness of the component. Formula to calculate natural frequency is ω=√(k/m). So natural frequency is directly proportional to square root of stiffness (geometric characteristics such as thickness, length, breadth and width etc, stiffness change due to loading) and inversely proportional to square root of mass.

So if you go on increasing the thickness of membrane, the natural frequency is going to increase in proportion of square root of thickness. If there is decrease in thickness then there will be similar decrease in proportion of square root of thickness.

Pre-stretch can increase natural frequency since the stiffness due to stretching is increased (by pre-stretching membrane the looseness of membrane vanishes and membrane becomes stiffer and can marginally take in-plane load than loose membrane) although thickness is reduced. So any change in stiffness due to geometric or load conditions, the natural frequency of components can be altered.