The key equation for SHM is: #a=–ω^2x#
Where the angular frequency, #ω=2πf# #a# is the acceleration,
and #x# is the displacement.

As you can see the frequency is not related to the amplitude. Actually #ω# is a constant for this equation. It is the proportionality constant for #aprop-x#. Thus frequency is dependent only on the dimensions of the oscillator (e.g. mass of bob and length of string for pendula or force constant of spring and mass for mass-spring systems). This is the important point here, frequency is independent of amplitude only for a given oscillator. All SHM oscillators do not have the same frequency!

The equation, #a=–ω^2x#, can be proven mathematically but it is quite a long derivation. I will add it below when I have time.

In practical situations the presence of friction and drag do cause the frequency to change if amplitude changes, especially at large amplitudes.