Revised Harmonic Ocillator, Period and Why gravity doesn't matter

The Harmonic oscillator period is proportional to the square root of the mass, because the acceleration is inversely proportional to the mass, and period is proportional to 1/the square root of the acceleration. This is just as true in vertical and as in horzontal oscillators.

The period's relationship to the mass has nothing to do with the spring having to overcome greater gravitational force. The gravitational force is "automatically" canceled by the spring being stretched to its NEW EQUILBRIUM position. In other words the force the spring exerts on the mass while in the new equilibrium position already is equal to mg upward. Thus , if we add this +mg from the spring into our equations and also add the -mg due to the force of gravity, they cancel and we are left with exactly the same equations about the new equlibrium position that we would have for a horizontal spring/mass about the original relaxed (equilbrium) position of the spring. See me if this not yet clear to you.