Why the speed of sound does not depend upon Pressure??

1. The problem statement, all variables and given/known data

Why the speed of sound doesn't depend upon Pressure When it does depend on Density???

2. Relevant equations

c =sqrt(gamma p/rho)

3. The attempt at a solution

I honestly would've thought that higher pressures mean that particles are packed closer together, meaning that the wave would travel faster, but wikipedia says otherwise:

This appears to indicate that density doesn't play a role in things either.... which really baffles me. It says humidity DOES affect it, so its definitely in. Put in or leave out density, it's up to you. Maybe someone more learned could clarify this?

In an ideal gas, intermolecular gaps are always taken to be very large, regardless of pressure. Pressure is solely due to the interaction with (imaginary) walls, not due to the interaction between molecules.

His premise is spot on !!! Speed of sound depends on the speed of molecules i.e. temperature.
Pressure and density are related and the speed of sound does not depend on pressure or density.

It depends on what you think is being held constant and what being varied. If you manage to vary the pressure while keeping the density constant then, as shown in the formula, you will see the speed as depending on the pressure. But how would you manage to do that? ... by varying the temperature.
From the same link:
## c = \sqrt{\frac{\gamma p}{\rho}} = \sqrt{\frac{\gamma R T}M}= \sqrt{\frac{\gamma k T} m}##

I think the OP is really asking something about the physical mechanism for sound transport. To me, his question is somewhat analogous to "why is the acceleration of a falling body in a vacuum independent of this mass?"

In the case of sound pulses, you have a differential balance between pressure gradient forces and inertial forces (which depend on density), and you need to take into account the adiabatic compression of the gas via the ideal gas law. The net result is that the speed of sound turns out to be proportional to the square root of the pressure divided by the density (which is proportional to the absolute temperature). Formulating the partial differential wave equation to capture all these mechanisms is very straightforward to do, and automatically delivers the speed of the wave as a key parameter in the equation.