Physical experiments on wave propagation on granular material control two macroscopic parameters that are assumed to characterize the response of a given assembly: the isotropic pressure p0 and the solid volume fraction . Here, by means of numerical simulation, we investigate the effect of the coordination number (the average number of contacts per particle) and the fluctuation of the number of contacts per particle Z′. We adopt a numerical protocol to create several initial packings characterized by the same volume fraction and, for a given isotropic pressure, we are able to obtain different coordination numbers. That is, pressure and coordination number, for a given volume fraction, are independent. The result is that packings with fixed volume fraction and isotropic pressure exhibit a different elastic response. We attribute this behavior only to the coordination number as we find, surprisingly, that is linked to Z′.