The elastic response of disordered mixtures of granular and soft matter is investigated by means of wave-propagation, both experimentally and numerically. This allows inferring fundamental properties of granular and soft disordered materials such as elastic moduli and dissipation mechanisms. Mixtures are prepared with different volumes of soft matter mixed with hard matter to identify the transition from a rigid to a soft granular skeleton. We compare physical experiments in a triaxial cell equipped with piezoelectric wave transducers and Discrete Element Method simulations (DEM). DEM simulations offer deeper insights into the micromechanics in and at the transition between the stiff- and soft-dominated regimes. Interestingly, we find that the behavior is highly non-linear and also non-monotonic with increasing the percentage of soft content. The presence of soft particles alters the formation of force chains. The qualitative agreement between experiment and simulation is well captured despite of complex interaction between soft and stiff particles that render several ongoing challenges like, e.g., the presence multi-body-interactions instead of pairwise contacts.