Theoretical and experimental determination of sonic band structures of two-dimensional (2-D) arrays of rigid cylinders in air is reported. We present measurements for square and triangular lattices. A variational method is employed to calculate the acoustic dispersion relation. Experimentally, a transmission technique and the analysis of the phase delay between the incident and scattered waves by the structure are used to construct the acoustic bands. The comparison between theory and experiments allows to fully characterize the band gaps and it has also demonstrated the existence of deaf bands; i.e., bands which cannot be excited due to symmetry reasons. For the case of square lattice we show that the structure with a filling fraction of 0.41 has a full acoustic gap