Extensive band structure computations are performed for two-dimensional periodic arrays of rigid cylinders in air, with the Bloch vector being perpendicular to the cylinders. It is found that for any value of the period (of the square lattice), there is no acoustic gap for filling fraction f<30%; the magnitude of the gap, for f >30%, is inversely proportional to the period. Moreover, if the filling fraction exceeds 40%, a second gap, higher in frequency, opens up. Within these gaps (or stop bands) sound and vibrations are forbidden, irrespective of the direction of propagation. It is noted that for cylinders 2.9 cm in diameter and period of 10 cm (i.e., f=0.066), there is no acoustic gap for frequencies below 6.4 kHz. However, the density of states reveals prominent minima at 1.7 and 2.4 kHz. These frequencies do agree with those of the first two attenuation maxima observed in a recent experiment [Nature 378, 241 (1995)] and are indeed related to diffraction from the  and  planes. Thus, even with idealization, Sempere's sculpture (investigated in the above reference) exhibits pseudogaps---not full gaps. A proposed application of the above-mentioned band structure calculations is presented in the following abstract.