We have analyzed the properties of oscillations in a sunspot based on SDO observations with a duration of 6 h. The 3-min oscillation spectrum has turned out to consist of dozens of spectral lines. The line widths are at the spectral resolution limit. The oscillations in a sunspot have been broken down into individual areas with a size of a few arcseconds, each having its own oscillation spectrum. These oscillation properties cannot be explained in terms of the existing models based on the assumption that the sunspot oscillates as a whole. We propose a model of local oscillations that explains the complex spectrum of the oscillations and their locality. We show that in addition to the chromospheric resonator, there exists a subphotospheric resonator for slow MHD waves. The existence of this resonance layer allows the locality of the oscillations and their complex spectrum to be explained in terms of Parker's model.