Interleaving spirals are a natural setting for attaining fast MRI signal reconstruction. Using results of Beurling and Landau, as well as quantitative coverings of the spectral domain by translates of the polar set of the target disk space E, harmonics for Fourier frames F are constructed on interleaving spirals to reconstruct signals on E. Because of weak Fourier frame bound estimates by standard calculations, implementation is addressed in terms of finite frame approximants of F and convex symmetric polygonal approximants of E. The methods are extended to motion problems in MRI, and to finite tight frames on multidimensional spheres.