We have studied the spin excitations of ferromagnetic rings and observed a distinct series of quantized modes in the vortex state. We attribute them to spin waves that circulate around the ring and interfere constructively. They form azimuthal eigenmodes of a magnetic ring resonator which we resolve up to the fourth order. The eigenfrequencies are calculated semianalytically and classified as a function of magnetic field by a quantization rule which takes into account a periodic boundary condition. Strikingly each mode exists only below a characteristic field.