A theoretical framework is presented for designing birdcage resonators for MRI and MR spectroscopy. The analogy between the birdcage problem and the phonon problem in solid-state physics is used to achieve multiple tuning. Allowing that the capacitances in the columns of the cage assume unequal values, it is possible to achieve multiple tuning and simultaneously preserve the sinusoidal current distribution necessary to set a homogeneous magnetic field. Given the physical dimensions of the columns and branches of the cage as well as the desired resonant frequencies, the corresponding values of the capacitances can be calculated exactly. Closed-form expressions for the capacitances are given in terms of the mutual inductances and the desired resonant frequencies. A detailed analysis for a symmetrical low-pass birdcage is presented. The expressions for the resonant frequencies reduce to those given by other authors when only nearest-neighbor mutual inductances are included.