A vibrating system is constructed such that its natural frequencies are exact integer multiples of a base frequency. This system requires little energy to produce a periodic motion whose period is determined by the base frequency. The ability to amplify integer multiples of a base frequency makes this device an effective mechanical Fourier series generator. The proposed topology makes use of symmetry to assign poles and zeros at optimal frequencies. The system zeros play the role of suppressing the energy at certain frequencies while the poles amplify the input at their respective frequencies. An exact, non-iterative procedure is adopted to provide the stiffness and mass values of a discrete realization. It is shown that the spatial distributions of mass and stiffness are smooth; thus it is suggested that a continuous realization of a mechanical Fourier series generator is a viable possibility. A laboratory experiment and numerical examples are briefly described to validate the theory.