[en] An undulator has two properties which make it an extremely attractive source of electromagnetic radiation. The first is that the radiation is concentrated in a number of narrow energy bands known as harmonics of the device. The second characteristic is that under favorable operating conditions, the energy of these harmonics can be shifted or open-quote tunedclose quotes over an energy interval which can be as large as two or three times the value of the lowest energy harmonic. Both the photon energy of an undulator as well as its tunability are determined by the period, λ, of the device, the magnetic gap, G (which is larger than the minimum aperture required for injection and operation of the storage ring) and the storage ring energy E_R. Given the photon energy, E_p, the above parameters ultimately define the limits of operation or tunability of the undulator. In general, the larger the tunability range, the more useful the device. Therefore, for a given required maximum photon energy, it is desirable to find the operating conditions and device parameters which result in the largest tunability interval possible. With this in mind, we have investigated the question of undulator tunability with emphasis on the role of the ring energy in order to find the smallest E_R consistent with the desired tunability interval and photon energy. As a guideline, we have included a preliminary criteria, concerning the tunability requirements for the Advanced Photon Source (APS) to be built at Argonne. The analysis is aimed at X-ray undulator sources on the APS but is applicable to any storage ring