This paper considers the dynamic response and performance characteristics of a special class of centrifugal pendulum torsional vibration absorbers. The absorbers of interest are designed by selection of the path that their center of mass follows, such that their dynamics are linear or nearly so, out to large amplitudes of motion, thereby avoiding the nonlinear-induced detuning that typically accompanies such responses. These order-tuned, tautochronic or isochronic, absorbers have been the subject of previous investigations, including analyses of the synchronous and certain nonsynchronous responses of systems comprised of a set of identical absorbers. The analysis and experiments have demonstrated that the synchronous response of such absorber systems can experience an instability that results in nonsynchronous responses in which a subset of absorbers have significantly larger amplitude than the corresponding synchronous response. In this study, we present results that generalize these stability results to include absorbers whose dynamics differ slightly from tautochronic by varying the absorber path such that both linear and nonlinear perturbations of perfect tuning are included. It is shown by analysis and verified by simulations that the perfect tuning case is quite special, specifically that the instability described above occurs for tunings very close to ideal and that the synchronous response can be made stable over the entire feasible operating range by employing small levels of linear and/or nonlinear detuning. Such detuning is known to have the additional benefit of resulting in smaller absorber responses and an attendant larger operating range albeit at the expense of absorber performance in terms of attenuating rotor torsional vibrations. The main conclusion of these results is that one can select a very small amount of detuning to avoid this undesirable instability and that such detuning does not have a significant effect on absorber effectiveness. The analytical results derived also provide a quantitative means of predicting synchronous absorber response amplitudes and the associated rotor torsional vibration levels, as well as the stability properties of these responses, results are very useful for the design of absorber systems.