This communication addresses the tuning of PI and PID controllers on the basis of the IMC approach. The tuning is based upon a first order plus time delay (FOPTD) model and aims to achieve a step response specification. Through analysis it has been found that by using the IMC approach we get a PI or a PID depending on the rational approximation used for the time delay term. This article raises the question that the use of a PID instead of a PI controller should be based on another reason more related to the control objectives rather than the use of a better approximation for the time delay. An alternative tuning is presented here, from within the IMC formulation, based on a min-max optimization. From the tuning rule provided by this approach the optimum settings from an integral squared error criterion point of view are derived. The optimal controller results in being a PI controller. From this optimal controller as the starting point, the introduction of the derivative action can be seen as a detuning procedure that can increase the robustness of the controller. This approach provides further insight into the tuning of PI and PID controllers giving the (alternative) parameters a precise engineering meaning.