In hard real-time systems, a significant disparity in schedulability exists between EDF-based scheduling algorithms and Pfair scheduling, which is the only known way of optimally scheduling recurrent real-time tasks on multiprocessors. This is unfortunate because EDF-based algorithms entail lower scheduling and task-migration overheads. In work on hard real-time systems, it has been shown that the disparity in schedulability can be lessened by placing caps on per-task utilizations. In this paper, we show that it can also be lessened by easing the requirement that all deadlines be met. Our main contribution is a new EDF-based scheme that ensures bounded deadline tardiness. In this scheme, per-task utilizations must be capped, but overall utilization need not be restricted. The required cap is quite liberal. Hence, our scheme should enable a wide range of soft real-time applications to be scheduled with no constraints on total utilization. We also propose techniques and heuristics that can be used to reduce tardiness.