Cache memories have been traditionally precluded in real-time systems because of their unpredictable behavior. The needs of better performance have motivated the development of tools to obtain tighter bounds on the worst-case execution time (WCET) of cached programs. However, they do not allow preemption, because from the point of view of program analysis, the number of preemptions is unknown. To face this problem, the cache-related preemption cost can be considered in the schedulability analysis, or annulled by the use of private cache partitions. Regarding the first approach, in a previous paper we described how to incorporate the effect of the cache to the Response Time schedulability Analysis (RTA). RTA is an efficient analysis for preemptive fixed-priority schedulers. In this paper, we have improved such a technique by taking into account the harmonic relations amongst task frequencies. The resulting analysis is applied to sample task-sets to illustrate the potential gains in terms of schedulable utilization, particularly for highly demanding task-sets.