Over the last two decades, a large variety of type I error rates and control procedures have been proposed in the field of multiple hypotheses testing. This paper proposes a framework that includes many existing proposals by investigating procedures in which the orderedp-values are compared to an arbitrary positive and non-decreasing threshold sequence. For this case, we derive the error rate being controlled under different assumptions on thep-values. Our focus will be on step-up procedures. The new formulation gives insight into the relations between existing error rates and opens new perspectives for the whole field of multiple testing.