The intent-oriented search diversification methods developed in the field so far tend to build on generative views of the retrieval system to be diversified. Core algorithm components –in particular redundancy assessment– are expressed in terms of the probability to observe documents, rather than the probability that the documents be relevant. This has been sometimes described as a view considering the selection of a single document in the underlying task model. In this paper we propose an alternative formulation of aspect-based diversification algorithms which explicitly includes a formal relevance model. We develop means for the effective computation of the new formulation, and we test the resulting algorithm empirically. We report experiments on search and recommendation tasks showing competitive or better performance than the original diversification algorithms. The relevance-based formulation has further interesting properties, such as unifying two well-known state of the art algorithms into a single version. The relevance-based approach opens alternative possibilities for further formal connections and developments as natural extensions of the framework. We illustrate this by modeling tolerance to redundancy as an explicit configurable parameter, which can be set to better suit the characteristics of the IR task, or the evaluation metrics, as we illustrate empirically.