Obtaining meaningful quantitative descriptions of the statistical dependence within multivariate systems is a difficult open problem. Recently, the Partial Information Decomposition (PID) was proposed to decompose mutual information (MI) about a target variable into components which are redundant, unique and synergistic within different subsets of predictor variables. Here, we propose to apply the elegant formalism of the PID to multivariate entropy, resulting in a Partial Entropy Decomposition (PED). We implement the PED with an entropy redundancy measure based on pointwise common surprisal; a natural definition which is closely related to the definition of MI. We show how this approach can reveal the dyadic vs triadic generative structure of multivariate systems that are indistinguishable with classical Shannon measures. The entropy perspective also shows that misinformation is synergistic entropy and hence that MI itself includes both redundant and synergistic effects. We show the relationships between the PED and MI in two predictors, and derive two alternative information decompositions which we illustrate on several example systems. This reveals that in entropy terms, univariate predictor MI is not a proper subset of the joint MI, and we suggest this previously unrecognised fact explains in part why obtaining a consistent PID has proven difficult. The PED also allows separate quantification of mechanistic redundancy (related to the function of the system) versus source redundancy (arising from dependencies between inputs); an important distinction which no existing methods can address. The new perspective provided by the PED helps to clarify some of the difficulties encountered with the PID approach and the resulting decompositions provide useful tools for practical data analysis across a wide range of application areas.