We show that for uniform domains Ω ⊆ ℝd+1 whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to α-dimensional Hausdorff measure unless α ≤ d. We employ a lemma that shows that, at almost every non-degenerate point, we may find a tangent measure of harmonic measure whose support is the boundary of yet another uniform domain whose harmonic measure resembles the tangent measure. © 2018 European Mathematical Society.