The dominant minerals in Earth's lower mantle are thought to be Fe- and Al-bearing MgSiO3 bridgmanite and (Mg, Fe)O ferropericlase(1). However, experimental measurements of the elasticity of these minerals at realistic lower-mantle pressures and temperatures remain impractical. As a result, different compositional models for the Earth's lower mantle have been proposed(2-4). Theoretical simulations, which depend on empirical evaluations of the effects of Fe incorporation into these minerals, support a pyrolitic lower mantle that contains a significant amount of ferropericlase(5,6), much like the Earth's upper mantle. Here we present first-principles computations combined with a lattice dynamics approach that include the effects of Fe2+ and Fe3+ incorporation. We calculate the densities and elastic-wave velocities of several possible lower-mantle compositions with varying amounts of ferropericlase along a mantle geotherm. On the basis of our calculations of aggregate elasticities, we conclude that neither a perovskitic composition (about 9:1 bridgmanite to ferropericlase by volume) nor an olivine-like composition (about 7:3) reproduces the seismological reference model of average Earth properties. However, an intermediate volume fraction (about 8:2) consistent with a pyrolitic composition can reproduce the reference velocities and densities. Bridgmanite that is rich in ferric iron produces the best fit. Our findings support a uniform chemical composition throughout the present-day mantle, which we suggest is the result of whole-mantle convection.