Turbulent convection of the liquid iron alloy outer core generates Earth's magnetic field and supplies heat to the mantle. The exact composition of the iron alloy is fundamentally linked to the processes powering the convection, and can to be constrained by its seismic properties. Discrepancies between seismic models determined using body waves and normal modes show that these properties are not yet fully agreed upon. Additionally, technical challenges in experimentally measuring the equation-of-state (EoS) parameters of liquid iron alloys at high pressures and temperatures further complicate compositional inferences. We directly infer EoS parameters describing the Earth's outer core from normal mode center frequency observations, and present the resulting Elastic Parameters of the Outer Core (EPOC) seismic model. Unlike alternative seismic models, ours requires only three parameters and guarantees physically realistic behavior with increasing pressure for a well-mixed homogeneous material along an isentrope, consistent with the outer core's condition. We show that EPOC predicts available normal mode frequencies better than the Preliminary Reference Earth Model (PREM) while also being more consistent with body-wave derived models, eliminating a longstanding discrepancy. The velocity at the top of the outer core is lower, and increases with depth more steeply, in EPOC than in PREM, while the density in EPOC is higher than in PREM across the outer core. The steeper profiles and higher density imply the outer core comprises a lighter but more compressible alloy than that inferred for PREM. Furthermore, EPOC's steeper velocity gradient explains differential SmKS body wave travel times better than previous 1D global models, without requiring an anomalously slow $\sim$90-450 km thick layer at the top of the outer core.