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A coherent transferable tight-binding (TB) parametrization including magnetism has been developed for the Fe-C interaction. We use a downfolding procedure to obtain continuous and transferable Fe-C bond integrals from density functional theory. A TB model is constructed using these bond integrals and a parameterized interatomic repulsion based on simple exponentials fit to the excess energy of interstitial carbon in α, γ, and e-iron. An accurate description of the energy hierarchy of these structures can be achieved with only three fitting parameters. The model is used to calculate bulk properties and energies of a variety of Fe carbides, and good agreement is found for a number of properties. Excellent agreement with the individual elastic constants of cementite is achieved. We obtain a satisfactory migration energy barrier of carbon in α iron and confirm the instability of carbon in the tetrahedral interstitial site. Defect binding energies are calculated for a number of defects and largely agree with density functional theory calculations. This simple model based on physical insights may be used to study systems containing thousands of atoms. Furthermore, it may be employed as the basis for O(N) bond-order potentials.