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An analytic interatomic bond-order potential is derived that depends explicitly on the valence of the transition-metal element. It generalizes the second-moment Finnis-Sinclair and fourth-moment Carlsson potentials to include higher moments. We find that the sixth-moment approximation predicts not only the structural trend from hcp → bcc → hcp →fcc that is observed across the nonmagnetic 4d and 5d transition-metal series, but also the different ferromagnetic moments of the bcc, fcc, and hcp phases of the 3d transition-metal iron. An analytic expression for the force is obtained and proved to converge to the Hellmann-Feynman force as higher moments are included.