Odd mass density: Solids built out of active components have exhibited odd elastic stiffness tensors whose active moduli appear in the antisymmetric part and which give rise to non-Hermitian static and dynamic phenomena. Here we present a new class of active metamaterial featured with an odd mass density tensor whose asymmetric part arises from active and non-conservative forces. The odd mass density is realized using metamaterials with inner resonators connected by asymmetric and programmable feed-forward control on acceleration and active forces along the two perpendicular directions. The active forces produce unbalanced off-diagonal mass density coupling terms, leading to 1D nonreciprocal wave coupling and 2D directional wave amplification. Contributors: Guoliang Huang and Qian Wu