Force control implemented by a passive mechanical device (perhaps a wrist) has inherent advantages over an active implementation of the same force control law. Such a passive device can regain some of the versatility of its active counterpart if it comprises mechanical elements with programmable parameters, e.g., damping coefficients, spring stiffnesses, etc. The authors characterize the range of admittance matrices that a passive device may be programmed to possess. They stress that the admittance matrices that can be achieved belong to a much broader class than just ones having a 'center'. The authors describe the topological properties of the set of attainable matrices in task-space, and show that each matrix can be composed of positive linear combinations of a fixed set of basis matrices. They compare the volume of the space of attainable matrices to the volume of all positive real matrices, and suggest a method of visualizing the spaces in low-dimensional examples.