We show that Harmonic Grammars (HGs) translate into linear systems and are thus solvable using the simplex algorithm, an efficient, widely-deployed optimization algorithm that is guaranteed to deliver the optimal solution if there is one and to detect when no solution exists. Our associated software package HaLP provides a practical tool for studying even large and complex HGs. We provide an initial comparison of HG with standard Optimality Theory and with the enriched version allowing local constraint conjunction. This comparison shows that HG has considerable potential as a framework for the study of typology. The availability of HaLP can facilitate the future evaluation of that potential.