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The response of masonry structures to explosions can be hardly investigated relying only on numerical and analytical tools. Experimental tests are of paramount importance for improving the current comprehension and validate existing models. However, experiments involving blast scenarios are, at present, partial and limited in number, compared to tests under different dynamic conditions, such as earthquakes. The reason lies on the fact that full-scale blast experiments present many difficulties, mainly due to the nature of the loading action. Experiments in reduced-scale offer instead greater flexibility. Nevertheless, appropriate scaling laws for the response of masonry structures under blast excitations are needed before performing such tests. We propose here new scaling laws for the dynamic, rigid-body response and failure modes of masonry structures under blast loads. This work takes its roots from previous studies, where closed-form solutions for the rocking response of slender blocks due to explosions have been derived and validated against numerical and experimental tests. The proposed scaling laws are here validated with detailed numerical simulations accounting for combined rocking, up-lifting, and sliding mechanisms of monolithic structures. Then, the application to multi-drum stone columns is considered. In particular, we show that, whilst the presence of complex behaviors, such as wobbling and impacts, similarity is assured. The developments demonstrate their applicability in the design of reduced-scale experiments of masonry structures.