This work is aimed at theoretically and numerically investigating the presence of frequency band gaps in the wave dynamics of 1D tensegrity-based composite metamaterials, and to exploit the possibility of their tuning for the design and test of novel waveguides, soundproof layers and vibration protection devices. Building on established results for structures consisting of periodic lattice geometries, this work shows how it is possible to design band gap metamaterials formed by chains of tensegrity units and lumped masses, which will be tunable by varying the unit’s parameters for both the initial static pre-compression of the constituent units and the whole structure.