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The research field of linear and weakly nonlinear wave dynamics of acoustic metamaterials isdevoting considerable attention to so‐called “phononic band gap” theory (refer, e.g., to [1-3]and references therein). It is well known that composite materials with periodic variations ofthe density and/or wave velocity can display band gaps where the propagation of mechanicalwaves is forbidden [4].The present work aims to reveal band gaps in tensegrity-based metamaterials and to exploitthe possibility of their tuning for the design and test of novel bandgap devices. Attention isfocused on the optimal design of 1-D chains of tensegrity units and lumped masses, which aretunable by varying the unit’s parameters and initial static precompression of the constituentunits (internal self-stress) and the whole structure (external prestress).The bandgap response of such metamaterials is investigated through analytic and numericalstudies. By suitably designing the topological arrangements of soft and hard tensegrity units,the given results highlight that the analysed systems can be effectively employed as novelwave steering devices, sound proof layers and/or vibration isolation devices [5-6].