In this work, we exploit the two-scale homogenization approach to compute explicitly the band gaps for out-of-plane wave propagation in ternary locally resonant metamaterials (LRM) with two-dimensional periodicity. The homogenization approach, recently developed by the authors for binary LRM, leads to the definition of the dynamic effective mass density, depending on the frequency, that becomes negative near the resonant frequencies of the inclusions. The intervals of negative effective mass give the band gaps. These explicit...