We consider a consistent linear effective vielbein matter coupling without
introducing the Boulware-Deser ghost in ghost-free massive gravity. This is
achieved in the partially constrained vielbein formulation. We first introduce
the formalism and prove the absence of ghost at all scales. As next we
investigate the cosmological application of this coupling in this new
formulation. We show that even if the background evolution accords with the
metric formulation, the perturbations display important different features in
the partially constrained vielbein formulation. We study the cosmological
perturbations of the two branches of solutions separately. The tensor
perturbations coincide with those in the metric formulation. Concerning the
vector and scalar perturbations, the requirement of absence of ghost and
gradient instabilities yields slightly different allowed parameter space.