  We study the matter density perturbations in modified teleparallel gravity
theories, where extra degrees of freedom arise from the local Lorentz violation
in the tangent space. We formulate a vierbein perturbation with variables
addressing all the 16 components of the vierbein field. By assuming the perfect
fluid matter source, we examine the cosmological implication of the 6
unfamiliar new degrees of freedom in modified $f(T)$ gravity theories. We find
that despite the new modes in the vierbein scenario provide no explicit
significant effect in the small-scale regime, they exhibit some deviation from
the standard general relativity results in super-horizon scales.
