In previous work (arXiv:1602.01166, arXiv:1707.06243) it has been shown there is a close connection between MERA and wavelets. This correspondence was made explicit for massless free fermions (or equivalently the critical Ising model). In this talk I will explain an analogous construction for free bosonic chains. In this setting it is natural to use biorthogonal wavelets. The dispersion relation of the Hamiltonian translates to a wavelet filter design problem. In contrast to the fermionic model this construction is not restricticed to critical Hamiltonians. The field theory limit has a natural interpretation in terms of the wavelet functions. Based on arXiv:2004.11952, joint work with Michael Walter.