  Let $(X,J,\omega)$ be a compact $2n$-dimensional almost K\"ahler manifold. We
prove primitive decompositions for Bott-Chern and Aeppli harmonic forms in
special bidegrees and show that such bidegrees are optimal. We also show how
the spaces of primitive Bott-Chern, Aeppli, Dolbeault and $\partial$-harmonic
forms on $(X,J,\omega)$ are related.
