  Given a noetherian abelian category $\mathcal Z$ of homological dimension two
with a tilting object $T$, the abelian category $\mathcal Z$ and the abelian
category of modules over $\text{End} (T)^{\textit{op}}$ are related by a
sequence of two tilts; we give an explicit description of the torsion pairs
involved. We then use our techniques to obtain a simplified proof of a theorem
of Jensen-Madsen-Su, that $\mathcal Z$ has a three-step filtration by
extension-closed subcategories. Finally, we generalise Jensen-Madsen-Su's
filtration to a noetherian abelian category of any finite homological
dimension.
