  In 1965 Tauer produced a countably infinite family of semi-regular masas in
the hyperfinite $\mathrm{II}_1$ factor, no pair of which are conjugate by an
automorphism. This was achieved by iterating the process of passing to the
algebra generated by the normalisers and, for each $n\in\mathbb N$, finding
masas for which this procedure terminates at the $n$-th stage. Such masas are
said to have length $n$. In this paper we consider a transfinite version of
this idea, giving rise to a notion of ordinal valued length. We show that all
countable ordinals arise as lengths of semi-regular masas in the hyperfinite
$\mathrm{II}_1$ factor. Furthermore, building on work of Jones and Popa, we
obtain all possible combinations of regular inclusions of irreducible
subfactors in the normalising tower.
