  In this paper, we propose a modification to the density approach to Stein's
method for intervals for the unit circle $\mathbb{S}^1$ which is motivated by
the differing geometry of $\mathbb{S}^1$ to Euclidean space. We provide an
upper bound to the Wasserstein metric for circular distributions and exhibit a
variety of different bounds between distributions; particularly, between the
von-Mises and wrapped normal distributions, and the wrapped normal and wrapped
Cauchy distributions.
