  In the language of $L^\infty$-modules proposed by Gigli, we introduce a first
order calculus on a topological Lusin measure space $(M,\mathfrak{m})$ carrying
a quasi-regular, strongly local Dirichlet form $\mathscr{E}$. Furthermore, we
develop a second order calculus if $(M,\mathscr{E},\mathfrak{m})$ is tamed by a
signed measure in the extended Kato class in the sense of Erbar, Rigoni, Sturm
and Tamanini. This allows us to define e.g. Hessians, covariant and exterior
derivatives, Ricci curvature, and second fundamental form.
