We show by constructing explicit homotopy operators that the Hochschild (co)homology of an $A_{\mathrm{\infty }}$-algebra of Stasheff admits a differential calculus structure. As an application, we reproduce a result of Tradler which says that the Hochschild cohomology of a cyclic $A_{\mathrm{\infty }}$-algebra admits a Batalin-Vilkovisky algebra structure.