in 1998, p. biane and r. speicher developed a theory of stochastic integration against free additive self-adjoint brownian motion, the large-$n$ limit of $n \times n$ hermitian brownian motion. this development included a kind of it\={o} formula for certain functions of free it\={o} processes. in this talk, we discuss -- for motivation -- the finite-$n$ version of this it\={o} formula and use objects called \textit{double operator integrals} to give a new analytic interpretation of the quantities therein, which to date have been understood and interpreted mostly in combinatorial ways. this analytic interpretation yields an extension, which we shall also discuss, of biane and speicher's original formula.