Let $A$ be a dg algebra, and $x, z \in A$ cocycles. Let's consider the maps $$ A \to A \oplus A \to A$$ given by $y \mapsto (xy,yz)$ and $(u,v) \mapsto uz-xv$, respectively. We think of this as defining a double complex with three nonzero columns. (Here we need either to assume that $x,z$ are of degree $0$, or shift the grading in the second and third column.) The spectral sequence of a double complex will have an $E_1$-page given by $H(A) \to H(A) \oplus H(A) \to H(A)$. The $E_2$-page will have as its first column the kernel $$ \{ y \in H(A) : xy = yz = 0\},$$ and the $E_2$-differential gives a map to the third column, i.e. the cokernel $$ H(A)/\left( x H(A) + H(A)z\right).$$ This map is exactly the triple Massey product $\langle x,y,z\rangle$.
Question Is there a generalization of this construction to higher Massey products? I.e. can one write down a double complex with $n$ columns, such that the $E_{n-1}$-differential could have been used as an alternate definition of the $n$-fold Massey product?
To clarify what I'm after, there are of course many spectral sequences in which the differentials are known to be given by Massey products. What is special with the above construction is that it also produces the right indeterminacy: the triple Massey product $\langle x,y,z\rangle$ is in general well defined precisely as an element of $ H(A)/\left( x H(A) + H(A)z\right),$ which is also what you get from the spectral sequence. One tricky thing about trying to cook up a similar thing for higher products is that there is no similarly simple description of the indeterminacy for the higher products.
