  Motivated by the cohomology theory of loop spaces, we consider a special
class of higher order homotopy commutative differential graded algebras and
construct the filtered Hirsch model for such an algebra $A$. When $x\in H(A)$
with $\mathbb{Z}$ coefficients and $x^{2}=0,$ the symmetric Massey products $%
\langle x\rangle ^{n}$ with $n\geq 3$ have a finite order (whenever defined).
However, if $\Bbbk $ is a field of characteristic zero, $\langle x\rangle ^{n}$
is defined and vanishes in $H(A\otimes \Bbbk )$ for all $n$. If $p$ is an odd
prime, the Kraines formula $\langle x\rangle ^{p}=-\beta \mathcal{P}_{1}(x)$
lifts to $H^{\ast }(A\otimes {\mathbb{Z}}_{p}).$ Applications of the existence
of polynomial generators in the loop homology and the Hochschild cohomology
with a $G$-algebra structure are given.
