independent $\Z$ filtrations allows one to define many auxilary knot invariants and in this paper we deal with a less robust invariant - the filtered chain homotopy type of the $\Z$-filtration of $\CFa(Y,\spinc)$ obtained by setting one of the $\Z$ filtrations equal $0$. We denote this filtration by $\FiltY(K)$ so that we have the sequence of inclusions: $$0=\FiltY(K,-i) \subseteq \FiltY(K,-i+1)\subseteq \ldots \subseteq
\FiltY(K,n)=\CFa(Y,\spinc).$$

The associated graded complexes of this filtration, $\frac{\FiltY(K,j)}{\FiltY(K,j-1)}$, will be denoted by $\CFKa_\spinc(Y,K, j)$, and their homology by $\HFKa_\spinc(Y,K, j)$. The homology groups $\HFKa_\spinc(Y,K,j)$ are commonly referred to as the [*knot Floer homology groups of $K\subset Y$*]{}. These groups were studied for the $(p,pn\pm1)$ cables of an arbitrary knot, $K\subset S^3$, in [@Cabling; @MyThesis]. In that paper a stabilization theorem was proved which provided a formula for the groups $\HFKa(S^3,K_{p,pn\pm1},i)$ in the case when the parameter $n$ was sufficiently large. The formula required $H_*(\Filt(K))$ as input, while the output was merely the associated graded object; hence, there was a loss of information. Despite this loss of information the formulas and techniques of [@Cabling] have proved to be quite useful and, in particular, were implemented by  [@OSThurston] and Ni [@NiThurston] in the proof that link Floer homology detects 