Let $R$ be a graded ring (concentrated in nonnegative dimensions and maybe bounded from above). For every positive natural number $n$, denote by $R\to\tau_{\leq n}R$ the $n$-truncation and by $\tau_{\geq n}R \to R$ the analogous procedure killing all dimensions below n. These two rings may both be viewed as $R$-modules.

My question is the following: Is there a simple procedure computing $Tor_*^R(\tau_{\leq n}R, \tau_{\geq n+1}R)$ (in the category of graded modules)?