Given an abelian category $\mathcal C$ of finite global dimension and a chain complex $Y^\bullet\in D^b(\mathcal C)$ in the derived category. As a triangulated category, $D^b(\mathcal C)$ of course is generated by the projective objects of $\mathcal C$. Now, I am interested in the set
$${}^\bot Y := \{X^\bullet\in D^b(\mathcal C)\colon\operatorname{Ext}^\bullet(X^\bullet, Y^\bullet)=0\}$$
It is true that this set is a triangulated subcategory of $D^b(\mathcal C)$, so I would like to find a generating set of it.
Question: Is there a general strategy how to do this? My problem is the following: Assume that some of the projectives of $\mathcal C$ lie in ${}^\bot Y$ while others don't. However, I am not convinced that I cannot form a complex of the latter ones which lies in ${}^\bot Y$. Conversely, is it immediate that any chain complex built of the former ones also lies in ${}^\bot Y$?
How to cope with this problem?