Given a lattice path $ν$, the $ν$-Tamari lattice and the $ν$-Dyck lattice are two natural examples of partial order structures on the set of lattice paths that lie weakly above $ν$. In this paper, we introduce a more general family of lattices, called alt $ν$-Tamari lattices, which contains these two examples as particular cases. Unexpectedly, we show that all these lattices have the same number of linear intervals.