  We discuss $\beta$-deformed geometries on two types of $T^3$'s where the
direction along the third coordinate is not orthogonal to the direction along
the second coordinate or the direction along the first coordinate. We show that
the intersection angle between the direction along the third coordinate and the
direction along the second coordinate corresponds to the parameter of the
S-duality of the $\beta$-deformation while the intersection angle between the
direction along the third coordinate and the direction along the first
coordinate generalizes the $\beta$-deformed geometry.
