  We show the existence of a family of frames of $L^2(\mathbb{R})$ which depend
on a parameter $\alpha\in [0,1]$. If $\alpha=0$, we recover the usual Gabor
frame, if $\alpha=1$ we obtain a frame system which is closely related to the
so called DOST basis, first introduced by Stockwell and then analyzed by
Battisti and Riba. If $\alpha\in (0,1)$, the frame system is associated to a so
called $\alpha$-partitioning of the frequency domain. Restricting to the case
$\alpha=1$, we provide a truly $n$-dimensional version of the DOST basis and an
associated frame of $L^2(\mathbb{R}^d)$.
