  In the present investigation we are linking noncommutative geometry over
noncommutative tori with Gabor analysis, where the first has its roots in
operator algebras and the second in time-frequency analysis. Therefore we are
in the position to invoke modern methods of operator algebras, e.g. topological
stable rank of Banach algebras, to exploit the deeper properties of Gabor
frames. Furthermore we are able to extend results due to Connes and Rieffel on
projective modules over noncommutative tori to Banach algebras, which arise in
a natural manner in Gabor analysis. The main goal of this investigation is
twofold: (i) an interpretation of projective modules over noncommutative tori
in terms of Gabor analysis and (ii) that the Morita-Rieffel equivalence between
noncommutative tori is the natural framework for the duality theory of Gabor
frames. More concretely, we interpret generators of projective modules over
noncommutative tori as the Gabor atoms of multi-window Gabor frames for
modulation spaces. Moreover, we show that this implies the {\it existence} of
good multi-window Gabor frames for modulation spaces with Gabor atoms in
Feichtinger's algebra and in Schwartz space.
