  We study the Landau-problem on the $\theta$-deformed two-torus and use
well-known projective modules to obtain perturbed spectra. For a strong
magnetic field B the problem can be restricted to one particular Landau-level.
First we represent generators of the algebra of the non-commutative torus as
finite dimensional matrices. A second approach leads to a reducible
representation with a $\theta$-dependent center. For a simple periodic
potential, the rational part of the Hofstadter-butterfly spectrum is obtained.
