  We study the noncommutative geometry of the Moyal plane from a metric point
of view. Starting from a non compact spectral triple based on the Moyal
deformation A of the algebra of Schwartz functions on R^2, we explicitly
compute Connes' spectral distance between the pure states of A corresponding to
eigenfunctions of the quantum harmonic oscillator. For other pure states, we
provide a lower bound to the spectral distance, and show that the latest is not
always finite. As a consequence, we show that the spectral triple [20] is not a
spectral metric space in the sense of [5]. This motivates the study of
truncations of the spectral triple, based on M_n(C) with arbitrary integer n,
which turn out to be compact quantum metric spaces in the sense of Rieffel.
Finally the distance is explicitly computed for n=2.
