A spatial periodic adaptive control (SPAC) approach is developed to deal with rotary machine systems performing speed tracking tasks. Since the angular displacement is periodic when rotating by 2 pi radians, most rotary machine systems present certain cyclic behaviors with a fixed periodicity which is either a fraction or multiple of 2 pi. As a consequence, unknown system parameters and disturbances that characterize the system behaviors are also periodic in nature. By utilizing the spatial periodicity, the SPAC aims at improving the system performance. In the SPAC design, the dynamics of the rotary machine systems is first converted from the temporal to spatial coordinates in canonical form using the feedback linearization method. Then the new adaptive controller updates the parameters and the control signal periodically in a pointwise between two consecutive spatial cycles. Using a Lyapunov-Krasovskii functional, the convergence property of the SPAC can be analyzed for high order rotary systems and the periodic adaptation can be applied to rotary systems with pseudo-periodic parametric uncertainties. The effectiveness of the SPAC is verified through rigorous analysis and two numerical examples.