Many processes are characterized by their oscillating or cyclic time behavior. This holds for rotating machines or alternating currents. The resulting signals are then periodic signals or contain periodic parts. It can be used for fault detection of rotating machines. In this paper, we studied the periodic time series of the superposition of two oscillations from the multifractal point of view. The wavelet transform modulus maxima method was used for the singularity spectrum computations. The results show that the width and the peak position of the singularity spectrum changed significantly when the amplitude, frequency, or the phase difference changed. So, the width and the peak position of the singularity spectrum can be used as a new measure for periodic signals.