Our experiments on doubly periodic wavy vortex flow show that the second spectral frequency corresponds to a modulation of the m azimuthal traveling waves. We use spacetime symmetry to classify the possible modulation patterns and to predict a basic period ? such that in the frame rotating with the angular velocity of the waves the wave pattern at times t and t+? will differ only by a rotation through an integral multiple of 2?/m. Measurements on the twelve observed modulation patterns agree with this prediction.