Recent research suggests that the different components of eye movements (fixations, saccades) are not strictly separate but are interdependent processes. This argument rests on observations that gaze-step sizes yield unimodal distributions and exhibit power-law scaling, indicative of interdependent processes coordinated across timescales. The studies that produced these findings, however, employed complex tasks (visual search, scene perception). Thus, the question is whether the observed interdependence is a fundamental property of eye movements or emerges in the interplay between cognitive processes and complex visual stimuli. In this study, we used a simple eye movement task where participants moved their eyes in a prescribed sequence at several different paces. We outlined diverging predictions for this task for independence versus interdependence of fixational and saccadic fluctuations and tested these predictions by assessing the spectral properties of eye movements. We found no clear peak in the power spectrum attributable exclusively to saccadic fluctuations. Furthermore, changing the pace of the eye movement sequence yielded a global shift in scaling relations evident in the power spectrum, not just a localized shift for saccadic fluctuations. These results support the conclusion that fixations and saccades are interdependent processes.