Rhythmic and sequential segmentation of the growing vertebrate body relies on the segmentation clock, a multi-cellular oscillating genetic network. The clock is visible as tissue-level kinematic waves of gene expression that travel through the presomitic mesoderm (PSM) and arrest at the position of each forming segment. Here, we test how this hallmark wave pattern is driven by culturing single maturing PSM cells. We compare their cell-autonomous oscillatory and arrest dynamics to those we observe in the embryo at cellular resolution, finding similarity in the relative slowing of oscillations and arrest in concert with differentiation. This shows that cell-extrinsic signals are not required by the cells to instruct the developmental program underlying the wave pattern. We show that a cell-autonomous timing activity initiates during cell exit from the tailbud, then runs down in the anterior-ward cell flow in the PSM, thereby using elapsed time to provide positional information to the clock. Exogenous FGF lengthens the duration of the cell-intrinsic timer, indicating extrinsic factors in the embryo may regulate the segmentation clock via the timer. In sum, our work suggests that a noisy cell-autonomous, intrinsic timer drives the slowing and arrest of oscillations underlying the wave pattern, while extrinsic factors in the embryo tune this timer's duration and precision. This is a new insight into the balance of cell-intrinsic and -extrinsic mechanisms driving tissue patterning in development.