A method and apparatus for positioning a magnetic servo head relative to a servo track on a traveling magnetic tape both make use of the observation that a given tape in a given tape drive unit exhibits transverse tape movements during operation which constitute a "signature" waveform for the coupled tape cartridge and drive unit. In the method and apparatus, the transverse movements of a traveling tape in a drive unit are monitored to obtain the aforementioned waveform which is characteristic of the transverse movements for that tape, and the waveform is stored. The positioning of the servo head is controlled relative to a servo track on the tape using a control signal generated according to any known track seek algorithm. The stored waveform is recalled and is superimposed on the control signal, thereby significantly increasing the accuracy of the positioning of the servo head, which in turn increases the accuracy of positioning of the write/read head.