Accurate control of tape tension and velocity will be required in advanced tape drives, as tape thickness is reduced and track density is increased towards target tape capacities of 100 TB and beyond. In this paper, a novel method is introduced for the feedback control of the tape transport in tape drives without tension transducers. The method relies on estimating tape tension variations from the difference of the measured lateral positions of servo read elements on adjacent servo bands. Furthermore, a technique for the suppression of periodic tape tension disturbances, which are caused by reel eccentricities and other once-around effects, is developed and presented. The technique uses a time-varying narrow-band bi-quad filter with variable center frequency to suppress the slowly time-varying disturbances. To avoid the noise enhancement that would occur when the disturbance frequency moves outside of the loop bandwidth, the filter parameters are determined by H∞-norm minimization. Using the proposed tape transport system, we experimentally demonstrate a reduction in the standard deviation of the tension of about 40% and a reduction in the standard deviation of the velocity of about 23% with respect to state-of-the-art tape drives.