This paper focuses on a novel harmonic movable tooth drive as an alternative of currently used harmonic gear drives. Different from the harmonic gear drives, the novel drive removes small module teeth from thin-walled flexible rim, instead, adopts several rigid movable teeth with logarithmic spiral tooth profile. Such tooth geometry makes almost surface contact possible under load condition, in comparison with the conventional involute tooth profile creating a linear contact. Aiming at the issue of cusp meshing between movable tooth and ring gear during the meshing process, this paper proposes a modification design method for the conjugate tooth profiles. First, the radial motion law of movable tooth is modified, in order to avoid rigid impacts at the exchange point of the rise travel and return travel. Then, the tooth profile of ring gear is generated by the envelope of the modified profile of movable tooth. Based on the formulated modification scheme, the equations of the conjugate tooth profiles are derived, and the influences of spiral angle and modification parameters on the meshing characteristics are systematically investigated. Finally, the wave generator cam profile and pivoted segment profile are established considering the strength consistency. The outcomes can provide significance guidance for the parametric design and optimization of this novel drive.