There are several studies focusing on rapid tapping motion for a robotic finger, including not only a theoretical approach of musculoskeletal modeling of a human finger but also an empirical challenge with robotic finger. Shourijeh had made a success of 6Hz-periodic tapping motion by an index finger model for forward dynamic simulation, however the contribution is restricted to analyzing the validity of a musculoskeletal model of finger and is based only on simulation of not a force control but an angular control of the finger. This paper, therefore, presents a novel three-DOF robotic finger that enables fast tapping motion applicable in piano playing and keyboard typing, which consists of metacarpophalangeal(MCP), proximal interphalangeal(PIP), and distal interphalangeal(DIP) joints. In previous study, we developed Twisted Round-belt Actuator(TbA) capable of producing contraction force by twisting a small-diameter round-belt having high elasticity, and empirically derived a contraction force model applicable only to two-DOF robotic finger. This study, therefore, derives such force model available for an extended three-DOF finger mechanism that is containing Variable-pitch Screw Module(VpSM) newly designed in this paper. This module is able to eliminate irregular twist phenomenon resulting in unexpected discontinuous movement in finger joints. Thus, we first describe the detail mechanism of the three-DOF robotic finger and indicate great advantage of the finger, which is obtained by combining the TbA and VpSM. Finally, we demonstrate a new force feedback method based on fingertip force estimation without any force/pressure sensors, in which the fingertip force can be accurately estimated by using mathematical model of the contraction force by TbA with VpSM. In addition, this paper shows successful experimental results of fast tapping motion by the robotic finger, and clearly reveals that the robot is able to move against sinusoidal force reference up to 5Hz interval.