Abstract: A tapered thread cutting machine includes a multi-part rotary machining head, one part including a through-recess and another being a separable interior wear plate. Interacting chaser holders are mounted in the recess and include angled V-shaped slots in which block type chasers may be secured between clamps and locating pins enabling the chasers to be removed and replaced without removing the holders from the recess. The holders are driven by a linear cam for opening and closing movement through a thrust block mounted on a guide rod, such thrust block being fitted in a slot in one holder but otherwise not secured thereto so that binding in the opening and closing of tool holders is minimized.