A multiple head tape placement system includes several tape heads. Each tape head includes: a guide chute and a compaction roller for delivering a composite material to a mandrel. A backing is removed from the composite tape material before it reaches the compaction roller. Each tape head also includes a material cutter disposed to cut the composite tape material after the backing is removed and before the material reaches the compaction roller. The material cutter includes a curved blade with a convex cutting surface and a flat blade that contacts the curved blade in at most two contact points along a cutting edge of the flat blade as the flat blade moves vertically up and down past the curved blade with a horizontal rocking motion. The curved blade and the flat blade cut the composite tape material simultaneously in two opposing directions without laterally misaligning the composite tape material.