An attachment system for a turbine airfoil of a turbine engine, comprising: at least one root configured to support a turbine airfoil within a turbine engine; at least one supporting disc; wherein the at least one root includes a first side first axially extending tooth extending from a first side of the at least one root; wherein a radially outer bearing surface of the first side first axially extending tooth extends at a first acute angle inwardly relative to a radially extending longitudinal axis of the at least one root; and wherein a laterally extending outer edge of the first side first axially extending tooth that extends generally along an axis aligned with a centerline of the turbine engine is curved about the axis such that the curved laterally extending outer edge is viewed orthogonally to the centerline of the turbine engine, radially outward of the turbine airfoil looking into the centerline of the turbine engine, whereby the curved laterally extending outer edge of the first side first axially extending tooth substantially reduces circumferential motion of the turbine airfoil attached to the at least one root relative to the supporting disc; wherein the supporting disc includes at least one first side first axially extending tooth recess that is curved and configured to receive the first side first axially extending tooth, characterized in that a straight line distance measured in a direction orthogonal to the centerline of the turbine engine between an upstream end of the curved laterally extending outer edge of the first side first axially extending tooth and a lateral high point of the curved laterally extending outer edge is greater than a tooth clearance dimension measured between the first side first axially extending tooth and the supporting disc, wherein the tooth clearance dimension is measured between the first side first axially extending tooth and the supporting disc is measured between a radially inward surface of the first side first axially extending tooth and a closest surface of the supporting disc and the tooth clearance dimension is between 0.001 mm and 1 mm. at least one root configured to support a turbine airfoil within a turbine engine; at least one supporting disc; wherein the at least one root includes a first side first axially extending tooth extending from a first side of the at least one root; wherein a radially outer bearing surface of the first side first axially extending tooth extends at a first acute angle inwardly relative to a radially extending longitudinal axis of the at least one root; and wherein a laterally extending outer edge of the first side first axially extending tooth that extends generally along an axis aligned with a centerline of the turbine engine is curved about the axis such that the curved laterally extending outer edge is viewed orthogonally to the centerline of the turbine engine, radially outward of the turbine airfoil looking into the centerline of the turbine engine, whereby the curved laterally extending outer edge of the first side first axially extending tooth substantially reduces circumferential motion of the turbine airfoil attached to the at least one root relative to the supporting disc; wherein the supporting disc includes at least one first side first axially extending tooth recess that is curved and configured to receive the first side first axially extending tooth, characterized in that a straight line distance measured in a direction orthogonal to the centerline of the turbine engine between an upstream end of the curved laterally extending outer edge of the first side first axially extending tooth and a lateral high point of the curved laterally extending outer edge is greater than a tooth clearance dimension measured between the first side first axially extending tooth and the supporting disc, wherein the tooth clearance dimension is measured between the first side first axially extending tooth and the supporting disc is measured between a radially inward surface of the first side first axially extending tooth and a closest surface of the supporting disc and the tooth clearance dimension is between 0.001 mm and 1 mm.