A method of forming a horological spiral hairspring by increasing its concentricity; the hairspring having an inner terminal end portion for engagement with a collet and an outer terminal end portion for engagement with a stud, a limb portion extending from the inner terminal end portion towards the outer terminal end portion, said limb portion having a first modified portion, an outer turn and a stiffening portion positioned at the outer turn of the hairspring and having a second modified portion having cross-sectional second moment or moments of area different to that or those of the limb portion, such that the bending stiffness of the stiffening portion has a greater bending stiffness than that of the limb portion, said method including the steps of: A) receiving, as inputs, design parameters for each of said first and second modified portions (41a, 41b) to define their geometry, the design parameters including: - the cross-sectional second moment of area I b, respectively I a, at a location θ b, respectively θ a, of a modified cross section of the limb portion, respectively the stiffening portion, - the arc lengths L b and L a of respectively said first and second modified portions (41a, 41b), the parameter I a, respectively I b, is defined as a ratio compared to the second moment of area of the rest of the hairspring strip, the parameter L a, respectively L b, is defined as the length of each respective modified portion or as the angular span in polar coordinates, and the parameter θ a, respectively θ b, is measured relative to the stud or the collet as the arc distance or as the angular distance in polar coordinate; B) modifying, using said design parameters, said cross-sectional second moments of area I a and I b, by way of minimization of a cost function throughout the amplitude of the rotation of the hairspring in use, wherein the cost function is correlated to the net concentricity of the hairspring, wherein the metric of the concentricity is associated to the variation in the position of the centre of mass of the hairspring to the axis of rotation of the hairspring over the operating range of the hairspring, wherein the operating range of the hairspring is the amplitude of rotation of the hairspring in use; and C) forming a spiral hairspring for a mechanical timepiece in accordance with the modified cross-sectional second moments of area of the modified portion of the limb portion and of the modified portion of the stiffening portion. A) receiving, as inputs, design parameters for each of said first and second modified portions (41a, 41b) to define their geometry, the design parameters including: - the cross-sectional second moment of area I b, respectively I a, at a location θ b, respectively θ a, of a modified cross section of the limb portion, respectively the stiffening portion, - the arc lengths L b and L a of respectively said first and second modified portions (41a, 41b), the parameter I a, respectively I b, is defined as a ratio compared to the second moment of area of the rest of the hairspring strip, the parameter L a, respectively L b, is defined as the length of each respective modified portion or as the angular span in polar coordinates, and the parameter θ a, respectively θ b, is measured relative to the stud or the collet as the arc distance or as the angular distance in polar coordinate; - the cross-sectional second moment of area I b, respectively I a, at a location θ b, respectively θ a, of a modified cross section of the limb portion, respectively the stiffening portion, - the arc lengths L b and L a of respectively said first and second modified portions (41a, 41b), B) modifying, using said design parameters, said cross-sectional second moments of area I a and I b, by way of minimization of a cost function throughout the amplitude of the rotation of the hairspring in use, wherein the cost function is correlated to the net concentricity of the hairspring, wherein the metric of the concentricity is associated to the variation in the position of the centre of mass of the hairspring to the axis of rotation of the hairspring over the operating range of the hairspring, wherein the operating range of the hairspring is the amplitude of rotation of the hairspring in use; and C) forming a spiral hairspring for a mechanical timepiece in accordance with the modified cross-sectional second moments of area of the modified portion of the limb portion and of the modified portion of the stiffening portion.