We show that a one-dimensional aperiodic Delaunay set of points together with the Fourier transform of its autocorrelation measure (square modulus of its structure factor) at a wavevector k = 2/, can be associated with a generalized Meyer set under some assumptions: (a) that the internal space is toric, /, with a window, assumed finite, equal to the set of affine lattices of period which have a non-empty intersection with and rarefaction laws at infinity, a selection rule based on a congruence mode with respect to ; (b) a scaling exponent function, having values in [0;1], can be uniquely defined on the window from rarefaction laws, which is related to the scaling properties of the intensity function; (c) the projection mappings are adapted to the average lattice of and are not orthogonal. The case of Bragg peaks of the Thue-Morse sequence spectrum is developed explicitly in this context.