Let (X,0) be a reduced, equidimensional germ of analytic singularity with reduced tangent cone (C_{X,0},0). We prove that the absence of exceptional cones is a necessary and sufficient condition for the smooth part \X^0 of the specialization to the tangent cone ϕ: \X \to \C to satisfy Whitney's conditions along the parameter axis Y. This result is a first step in generalizing to higher dimensions Lê and Teissier's result for hypersurfaces of \C^3 which establishes the Whitney equisingularity of X and its tangent cone under this conditions.