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Let $(X,0)$ be a reduced, equidimensional germ of an 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 $\varphi :𝔛\to ℂ$ 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...