Within the semi-classical three-step model for high harmonic generation (HHG), the shape of the atomic/molecular potential and its eigenstates are completely neglected, and are instead parameterized through the target's ionization potential. Still, in some cases the effects of these physical quantities are non-negligible, and may contribute to the harmonic response. Here we show that HHG driven by ω-2ω bi-elliptical pumps (i.e. co-linear counter-rotating elliptically polarized beams of equal ellipticity[3,4]) is highly sensitive to the shape of the atomic potential, and that this sensitivity is especially accentuated in HHG spectrograms that measure the harmonic response vs. the full range of pump ellipticities. We theoretically explore this phenomenon in the noble gases He, Ne, Ar, and Kr, and identify the responsible physical mechanisms. The effect is experimentally demonstrated by observing clear disparities in the bi-elliptical HHG spectrograms from atomic Ar and Kr gases, which have relatively similar ionization potentials (Ip), but substantially differ in their valence p-shell widths and potential structures.