The notion of a pseudo-natural SSI frequency was introduced in a recent publication by the authors, as the frequency where foundation motion is minimized with respect to the free field surface motion. This frequency is determined analytically in this paper, for a single-degree-of-freedom structure supported on a pile foundation. The analytical solution is compared to numerical results from rigorous finite element analyses for different pile and structural configurations. The relationship between pseudo-natural (fpSSI) and effective natural SSI frequency (fSSI) of the coupled system is also analytically quantified. It is concluded that fpSSI may deviate substantially from fSSI when a stiff squatty structure is founded on a stiff and/or short end-bearing pile for which foundation translation prevails. Conversely, when a flexible tall structure is supported on a flexible pile, fpSSI and fSSI nearly coincide due to dominant base rocking effects. In the latter case the effective natural SSI frequency can be predicted by standard identification procedures even when free-field recordings are missing. Effective damping effects are also discussed.