We revisit the problem of constructing finite state ρ/μ approximations of systems over finite alphabets for the purpose of certified-by-design control synthesis. In this context, we focus on a specific class of systems, namely second order linear time-invariant (LTI) systems with finite inputs and binary quantized outputs. Specifically, we investigate the problem of picking the `initial partition' for generating state-space based ρ/μ approximations enabling control design. We propose and analyze a new algorithm that is tailored to the dynamics and the sampling period of the system, and we illustrate its use via a set of numerical examples.