  We investigate the interplay between numerical relativity (NR) and adiabatic
point-particle black hole perturbation theory (ppBHPT) in the comparable mass
regime for quasi-circular non-spinning binary black holes. Specifically, we
reassess the $\alpha$-$\beta$ scaling technique, previously introduced by Islam
et al, as a means to effectively match ppBHPT waveforms to NR waveforms within
this regime. In particular, $\alpha$ rescales the amplitude and $\beta$
rescales the time (and hence the phase). Utilizing publicly available long NR
data (\texttt{SXS:BBH:2265}~\cite{sxs_collaboration_2019}) for a mass ratio of
$1:3$, encompassing the final $\sim 65$ orbital cycles of the binary evolution,
we examine the range of applicability of such scalings. We observe that the
scaling technique remains effective even during the earlier stages of the
inspiral. Additionally, we provide commentary on the temporal evolution of the
$\alpha$ and $\beta$ parameters and discuss whether they can be approximated as
constant values. Consequently, we derive the $\alpha$-$\beta$ scaling as a
function of orbital frequencies and demonstrate that it is equivalent to a
frequency-dependent correction. We further provide a brief comparison between
post-Newtonian (PN) waveforms and the rescaled ppBHPT waveform at a mass ratio
of $q=3$ and comment on their regime of validity. Finally, we explore the
possibility of using PN theory to obtain the $\alpha$-$\beta$ calibration
parameters and still provide a rescaled ppBHPT waveform that matches NR.
