  We present a new grid of non-adiabatic, linear pulsation models of
Long-Period Variables (LPVs), including periods and growth rates for radial
modes from the fundamental to the fourth overtone. The models span a wide range
in mass, luminosity, metallicity, C/O ratio and helium abundance, effectively
covering the whole thermally-pulsing asymptotic giant branch (TP-AGB)
evolution, and representing a significant update with respect to previous
works. The main improvement is the inclusion of detailed atomic and molecular
opacities, consistent with the models chemical mixture, that makes the present
set of models the first to systematically account for variability in C-stars.
We examine periods and growth rates in the models, and find that, while the
fundamental mode is affected by the structure of the envelope, overtones are
less sensitive to the interior and largely determined by the global properties.
In the models, the frequency of the overtone with the largest degree of
excitation is found to scale with the acoustic cut-off frequency at the stellar
surface, a behaviour similar to that observed for the frequency of maximum
oscillation power for solar-like oscillations in less evolved red giants. This
allows us to provide a simple analytic prescription to predict the most-likely
dominant mode as a function of stellar parameters. Best-fit relations for
periods are also provided. By applying results of pulsation models to
evolutionary tracks, we present a general picture of the evolution of
long-period variability during the TP-AGB, that we find consistent with
observations. Models are made public through a dedicated web interface.
