  We apply neutron spectroscopy to measure the magnetic dynamics in the S=3/2
magnet $\beta$-CaCr$_2$O$_4$ (T$_N$=21 K). The low-energy fluctuations, in the
ordered state, resemble large-S linear spin-waves from the incommensurate
ground state. However, at higher energy transfers, these semi-classical and
harmonic dynamics are replaced by an energy and momentum broadened continuum of
excitations. Applying kinematic constraints required for energy and momentum
conservation, sum rules of neutron scattering, and comparison against exact
diagonalization calculations, we show that the dynamics at high-energy
transfers resemble low-S one-dimensional quantum fluctuations.
$\beta$-CaCr$_2$O$_4$ represents an example of a magnet at the border between
classical N\'eel and quantum phases, displaying dual characteristics.
