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We apply neutron spectroscopy to measure the magnetic dynamics in the S=3/2 magnet β−CaCr2O4 (TN=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 semiclassical 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. β−CaCr2O4 represents an example of a magnet at the border between classical Néel and quantum phases, displaying dual characteristics.