In this paper we present the first results from a high-precision radial velocity (RV) study of the rapidly oscillating Ap (roAp) star
HR 1217. Data spanning a complete rotation period were acquired on 9 nights in late 1997 and early 1998 using the Harlan J. Smith 2.7m telescope at McDonald Observatory. The RVs were measured using the wavelength range of each spectral order (≃100Å) of the ``2dcoude'' echelle spectrograph. Most of the pulsational modes can be seen in all spectral regions but the amplitude varies by factors of ten or more between the different wavelength intervals. A detailed analysis of the spectral order centered on 5825Å, which had the largest amplitudes, showed the presence of all published photometric frequencies f
1-f
7. The multi-mode oscillations were also found in the RV variations of the high amplitude NdIII 5294Å feature. Our RV data reveal two new excited modes at frequencies f
0=223.37c/d (2585.3µHz) and f
–1=220.58c/d (2553.0µHz) that follow the odd and even alternative mode spacing with Δν
0/2≃34µHz. An analysis of individual nights shows strong amplitude and phase modulation of all excited modes, including the two newly discovered ones. The amplitudes of the f
2=229.21c/d (2652.9µHz) and f
4=235.08c/d (2720.85µHz) modes are modulated with the published magnetic field variations and reach their maximum and minimum at magnetic extrema. However, the phase variability is in disagreement with that expected from the standard oblique pulsator model. The phase jumps for the f
2 mode occur exactly at magnetic maximum and close to the phase of magnetic minimum, while the f
3 and f
4 modes have a continuous change. The peculiar phase variability is attributed to strong vertical phase changes in the line-forming layers of atmosphere. An echelle-diagram for all known excited modes in
HR 1217 is constructed. We interpret the f
7=242.41c/d (2805.7µHz) mode with the ``peculiar'' spacing as due to a mode of degree l=4 that is the only observed member of another system of equally-spaced frequencies. We predict the existence of modes belonging to this system that should be equally spaced at 68µHz with f
7.