  We present a dynamical study of the Galactic black hole binary system Nova
Muscae 1991 (GS/GRS 1124-683). We utilize 72 high resolution Magellan
Echellette (MagE) spectra and 72 strictly simultaneous V-band photometric
observations; the simultaneity is a unique and crucial feature of this
dynamical study. The data were taken on two consecutive nights and cover the
full 10.4-hour orbital cycle. The radial velocities of the secondary star are
determined by cross-correlating the object spectra with the best-match template
spectrum obtained using the same instrument configuration. Based on our
independent analysis of five orders of the echellette spectrum, the
semi-amplitude of the radial velocity of the secondary is measured to be K_2 =
406.8+/-2.7 km/s, which is consistent with previous work, while the uncertainty
is reduced by a factor of 3. The corresponding mass function is f(M) =
3.02+/-0.06 M_\odot. We have also obtained an accurate measurement of the
rotational broadening of the stellar absorption lines (v sin i = 85.0+/-2.6
km/s) and hence the mass ratio of the system q = 0.079+/-0.007. Finally, we
have measured the spectrum of the non-stellar component of emission that veils
the spectrum of the secondary. In a future paper, we will use our
veiling-corrected spectrum of the secondary and accurate values of K_2 and q to
model multi-color light curves and determine the systemic inclination and the
mass of the black hole.
