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 \pm 2.7 km s ^ { -1 } , 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 \pm 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 \pm 2.6 km s ^ { -1 } ) and hence the mass ratio of the system q = 0.079 \pm 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 .