Laser-induced phonon spectroscopy has been applied to the determination of the elastic constants and equation of state at 25 ¿C of a natural garnet of composition Py51.6Al31.7Gr15.8Sp0.9 to a pressure of 20 GPa. Moduli are linear with pressure. The adiabatic bulk modulus in GPa is given by K=170.8+4.09 P and the shear modulus is &mgr;=94.7+1.76 P. The derivative of the shear modulus with respect to temperature at constant density is derived from the temperature and pressure dependencies of the elastic constants. The constant pressure derivatives of the mean longitudinal (Vp) and shear (Vs) velocities are tabulated. At 298 K and 1 bar (∂ ln Vp/∂ ln &rgr;)p=1.7, (∂ ln Vs/∂ ln &rgr;)p=1.5 and (∂ ln Vs/∂ ln Vp)p=0.9. The corresponding values calculated at 1000 K and 20 GPa are 1.6, 1.7, and 1.1.¿ 1997 American Geophysical Union