A formal connection is made between the vibrational density of states (DOS) of a liquid and its approximation by way of instantaneous normal modes (INMs). This analysis leads to a quantum generalization of the INM method (QINM), and to the possibility of evaluating the classical DOS exactly. Further, INM approximations to spectroscopic quantities (e.g., infrared absorption and Raman scattering) follow in a consistent manner by evaluating the appropriate golden rule expressions for harmonic oscillators, using the INM or QINM DOS in place of the true DOS. INM and QINM methods are then applied along with traditional time correlation function (TCF) methods to analyze the entire infrared (IR) spectrum of ambient water. The INM and TCF approaches are found to offer complimentary information. TCF methods are shown to offer an unexpectedly accurate description of the O-H stretching line shape. Further, the 19-fold enhancement in liquid phase absorption compared to the gas phase is also reproduced. INM and QINM methods are used to analyze the molecular origin of the water spectrum, and prove especially effective in analyzing the broad O-H stretching absorption. Further, it is argued that a motional narrowing picture is qualitatively useful in analyzing INM approximations to spectroscopy.