Turbulent combustion processes are inherently unsteady and, thus, a source of acoustic radiation. While prior studies have extensively characterized their total sound power, their spectral characteristics are not well understood. This work investigates these acoustic spectral features, including the flame's low- and high-frequency characteristics and the scaling of the frequency of peak acoustic emissions. The spatiotemporal characteristics of the flame's chemiluminescence emissions, used as a marker of heat release fluctuations, were measured and used to determine the heat release spectrum, spatial distribution and spatial coherence characteristics. These heat release characteristics were then used as inputs to an integral solution of the wave equation and compared to measured acoustic spectra obtained over a range of conditions and burners and at several spatial locations. The spectral characteristics of the flame's acoustic emissions are controlled by two processes: the underlying spectrum of heat release fluctuations that are ultimately the combustion noise source, and the transfer function relating these heat release and acoustic fluctuations. An important result from this work is the clarification of the relative roles of these two processes in controlling the shape of the acoustic spectrum. This transfer function is primarily controlled by the spatiotemporal coherence characteristics of the heat release fluctuations which are, in turn, strongly influenced by burner configuration/geometry and operating conditions. Low-frequency acoustic emissions are controlled by the heat release spectrum essentially independent of flame geometry. Both the heat release spectrum and heat release-acoustic transfer function are important at intermediate and high frequencies. An important feature of the investigated geometry that controls the heat release phase dynamics is the high-velocity flow relative to the flame speed and anchored character of the flame. This leads to convection of flame sheet disturbances (i.e. heat release fluctuations) along the front that dominates the high frequency and peak frequency scaling of the flame's acoustic emissions.