We investigate short-range order and local atomic configuration in charge-balanced aluminosilicate glasses as functions of composition, using O-17 and Al-27 MAS and triple-quantum magic angle spinning (3QMAS) NMR spectroscopy. Enhanced resolution in O-17 and Al-27 3QMAS spectra, compared to MAS NMR, allows the quantification of the spectra and the extent of disorder using a semiempirical function relating 3QMAS efficiency to a quadrupolar coupling constant (C-q). The variations with the Si/Al ratio (R) in peak positions and widths in the isotropic dimension of Al-27 3QMAS NMR spectra in both Na- and Ca-aluminosilicate glasses can be ascribed to variations in the populations of Al sites with varying numbers of Al vs Si neighbors with composition. In the O-17 3QMAS spectra, variations of populations of three clearly resolved oxygen sites (Al-O-Al, Si-O-Al, and Si-O-Si) with R and cation field strength are consistent with the predictions given in our previous results from Si-29 MAS NMR. The quadrupolar coupling product (P-q) of each oxygen site does not vary significantly with R, but it increases with cation field strength. On the other hand, isotropic chemical shifts (delta(iso)(CS)) increase with decreasing R and increasing cation field strength. These trends suggest that the configuration and framework connectivity in aluminosilicate glasses and melts are relatively constant with R but can be perturbed by high field strength cations with increased Al-O-Al and angular disorder, manifested by the increased variation of delta(iso)(CS) and the formation of non-bridging oxygen (NBO). The extent of disorder in aluminosilicate glasses is reflected in calculated configurational enthalpy, which increases with increasing cation field strength, consistent with the excess enthalpy of mixing data from calorimetry. The method and results given here provide improved prospects for the quantitative application of 3QMAS NMR and add to a more complete understanding of framework site connectivity in aluminosilicate glasses.