We apply the x-ray extended-range technique (XERT) to measure mass attenuation coefficients over one order of magnitude more accurately than previously reported in the literature. We describe the application of the XERT to the investigation of systematic effects due to harmonic energy components in the x-ray beam, scattering and fluorescence from the absorbing sample, the bandwidth of the x-ray beam, and thickness variations across the absorber. The high-accuracy measurements are used for comparison with different calculations of mass attenuation coefficients, and to identify particular regions where these calculations fail.