We report new measurements of the mass composition of cosmic-ray Ne, Mg, Si, and S nuclei made on the Voyager spacecraft. These measurements have ~4 times the statistical accuracy of previously published measurements covering these four charges. With the new cosmic-ray source mass fractions of these elements that we obtain, only the isotope 22Ne shows a cosmic-ray source abundance that is significantly different from the solar abundances. The limits of +/-15% that we set on the cosmic-ray source-to-solar abundance ratios of 25Mg/24Mg and 26Mg/24Mg as well as the heavier Si isotopes place severe limits on the models that have been proposed to explain compositional differences between galactic cosmic rays and solar system abundances. The only two statistically significant isotopic differences between the cosmic-ray source and the solar system in the charge range Z = 6-16 now appear to be the underabundance of 14N and the overabundance of 22Ne. This suggests to us that the helium-burning process in which 14N is turned into 22Ne plays an important role in at least some of the sources of those particles ultimately accelerated as cosmic rays.