  We prove the validity of Schwinger's quartic nonet mass formula for a general
nonet structure with the isoscalar singlet mass being shifted from its
``ideal'' value. We apply this formula to the problem of the correct q\bar{q}
assignment for the scalar meson nonet. The results favor scalar isoscalar
singlet and isoscalar octet masses in the vicinity of 1 GeV and 1.45 GeV,
respectively. We explain the failure of Schwinger's formula for the
pseudoscalar meson nonet, and suggest a new version of this formula, modified
by the inclusion of the pseudoscalar decay constants, which holds with improved
accuracy for this nonet. We also rederive the new Gell-Mann-Okubo mass formula
3M^2(n\bar{n},I=1)+2M^2(s\bar{s})=4M^2(s\bar{n})+M^2(n\bar{n},I=0) (n=u,d),
suggested in our previous publications.
