Known two-quark or unmixed mesons are classified according to the recently proposed spinor strong interaction theory, which also aids in the tentative classification of four-quark mixed mesons. No approximation is made in the derivation of the mass formula from the covariant basic equations, in which linear confinement is inherent. The ground-state scalar and axial vector mesons are not confined and cannot exist. The forbidden states include the unseen . Six unmixed pseudoscalar meson masses are used to determine the five quark masses and the eigenvalue of the nonlinear differential equation of the theory. The predicted masses of the remaining unmixed pseudoscalar mesons agree with data within experimental error or very close to it. The predicted vector meson masses agree fairly well with data. Predicted masses of purely radially excited mesons also agree with data within experimental error, using three data points as input.