The symmetric quark model of baryon resonances is applied to the (56,${0}^{+}$), (70,${1}^{\ensuremath{-}}$), and (20,${1}^{+}$) supermultiplets. Using a systematic $\mathrm{SU}(6)$ analysis, octet dominance, and dominance of two-body contributions to the mass operator, the G\"ursey-Radicati mass formula for the (56,${0}^{+}$) is derived without use of perturbation theory. An equal-spacing relation is derived for the $\mathrm{SU}(6)$-symmetric mass contribution for the (56,${0}^{+}$), (70,${1}^{\ensuremath{-}}$), and (20,${1}^{+}$) in ascending order, with the (20,${1}^{+}$) lying above 2 BeV. A detailed analysis of the (70,${1}^{\ensuremath{-}}$), which yields a quantitative fit for the spin-orbit-split negative-parity resonances, is made, with the result that the magnitude of the octet spin-orbit term is about six times greater than the singlet one. The masses and mixing amplitudes for all the resonances in the (70,${1}^{\ensuremath{-}}$) are calculated, and it is pointed out that because of the large mixings for ${J}^{P}={\frac{1}{2}}^{\ensuremath{-}} \mathrm{and} {\frac{3}{2}}^{\ensuremath{-}}$, the Gell-Mann-Okubo mass formula cannot be expected to hold for these resonances, so that there is no sense in trying to group them into octets and decuplets.