An even-wave harmonic oscillator (h.o.) model for the quark-quark interaction proposed recently for the baryon spectrum is described with a detailed mathematical formulation. The mechanism, which formally admits of a relativistic extension of the Feynman-Kislinger-Ravndal type, leaves unchanged the usual h.o. predictions for 56 states (symmetric) for all $L$ values even and odd, but totally keeps out the 20 states (antisymmetric). It changes the structure of the 70 states considerably, while retaining the principal feature of linear rise of ${(\mathrm{mass})}^{2}$ with $J$ through the interplay of two reduced slopes of magnitudes $\frac{1}{2}\ensuremath{\alpha}$ and $\frac{1}{2}\sqrt{3}\ensuremath{\alpha}$, compared to $\ensuremath{\alpha}$ for the 56 spectrum. The new features of the 70 states are (i) a dual spectrum leading to considerable mass splitting compared to the usual h.o. model without SU(6)-breaking effects, (ii) prediction of a unique (70, ${0}^{+}$) supermultiplet lower than the (70, ${1}^{\ensuremath{-}}$), and (iii) the prediction of low radial excitations because of the reduced slopes. The immediate experimental successes are (i) an understanding of ${P}_{11}(1470)$ together with possible $\ensuremath{\Delta}$, $\ensuremath{\Sigma}$, $\ensuremath{\Lambda}$ counterparts, (ii) two distinct mass groupings manifest in (70, ${1}^{\ensuremath{-}}$) states, and (iii) plausible explanation of ${P}_{11}^{\ensuremath{'}}(1750)$ as a radial excitation of ${P}_{11}(1470)$. The mass splittings of $\ensuremath{\Delta}$, $\ensuremath{\Sigma}$, $\ensuremath{\Lambda}$ from their $N$ counterparts, compared for 56 and 70 states, conform extremely well to the ratio of the average slope $\ensuremath{\delta}=\frac{1}{4}\ensuremath{\alpha}(1+\sqrt{3})\ensuremath{\approx}0.68 \ensuremath{\alpha}$ for 70 states to that ($\ensuremath{\alpha}$) for the 56, thus facilitating the prediction of $\ensuremath{\Delta}$, $\ensuremath{\Sigma}$, $\ensuremath{\Lambda}$ positions from those of $N$ states for different quantum numbers. Extra predictions of states are discussed in terms of an extended classification scheme given by an ordered set of four quantum numbers (${n}_{x}{l}_{x}{n}_{y}{l}_{y}$) defined in the text.