I present evidence that the strong electron-electron interactions in   gapped carbon nanotubes lead to finite hierarchies of excitons within   a given nanotube subband. I study these hierarchies by employing a   field theoretic reduction of the gapped carbon nanotube permitting   electron-electron interactions to be treated exactly. I analyze this   reduction by employing a Wilsonian-like numerical renormalization   group. I am so able to determine the gap ratios of the one-photon   excitons as a function of the effective strength of interactions. I   also determine within the same subband the gaps of the two-photon   excitons, the single particle gaps, as well as a subset of the dark   excitons. The strong electron-electron interactions in addition lead   to strongly renormalized dispersion relations where the consequences   of spin-charge separation can be readily observed.