The photophysics of single-wall carbon nanotubes (SWCNTs) is intensively studied due to their potential application in light harvesting and optoelectronics. Excited states of SWCNTs form strongly bound electron-hole pairs, excitons, of which only singlet excitons participate in application relevant optical transitions. Long-living spin-triplet states hinder applications, but they emerge as candidates for quantum information storage. Therefore, knowledge of the triplet exciton energy structure, in particular in a SWCNT chirality dependent manner, is greatly desired. We report the observation of light emission from triplet state recombination, i.e., phosphorescence, for several SWCNT chiralities using a purpose-built spectrometer. This yields the singlet-triplet gap as a function of the SWCNT diameter, and it follows predictions based on quantum confinement effects. Saturation under high microwave power (up to 10 W) irradiation allows the spin-relaxation time for triplet states to be determined. Our study sensitively discriminates whether the lowest optically active state is populated from an excited state on the same nanotube or through Forster exciton energy transfer from a neighboring nanotube.