Using a collection of 12 semiconducting carbon-nanotube samples, each highly enriched in a single chirality, we study the chirality dependence of the K-momentum dark singlet exciton using phonon sideband optical spectroscopy. Measurements of bright absorptive and emissive sidebands of this finite momentum exciton identify its energy as 20–38 meV above the bright singlet exciton, a separation that exhibits systematic dependencies on tube diameter, 2n+m family, and chiral index. We present calculations that explain how chiral angle dependence in this energy separation relates to the Coulomb exchange interaction and elaborate the dominance of the KAA; phonon sidebands over the zone-center phonon sidebands over a wide range of chiralities. The Kataura plot arising from these data is qualitatively well described by theory but the energy separation between the sidebands shows a larger chiral dependence than predicted. This latter observation may indicate a larger dispersion for the associated phonon near the K point than expected from finite distance force modeling.