It is well known that carbon nanotube (CNT) optical resonances and electrical characteristics are strongly dependent on the atomic structure of the CNT. However, the relationship between electrical properties and CNT atomic structure has been challenging to fully resolve due to the complicating factor of dielectric environment. Early studies of CNTs on gold surfaces (strong dielectric screening) reveal only one slice of a bigger relationship. Here we describe a combination of nanoscale spectroscopy and transport measurements to explore the full relationship between CNT structure, electrical properties, and dielectric environment. We use spectrally-resolved scanning photocurrent spectroscopy to determine the chiral index of individual-contacted suspended CNTs. Dielectric environment is controlled by using various dielectric liquids. In semiconducting CNTs we observe band gap renormalization of approximately 30%. In metallic CNTs we observe a transport gap that is consistent with the theoretical predictions of a Mott gap. The gap is independent of chiral angle, inversely proportional to CNT diameter, and scales inversely with the dielectric constant of the environment. Our results emphasize the importance of spectrally-resolved scanning probe techniques in nanometrology, and highlight the important effect of dielectric environment on nanoscale physical properties.