We have generated and detected coherent lattice vibrations in single-walled carbon nanotubes corresponding to the radial breathing mode (RBM) using ultrashort laser pulses. Because the band gap is a function of diameter, these RBM-induced diameter oscillations cause ultrafast band gap oscillations, thereby modulating the interband excitonic resonances at the phonon frequencies (3-9 THz). Excitation spectra show a large number of pronounced peaks, allowing the determination of the chiralities present in particular samples and relative population differences of particular chiralities between samples.