This work revisits the physics giving rise to the carbon nanotube phonon bands in the midinfrared. Our measurements of doped and undoped samples of single-walled carbon nanotubes in Fourier transform infrared spectroscopy show that the phonon bands exhibit an asymmetric line shape and that their effective cross section is enhanced upon doping. We relate these observations to electron-phonon coupling or, more specifically, to a Fano resonance phenomenon. We note that the dopant-induced intraband (not interband) continuum couples strongly to the phonon modes, and that defects created on the sidewall are scattering centers that increase the spectral amplitude of the resonance.