Spectral tones observed in the near field of isothermal, high-Reynolds-number turbulent jets are shown to be due to trapped acoustic modes. These are confined to the potential core, have negative phase velocity and exist in a frequency band that stability analysis shows to be bounded by a pair of saddle points in the complex wavenumber plane. The physical nature of the modes is clarified by analogy with a soft-walled cylindrical duct. A space-time stability analysis suggests that they correspond to marginally stable global modes.