Enhanced magnetic fluctuations with frequencies peaking at the ion cyclotron frequency and its harmonics have been observed at comet Halley and also in the upstream regions of planetary bow shocks. It is thought that these waves are generated at comets by pickup ions that are unstable to the generation of waves at harmonics of the ion cyclotron wave number. In the spacecraft frame of reference these waves are observed as harmonics of the ion cyclotron frequency. In this report we show that the ring-beam distributions of pickup ions observed in the cometary environment are capable of generating these waves if the beam speed component of the distribution (parallel to the ambient magnetic field) is much larger than the ring speed (perpendicular to the magnetic field). As the ring speed increases relative to the beam speed, other instabiliteis occcur at the same wave number which have even larger growth rates. These additional instabilities do not lead to generation of harmonics. ¿American Geophysical Union 1991