The absorption of fast Alfvén waves (FW) by ion cyclotron harmonic damping in the range of harmonics from fourth to eighth is studied theoretically and with experiments in the DIII-D tokamak. A formula for linear ion cyclotron absorption on Maxwellian ion species is used to estimate the single-pass damping for various cases of experimental interest. It is found that damping on fast ions from neutral beam injection can be significant even at the eighth harmonic if the fast ion beta and the background plasma density are both high enough. The predictions are tested in several L-mode experiments in DIII-D with FW power at 60 MHz and at 116 MHz. It is found that 4th and 5th harmonic absorption of the 60 MHz power on the beam ions can be quite strong, but 8th harmonic absorption of the 116 MHz power appears to be weaker than expected. Possible explanations of the discrepancy are discussed. © 2005 American Institute of Physics .