Phantom beats in mistuned chords
J. L. Hall, M. V. Mathews, Lynne Roberts · The Journal of the Acoustical Society of America · 1983
Beats are heard in slightly mistuned musical intervals. For example, a mistuned fifth with frequencies f1 = 804 Hz and f2 = 1204 Hz will produce 4 beats per second. These beats have been attributed to interaction between the 3rd harmonic of f1 at 2412 Hz and the 2nd harmonic of f2 at 2408 Hz. However, 4 beats per second can be heard when the mistuned fifth is played with pure sinusoids which have no harmonics. A second hypothesis attributes these phantom beats to interaction between combination tones f2 − f1 at 400 Hz and 2f1 − f2 at 404 Hz. Experimental results supporting the second hypothesis are presented. If one combination tone is removed by a cancellation tone of appropriate amplitude and phase, the beats can no longer be heard. If tones are added to a perfectly tuned fifth so as to approximate the combination tones produced by a mistuned fifth, the resulting stimulus is difficult to distinguish from the mistuned interval. Similar results were obtained for mistuned fourths, attributable to interaction between combination tones f2 − f1 and 3f1 − 2f2.