A sensation of beats also occurs when the frequencies of two tones f1 and f2 are nearly, but not quite, in a simple ratio. If f2 = 2f1 + Δ (mistuned octave), beats are heard at a frequency d. In general, when f2 = (n/m)f1 +Δ, mΔ beats occur each second. These are called second-order beats or beats of mistuned consonances, because the relationship f2 = (n/m) f1, where n and m are integers, defines consonant musical intervals, such as a perfect fifth (3/2), a perfect fourth (4/3), a major third (5/4), etc.
Primary beats can be easily understood as an example of linear superposition in the ear. Second-order beats between pure tones are not quite so easy to explain, however. Helmholtz (1877) adopted an explanation based on combination tones (Demonstration 34), but an explanation by means of aural harmonics was favored by others, including Wegel and Lane (1924). This theory, which explains second-order beats as resulting from primary beats between aural harmonics of f1 and f2, predicts the correct frequency m f2 - n f1, but cannot explain why the aural harmonics themselves are not heard (Lawrence and Yantis, 1957).
An explanation which does not require nonlinear distortion in the ear is favored by Plomp (1966) and others. According to this theory, the ear recognizes periodic variations in waveform, probably as a periodicity in nerve impulses evoked when the displacement of the basilar membrane exceeds a critical value. This implies that simple tones can interfere over much larger frequency differences than the critical bandwidth, and also that the ear can detect changing phase (even though it is a poor detector of phase in the steady state).
Beats of mistuned consonances have long been used by piano tuners, for example, to tune fifths, fourths, and even octaves on the piano. Violinists also make use of them in tuning their instruments. In the case of musical tones, however, primary beats between harmonics occur at the same rate as second-order beats, and the two types of beats cannot be distinguished.
In the first example, pure tones having frequencies of 1000 and 1004 Hz are presented together, giving rise to primary beats at a 4-Hz rate.
In the next example, tones with frequencies of 2004 Hz, 1502 Hz, and 1334.67 Hz are combined with a 1000-Hz tone to give secondary beats at a 4-Hz rate (n/m = 2/1, 3/2, and 4/3, respectively).
It is instructive to compare the apparent strengths of the beats in each case.
References
H.L.F. von Helmholtz (1877), On the Sensations of Tone as a Physiological Basis for the Theory of Music, 4th ed. Transl. A.J.Ellis (Dover, New York, 1954).
M.Lawrence and P.A.Yantis (1957), "In support of an 'inadequate' method for detecting 'fictitious' aural harmonics," J. Acoust. Soc. Am. 29, 750-51.
R.Plomp (1966), Experiments on Tone Perception, (Inst. for Perception RVO- TNO, Soesterberg, The Netherlands).
T.D.Rossing (1982), The Science of Sound (Addison-Wesley, Reading, MA). Chap. 8
R.L.Wegel and C.E.Lane (1924), "The auditory masking of one tone by another and its probable relation to the dynamics of the inner ear," Phys. Rev. 23, 266-85.
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