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Vassilakis, P.N. (1997).  Beats, the Difference Tone, and the Perception of the Missing Fundamental.  Master's Thesis.  University of California, Los Angeles.  Advisor: R.A. Kendall.  Committee: E.C. Carterette, T. Rice,  R.W.H. Savage. 

Hypothesis: the beating sensation and the difference tone have common physical bases and are responsible for the perception of the missing fundamental.

Qualitative evidence supporting this hypothesis:

  1. While slow amplitude fluctuations describe an objective phenomenon, related to interference and occurring in the medium of propagation, their perceptual manifestation, beating, cannot be explained in terms of the principle of linear superimposition. With the beating frequency being equal to the difference frequency, beats must be introduced by a quadratic term of displacement and may therefore have nonlinear bases. Considering the difference tone a high frequency beat phenomenon, objective in nature, implies that air may be partly be a nonlinear medium with regards to sound propagation. This suggestion is supported by the observation that the nonlinear transfer characteristic introduced to account for both beats and the difference tone describes an asymmetry which is present in air. For a graphic representation of the argument in favor of beats' nonlinearity click here.

  3. Amplitude fluctuations/modulations, resulting from the interference of two (or more) simple harmonic motions propagating in a medium (air) as sound waves, behave as waves themselves. They propagate in the same medium as the individual components, with a 'group velocity' (vg) that is different in general from the velocity of the components, or 'phase velocity' (vph), and they carry energy as they propagate.
  4. In the case of complex harmonic tones with no energy at the fundamental frequency, the existing components interact and result in modulations (or, more generally, amplitude fluctuations) that propagate in a medium and transfer energy at a rate equal to the frequency difference between successive components.  For such tones the difference frequency (rate of amplitude fluctuation) matches in value the frequency of the missing fundamental. Therefore, even if no energy is supplied at the fundamental frequency, the interaction of the existing components within the medium of propagation result in modulations that may introduce a component at that frequency.
  5. Regarding the two phenomena known as the 1st and 2nd pitch shift effects, it is argued that they are simply alternative manifestations of a single phenomenon, the 1st pitch shift effect (see the relevant paper). 
    The pitch shift effect is explained in terms of changes in the relationship between the effective group (vg) and phase (vph) velocities for the complex tones used to demonstrate it. These changes -which relate directly to the way the said tones are constructed- influence the effective frequency of the resulting modulations, making the 1st pitch shift effect partially analogous to the Doppler effect (see the relevant paper). For a related illustration click here.
  6. Various studies claiming that no spectral information corresponding to the fundamental is necessary at the cochlear level for the perceived pitch of complex tones to match that of the missing fundamental are addressed. With regards to the studies of Schouten (1938) and Licklider (1954), their claim regarding the absence of spectral information at the fundamental has been challenged. In the case of the study by Houtsma & Goldstein (1972), where dichotic presentation ensured the absence at the cochlear level of any component corresponding to the fundamental, the authors' claim that their experimental results indicate processing of fundamental information has been questioned.

The general inability of place (spectral) theories of pitch perception to account for residue phenomena has been the main justification for the introduction of various periodicity theories, which are often favored despite their own inability to explain phenomena such as:
   a)  Binaural diplacusis, describing frequency-pitch differences for a person's two ears (Bekesy, 1963a; van den Brink, 1970);
   b)  Pitch shift of pure tones as well as of complex tones (with or without energy at the fundamental) resulting from the introduction of white noise, band-filtered noise or high/low-pass-filtered noise (Bekesy 1963b; van den Brink, 1970);
   c)  Complex tones with no periodicity but prominent pitch (narrow bands of noise). Complex signals with the same periodicity and different pitch, or the same pitch and different periodicity. Many examples of such tones can be found in experiments that examine the pitch shift effect.  For example, the following 3-component tones have the same periodicity (their signals repeat themselves 40 times per second ) while their perceived pitch differs [pitch (1) ~ 212 Hz, pitch (2) ~ 209 Hz.]:
                                     (1): 640 Hz - 840 Hz - 1040 Hz     &     (2): 840 Hz - 1040 Hz - 1240 Hz
At the same time, the perceived pitch of the following 3-component tones is the same [pitch (1') = pitch (2') ~ 212 Hz] while their periodicity differs [the signal of (1') repeats 40 times per second and that of (2') 212 times per second]:
                                    (1'): 640 Hz - 840 Hz - 1040 Hz      &     (2'): 636 Hz - 848 Hz - 1060 Hz

The place theory of pitch perception can explain the above phenomena and, as the present study has indicated, it may also explain residue phenomena. In its most common form (von Bekesy, 1963a,b) this theory states that pitch is determined by the place of maximum excitation along the basilar membrane and is not dependent on the firing rate of neural discharges. This independence on firing rate allows the place theory to eliminate one of the greatest problems faced by periodicity theories: their inability to explain how pitch and loudness are separated in the auditory nervous system. On the other hand, the dependence on the place of maximum excitation seems to be challenged by the fact that the pitch of complex tones does not necessarily correspond to the component of a vibration that has maximum energy. Von Bekesy (1963a,b; 1972) has answered this challenge by arguing that the place of maximum excitation along the basilar membrane differs from the vibration maximum as a consequence of time delay and nervous inhibition, nonlinearities that are introduced by the inner ear and result in the amplification of the lowest component present in the cochlea (whether it was present in the original stimulus or it was introduced as the difference tone). An alternative answer to this challenge is suggested below.

The hypothesis is that the pitch is determined not by the place of maximum excitation along the basilar membrane but by the place along the basilar membrane where the change in the slope of excitation changes direction [a somewhat similar suggestion has been made by Evans, (1975; in Campbell & Greated, 1987)]. For complex stimuli there are more than one such places (corresponding to successive components lying within separate critical bands) fact that may explain the difference between synthetic and analytic pitch. According to the above hypothesis, synthetic and analytic modes of listening may be seen as describing the same operation, with the only difference depending on whether a listener focuses on the place with the most prominent change in slope direction (lowest component) or on one of the other available such places (first 4-5 components). The advantage of such an explanation over the one based on the maximum-excitation hypothesis is that it explains the importance of the fundamental (lowest component) to the perceived pitch of harmonic complex tones.

Studies on the pitch of pure tones seem to support the above hypothesis. Von Bekesy (1960, in von Bekesy 1963a) has shown that there are consistent deviations between the anatomically determined frequency localization (place of maximum excitation along the basilar membrane) and the pitch localization deriving from psychological experiments. More specifically, when compared with the anatomically determined frequency localization, pitch localization along the basilar membrane is shifted towards the helicotrema, with a shift that decreases as we move from low to high frequencies. Since the excitation pattern along the basilar membrane in response to low frequencies spreads over a larger area than it does in response to high frequencies, the observed localization shift may be explained in terms of the place difference between the point of maximum excitation and the point of change in the slope direction, a difference that is greater for low frequencies.

A modified place theory of pitch perception based on changes in the slope direction of the excitation pattern along the basilar membrane has interesting implications regarding the pitch of inharmonic complex tones. In such tones, interference between the original components introduces (according to the present study) new components that do not in general match the frequency of the original components or of each other. The resulting concentration of more than one components on each critical band may smooth out the slopes of the excitation pattern along the basilar membrane, making pitch determination difficult, ambiguous, or impossible, depending on the total number of components and on the uniformity of energy/frequency distribution [for example, infinite number of components and complete uniformity (i.e. white noise) yields no pitch].



von Békésy, G. (1963a). Hearing theories and complex sounds. J. Acoust. Soc. Am., 35(4): 588-601.
                     (1963b). Three experiments concerned with pitch perception. J. Acoust. Soc. Am., 35(4): 602-606.
                     (1972). The missing fundamental and periodicity detection in hearing. J. Acoust. Soc. Am., 51(2/2): 631-637.

van den Brink, G. (1970). Two experiments on pitch perception: diplacusis of harmonic AM signals and pitch of inharmonic
                     AM signals. J. Acoust. Soc. Am., 48(6/2): 1355-1365.

Licklider, J. C. R. (1954). "Periodicity" pitch and "place" pitch. J. Acoust. Soc. Am., 26: 945.

Campbell, M. and Greated, C. (1987). The Musician's Guide to Acoustics. New York: Schirmer Books.

Houtsma, A. J. M. and Goldstein, J. L. (1972). The central origin of the pitch of complex tones: evidence from musical
                     interval recognition. J. Acoust. Soc. Am., 51(2/2): 520-529.

Schouten, J. F. (1938). The perception of subjective tones. Proceedings of Koninklijke Nederlandsche Akademie van
, 41(10).