When one listens to a pair of successive musical tones, one can ordinarily tell whether or not the tones are equal in pitch; or if the first is higher in pitch than the second; or vice versa. However, even for ordinary musical tones there is octave equivalence, which means that tones may be confused with one another although their oscillation frequencies differ by a factor of two. This implies that for harmonic complex tones there exists a certain ambiguity of pitch which naturally emerges from the multiplicity of pitch (see topics pitch perception, definition of pitch, virtual pitch).
The ambiguity of pitch can be much amplified by suppressing certain harmonics from the Fourier spectrum of a "natural" harmonic complex tone. Shepard (1964a) has described observations on harmonic complex tones whose Fourier spectrum consisted only of harmonics that were in an octave relationship, i.e., the 1st, 2nd, 4th, 8th, 16th, etc. While the musical pitch class (the chroma) of such tones is well defined, the absolute height of pitch is quite ambiguous; that is, octave confusions are very likely to occur. This is particularly true when the frequency of the first harmonic is near the lower limit of the hearing range while the upper part tones extend up to the high end of the hearing range. In that case, there indeed is little - if any - information available to the ear about what actually is the fundamental frequency (oscillation frequency).
When, for instance, the oscillation frequency of the above type of tone is 10 Hz and the number of part tones chosen is 11, the listener is exposed to a spectrum of part tones with the frequencies 10, 20, 40, 80, ... 10240 Hz. When that tone is followed by another having twice the oscillation frequency of the first, the listener gets exposed to 20, 40, 80, ... 20480 Hz, and it is not surprising that one will not perceive much of a difference, if any. So, under these conditions there is "perfect" octave equivalence.
From this notions it is easy to understand that when the ratio between the oscillation frequencies of the two tones is 1.414 (square root of two; half and octave; tritone interval), the listener on first sight cannot be expected to be able to tell whether the second tone is higher in pitch than the first or vice versa. The so-called tritone paradox, first described by Deutsch (1986a), Deutsch et al. (1987a), originates from the observation that listeners in fact do make fairly consistent decisions on which of the two tones is higher in pitch, i.e., whether they heard an upward or downward step of pitch. However, while the responses of individual listeners are fairly consistent and reproduceable, different listeners may give opposite responses. Moreover, the responses of individual listeners turn out to be dependent on the absolute height of the oscillation frequencies. That is, when the listening experiment is made with a base frequency of, e.g., 12 Hz instead of 10 Hz, the individual responses may systematically change. This was regarded as a particularly "paradox" outcome.
The basic aspects of the tritone paradox can be fairly well explained by the theory of virtual pitch [82], [93], [104] p. 376. However, the theory cannot account for the observed individual differences, as the factors governing those differences as yet are unknown.
Author: Ernst Terhardt terhardt@ei.tum.de Mar 1 2000