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