The following discussion assumes familiarity with the Shepard's Tones illusion, so if you haven't yet explored it you should do so now.
Although pitch discrimination cues have been carefully removed from Shepard's Tones, proximity information remains. Two consecutive tones are always separated by a single semitone. So although you can't really determine which is higher based on the tones alone, your choice is that the second tone is either one semitone higher or eleven semitones lower in pitch than the first. It is natural for the smaller distance to be automatically selected.
This can be thought of as an auditory counterpart of the apparent motion generated by the colour bar illusion. In that illusion, the cue used to determine the direction of motion can be eliminated, resulting in a pattern which doesn't appear to move at all. In the case of Shepard's Tones, what would happen if the proximity cue was removed? In other words, what would you hear if the second tone played was either half an octave higher or half an octave lower than the first? (The midpoint of the octave is called the tritone, hence the name of the current illusion.)
In 1987, the psychologist Diana Deutsch published a paper entitled The tritone paradox: Effects of spectral variables [Deutsch-87] that explored exactly this question. Her findings were surprising in a number of respects.
The applet window contains several different controls that can be used to explore the tritone paradox. The basic demonstration, however, requires only the Start button. When pressed, this button will cause two of Shepard's Tones, separated by half an octave, to be played alternately and repeatedly. You should attempt to judge which one is higher in pitch. Once you have made this decision, you should also spend a few moments to determine whether you can force yourself to hear the other as higher (in much the same way that you may have been able to switch at will between the two possible orientations of Necker's cube).
Next, select the Display switch. This will display the currently sounding tone using the same format as for Shepard's Tones. You can use the label at the top of the vertical axis to name the tone which sounds higher.
Make note of the label of the higher sounding tone, then switch the Base frequency selection from 110 Hz to 160 Hz. Which tone sounds higher in pitch now?
It is often quite interesting to show this demonstration to a large audience. At a base frequency of 110 Hz, most people will perceive Tone 1 as being higher than Tone 2. But a small percentage of people will hear the opposite order (and they may be reluctant to oppose the apparent consensus of the group, so you may need to do some gentle coaxing to identify them). When the base frequency is switched, most people will perceive the opposite ordering.
In actual fact, Tone 2 is higher than Tone 1 at both base frequencies (higher in the sense that each component of Tone 2 is half an octave higher than the corresponding component of Tone 1). Nevertheless, most people hear Tone 1 as higher in at least on of the conditions.
There are at least two mysterious aspects of this illusion. First, the direction in which the ambiguity is resolved does not appear to be random but depends instead on the absolute frequencies involved. If you heard the first tone as higher at a base frequency of 110 Hz, you can be confident that if you try return to this illusion next week or next year you will hear it the same way. This is surprising because perfect pitch (the ability to identify the absolute pitch of a tone) is rare, but it appears to be the absolute pitch which determines how the ambiguity is resolved. As Deutsch points out in a separate article [Deutsch-92], this is analogous to having the perception of an image (in isolation) reliably change as it is shifted in space. Clearly this is a unique aspect of audition.
Perhaps more unusual still is the result of a study, reported by Deutsch in her second article, which found that British subjects reliably resolved the ambiguity in one direction while Californian subjects reliably resolved it in the other!