The tones used in the tritone paradox have 6 components (octaves) and are presented through a bell-shaped spectral envelope (see Figure to the left). The filtering results in the frequency components being attenuated about the centre frequency. These Deutsch tones are perceptually similar to Shepard tones, although to date, there has been no systematic investigation of perceptual effects arising from the differences between the two types of tones.

Illusions arise from Deutsch and Shepard tones when the spectral envelopes are held constant and the frequencies components are raised or lowered in pitch. This raising or lowering in pitch can be done in a continuous fashion creating a glide, or in discrete steps. Theoretically, the interval distance of each step could be any value within the span of an octave - the limit of which results in an identical tone. Most studies have employed the 12 equal logarithmic steps that span an octave in traditional Western Music called semitones. The resulting 'chromatic' tones are represented on a piano by the 12 black and white keys that span an octave interval.

For any given spectral envelope, individual tones are ambiguous in terms of their pitch height because multiple octaves are presented simultaneously. However, one can vary the relative height of a given complex tone by shifting the spectral envelope to be centered on a higher or lower pitch. There is no change in the perceived pitch as a result of such spectral envelope shifts since the fundamental frequency, which determines the perceived pitch of complex harmonic tones, remains the same. Nevertheless, such changes result in a shift in the proportional amount of energy for each of the frequency components, resulting in a perceived difference in the timbre of the tones. In addition to the temporal characteristics of a tone's individual components, it is such a proportional spectral weighting of energy that allows us to hear a difference between two instruments that are playing the same pitched tone.

The result of filtering is a tone that is ambiguous in terms of its pitch height. If you were to play the 12 chromatic tones that make up an octave over and over again, you will create an illusion of a continuous ascending or descending pitch sequence depending upon whether you play the chromatic series as ascending or descending respectively. This is the auditory equivalent to the Penrose visual illusion of a continuous staircase made famous by Escher. In the demos below, the same 12 tones are repeated, but the point at which the sequence starts over is not perceived.

Because there are multiple octaves "being played" within each tone, the pitch height of each tone becomes ambiguous. These tones represent the "chroma" or "pitch class" of the 12 musical tones than span an octave in traditional Western music and that are given the letter names C, C#, D, D#, E, F, F#, G, G#, A, A#, B.

Playing pairs of these tones will create the perception of ascending or descending intervals on the basis of pitch proximity. That is, when the second tone is less than one half an octave (an interval called the tritone) "above" the first, subjects will hear the second tone as higher in pitch compared to the first (e.g., C-E). If the second is more than a tritone "above" the first, subjects perceive the second tone as lower in pitch compared to the first (e.g., C-G). It is the perception of the half-octave intervals (e.g., C-F#) that serves as the basis for the tritone paradox.

Since there are 12 different Shepard tones in the chroma circle, there are 12 tritone pairs. To find the tritone interval for any given tone, simply go directly across the chroma circle, as illustrated below. The interesting question is whether the interval is perceived as higher or lower in pitch. Thus in the Figures below, is the C (colored blue) above or below the F# (colored red)?

Diana Deutsch and her colleagues have conducted many studies of how people perceive the 12 tritone pairs. Since this interval is maximally ambiguous in terms of pitch proximity, one could expect subjects to be able to hear it both ways. That is, sometimes they could hear it as ascending, other times as descending. Surprisingly, Dr. Deutsch found that for some of the tritone pairs, there is very little ambiguity in how they are heard. A subject will almost always hear them as ascending, or descending, but not both ways. Other tritone pairs can easily be perceived both ways.

The pairs with little ambiguity tend to have their first tones in the pair on opposite sides of the chroma circle. Those that can be heard both ways are found in between. If you were to envision the 12 chromatic pitches in series as the numbers on the face of a clock, pairs in which the first member is in the upper half of the clock will result in more descending perceptions, and those in the bottom with more ascending perceptions. Pitch classes at the top or bottom of "the clock" are the least ambiguous. This can be seen in the figure below which represents the proportion of times each interval pair is heard as descending for a hypothetical subject. Which pitch one hears as the "highest" (i.e., at 12 oclock) differs across subjects, but in a very interesting way.

Dr. Deutsch and her colleagues have found an amazing consistency across individuals within geographical areas in the orientation of the pitch class circle. Testing subjects from Southern Britain, she found that the tritone pairs beginning with pitches F#-A tend to generate the most number of descending interval judgments suggesting that subjects from Britain orient their pitch class circle with these values at the top. Subjects from California tend to perceive the same tritone pairs as generating the fewest descending judgments. That is, they have pitch class circles oriented exactly opposite, with F#-A at the bottom. Dr Deutsch and her colleagues have found a positive correlation between the peak pitch class and the upper limit of the main octave of their speech. In addition, there is some evidence of a relationship between how children and their mothers perceive the tritone pairs.

Lloyd Dawe, John Platt, and Eydra Welsh tested 20 students at McMaster University who were residing in South Western Ontario, Canada. Many of the subjects were immigrants, and most (about 80%) were multilingual. In addition, the subjects indicated on a survey, that they spent considerable time watching American television stations and listening to American media. On the basis of this information, one would expect the response pitch profiles to be either quite variable or similar to those found by Dr. Deutsch for most Americans. On the other hand, Canada maintains closer ties with the British, and often uses British spelling and pronunciation, which would lead one to expect subjects to give British-like response profiles.

The results were somewhat surprising. All subjects gave profiles indicating peak pitch classes similar to the British. This may indicate that the orientation of the pitch class circle is malleable, and changes to match a prevalent mode within a geographical area.

Lloyd Dawe, from Cameron University in collaboration with John Platt and Eydra Welsh from McMaster University found spectral-motion aftereffects following a long presentation of a Shepard scale. Motion aftereffects are very well documented in vision. After prolonged presentation of a movement in a particular direction, such as a rotating spiral, stationary objects (e.g., static spirals) appear to "drift" in the opposite direction.

Why does this happen? There are cells within our nervous system that are responsible for encoding movement in particular directions. These cells are "leaky" in the sense that they have a spontaneous rate of responding even when there is no objective movement within one's environment. The reason we do not perceive movement under such static conditions is because no single population of cells tends to dominate. Spontaneous responses from a right-to-left motion cell would cancel out with a spontaneous response from a left-to-right motion cell. Following a long exposure to movement in a particular direction, the cells responsible for encoding that movement become fatigued and gradually stop responding (a process called adaptation). When people then view a static scene, the spontaneous responses from cells responsible for encoding movement in the opposite direction are "unopposed" and create the perception of motion in that direction.

After listening to a prolonged presentation of a Shepard scale, perception of the tritone pairs changes depending upon whether the scale was ascending or descending. If you listen to an ascending scale, there will be more descending responses, and a shift in the orientation of the chroma circle in a counter clockwise direction.

After listening to a descending scale, there will be fewer descending responses and a shift in the orientation of the chroma circle in a clockwise direction.

These effects are the auditory equivalent to spiral-motion aftereffects in vision, and indicate that spectral-motion specific cells within the auditory nervous system are involved in perception of the tritone paradox.

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Created by Lloyd A. Dawe. Last updated October 7, 1997.