The tritone paradox was discovered by Deutsch in 1986, and first described at a meeting of the Acoustical Society of America (Deutsch, 1986) and first published by Deutsch, Music Perception, 1986. The basic pattern that produces the this illusion consists of two computer- produced tones that are related by a half-octave. (This interval is called a tritone). When one tone of a pair is played, followed by the second, some people hear an ascending pattern. But other people, on listening to the identical pair of tones, hear a descending pattern instead. This experience can be particularly astonishing to a group of musicians who are all quite certain of their judgments, and yet disagree completely as to whether such a pair of tones is moving up or down in pitch.
The tritone paradox has another curious feature. In general, when a melody is played in one key, and it is then transposed to a different key, the perceived relations between the tones are unchanged. The notion that a melody might change shape when it is transposed from one key to another seems as paradoxical as the notion that a circle might turn into a square when it is shifted to a different position in space.
But the tritone paradox violates this rule. When one of these tone pairs is played (such as C followed by F#) a listener might hear a descending pattern. Yet when a different tone pair is played (such as G# followed by D), the same listener hears an ascending pattern instead. (Another listener might hear the C-F# pattern as ascending and yet hear the G#-D pattern as descending.)
The tones that are used to create the tritone paradox are so constructed that their note names (C, C#, D and so on) are clearly defined, but they are ambiguous with respect to which octave they are in. For example, one tone might clearly be a C, but in principle it could be middle C, or the C an octave above, or the C an octave below. This ambiguity is built into the tones themselves. So when someone is asked to judge, for example, whether the pair of tones D-G# is ascending or descending in pitch, there is literally no right or wrong answer. Whether the tones appear to move up or down in pitch depends entirely on the mind of the listener. (Ambiguous tones such as these were used by others, particularly Roger Shepard and Jean-Claude Risset, to create illusions of endlessly ascending or descending pitches.)
The way that any one listener hears the tritone paradox varies depending on the names of the notes that are played. The musical scale is created by dividing the octave into twelve semitone steps, and each tone is given a name: C, C#, D, D#, E, F, F#, G, G#, A, A# and B. The entire scale, as it ascends in height, consists of the repeating occurrence of this succession of note names across octaves. So when you move up a piano keyboard in semitone steps beginning with C, you go first to C#, then D, then D#, and so on, until you get to A#, then B, and then C again. At this point you have reached an octave, and you begin all over, repeating the same series of note names in the next octave up the keyboard.
Because all Cs sound in some sense equivalent, as do all C#s, all Ds, and so on, we can think of pitch as varying both along a simple dimension of height and also along a circular dimension of pitch class - a term that musicians use to describe note names. So, for example, all Cs are in pitch class C, all C#s are in pitch class C#, and all Ds are in pitch class D.
Let us suppose that listeners mentally arrange pitch classes as a circular map, like a clockface. To explain different listeners' perceptions of the tritone paradox, Deutsch conjectured that one person might orient his or her clockface so that C is in the 12 o'clock position, C# is in the 1 o'clock position, and so on around the circle. This listener would tend to hear the pattern C-F# (as well as B-F, and C#-G) as descending, and the pattern F#-C (as well as F-B and G-C#) as ascending. But another person might orient is or her clockface so that F# is in the 12 o'clock position, G is in the 1 o'clock position, and so on. This listener would instead tend to hear the pattern C-F# (as well as B-F, and C#-G) as ascending, and the pattern F#-C (as well as F-B and G-C#) as descending. In other words, differences between listeners in perception of the tritone paradox could be due to differences in the way they orient their maps of the pitch class circle.
In one experiment, listeners were played many such pairs of tones, and they judged in each case whether they heard an ascending or a descending pattern. Then the proportion of times that each listener heard a descending pattern was plotted as a function of the pitch class of the first tone of the pair.
The results supported Deutsch’s conjecture - the judgments of most listeners varied systematically depending on the positions of the tones along the pitch class circle: Tones in one region of the circle tended to be heard as higher, and tones in the opposite region as lower.
In addition, the orientation of the pitch class circle varied strikingly from one listener to another. To illustrate these differences, the judgments of two subjects are shown below. These subjects heard the tritone paradox in a very pronounced fashion. The first subject clearly heard tone pairs C#-G, D-G#, D#-A and E-A# as ascending, but F#-C, G-C#, G#-D, A-D#, A#-E, and B-F as descending. The second subject, in contrast, heard tone pairs B-F, C-F#, C#-G, D-G#, D#-A, and E-A# as descending, and F#-C, G-C#, G#-D, and A-D# as ascending. So for the most part when the first subject heard an ascending pattern the second subject heard a descending one, and vice versa. The upper part of the figure shows the two orientations of the pitch class circle with respect to height which were derived from the judgments of these subjects. For the first subject, pitch classes G# and A stood at the top of the circle, but for the second subject, C# and D stood in this position instead. To further illustrate the differences between listeners in perception of the tritone paradox, the judgments of four more subjects are shown below.
Another surprising consequence of the tritone paradox concerns absolute pitch - the ability to name a note just from hearing it. This faculty is generally considered to be very rare. But the tritone paradox shows that the large majority of people in fact possess a form of absolute pitch, since we hear tones as higher or as lower depending simply on their pitch classes, or note names.
Why do people orient their maps of the pitch class circle in different ways? Deutsch conjectured that the answer might lie in the speech patterns that we hear. When people from other countries visited her laboratory in California, they often heard this pattern differently from native Californians. And when the effect was demonstrated to audiences in other countries and they were asked to indicate their judgments with a show of hands, audiences appeared to differ from each other in what they heard.
So on the basis of these observations, Deutsch (1991) compared two groups of subjects. Those in the first group had grown up in California, and those in the second group had grown up in the south of England. These two groups tended to differ in how they heard the tritone paradox: Frequently when a Californian subject heard a pattern as ascending, a subject from the south of England heard the identical pattern as descending, and vice versa.
In another study, Deutsch, North, and Ray, 1990, obtained a significant correspondence between the pitch range of a person's speaking voice and how he or she perceived this pattern. This study provided a further indication that speech patterns influence the the way the tritone paradox is heard.
Other studies have uncovered regional differences within the United States and Canada in the perception of the tritone paradox (Ragozzine and Deutsch, 1994; Treptoe, 1997, Giangrande, 1998, Dawe, Platt and Walsh 1998; Chalikia, 1999, 2000 . Because there are regional dialects within the United States, it seems that speech patterns are likely to lie at the root of these differences also. It even appears that the way a person hears the tritone paradox is related, not only to the geographical region in which he or she grew up, but also to the regions in which his or her parents grew up (Ragozzine and Deutsch, 1994; Deutsch, 1996; Deutsch, Henthorn, and Dolson, 2000). So it seems that the speech to which we were exposed as children influence the way we hear the tritone paradox as adults. In addition, people who have spent considerable time in more than one geographical region sometimes produce mixed results. For example, their plots may have two peaks, reflecting their exposure to different languages or dialects.
This compact disc includes the signals for a full experiment on Deutsch's tritone paradox. This experiment contains 16 blocks with 12 tone pairs in each block.
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Deutsch, D. Mothers and their offspring perceive the tritone paradox in closely similar ways. Archives of Acoustics, 32, 3-14, 2007. [PDF Document]