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The Tritone Paradox

From Deutsch, Diana (1995), Musical Illusions and Paradoxes [CD Recordings], La Jolla: Philomel. More info available at <http://philomel.com/>

“The basic pattern that produces the tritone paradox 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… 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 is 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 a listener might hear a descending pattern; yet when a pair that differs by transposition is played, the same listener might hear an ascending pattern instead.” (Deutsch 1995, 10)

Carrying out the experiment

“To do the experiment, find a time when you will not be disturbed, so that you can relax and listen to the entire set of tones without interruption. For best results, avoid listening to music immediately before judging the patterns. If you have time, run through the experiment at least twice, on separate days, and average the results. If you are testing a group of subjects at the same time, make sure that they do not have the opportunity to compare notes before they have completed the experiment.” (Ibid. 17)

The first track pertaining to the tritone paradox is an explanation of the experiment that includes six practice examples. These are important especially for academic musicians, I find—most of us upon first hearing the tone-pairs will profess to hear something more complex than a simple up or down contour, because our analytical minds detect a kind of polyphony in each tone, and hear both parallel and contrary motion. However, after 2-3 examples, you’ll realize that on a deeper level of listening, you’re making more immediate decisions about how fundamental frequencies connect between the two tones. Try to let go of your analytical approach, and just let your mind make an instinctual judgment.

Practice Track

Now you’re ready to continue with the experiment. To help organize your responses (and evaluation of data), please download this reporting sheet, which will be explained below.

“There are 16 groups of tone pairs, with four groups on each of the four tracks below, labeled 1, 2, 3, and 4. (These are tracks 15-18 on Deutsch 1995. On some browsers, the mp3 players below might take a while to load.) Each group consists of twelve tone pairs, and there are pauses of thirty seconds between the different groups. These pauses are important, so it is best not to shorten them in any way. Column 1 refers to the first group of of twelve tone pairs in the experiment. When you hear the first tone pair of this group, indicate your judgment by drawing an upward or a downward arrow in the topmost box of column 1. When hear the next tone pair, indicate your judgment by drawing an upward or a downward arrow in the next box down; continue in that way until you’ve judged all 12 pairs in the first group.

Following a 30-second pause, the second group of tone pairs begins. Make your judgments for this group in the second column, in the same manner as for column 1. Continue in this way, column by column, until you have made judgments on all sixteen groups of tone pairs. You may take a break approximately half-way through the experiment if you like.

1: Groups 1-4

2: Groups 5-8

3: Groups 9-12

4: Groups 13-16

Interpreting the Results

In the answer key to the stimuli you just heard, you’ll find note-names in an array of boxes matching those that you used to report upwardly- and downwardly-perceived tone pairs. The answer key contains the first pitch-class of each tone pair separated by a tritone. (G = the pair G-C#, A = the pair A-Eb, etc.) Looking down your reporting sheet, and referring to the answer key, write the note-names corresponding to each box containing a downward arrow. Your report sheet should then contain, in each box, either upward arrows alone, or downward arrows accompanied by the note-names of the initial members of tone pairs.

Next, count the number of Cs in downward-arrow boxes, the number C#s, the number of Ds, etc. By doing this, you are computing the number of times each pitch-class was heard as higher when it was the first in the pair. Then, we tally the number associated with each pitch-class by number, in a circular display, so that the quantities associated with each pitch are placed on a different element in a 12-point “clock-face.” By trial and error, we then bisect this circle in such a way as to maximize the difference between the two halves; it may take a few tries to determine which bisection produces the maximum difference between the halves. Finally, we use this bisection to describe a pitch-class “peak” associated with each listener, and this in turn is the data that we attempt to correlate with regional linguistic experience.

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