Regression to the Mean in Music
Recall that encountering a tall person does not cause the next person to be shorter. It is simply that most people tend to be of average height.
For over 400 years, it has been frequently observed that large melodic intervals tend to be followed by a change in melodic direction. Statistical studies show that this is indeed the case. The following folksong melody provides a typical example. In this melody there are 13 melodic leaps—if a leap is defined as any interval larger than a diatonic step. Of these 13 melodic leaps, 11 are followed by a change of direction, 1 leap continues in the same direction, and 1 is followed by a repeat of the same pitch. This example is consistent with the notion that melodic leaps are followed by a change in direction. However, there are other possibilities.
Suppose that a melody made use of just three pitches: A, B and C. Notice that there is only one possible leap: between A and C. Notice also that if we leap up to the C, the only possible continuations are a unison repetition of the C, or descending to B or A. Similarly, if we leap down to the A, the only possible continuations are a unison repetition or an ascending contour. Simply due to the constraints on range, melodic leaps will tend to be followed by a change of direction.
If we expand our melodic range from three pitches to four pitches, the effect is reduced, but there will be a tendency for leaps to be followed by a change of direction. If most of the pitches in a melody tend to be in the center of the tessitura, and if large leaps have a tendency to take the melody to the extremes of the range, then most leaps will tend to be followed by a change of direction.
We can now propose two competing theories regarding the melodic contours following a leap. The post-skip reversal theory proposes that melodic leaps tend to be followed by a change of direction. The regression-to-the-mean theory proposes that there is a tendency for melodies to stay in the central portion of the tessitura.
Notice that the regression-to-the-mean theory implies that melodic behavior following a leap depends on whether the consequent pitch is high, medium, or low in tessitura. By contrast, the post-skip reveral theory makes the same prediction no matter where in the tessitura the leap occurs.
Let’s distinguish four possible types of melodic leaps. A median-departing leap is a leap that begins above (or below) the median pitch, and moves even further away from the median. A median-crossing leap is a leap that begins above (or below) the median pitch, and crosses over the median in the other direction. A median-landing leap is a leap that land on the central or median pitch itself. Finally, a median-approaching leap is a leap that moves closer to the median, but both pitches are above (or below) the median. These different types of leaps are schematically illustrated below. The illustration pertains only to ascending leaps, but the concepts apply to descending leaps as well.
Notice that post-skip reversal and regression-to-the-mean make different predictions. Post-skip reversal predicts a change in direction for intervals in all four conditions. However, regression-to-the-mean makes different predictions depending on the condition. In the case of median-departing and median-crossing leaps, regression-to-the-mean predicts a change of direction (back toward the median pitch). However, in the case of median-landing leaps, regression-to-the-mean predicts that the subsequent direction doesn’t matter: a melodic contour is just as likely to continue in the same direction or reverse direction. Finally, in the case of median-approaching leaps, regression-to-the-mean predicts that there will be a tendency for the melodic contour to continue in the same direction—exactly contrary to the post-skip reversal theory.
So what do actual melodies do? The following graph represents data from over 3,000 melodic leaps. As can be seen, for median-departing leaps there is a clear tendency for melodies to continue with a change of direction. Similarly, for median-crossing leaps, leaps tend to be followed by a change of direction. These results are consistent with both post-skip reversal and regression-to-the-mean. In the case of median-landing leaps, reversing direction or continuing in the same direction are equally likely—as predicted by regression-to-the-mean. Finally, in the case of median approaching leaps, it is more likely to melodies to continue in the same direction—as predicted by regression-to-the-mean. After a study by von Hippel and Huron (2000), further research published by von Hippel (2000a, 2000b) confirmed that melodic organization is consistent with regression-to-the-mean, and is not consistent with post-skip reversal.
References:
Paul von Hippel, & David Huron (2000). Why do skips precede reversals? The effect of tessitura on melodic structure. Music Perception, Vol. 18, No. 1, pp. 59-85.
Paul von Hippel (2000). Redefining pitch proximity: Tessitura and mobility as constraints on melodic intervals. Music Perception, Vol. 17, No. 3, pp. 315-327.
Paul von Hippel (2000). Questioning a melodic archetype: Do listeners use gap-fill to classify melodies? Music Perception, Vol. 18, No. 2, pp. 139-153.