A friend of mine and I were having a conversation today, and one of us (I don’t remember who) brought up a poster that we’d seen at a conference a few years ago. Later, I wondered what had become of the work in the poster (it’s about negative numbers being represented on a mental number line). Apparently, nothing. But in the process of looking for more information, I came across another paper that might be even more interesting.
The poster was inspired by work showing that we may represent positive numbers on a “mental number line.” In one experiment testing the mental number line hypothesis, participants who were asked to indicate whether a number was positive or negative did so faster for large numbers when the response key was on the right side of the keyboard, and small numbers when the response key was on the left side of the keyboard1. Since the concept of a mental number line presumably involves representing small numbers on the left side of the mental number line, and large numbers on the right side, this result is consistent with the existence of such a line.
The paper I discovered while looking for the poster isn’t about math, though. Instead, it’s about music. The work, by Rusconi et al.2, was designed to show a connection between how we represent music with how we represent number. They note that across many languages (they list Chinese, English, French, German, Italian, Polish and Spanish), the words use to denote differences in pitch are spatial (e.g., high pitches and low pitches). The idea, then, is that like number, pitch may be represented on a mental line, with (when the terms for pitch are vertical terms) high pitches represented higher on the line than low pitches.
In their first experiment, Rusconi et al. first presented Chinese participants (most of whom spoke both Cantonese and English) who had no musical experience with a reference pitch (C4, for you music geeks), followed by a target pitch that was either higher or lower than the reference pitch (E3, F3#, G3#, A3#, D4, E4, F4#, G4#, again, for the music geeks). Participants were asked to indicate whether the pitch of the target was higher or lower than that of the target pitch. In one condition, the response keys, the spacebar and the 6 key, were above and below each other, and in a second condition, the response keys, Q and P, were across from each other on the same keyboard row. In both conditions, half of the participants were told to to higher pitches with one key (e.g., 6 or P) and lower pitches with the other (spacebar and Q), and the other half were told to respond in the opposite way. The prediction, then, is that when the response keys are on a vertical axis (spacebar and 6), people will respond to higher pitches faster when the response key is the upper one (6), and lower pitches when the response key is the lower one (spacebar). There should be no difference between the two response key configurations when the response keys are on the same horizontal axis (Q and P).
First, they found that bigger differences between the target and reference pitches resulted in faster response times than smaller pitches. No surprise there. They also found that participants were faster to respond to higher pitches with the P key (on the right), and lower pitches with the P key, though this difference wasn’t quite significant. I’m not exactly sure what to make of this difference. Consistent with their predictions, though, when the response keys were on the vertical axis, their responses were faster when the upper key (6) was used to respond to higher pitches, and the lower key (spacebar) was used to respond to lower pitches.
In their second experiment, a second group of musical novices (again, Cantonese as first language and English as second language) were presented with different tones (F3#, G3#, A3#, C4, E4, F4#, G4# and A4#) played by either wind instruments (french horn or tenor trombone) or percussion instruments (marimba or vibraphone). Their task was to indicate the instrument family (wind or pecussion). Once again, the response keys were either on a vertical axis (spacebar and 6) or a horizontal one (Q and P), and as in the first experiment, the prediction was that participants would be faster to respond to high pitches with the upper key than the lower, and for low pithces, responses would be faster with the lower key than the upper. This time, they didn’t find a difference between the horizontal response keys (maybe the difference in the first experiment was a fluke?), but as in the first experiment, they did find the predicted difference for the horizontal keys: responses to high pitches was faster when the response key was the upper one (6), and they were faster for low pitches when the response key was the lower one (spacebar), despite the fact that the task didn’t involve making any distinctions between pitches.
Their third experiment was identical to the second, but this time their participants were musicians. Once again, they found that high pitches led to faster response times with the upper key than the lower one, and low pitches resulted in faster response times with the lower key. In fact, the difference between consistent keys (6 for high pitches and spacebar for low keys) and inconsistent keys (spacebar for high pitches and 6 for low pitches) was even greater for the musicians than it was for the novices in the second experiment. Furthermore, the musicians also responded to high pitches faster with the right key than the left, and lo pitches with the left key than right (OK, so maybe it wasn’t a fluke).
It seems, then, that to some extent, we do represent pitch on a mental line, with high pitches at the top and low pitches at the bottom, at least in languages where the terms for pitch are vertical terms. We may also represent pitch on a horizontal axis, too, as evidenced by the differences in response times for the horizontal keys in the first and third experiments. Rusconi et al. argue that the horizontal effect may be due to a “remapping” of the vertical dimension onto the horizontal dimension, which musical training somehow facilitates (the horizontal effect was strongest in the experiment with experts). I think that explanation can be translated as, “We haven’t the slightest idea why we got this result.” Perhaps future work will figure it out, though.
1Dehaene, S., Dupoux, E., & Mehler, J. (1990). Is numerical comparison digital: Analogical and symbolic effects in two-digit number comparison. Journal of Experimental Psychology: Human Perception and Performance, 16, 626-641.
2Rusconi, E., Kwan, B., Giordano, B.L., Umilta, C., & Butterworth, B. (2006). Spatial representation of pitch height: The SMARC effect. Cognition, 99(2), 113-129.