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.
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Isn't a possible alternate explanation for all of these that the spatial linguistic concepts simply augment the decision making? For instance imagine if higher pitches were termed "yellower" and lower pitches were termed "redder." I can imagine that people would respond faster if the higher-pitch key is yellow and the lower pitch key is red, even though nobody would argue that the pitches are in fact represented as colors in the brain. Perhaps this experiment should be repeated with animals, or with linguistically impaired humans, to eliminate this extra dimension.
I am a musician (cello) and a molecular biology student. Perhaps as a follow-up experiment, they should test the response times for moving tones. For example, are musicians faster than non-musicians to classify a rising tone or falling tone as left or right or up or down? Especially when playing in chamber ensembles, musicians are, at least in my experience, encouraged to think about the shape of the music that they're playing. For example, I recently played in a chamber orchestra that was peforming Bach's "Double Violin Concerto in D Minor", and my part involved a lot of rapidly rising and falling lines to provide movement behind the soloists. It might also be interesting to investigate whether musicians peform differently than non-musicians in assortative tasks and finding the relationships between things. As musicians, we are often told to think of and feel the dynamics between our instrument and others in the ensemble. Therefore we must not only execute our own part, but also be aware of and support others' parts. Furthermore, it would also be interesting to see how musicians vs. non-musicians process the concept of color and texture in response to musical cues. With cello, I can make the texture of the music smooth and soothing or rough and jagged depending on the line and the appropriate bowings. And when playing a D melodic minor scale over a low D drone, I find myself thinking of a stormy grey texture, whereas an F major scale makes me think of royal yellow. Also, it might be interesting to see if there are differences among musicians, e.g., do vocalists process harmony differently than percussionists or string instrument musicians?
That kind of result could also be explained by visual representationo of the keys. Up is automatically higher, while left or right doesn't mean higher or lower.
It's worth noting that in musical notation, higher pitches are positioned higher on the staff than lower pitches. Also, on a piano, keys on the right produce higher pitches than keys on the left. It's not surprising then that trained musicians associate higher pitches with both up and right, independent of language.
Do you know if these uses of "higher" and "lower" are historically independent? It seems to me that longer and shorter or lighter and heavier would be more natural, since although in the olden days people wouldn't have known that sound waves have a literal length, still it would have been obvious that longer and heavier strings have lower pitches than shorter and lighter ones, you lower the note of a flute by covering holes increasing its effective length) and so on.
why should the map be (right-up left-down) instead of (right-down left-up)? I'd have assumed that this was completely arbitrary - after all, it's the difference between an right handed and left handed coordinate system. My guess would have then been that (say) English speakers and Arabic speakers respond oppositely, but you say Chinese speakers respond like we do. Weird.
I have to admit I didn't read your whole post, but I wanted to respond to the slight difference the authors found for the right-left key mapping for higher vs lower pitches.
I don't remember the exact paper for this, but it was written by Diana Deutsch. In any case, she presented high and low pitches that moved sort of stepwise, with the lower pitches coming up and the higher pitches going down. Both stimuli were presented to both ears simultaneously. Well, people totally segregated the sounds. Most right-handed people put all of the high pitches in their right ears and low pitches in their left ears. Left-handed people sometimes did the opposite. So, I'm frankly surprised that the authors' difference wasn't even bigger. I'm also surprised the authors didn't know about this finding or remark on it.
Well, more to do with pitches and Chinese (although it's Mandarin in this case, not Cantonese):
http://www.sciencedaily.com/releases/2006/12/061212213436.htm
I presume that everyone here knows about:
(1) asymmetry between human left-ear and right-ear sound processing, one brain hemisphere several decibels better for music, other several decibels better for speech.
(2) difference in brain hemispheres used by music students in first (roughly) half year of intense study; then hemispheriic transfer of control; then opposite himisphere. Crudely, reading and playing music is initially a logical, sequential, analytic process. Then it becomes an intuitive gestalt process.
(3) 2-D math for music, first Music Theory in the journal Science in over a century:
http://www.sciencemag.org/cgi/content/summary/313/5783/49
Science 7 July 2006:
Vol. 313. no. 5783, pp. 49 - 50
DOI: 10.1126/science.1129300
Perspectives
MATHEMATICS:
Enhanced: Exploring Musical Space
Julian Hook
New mathematical approaches can elucidate abstract musical spaces and help our understanding of harmonic processes at work in musical compositions.
The author is at the Jacobs School of Music, Indiana University, Bloomington, IN 47405, USA. E-mail: juhook@indiana.edu
REPORTS
The Geometry of Musical Chords
Dmitri Tymoczko (7 July 2006)
Science 313 (5783), 72. [DOI: 10.1126/science.1126287]
| Abstract � | Full Text � | PDF � | Supporting Online Material �
See also
Mathematics and Music
http://www.math.niu.edu/~rusin/uses-math/music/
including page on Psychology of Music
http://www.math.niu.edu/~rusin/uses-math/music/psych
haer the podcast
Science/AAAS | Science Magazine: About the Journal: E-Mail Alerts ...
7 July 2006. Science Podcast: ... the Math of Music, and More [Listen to MP3] ... ...
www.sciencemag.org/about/podcast.dtl
The below is hereby resubmitted. Was it filtered out, or do you dislike the references, or what?
===================
JVP blogs on
Music In the Mind: Is Pitch Represented On a Mental
Line?
thread of
Mixing Memory science blog
http://scienceblogs.com/mixingmemory/2006/12/music_in_the_mind_is_pitch…
Previewing your Comment
I presume that everyone here knows about:
(1) asymmetry between human left-ear and right-ear
sound processing, one brain hemisphere several
decibels better for music, other several decibels
better for speech.
(2) difference in brain hemispheres used by music
students in first (roughly) half year of intense
study; then hemispheriic transfer of control; then
opposite himisphere. Crudely, reading and playing
music is initially a logical, sequential, analytic
process. Then it becomes an intuitive gestalt process.
(3) 2-D math for music, first Music Theory in the
journal Science in over a century:
http://www.sciencemag.org/cgi/content/summary/313/5783/49
Science 7 July 2006:
Vol. 313. no. 5783, pp. 49 - 50
DOI: 10.1126/science.1129300
Perspectives
MATHEMATICS:
Enhanced: Exploring Musical Space
Julian Hook
New mathematical approaches can elucidate abstract
musical spaces and help our understanding of harmonic
processes at work in musical compositions.
The author is at the Jacobs School of Music, Indiana
University, Bloomington, IN 47405, USA. E-mail:
juhook@indiana.edu
REPORTS
The Geometry of Musical Chords
Dmitri Tymoczko (7 July 2006)
Science 313 (5783), 72. [DOI: 10.1126/science.1126287]
| Abstract � | Full Text � | PDF
� | Supporting Online Material �
See also
Mathematics and Music
http://www.math.niu.edu/~rusin/uses-math/music/
including page on Psychology of Music
http://www.math.niu.edu/~rusin/uses-math/music/psych
haer the podcast
Science/AAAS | Science Magazine: About the Journal:
E-Mail Alerts ...
7 July 2006. Science Podcast: ... the Math of Music,
and More [Listen to MP3] ... ...
www.sciencemag.org/about/podcast.dtl
Posted by: Jonathan Vos Post | January 1, 2007 01:11
AM
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