Can a musician’s superior maths ability be explained by superior timing ability?

Title of paper under discussion

Musicians outperform nonmusicians in magnitude estimation: evidence of a common processing mechanism for time, space and numbers


Christian Agrillo and Laura Piffer


The Quarterly Journal of Experimental Psychology, vol 65, issue 12, pp 2321-2332

Link to paper (free access)


Could it be that the human brain uses the same magnitude system to measure time, space and numbers? Two researchers from Padua, Italy reasoned that if this were true then musicians, who are known to excel at temporal discrimination tasks (“which of two notes is the longer?”), would be able to transfer this skill to spatial discrimination tasks (“which of two lines is the longer?”) and numerical discrimination tasks (“which of two group of dots is the more numerous?”). Agrillo and Piffer discovered that musicians did indeed perform better than non-musicians not only at temporal discrimination but also at spatial and numerical discrimination – though the latter only if the numbers were large. Perhaps then, the authors suggest, a common magnitude system explains the often reported crossover between musical and mathematical ability.


British neuroscientist Vincent Walsh published a paper back in 2003 suggesting that “time, space, and numbers are represented in the brain by a common magnitude system”, a theory called ATOM (“A Theory Of Magnitude”). It has been suggested that such a system “would be mainly located in the parietal cortex, especially in the inferior parietal cortex, because of the need to integrate temporal, spatial and numerical information…”

Above: inferior parietal cortex, perhaps the seat of a ‘common magnitude system

Some studies have already hinted that Walsh is correct – that there is a common mechanism to calculate magnitude, whether it be magnitude of time, space or numbers. Typically such studies have looked at what happens when a person is presented with objects defined by two different species of magnitude (say, duration and size); if one magnitude (e.g. size) confuses the person’s judgment of another magnitude (e.g. duration), it would seem reasonable to suspect that both magnitudes are being calculated by a common system, and hence the two judgments are confusing one another.

For instance a musical note of a certain duration might be simultaneously presented visually in a certain font size. In this case, a note in a larger font size might be judged as lasting longer than a note in a smaller font size, even though in reality they are of identical duration.

Several experimenters, quoted by the authors of this paper in their introduction, have investigated this ‘magnitude confusion’ paradigm and found it to be real; but others have found no such confusion.

Also referenced are brain imaging experiments – the idea being that if there is a common system of judging magnitude then it seems reasonable to expect to find a common area of brain activity underlying these judgments. Again, some experiments point to such a common brain area, whereas others find separate brain areas for processing the different magnitudes (of space, time and numbers).

Walsh himself acknowledges that “not all aspects of time, space and numbers have a common origin, and therefore some tasks are expected to be entirely solved using temporal [time]/spatial/numerical systems outside of the common mechanism”. In other words, it is to be expected that some experiments will find a common system of magnitude and others won’t.

Our authors, Agrillo and Piffer, chose not to use the ‘confusion paradigm’. Instead they “adopted a new approach”, choosing instead to investigate people – musicians – who have expertise in judging a certain type of magnitude – temporal magnitude [note duration] – and seeing if they thereby ‘automatically’ had expertise in judging two other types of magnitude: spatial and numerical.


27 volunteers – 13 musicians and 14 non-musicians – took part in the study. Three different tasks were presented to each: temporal [duration] discrimination, spatial discrimination, and numerical discrimination. In short, each task consisted of two stimuli, one after the other, and the participant had to say which was longer in duration (for the temporal tasks), longer in length (for the spatial tasks) , or more in number (numerical tasks). In more detail:

Temporal discrimination task

84 separate trials were taken in turn by the participant, each trial consisting of a consecutively presented pair of tones of different durations. In each trial the participant heard the first tone, waited half a second, then heard the second tone; s/he was then asked to judge if the second tone was longer or shorter than the first, and all the answers were recorded and marked for accuracy. Each tone had a duration of somewhere between 0.25 and 2 seconds.

Spatial discrimination task

80 separate trials were taken in turn by the participant, each trial consisting of a consecutively presented pair of black lines of different lengths viewed on a screen. In each trial the participant saw the first line, waited half a second, then saw the second line; s/he was then asked to judge if the second line was longer or shorter than the first, and again all the answers were recorded and marked for accuracy. Each line had a length of somewhere between 1.5 and 3 centimetres.

Numerical discrimination task

80 separate trials were taken in turn by the participant, each trial consisting of a consecutively presented pair of ‘arrays of black dots’ of different ‘dot number’ viewed on a screen. In each trial the participant saw the first array, waited half a second, then saw the second array; s/he was then asked to judge if the second array contained more or fewer dots than the first, and again all the answers were recorded and marked for accuracy. Each array had a dot number of somewhere between 1 and 24.

Here is the authors’ figurative summary of the three types of task:

In each task (whether it were temporal, spatial or numerical) the experimenters were keen to learn how good the participants were at discriminating between the magnitudes of the first and second stimuli in each pair. They made sure that a wide variety of magnitude ratios was presented. So, for example, one temporal task might consist of a 0.5 second tone followed by a 1 second tone, a ratio of 0.5. Another might consist of a 0.45 second tone followed by a 0.5 second tone, a ratio of 0.9. The expectation was that the higher the pair’s magnitude ratio (i.e. the more similar the two stimuli were in magnitude), the more difficult it would be to discriminate between them.

[One more thing about numerical discrimination before we look at the results: it’s important to note that humans seem to use two different mechanisms for numerically discriminating between pairs of ‘dot arrays’ depending on how many dots in those arrays: pairs of ‘many-dotted’ arrays are discriminated using an ‘analogue mechanism’, and pairs of ‘few-dotted’ arrays are discriminated using a ‘subitizing mechanism’. The ‘analogue mechanism’ can deal with any number of dots in its arrays – but the smaller the difference in dot number between two arrays, the less accurately it can discriminate between them. In contrast the ‘subitizing mechanism’, though only able to deal with arrays of a few dots, is as accurate at discriminating a 1 dot difference as it is a 4 dot difference. In other words, the accuracy of the ‘analogue mechanism’ is ratio-dependent, but the accuracy of the ‘subitizing mechanism’ is ratio-independent.]


Temporal discrimination task

As expected, musicians performed better than non-musicians at judging which of two tones had the longer duration. In the results graph below, musicians are in blue and non-musicians are in red. Note also that as the ratio rose (i.e. as the two durations became closer in duration) the accuracy of answers went down, in both groups.

Above: results of the temporal discrimination task

Spatial discrimination task

Just as in the temporal discrimination task, musicians outperformed non-musicians when judging which of two lines was longer. And again both groups performed less accurately the smaller the difference (higher the ratio) between the two lengths.

Above: results of the spatial discrimination task

Numerical discrimination task

For arrays with large numbers of dots (6-24), musicians yet again outperformed non-musicians. But for arrays with small numbers of dots (1-4) the two groups performed pretty similarly – the statistical analysis decreed that with these smaller-numbered arrays there was no significant difference between the discriminatory accuracy of musicians and non-musicians. A look at the graph below shows one more intriguing thing: in the small number range, musicians’ discrimination accuracy worsens the higher the magnitude ratio (i.e. the more similar in dot number are the pair of arrays); but non-musicians’ discrimination accuracy stays level…


The authors had set out to discover “whether time, space, and numbers are processed in a common system for measuring magnitude.” They compared experts [musicians] in one of these domains [time] with a control group [non-musicians]. As they explain in the paper: “The idea was simple: If one general system is used for processing time, space, and numbers, experts should exhibit a better performance in tasks that are not directly related to their domain of expertise.”

Having shown that musicians, as expected, were better at discriminating time magnitude, they also showed they were better at discriminating spatial magnitude – and numbers magnitude, so long as those numbers were ‘large’ (from 6 – 24).

Referring to these results, the authors suggest that if, as it seems, musicians can develop a better sense of magnitude of space and (large) numbers by developing expertise in time magnitude then perhaps this reveals that magnitudes in time, space and (large) numbers are indeed all being processed by a common system in the human brain.

They further speculate that apprehending the magnitude of small numbers – called ‘subitizing’, a process that is normally ratio-independent when discriminating between pairs of arrays – does NOT use this common system. But that musicians, whose “time, space and (large) number” common system is so expert, instead use that “analogue” system to discriminate between small numbers, rather than the “subitizing” mechanism. This leads to ratio-dependent (as opposed to ratio-independent) accuracy, as exhibited by the downward slope of the (blue trace) musicians compared with the level gradient of the (red trace) non-musicians in the “small number range” graph above.

Finally, Agrillo and Piffer conclude that “this study may have pedagogical implications”, claiming that their data suggest “that musical training per se can improve mathematics achievement by increasing the precision of our analogue magnitude system”. Although “the beneficial effects of musical training on mathematical proficiency” are normally interpreted as “the results of the superior working memory of musicians, or a general cortical reorganisation of cognitive skills”, the authors propose that “[in] the light of the results presented here, the higher mathematics achievement of musicians sometimes reported in the literature may also be explained as a consequence of having a better shared mechanism processing temporal, spatial, and numerical information.”



from Le nozze di Figaro by W. A. Mozart

Karl Bohm/Vienna Philharmonic Orchestra

Hermann Prey … Figaro

Mirella Freni … Susanna

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