It might seem silly that your handedness has anything to do with your musicality, but back in 1980 an interesting study was done by psychologist Diana Deutsch that showed a significant correlation between the two. The study consisted of two experiments, both of which resulting in interesting findings.
Experiment 1
"The following task was employed. A test tone was presented followed by a sequence of six interpolated tones, and then by a second test tone. The test tones were either identical in pitch or they differed by a semitone, and subjects were instructed to judge whether they were the same or different. (...) The subjects were 76 right-handed and 53 left-handed university undergraduates. Handedness was assessed by the short form of the Edinburgh Handedness Inventory (Oldfield, 1971).".
Interestingly enough, the right-handed group produced an average error rate of 38.1%, while the left-handed group produced one of 32.5%. However, it was also found that the variance in error rate was a lot higher in the left-handed group than in the right-handed group, indicating that a more specific group of left-handed people was causing the lower error rate. It was hypothesized that a difference might emerge between people who were strongly left-handed and those with mixed hand preference, "as individuals in this latter group would be expected to have more bilateral representation of function (Gillies, MacSweeney, & Zangwill, 1960; Hécaen & Sauget, 1971; Zangwill, 1960).".
To clarify: people with a mixed hand preference have a dominant hand, but they prefer to do certain tasks with their non-dominant hand. It is estimated that around 1% of the global population has this type of handedness.
Based on the same Handedness Inventory as before, the two groups were divided even further into four groups: pure left-handers, mixed left-handers, pure right-handers and mixed right-handers. The same experiment was now repeated, giving the following results:
Given these error rates, the performance level of mixed left-handers was found to be significantly higher than that of any of the other groups! "It was concluded that the type of brain organization characteristic of mixed left-handers is associated with enhanced levels of performance on this task.".
Experiment 2
"This experiment was undertaken to test the generality of the findings obtained in Experiment 1. A different pitch recognition task was used. Subjects were presented with a standard five-tone sequence, and then, after a pause, with a probe tone. They were required to judge whether or not a tone of the same pitch as the probe had been included in the sequence. On half of the sequences such a tone was included, and on the other half it was not. The included tones occurred an equal number of times at each of the first four serial positions of the sequence, and the pitches of these tones were strictly counterbalanced across serial position. (...) This experiment employed 74 right-handers and 30 left-handers.".
Like the previous experiment, the participants were divided into four groups (the same ones as before). The results were as follows:
Again, the performance of mixed left-handers was found to be significantly higher than that of any of the other groups, reinforcing the findings of experiment 1.
Conclusion
In short, being left handed does not grant you any musical superpowers compared to being right handed. However, being mixed left-handed (having your left as your dominant hand but doing certain tasks with your right) does seem to significantly benefit your pitch memory, therefore making you somewhat more musically gifted. This has to do with the way our brains are organized. It is hypothesized that the brains of mixed left-handers process and storage pitches in both hemispheres, making the chance of correct judgement more likely. On the contrary, people with other types of handedness only seem to use one side of their brain for pitch memory, making judgement less reliable.
At the end of the day, there is much more to being a musician than having a great pitch memory. If you're not a mixed left-hander, don't worry about it. You probably have other advantages that make you stand out. Besides, hard work beats talent any day!
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Source material: Handedness and Memory for Tonal Pitch [DIANA DEUTSCH] - https://deutsch.ucsd.edu/pdf/Ch-Handedness-1980.pdf