The answer is partial yes, as most things are the result of an interaction between genes and the environment. Still, a new Harvard study by Jennifer Zuk and colleagues has examined the neurobiological predispositions for musical talent — in infancy.

These are their findings in short taken from the press release:

• Early aspects of brain structure in babies can set the stage for later success with music.
• The authors, who conducted their study at the Gaab Lab at Harvard, found structural networks in infants’ brains linked to some of the differences that were previously only observed in the brains of musically trained adults or much older children.
• Both nature (a predisposition for music) and nurture (musical training) are believed to “establish a neural foundation for musicality,” according to the report. The researchers write that they observed structures in the infant brain that “may serve as a scaffold upon which ongoing musical experience can build.”

But while the study has its merits, there are important limitations:

The researchers relied on a small sample size, with minimal socioeconomic variation. Twenty-five infants from the Greater Boston area were selected for neuroimaging, from a larger existing longitudinal study that was already tracking brain and language development. Follow-up assessments of musical aptitude — tonal and rhythmic abilities — were conducted when most of the children reached kindergarten.

Abstract of the study:

Musical training has long been viewed as a model for experience-dependent brain plasticity. Reports of musical training-induced brain plasticity are largely based on cross-sectional studies comparing musicians to non-musicians, which cannot address whether musical training itself is sufficient to induce these neurobiological changes or whether pre-existing neuroarchitecture before training predisposes children to succeed in music. Here, in a longitudinal investigation of children from infancy to school age (n = 25), we find brain structure in infancy that predicts subsequent music aptitude skills at school-age. Building on prior evidence implicating white matter organization of the corticospinal tract as a neural predisposition for musical training in adults, here we find that structural organization of the right corticospinal tract in infancy is associated with school-age tonal and rhythmic musical aptitude skills. Moreover, within the corpus callosum, an inter-hemispheric white matter pathway traditionally linked with musical training, we find that structural organization of this pathway in infancy is associated with subsequent tonal music aptitude. Our findings suggest predispositions prior to the onset of musical training from as early as infancy may serve as a scaffold upon which ongoing musical experience can build.