Involvement in all things musical like playing instruments, composition, technical skill, ear training, or music theory, has shown to greatly improve cognitive and emotional development for children (and adults).
Music boosts creativity, language skills and language acquisition, logic, reasoning and abstract thinking.
Music and playing an instrument (or singing) builds resiliency. Yes! Music can actually help trauma-proof your child by giving him/her another outlet to process overwhelming experiences, and express emotions that are sometimes outside of your child's comfort zone.
Music boosts social interaction and adjustment as many of the processing centers for music are in the limbic (emotional) region of the brain that is responsible for functions of attachment (bonding) and interaction with others. It also helps improve emotional processing and appropriate emotional responsivity to overwhelming events.
Playing an instrument involves touch, then hearing and vision, motor control and planning, emotional attunement, pattern recognition, abstract symbol interpretation, personal responsibility and discipline, activates language processing centers in the brain, takes decisional balancing, requires concentration and active communication between hemispheres, and so on and so forth. All of these functions activate drastically different brain systems and allow for growth and potentiation.
This is why some Alzheimer's patients can perform and remember musical passages long after they have forgotten most other things.
There are thousands of cases of music bringing back the memories of patients affected by dissociative disorders or amnesia.
It's been proven that children that participate in musical instruction have an easier time learning non-native languages later in life, as well as a better memory recall, and long-term skill retention.
Consider Bach, one of the most prolific composers in history: Mathematicians look to his works for examples of incredibly logical and mathematical structures. Bach put whole equations into his music, equations that we still find fascinating today.