This question is inspired by a set of questions at the end of Timothy Gowers' book:
The possible relation between mathematics and music is an old open question. In the ancient Greece some philosophers like Pythagoras and Plato strongly believed that there is a mysterious relation between these two completely different subjects. Their arguments were mainly based on the abstract nature of mathematics amongst different fields of science and abstract nature of music amongst different fields of art. Also they argued on the possible similarity between "mathematical beauty" and "beauty of music" based on the philosophical similarity between notions of "pattern" and "harmony" in math and music respectively.
In the recent years the new approach to this old question is mainly focused on the possible positive impacts of listening and training music on improving mathematical abilities of children and adults.
Question 1. Are there any scientific math education research which proves a direct relevance between listening/training music and a meaningful improvement in children/adults' mathematical abilities?
Question 2. If the answer of the question 1 is positive and there is such a relation, do different genres of music (e.g. classic, without vocal, pop, rock, etc.) have different impacts? What the statistics shows?
Question 3. Does listening/training music regularly have a notable impact on notable mathematical and scientific discoveries? Precisely, is there any statistical research which determines the frequency of musical interests of great mathematicians and scientists? e.g. Both Cantor and Einstein were good violinists.