Look carefully and you’ll find musicians at the top of almost any industry. Woody Allen performs weekly with a jazz band. The television broadcaster Paula Zahn (cello) and the NBC chief White House correspondent Chuck Todd (French horn) attended college on music scholarships; NBC’s Andrea Mitchell trained to become a professional violinist. Both Microsoft’s Mr. Allen and the venture capitalist Roger McNamee have rock bands. Larry Page, a co-founder of Google, played saxophone in high school. Steven Spielberg is a clarinetist and son of a pianist. The former World Bank president James D. Wolfensohn has played cello at Carnegie Hall.

“It’s not a coincidence,” says Mr. Greenspan, who gave up jazz clarinet but still dabbles at the baby grand in his living room. “I can tell you as a statistician, the probability that that is mere chance is extremely small.” The cautious former Fed chief adds, “That’s all that you can judge about the facts. The crucial question is: why does that connection exist?”

Paul Allen offers an answer. He says music “reinforces your confidence in the ability to create.” Mr. Allen began playing the violin at age 7 and switched to the guitar as a teenager. Even in the early days of Microsoft, he would pick up his guitar at the end of marathon days of programming. The music was the emotional analog to his day job, with each channeling a different type of creative impulse. In both, he says, “something is pushing you to look beyond what currently exists and express yourself in a new way.”