Let's put this thread back into some perspective and context.
Harvard (est. 1636 - oldest university in US) and Boston Latin (est. 1635 - oldest public school and grammar school in US) were originally, primarily established as a training ground for the ministry and thus an educated populace since books were not readily available or accessible. Wealthy merchants contributed to both of these institutions due to the overall benefits to society at large. Bright, poor boys who qualified based on their talents attended these schools and were supported. For many years, it was no different at many boarding schools in New England, including Exeter, or other universities, including other ivies.
Schools, such as Juilliard, were established for different reasons than the ivies or boarding schools such as Exeter. Juilliard was founded in 1905 because the US lacked a premier music school and many US students were going to Europe. Juilliard (and Vanderbilts) bequeathed a substantial amount of money for the advancement of music in the United States for the school. Juilliard wouldn't have been created IF exclusive boarding schools or other schools supported such training - but they did not.
Remember, this thread is about prodigies. Juilliard attracts prodigies because it's one of the few schools in the country where they can receive appropriate world-class musical training. Not everyone is a prodigy there, but they probably have a disproportionate just based on the fact that Juiliard is a singular type of place for such unique training. You don't go to Exeter or Eton for classical musical training; you go to Juilliard! And IF you're a child prodigy, you tend to go early too.
I'm sure there are various ranges or levels of prodigies like there is with giftedness. But if you're talking the extreme end, this is very small number of the population - perhaps fewer than the number of pg kids and/or those in DYS. It's not like Exeter or other boarding schools are getting flooded by them. Hardly.
Juilliard, for example, wanted to train Marvin Hamlisch to be the next Vladmir Horowitz and was pretty bent on it. They recognized his amazing talent. Hamlisch, however, didn't want to be the next Horowitz but he accepted that he could learn how to play well at Juilliard and then later on apply such training to things he wanted to do -
http://www.pbs.org/wnet/americanmas...rn-between-classical-and-pop-music/2782/Hamlisch was declared a child prodigy at 4. His father was a accordionist and bandleader so there was a genetic component with music. The family also lived within a relatively short distance to Juilliard too so they didn't have to move across the country either. He is one of only thirteen people to have been awarded Emmys, Grammys, Oscars, and a Tony (those four together are known as an EGOT). Extreme talent here.
FYI - there's a children's book by him, Marvin Makes Music. NOTE: Marvin never wanted to practice either.
