My first intermediate flute was a Gemeinhardt 3SHB. Now Gemeinhardt's take a lot of heat by professional flutists, claiming they have a shrill sounding tone. I did not notice this on mine but of course I was in High School.
I bought my intermediate flute when I started taking private lessons, 10th grade. I would suggest buying an intermediate flute when the your child enters high school and shows an interest in the flute. If your child does not show a serious interest in the instrument, then you shouldn't buy them one.
I bought my intermediate flute in the 10th grade, which means that I had been playing for 5 1/2 years. I would suggest that you buy it no sooner than 3 years. Make sure you need a better flute. At a recent masterclass with James Galway he picked up a beginner flute from the audience and played it just as good as his $20K flutes.
At this level, as long as you buy a flute that has no mechanical problems. You don't need to spend over $1000.
I would never suggest buying a flute from a music store. I do not want to see local music stores get shut down, but when buying an instrument, they usually mark up the price by at least 200%. My Gemeinhardt was listed at $1000 at a local music store; I was able to buy it for approximately $500.
I would test out the flute you want at a local music store and then look online for a company with the same model. Here is why: Since intermediate flutes are machine made they will all sound about the same. I have suggested this a few times on web forms and have gotten the response that this will put local music stores out of business. These local music stores make a lot more money off guitars than flutes, so they will not be losing as much profit as it seems.
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