. . .
A very good thing to pick up and give the young student is one of those small quartz tuners. For a Trumpet player, make sure that the Tuner is a Chromatic type and it is easier to use one that automatically changes pitches rather than one that is manually set to only one note at a time. This kind of visual feedback, if used for ten minutes a day for awhile in their regular practice regimen can really speed up the process that a Trumpet player must develop, which is the ability to hear the next tempered note in a scale in their head *before* attempting to play said note. . . .
I'm taking singing lessons (not an easy thing to start at age 67). I was bitching about my difficulty in staying "on pitch", and my teacher suggested a tuner. They're priced around $20, now.
I got one, and started using it. Came back to my lesson next week really discouraged -- the needle was wandering all over the scale, and I couldn't control it.
. . . "Yes", she said, "I cried for a while after I got mine."
It's a lot easier if you start in high school.<g>
PS -- I know this is off-topic . . .
PPS -- the "beats" that a piano tuner hears -- or that anyone can hear, if two instruments are playing the same note _slightly_ out of tune -- are a _linear_ effect.
The Fourier analysis of the "two trumpets" tone shows two sine waves at slightly different frequencies.
If you graph the sound pressure level, you'll see a fast wave (at the played frequency) that _appears to be_ modulated by a slow wave (at the "difference frequency"). And your ear picks up that _apparent_ modulation, even though the Fourier analysis says it doesn't exist.
If you were to record the two trumpets, and play the signal back through a loudspeaker, you could put your finger on the speaker cone and _feel_ the signal go from "loud" to "nothing".
There's no "non-linearity" involved, but we _hear_ differently than we _analyze_.