When, at age 12 or so, I first became aware of overtones on the piano, I was so confused by them that I played always with the soft pedal, effectively blocking them.
Many years later my first truly great teacher, Hans Neumann, insisted that I play always with the piano lid up so as to become increasingly involved with overtones. (Of course, the overtones on his piano in no way matched those of mine, thus they were incredibly confusing, though admittedly still overtones.)
Yet again many years later I had a student who did the same thing, and was able to address the issue in terms of welcoming the difference between "mine" and "hers."
Now I find myself describing to people that this is exactly what they have done, but to their hearing. No matter what instrument you play you probably have a tendency to block the complicating factors of its sound so as to match the overtone-poor quality of most commercial recordings.
Allowing the overtones into your sound, into your awareness of sound is empowering beyond belief; for along with it goes increased range of expression, of dynamics, even of speed.
Music Inside and Out. Check it out.
