Bilbo (1340 points)
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Hi, Hope that this doesen't end up being too technical but here goes.
Not sure what the judge was getting at but harmonics are one of the oldest and most common "special effects" in music next to embellishments like trills. The harmonic is used correctly when the composer wants you to play one. This is generally indicated by a small circle above the desired pitch. However, the music doesen't always tell you what fingering to use for that pitch and to keep it simple, I generally use the one that produces the most pleasing sound.
But to go on...
Harmonics are a natural phenomenon in music. They are used on instruments of the brass family extensively and on other woodwinds such as the flute to obtain our full chromatic range in a decent enough tone quality.
Here's a harmonic (Aka overtone series) on low C:
Finger a low C (flutists often call it C1) The fundamental is the low C but if you adjust your airstream a bit you can play the Octave -C2, or the G above -G2, or the next Octave C -C3, or on up to E3, G3, Bb3 or even the very high C4 and above if you are.... special. This overtone series is based on the mathematics of our aural wave forms and theoretically (at least) can be created above any fingering.
So as an example of where the confusion can come in, if you have a small circle above a high E3 natural you can finger a low C1, or a low E1 or even a low A1 and still sound the high E3 but when you do this, note that each fingering produces a slightly different sound to the harmonic although you are still playing the high E3 with them.
One final point, if the judge is telling you to use a harmonic at your own descretion to "color" the note and there's no indication to do so in the score, I generally wouldn't. He was just probably trying to entertain himself with his vast amounts of knowledge.
~Bilbo
N.E. Ohio
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