Harmonic Frequencies

A string or open pipe doesn't resonate at just one frequency but at a whole ladder of them: every whole-number multiple of the fundamental. A guitar string tuned to 110 Hz simultaneously supports 220, 330, 440 Hz and beyond, all ringing at once in proportions that define the instrument's timbre. Touch the string lightly at its midpoint and you silence every odd harmonic, leaving the pure 220 Hz second harmonic: the bell-like "harmonic" trick guitarists use.

The integer ladder is also the foundation of musical harmony. The second harmonic is exactly an octave up; the third is an octave plus a fifth; brass players get every note without moving a valve by "overblowing" up the harmonic series, which is why a bugle can play Taps at all. Remarkably, your brain uses the ladder in reverse: play harmonics 2 through 6 of a 100 Hz tone with the fundamental removed, and you still perceive a 100 Hz pitch, the "missing fundamental" that lets tiny phone speakers imply bass they cannot physically produce.