Harmonic Overtones and Resonance

As is the case with stereo equipment, all musical instruments have a limited range of frequencies they can reproduce. The cutoff point for the low end of an instrument’s frequency range has a high-pass resonance. The cutoff point for the high end of the instrument’s range has a low-pass resonance. In between these two points there will be band-pass resonances as well. Pass means frequencies that are not filtered out of the sound.

A horn is a high-pass filter with 12 dB per octave attenuation below the cutoff frequency, and a vented box is a high-pass filter with 24 dB per octave attenuation below the cutoff frequency. The falloff of low frequencies with a vented box is faster than that of horn, because the vented box has two low-pass resonances rather than one. The first resonance is that of the air in the box and the second is that of the air in the port. The size of the port is defined by both its area and its depth as defined by the thickness of the walls of the box.

The intensity of a resonance is measured in Q or its inverse, damping. A saxophone has high damping, while a trumpet has low damping. This is because a saxophone flares at conical ratio, while most of length of a trumpet is tubular. The longer a horn is the lower the frequency of its low-pass resonance. It is the ratio of the flare, not how long the horn is that determines its damping. The resonant frequency and the damping of a vented box, such as a violin or guitar, is determined by the sizes of the box and of the opening or openings in the box.

The frequency of notes on a string instrument is determined by which string is excited and by varying the length of the area of the string that is allowed to vibrate. The frequency of notes on a horned instrument is based on varying the resonant frequency of the horn by opening and closing holes along its length. This introduces band-pass resonances which filter out most of the frequencies in complex sound waves.

Wood is rigid and brass is springy. When a brass horn is subjected to sound waves it stretches outwards and inwards in response to the acoustic energy. This causes its resonant frequencies to fluctuate, whereas with woodwinds this modulation does not occur at an audible level.