October 6, 2006
Resonance

Any sort of vibration can live on a guitar string, or a kalimba tine, or in the column of air in a flute, or in the resonant box of a kalimba--for a short time. Most frequencies of vibration will not live for very long. It is only in the first 0.05 seconds or so, in the so-called "attack" phase of the note, where this disorganized jumble of discordant frequencies exists.

The energy in all those different vibrations quickly gets organized: a vibration on the tine will bounce off the bridge mechanism and come back to the end of the tine, and be reflected off the end of the tine, back and forth. Most vibrations tend to destroy themselves when they bounce back and forth, but at certain frequencies the vibrations are self-sustaining. It is like when you are on a swing - you can't just go back and forth at any old rate, but only at the resonant frequency of the swing.

A kalimba box also has resonant frequencies. It turns out that the frequency of a vibration is inversely related to the wavelength of the vibration: if you double the wavelength, you cut the frequency in half, and it is perceived as a lower pitch. Now, the wavelength of a vibration is related to the physical size of the thing that is vibrating. For the fundamental (i.e., the main note, not the overtones) a guitar string ends up being the length of half the wavelength of the vibration. The frequency also depends upon the speed of sound, which will depend upon the density and tension of the string, which is why the different strings have different pitches even if their lengths are all the same.

The resonant frequency of the kalimba box depends upon the size of the kalimba box. The larger ALTO kalimba box is very close to A 440 Hz, while the resonant frequency of the smaller TREBLE kalimba box is very close to B 493 Hz or C 523 Hz. The ratio of the resonant frequencies of the Treble to the Alto is then something like 493/440 = 1.12 or 523/440 = 1.19. What is the ratio of the box sizes?

The Treble box is 7.1" x 5" wide at the front sound hole; the Alto box is 8.1" x 5.5" across at the front sound hole. The wood is about 0.2" thick, so the inside cavities are about 6.7 x 4.6 and 7.7 x 5.1. What are the ratios of the Alto to the Treble sizes? In length, 7.7/6.7 = 1.15 and in width, 5.1/4.6 = 1.11. This length ratio 1.15 is right in between the 1.12 and 1.19 ratios of the resonant frequencies of the box, and the width ratio 1.11 is right near 1.12.

So, our logic is matched by our measurements.