The timbre of woodwind - unique sound and shape of each woodwind instrument

 In the string family, each member has a similar look to each others. People who do not have deep knowledge of music would call every string instrument a violin. The vibration of a string follows the harmonic series pattern. In the woodwind family, every instrument's shape and timbre is very unique, the spectrum may or may not follow the harmonic series. In short, to break down the shape of the woodwind instrument, there are three main types of shape: open cylindrical, close-ended cylindrical, conical. 

For cylinder instruments like flute and recorder with both open ends, the spectra follow the harmonic series, ideally. The vibration stops at both ends resulting in a string with both ends fastened. But the vibration would still fade out after leaving the end, in other words, the length of vibration is slightly longer than the pipe itself, that is the topic of end correction. fn=n*(v/2L)

For closing one end in cylindrical instruments like the clarinet, one end is stopped by a reed. The vibration bonce back at the stopped end, right at the reflection surface, the oscillation turns 180degree of the phrase. Based on the fact, just before the reflection, the vibration is at the highest cycle and after the reflection, it is in the highest anti cycle. (The reflection point must be at the amplitude, the peak of the vibration) The cycle is completed by reflection. Thus, the pipe length only vibrates in the period at 0.5, 1.5, 2.5, 3.5, etc, the reflection completes the cycle of, at the same time, the spec f, 3f, 5f, etc. Only the odd number of the harmonic series. The frequency calculation is fn=(2n-1)*(v/4L) At the same time, the length of the instrument is double, the reason why the clarinet can play lower range with the similar length of oboe and flute.

Doubling clarinets in octave f:2f, instead of the tones stacking nicely, no harmonic parties line up, they tend to have a certain edgy. Mahler uses it for special timbre, to mimic sailor’s lusty hornpipe in the first movement of his Fourth Symphony, by adding a melody by exposed octave clarinets near the start.

For bassoon, saxophone, and oboe, the shape of the instrument is conical, and the vibration model is inharmonic. The making of the reed is very consumable, making the spectra even harder to predict. Overall, the conical shape makes the closed end-pipe harmonic only odd number harmonic sequence to full integer harmonic series like a flute. fn=n*(v/2L). However, the wider diameter of the open cone end cuts off the high harmonic partials. This model is also applicable on a brass instrument, with lips as the closed-end reed and conical shape of the bore. The formula for critical mode number is N=L/D (Hall, 2002).

v = volocity; N = number of partials; L the length of the pipe; D = diameter.

  Roederer, J. G. (2008) The physics and psychophysics of music : an introduction. 4th ed. New York: Springer.

Hall, D. E. (2002) Musical acoustics. 3rd ed. Pacific Grove, Calif: Brooks/Cole Pub. Co.

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