## Clarinet

A clarinet is an example of a cylindrical bore instrument closed at one end. Hence, the normal resonant modes must have a pressure maximum at the closed end (the mouthpiece) and a pressure minimum near the first open key (or the bell). These conditions result in the presence of only odd harmonics in the sound. This contrasts to the saxophone or oboe, which have a conical bore and hence include the even harmonics.

A snapshot of the sound of a clarinet (playing Bb) is shown below:

and the absence of the even harmonics in the spectrum is clearly evident.

The absence is even harmonics is (part of) what is responsible for the "warm" or "dark" sound of a clarinet compared to the "bright" sound of a saxophone.

"The conical bore in musical acoustics," by R. D. Ayers, L. J. Eliason, and D. Mahgerefteh, American Journal of Physics, Vol 53, No. 6, pgs 528-537, (1985).

Here is a WAV file with a recorded clarinet sound: Clarinet.

You can create your own simulated clarinet sound s(t) as follows:

with w1 = frequency of fundamental (in Hz) times 2 pi, the simulated clarinet waveform as a function of time, t (in seconds) is:

s(t) = sin(w1t) + 0.75*sin(3*w1t) + 0.5*sin(5*w1t) + 0.14*sin(7*w1t) + 0.5*sin(9*w1t) + 0.12*sin(11*w1t) + 0.17*sin(13*w1t)

and then multiply s(t) by a constant to change the amplitude. The waveform for the simulated clarinet will look a little different than that of the recorded clarinet since no phase shifts are included (e.g. only sine functions are used, and no cosine functions), but it sounds remarkably similar. Note that the cosine functions are the same as sine functions shifted in time and for sustained tones, these "phase shifts" do not change the sound.
Check it out: Simulated Clarinet WAV file (fundamental = 235.5 Hz).

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