The clarinet and the oboe are almost the same length and are both played with a reed, yet the clarinet has much lower notes and a different timbre. In these experiments with TubeLabLight we find out why: the difference lies in the shape of the bore, cylindrical in the clarinet and conical in the oboe. It is best to follow the path on cylindrical pipes first. Remember that TubeLabLight does not simulate holes: the resonances it computes correspond to the lowest note of the instrument, with all holes closed.

The clarinet: a closed cylindrical pipe

1. Instructions
   • Open TubeLabLight with the Clarinet project (B♭ clarinet): a cylindrical bore 575 mm long with a radius of 7.5 mm, which widens only in the last 10 cm (the bell). The Mouthpiece is closed by the reed.
   • Read in the table of modes the frequencies and the ×f₁ column (ratio to the fundamental).
   • Click the dot of the first mode and look at the shape of the pressure along the tube. Then listen with ▶ Listen.
2. Remarks
   • The fundamental is about 161 Hz and the following modes, at least the first ones, are in the ratios 3 and 5 to the fundamental: the series of odd multiples of the pipe closed at one end. The reed, resting on the mouthpiece, in fact closes the bore at the mouthpiece end.
   • The pressure is largest at the mouthpiece, against the reed, and vanishes near the bell.
   • The second mode is at about three times the fundamental: this is why, when the clarinettist "moves to the upper register", the note rises by an octave plus a fifth (a twelfth), and not by an octave as in the other wind instruments.
3. Related topics: see Tuning and resonances of the clarinet, Cylindrical bore, Normal modes of a cylindrical pipe
4. To find out more: see Timbre and registers of the clarinet, Acoustics of model wind instruments

What is the bell for?

1. Instructions
   • With the Clarinet project loaded, note the frequencies of the first five modes.
   • Press Remove the bell: the radius of the last four points (those with x from 505 to 575 mm) becomes 7.5 mm, as in the rest of the bore. The bell disappears and a perfect cylinder remains. Compare the new frequencies with the ones you noted.
   • Restore the bell with Put the bell back (or by reloading the project).
2. Remarks
   • Without the bell the modes are almost exactly 1, 3, 5, 7, 9 times the fundamental (148 Hz). With the bell the high modes are lower than the odd multiples: the fourth mode, for example, is at 6.86 times the fundamental instead of 7.
   • Low-frequency waves are reflected already at the start of the bell: for them the tube is a little shorter, and indeed with the bell the fundamental is higher (161 Hz instead of 148 Hz). High-frequency waves, on the other hand, go deeper into the bell before being reflected: for them the tube is longer and their resonances drop. They are also radiated better into the surroundings. The bell makes the transition between the bore and the outside air more gradual (an impedance matching) and changes both the tuning of the high modes and the timbre.
3. Related topics: see Mixed pipe, Impedance matching, Acoustic impedance
4. To find out more: see Reflection in a horn (path with the 2D Waves app)

The same tube, played like a flute

1. Instructions
   • Reload the Clarinet project and change the Mouthpiece to open (as in the flute): now the mouthpiece is open, as in the flute. The tube does not change.
2. Remarks
   • The fundamental rises to about 322 Hz, twice the previous value, and the modes are now in the ratios 1, 2, 3, ... (until the bell makes itself felt): the complete harmonic series of the open-open pipe.
   • What decides the notes of a tube is not only its shape but also how the mouthpiece is made: the reed closes it, the air jet of the flute leaves it open.
3. Related topics: see Cylindrical bore, Flute, Natural frequencies of pipes

The oboe: a conical bore

1. Instructions
   • Choose the Oboe project from the Project menu: a conical bore 580 mm long, which widens steadily from a radius of 1.8 mm at the mouthpiece to 10 mm at the bell. The Mouthpiece is again closed.
   • Read the frequencies of the modes and the ×f₁ column, look at the shape of the modes and listen with ▶ Listen. Compare with the clarinet.
   • Now change the Mouthpiece to open (as in the flute) and see how the fundamental changes.
2. Remarks
   • The fundamental is about 243 Hz (the app shows B3, just above B♭3, the lowest note of the real oboe). The following modes are in the ratios 2.07, 3.21, 4.38, ...: they are close to the complete harmonic series 2, 3, 4, ..., even though the bore is closed by the reed like that of the clarinet. This is why the oboe moves to the upper register by rising an octave.
   • This is the "paradox" of the conical bore: a complete cone, closed at the apex and open at the base, has the same frequencies as a cylindrical pipe open at both ends and as long as the cone.
   • However, the fundamental of the oboe with the mouthpiece closed is lower than with the mouthpiece open (about 293 Hz). The reason is that our bore is a "truncated" cone: the apex is missing. Extending the cone up to its apex gives a complete cone about 707 mm long, and c/2L = 348/(2 × 0.707) = 246 Hz, very close to the 243 Hz computed. In the real oboe the volume of the reed and of the small tube on which it is mounted takes the place of the missing tip.
   • The clarinet and the oboe are almost the same length, but the clarinet, with its closed cylindrical bore, sounds a fifth lower (161 Hz against 243 Hz).
3. Related topics: see The paradox of the semi-closed bore, Normal modes of a conical pipe, Oboe
4. To find out more: see Conical bore, Natural frequencies of pipes, Acoustics of model wind instruments
5. Other experiments
   • Turn the oboe into a cylinder of the same length: press Make it cylindrical (the radius of all points becomes 5.9 mm). With the mouthpiece closed the fundamental drops to about 146 Hz and the series of odd multiples reappears: you have built a "clarinet".