Cylindrical stopped pipes support only odd-numbered modes, making instruments sound an octave lower

Evidence

The musically useful bores in this context are based on tubing that is either cylindrical, as in the clarinet, or conical, as in the oboe, cor Anglais and members of the saxophone and bassoon families. The cylindrical resonator of a clarinet acts as a pipe that is stopped at the reed end (see earlier) but is open at the other. Oddnumbered modes only are supported by such a resonator (see Figure 4.18), and its f0 is an octave lower (see Example 4.2) than that of an instrument with a similarlength pipe that is open at both ends, such as a flute (see Figure 4.3). The first overblown mode of a clarinet is therefore the third mode, an interval of an octave and a fifth (see Figure 3.3), and therefore, unlike a flute or recorder, it has to have sufficient holes to enable at least 19 chromatic notes to be fingered within the first mode prior to transition to the second. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1213

Categorization

• XREF: The stopped-pipe acoustics here connect directly to organ pipe physics and to the earlier chapters’ treatment of pipe open/closed boundary conditions. — connects to: A stopped cylindrical pipe supports only odd-numbered modes — David Howard, Jamie Angus, coustics and Psychoacoustics, 1241
• QUESTION: Why exactly does a conical pipe shrink to a very small bore slow down the wave enough to suppress reflection at the narrow end? That mechanism deserves chasing further.