A stopped cylindrical pipe supports only odd-numbered modes

Evidence

The cylindrical resonator of a clarinet acts as a pipe that is stopped at the reed end (see above) but is open at the other. Odd numbered modes only are supported by such a reso­ nator (see Figure 4.18), and its f0 is an octave lower (see Example 4.2) than that of an instrument with a similar length pipe which 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 Howard, Jamie Angus, coustics and Psychoacoustics, 1241

Categorization

• DEFINE: The stopped-cylinder rule (only odd harmonics, an octave lower than an open pipe) is a core psychoacoustic principle about how bore geometry conditions an instrument’s mode structure.
• XREF: Connects to why the clarinet’s low register has its characteristic nasal, hollow tone compared to flutes ’ which produce all harmonics. — connects to: Cylindrical stopped-pipe bores support only odd-numbered modes, forcing extra finger holes; Cylindrical stopped pipes support only odd-numbered modes, making instruments sound an octave lower — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1213; A stopped flue pipe’s odd-mode support creates its characteristic chiff at onset — David Howard, Jamie Angus, coustics and Psychoacoustics, 1474 (+4 more)

Broader Connections

• A pipe closed at one end produces only odd-numbered harmonics. #corroborates (0.98) — David Howard, Jamie Angus, coustics and Psychoacoustics, 416
• Cylindrical stopped-pipe bores support only odd-numbered modes, forcing extra finger holes #corroborates (0.97)
• Cylindrical stopped pipes support only odd-numbered modes, making instruments sound an octave lower #corroborates (0.97) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1213
• Standing waves in pipes occur only at odd multiples of the lowest frequency #corroborates (0.95) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 410
• Stopped pipes save space but sacrifice even-numbered harmonics #expands (0.90)
• A stopped flue pipe’s odd-mode support creates its characteristic chiff at onset #expands (0.85) — David Howard, Jamie Angus, coustics and Psychoacoustics, 1474
• Overblowing a pipe jumps its pitch to the next resonator-supported harmonic #expands (0.75) — David Howard, Jamie Angus, coustics and Psychoacoustics, 1211