Percussion bars are shaped to tune their modes toward harmonic ratios

Evidence

In order that notes can be played which have a clearly perceived pitch on percussion instruments such as the xylophone, marimba and vibraphone—with playing ranges from C5 (523 Hz) to C8 (4186 Hz), A2 (110 Hz) to C7 (2093 Hz) and F3 (175 Hz) to F6 (1397 Hz), respectively—the bars are shaped with an arch on their undersides to tune the modes to be close to harmonics of the first mode. In the marimba and vibraphone, the second mode is tuned to two octaves above the first mode, and in the xylophone, it is tuned to a twelfth above the first mode. These instruments have resonators, which consist of a tube closed at one end, mounted under each bar. The first mode of these resonators is tuned to the f0 of the bar to enhance its loudness, and therefore the length of the resonator is a quarter of the wavelength of f0 (see Equation 1.36). — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1287

Categorization

• DEFINE: Explains why xylophone/marimba/vibraphone bars have arched undersides and why resonator tubes are quarter-wavelength: both are structural tuning actions.
• ADDENDUM: Connects to the trees/wood infrastructure article - mallet percussion shows how the same shaped-wood resonant principle appears in instruments, alongside musical-instrument construction.
• corroborated: 2 independent passages support this claim — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1287
• XREF: connects to: Higher drum modes self-compensate and leave kettle volume unchanged; A wind instrument’s playability depends on how many of its harmonics align with resonances.; A tapping bridge adds trumpet-like harmonics to a bowed monochord (+7 more)