Modal frequencies

We do know something about the general acoustics of bells, dating back at least to a seminal 1987 paper by Thomas D. Rossing and Robert Perrin exploring the forms and vibrational modes of church bells, carillon bells, and handbells. A given bell’s sound is determined by relatively simple geometry—which determines the vibration patterns radiating sound to listeners’ ears—and the material of which it is made (bronze), which determines the exact pitch and duration of the ring.

Per Sean Collier, co-author of this latest article, relying solely on the meager online information about the Liberty Bell to replicate its sound produces “a terrible sounding bell that in presence does not match the silhouette of the iconic bell.” There are several full-size replicas of the Liberty Bell made in the 20th century, including one housed on the Penn State Behrend campus, and one can tap that bell to produce a ring. But scientists desire more quantitative information, such as the specific notes, which notes ring the longest, and how the bell deforms as it vibrates. And that requires modal analysis, a technique used by bell manufacturers to assess tuning and make any necessary corrections.

For instance, Western church bells are typically tuned to the first five normal frequencies, according to the authors—the hum note, fundamental (the note that is most dominant to the human ear), tierce, quint, and nominal—following well-defined ideal ratios. “Musically, the frequencies sound as a minor triad, which is why church bells are commonly perceived as having a somber tone,” they wrote. Further, “The human ear perceives the pitch of a bell as a subjective tone called the strike note,” arising from the combination of how the mode shapes interact, and the listener’s own expectations.