What did the Liberty Bell sound like?

When the Liberty Bell is brought up, it is usually discussed as a symbol of the US rather than a musical instrument. The weathered bronze and famous crack are reminders that the iconic bell has borne witness to the history of the US in its entirety. Given its current state, it’s easy to forget that it once rang aloud in the Philadelphia building now known as Independence Hall (then called the Pennsylvania State House). Although its chime has been silenced for almost two centuries, it is interesting to consider what it might have sounded like ringing all those years ago.

Many full-size replicas of the bell were made during the 20th century, one of which ended up in the far northwest corner of Pennsylvania on the campus of Penn State Behrend. If you walk up to the bell and give it a tap, you will hear the bright timbre of the Liberty Bell. So, that’s it, case closed—that’s what it would have sounded like, right?

Not quite. Human ears are good at hearing if a bell sounds in or out of tune qualitatively, but it would be nice to have a more quantitative description of the sound. In honor of the 250th anniversary of US independence, the three of us decided to create an acoustic model of the famous bell. We aimed to determine what notes you hear when the bell rings, how the bell deforms as it vibrates, and which notes ring the longest. Doing so in a more scientific manner requires an experimental tool that has been used for many decades: modal analysis.

When a bell is struck by its clapper—the metal ball on a stick that hangs inside of the bell—the bell is excited into vibration, which listeners hear as its chime. Like all structures, the bell vibrates in specific patterns at discrete frequencies. Those patterns are called mode shapes, and the frequencies at which those shapes exist are called natural—or modal—frequencies. They are the ones heard when the bell rings. Finally, each mode has a loss factor associated with it that governs how long it will ring. The goal of modal analysis is to experimentally measure those mode shapes, natural frequencies, and loss factors.

Bell dynamics

When Western church bells are made, foundries attempt to tune the first five natural frequencies of the bell—known as the hum note, fundamental, tierce, quint, and nominal—to have very specific ratios. As its name indicates, the fundamental is the dominant note heard by the human ear and is thus the frequency around which the others are measured: The hum note is approximately an octave below the fundamental, and the tierce, quint, and nominal are approximately a minor third, perfect fifth, and octave above it, respectively. Relative to the fundamental of an ideally tuned bell, the first five frequencies have ratios of 0.5, 1.0, 1.2, 1.5, and 2.0. Musically, the frequencies sound as a minor triad, which is why church bells are commonly perceived as having a somber tone.

The acoustics of bells also involves the human listener. The human ear perceives the pitch of a bell as a subjective tone called the strike note, which results from the complex interaction between the mode shapes and the listener’s learned expectation of the fundamental’s harmonic relationship with the other natural frequencies. Moreover, because bells have a symmetrical structure, each mode comes as a degenerate pair of modes that are often slightly out of tune with one another. That further complicates the sound of the bell, because the two near-matched pitches beat with one another and contribute a subtle harshness to the perceived sound.

Experimental modal analysis

We needed two types of hardware to measure the necessary data to determine the bell’s mode shapes, natural frequencies, and loss factors: accelerometers and a specialized hammer equipped with a force gauge. After gluing the accelerators into the bell in various locations, we tapped the bell in many locations equally spaced across its surface. The force gauge on the hammer measured the force of impact as a function of time when striking the bell, and the accelerometers measured the response. Together, the data allowed us to characterize the inherent dynamics of the bell on the basis of its shape, size, material, and the way in which it is supported. Using some signal processing and rational fraction approximations, we extracted from the experimental data the natural frequencies at which the bell vibrated, the corresponding mode shapes, and their respective loss factors.

For the Liberty Bell replica at Penn State Behrend, we measured frequencies of 174 Hz for the hum note, 323 Hz for the fundamental, 381 Hz for the tierce, 510 Hz for the quint, and 623 Hz for the nominal. At 323 Hz, the fundamental of the replica rings somewhere between an E-flat and E above middle C. Normalizing those first five frequencies with respect to the fundamental, we obtained frequency ratios of 0.54, 1.00, 1.18, 1.58, and 1.93—which are very close to the tuning of an ideally tuned bell. The figure below shows side and end views of mode shapes for the fundamental frequency of a bell. The diagrams in the left column are adapted from a study of Western church bells; those in the right column are derived from our Liberty Bell test. The measured mode shapes closely approximate those found in the literature.

The final modal parameters to discuss are the loss factors, which are arguably the most difficult to predict. Although the values we measured are well within the range of those found in the literature, that only tells us so much about the true bell. Generally, the type of bronze used in the bell and any imperfections present in the bell’s cast tend to govern the loss factors. Our experimental results suggest that the real bell likely had loss factors similar to the replica’s, but given the differences between 18th-century and 20th-century manufacturing processes, the exact loss factors are difficult to predict.

Bell manufacturers use modal analysis to improve their instruments’ sound by assessing tuning and correcting problematic modes. But the technique is also used by structural engineers to gain insight into how an object is vibrating. With that knowledge, they can design solutions to mitigate vibrations, which can create noise pollution and, over time, cause severe damage to structures. Bolts loosen, materials weaken, and eventually, vibrations can cause a once-strong building, bridge, car, airplane—or historic bell—to crack!