@TruthSandwich got me interested in the vibrational modes of bells. They're not harmonics with frequencies 1, 2, 3, 4, ... times the lowest frequency: they're much more complicated! That's why bells sound clangy. This chart shows how they sometimes work.
The lowest frequency vibrations are called:
• the 'hum' (the lowest frequency)
• the 'prime' (with frequency roughly 2 times that of the hum)
• the 'tierce' (roughly 2.4 times the hum, so a minor third above the prime)
• the 'quint' (roughly 3 times the hum, so a major fifth above the prime)
• the 'nominal' (roughly 4 times the hum, so an octave above the prime)
and so on. If you think these names are illogical, join the club! One reason it's tricky is that the loudest vibration is not the lowest one: it's the 'prime'.
The numbers I just gave you should be taken with a big grain of salt. They really depend on the shape of the bell, and you'd have to be great at designing bells to make them come out as shown here. It's not like a violin string or flute, where the math is on your side.
This quote helps explain the chart:
"Modern theory separates the modes of vibration into those produced by the "soundbow" and those produced by the remaining bell "shell". The bell vibrates both radially and axially and the principal vibrational modes are shown in the diagram together with their classification using the scheme proposed by Perrin et al. This scheme consists of the mode of vibration (RIR - Ring Inextensional Radial, RA - Ring Axial, R=n - Shell driven), the number of meridians (where “m” is half the number of meridians) and the number of nodal circles (n)."
Starting to sound like orbitals in quantum mechanics!
