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English-style full-circle bell-ringing.

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Sound of Bells – The sound of a bell

Unlike Wind & String instruments, the harmonics of many Percussion instruments do not have a simple mathematical relationship to the lowest frequency. Canon Simpson was aware of this fact when his famous article "On Bell Tones1" was published. His hypotheses and its impact on bell tuning will be addressed later in this article.

When a bell is struck the whole body of the bell vibrates except the upper crown of the bell which is secured to the headstock. Modern theory separates the modes of vibration into those produced by the soundbow and those produced by the remaining bell "shell"2,3,4,5. 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 al3. 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). For the purpose of this article the principal vibrational modes are the lower order RIR and R=n modes. These modes have nodes (points of minimum movement) and antinodes (points of maximum movement) equally positioned along the perimeter of the bell's mouth. At each node location there is a nodal meridian between the rim and the upper crown. Most of the vibrational modes have one or more nodal circles parallel to the rim. The precise location is dependent on the partial and the shape of the bell. The simplest vibrational mode is the Hum, which has 4 nodal meridians and no nodal circles (classification RIR, m=2, n=0). If we consider the bell to be split into four equal segments, the boundary of each segment is the nodal meridian. If the bell is struck it receives an impulse which starts the bell vibrating and providing the clapper does not rest on the bell, all four segments will vibrate equally. The segments will alternate between vibrating inwards and outwards. Whenever a segment vibrates inwards, its neighbours will vibrate outwards and vice versa. The same theory applies to all the principal vibrational modes, the Prime for example behaves similarly to the Hum except the bell is split into eight segments, four in the waist and four below the waist. If a lower segment is vibrating inwards, its three neighbouring segments - including the upper segment will be vibrating outwards and vice versa. The Prime partial is a shell driven vibrational mode and its classification is R=1, m=2.

When the bell vibrates it imparts energy into the surrounding air. The larger the amplitude of vibration, the louder the individual partial will sound. The larger the vibrating area, the louder the bell will sound (assuming the diameter/thickness ratio and metal mix remains constant). The quicker the vibration, the higher the partial's frequency.

The first five partials are known as the hum, prime (also confusingly known as the fundamental), tierce, quint and the nominal. It is often incorrectly assumed that these are the only partials of a bell. This is not so, and one research project3 has documented over 100 individual partials. Some of these do not occur in practice as the crown of the bell is usually fixed to a headstock. The vast majority of the remaining partials generate sound of such low amplitude and short duration that they can be ignored. However, as will become clearer in later sections of this article, a bell's sound is dependent on more than the first five partials.