The Science Behind Cello

The Science Behind Cello

The violoncello, or cello, is a member of the violin family of string instruments, but it is distinctly different from the violin. Its strings are tuned to C2, G2, D3, and A3, an octave below the viola and a twelfth (octave plus a fifth) below the violin.

Although the strings vibrate at one-third the frequency of those on a violin, the length and width of its body are closer to twice rather than to three times those of the violin. Increased rib height and relatively thinner construction help to keep the resonances sufficiently low for bass enhancement. An exploded view of a cello is shown in Fig. 14.1.

The cello has a rich history. The oldest surviving cello is thought to be the work of Andrea Amati, and is dated 1572.

It was made as part of a group of instruments for Charles IX, King of France, and shows that the cello was established as a member of the violin family at an early time (Dilworth 1999). The early instruments made in Cremona by Amati and his family were 79 cm in length, whereas smaller cellos of about 71 cm (twice the length of the violin) were made in Brescia.

These two sizes persisted well into the eighteenth century.

The standard length today is about 75 cm (Dilworth 1999). A comparison of cello and violin dimensions is given in Table 14.1. Most acoustical research on bowed string instruments has been concentrated on the violin. Although in a typical orchestra there are about one-third as many cellos as violins, published material on cello acoustics is relatively scarce (see Firth 1974, Egger 1991, Langhoff 1995, Woodhouse and Courtney 2003).

Modal analysis of cello

Probably the most important determinant of the sound quality and playability of a string instrument is the vibrational behavior of its body. The rather complex vibrations of the body are conveniently described in terms of normal modes of vibration or eigenmodes.

Violin body vibrations have been studied for 150 years or more, although the development of optical holography and digital computers has greatly contributed to our understanding in the past 20 years.

Vibrations of cello bodies are equally complex, and their vibrations have been studied in more recent years. The normal modes of vibration of violins and cellos are determined mainly by the coupled motions of the top plate (table), back plate, and enclosed air.

Smaller contributions are made by the ribs, neck, fingerboard, and other parts. The coupling between the various oscillators is more difficult to model than in the guitar, for example, because of the soundpost. Although the free plates of violins have been successfully modeled by finite-element methods, most of our knowledge about cello vibrations is based on experimental studies.

Modes of vibration

The complex vibrations of a complex structure, such as a musical instrument, can be described in terms of normal modes of vibration. A normal mode of vibration represents the motion of a linear system at a normal frequency.

Each mode is characterized by a natural frequency, a damping factor, and a mode shape. Normal implies that each shape is independent of all other mode shapes for the structure. The vibration of a structure when an oscillating force is applied is often called the operating deflection shape (ODS). The operating deflection shape at each frequency will generally be a combination of normal mode shapes.

If the normal mode frequencies are well separated and the mode damping is small, the ODSs correspond closely to the normal modes of vibration. In a cello, they represent the normal modes fairly well at low frequency, but at higher frequencies rather special techniques are required to observe the normal modes of vibration.

In a cello, sound radiation from the large top and back plates dominates, but sound is also radiated through the f-holes, and so the motion of the air inside the Fig. 14.2 Mobility curves for a violin (top) and a cello (bottom).

Note the different frequency scales (Askenfelt 1982) 248 E. Bynum and T.D. Rossing cello is of considerable interest. By applying appropriate constraints it is possible to study the vibrations of the top plate, back plate, and enclosed air alone, but in general when a force is applied to any part of the cello, or when the cellos is played, the resulting vibrations are due to coupled vibrations of many parts of the cello.

Although we sometimes speak of plate modes or air modes at frequencies where the motion of one part of the cello is especially strong, most modes of vibration involve the motion of several different parts of the instrument.

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