When you are listening to an orchestra, physics is probably the last thing on your mind. However, it does explain a lot of what you are hearing and why the viola and other larger string instruments may be too small to make the best sound.
In a string instrument, there are two main resonances: the fundamental vibration of the body or ‘wood resonance’, and the fundamental vibration of the air inside the body or ‘cavity resonance’. If the pitch of the note played on a violin matches frequency of one of these resonances, this note sounds louder because the wave is amplified by this resonance. Experimentation has shown that the best positions are for the wood resonance to be close to the note of the third string, A4 (440 Hz), and the cavity resonance to be close to the note of the second string, D4 (293.7 Hz). The extensive development of the violin means that the correct shape to cause these resonances has been determined.
When a stretched spring is divided into two lengths, and this string is played, the two notes form a chord. Pythagoras discovered that if these two lengths are divided in a simple integer ratio, the chords formed are those which are the most pleasing to hear – harmonious chords. For example, a ratio of 1 : 2 forms an octave and 2 : 3 a perfect fifth. It is these ratios which mean that violas and cellos are not the optimal size.
The viola plays a perfect fifth below a violin and a cello an octave below the viola. To keep the wood and cavity resonances in the same relative place for the viola, thus making the best instrument, we have to use the ratio for a perfect fifth calculated by Pythagoras, 2:3. Therefore, the linear dimensions of a violin compared to a viola should be in the ratio of 2 : 3. However, the length of the body of a typical violin is 35.5cm and viola 39-42cm. This is roughly a ratio of 1 : 1.14. To create a viola with the best acoustics, it should have a body length 50% greater than a violin, 53.25cm. This would make an instrument too large to play under the chin but one with a better sound.
The same is true of a cello. As it plays an octave lower than a viola, the linear dimensions should be in the ratio of 1 : 2. Therefore, a cello should be twice the length of a viola or three times the length of a violin. However, the length of a cello body is about 76cm or just over two times the length of a violin body. The optimal length for a cello should be 106.5cm.
Maybe making all viola players learn to play like cellists is a step too far but a set of eight instruments that match these optimal sizes has been developed by the Catgut Acoustical Society and they even have a small repertoire of music written for them.
Saffron Walden County High School