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Harmonics in Strings

Author: N.H. Crowhurst

That it is differences in the overtone structure that account for the different "sound" of various instruments can be verified with an electric guitar. With this instrument, the strings themselves produce relatively little sound. An electrical pickup placed close to the strings is used to pick up their vibration, which is amplified electronically, the sound being heard from a loudspeaker. If you take one of these guitars and try moving the pickup to different parts of the string, you will find that the timbre of the note changes.

Assume that after the string has been set vibrating, the amplitude of vibration of the string is 1/10 inch at the lowest frequency, 1/20 inch at the next above this, which is twice the frequency (the second harmonic or first overtone), 1/30 inch at the third harmonic or second overtone, 1/40 inch at the fourth harmonic or third overtone, and so on. Now suppose that we move the pickup to positions 1/10, 1/5, 1/4, and 1/3 of the length of the string from one end. The differences in timbre will be clearly audible. From the drawings we can find the magnitude of vibration at each point along the string for the fundamental and each harmonic. These can be made into a table:

Maximum Vibration at pickup points in inches
Tone Max. Vibration
Fundamental 0.100 0.031 0.059 0.071 0.087
2nd harmonic 0.050 0.029 0.048 0.059 0.043
3nd harmonic 0.033 0.027 0.032 0.024 0.000
4nd harmonic 0.025 0.024 0.015 0.000 0.022
5th harmonic 0.020 0.020 0.000 0.014 0.017

Extent of vibration at various pickup points for the first five harmonics

Of course there is less output from the positions nearer the end, but this can be compensated for by extra amplification, so the tonal quality, as judged by the ear, will depend on how much of each overtone there is, compared to the fundamental. Let us adjust the table to give this information, by expressing the strength of each harmonic as a percentage of fundamental. We do this by dividing each vibration figure by that for the fundamental at the same point, and multiplying by 100%. For example, when the fundamental vibrates 0.031 inch and the harmonic is 0.029 inch, the harmonic is (0.029 X 0.031) X 100% = 93.7%. The table, completed in this way, then looks like this:

Vibration percentages
Pickup Point Fundamental 2nd
one tenth 100 93.7 87.1 77.5 64.5
one fifth 100 81.4 54.3 25.4 0.0
one fourth 100 70.5 33.8 0.0 19.8
one third 100 50.0 0.0 25.0 20.0

Now notice the difference in composition: the first position, 1/10 from the end, gives a large proportion of all harmonics; even the fifth harmonic is more than half as strong as the fundamental. The second position, 1/5 from the end, eliminates fifth harmonic, and leaves the others in different strengths. Each position gives different proportions of harmonics. But the most important difference is that the further we go from the end, the weaker all the harmonics get, compared with the fundamental. At 1/10 from the end, all the harmonics have more than half the amplitude of the fundamental. At 1/3 from the end, only the second harmonic is even half as strong as the fundamental; the others are much weaker.

Percentage distribution of harmonics at the pickup points

From this we can see that the tone quality, or timbre, of a stringed instrument will vary considerably if we change the point along the string where the sound is picked off. In most stringed instruments, the sound is taken from the bridge, which transmits it to the body of the instrument. This means that the pickup point is fixed.

When the pickup point is fixed, the plucking or bowing point and the instrument's resonance control timbre.

In this case, the precise point at which the string is plucked or bowed influences the harmonics or overtones of the string that are set in vibration together with the fundamental, thus changing the tone quality. In addition, the natural vibration properties of the body of the instrument (called resonances) influence the relative strength of the various frequency components as they are radiated into the air as sound. This accounts for the characteristic differences between different instruments using the same kind of strings.

We can extend the same general idea of overtones, and the way they are excited, to other kinds of instruments, using pipes, reeds, vibrating bars or rods, or other basic sound generators - even triangles, bells or drums. Variations in the complex pattern of harmonics give each instrument its own character.

Last Update: 2010-11-03