The Undertone Series


By Jay Jaganaath


Music. We all listen to it in some form. Some of us even engage in playing it. However, few of us really know of the extensive science behind the creation of these melodic tones. In the most basic terms, music consists of sound waves of different frequencies created in rhythmic intervals. Let us understand what takes place in the smaller scales of reality where music is propagated.

A sound wave is the pattern of disturbance caused by the movement of energy traveling through a medium (such as air, water, or any other liquid or solid matter) as it propagates away from the source of the sound.

But what is this ‘disturbance’? When sound is emitted from the source, the particles of the source vibrate and disturb the particles adjacent to it, which vibrate and disturb particles adjacent to it and so on until the energy required for the disturbance runs out.

Now, sound mainly propagates itself in the form of longitudinal waves. longitudinal sound waves are waves in which the particles vibrate in the same direction as the direction that the wave moves in. The best example of this is the slinky, which clearly exhibits longitudinal waves when disturbed.

 For the current topic of discussion, we’ll discuss sound as it travels through the string of a violin.

A violin string vibrates transversely, as pictured in the diagram above

The violin consists of 4 different strings of different pitches and a ‘bow’, or a stick like instrument, is slid across the strings to generate sounds. However, the sound generated by a violin is different when compared to the sound generated by another instrument, like a guitar, even when both the instruments are playing the same notes. This difference in sound quality is generally attributed to the number of overtones than can be heard on the instrument.

Overtones – here’s a new term. To properly explain it, I will have to explain the mechanics of sound propagation in a string.

Upon creating a disturbance in a string, the string vibrates with a wave motion which is a sum of a series of distinct motions, the lowest of which is known as the fundamental frequency and is often the loudest of all the other wave motions generated. The succeeding wave motions are known as overtones and are integer multiples of the fundamental frequency. It is the variation in the loudness or amplitude of these overtones that determines the sound’s quality.

As I stated before, the fundamental frequency is the lowest frequency at which the string can vibrate at. Ergo, it should not be possible for the string to vibrate at lower frequencies, but some facts suggest the contrary.

Like the overtone series, a theoretical series known as the undertone series exists, with a frequency series that are an inverse of the harmonic series eg. if the harmonic series consists of f, 2f, 3f, 4f… with f denoting the fundamental frequency, the undertone series is f, f/2, f/3, f/4.

While these lower frequencies can be produced by electronic methods, it is not part of the natural sound of a string and hence cannot be extracted through acoustic means.

However, there exists a method to extract such a sound out of the violin through unusual use of the violin bow or in layman terms, ‘that long stick used to play the violin’. To understand this method, let us better understand the mechanics behind the role of the bow in sound production in a violin.

Unlike a guitar, which relies on striking the strings to produce a sound, a violin relies on the principle of Helmholtz motion to produce its distinct, drawn out tone. Helmholtz motion is an accurate description of the processes taking place when the bow generates a sound out of a string. This method of sound production can be described as a ‘stick and slip’ method. In this method, the violin string first sticks to the horsehair on the bow as it moves till a certain point, before slipping back just to get stuck on the hair again, this cycle taking place a few hundred times per second.

When the amount of force exerted on the bows is increased, the point till where the string slips is extended, reducing the number of cycles it can have per second as each cycle now takes longer time.  we have obtained string vibrations at lower frequencies than possibly thought. While most of this sound emitted it displeasing to the ears (known as raucous sound), there exists a sweet spot that, when obtained, can produce sounds of the tones belonging to the first few undertones of the undertone series.

As is deduced from the course of this article, even the most mundane of things can reveal the most astonishing phenomena of the physical world.