Jones (1.1.2) Music Theory, “Vibrating Strings”

 

by Corry Shores

 
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George Thaddeus Jones

Music Theory

Part I
Notation, Terminology, and Basic Theory

Chapter 1
Elementary Acoustics and the Properties of Sound

1.1.2
Vibrating Strings

Brief Summary:
A stringed instrument is one whose sound is produced primarily by vibrating strings. They are further classified according to the action setting the string in motion: {1} bowed instruments (including violin) {2} plucked instruments (including harp and guitar); and {3} struck string instruments (including piano). The string vibrates on the whole by swinging back and forth. Throughout this main swing are smaller ones, dividing the string into halves, thirds, fourths, and so on, providing the overtone series. The way that an instrument accentuates certain overtones gives it its timbre.

 

 

Summary

 

 

1.1.2.1

[Stringed instruments produce their sounds with vibrating strings. The means of setting the strings in motion subclassifies them into: {1} bowed instruments (including violin) {2} plucked instruments (including harp and guitar); and {3} struck string instruments (including piano)]

 

In stringed instruments, the tones are made by making strings vibrate. There are different ways the strings can be set in motion, and the stringed instruments can be classified accordingly: {1} bowed instruments (including “violin, viola, cello, bass, and the obsolete family of viols”); {2} plucked instruments (including harp, harpsichord, guitar, lute, mandolin, and banjo); and {3} struck string instruments (piano, clavichord, and cimbalom). But “Whatever the method of setting the string in vibration, it reacts acoustically in substantially the same way for all of these instruments” (4).

 

 

 

1.1.2.2

[The string moves by making cycles from the resting position to a displaced position some distance away, rebounding to the opposite displaced position, and continuing back and forth. The ends of the string are the nodes of the wave, and the center is the loop. One cycle from the resting position through both opposite positions and back to the resting position is one wave or vibration.]

 

Jones will now explain how the physical motion of the string is involved in the physical properties of the sound waves it produces. We first consider an “elastic material” tied to two ends [perhaps we think of an elastic material generally so that these principles can apply to other sorts of things like drum heads. But it seems in this application we need to think of the elastic material as being shaped like a string.] It is relatively taut, and so it occupies the horizontal position shown in the diagram.

[Figure 1]

We then suppose that we displace it from that base-line position, drawing the center up to point A. As a result, it will rebound down to point B: “If it is moved out of its position of rest to point A by being struck, plucked, or bowed, the elasticity of the string and its momentum will carry it to point B, a distance past the point of rest approximately equal to the original displacement (A)” (4). Then the resistance from the air will gradually dampen the motion: “If it is then left free to vibrate, it will eventually be brought again to a state of rest by the friction of the medium, in this case air” (4). If the string is pulled back more, its own motions will have a greater amplitude, and the sound waves it produces will too: “The degree of displacement, which is the amplitude, determines the loudness of the sound” (4).  So the string begins at rest, moves to A, rebounds to B, then in its rebounding back toward A, will pass through its rest position, thereby making one cycle, also called one vibration or wave: “One entire cycle, from point of rest to A, then to B, and back to point of rest, is considered one vibration or wave; the ends of the string are the nodes of the wave, the center point is called the loop” (4). Certain physical properties can have some influence on the pitch of the string, like its material, thickness, and tension. However, what primarily determines the pitch is the string length (5).

 

 

 

1.1.2.3

[The full back and forth motion motion is the fundamental frequency. All the while, a series of additional motions moves through the same string. These are partials or overtones, and they bear frequencies increasing with the natural numbers.]

 

[First I will quote, then comment.]

The vibration of the entire length of the string as shown in figure 1 produces the fundamental, that is, the basic pitch we assign to this string length. However, being flexible the string vibrates also in parts of halves, thirds, quarters, and so on, and each of these segments produces a sound. These sounds are called partials, or overtones.

(5)

[There is a lot that is interesting about wave movement and overtones. See this entry where the physics of it is discussed at greater length. I will share some comments, but please consult a more reliable source on this topic. The image at the bottom of the above figure looks a lot more like what we actually see when viewing a vibrating string, and as the diagram suggests, what we normally see in all that blurry, dynamic motion is the combination of the main wave with its harmonic partial waves. All of them move through the string simultaneously, but with decreasing influence, normally, as we go down the series of partials. Were we to take an instantaneous snapshot of the string, however, it would not look like that blurry one on the bottom and most likely not like any of the ones pictured above it. It would rather be a deformation with no regularity, probably. The diagram below shows this. (Please contact me if you know the print source.)

(Image obtained from: http://nexusilluminati.blogspot.com. Many thanks to this source, and seeking the original print source.)

I am not sure of this however. But as far as I understand, at any one instant, the string takes on a deformation which expresses all the wave forces moving through it, but only over time, as those forces have periodic effect, are we able to discern the distinct wave frequencies.]

 

 

1.1.2.4

[The set of partial vibrations accompanying the fundamental is called the overtone series.]

 

The composite sound of the additional frequencies accompanying the fundamental constitutes the overtone series. For C, we obtain the “following series of pitches” resulting “from the partial vibrations” (5):

 

 

 

1.1.2.5

[The way that an instrument accentuates certain overtones determines its timbre (characteristic tone color).]

 

[I do not follow the next point about the tempered scale, but we return to it later anyway. The point might be that the notes as they have been assigned in the tempered scale have frequencies that are not precisely the doublings, triplings and so on of the original tone, but are rather approximations. As we noted above, the fundamental has the greatest intensity, with the next partials generally speaking decreasing in intensity with each one. However, that is not how it works really, because different instruments accentuate certain overtones. Let me quote:]

The pitches are shown in our “tempered scale” notation, and are only approximate; the space between the partials decreases proportionately as the series ascends. The series does not stop at the sixteenth partial, but this segment is, for practical purposes, all that we need be concerned with. The fundamental and the lower partials have greater intensity and are therefore easier to | hear than some of the more remote overtones, but it would be an oversimplification to say that the series gradually diminishes in intensity as it ascends. In the timbre or characteristic tone color, of some instruments certain of the upper partials are stronger than certain others, and it is due partly to this fact that we are able to distinguish one instrument from another – an oboe from a flute, for example.

(5-6)

 

 

 

1.1.2.6

[The fundamental is numbered as 1 and the first overtone as 2.]

 

We give the fundamental the number 1, and the first overtone is numbered 2. (6)

 

 

 

From:

Jones, George Thaddeus. Music Theory. New York: Barnes & Noble Books / Harper & Row, 1974.

 

Other images sources:

String wave overtone synthesis:

http://nexusilluminati.blogspot.com.tr/2011/06/philosophers-stone-how-to-transmute.html