A String in Vibration

For thousands of years people have been deriving pleasure from the sounds of strings being vibrated under tension.

Scholars often credit the ancient philosopher and sage, Pythagoras, with first exploring the harmonic relationships between strings and their tones. His studies form the backbone of our modern tonal music system, so some of the basics it would profit you to know follow

Monochord - a string in tension

The diagram represents a string in tension. It is just like any of the strings on your guitar. If it is plucked it will vibrate with a dominant frequency, plus other harmonics. These harmonics effect the timbre and tonal qualities of the instrument. For now though lets just consider the dominant frequency which is caused by the fundamental vibration.

The fundamental vibration mode involves the whole string vibrating side to side and held at the ends. The frequency of this vibration determines the pitch of the note. So in acoustics, the term pitch really means the frequency of the fundamental vibration.

There are other harmonic modes in the string, which is another way of say that the string can vibrate between its fixed ends in a number of ways. For example, the 2nd and 3rd Harmonics are shown below. You will notice that the ends are always fixed and form what is called a node. A node is just a fancy way of saying a part of the string that is not moving (much). On a monochord, the ends are always held in place and so MUST form nodes at the ends, but you can see that for higher harmonics, nodes can also appear along the string.

The nature of harmonics and vibration is not important at this early stage, other that to show you that the nature of even a simple string is not so simple. These variations in harmonics are hard to predict, but they do account for the characteristics of the guitar sound that we love.

Changing tension in the string changes the pitch of the string. A tight string vibrates faster and sounds higher.
Shortening the string also affects the pitch. A short string vibrates faster than a longer one.
2:1 Ratio - The Octave

In fact, now we come to one of Pythagoras' interesting discoveries. Halving the length of the string doubles the frequency. This note produced by the half string has a strong harmonous relationship with the original note and is said to be an octave higher. The frequency of a note one octave higher than some starting note will always be twice the frequency of the starting note . Like magic, except this is the beautiful harmony of the physics of acoustics.

You can see this relationship on your guitar. The position of the 12th fret is the halfway point between bridge and the nut. (Go on - measure it out and see for yourself.) Holding a note at the 12th fret is one octave higher than the same string played open.

The monochord can be divided into octaves by repeatedly halving the string...[ad infinitum]

As you are starting to see, this process is just a way of dividing the length of the monochord to make similar sounds that have a pleasant relationship to each other. There are, of course, other elegant mathematical ratios that can be used to divide the string. They are covered in the next section.

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