This post is one of a growing series of holistic investigations into various aspects of music theory. The full list can be found in the Posts page under the category Music Theory De-Mystified.
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A plucked guitar string is a good physical representation of half a sound wave.
Sound waves, like ripples in a pond, are wave shaped pulses that travel and spread away from the source. Single frequencies have an evenly-curved shape called a sine wave. A complete wave, from the start to where it begins to repeat, is called a cycle.
Unlike ripples in a pond, a string on a guitar (or any string instrument) is fixed and doesn’t travel. A vibrating string produces half a sine wave at a time, moving gradually upward then downward for each wave cycle. (The full sine wave is twice the length of the string.)
When you lightly touch the string above the 12th fret (half-way along its length) and pluck the string, we hear a pure sound called a harmonic. By not pressing all the way down, both halves of the string are free to vibrate: only the middle is blocked, allowing a complete sine wave of half the string length.
The sound we hear is exactly one octave above the sound of the open (whole) string.
- One octave higher = half the string length.
- In other words, one octave higher = half the wavelength.
By the way, you can check the accuracy of a guitar’s intonation by comparing just touching the string at the 12th fret to pressing all the way down at (behind) the 12th fret. The pitch should sound the same.
In Why Are Octaves Special? we saw that one octave higher = double the frequency, so:
- double the frequency = half the wavelength. As the frequency goes higher, the sound wave becomes shorter.
You can also place a finger lightly over the 5th fret, 1/4 of the string length, and hear a note 2 octaves above the open string, at 4x the frequency.
This is just another way of demonstrating the close relationship that exists between notes one or more octaves apart. The octave is fundamental to how music behaves. It is a universal musical phenomenon, independent of genre or culture.
Even though we don’t think of sound waves when playing or listening, I suspect that we are innately aware of them. We tend to think of bass notes as big and piccolo or tin whistle notes as little…
Bear with me- there’s a little more in the next post, (Guitar) String Theory 2: Why Do Frets Get Closer Together?