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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Frets on a guitar are placed 1 semitone apart. The 12th fret produces a note one octave above the open (full-length) string.
The Relationship Between Pitch And Frequency
The frequency of a note is the speed at which a sound wave vibrates in order to produce a given pitch. The lower the frequency, the lower the pitch.
The common factor between the pitch of a note and its frequency is the octave. One octave equals 12 semitones, where each semitone sounds the same distance apart as the next, like centimetre or inch markings on a ruler.
An octave is also the frequency ratio of 2:1. Every 12 semitones higher, the frequency doubles. We can look at the relationship between sound waves and what we hear by creating a graph with pitch on one axis and frequency on the other. It would look something like this:
The above frequencies are based on a guitar A string, A = 110Hz.
- One octave higher = double the frequency.
- Double the frequency = half the wavelength and thus half the string length.
- One octave higher than the open (full-length) string is half the string length, half-way from the nut to the saddle.
- The next octave higher is half of the remaining string length = 3/4 of the string away from the nut.
In other words, the first half of the string has 12 frets and the next quarter of the string also has 12 frets.
The effect of this relationship is that for every semitone higher in pitch, the frequency increases by a little bit more than the last semitone.
The Relationship Between Frequency And String Length
Frequency and wavelength are inversely related: as one goes up, the other goes down. As the frequency increases, the wavelength, and thus the string length, becomes smaller, a little less so for each semitone.
Strings are effectively half a wave. Higher notes are produced by making the playing part of the string, and thus the wave length, shorter. For each semitone higher, the adjustment is a little less than the previous semitone. The frets mark these positions.
Why do we care? Maybe we don’t need to, but isn’t it nice to know why frets are laid out differently from piano keys?