How to tune a guitar or bass using harmonics
This should work for any stringed instrument tuned in fourths (except for that pesky B string on the guitar). The nth harmonic of an open string is n times the frequency of its open fundamental note, and can be generated by lightly touching a string at m/n of its length, where m and n are integers forming a simple fraction, then plucking it. For example, touch the string above the twelfth fret (1/2 of the length) and play to get the second harmonic; touch above the fifth or nineteenth fret (1/3 and 2/3) to get the third harmonic.
A semitone is an increase in frequency of the twelfth root of two above the previous note. Using this, we can determine how many semitones up from the fundamental (x) each harmonic is:
x = log(n) / log(12th root of 2)
All power-of-two harmonics are an exact number of octaves up on the fundamental. Thus, the fourth harmonic is exactly two octaves up. The third harmonic turns out to be 19.02 semitones up, which is almost exactly an octave-and-fifth.
Using this information, its possible to tune using harmonics:
- Touch the bottom string above the fifth fret and pluck the string - it should sound two octaves above open;
- While that is still sounding, touch the next string up, which needs to be tuned a fourth higher, at the seventh fret and pluck to get the third harmonic - an octave-and-fifth above open;
- These should be the same note, so adjust the upper string to match the frequencies: listen for the beats and cancel them out;
- Repeat on the second and third strings, and so on.
This method has the potential to be a more accurate method of tuning than using fretted notes, as the tuning is independent of the shape of the neck or the accuracy of the fret positioning.
On a guitar, use the seventh fret harmonic on the A string to tune top E, and then use the above method on the B and top E strings, but adjust the B to correct.
Note that this method only tunes to an interval of a fourth, although for fifths you could use the seventh and twelfth frets. It also only tunes the instrument to itself, so make sure the bottom note is right before you start!