When a guitarist plucks a string while touching it lightly with his finger right above the 12th fret (without pressing the string to the fingerboard), the resulting note will be exactly one octave above the note the open string produces. This happens because the finger blocks the string from vibrating at its full length, but since the 12th fret is placed exactly at the middle of the string, the finger allows the string to vibrate at half of its length, generating a frequency which is twice the frequency of the open string, and thus a note which an octave above the note of the open string.
A similar thing happens when the guitarist plucks the string while touching it right above the 7th or the 5th fret, which are placed exactly at one third and one fourth of the string’s length, respectively. The string is then forced to vibrate at three times or four times the frequency of the open string, generating notes that are an octave and perfect fifth, or two octaves, respectively, above the note of the open string. Such notes are called harmonics, or overtones, and they are strongly related to Fourier analysis, which we discuss here. The fundamental frequency of a string, corresponding to its full length, is the first harmonic; the harmonic produced by touching the string at the 12th fret (1/2 of the string's length) is called the second harmonic; the one at the 7th fret (1/3 of the string's length) is the third harmonic; the one at the 5th fret (1/4 of the string's length) is the fourth harmonic; and in general, the one at 1/n of the string's length is the nth harmonic.
Guitarists often use these harmonics to tune their guitars. For example, the fourth harmonic of the low E string (at the 5th fret) is the E two octaves above the low E; the third harmonic of the A string (at the 7th fret) is also the E two octaves above the low E, as it is an octave and a perfect fifth above the A. By playing these two harmonics together, it is fairly easy to tune the A string relative to the E string.
But here is the catch: the third harmonic from the A string gives us an E according to the Pythagorean temperament (relative to the A) – its frequency is exactly three times that of the A, instead of being 2.9966 times, as the equal temperament requires. That means that strictly speaking, it is simply wrong to tune the guitar this way.
However, as we saw (and heard) earlier, the difference between a Pythagorean perfect fifth and an equal perfect fifth is tiny, and very difficult to detect by our ears, so perhaps we can neglect it. The problem becomes more serious, though, if we continue using this method and tune the D string according to the A string, and the G string according to the D string; this way we “accumulate” errors that may add up to a more substantial discrepancy. This is why some people say that it is a bad practice to tune the guitar using harmonics.
You can learn more about temperament problems of guitarists and of the guitar
here.
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