Harmonic series

What sounds along in every note

On the page Intervals there is a sentence that hits the core: a plucked string does not produce only its fundamental, it sets a whole series of overtones vibrating. This very overtone structure, according to the page, distinguishes a violin from a trumpet or a voice. The open string is itself the first natural harmonic. All higher natural harmonics, octave, fifth, major third, minor seventh and so on, are already contained in the fundamental as upper vibrations, even if you do not hear them individually, and shape its timbre.

The frequencies of the partials are whole-number multiples of the fundamental. If you know the fundamental, you can work out every partial: the second has twice the frequency, the third three times and so on. The table works this out for the fundamental E3, which is the twelfth fret of the low E string.

Where the pure intervals come from

Every pure interval is a distance between two partials. The Intervals page describes it step by step:

• Octave (2:1): the second partial, the octave above the fundamental. Notes an octave apart carry the same name, because the ear experiences them as the same note at a different height.
• Fifth (3:2): the third partial lies an octave plus a pure fifth above the fundamental and is in the ratio 3:2 to the second partial. The article Why the fifth sounds hollow concludes from it: if you play a fifth, you only double what is already sounding, so little new is added.
• Major third (5:4): the fifth partial, to the fourth in the ratio 5:4. The article Major and minor: the third as a switch concludes: the major triad is already laid down in every single note. The minor third 6:5 does not occur in the series at a comparable place.
• Major sixth (5:3): the distance between the third and fifth partials. On the guitar you hear it when you play in succession the third partial above the 7th fret and the fifth above the 4th fret.
• Major second (9:8): the ninth partial lies in the ratio 9:8 to the eighth and describes the pure whole tone as an upper vibration.

What is not in the series

Not every interval of the scale occurs directly in the harmonic series. The Intervals page names two examples. The minor second (16:15), according to the page, arises only from the fifth and the major third and therefore counts as an artificial interval. The tritone with the ratio 45:32 does not lie in the natural harmonic series; the nearest note would be the eleventh partial (11:8, about 551 cents), almost a quarter tone lower than the tempered tritone (600 cents).

The seventh partial lies, according to the page, about 31 cents, just under a third of a semitone, lower than the tempered minor seventh. This very “blue seventh” gives the blues part of its yearning character. The piano cannot play these notes, because its twelve notes are evenly distributed, as the article on tuning describes. The equal-tempered major third lies there 14 cents above the fifth partial.

Common mistakes

• Confusing overtone and partial: The first partial is the fundamental, the first overtone is the second partial. The count shifts by one. The Intervals page counts in natural harmonics, like the table above.
• Pressing too firmly: For a harmonic you touch the string only lightly above the fret. If you press the fret down, you play an ordinary note.
• Taking the harmonic series for equal-tempered: Partials 3 and 5 are pure, the piano tunes them equal-tempered and is thereby off by 2 and 14 cents.