The Harmonic Series and Overtones: Why Notes Have Timbre
Every musical note contains a stack of overtones at whole-number multiples of its frequency. Learn the harmonic series and how it shapes timbre and intervals.
The harmonic series is the set of frequencies that are whole-number multiples of a fundamental frequency. If a note has a fundamental of 110 hertz, its harmonics lie at 220, 330, 440, 550 hertz, and so on. A real musical note is never one pure frequency: it is a blend of the fundamental and many of these harmonics, called overtones. The relative loudness of the harmonics gives each instrument its tone colour, or timbre, and the intervals between neighbouring harmonics are the source of the octave, the fifth, and the major third in music.
What is a harmonic?
A vibrating string or air column can vibrate in several patterns at once. A string fixed at both ends can vibrate along its whole length, which gives the lowest frequency, the fundamental. It can also vibrate in two halves at twice the frequency, in three thirds at three times the frequency, and so on. Each of these standing-wave patterns is a harmonic. The terminology is a common source of confusion: the fundamental is the first harmonic, and the first overtone is the second harmonic.
| Harmonic | Frequency (fundamental 110 Hz) | Interval above the fundamental | Nearest note |
|---|---|---|---|
| 1 | 110 Hz | unison | A2 |
| 2 | 220 Hz | octave | A3 |
| 3 | 330 Hz | octave and a perfect fifth | E4 |
| 4 | 440 Hz | two octaves | A4 |
| 5 | 550 Hz | two octaves and a major third | C#5 |
| 6 | 660 Hz | two octaves and a perfect fifth | E5 |
| 7 | 770 Hz | two octaves and a minor seventh | G5, noticeably flat |
| 8 | 880 Hz | three octaves | A5 |
Why do instruments sound different?
A flute and a violin can play exactly the same pitch, because they share the same fundamental, but they sound different because they emphasise different harmonics. A flute is close to pure, with weak upper harmonics. A clarinet, which behaves like a pipe closed at one end, strongly favours the odd-numbered harmonics, which gives it a hollow, woody quality. A violin or a trumpet has many strong upper harmonics and sounds bright. The way a note starts and fades, called its envelope, matters too. The mathematician Joseph Fourier showed in the early 1800s that any repeating wave can be built from sine waves of harmonic frequencies, which is the basis of additive synthesis in electronic music. Human vowels work the same way: the shape of the mouth boosts certain groups of harmonics, called formants, and those boosts are what we hear as different vowels (see the International Phonetic Alphabet).
How does the harmonic series create musical intervals?
The ratios between the early harmonics are the basic intervals of music: 2 to 1 is the octave, 3 to 2 the perfect fifth, 4 to 3 the perfect fourth, 5 to 4 the major third, and 6 to 5 the minor third. Pythagoras and his followers connected such ratios to string lengths in the sixth century BC. A major chord is built directly from the series, because the fourth, fifth, and sixth harmonics are in the ratio 4 to 5 to 6, which is a major triad (see major and minor scales).
The series also explains a famous problem in tuning. Stacking pure 3 to 2 fifths is how the circle of fifths is built, but twelve pure fifths overshoot seven octaves by a small gap known as the Pythagorean comma, about a quarter of a semitone. Equal temperament, the tuning of the modern piano, spreads that gap evenly, so every fifth is very slightly narrow and every chord is very slightly impure, a trade made so that every key can be played.
Brass instruments and the natural harmonic series
A bugle with no valves can play only the notes of one harmonic series, which is why bugle calls such as Reveille and Taps use only a handful of pitches. The player picks a harmonic by changing the tension of the lips, and the valves and slides on other brass instruments change the length of the tube to switch to a different series. String players can sound harmonics by lightly touching the string at its midpoint or one third of the way along, which damps the fundamental and leaves a pure, flute-like tone.
What is the missing fundamental?
The brain does not need to hear the fundamental to perceive a pitch. If you play only the harmonics at 220, 330, and 440 hertz, listeners still hear a note at 110 hertz, because the spacing between the harmonics implies it. This is the missing fundamental, or residue pitch. It explains why small phone and laptop speakers, which cannot reproduce deep bass (see how loudspeakers work), can still seem to play bass notes: they produce the harmonics, and the brain fills in the rest.
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acoustics harmonics music theory overtones sound