Harmonics: the overtones you already play
Every note you play is a chord in disguise. Inside one key sit dozens of quieter notes, the harmonics, and their balance is what makes a piano sound like a piano. You have been playing them for years.
- Describe a note as a stack of harmonics
- Find the harmonic series of any key on the keyboard
- Hear the harmonics inside a single note
01One note, many frequencies
A sine wave is the only sound with a single frequency. Every other pitched sound, from a piano string to your voice, vibrates at many frequencies at once.
The lowest of them is the fundamental: it sets the pitch you hear. Above it sit the harmonics, at exactly 2, 3, 4, 5… times the fundamental’s frequency. For C2 (65.4 Hz) that is 131 Hz, 196 Hz, 262 Hz, 327 Hz and on upward. The ear doesn’t hear them as separate notes. It fuses them into one sound and hears their balance as its color.
The best way to understand this is to build a sound yourself. Here, KiMiA’s FM engine plays four pure sine waves at 1, 2, 3 and 4 times the note’s frequency, added one at a time:
Building a sound one harmonic at a time
Why whole-number multiples? Deeper theory
A string fixed at both ends can only vibrate in patterns that fit its length exactly: the whole length (the fundamental), two halves (twice the frequency), three thirds (three times), and so on. A column of air in a pipe behaves the same way. That is why harmonics are whole-number multiples: they are the only vibrations the instrument allows, and any repeating waveform can be built from them (Fourier’s theorem).
02The harmonic series on the keyboard
Where do the harmonics of a note fall on the keyboard? Doubling is an octave, three times is an octave and a fifth, four times is two octaves, five times adds a major third. The whole pattern is the harmonic series, and it is the same for every note:
The first 16 harmonics of C2 on the keyboard
| Harmonic | Frequency (C2) | Nearest key | Off by |
|---|---|---|---|
| 1 | 65.4 Hz | C2 | in tune |
| 2 | 130.8 Hz | C3 | in tune |
| 3 | 196.2 Hz | G3 | +2 cents |
| 4 | 261.6 Hz | C4 | in tune |
| 5 | 327 Hz | E4 | −14 cents |
| 6 | 392.4 Hz | G4 | +2 cents |
| 7 | 457.8 Hz | B♭4 | −31 cents |
| 8 | 523.3 Hz | C5 | in tune |
| 9 | 588.7 Hz | D5 | +4 cents |
| 10 | 654.1 Hz | E5 | −14 cents |
The first few harmonics land almost exactly on keys: the 3rd is only 2 cents sharp of G3. Higher up, nature and the keyboard part ways: the 5th harmonic is 14 cents flat of the E, and the 7th is 31 cents flat of the B♭. A piano tuned in equal temperament can only approximate the harmonic series; a brass player or a singer, who can bend pitches, often follows it.
Try this on an acoustic piano. Silently press and hold C4, so its damper lifts without the note sounding. Then strike C2 hard and let go at once. You will hear C4 keep ringing: it is the 4th harmonic of C2, and the string of C4 picks it up. Try holding G3, E4 or G4 instead. Each one rings, because each one is a harmonic of C2. You have just heard the harmonic series with your own hands.
03Hearing the harmonics inside a note
Harmonics are hard to hear on their own because the ear fuses them so well. A filter can separate them. Here one key, C2, is held for eight seconds while a very narrow filter slides up through its spectrum, letting one harmonic through at a time:
The harmonics inside one note, picked out one by one
What you hear is not a melody someone played. It is the inside of a single held note, harmonic by harmonic. Notice how the steps get smaller as you go up: the harmonics sit evenly in hertz, but the ear hears pitch in ratios, so higher harmonics crowd closer together on the keyboard.
The harmonic series explains harmony. The octave (2:1) and the fifth (3:2) sound smooth because their harmonics line up with each other’s. The major triad sits in the series too, as the 4th, 5th and 6th harmonics. When you voice a chord with the wide intervals at the bottom and the close ones at the top, you are following the harmonic series, which is why that voicing sounds so full.
04Try it
- Build a sound. Load 01 · Harmonic Builder. In Edit›FM · layer A, set the Level of Op 2, Op 3 and Op 4 to 0, then bring them back one by one while you hold a note. Listen to the color change at each step.
- Find the harmonics. Load 01 · Harmonic Finder. Hold C2 and move Cutoff slowly upward. Name the notes you hear, and check them against the keyboard figure above.
- Odd only. Load 01 · Odd Harmonics and compare it with the Harmonic Builder at the same note. One has harmonics 1, 2, 3, 4, the other 1, 3, 5, 7. Describe the difference in one word.
- The piano test. If you have an acoustic piano, do the silent-key experiment from this lesson with C2 and four different held keys.