The Harmonic Series: Why One String Sounds Like Many
Pluck a single guitar string, or press a single piano key, and you would say you heard one note. You heard six or more. The extra ones are quiet, they are all above the note you asked for, and your ear folds them into a single sound rather than reporting them separately — but they are there, they are measurable, and once you know what they are, several things about music that look arbitrary stop looking arbitrary.
A string vibrates in several ways at once
A stretched string held at both ends can swing as one long arc, and that slowest motion is the note you name — the fundamental. But nothing prevents it from also vibrating in two halves at the same time, with a still point in the middle, or in three parts, or in four. It does all of them simultaneously, and each of those faster motions makes its own sound.
Those extra tones are the harmonics, and the whole stack is the harmonic series. What makes the series worth learning is that the frequencies are not scattered: the half-length motion vibrates exactly twice as fast as the whole, the third exactly three times as fast, the quarter four times. Whole-number multiples, no exceptions, all the way up.
Which notes come out, and in what order
Because the multipliers are whole numbers, the notes are always the same relative to the fundamental, whatever note you start on. Here is the bottom of the series over the guitar’s open fifth string, A at 110 Hz:
| Partial | Frequency | Note | Distance above the fundamental |
|---|---|---|---|
| 1 | 110 Hz | A | — it is the fundamental |
| 2 | 220 Hz | A | one octave |
| 3 | 330 Hz | E | an octave and a fifth |
| 4 | 440 Hz | A | two octaves |
| 5 | 550 Hz | C♯ | two octaves and a major third |
| 6 | 660 Hz | E | two octaves and a fifth |
Read the first few and something should jump out. The series hands you, in order and unasked: the octave, then the fifth, then the major third. Those are exactly the notes of a major chord, and they arrive in that order out of the physics of a single string, before anyone has decided anything.
Two honest caveats, because the table is tidier than the world. The higher partials drift away from the notes on a keyboard — the fifth partial is about a seventh of a semitone flat of the C♯ you would play, and the seventh partial, left out above, is so flat of G that no key on the instrument is really it. And the intervals your instrument is actually tuned to are a hair off these ratios on purpose, for reasons that belong to a much later lesson.
Hearing it
You do not have to take any of this on faith, and there are two ways to check it today.
If you have a guitar or a bass, this takes one finger. Rest a fingertip lightly on the fifth string directly over the twelfth fret — touching the string, not pressing it down to the wood — pluck, and lift the finger:
What rings is a thin, bell-like A an octave above the open string. Your finger sat at the exact midpoint and stopped the whole-length motion, leaving the halves-motion — the second partial — to sound on its own. It was already there before you touched it. The same trick works over the seventh fret, where you get the third partial, and over the fifth fret for the fourth.
The other way is to listen to intervals on this site. Every note you hear in these exercises is built rather than recorded, and the piano in particular is assembled from as many as twenty partials at once, each one decaying a little faster than the one below it — which is a harmonic series, put together on purpose because that is what a piano is. The ear training lesson plays intervals for you to name, and an octave and a fifth are two of the first things it asks for.
Why some intervals sound consonant
Now the payoff, and it is the reason this lesson sits in the theory module rather than a physics one.
Play two notes an octave apart, A at 110 and A at 220. The upper note’s partials are 220, 440, 660 — every one of which is already in the lower note’s series. Nothing new arrives and nothing collides. That is about as much agreement as two different notes can have, which is why the octave is the one interval every musical culture appears to have found.
Now the fifth, A at 110 against E at 165. The E contributes 165, 330, 495, 660; the A already had 330 and 660. Fewer shared partials than the octave, still a lot, and the ones that are not shared sit comfortably far apart. It sounds stable and open, and it is the interval the intervals lesson called a stable frame.
Now two notes a semitone apart, 110 and about 116.5. Their partials come out 110 against 116.5, 220 against 233, 330 against 350 — pairs sitting a few hertz apart, all the way up the stack. Two tones that close do not blend; they beat against each other, and a great many of those beats at once is heard as roughness. That roughness is most of what the word dissonant is pointing at.
The rule of thumb that falls out: the simpler the frequency ratio between two notes, the more partials they share, and the smoother they sound. The octave is 2:1, the fifth 3:2, the major third 5:4, and a semitone is a ratio with nothing simple about it.
This is the standard physical account, and it explains a great deal without explaining everything — what a listener has grown up hearing decides a lot too, and intervals that one era treated as unusable are ordinary now. Take it as the reason consonance is not arbitrary, not as a ranking of which chords are allowed.
Common mistakes to avoid
- Counting overtones and partials the same way. The fundamental is the first partial but not an overtone at all, so the first overtone is the second partial. Two systems, one off by one; say which you mean.
- Expecting to hear the harmonics as separate notes. You do not, and that is the point — your ear fuses them into one sound with a colour. The previous lesson called that colour timbre, and this is where it comes from.
- Reading the series as the origin of the major scale. It gives you an octave, a fifth and a major third, and that is genuinely suggestive. It does not hand over seven notes, and scales that do not match it are not mistakes.