Why chords sound good

01 — Vibration

A note is a pattern that repeats

When a string, a reed, or your vocal cords make a note, they push the air back and forth in a steady rhythm. Your eardrum picks up that rhythm. The number of back-and-forth cycles per second is called the frequency, and you hear it as pitch.

The A below middle C vibrates 220 times a second. Double that, and you hear the same note, only higher. Halve it, and it drops an octave. Keep that doubling in mind, because it's the first clue.

Fig. 1. One fiftieth of a second of a pure tone. Click a note to hear it. Every time the frequency doubles, twice as many cycles fit in the same window.

02 — Ratios

Two notes add up to one new pattern

Play two notes at once and your ear doesn't get two separate waves. The air can only move one way at a time, so it gets their sum: one combined wave.

What that combined wave looks like depends on the ratio between the two frequencies. If the high note vibrates exactly 3 times for every 2 of the low note, their peaks line up again and again, and the sum settles into a short, tidy loop. If the ratio is messy, the peaks rarely meet, and the pattern takes a long time to repeat.

Fig. 2. Eight vibrations of a low note (220 Hz) and a higher one. Dots mark each wave's peaks. Dotted lines mark the moments both peak together, where the shared pattern starts again.

People noticed this more than 2,500 years ago. Pythagoras's followers found that strings whose lengths form simple ratios (1:2, 2:3, 3:4) sound pleasing together. That's the octave, the fifth and the fourth, still the backbone of most music.

But "simple numbers" is a description, not a reason. Your ear doesn't count. To see why simple ratios sound smooth, we have to look at what happens when two frequencies come close to each other without quite matching.

03 — Roughness

Close frequencies rub against each other

Play two pure tones that are almost the same and something odd happens. Their peaks drift in and out of step, so the sound swells and fades. Musicians call this beating; it's how guitarists tune by ear.

Now pull them further apart. Once the wobble passes about fifteen times a second, you stop hearing a pulse. It turns into a grainy, buzzing texture called roughness. Pull further still and the buzz clears: you hear two clean, separate notes.

Gap between the two tones 3 Hz

Fig. 3. Top: the loudness of a 220 Hz tone plus a second tone, over one second. Bottom: how rough the pair sounds as the gap grows. Roughness peaks at around 20 Hz apart for notes in this range, then fades. Press Listen and drag the slider.

Why does this happen? Your inner ear sorts sound by frequency along a coiled membrane, a bit like a piano keyboard laid out in flesh. Two frequencies close enough to excite the same patch can't be told apart, so they interfere. That patch is called a critical band. Roughness is the sensation of two tones fighting over the same stretch of keyboard.

04 — Harmonics

Every note is secretly a chord

Here's the piece that ties it all together. Real notes aren't pure tones. A plucked string vibrates as a whole, and at the same time in halves, thirds, quarters, and so on. Each of those adds a quieter, higher frequency at exactly 2×, 3×, 4×… the main one.

These extra frequencies are called harmonics. Your brain fuses them into a single note. They're why a piano and a clarinet playing the same A sound different: the same ladder, with different rungs emphasised.

One note, A at 220 Hz, split into its first six harmonics

Fig. 4a. A note's harmonics sit at whole-number multiples of its frequency, and each one is a little quieter than the last. Together they sound like one note.

So two notes played together are really two ladders of harmonics. That changes the question. Wherever a rung from one ladder lands exactly on a rung of the other, the two reinforce each other and nothing clashes. Wherever rungs land close but not equal, they rub, and we're back to the roughness from Fig. 3.

Fig. 4b. The low note's harmonics and the high note's, on a shared frequency scale. The bottom strip shows them as your ear hears them: all at once.

Simple ratios make the ladders interlock. That's the whole secret.

With a fifth (3:2), every third rung of the low ladder lands exactly on every second rung of the high one. The rest are spaced well apart. With a semitone (16:15), almost no rungs match, and nearly every pair lands in each other's critical band. The ratio isn't magic. It just predicts how cleanly the harmonics fit together.

05 — The map

Add up the friction for every interval

If you total up the rubbing between every pair of harmonics, for every possible gap between two notes, you get a map. In the 1960s the researchers Reinier Plomp and Willem Levelt measured how rough people found pairs of pure tones. Run their data through the harmonic ladders and this curve falls out.

The valleys land exactly where musicians have always placed their favourite intervals.

Fig. 5. Predicted roughness of two notes, from unison (left) to an octave apart (right), for notes with six harmonics. Valleys are consonant intervals. Hover to explore and click anywhere on the curve to hear that interval. Ticks along the bottom mark the twelve piano semitones.

View as a table

Interval	Ratio	Semitones	Roughness (0–100)

The best evidence that harmonics are behind this: change the harmonics and the valleys move. Bells, gongs and metal bars have harmonics that aren't neat whole-number multiples. Music made for them, like Indonesian gamelan, often uses intervals that would sound "off" on a piano but blend beautifully on those instruments.

06 — Chords

A chord is three ladders that fit

A chord is the same idea with more ladders. The major chord is the textbook example: its three notes vibrate in the ratio 4 : 5 : 6. That means they are exactly the 4th, 5th and 6th harmonics of one lower note that isn't even being played.

So a major chord isn't three strangers. It's three neighbouring rungs from a single ladder. Their harmonics interlock, the friction stays low, and your brain can hear the whole thing as one rich, unified sound. Researchers call this property harmonicity. In listening experiments it's one of the strongest predictors of which chords people like.

Harmonicity also fills a gap that roughness leaves. In this model a tritone rubs about as much as a minor third, yet it sounds far more unsettled. The difference is that its notes share no simple ladder (45 : 32), so the brain can't fuse them into one sound.

Try the presets, or play your own three notes on the keyboard.

Closest simple ratio

4 : 5 : 6

Each pair of notes

Fig. 6. Chords are played in piano tuning. Roughness is measured against a tight cluster of neighbouring piano keys in the same spot. Try a chord low on the keyboard and then high: the same chord gets muddier lower down, because low harmonics crowd into the same critical bands.

The minor chord is a nice contrast. Its friction is nearly as low as the major chord's, but its notes fit a more distant common ladder (10 : 12 : 15). It's smooth but less fused, which is part of what gives it its darker, more complex colour. The diminished chord's numbers balloon, and so does its tension.

07 — The other half

Physics sets the stage. People write the play.

Everything so far explains why some combinations are smooth and others rough. Whether smooth sounds good is a different question, and the answer is more interesting.

Your piano is slightly out of tune, on purpose. A perfect 5:4 major third only works in one key. To make every key usable, modern instruments split the octave into twelve exactly equal steps. Fifths stay almost perfect. Thirds come out a little wide, so two of their harmonics beat gently about ten times a second. We've grown so used to that shimmer that most of us never notice it.

Pure major third · 5 : 4

The 5th harmonic of C and the 4th harmonic of E land on the same frequency: 1308 Hz.

Piano major third · 2^4/12

They miss by about 10 Hz, so where they overlap the sound swells and fades about ten times a second.

Fig. 7. Loudness, over one second, of the two harmonics where C and E overlap. Listen closely to the upper part of the sound for the slow shimmer in the piano version.

Tension is a feature, not a flaw. Music needs friction to go anywhere. A slightly rough chord creates a sense of leaning, an expectation, and the smooth chord after it feels like arriving home. Nearly every pop song, hymn and symphony runs on this pull.

G7, a chord with a built-in tritone, resolving to C major

And taste is learned. In 2016, a team led by Josh McDermott played consonant and dissonant chords to the Tsimane', a community in the Bolivian Amazon with little exposure to Western music. They could tell smooth chords from rough ones, but they didn't prefer one over the other. Their preferences tracked what music they knew. The physics tells you which notes blend. Culture and experience decide how you feel about it.

In one breath