Press middle C and the E above it on a piano and the result sounds bright and settled. Now press C with the C sharp right next to it. Same instrument, same loudness, but the second pair sounds sour, almost like it's buzzing. Nothing about the individual notes is wrong. The clash only exists when they sound together. So where does it come from?
When a string or a column of air vibrates, it doesn't just wobble at one frequency. It also vibrates in halves, thirds, quarters and so on, all at once. The result is a stack of frequencies at whole-number multiples of the lowest one: if the note is 220 Hz, you also get 440, 660, 880, 1100 Hz and upward. These are called harmonics (or overtones). You rarely hear them as separate pitches; your brain fuses them into one note and a sense of tone colour. That's why a violin and a flute playing the same pitch still sound different: the heights of their harmonic stacks differ.
So when you play two notes, you're really playing two ladders of frequencies, and every rung of one ladder can interact with every rung of the other.
Put two pure tones close together in frequency and they drift in and out of step. When their peaks line up they add; half a cycle later they cancel. You hear this as a pulsing loudness called beating, and the pulse rate is simply the difference between the two frequencies. Tones 2 Hz apart throb twice a second, which guitarists use to tune strings by ear.
Push the frequencies further apart and the throb speeds up until you can no longer follow it. It turns into a grainy, rattling quality called roughness. Push further still and the roughness fades: the two tones separate into two clean, distinct pitches.
In 1965, Reinier Plomp and Willem Levelt measured this carefully by asking listeners to rate pairs of pure sine tones. The unpleasantness depended mainly on how close the tones were relative to the ear's critical bandwidth, the frequency range over which the inner ear can't cleanly separate two sounds. Tones well inside one band clashed; tones more than a band apart sounded fine. Crucially, for pure tones there was nothing special about a perfect fifth or a major third. Any wide-enough gap sounded smooth.
The sweetness of a fifth isn't in the two notes. It's in their overtones.
Below, the lower note is fixed at A (220 Hz). Drag across the picture, or tap an interval, to move the upper note. The top half shows both harmonic ladders on a frequency axis; red zig-zags connect harmonics close enough to rub against each other. The bottom half is a roughness curve across a whole octave, computed from Plomp and Levelt's data using a model by the music theorist William Sethares. Then switch to pure tones and watch the valleys disappear.
Look at where the valleys fall: an octave (frequency ratio 2:1), a fifth (3:2), a fourth (4:3), a major third (5:4), a major sixth (5:3). When two notes are in a simple ratio, many of their harmonics land on exactly the same frequencies. Coinciding harmonics can't beat, and the ones that don't coincide are mostly far apart. Nudge the interval a little off the ratio and those near-matches start to throb, which is why a slightly out-of-tune fifth sounds worse than a clearly different interval.
A semitone (roughly 16:15) is the opposite case. The two fundamentals themselves sit inside one critical band, and harmonic after harmonic lands just close enough to rattle. That's the buzz you hear from C and C sharp: their fundamentals alone beat about 16 times a second.
This idea goes back to Hermann von Helmholtz, whose 1863 book On the Sensations of Tone argued that dissonance is the roughness of beating overtones. Plomp and Levelt's experiments a century later put numbers on it.
A modern piano doesn't use pure ratios. It uses equal temperament, which divides the octave into 12 identical steps so music can move freely between keys. That makes some intervals slightly off their ideal ratios:
| Interval | Pure ratio | Piano (equal) | Beat near middle C |
|---|---|---|---|
| Fifth C–G | 702.0 cents | 700 cents | ≈ 0.9 per second |
| Major third C–E | 386.3 cents | 400 cents | ≈ 10 per second |
(A cent is a hundredth of a semitone.) The fifth is only 2 cents narrow, so the third harmonic of C and the second harmonic of G drift past each other less than once a second, a slow shimmer. The piano's major third is almost 14 cents wide, so the fifth harmonic of C and the fourth of E beat about ten times a second. Listen closely to a sustained C–E on a piano and you can hear that faint flutter. Choirs and string quartets, who can adjust pitch freely, often drift toward the purer third.
The roughness itself is physics and physiology: beating harmonics happen in any ear. But whether people dislike it is partly learned. In a 2016 study in Nature, Josh McDermott and colleagues found that Tsimane' people in the Bolivian Amazon, who at the time had little exposure to Western music, rated consonant and dissonant chords as about equally pleasant, even though, like Western listeners, they disliked acoustic roughness in other tests. A 2025 follow-up from the same lab found that preferences for consonance showed up in Tsimane' communities with more contact with the wider world.
So the clash is real, and you can see it in the ladders above. What your brain makes of that clash depends a lot on the music you grew up with. Plenty of styles, from blues to Balkan singing, lean into the buzz on purpose.