Why a Fifth Sounds Sweet

For two and a half thousand years the story was Pythagoras's: a fifth sounds sweet because its string lengths are in the ratio 3:2, and small whole numbers are pleasing to the soul. It is a beautiful story and it explains nothing — why should the soul care about small integers?

In 1965 Reinier Plomp and Willem Levelt gave the answer the Greeks couldn't: the ear has a resolution limit. Two pure tones close in frequency beat — a slow throb you can count. Push them apart and the beats speed up until, somewhere around a tenth of a second apart in pitch, they fuse into a rough, grating buzz. Push further and the roughness dissolves into two clean, separate tones. The whole drama happens inside one critical band — the width of a single filter on the basilar membrane. Roughness is two partials fighting over one filter.

Everything below is computed live in your browser from the published model. Nothing is prerecorded. Turn your volume to a comfortable level and press a play button.

That was a single pair of pure tones — but no real instrument is a pure tone. A struck string, a bowed string, a sung vowel, a reed: each is a stack of harmonics, partials at 1×, 2×, 3×… the fundamental. When you play two such tones together, every partial of the first can beat against every partial of the second. The total roughness is the sum over all those pairs — and now something remarkable happens.

Slide one harmonic tone past another and total roughness rises and falls. Its valleys — the intervals where the fewest partials are caught fighting in one critical band — fall almost exactly on the ratios the Greeks revered. Not because the numbers are magic. Because at 3:2 the upper tone's partials land on top of the lower tone's, sharing filters instead of crowding them.

This is William Sethares's 1993 result, and it carries a sting in its tail. The valleys are not a property of the ratios — they are a property of the spectrum. The just intervals are special only because harmonic tones have partials at integer multiples, so those partials coincide at small-integer ratios.

Change the spectrum and the consonant intervals change with it. Stretch the partials apart — as a stiff piano string or a struck metal bar genuinely does — and the deepest valley walks off the 3:2 fifth and off the 2:1 octave entirely. Pick the “stretched” timbre above and drag toward the right edge: the consonant “octave” for that sound is near 2.1, not 2.0. The Indonesian gamelan, tuned to the inharmonic spectra of bronze, lands on scales no Western keyboard contains — and is in perfect tune with its own timbre. Consonance is timbre × interval, never the integer alone.

Below, the page re-runs the model in front of you and reports what it finds — including the parts where roughness is not the whole story.

Where the model stops being the whole truth

Roughness is real and measurable, but it is only one ingredient of consonance — and naming the rest honestly is the point of this room. A second ingredient is harmonicity: how well two tones together fuse into a single harmonic series. McDermott, Lehr and Oxenham (2010) tested over 250 listeners and found that liking for consonant chords tracked a preference for harmonic spectra — and not, for most people, an aversion to beating. Roughness alone may not be the lever it looks like here.

And the deepest caveat is cultural. McDermott and colleagues (2016) tested the Tsimané, an Amazonian people with little exposure to Western music, and found they rated consonant and dissonant chords as equally pleasant — while still disliking acoustic roughness. The roughness in this instrument is a genuine, near-universal sensation. The judgement that smooth is better is, at least in part, something a culture teaches. The curve below is true about the cochlea. It is not the last word on beauty, and it does not pretend to be.