Consonance is not a fact about the integers. It is a fact about the ear, and two sliding tones let you hear one ingredient: roughness.
verify-audio.mjs) calculates roughness from the rendered audio's spectrum and finds a Pearson correlation of 0.943 with the model curve. The screen combines verifier quotations with explanatory prose and the page's frontier statement. Its ALL PASS and 24 ok, 0 fail record the June 21 run; Plomp–Levelt 1965 / Sethares 1993 / Glasberg–Moore 1990 are cited, not measured. Source: research/critical-band-consonance/film.
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. The Pythagoreans linked musical harmony to number. 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 20 or 30 beats per second, 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 most musical instruments have richer spectra. A struck string, a bowed string, a sung vowel, a reed: each can be approximated by 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, with scale steps at or near roughness minima for 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.
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 preference for consonant chords appears to depend in part on musical culture. The curve below is true about the cochlea. It is not the last word on beauty, and it does not pretend to be.
Corrected 2026-10-14: The page and film attributed a cultural difference to preference for smoothness. McDermott et al. (2016) found differences in consonance preference while all tested groups preferred smooth tones to rough ones (found by the assay line, 2026-10-14). The page text and film source are corrected; the published film still carries the old claim.
Corrected 2026-10-14: The film caption said every word on screen came verbatim from the verifier. The film combines verifier quotations with explanatory prose and a frontier statement taken from the page's dek (found by the assay line, 2026-10-14).
Corrected 2026-10-14: The page described tones as 'a tenth of a second apart in pitch'. Pitch separation is not measured in seconds; Sethares describes the change from beats to roughness around 20 or 30 beats per second (found by the assay line, 2026-10-14).
Corrected 2026-10-14: The Deep Scales relation said the diatonic scale's intervals were the roughness valleys of a harmonic timbre. This seven-partial model has minima near several prominent consonant intervals, not at every diatonic step (found by the assay line, 2026-10-14).
Corrected 2026-10-14: The Magenta relation described consonance largely as absence of roughness. Smoothness is the sensory contribution modeled here; harmonicity also matters to consonance, and McDermott et al. (2010) found that only harmonicity preference consistently tracked consonant-chord preference (found by the assay line, 2026-10-14).
Corrected 2026-10-14: The film metadata gave 2026-07-09 as the upload date. The shipping log and first film commit record publication on 2026-06-21 (found by the assay line, 2026-10-14).
Corrected 2026-10-14: The film calls the fifth the deepest valley in the whole curve, and the live check omitted 'interior'. The verifier identifies the fifth as the deepest interior minimum for 1.02 < r < 1.98; the octave and unison have lower modeled roughness (found by the assay line, 2026-10-14). The film source is corrected; the published film still carries the old headline.
Updated 2026-10-14: The footer's 24-check total was recorded on 2026-06-21. The suite has 26 checks after the 2026-09-26 page checks were added; all 26 pass in this run (found by the assay line, 2026-10-14). The film retains its June 21 result.
Written 2026-06-21. Claims re-read against their sources on 2026-10-14: 80 checked, 61 confirmed, 17 wrong, 0 unverifiable, 2 true when written and overtaken since (dated, not corrected). By the assay line (research/assay-line/queue-1014c/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex.
This was the directed queue-1014c repair pass for critical-band-consonance, read at c8289bd340. I re-read the evidence for all 19 triage items before editing and retained the 61 CONFIRMED source comparisons from the Codex checker's N2 record, with their stated limits. The 17 wrong-count entries comprise nine WRONG and eight MINOR claims; the two STALE entries are separate. The page is authored directly in its HTML and markdown, with no layer page generator. Film source corrections are made, but the unchanged published film and two relations preserved in another layer still require follow-up. Search copies are rebuilt at landing from the corrected markdown. Pending note placeholders are explicit and must not be mistaken for a passing claims gate.
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