Consonance and Dissonance: The Physics of Rough and Smooth

Two competing explanations, one clean experiment that separates them, and the cross-cultural result that complicates both.

Short answer: when two notes sound together, their partials either fall far enough apart to be heard separately or close enough to interfere. Interference inside a single auditory filter produces roughness, and roughness is the classic explanation for dissonance. But when researchers tested which acoustic factor actually predicts what people like, the winner was not the absence of roughness; it was the presence of a harmonic spectrum. And a study in an Amazonian society with little exposure to Western music found no preference for consonance at all, which means part of what we call dissonance is a habit rather than a fact about ears.

Two different questions

Consonance gets used for two different things, and the argument is much shorter once you separate them.

Sensory consonance is about the sound itself, independent of any musical context: is this combination smooth or rough? Play two sine tones a semitone apart and the roughness you hear does not depend on knowing anything about keys. Musical consonance is about function: is this combination stable, does it want to resolve? A dominant seventh chord is sensorially quite smooth and musically very unstable. A tritone in one style is a crisis; in another it is a colour.

This page is mostly about the first. The second is a matter of style and training, and it changes across centuries and traditions in ways no acoustic measurement will predict.

The classic explanation: critical bands and roughness

The cochlea behaves like a bank of overlapping filters. Two frequency components that land inside the same filter are not resolved separately; they interact, and their sum fluctuates in amplitude. If the two are within a few hertz you hear that fluctuation as slow beating. Push them further apart and the fluctuation speeds up until, somewhere above roughly fifteen to twenty pulses per second, you stop counting and start hearing a harsh, buzzy quality. That is roughness.

The width of the filter is the critical band. Below about 500 Hz it stays roughly 100 Hz wide; above that it is approximately 20 percent of the centre frequency. Two tones separated by more than a critical band are resolved cleanly and the roughness disappears.

Plomp and Levelt turned this into a quantitative account in 1965. Testing pairs of simple tones, they found the transition between consonant and dissonant judgements tracked the critical band as a function of frequency, with maximum dissonance at roughly a quarter of the critical bandwidth and smoothness returning once the separation exceeded it. Extend that to complex tones, where each note contributes a whole series of partials, and you can predict the classic ranking of intervals: the more partial pairs that land in the rough zone, the more dissonant the dyad.

Why that predicts the interval ranking

Take two complex tones an octave apart. Every partial of the upper note lands exactly on a partial of the lower one, so nothing is close-but-not-equal, and there is nothing to beat. A perfect fifth in just intonation aligns every third partial of the lower note with every second of the upper. A major third aligns fewer of them, and further up the series where they are weaker. A minor second aligns almost nothing and puts many pairs squarely inside one critical band.

Two practical consequences follow immediately. Register matters. A major third near 100 Hz spans only about 25 Hz, which sits inside one critical band and therefore sounds muddy; the same third two octaves higher spans four times the frequency distance and is resolved cleanly. That is the acoustic reason arrangers open out low chords and keep close voicings in the middle register. Timbre matters. Roughness is generated by partials, so a nearly pure tone produces very little of it, while a partial-rich tone produces a lot. The same written interval is a different sound on a flute and on a distorted guitar, which is a point our page on timbre develops further.

The complication: roughness is not what people prefer

Roughness theory explains what sounds rough. It turns out to explain less well what sounds good. In an experiment designed to pull the factors apart, listeners rated musical chords and also non-musical sounds that isolated either beating or harmonic spectra. People disliked beating and liked harmonic spectra, as expected. But only the preference for harmonic spectra was consistently correlated, across individuals, with preferring consonant chords over dissonant ones. The aversion to beating was not.

That is the value of individual differences as a method: if two candidate causes always travel together in the average listener, look for people in whom they come apart. The suggestion is that consonance is largely about whether a combination resembles a single harmonic sound, the kind of spectrum a voice or a resonant instrument produces, rather than about the mere absence of interference.

A study of listeners with congenital amusia pushed the same wedge in from the other side. People with amusia showed no preference for consonant over dissonant sounds, yet detected beating and roughness normally. If roughness detection were sufficient for the pleasure of consonance, they should have shown the usual preference. They did not, which points again at pitch and harmonicity processing as the load-bearing part.

The bigger complication: it is partly learned

Then there is culture. Researchers tested the Tsimane, an Amazonian society with minimal exposure to Western music, alongside Bolivian town dwellers, Bolivian city dwellers and US listeners. The Tsimane rated consonant and dissonant chords and vocal harmonies as equally pleasant. The other groups preferred consonance, and did so more strongly the more exposure they had. All groups discriminated the sounds perfectly well and shared other basic preferences, such as disliking harsh synthetic timbres.

The conclusion is careful and worth stating precisely. The perceptual machinery that separates rough from smooth appears to be shared. The preference for smooth over rough, in the specific case of musical chords, tracks exposure to polyphonic music. Dissonance is not an assault on the ear. It is a category we are trained into, built on a real acoustic distinction.

What to listen for, in practice

Once you know the mechanism you can hear it deliberately rather than vaguely. Sustain a major third on a rich instrument and listen for the shimmer sitting on top of the sound rather than for the chord as a whole; that shimmer is a pair of partials beating. Play the same third two octaves lower and hear it thicken. Play it on a flute-like tone and hear the roughness mostly disappear. Detune one note by a few cents and hear the shimmer speed up. None of this requires naming chords. It requires attending to the surface texture of a sustained sound, which is a trainable habit and one most listeners have never been asked to develop.

Our games work with single tones rather than chords, so they train the ingredients rather than the finished dish. Heartween asks you to judge one pitch against two others, which is the relational listening you need before roughness cues can mean anything. Hearddle gives you repeated attempts at a target with feedback, which is where you learn what a few cents of mistuning actually feels like, and a few cents of mistuning is exactly what turns a locked interval into a beating one.

Keep reading

• Beats and tuning by ear: the slow end of the same interference, put to work.
• Equal temperament versus just intonation: why a tempered third never quite locks.
• What timbre is: why the same interval is smooth on one instrument and harsh on another.
• Musical intervals explained: the names attached to all these ratios.

Sources: Plomp & Levelt (1965), Tonal consonance and critical bandwidth, Journal of the Acoustical Society of America 38(4); McDermott, Lehr & Oxenham (2010), Individual differences reveal the basis of consonance, Current Biology 20(11); Cousineau, McDermott & Peretz (2012), The basis of musical consonance as revealed by congenital amusia, PNAS 109(48); McDermott, Schultz, Undurraga & Godoy (2016), Indifference to dissonance in native Amazonians reveals cultural variation in music perception, Nature 535; Truax (ed.), Critical band and Beats, Handbook for Acoustic Ecology, Simon Fraser University; Moore, An Introduction to the Psychology of Hearing, Routledge; Promenade around the cochlea, psychoacoustics of pitch.