Why Two Notes Fight

Why do some pairs of notes sound smooth together and others grate?

Half of the answer is physics, and it has been understood since Helmholtz published it in 1863.

Every note carries its harmonic series. When two notes stand in a simple ratio (2:1 for an octave, 3:2 for a fifth) a great many of their harmonics land on precisely the same frequencies, and the rest are comfortably far apart. When the ratio is complex, harmonics fall close to each other without coinciding.

Two frequencies a few hertz apart produce beating: a pulsing as they drift in and out of phase. Slow beating is the sound a piano tuner listens for. Faster beating, in the region of twenty to thirty times a second, is not heard as pulsing at all but as a grating quality: roughness.

So consonance is the absence of roughness, and it depends on the spectrum as much as on the interval. The same interval sounds rougher low down, where the ear's critical bands are proportionally wider, which is why a close-voiced chord in the bass sounds muddy and the same chord an octave up is clear. Arrangers know this as a rule of thumb; it is a fact about the cochlea.

That is the physical half, and it is real and measurable and does not vary between cultures.

The other half is the threshold (how much roughness a musical culture treats as acceptable, and where it may occur) and that has moved continuously.

Early medieval theory treated the third as a dissonance. The third is the basis of all subsequent Western harmony. Sevenths required careful preparation and resolution, and then became the ordinary colour of jazz. Each admission was fought over by people who could hear perfectly well and disagreed about what was permissible.

The best example is the tritone. Medieval theorists called it diabolus in musica, the devil in music, which sounds like a prohibition and was really a teaching mnemonic: it is hard to sing and awkward to resolve. The story that the Church banned it is later invention.

That same interval is the flattened fifth of the blues, the axis of jazz chord substitution, and the standard cue for menace in film scoring. The physics has not altered by a single hertz. The listeners have.

There is a stronger demonstration that consonance is a relationship rather than a property.

Gamelan instruments are tuned metallophones (struck bars and gongs) and their overtones are not whole-number multiples of a fundamental. The spectrum is inharmonic. So the intervals that minimise roughness for those instruments are not the ones that minimise it for strings and pipes, and gamelan tuning systems are internally consistent by a different logic.

Play a gamelan scale on a piano and it sounds out of tune. Play Western intervals on gamelan instruments and they sound out of tune. Neither is wrong; consonance is a relationship between a spectrum and an interval, and the instruments carry different spectra.

Which suggests an experiment nobody has quite settled. If you built instruments with deliberately stretched harmonics, the consonant intervals would be stretched to match, and listeners do adapt to such systems, at least partially, within an experimental session.

The physics sets what is rough. Everything about which roughness is meaningful, expressive or unbearable is learned, and by the time a device has become the ordinary furniture of advertising music, it is genuinely difficult to reconstruct what the fuss was about.