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Why some intervals sound consonant

Simple ratios, shared harmonics and beating explain much of why a fifth sounds smooth and a half step rough. Culture and familiarity shape the rest.

· 7 min read · a drill to try, 4 sounds to play

Play a perfect fifth and a minor second, each with both notes at once. To most Western ears the fifth sounds smooth and settled, and the minor second sounds harsh, as if the two notes were rubbing against each other. Musicians call the first kind of sound consonant and the second dissonant. Open Music Theory defines them by what they seem to want: consonant intervals are “considered more stable, as if they do not need to resolve”, and dissonant ones “less stable, as if they do need to resolve”.

Perfect fifth 3:2

C4 and G4 together: a perfect fifth, 7 semitones.

C4 with G4, then C4 with C♯4, both notes at once. The fifth’s figure closes after a few swings; the minor second’s fills the plate.

Where does the difference come from? Part of the answer is physics: simple frequency ratios, the harmonics inside every note, and the beating between them. Part of it is what you’re used to hearing.

Simple ratios

Every interval has a frequency ratio: how many times the upper note vibrates for each vibration of the lower one. The intervals most often called consonant have the simplest ratios. The octave is 2:1, the fifth 3:2 and the fourth 4:3. The thirds and sixths come next, with slightly bigger numbers: 5:4, 6:5, 5:3. The minor second is 16:15, and the tritone 45:32.

The link between consonance and simple ratios is old. In classical antiquity the ratios were measured as lengths of string, and Pythagorean tuning built its scale from the octave and the fifth. These are the ratios of just intonation. A piano plays all of them but the octave very slightly off, as you’ll see further down.

But a ratio is only arithmetic. Why would your ear care how two frequencies divide?

Every note is a stack of harmonics

A note sung or played on a piano isn’t a single frequency. It’s a fundamental, the pitch you name, plus quieter harmonics at whole-number multiples of it: twice the frequency, three times, four times and on up. That series already holds the simple intervals, in order. From the first harmonic to the second is an octave, from the second to the third a fifth, then a fourth, then a major third, and a minor third between the fifth and sixth harmonics.

Now play two notes at once. When their fundamentals make a simple ratio, many of their harmonics coincide. Take a fifth on 200 and 300 Hz. The lower note’s harmonics run 200, 400, 600, 800, 1,000, 1,200 and up; the upper note’s run 300, 600, 900, 1,200. Every second harmonic of the upper note lands exactly on every third harmonic of the lower one. In an octave the match is complete: every harmonic of the upper note is also a harmonic of the lower note.

Ears tend to group harmonically related frequencies into a single sensation. When two notes share many harmonics, they can blend so well that a listener hears them as one tone, which researchers call fusion.

Beats and roughness

When two frequencies are close but not equal, they interfere. You hear one tone that swells and fades, called beating, at a rate equal to the difference between the two frequencies: tones at 220 and 222 Hz beat twice a second. Slow fluctuations, up to about 20 a second, sound like beating. Faster ones, from about 20 up to somewhere between 75 and 150 a second, blur into a buzzing roughness.

In On the Sensations of Tone, first published in 1863, Hermann von Helmholtz proposed that dissonance is this beating. When two notes are too far apart for their fundamentals to beat, he argued, their upper harmonics still can, so notes that share their low harmonics make fewer beats and sound more consonant. He ranked the consonances in this order: octave, fifth, fourth, major sixth, major third, minor third.

Helmholtz’s six consonances, from most to least consonant in his ranking. The numbers in the ratios grow in the same order, and the figures grow busier with them.

About a century later, R. Plomp and W. J. M. Levelt tested the idea with simple tones, pure sine waves with no harmonics. Their listeners judged a pair of tones consonant once the two frequencies were further apart than a critical bandwidth, the band of frequencies within which the inner ear can’t fully separate two tones. The pairs judged most dissonant were about a quarter of a critical bandwidth apart. For low notes a critical band is about as wide as a minor third; for high notes it’s as narrow as a minor second. With real notes, roughness comes at least in part from harmonics that fall within a critical band of each other.

Minor second 16:15

B3 and C4 together: a minor second, 1 semitone.

B3 with C4, a half step, then C4 with B4, a major seventh: the same two note names, spread apart. Many listeners hear the seventh as less rough. Its fundamentals are no longer within one critical band, though some of its harmonics still are.

Beats are also how instruments get tuned. Musicians listen for them to check a unison or a fifth, and piano tuners count them, aiming at a particular number of beats for each interval. Here is a major third tuned two ways.

C4 and the major third above it. Tuned just, 5:4, the pair is steady. Tuned as a piano tunes it, the major third is 14 cents wider than 5:4, and you hear about 10 beats a second.

Two synthesized tones with harmonics, like a piano’s. Tuned to a just 5:4, their matching harmonics hold steady. In the piano’s tuning those harmonics drift slightly apart, and the drift is the beating you hear.

How a ratio draws a figure

The figures on this page are drawn the way a harmonograph draws. The harmonograph is a machine of pendulums that appeared in the mid-19th century and was most popular in the 1890s. In its simplest form, one pendulum swings a pen along one axis while a second swings the paper along the other. Set the two swinging at an interval’s ratio, and the pen traces a Lissajous figure. When the ratio is made of whole numbers, the figure closes on itself, and the bigger the numbers, the more loops it makes before it does.

So simple ratios close quickly into simple figures, and complicated ones fill the plate. The octave closes almost at once; the minor second and the tritone keep winding. That’s why Earstep draws every interval this way: the figure is a picture of the ratio behind the sound (more on the home page).

Octave 2:1

Two swings of one pendulum against one of the other. On a piano it’s exactly 2:1.

Pick an interval to hear it on the piano and watch the pendulums draw its ratio. The bigger the numbers, the more swings the figure needs before it closes.

A piano doesn’t play these ratios exactly. Equal temperament divides the octave into twelve equal semitones, which leaves every interval except the octave slightly off its just ratio. A ratio that is slightly off makes a Lissajous figure precess, turning slowly instead of standing still. That’s a story for Equal temperament and cents.

Culture and familiarity

Physics doesn’t settle the whole question. Which intervals count as consonant has “varied with milieu”, as Open Music Theory puts it. In medieval music, thirds were treated as dissonances, unusable in a stable final chord, while the fourth was ranked with the consonances until the 15th century. Some traditions seek out the very roughness described above: Bosnian ganga singing, music built on the buzzing drone of the Indian tambura and Indonesian gamelan all treat a slight roughness as an attractive part of the sound.

A direct test came in 2016. Josh McDermott and his colleagues played chords and vocal harmonies to the Tsimane’, a farming and foraging society in the Bolivian Amazon with little exposure to Western music, and to listeners in Bolivian towns and cities and in the United States. The Tsimane’ rated consonant and dissonant chords as equally pleasant, even though they could tell them apart and showed the same dislike of acoustic roughness as Westerners. Town and city dwellers in Bolivia preferred consonance, though less strongly than US residents, and MIT’s report of the study adds that among Americans the preference was bigger in musicians than in nonmusicians. The authors concluded that the preference is most likely shaped by exposure to musical harmony, with culture playing the dominant role.

A 2020 study by Malinda McPherson, Josh McDermott and their colleagues added the other half of the story. When they tested whether people heard two notes as one, both Tsimane’ and Western listeners were more likely to mistake notes in simple ratios for a single sound. But that fusion didn’t predict which intervals anyone liked. The authors read this as a shared way of hearing that combines with culture to produce what we call consonance.

What it means for your ear

For ear training, smoothness is a real cue and a useful first sort. Many listeners hear the octave and the fifth as the most blended, the fourth nearly as much, thirds and sixths in between, and seconds and sevenths as rough. Treat that as something to notice and name, not as a verdict on which sounds are good. Every interval has its uses in music, and the rough ones are worth hearing as clearly as the smooth.

Try it with two intervals played together. The drill plays a major third or a perfect fifth from a different note each time. Many listeners hear the fifth as hollow and the third as fuller and sweeter.

Third or fifth, together?

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