Intervals and ratios: why does 3:2 sound consonant?
An interval is a ratio
If the lower sound is 220 Hz and the higher one 330 Hz, the ratio is 330 ∶ 220 = 3:2, and this interval is called a perfect fifth. The ratio does not depend on pitch: 100 and 150 Hz is a fifth too, and so is 440 and 660 Hz. The octave is 2:1, the perfect fourth 4:3, the major third 5:4 and the minor second 16:15.
The Intervals screen sets the twelve ratios side by side and shows how simple each one is with a single number: log₂(n·d). For the fifth it is 2.58, for the minor second 7.91. The smaller the number, the simpler the ratio.
When harmonics coincide
A string, a voice or a wind instrument does not produce a single frequency: alongside the fundamental you also hear harmonics at 2, 3, 4… times it. The harmonics of a 220 Hz sound are 440, 660, 880, 1100…; those of a 330 Hz sound are 660, 990, 1320… The third harmonic of the lower sound coincides exactly with the second harmonic of the higher one (660 Hz), and again at 1320 Hz.
Because the ratio is 3:2, this coincidence repeats at every third harmonic of the lower sound and every second harmonic of the higher one; the harmonics in between stay well apart. The Intervals diagram draws this: the harmonics of the lower sound on top, of the higher sound below; solid lines mark the ones that coincide.
Close but apart: roughness
When two harmonics do not coincide but fall close together, their sum grows stronger and weaker as often as their difference. If the difference is once or twice a second you hear a slow wavering; if it is between a few and a few dozen times a second, the sound comes across as rough or 'grainy'.
In a minor second (16:15), 234.7 Hz sounds next to 220 Hz; the second harmonics are 440 and 469 Hz, about 29 Hz apart. Intervals does not play such combinations together; it plays the two sounds one after the other and says why: in Frekans Studio, beats between 4 and 40 Hz are used only in labelled techniques.
Just and equal tuning
Just tuning builds intervals from exact ratios. Equal tuning, used on the piano, divides the octave into twelve equal steps; each step is 100 cents (an octave is 1200 cents). This lets you play the same intervals from every key, but the intervals drift slightly from the exact ratio.
An equal-tempered fifth is 1.96 cents narrower than a just fifth: above 220 Hz it sounds 329.6 Hz instead of 330, and a very slow wavering of about 0.7 a second appears between the coinciding harmonics. An equal-tempered major third is 13.7 cents wider than a just third; in sounds with harmonics, its upper harmonics beat a few times a second. The 'Just / Equal' choice in Intervals shows this difference in pitch and in cents.
The Lissajous figure
Move a point horizontally with one sound's wave and vertically with the other's, and it draws a figure. If the ratio is exact, the figure closes and stands still; at 3:2 a knot with three and two loops appears. If the ratio drifts a little, the figure turns slowly.
The physicist Jules Lissajous used these figures to compare tuning forks; he also sat on the commission that fixed A at 435 Hz in France in 1859 (more in the 'History of tuning' article). The figure in Intervals is worked out from two sine waves, not from a recording; it is slowed down for display and stands still with reduced motion.
Bells and bowls
The modes of bowls and bells are not whole multiples of the fundamental, so 3:2 does not make them coincide the way it does on a string instrument. Even so, two bowls a fifth apart keep their modes well away from one another and ring smoothly together. Intervals checks every pair of modes: if no pair meets 4-40 Hz apart, the two bowls play together; if one does, they play one after the other.
Real instruments rarely land on an exact ratio. The 'two orins' example in Intervals is 12 cents below 3:2; that is why its figure turns slowly.
Frequently asked questions
- Why do some intervals sound 'consonant' and others 'tense'?
- In simple ratios the harmonics either coincide exactly or stay well apart; in complex ratios they fall close together and beat. Habit and musical culture play a part too: which interval counts as tense varies from tradition to tradition.
- Is equal tuning wrong?
- No. Equal tuning knowingly accepts small drifts so that the same intervals can be played from every key. Just tuning is smoother in a single key, but some intervals go badly out of tune when the key changes. They are answers to different needs.
- Do these intervals have an effect on the body or mind?
- This article only describes the physics and arithmetic of sound; it makes no claim about any effect of intervals.
Listen and try
Last updated: 6 October 2026 · This article is for information only; it is not medical advice.