Interval
Tuning
What's going on
Part of the answer is simple arithmetic. Intervals that sound pleasant vibrate in simple ratios: while one tone vibrates twice, the other vibrates exactly three times. Tense intervals have complicated ratios, or none at all. Draw the two tones against each other and you can see it; the rest of the answer lies in how we hear, and in the music we grew up with.
One tone moves a dot up and down, the other moves it left and right. When the two vibrate in a simple ratio, for example one vibrates three times while the other vibrates twice, the dot returns to where it started and draws a closed figure that stands still. Simple ratios give simple figures, and the intervals that sound smoothest, the octave and the fifth, have the simplest ratios.
A piano bends the rules slightly: so that every key works in every key signature, it tunes all intervals a tiny bit away from these pure ratios. That is why the figure drifts in equal temperament. The tritone is the extreme case: on a piano it matches no whole-number ratio at all.
Why it matters: tuning is a trade-off between pure intervals and freedom to change key, and it is still a live topic in music technology, from synthesizers to software that retunes recordings.
The maths behind itFor the curious · formulas and details
The picture is an oscilloscope in X-Y mode, with the lower tone f₁ on the vertical axis and the upper tone f₂ on the horizontal one:
x(t) = sin(2π·f₂·t) y(t) = sin(2π·f₁·t)
The curve closes exactly when f₂ / f₁ = p / q is rational; it then repeats after q cycles of the lower tone. Otherwise it never closes and, over time, fills the square.
Equal temperament divides the octave into 12 equal steps of 21/12. Interval sizes are measured in cents, 1/100 of a semitone: 1200·log₂(ratio). The equal-tempered fifth, 27/12 ≈ 1.4983, is about 2 cents narrow of 3 : 2; the major third is about 14 cents wide of 5 : 4. The tritone 26/12 = √2 is irrational. The figure's drift is shown at a quarter of the real beating speed.