12-Tone Equal Temperament, Quickly

How different are "pure" whole-number ratios from the tuning inside every modern instrument? Skip the explanation for now — compare them by ear first.

Whole-number ratios sound sweetest

Two tones whose frequencies form a simple ratio lock together beautifully: 2:1 is the octave, 3:2 the perfect fifth, 5:4 the major third. In the comparison above you may have caught it — the pure version sits perfectly still, while its neighbor shimmers with a slow "beating".

But pure ratios do not tile the octave

Here is the catch: stack 3:2 fifths on top of each other and you never land exactly some whole number of octaves up. Pure ratios simply cannot slice the octave into neat, matching pieces. Tune an instrument to pure ratios and one key sounds gorgeous — change key, and everything drifts out of tune.

The fix: 12 equal slices

Modern tuning divides the octave into 12 equal steps; each step multiplies the frequency by 2^(1/12) (about 1.0595). Every interval ends up a tiny bit off pure — the fifth is narrow by only about 0.1% — but the payoff is huge: all 12 keys work equally well, so you can change key whenever you like. Pianos, guitars, and synths all ship tuned this way. That compromise is 12-tone equal temperament.

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