The sweetest-sounding chords use PURE whole-number ratios — a major third of exactly 5:4. But tune a keyboard that way and it only sounds right in ONE key; move to another and a horrible ‘wolf’ interval appears. So pianos use EQUAL temperament, spreading the error evenly so every key is slightly-but-equally out of tune. Slide between pure and even and hear the trade.
Just intonation tunes intervals to small whole-number frequency ratios — the octave 2:1, the perfect fifth 3:2, the major third 5:4 — which beat-free ratios the ear hears as maximally CONSONANT. But those pure intervals don’t stack into a closed 12-note system: the major third 5:4 = 1.2500 differs from the equal-tempered 2^(4/12) = 1.2599 by ~14 cents (a syntonic-comma-scale error), and pure tuning leaves one key sweet and others with a ‘wolf’. Equal temperament (2^(n/12)) sacrifices purity so EVERY key is equally usable — the compromise that made the keyboard modulate. A fail-loud self-check throws unless the just and equal thirds differ by ~14 cents. ◆ real music theory, node-verified.
The just vs 12-tone-equal major third comparison (the exact 14-cent gap); real historical temperaments (meantone, well-tempered) sit between them — the pure-is-sweeter-but-can’t-transpose tension is exact.