Why do F-sharp and G-flat sound the same on an equal-tempered keyboard?

F-sharp and G-flat sound the same on an equal-tempered keyboard because of enharmonic equivalence, a fundamental feature of the equal temperament tuning system.12

In this system, the octave is divided into 12 semitones of equal size, with each semitone measured at exactly 100 cents. Because the keyboard is designed to accommodate this specific division, a single physical key is used to represent both F-sharp and G-flat. While these notes are spelled differently in musical notation, they are functionally identical in pitch within this tuning framework.132

This differs from earlier tuning systems, such as just intonation or meantone temperament, where enharmonic tones were not identical in pitch. In those older systems, C-sharp, for example, would sound lower than D-flat by about one-fifth of a tone. Equal temperament replaced these systems to allow keyboard instruments to play in all keys without the intonational flaws that would otherwise occur when moving between different musical keys.213