How does a metronome marking such as quarter note equals 120 set a tempo?

In musical notation, a metronome marking such as "quarter note = 120" provides a precise, objective measurement of tempo by defining the speed of the music in beats per minute (BPM).12

Specifically, this notation indicates that the quarter note serves as the beat unit and that there are 120 of these beats in one minute. Mathematically, this means each quarter note lasts for $1/120$ of a minute, which is equivalent to one-half second.12

By specifying both the note value and the frequency, this marking removes the ambiguity often associated with traditional verbal tempo instructions (such as allegro or andante), allowing performers to achieve a consistent and accurate pace as intended by the composer. This system was popularized by the invention of the metronome in the early 19th century, providing a reliable tool for maintaining this steady pulse throughout a performance.123