Those two stacked numbers at the start of a score quietly decide how music feels
A time signature looks like a fraction but works like a recipe: the bottom number names a note length, the top says how many fit in a bar. Mathematically 3/4 and 3/8 are interchangeable, yet writing a rock song in anything but 4/4 would look deeply odd.
Also called a meter or measure signature, the time signature sits just after the key signature, or after the clef if there is none. The lower figure is nearly always a power of two: 2 means a half note, or minim, 4 a quarter note, or crotchet, 8 an eighth note, or quaver, and 16, seen mostly in Baroque music, a sixteenth. The upper figure counts how many of those notes make a bar. So 2/4 means two crotchets per bar, while 4/8 holds four quavers. A new signature partway through, usually right after a barline, marks a change of meter.
Two shorthand symbols survive from older mensural notation. A C-like sign can stand for 4/4, called common time or imperfect time, and the same sign struck through stands for 2/2, known as alla breve or simply cut time. The workhorse signatures are 2/4, 3/4 and 4/4.
Meters fall into families. Simple ones group notes in pairs, with a top number of 2, 3 or 4 giving duple, triple or quadruple time. Compound ones group notes in threes, with a top number such as 6, 9 or 12 and usually an 8 below. That changes where the beat falls: 3/4 is felt as three beats, each split in two, while 6/8 is normally felt as two beats of three quavers each. Other top numbers produce irregular meters.
Tempo complicates the arithmetic. Most music pulses somewhere between 60 and 140 beats a minute, so a quick waltz in 3/4 may be felt as one beat a bar, and a very slow piece may pulse on notes smaller than the signature names. Because 3/4 can be rewritten as 3/8 simply by halving every note, choosing between equivalent signatures is largely tradition, which is why certain styles cling to particular ones.
Source: Time signature
