The top number counts the beats in a bar. The bottom number names the note that gets one beat. Put them together and you have measured the bar as a fraction of a whole note.
The two numbers at the start of the music look like a fraction because they very nearly are one. The top number counts the beats in a bar. The bottom number names the note that gets one beat — 4 means a quarter note, ¼ of a whole note. Pick two numbers and watch the bar change size.
top — how many beats in a bar
bottom — which note is one beat
one bar of 3/4, drawn against one whole note
how much of a whole note is that?
3 × ¼=3⁄4=¾
3 quarter notes make one bar of 3/4. Laid end to end that is ¾ of a whole note. 3/4 and 6/8 both come to ¾ of a whole note. Same amount of time, completely different feel — that comparison is two panels down.
84 beats a minute — and here a beat is one quarter note. The big drum lands on beat 1; the hi-hat ticks the rest.
and this is how it is written down
Real notation, on one pitch so the only thing to read is time: the 3/4 at the start, then 3 quarter notes filling the bar. Press play above and the note you are hearing lights up.
the big one — 3/4 against 6/8
Both bars are built from the same six eighth-notes, so both last exactly the same amount of time: ¾ of a whole note. The only thing that changes is where the strong hits land. Play them one after the other and you will hear two different pieces of music made out of identical arithmetic.
ONE-two-three — three beats, each split in two.
ONE-two-three-FOUR-five-six — two beats, each split in three.
3 × ¼ = ¾ and 6 × ⅛ = 6⁄8 = ¾. The fractions are equal. The bars are equal in length. But 3/4 groups the six as 2 + 2 + 2 and 6/8 groups them as 3 + 3 — so the accents land in different places, and your feet do different things. Equivalent fractions that do not feel equivalent.
From the NSM syllabus
2 means halves, 4 means quarters, 8 means eighths. It is the bottom of a fraction, doing exactly the job a denominator always does: saying how many equal pieces one whole is cut into.
It is not part of the fraction so much as a multiplier: take that many of that note. 7/8 means seven eighths — the bar is ⅞ of a whole note, and it ends a shade early. That is the lurch you can hear.
6⁄8 = ¾ is true in a maths book and true in music: the bars last the same time. Music adds one thing arithmetic does not care about — where the accent goes — and that is enough to make them sound like different countries.
For the classroom