Rates & divisionGrades 5–7

How fast is this song?

Beats per minute is a rate, exactly like kilometres per hour. One division tells you how long a beat lasts, how long a bar lasts, and how many bars are in a three-minute song.

Ratio & proportionGraphs & functionsMeasurement

BPM means beats per minute. It is a rate, exactly like kilometres per hour or pages per day: a number of things, divided by a lump of time. Once you know the rate you can work out everything else about a song with nothing but division — and every answer below shows its working.

tempo

120BPM

40200

Big drum on beat 1, so you can hear the bars

what 120 BPM actually means — with the working

one beat lasts60 ÷ 120=0.50 s
one beat, in milliseconds60000 ÷ 120=500 ms
one bar of 4/4 lasts4 × (60 ÷ 120)=2.00 s
beats in a 3-minute song120 × 3=360
bars in a 3-minute song(120 × 3) ÷ 4=90

A three-minute song at 120 BPM is 360 beats long. Split those into bars of four and you get 90 bars — which is roughly how a songwriter counts a song before a single note is written.

now run it backwards — tap your own tempo

Tap the button four times or more, at any speed you like. We measure the gaps between your taps, average them, and divide 60 by that — which turns your hand into a tempo.

0 taps

waiting for your first tap

double the number, halve the time

Speed up and each beat gets shorter. The two numbers pull in opposite directions, and they do it in an exact way: multiply the BPM by 2 and the seconds per beat get divided by 2.

BPM	60 ÷ BPM	seconds a beat
60	60 ÷ 60	1.00 s
90	60 ÷ 90	0.67 s
120	60 ÷ 120	0.50 s
180	60 ÷ 180	0.33 s

60 → 120 doubles the tempo and 1.00 s → 0.50 s halves the beat. 90 → 180 does it again. Multiply the two columns together and you always get 60, no matter which row you pick — that fixed product is what inverse proportion means.

seconds a beat, against BPM

A curve that dives fast and then flattens out: this is y = 60 ÷ x, the same reciprocal graph the trombone slide draws when you plot pitch against tube length. See it on the trombone →

From the NSM syllabus

“The “Metronome” and developing a strong sense of timing”
“Counting Subdivisions using Grid & Konnakol”
“Rhythmic Devices — Divide by 2”

A tempo is a speed

Kilometres per hour, words per minute, beats per minute — all the same shape of number. “Per” is a division sign that learned to talk.

60 is the whole trick

There are 60 seconds in a minute, so seconds per beat is always 60 ÷ BPM, and BPM is always 60 ÷ seconds. One division, run in either direction, answers every question on this page.

Faster means shorter

The bigger the BPM, the less time each beat gets. Multiply one and you divide the other, and their product never moves off 60. Two quantities tied together like that are in inverse proportion.

For the classroom

Learning goals

Reading BPM as a rate (beats per minute) and converting between a rate and its unit duration by dividing: 60 ÷ BPM seconds, 60000 ÷ BPM milliseconds.
Multi-step calculation with a rate — beats in 3 minutes = BPM × 3, then bars = beats ÷ 4 — and interpreting a non-whole answer (82.5 bars means the song ends mid-bar).
Inverse proportion made physical: BPM × seconds-per-beat = 60 always, so the graph of seconds against BPM is the reciprocal curve y = 60/x, not a straight line.

Try this

1.Set the click to 60 BPM and have the class count seconds out loud against it. Then jump to 120 and ask what happened to the length of one beat before showing the working.
2.Everyone taps their favourite song from memory, four taps or more, and reads off their BPM. Collect the numbers on the board — the spread between students is the lesson.
3.Give the class a 3-minute song at 96 BPM and ask for the number of bars (72). Then ask what tempo gives exactly 100 bars in 3 minutes, and let them rearrange the formula to find 133⅓ BPM.