Imagine I'm in a room that's silent apart from a clock whose second hand moves with an audible tick.

If I watch the second hand of the clock and clap every time it moves, I'll be clapping at 60 bpm.

I can easily clap at 120 bpm or 180 bpm or 30 bpm by looking at the second hand.

I can clap precisely 90 bpm by clapping three times for each two clock ticks.

I can clap at precisely 150 bpm by clapping five times every two clock ticks.

But if I wanted to construct other tempos, like 50 bpm, 70 bpm or 110 bpm, what would be the best way to do it?

What I'd really like is to have a collection of polyrhythms that I can perform against the second hand to give me a range of tempos, in ~10 bpm increments, starting at 30 bpm, say.

The reason for the question is that if I choose exact 10 bpm increments I might end up with a really awkward polyrhythm, whereas if I'm more relaxed I can get a polyrhythm that's much simpler to perform but still gets close enough to the required tempo.

And I do realise a metronome could give me the tempo. But I want to work for this - I want the challenge of constructing the tempo myself (in my otherwise silent room!)

  • 1
    @Tim I posted a question previously about Perfect Tempo and whether it exists or not. I don't have it, and often (usually!) I don't have my metronome with me either. Deriving tempo from a second hand seemed to me a useful musical skill to acquire. Apr 28, 2020 at 12:21
  • 1
    It certainly is, but like a lot of make-dos, it has serious limitations. It'll do for 60, 90, 120, easily, and then you'll have to guesstimate the in-betweens. Just as well to know a few songs. refer the other question to me! Piece of string with several knots and a small weight at one end could easily be made to give certain tempos... Pocketable, too.
    – Tim
    Apr 28, 2020 at 12:24
  • 1
    Brian, I bet there is an metronome app available for your mobile ;) Apr 28, 2020 at 12:58
  • 1
    @AlbrechtHügli My mobile is not a smart phone and doesn't run apps. So no metronome on my phone! Apr 28, 2020 at 13:38
  • 1
    4 on 3 polyrhythm is 80 BPM Apr 28, 2020 at 14:44

1 Answer 1


Well, this is a fun math problem.

If you clap n times for every m ticks of your clock, then this corresponds to 60*(n/m) beats per minute. It's not too hard to assemble a list of values between 30 bpm and 210 bpm using this principle. I've listed them below; exact multiples of 10 are in boldface.

  • Values for m = 1: 60 bpm (1:1); 120 bpm (2:1); 180 bpm (3:1)
  • Values for m = 2: 30 bpm (1:2); 90 bpm (3:2); 150 bpm (5:2); 210 bpm (5:2)
  • Values for m = 3: 40 bpm (2:3); 80 bpm (4:3); 100 bpm (5:3); 140 bpm (7:3); 160 bpm (8:3); 200 bpm (10:3)
  • Values for m = 4: 45 bpm (3:4); 75 bpm (5:4); 105 bpm (7:4); 135 bpm (9:4); 165 bpm (11:4); 195 bpm (13:4);
  • Values for m = 5: 36 bpm (3:5); 48 bpm (4:5); 72 bpm (6:5); 84 bpm (7:5); 96 bpm (8:5); 108 bpm (9:5); 132 bpm (11:5); 144 bpm (12:5); 156 bpm (13:5); 168 bpm (14:5); 192 bpm (16:5); 204 bpm (17:5)
  • Values for m = 6: 50 bpm (5:6); 70 bpm (7:6); 110 bpm (11:6); 130 bpm (13:6); 170 bpm (17:6); 190 bpm (19:6)

We see that:

  • You can get all multiples of 20 bpm relatively easily, by using an n:3 polyrhythm.
  • The n:4 polyrhythms are probably easier to perform, but the tempos that require n:4 (i.e., they aren't performable with a simpler n:2 polyrhythm) are all smack in the middle of your desired 10 bpm increments.
  • The n:5 polyrhythms give you some interesting options that couldn't be constructed in any other way, but...
  • The n:6 polyrhythms complete the set of 10 bpm increments (because 60/6 = 10). Of course, doing a 19:6 polyrhythm to get a 190 bpm beat would not be an easy task.

On the assumption that the difficulty of an n:m polyrhythm increases with the sum of n and m, you can also make the following graph:

enter image description here

(This graph doesn't include all of the options listed above; it also includes some options not listed above.) We can see that the hardest options to get are the ones that are "close" (but not equal to) multiples of 60 bpm: 110, 130, 170, and 190 bpm.

  • Feel free to request additional data or clarification on what I'm doing here; I can create additional graphs relatively easily. Apr 28, 2020 at 15:33
  • This is all very laudable, but it needs one to be able to keep an absolute constant timing at whatever bpm. There are many players who cannot do that well - including drummers! So it's pretty academic, and at the end of the day, the metronome seems to be the answer to everyone's dreams (and nightmares for some of us).
    – Tim
    Apr 28, 2020 at 16:00
  • @MichaelSeifert Thank you so much! This is absolutely brilliant! Can I check - if I construct 150 bpm then by dropping every second and third beat I should be able to produce 50 bpm, shouldn't I? But the graph suggests there's no easy way to construct 50 bpm. Or am I reading it wrong? Apr 28, 2020 at 17:06
  • 1
    @BrianTHOMAS Considering 150 is 5 claps every two clock ticks, it would indeed be difficult to drop every second and third beat - doing so results in 5:6 anyway which is definitely more difficult
    – Quintec
    Apr 28, 2020 at 18:17
  • Very cool, Michael. Using a two step process, it should also be pretty easy to transform derived tempo X to tempo 2X, 3X, X/2, and X/3. So, for example, getting 200bpm would probably be easier by getting 100bpm (5:3 from 60bpm) and then doubling it, than it would by doing 10:3 from 60bpm directly. Apr 28, 2020 at 23:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.