How would you derive accurate tempos if you have a clock with a second hand but no metronome?

Question

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!)

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:

(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.