Slow, Sarcastic Clap

I mentioned looking forward to coming back to Cardio Drumming at the gym where Marika and I go, but sadly they no longer offer it. Instead, they offer something called Pound Fitness that we've been trying out, which involves using weighted drumsticks without the ball drums. It's got rhythm on my mind, and made me think of another idea I've wondered about for a while: When large groups clap along to music, why does it seem like there are so many who are off-beat?

I suppose the uncharitable answer is that they're bad at keeping time, but I wondered whether the relatively slow speed of sound had anything to do with it. If we imagine the beat as a pulse that leaves the stage at 343 m/s, each audience member will clap when that pulse reaches them. That starts a new beat-pulse that will be out of sync when it reaches other audience members.

The equation for the error is fairly simple:

where R is the distance from the stage to the audience members, c is the speed of sound, and 𝜃 is the angle between the audience members.

To visualize this a bit better, I put together another HTML5 simulation, using dimensions from here. If you'd like to see the stand-alone code, it's here.

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Room Depth (meters)

Room Opening (degrees)

Number of Seats

Delay (milliseconds)