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TL;DR: My question is not about programming. Calculating BPM (beats per minute) using average is not working well. Any other ideas?

I'm studying a programming language and decided to develop a simple "Tap Tempo" app. It basically get user mouse clicks and calculate the BPM (beats per minute).

However, it's taking too long to get closer to the real BPM and even so it's an approximation. The more samples (clicks) I have, the more it gets near the real BPM, but more samples means the user has to click too many times to discover the BPM.

I'm basically starting a timer when the user clicks the first time, then divide the number of beats (samples/clicks) per the elapsed time since the last click/beat.

Something like: BPM = beats_count / elapsed_time.

The number of beats increases at every click and elapsed_time is not being reset periodically, so it grows "forever". I tried to reset the timer, but it doesn't have the expected effect.

Do you have any clues on other strategy other than using the average of beats per time?

0d4n 0d1n
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    What is the "real BPM"? BPM is not a single "true value" but distributed probabilistically. Do you want to infer the mean of BPM? The variance of BPM? – Shubham Johri Nov 03 '20 at 18:15
  • The BPM itself. I didn't mean "real" as in "real number". The final BPM can be a positive integer. – 0d4n 0d1n Nov 03 '20 at 18:18
  • I want to infer the mean of BPM. – 0d4n 0d1n Nov 03 '20 at 18:18
  • I know there's no exact BPM, but the integer part is great for my application. – 0d4n 0d1n Nov 03 '20 at 18:19
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    You're not including that first click, the one that starts the timer, in the "beats count", are you? – JonathanZ Nov 03 '20 at 18:20
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    Make sure you're not counting that first click. If you click 3 times and elapsed_time is 1 minute, that's (3-1)/1=2 beats per minute. If you count the first click, it would eventually approach the right value as the numbers get big but it would take a long time to get to the right BPM as you describe. – aquaticapetheory Nov 03 '20 at 18:20
  • Yes, actually I'm counting the first click. The behavior is exactly what you mean @aquaticapetheory. I'll try to reimplement it using this strategy. I will check also what Shubham said. – 0d4n 0d1n Nov 03 '20 at 18:24
  • @aquaticapetheory: I typed faster, but you gave a more complete answer. You should make it an actual answer and get the reputation. Maybe mention that this is a classic "fence post error". – JonathanZ Nov 03 '20 at 18:26
  • I see other software getting the BPM approximation in just a few clicks (3-5). Thank you. I provide feedback later. – 0d4n 0d1n Nov 03 '20 at 18:26
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    @JonathanZ Thanks, I will. I've not heard the fence post error term but I've made enough mistakes for that problem to jump to mind. .. Nevermind, someone beat me to posting an answer so I'll just upvote that one instead. – aquaticapetheory Nov 03 '20 at 18:31

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I think I have a solution to part of your problem: subtract $1$ from your number of beats. Let me explain why:

Imagine I am a perfect metronome. I click at $120$ bpm, and I use your app.

Tap. The time starts. $t=0$. Tap, $t=0.5$sec. Tap, $t=1sec$.

If you're keeping track so far, one second has passed between the first and last clicks. I've tapped $3$ times, so by the formula $\text{bpm}=\frac{\text{clicks}}{\text{time}}$ I have a tempo of $\frac{3\text{ clicks}}{1\text{ sec}}=180\text{ bpm}$.

If I do another $4$ clicks, now there have been $3$ seconds since the beginning, and $7$ total taps. Calculated value: $140$ bpm. After $11$ clicks, there have been $5$ seconds since the start, so calculated value: $132$ bpm. Even if I tap for a full $30$ seconds, I've tapped $61$ times so the calculated value is $122$ bpm!

This is what's known as a Fencepost error; if I tap $n$ times, there have only been $n-1$ "gaps" in between my taps. If we take $n$ taps and divide by $n-1$ gaps, our estimate is always too high. But if we subtract $1$ from our taps, then suddenly the error is resolved; taps=gaps and all is right with the world.

See if that helps your count.

Edit: I see that while I was typing this answer, a conversation about this was happening in the comments. I'll leave this up in case it helps anyone.

tromben
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