Up to this point we've done everything we would need to build a true, accurate, metronome. It's time to break out the maths and build a model for our metronalmost that can be proved to never return Hackaday's target 1 Hz tick rate.

A normal, or Gaussian, distribution can be used to build a mean-weighted random number generator. Numbers will be clustered around the middle and only rarely appear towards the edges of the range.

If we take a notch out of the middle, the distribution will not generate numbers within that smaller range.

If we look at this notched-out function's cumulative distribution function instead of its probability distribution function, we can see a flat spot over the notch. This means that numbers are not being generated within this flattened range. Transposing the axes allows us to turn this in to a mapping function that turns a uniform random number (0.0–1.0) in to a centrally weighted, but stepped, number generator.

This new number mapping function maps 0.0 to 0.0 and 1.0 to 1.0 as normal, but maps 0.49 to 0.45 and 0.50 to 0.54. The discontinuity means a value of 0.5 can never be generated. Applying this mapping over a period of between 0.5 and 1.5 seconds will give us our almost, but never quite, 1 Hz metronome.