The article for F-measure in Wikipedia says:

The traditional F-measure or balanced F-score (F1 score) is the harmonic mean of precision and recall: $F_1=2\times\frac{precision \times recall}{precision+recall}$

Why is the harmonic mean used in particular, and not the arithmetic mean or geometric mean or any other type of averages?

What exactly does it mean, to calculate an harmonic mean?