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It extends the note for exactly half its value.
For example, a dotted half note is worth three beats. Half of a half note is one beat, add that on to the two beats a half note is already worth, and there you have it. Three beats.
So, basically we have a half note: 2
We place a dot after it, this dot being worth half its value: 1
It becomes: 3
Half of a half note is not necessarily one beat. These dots have nothing whatsoever to do with beats. If you have a note with a dot after it, the length of that note is extended by 1/2 of that note's length. A second dot means it is extended a further 1/4. Completely irrespective of how many beats there are in a bar.
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