Dot of prolongation
The dotting of a note.
What it means
Verlängerung (extension) designates any notation that extends a note value beyond its basic shape. The most common case is dotting: a dot to the right of the notehead adds half of the value, a second dot again half of that. Alongside this, the tie achieves the same thing by another route, by combining two written values into a single note.
In the notation
The dot stands immediately to the right of the head, in the same space; if the head sits on a line, the dot moves into the space above it. The same holds for rests. The tie connects two heads of the same pitch. Which means is chosen is not arbitrary: within a bar, and for ratios that can be represented by halving, one writes the dot; across the barline, and for all other ratios, the tie. A third form of extension is the fermata, which, however, adds no fixed value but leaves the duration to the performer's decision.
Musical use
Dotting is also the simplest way of gaining a triple division from the notation system's ongoing duple division: a dotted quarter comprises three eighths. This is why it is the normal value for a whole beat in triple-metre time signatures. In combination with a short note, the dotted rhythm arises, one of the most defining figures of all.
Extension in the sense of note value is to be distinguished from augmentation, in which a whole theme is repeated in enlarged values — there it is not a single value that changes but the ratio of a whole section.
The extension of a theme is called augmentation.
Performance practice
Really hold out extended values; they are regularly taken too short, because the eye is already at the next notehead. With double dotting, count in the smallest occurring unit until the ratio sits. And keep counting across tied notes, instead of starting afresh on the second head — especially with ties across the barline, the beat is otherwise lost.