Terzdezime
A third plus an octave.
What it means
Read literally, the word describes a third extended by an octave — that is the tenth, which stands in the glossary under its own entry. The designation “Terzdezime” is uncommon; what is familiar is the thirteenth (German Tredezime), that is, the sixth plus an octave, which plays a role in jazz as a chord extension.
Musical use
Compound intervals — those that reach beyond the octave — correspond in their function to their respective simple interval, but sound more open and transparent, because the notes lie farther apart. A third in a low, close position sounds muddy; the same third set as a tenth stays clear. For voice-leading this is a practical rule: in low registers, wide spacing is advisable.
In part-writing theory and in figured bass, compound intervals are as a rule reduced to their simple form — for the rules of voice leading, the tenth counts as a third, the twelfth as a fifth. From this it follows directly that parallel tenths are allowed and parallel twelfths forbidden.
In jazz chord theory, by contrast, extensions are expressly named as ninth, eleventh, and thirteenth, because their position above the seventh chord is decisive for the sound. There, the distinction between a sixth and a thirteenth is not a mere matter of form.
Presumably what is meant is the thirteenth, the sixth plus an octave, known in jazz as the extension tone 13. It appears above all in dominant chords with an added thirteenth, whose sound seems full and lightly floating. If, on the other hand, the third plus an octave is to be designated, tenth is the usual term. Since both readings are possible and the name occurs in no standard theory of intervals, it is always worth looking, when reading a source, at the actual notes, to clarify which interval is meant.
Performance practice
Grasp wide intervals beyond the octave by recognizing the simple interval and adding the octave, rather than counting through lines. Use wide spacing when a low texture sounds muddy. And with chord extensions, take care that the added notes lie on top, so the sound stays transparent.