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→Why/When 7ths?
As noted for the Augmented 7th and Half Diminished 7th Chords, the 7th helps add consonance and lightens the feel of the basic triad, while adding a richer harmony to it and making it feel more complex. I recommend always using the Half Diminished 7th over the triad when diatonic to the current key, especially as the viiø7 chord of a Major key.
Sevenths can also be desirable for voice leading purposes. Or it may simply be implied by a melody line.
Fully Diminished and Minor/Major 7th Chords are particularly dissonant and create a particularly unresolved feeling. The Fully Diminished Chord can also be used to signal an atonal feel.