Rashi citing Brachos 3b #1: David would rise [at midnight to sing to Hashem]; this was half the night, i.e. two "Ashmuros" (before morning; the entire night is four Ashmuros).

Brachos 3b #2: (There are three Mishmaros in the night; each is four hours.) David rose at midnight, two Ashmuros (six hours of the night and two of the day) before other kings.

Brachos 3b #3: (Each Mishmar is four hours.) David rose at midnight, one and a half Ashmuros (six hours) before morning.

Eichah Rabah 2:22: (The entire night is three Ashmuros.) Sometimes David rose after a third of the night (two Ashmuros before morning), and sometimes at midnight - "Chatzos Laylah Akum" (62).

Rashi: David would rise after a third of the night, engage in Torah until midnight - "Lasi'ach b'Imrasecha", and then he sang to Hashem until morning - "Chatzos Laylah Akum Lehodos Lecha" (62).

Radak: David would rise after a third of the night, and talk about Hashem's words and ways and contemplate them.

Malbim: That night (affliction), when the salvation delayed one Mishmar (a third of the night), I rose a second time (refer to 119:147:1:1) to speak in Your words and promise, and investigate why You did not save us immediately.