If we look at the Hebrew words used in this passage, note that there are two different words used:
Hodesh = "new moon" or "month" (used over 280 times)
and
Keseh = "full moon" (used only 2 times)
Also, if we read Psalm 81, it's structure is poetry, not necessarily correlating the two words together but actually contrasting them as different moments in time. For example, imagine singing the first four verses to a 6/8 (doo-wop or fast waltz) beat for the effect:
[1 2 3 4 5 6...]
[1 2 3 4 5 6...]
"Sing-for-joy, to, God-our-strength;
Shout-joyfully-to-the-God-of-Ja, cob.
Raise a song,
strike the timbrel,
The-sweet-sounding-lyre, with the harp.
Blow the-trumpet at-the-new moon,
At-the-full moon,
on-our-feast day.
For-it-is-a-statute-for-Is-ra-el,
An, ordinance of-the-God-of-Ja, cob."
Verse 2 is a direct mirror in structure to verse 3, contrasting different instruments to be played. Then verse 3 follows through by contrasting different times to play those instruments: at the new moon, at the full moon, and on feast days (all three of which were times when Israel would celebrate), completing the poetic structure.
So a new moon (hodesh) is different from a full moon (keseh)...But is a new moon a dark moon (unseen) or a visible moon (seen)?