OK, thinking about the things I want to get and how many actions are required to get them...
1)Build the Great Note: If I understand the application of our relevant Songs correctly, requires 5 AP. Constructing a second Great Note would be handy, but non-essential.
2)Research Experimental Weapons And Armor + Anti-Psychic Materials: 2 AP. Some further research would be nice, but these are the ones I want most.
3)Symphonies: 1 AP can get us either 2 or 4 Symphonies, depending on how we spend it.
4)Titans: 1 action would give us a full legion, and that's great (though we gotta make a bunch of sketches for it). Every extra legion after that is a bonus. Since it's military, 2 AP result in 3 actions.
5)Sigils. 1 action should suffice.
6)Extra dreadnoughts. 4 actions (3 AP) can give us something even better than the Seyfert.
7)Filling up our ship construction queue: Either 1 or 2 AP, depending how much we grow in the remaining turns.
8)Melodies and Songs: 1 AP would be enough to get our fill of new Melodies (at 1) and still make some new Songs. 2 AP would raise everything, 3 AP would give us basically everything.
So, looking at that...
Minimum: 5 AP for Grand Note + 2 AP for Research + 1AP for Symphonies + 1AP for a Titan Legion + 1AP for Sigils + 3AP for dreadnought + 1AP for fleet construction + 1AP for Melodies and Songs = 15 AP, so, 5 turns (not counting Cry For The Future, which we might not be in a position to use due to new Symphonies).
Best: Throw 1 more AP for Research, 1 more for Titans, 1 more for ship construction, 2 more for Melodies and Songs, that's now 20 AP. Doable in seven turns with some room left over.
...Oh, actually, spending some AP on Replicate The Rubicon might be a very good idea.