What's significant about a full moon on the fifteenth?

Question

In Joshiraku, Marii enters the room dressed as a Yankee, proclaiming that she's gonna break a load of windows since it's a full moon. Tetora responds with:

What's the significance of the 15th and why would she break a load of windows?

Answer 1

The dialogue here goes:

で、窓硝子はどこだい？
So, where are the windows?

なぜ？
Why do you ask?

壊して回るんだよ。
I'm going to go around breaking them.

魔梨威さん、十五夜と十五の夜は違うから。
Marii, it might be a full moon night, but it's not the fifteenth.

The last quoted line is the one depicted in your screenshot. There's a difficult-to-capture joke in this sequence, which I will nonetheless try to capture below.

Here, 十五夜【じゅうごや】 (character-by-character "fifteen-night") is translated as "full moon night". It literally just means "full moon". This originated as a reference to the lunar calendar, but now is just an ordinary synonym for "full moon".

十五【じゅうご】の夜【よる】 (the same thing with the genitive particle の interposed) is translated as "the fifteenth". A more direct interpretation of this, however, would be "a night at the age of fifteen" (an age when people are prone to "Yankee" antics).¹ The joke, then, originates from the fact that 十五の夜 (Juugo no Yoru) is also the name of an 80s song, the debut single of popular singer OZAKI Yutaka, and one that has great affection for the Yankee ideology/ethos. The song's chorus reads:

盗んだバイクで走り出す 行き先も解らぬまま
暗い夜の帳りの中へ
誰にも縛られたくないと 逃げ込んだこの夜に
自由になれた気がした 15の夜

Translated loosely as prose:

Riding the bike I stole, I dash off into the curtains of the dark night, knowing not where I'm headed. I don't want anyone to chain me down as I run off into the night, feeling free at last - the night of the fifteenth.

This is why Tetora's backdrop changes from a full moon (十五夜) to a guy riding a bike when she says 十五の夜.

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Alternate possibility (and probably a more likely one): Marii's lines could be a reference to a different Ozaki Yutaka song: 卒業【そつぎょう】 (sotsugyou, "Graduation"; lyrics), which contains the following line:

夜の校舎 窓ガラス壊してまわった
I went around the school building, breaking all the windows.

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Note also that Gankyou's line a few seconds later may be related:

つまらない大人にはなりたくないね。
I don't want to become a boring adult.

This is an almost-direct quotation of the penultimate line from the song ガラスのジェネレーション ("Glass Generation"), by SANO Motoharu, on his album "Heart Beat" (lyrics).

つまらない大人には　なりたくない
(same as above, but without the concluding interjectory particle ね)

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Notes

For more information about Joshiraku obscurata, see vale's translation notes.

¹ This is probably an elision of the construction 十五歳【じゅうごさい】の夜【よる】, and is literary or poetic in nature.

Answer 2

This is conjecture on my part, but it probably has to do with the differences in the Western and Eastern Calendars. The western Gregorian calendar is solar based, whereas the eastern calendar is lunar based.

This is important, and answers your question because the lunar calendar is based on the phases of the moon. The 15th of the lunar calendar is when the moon is full.

So based on that, she was probably saying that the moon is full, but going by the western calendar, its not the 15th.

Its either that, or they are going by the lunar calendar, and so going by the calendar, the moon is not completely full, yet. Another explanation would be there is something significant about the 15th in the anime, but I have no idea what that might be.