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Happy to leave cresent off this one, although some JEs have it.

hyphen, you learn something every day...

This is the first of many moon edits. Lots of apparent issues with half-moons and crescent-moons.  (Note, this term on it's own is not super-common compared to 下弦　and 上弦）。

There are so many mismarked crescent-moons that I was starting to wonder if the Japanese thought 半月 included crescents... (I believe they do not)

弦月 is apparently tied up in a popular anime character which breaks the image search. try 上弦の月 and 下弦の月 instead.

Here's another a site with diagrams specifically using various formulations of 弦月
https://tukihosi-tan.com/moon/2004/

https://ja.wikipedia.org/wiki/弦月
弦月（げんげつ）は、半月（はんげつ）の文学的な表現である
Wikipedia xref for 弦月 is English "Half Moon".
In moon phases:
新月（朔。0°=0）
上弦（90°=7）
満月（望。180°=14）
下弦（270°=21）
半月（弦。上弦と下弦の総称） (半月 is a generic name for the waxing and waning half-moons)

nikk: either the waxing half-moon, the waning half-moon, or 弓張り月 (also a half-moon)
げん‐げつ【弦月】
〘名〙 上弦または下弦の月。ゆみはりづき。彎月(わんげつ)。

Another ref, straight to 半月:
https://kotobank.jp/word/弦月-491742
普及版 字通 「弦月」の読み・字形・画数・意味
【弦月】げんげつ
半月。
字通「弦」の項目を見る。

Since 半月 means 15 days/half-a-month, the ngrams crush this.  But I get the impression that this term in particular is common when talking about lunar cycles specifically. (I am sure 半月 is readily comprehensible though, perhaps moreso).


https://ja.wikipedia.org/wiki/弦月
英語でも、日本語での半月同様、一般的には half moon（または half-moon）というが、月相に基づく天文学用語としては quarter moon という。上弦の月は first quarter moon で、下弦の月は last quarter moon （または third quarter moon）である。

more notes on the xrefs.