Why does "blue mood" means "sad mood"? Why is the color blue associated with sadness?
According to Cambridge dictionary "blue" means sad or unhappy.
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From the English Language stack (https://english.stackexchange.com/questions/41804/origin-of-the-of-the-phrase-feeling-blue):
"Feeling blue" and "feel blue" begin to appear in Google Books publications from the 1830s. It seems highly likely that the wording arose naturally from earlier slang terms involving the word blue. Here are two early (and potentially relevant) entries from Francis Grose, A Classical Dictionary of the Vulgar Tongue (1785):
BLUE, to look blue; to be confounded, terrified, or disappointed.
BLUE DEVILS, low spirits.
John Barrett, Dictionary of Americanisms, first edition (1848) shows a similar range of meanings in U.S. English slang usage:
BLUE. Gloomy, severe; extreme, ultra.
BLUE DEVILS. To have the blue devils is to be dispirited.
Either term could reasonably be interpreted as providing the immediate referent for early use of the phrase "feeling blue."
It doesn't appear that there is consensus on the origin of a "blue mood" or "feeling blue." It likely has something to do with the idea of blue being a "cold" color -- the opposite of hot red.
Etymology online has this sense dating from 1400, and speculates that it could be from the "colour of bruises" sense of the word (that was introduced through Norse). Giving the notion of a "blue and bruised heart".
Figurative meaning "sad, sorrowful, afflicted with low spirits" is from c. 1400, perhaps from the "livid" sense and implying a bruised heart or feelings.