Why is a blue sky beautiful and having the "blues" such a sad thing?

The expression "the blues" has nothing to do with the colour blue.

Cultural Dictionary

blues

A kind of jazz that evolved from the music of African-Americans, especially work songs and spirituals, in the early twentieth century. Blues pieces often express worry or depression.

The American Heritage® New Dictionary of Cultural Literacy, Third Edition

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Word Origin & History

blues

as a music form featuring flatted thirds and sevenths, possibly c.1895 (though officially 1912, in W.C. Handy's "Memphis Blues"); meaning "depression, low spirits" goes back to 1741, from adj. blue "low-spirited," c.1385.

They are different kinds of blues :)

While the color blue.. is well, blue (personally, blue brings me a lot of happiness), the word or adjective "blue" has been associated with sadness and states of depression and melancholy since the 16th century. The color and the adjective has no connection.

The term blues is short for blue devils, which were believed to bring people sorrow and despair. Hence, having the blues is a sad thing, while the blue skies, the blue ocean, blue coffee mugs .. they're all beautiful.

Now there can be 'shades' of having the blues (blues - melancholy, sadness, depression), 'shades' here referring to stages/phases of depression.

blue skies are just as depressing as having the blues.

prolly the darker blues

Cuz blue skies have puffy white clouds, and when you have the blues, there are no puffy white clouds. Duh :P