Why does music written in a minor key sound "sad" or "downbeat," while music in a major key sounds "happy" or "upbeat"?
Answers
You can write sad music in major and happy music in minor, but in my opinion minor, because of its added complexity, simply adds beauty and not sadness.
Music being downbeat or upbeat is technically a rhythmic phenomenon, and is largely unrelated to the tonality of the music. Most modern popular music is downbeat, regardless of whether it is happy or sad.
Some people say that our association of the minor scale with sadness and of the major scale with happiness is exclusively due to conditioning. That's possible. Doesn't feel quite right, though.
You can write sad music in major, or happy music in minor. However.
Consonance of intervals, i.e. simplicity of the ratios between the frequencies of notes in music, is a big part of what makes these notes sound musical. My current personal theory is that this is due to the biomechanical design of our ears. I will not go into that in detail here, but if someone asks me in a comment I'll elaborate. I think that in general more consonant and therefore 'more musical’ intervals give a feeling of 'everything being alright’, a feeling of - contentment, perhaps, and as intervals get less consonant it starts to feel like something is wrong, and the music begins to communicate contemplativeness or even trepidation.
Let's have a look at the (for chromatic instruments, approximate) frequency ratios corresponding to the minor and major variants of the intervals which are different between the minor and major scales.
Major third, 5/4. Minor third, 6/5
Major sixth, 5/3. Minor sixth 8/5
Minor seventh 9/5, Major seventh 15/8
As you can see two of the three intervals are less simple in the minor scale. Use the less simple, i.e. more dissonant version of all three of these intervals: the minor third, the minor sixth and the major seventh, and you have the harmonic minor scale - which is by most accounts sadder than the natural minor.
And that brings me to my next point; there are ‘sadder’ scales than the natural minor scale, and 'happier’ scales than the natural major scale. The main reasons these are used so extensively are more historical than anything else. They are modes of each other, and even among their other modes there are arguably happier and sadder/creepier sounding scales. Pentatonic major and minor scales are also widely used, again due to historical reasons.
I like to think of long melodic lines as ideally presenting a narrative - albeit perhaps a different kind of narrative than one might through language. I'm going to take this analogy beyond, perhaps, where it is most useful, and say that, just as it is possible to tell a sad story using 'happy’ words - often making the story seem even more sinister - is it possible to play sad music using happy notes. Or vice versa. However, this does not mean that the words themselves are not happy or sad.