Early theories took the fact that a basic sound was not just that, but made up from overtones or harmonics of that basic sound. It was sort of agreed that the first five of these harmonics were audible to a standard human ear, and others, of which there are many beyond, were only discernible to those with very sensitive ears.
The first five harmonics of a fundamental (root, if you like) note are: octave of root, fifth of root, second octave, major third, and second fifth.These, when blended together, made (and still do!) the major chord. To get a minor chord, one has to go further up the harmonic ladder, but at the time, it was deemed that those notes constituted Mother Nature's contribution - it was a natural progression that happened to sound good.A simple example may be blowing over an open end of a bottle. The sound is a fundamental, but you can hear at least a couple of harmonics in the sound, too.
This is not key specific, as it can be heard from any basic sound (apparently),and would even work for notes 'in the cracks'. Each musical instrument has its own set of harmonics, which bring out each in different proportions, thus producing different timbres. The Germans have a word for it - 'klang'. I guess we've borrowed it and spell it with a c, as an onomatopoeia.