Not just harmonics (which, technically, are based on a slightly different fundamental principle), but the tonality of Western music is, for the most part, based on these simple ratios. For example, medieval vocal music was often sung in parallel open fifths because the perfect 3:2 ratio was so consonant.
But as composers demanded more complex harmonies, the limitations of perfect intervals became apparent. One notorious example was the D-A chord in a typical tuning in C. D was typically tuned to be a perfect major second above C (9/8 ratio), and A was typically tuned to be a perfect major third above F (5/4) which was a perfect fourth over C (4/3).
Putting that together, the ratio of the frequencies of D and A is 40/27 = 1.48148... when it should be 3/2 = 1.5; a very large difference (21 and a half cents) which makes the interval sound awful. Because of this, music of that era tended to avoid modulations that would force these chords.
Musicians worked on more sophisticated tuning methods, and musical tastes leaned more towards having better sounding major third intervals rather than trying to keep the other intervals pure (the major fifth in particular), and tuning systems developed that could have good sounding thirds in many keys, and while the fifth may not have sounded that great, blending it into the major triad smoothed it over.
Different keys sounded different, an effect used to good effect by composers, but there were still restrictions that certain chords simply couldn't work in certain keys, and some keys simply didn't exist (such as Ab major)
(Of course, keep in mind that all of this applies only to keyboard-type instruments and fretted instruments; other strings and all winds have always had the ability to modify their pitch to play with whatever tuning their ear demanded)
This last fact led J.S. Bach to demand a tuning system that allowed him to play in all keys, and he had a system developed that allowed all keys to sound good enough, as evidenced by the well-tempered clavier. However, still not all keys sounded the same; different keys had different qualities which Bach used to good effect, such as using brief hints sharp major thirds to add a brightness to music played in some keys while settling into the highly consonant chords of other keys for a more melodic sound.
Tuning systems continued to develop, each having varying degrees of success and persistence. A true equal tempered scale (modern "12 tone" music) is fairly recent over the course of music history. The simplicity of having every key sound reasonably good has allowed it to become predominant, but even still it is not accepted universally, as some keyboard performers keep their instruments tuned to alternate scales, and wind/string performers continue, as always, to employ their freedom of pitch to alter the harmonies to give them different qualities.
To keep this post from getting too long (too late?), I've omitted some things, like microtones and the fact some intervals (like the minor third) didn't have a standardized frequency ratio.
