For western music the greeks were the first to figure it out. It started off by the natural harmonics series generated by trumpets or horns and then they applied the same mathematical ratios (golden ratio) to strings. Pythagoras invented the pythagorean tuning of (3:2) perfect fifths and Octaves (2:1) to match naturaly occurring harmonic overtones. Later the greeks invented 7 modal scales based on pythagorean tuning. 7 Modes with 8 notes in a scale. These scales were Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian and Locrian. We still use Ionian (Major) and Aeolian(Minor). Like you mentioned the flaw with natural harmonics is that the octaves between each mode were slightly off from each other. Aristoxenus in the 4th century BC invented the 12 tones between octaves. In the early 1700's J.S. Bach was a big proponent of the using the (Tempered Scale equalizing the 12 tones by going sharp or flat between each tone to equalizing the natural gaps between each of the 12 semitones. Brass instruments in the baroque period had a bag of different sized crooks to adjust for each Key that they played in. String Instruments also had to retune for each key change. By using the tempered scale you could switch between all of the different keys without re-tuning.

For western music the greeks were the first to figure out the math that occurs naturally in the harmonics overtones generated by horns and other wind instruments. The Greeks applied the same mathematical ratios (golden ratio) to strings. Pythagoras invented the pythagorean tuning of (3:2) perfect fifths and Octaves (2:1) to match naturally occurring harmonic overtones. Later the greeks invented 7 modal scales based on pythagorean tuning. Seven Modes with eight notes in a scale. These scales were Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian and Locrian. We still use Ionian (Major) and Aeolian(Minor). The flaw with natural harmonics is that the octaves between each mode were slightly off from each other. Aristoxenus in the 4th century BC invented the 12 tones between octaves in an attempt to use the same ratio between each note. Later Keys were invented to use these 12 tones as a home base for each scale. The problem was that by nature these keys are slightly off from each other. To solve this J.S. Bach in the early 1700's promoted the use of the Tempered Scale. He equalized the natural occurring gap between each of the twelve semitones. Brass instruments in the baroque period had a bag of different sized crooks to adjust for each key that they performed in. String instruments also had to retune for each key change. By using the tempered scale a performer could switch between all of the different keys without re-tuning.