tuning systems.

1. Pythagorean Scale

this scale has actually been in use since Ancient Mesopotamia! there are two main intervals that correspond to frequency ratios.

• an octave leap = multiplying the frequency by 2
• a perfect fifth up = multiplying the frequency by 3/2.

essentially, the whole scale is tuned to the first note. you move up or down in perfect fifths (3/2), and adjust the octaves (2) to keep the notes in the range of one octave.
the problem with this tuning system is that it creates something called the "Pythagorean Comma", which is a small gap that is made quite obvious when you close the circle of fifths; if you move up 12 perfect fifths, you're supposed to be 7 octaves above your starting note. however, (3/2)^12 != 2^7. some specific intervals will sound just slightly out of tune because of this discrepancy. compare tonic with octave to hear the difference.
this is all caused by our two main frequency ratios. we are trying to make an equation that is based on tripling (perfect fifths) be equal to an equation based on doubling; we're trying to solve 2^x = 3^y, where x and y are rational.

2. Helmholtz's scale (an example of Just Intonation)

so after the shenanigans with the pythagorean, musicians tried adding extra ratios such as 5/4 (major third), and 6/5 (minor third). this gave sweeter sounding chords. these specific ratios make up the Helmholtz's scale.
however, this created new problems. following this system, a D is 9/8, and an A above that is 5/3. so the interval between them is 5/3 ÷ 9/8 = 40/27. this interval is a perfect fifth, so ideally this should be = to 3/2. the distance is off by 21.5 cents (the syntonic comma)

3. Equal temperament, the tuning we use today

there was a lot of playing around with ratios to try and even out this error across the notes, creating meantone temperament (makes fifths slightly smaller than 3:2, so major thirds sound nicer; this worked until you tried to play in different keys), and well temperament.
eventually, we got to equal temperament where the octave is divided equally into 12 notes. this means that the ratio 2:1 is equally split into 12 parts. therefore, the ratio between two adjacent notes is the 12th root of 2, 2^1/12
The two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu in 1584 and Simon Stevin in 1585.