Meantone resolves the 5th of D-A (which is out of tune) of just intonation, and set pure 3rd in 12 tones of the keyboard as many as possible.
This tuning system is for pipe organ in time of Renaissance, in order to get strict dignified sound of church.
Pythagorean system was used before mean tone.
But in the music of Renaissance,the harmony started to be used.
Pythagorean system constructs a strong howling on this 3rd.
This makes a feeling of dissonance.
In order to construct triad with a pure 3rd (386 cent),total of 4 successive 5th (ex.C-G-D-A-E) have to be 2786 cent (1200*2+386 cent)
(reference : syntonic comma)
Divide this by 4 (2786/4=696.5) and make a narrow 5th and make a pure 3rd and this is the system of meantone.
Continuation of this narrow meantone 5th in circle of 5th, you can make 8 pure 3rd,
but can not avoid to make a “wolf” at the end, and also 4 major 3rd out of tune.
In order to fit many keys, Cembalo with two divided keys of G♯/A♭ and D♯/E♭ was made.
From this fact, we can see that meantone system was widely used.
This is the time when basic of Western Music was made.
Meantone could be the root of classical temperaments of Baroque Music.
Refrence : Transition from pipe organ tuning to classical temperament.
Chisako Murakami translated it into English from pianotuning.jp
Special thanks to Ms. Midori Kita.