The Syntonic Comma

Just Major Third: Ratio 5:4, 386.3 cents, 115.9 degrees
Pythagorean major third: Ratio 81:64, 407.8 cents, 122.3 degrees
12-TET Major third: Ratio cubic root of 2, 400 cents, 120 degrees

Apart from the octave the two strongest interval functions are the perfect fifth (and its inverse, the perfect fourth) and the major third. Mathematically, these are the reflections of the first three prime number in the harmonic series. That is why the so-called Just Intonation builds upon the octave, factor 2; the perfect fifth, factor 3 and the just major third, factor 5.

The inherent structure that gives rise to the 12 semitones, however, is the so-called Pythagorean Tuning, which is based upon factor 2, the octave and factor 3, the perfect fifth. 12 consecutive fifths approximately correspond to 7 consecutive octaves. However, there is a difference between the result of the generation of fifths and that of the octaves as
(3/2) to the power of 12 = 129.746, while 2 to the power of 7 = 128th
This difference is called the Pythagorean comma and has been described in this article: http://kortlink.dk/f38y

The two tonal systems – Just Intonation and Pythagorean Tuning – have been widely used until the Renaissance.
The Just Major Third must be considered as the ear's reference for purity, since, as indicated it is found directly in the harmonic series as partial #5 (= 80/16).
In Pythagorean Tuning the major third is generated by stacking four perfect fifth intervals:
(3/2) x (3/2) x (3/2) x (3/2) = 81/16.

The ratio between the two versions of the major third is therefore 81:80=1.0125, so there is a discrepancy between the two tones at 1.25%, or 21.5 cents, as it is called in music theory terms. The 12-TET semitone is 100 cents, so the deviation is somewhere in between 1/4 and 1/5 semitone. It is about the same amount of impurity, we find in the Pythagorean Comma (23.46 cents).

In practical terms this means that the Pythagorean Tuning will be unsuitable for polyphonic music containing third intervals. This is the primary reason that the musicians of the Renaissance brought just tuning into play, but were faced with other problems as the Just Intonation system doesn't sound pure in keys of some distance from the chosen starting point.

These issues formed the basis of experiments with musical temperaments, which in particular flourished in large numbers in the Baroque period, and where we have ended up with total domination of equal temperament/ 12-TET (which was not developed in the Baroque period).

This development has many advantages, primarily the possibility to seamlessly modulate from one key to another, but it has not been a developement without costs. The thirds and sixths of modern keyboards are in the vicinity of 1/7 semitone off, almost as false as the Pythagorean Major Thirds, which were deemed useless by Renaissance musicians.

We have become accustomed to musical impurity or, in other words we have lost the sense of purity.