Pythagorean, Equal, Meantone, and “Well” Temperaments Historical
developments are central to discussions of temperament because
the music of the time is tied to the temperament of the time.
Pythagoras is credited with inventing the Pythagorean
Temperament at around 550 BC, in which the chromatic scale is
generated by tuning in perfect 5ths, using the circle of 5ths.
The twelve perfect fifths in the circle of fifths do not make an
exact factor of two. Therefore, the final note you get is not
exactly the octave note but is too high in frequency by what is
called the "Pythagorean comma", which is about 23 cents (a cent
is one hundredths of a semitone). Since a 4th plus a 5th make up
an octave, the Pythagorean temperament results in a scale with
perfect 4ths and 5ths, except at the end where you get a very
bad dissonance. It turns out that tuning in perfect 5ths leaves
the 3rds in bad shape. This is another disadvantage of the
Pythagorean temperament. Now if we were to tune by contracting
each 5th by 23/12 cents,