José Rodríguez Alvira
The Pythagorean tuning favors the fifth and runs into problems with thirds. The meantone temperament favors the third and the fifth bites us like a wolf. The dialogue begins to look like a dispute ... What choice do we have?
We answer with a new tuning system: well temperament. These temperaments spread the Pythagorean comma or the syntonic comma between several fifths and make possible to play in any key.
What is a Pythagorean comma? If we follow a cycle of fifths ( C - G - D - A ...F - C), the last note will be a little higher than the original note. The Pythagorean comma is the difference in pitch between these two notes.
What is a syntonic comma? The difference between the major third in the harmonic series and the Pythagorean major third.
Some of these temperaments:
Kirnberger divides the syntonic comma between four fifths (in yellow). The fifth between F sharp and C sharp is a little bit larger than the reduced fifths and it is the same size as the equal temperament fifth:
Werkmeister divides the Pythagorean comma between four fifths (in yellow):
Listen again Prelude in A flat major BWV 862 by J. S. Bach, this time with the Werckmeister III temperament:
And again with the meantone temperament: