José Rodríguez Alvira
The Pythagorean temperament favours the fifth and has problems with thirds. The meantone temperament favours the third and the fifth bites us like a wolf. The dialogue is starting to look like an argument ... What choice do we have?
We respond to nature with new tuning systems: well temperament tuning systems. These temperaments spread the Pythagorean comma or syntonic comma over several fifths. They allow us 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 slightly 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 include:
Kirnberger divides the syntonic comma into four fifths (in yellow). The fifth between F# and C# is slightly larger than the reduced fifths and is the same size as the equal temperament fifth:
Werkmeister divides the Pythagorean comma into four fifths (in yellow):
Listen again to J. S. Bach's Prelude in A flat major BWV 862, this time using the Werckmeister III temperament:
And again with the meantone temperament:
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