Well temperament

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:

• Werckmeister III (1691), Andreas Werkmeister (1645-1706)
• Rameau (1726), Jean-Philippe Rameau (1683-1764)
• Kirnberger II (1771), III (1779), Johann Philip Kirnberger (1721-1783)

Examples of temperaments

Kirnberger III

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:

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Werkmeister III

Werkmeister divides the Pythagorean comma into four fifths (in yellow):

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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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