Kirnberger temperaments

Kirnberger I temperament

Kirnberger I tunes D/A a syntonic comma flat, creating pure major thirds F/A, C/E, G/B, and pure minor third D/F.

Kirnberger II temperament

The syntonic comma is split in two and subtracted from the D/A and A/E fifths.

Kirnberger III temperament

Kirnberger III tunes C/E as a pure major third and flattens the pure perfect fifths between C/E by 1/4 of the syntonic comma (which is the difference between 5 pure perfect fifths and a pure major third), with the rest of the perfect fifths tuned pure, with break placed at D-flat/F-sharp.

Ratio-oriented description of Kirnberger III

Playing all descriptions of Kirnberger III simultaneously (there should be no beating between the notes):

References:

• Kirnberger temperaments, Carey Beebe Harpsichords
• Kirnberger temperament article on Wikipedia.
• Kirnberger III Well-Temperament article in Tonalsoft encyclopedia of microtonal music theory.