Three irregular temperaments developed by Johann Kirnberger
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The Kirnberger temperaments are three irregular temperaments developed in the second half of the 18th century by Johann Kirnberger. Kirnberger was a student of Johann Sebastian Bach who greatly admired his teacher; he was one of Bach's principal proponents.[1]
Kirnberger's tuning systems, or well temperaments are a way to artificially splice together two arcs on the "natural" spiral of fifths to turn it into an "unnatural" circle. In Kirnberger's and his teacher Bach's time, keyboard musicians were experimenting with different unobtrusive ways to alter the spacing of notes around the spiral of fifths to close it into a circle, so that every note needed for every key was at hand, even if some rarely used key signatures might be very dissonant, but tolerable.
The first Kirnberger temperament, Kirnberger I, had similarities to Pythagorean tuning, which stressed the importance of perfect fifths all throughout the spiral of fifths. His later tuning system(s), Kirnberger II and Kirnberger III, dispensed with perfectly tuned 3 / 2 Pythagorean fifths and instead improve the harmony of major minor thirds in chords, which are necessarily spoiled by adhering to perfectly tuned fifths (unless there are an unworkably huge number of distinct pitches in each octave: at least 31, and perhaps 53).
^Bach, J.S. (2002). Ledbetter, David (ed.). Bach's Well-Tempered Clavier: The 48 preludes and fugues. p. 48. ISBN 0-300-09707-7.
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McClain & Ming 1979, pp. 214–215. Barbour, James Murray (2004). Tuning and Temperament: A Historical Survey. North Chelmsford: Courier Corporation. pp. 55–56...
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