Kirnberger temperament is an irregular temperament developed in the second half of the 18th century by Johann Kirnberger. Kirnberger was a student of Johann Sebastian Bach, held great admiration for his teacher and was one of his principal proponents, though it is rumored that they had many disagreements concerning tuning systems at the time. Eventually, they went their separate ways, and each developed his own temperaments as time passed.
The first Kirnberger temperament, "Kirnberger I", had similarities to Pythagorean temperament, which stressed the importance of perfect fifths all throughout the circle of fifths. A complete circle of perfect fifths is not possible, because when the circle comes to an end at the tone it began, it will have overshot its original pitch. Thus, if one tunes C-G, G-D, D-A, A-E, E-B, B-F♯, F♯-C♯, C♯-G♯ (A♭), A♭-E♭, E♭-B♭, B♭-F, F-C... the new "C" will not be the same frequency as the first. The two "C"s will have a discrepancy of about 23 cents (a comma), which would be unacceptable. This difference between the initial "c" and final "c" that is derived from performing a series of perfect tunings is generally referred to as the Pythagorean comma. Many tuning systems have been developed to "spread around" that comma, that is, to divide that anomalous musical space among the other intervals of the scale.
Read more about Kirnberger Temperament: Practical Temperaments: Kirnberger II, Kirnberger III
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