Kirnberger temperament
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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 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).
Closing the ends of the spiral of fifths into a circle
In almost all tuning systems, the so-called "circle" of fifths is not a circle: Randomly chosen fifth sizes, or fifths chosen to produce greater consonance among other notes in a chord almost always form a spiral. Some impractical but very
The number 12 or so notes per octave is commonly used because after stepping up 12 fifths in sequence (and dropping down a whole octave as needed to remain in the original octave) the 12th note is almost the same pitch as the note the spiral started on; the error in pitch is called a
Thus, if one tunes in fifths, matching by ear from C–G, G–D, D–A, A–E, E–B, B–F♯, F♯–C♯, C♯–G♯, then crosses over from G♯ to A♭ (G♯ and A♭ are different pitches in nearly every tuning system, but are also very close, enabling musical subterfuge; for example, both can be replaced by their only slightly out of tune average frequency), then from A♭–E♭, E♭–B♭, B♭–F, and finishing with F–C. However, the ending C will not be the same frequency as the starting C: The first and last Cs will have a discrepancy of about 23 cents (a
In Kirnberger I, the D–A fifth is reduced by a syntonic comma, making the major thirds F–A, C–E, G–B, and D–F♯ pure, though the fifth based on D is a ratio of 40 / 27 instead of 3 / 2 (680.4 cents instead of 702.0 cents). 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.
Practical Temperaments: Kirnberger II
Kirnberger's first method of compensating for and closing the circle of fifths was to split the "
C-----G-----D------A-----E-----B-----F♯-----C♯-----Ab(G♯)-----Eb-----Bb-----F-----C p p −½ −½ p p p p p p p p |__________pure 3rd______| |__________pure 3rd______| |_______pure 3rd________| |__________Pythag. 3rd_________| |_________Pythag. 3rd___________| |________Pythag. 3rd___________|
The above table represents Kirnberger II temperament. The first row under the intervals shows either a "p" for pure, or "−½" for those intervals narrowed to close the circle of fifths (D–A), (A–E). Below these are shown the pure 3rds (between C–E, G–B, D–F♯), and Pythagorean (very wide) 3rds (B–D♯, F♯–A♯(almost B♭), D♭–F.)
Tempering any musical scale, however, is always a give-and-take situation: No
Kirnberger III
After some disappointment with his sour, narrow fifths, Kirnberger experimented further and developed another possibility, later named the Kirnberger III.
This temperament splits the
See also
- Construction der gleichschwebenden Temperatur
- meantone temperament
- Friedrich Wilhelm Marpurg
- well temperament
Sources
- ISBN 0-300-09707-7.
Further reading
- Klop, G.C. (1974). Harpsichord Tuning. Raleigh, NC: The Sunbury Press. — called "Kirnberger 2" in Lieberman & Miller (2006). Lou Harrison. p. 80. ISBN 0-252-03120-2; presumably similarly naming the other Kirnberger temperaments.
- Kroesbergen, Willem; Cruickshank, Andrew (2014). 18th century quotes on J.S. Bach's temperament (Report) – via academia.edu.
- Eckersley, Dominic (April 2013). Rosetta revisited (PDF) (Report) – via wordpress.com.