Johann Philipp Kirnberger temperament
Encyclopedia
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 principle 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
. Unfortunately, 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 we tune 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.
" interval known to those who have used meantone temperaments in half between two different fifths. That is, to compensate for the one extra comma, he removed 1/2 a comma from two of the formerly perfect fifths in order to complete the circle. In so doing, he allowed the remaining fifths to stay pure. At the time, however, pure thirds were more valuable than pure fifths (meantone temperament had 8 pure thirds and sacrificed four entire chords to achieve this end.) So, Kirnberger allowed for 3 pure thirds, the rest being slightly wide and the worst being three Pythagorean
thirds (22 cents wider than pure ) on the opposite end of the circle from the pure thirds. To put it graphically:
C-----G-----D------A-----E-----B-----F#-----C#-----Ab(G#)-----Eb-----Bb-----F-----C
p p -1/2 -1/2 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 "-1/2" 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 (between B-D, F-A(B), D-F.)
Temperament, however, is a give-and-take situation: none is a perfect solution. It must also be remembered that temperament applies only to instruments with fixed pitch: any keyboard instrument, lutes, viols, and so forth. Wind musicians, singers and string players all have a certain degree of flexibility concerning the exact pitch and intonation of what they play and may therefore be free of such restrictive systems. When the two classes of players come together, it's important when evaluating a temperament to look at the tendencies of the instruments vs. those of the temperament. Kirnberger II would only have been applicable to keyboard instruments such as the harpsichord
and organ
. The perks of this system are its three pure thirds: certainly pleasing to especially singers and winds at the time and lovely to hear. In fact, after hearing pure thirds, one is shocked by the wide thirds of modern equal temperament
used on all pianos and many organs. Also, the temperament uses 10 perfect fifths, which is good news to string players whose strings are tuned in fifths. The drawbacks are, of course, the two narrow "half-wolf" fifths and the three Pythagorean, super-wide thirds. The chords are not unusable but certainly must not be used frequently or in close succession within the course of a piece.
This temperament splits the 1/2-comma wolves between four fifths instead of two, allowing for four 1/4-comma wolves to take their place. 1/4-comma wolves are used extensively in meantone and are much easier to tune and to listen to. This also eliminates two of the three pure thirds found in Kirnberger II. Therefore, only one third remains pure (between C and E), and there are fewer Pythagorean thirds. A greater middle ground is thus reached in this improvement, and each key is closer to being equal to the next. The drawback is an aesthetic one: fewer chords have pure thirds and fifths. But every temperament system has a give-and-take compromise; each has to find a way of dealing with the comma.
Musical temperament
In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...
developed in the second half of the 18th century by Johann Kirnberger
Johann Kirnberger
Johann Philipp Kirnberger was a musician, composer , and music theorist. A pupil of Johann Sebastian Bach, he became a violinist at the court of Frederick II of Prussia in 1751. He was the music director to the Prussian Princess Anna Amalia from 1758 until his death. Kirnberger greatly admired J.S...
. Kirnberger was a student of Johann Sebastian Bach
Johann Sebastian Bach
Johann Sebastian Bach was a German composer, organist, harpsichordist, violist, and violinist whose sacred and secular works for choir, orchestra, and solo instruments drew together the strands of the Baroque period and brought it to its ultimate maturity...
, held great admiration for his teacher and was one of his principle 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
Circle of fifths
In music theory, the circle of fifths shows the relationships among the 12 tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys...
. Unfortunately, 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 we tune 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
Comma (music)
In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. The word "comma" used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning...
), 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.
Practical Temperaments: Kirnberger II
Kirnberger's first method of compensating for and closing the circle of fifths was to split the "wolfWolf interval
In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament...
" interval known to those who have used meantone temperaments in half between two different fifths. That is, to compensate for the one extra comma, he removed 1/2 a comma from two of the formerly perfect fifths in order to complete the circle. In so doing, he allowed the remaining fifths to stay pure. At the time, however, pure thirds were more valuable than pure fifths (meantone temperament had 8 pure thirds and sacrificed four entire chords to achieve this end.) So, Kirnberger allowed for 3 pure thirds, the rest being slightly wide and the worst being three Pythagorean
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
thirds (22 cents wider than pure ) on the opposite end of the circle from the pure thirds. To put it graphically:
C-----G-----D------A-----E-----B-----F#-----C#-----Ab(G#)-----Eb-----Bb-----F-----C
p p -1/2 -1/2 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 "-1/2" 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 (between B-D, F-A(B), D-F.)
Temperament, however, is a give-and-take situation: none is a perfect solution. It must also be remembered that temperament applies only to instruments with fixed pitch: any keyboard instrument, lutes, viols, and so forth. Wind musicians, singers and string players all have a certain degree of flexibility concerning the exact pitch and intonation of what they play and may therefore be free of such restrictive systems. When the two classes of players come together, it's important when evaluating a temperament to look at the tendencies of the instruments vs. those of the temperament. Kirnberger II would only have been applicable to keyboard instruments such as the harpsichord
Harpsichord
A harpsichord is a musical instrument played by means of a keyboard. It produces sound by plucking a string when a key is pressed.In the narrow sense, "harpsichord" designates only the large wing-shaped instruments in which the strings are perpendicular to the keyboard...
and organ
Organ (music)
The organ , is a keyboard instrument of one or more divisions, each played with its own keyboard operated either with the hands or with the feet. The organ is a relatively old musical instrument in the Western musical tradition, dating from the time of Ctesibius of Alexandria who is credited with...
. The perks of this system are its three pure thirds: certainly pleasing to especially singers and winds at the time and lovely to hear. In fact, after hearing pure thirds, one is shocked by the wide thirds of modern equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
used on all pianos and many organs. Also, the temperament uses 10 perfect fifths, which is good news to string players whose strings are tuned in fifths. The drawbacks are, of course, the two narrow "half-wolf" fifths and the three Pythagorean, super-wide thirds. The chords are not unusable but certainly must not be used frequently or in close succession within the course of a piece.
Kirnberger III
After some disappointment with his sour, narrow fifths, Kirnberger experimented further and developed another possibility, Kirnberger III. (Remember, of course, that these names were not used at the time and are of modern invention.)This temperament splits the 1/2-comma wolves between four fifths instead of two, allowing for four 1/4-comma wolves to take their place. 1/4-comma wolves are used extensively in meantone and are much easier to tune and to listen to. This also eliminates two of the three pure thirds found in Kirnberger II. Therefore, only one third remains pure (between C and E), and there are fewer Pythagorean thirds. A greater middle ground is thus reached in this improvement, and each key is closer to being equal to the next. The drawback is an aesthetic one: fewer chords have pure thirds and fifths. But every temperament system has a give-and-take compromise; each has to find a way of dealing with the comma.
Further reading
- Klop, G.C. (1974) Harpsichord Tuning. The Sunbury Press, Raleigh, N.C. Designated "Kirnberger 2" and presumably other Kirnberger temperaments.