Kirnberger temperament
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Kirnberger temperament is an irregular temperament which was developed in the second half of the eighteenth century by Johann Kirnberger. Kirnberger was a student of Johann Sebastian Bach, and 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# (Ab), Ab-Eb, Eb-Bb, Bb-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 serious 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.
[edit] Practical Temperaments: Kirnberger II
Kirnberger's first method of compensating for and closing the circle of fifths was to split the "wolf" 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 in order 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 which were 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#(Bb), Db-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.
[edit] 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.
