Now I want to start talking about some important well-tempered tuning systems invented by Johann Philipp Kirnberger. But first: who was this guy? As I tried to answer this question for myself I became sort of fascinated with his personality.
Kirnberger was a German music theorist who played an important role in formalizing baroque harmony and counterpoint. He was born in 1721.
As a child he studied the violin and harpsichord at home. He then moved to another town to study the organ, and then at 17 moved again to start seriously studying the violin. At the age of 18, he went to Leipzig to study performance and composition with Johann Sebastian Bach. He did this intermittently for three years, but this seems to have been a pivotal period in his life. Bach was an energetic teacher, with about 300 students over his life, but unfortunately he never wrote down his thoughts on music. In the end, it largely fell to Kirnberger to systematize Bach’s ideas on harmony and composition.
Between the ages of 20 and and 30, Kirnberger worked in Poland and wrote a book on Polish dances. He then became a violinist at the court of Frederick II of Prussia, and from the age of 37 until his death at the age of 61 he was music director for the princess of Prussia.
But most of all, Kirnberger was a big fan of Bach. He called Bach “the greatest of all composers.” Around the age of 40 he published a book of Bach’s clavier pieces, and he worked hard to publish all of Bach’s chorales, which finally appeared after Kirnberger’s death. He preserved many of Bach’s manuscripts in his library. He even wrote some pieces that for a while were attributed to either J. S. Bach or C.P.E. Bach—like this concerto for harpsichord, written when Kirnberger was about 50:
Kirnberger was better as a music theorist than a performer—though as a theorist he was quite polemical. In 1794, ten years after Kirnberger’s death, a musician named Friedrich Nicolai wrote:
Kirnberger has many good musical ideas […] he deserves full credit as a theorist. But he is unable to bring any of his ideas to good musical fruition, perhaps because of insufficient ability. His aim is not to see good music performed, but merely to find music containing “errors” so that he may make learned-and often violent-statements about others’ mistakes. As a performer he has practically no skill at all, except when playing his own compositions; his sense of rhythm is especially uncertain.
His frustrations seem to have driven him to mathematics. When Kirnberger was 52, Charles Burney wrote that he
is said to be soured by opposition and disappointment; his present inclination leads him to mathematical studies, and to the theory of music, more than the practice […] In his late writings, he appears to be more ambitious of the character of an algebraist, than of a musician of genius.
His discovery of the ‘atom of Kirnberger’, which I explained last time, indeed seems like something only a person with a strong mathematical bent could do!
His three-part theoretical work Die Kunst des reinen Satzes in der Musik, or The Art of Strict Musical Composition, had a big impact in his day. However, he wrote poorly. Nicolai wrote:
Kirnberger considered himself to be a philosophical musician. In reality, he had pondered over his art more than other musicians. For all that, he did not have clear concepts about so many things, still less philosophically correct ones. Because he had no formal education at all and had read little, he lacked much necessary knowledge, which he could acquire only by considerable effort through association with scholars; therefore, he sometimes could not explain rather ordinary things clearly. Scholars who wanted to come to an understanding with him had to divine his meaning.
In fact, parts of Die Kunst des reinen Satzes in der Musik were actually written by a student of Kirnberger’s who could write more clearly. This student, Johann Abraham Peter Schulz, said as much:
I had just recently made a systematic reduction of his [Kirnberger’s] principles of harmony for my own benefit and satisfaction. And at his request I had applied this system practically to the analysis of two pieces by Joh. Seb. Bach, which are difficult to understand […] His student’s writing pleased the teacher and he permitted it to be published under his name.
That could get someone in trouble today.
Kirnberger was an argumentative man. He was quite harsh in his condemnation of two other important theorists, Rameau and Marpurg. In 1800, a fellow named Reichart wrote:
Kirnberger was a very passionate man who gave himself up to his impetuous temperament […] The cultivation of his art, as he saw it and believed to embrace it, went before everything. The few righteous musicians whom he acknowledged possessed him completely and absorbed his entire disposition. Everything that did not immediately further the higher part of the art […] he despised and considered repugnant.
All in all, a curious and interesting character.
Kirnberger’s tuning systems
As discussed in Part 1, in a ‘well temperament’ each key gives a scale with its own different flavor. Quite a number of well temperaments had been used since Andreas Werckmeister invented three of them starting in 1681. (I’ll explain those later.) By the time Kirnberger got involved, equal temperament was beginning to take over. In fact in 1760, at the age of 39, he published something called Construction der gleichschwebende Temperatur, about the construction of equal temperament. But in Die Kunst des reinen Satzes in der Musik he explained that he didn’t like equal temperament, because it reduced the diversity of scales down to just two: major and minor. And in a letter 19 years later, he
wrote:
Equal temperament is absolutely terrible, only being useful in the case of properly positioning the frets of a theorbo, lute or other such similar instrument such as a psaltry, zither etc., as a temperament of another type does not do each string
justice.
Today, Kirnberger is mainly known for two well-temperaments called Kirnberger II and Kirnberger III. But he seems to have put work into a least two more tuning systems. First, unsurprisingly, there’s his well temperament called Kirnberger I. Second, there is ‘rational equal temperament’—a system I explained last time. But this is so close to 12-tone equal temperament that nobody can hear the difference: its only advantage, if you can call it that, is having frequency ratios that are rational numbers—with, unfortunately, rather huge numerators and denominators. I doubt anyone has actually used it, except perhaps as an experiment.
What are Kirnberger I, II and III actually like?
First, as a point of comparison, recall the tuning system that all well-tempered systems are responses to. Namely, quarter-comma meantone:
This system has a lot of fifths that have been lowered by a quarter comma: that is, divided by σ1/4 where σ is the syntonic comma. These ‘quarter-comma fifths’ are just slightly smaller than the ‘just’ perfect fifth, namely 3/2. So that’s good. It has a lot of ‘just’ major thirds, with frequency ratios of exactly 5/4, shown as the blue arrows above. So that’s great. But to pay the price for all those quarter-comma fifth, it has a ‘wolf fifth’ that’s 128/125 times bigger than all the rest. And that’s noticeably ugly!
Well-tempered fifths seek to kill the wolf. Here’s how Kirnberger I does it:
Here’s Kirnberger II:
And here’s Kirnberger III:
Puzzle. Do you see the pattern?
I’ll examine these systems in more detail next time.
References
The quote of Kirnberger saying equal temperament is absolutely terrible comes from here:
• Dominic Eckersley, The Rosetta revisited: Bach’s very ordinary temperament, Berlin 2012.
It originated in a letter from Kirnberger to someone named Forkel, written around 1779.
I got all my other quotes from here:
• Beverly Jerold, Johann Philipp Kirnberger and authorship, Notes 69 (2013), 688–705.
Notes is a nice name for a music journal! This article is about Kirnberger and his student Schulz, analyzing how much each might have contributed to the writing of Die Kunst des reinen Satzes in der Musik and also the encyclopedia Allgemeine Theorie der schonen Kunste (General Theory of the Fine Arts). To help figure this out, Jerold investigates Kirnberger’s views and personality.
For more on Pythagorean tuning, read this series:
For more on just intonation, read this series:
For more on quarter-comma meantone tuning, read this series:
For more on well-tempered scales, read this series:
• Part 1. An introduction to well temperaments.
• Part 2. How small intervals in music arise naturally from products of integral powers of primes that are close to 1. The Pythagorean comma, the syntonic comma and the lesser diesis.
• Part 3. Kirnberger’s rational equal temperament. The schisma, the grad and the atom of Kirnberger.
• Part 4. The music theorist Kirnberger: his life, his personality, and a brief introduction to his three well temperaments.
• Part 5. Kirnberger’s three well temperaments: Kirnberger I, Kirnberger II and Kirnberger III.
For more on equal temperament, read this series:
Azimuth 
I don’t understand that.
It's an overstatement, because there are still things like Dorian mode. But in equal temperament, D major is exactly the same as C major scaled up. In any other system, D major is different from C major scaled up, because the commas appear in different places. So the individual scales all have their own character (although that character depends on what temperament you're using).
From Kirnberger’s perspective there probably weren’t things like Dorian mode. Western European composers for well over a century before him had used major and minor almost exclusively.
Yes, I think Kirnberger’s comment implicitly shows how major and minor had completely displaced all the other modes by around 1750! There’s a nice book
• Paul Walker, Fugue in the Sixteenth Century, Oxford U. Press, Oxford, 2020.
which shows how around 1550 great composers like Gombert, Crecquillon, and Clemens non Papa were developing the idea of a fugue but still using different modes as earlier church music did… thus writing fugues that sound nothing at all like Bach! Two centuries later all that diversity was gone, and equal temperament was well on its way toward crushing the difference between keys…. though Kirnberger resisted it.
I love Baroque music, but one good thing about modern music is the resurgence of at least some of these old modes (especially Dorian, my personal favourite). Losing the more subtle distinction between the keys, is a price that I'm willing to pay to get the other modes back, if that's what it comes down to.
I tend to agree here. Also, equal temperament allows one to play a fixed-fretted instrument in all keys.
I recently saw Roine Stolt play a guitar which had frets of the same number at different places on different strings. I didn’t completely figure it out, but it obviously can’t work in all keys.
Right, so I would have expected him to say that there is essentially only one left, not two.
If two, then seven, because there are seven modes, but they sound different from each other for a different reason than different keys sound different in unequal temperament.
Philip Helbig wrote:
As we’ve seen, Kirnberger is famous for being unclear. But I’m sure he meant that in any 12-tone system other than equal temperament there are multiple different-sounding major scales, and multiple different-sounding minor scales—while in equal temperament all major scales sound the same (up to a constant rescaling of the frequencies) and all minor scales sound the same.
The goal of well temperaments (like Kirnberger’s) is to avoid this boring feature of equal temperament while still having many major and minor scales that sound quite good. So when I discuss Kirnberger’s well temperaments, and others, I’ll say something about how the different major scales sound different. (I probably won’t have the energy to analyze the minor scales.)
Modes are alive and well and living in prog: https://www.loudersound.com/news/swirling-phrygian-runs-insistent-sax-full-on-vocals-and-blistering-guitar-its-steve-hacketts-new-single-circo-inferno
Actually, modes are not uncommon in rock music. Many hard rock/heavy metal songs are in Dorian.
https://www.fretjam.com/metal-guitar-scales.html
I only just now got around to looking at the puzzle closely enough to spot the pattern. One thing I find interesting is that in all three of Kirnberger’s well temperaments, the wolf fifth has been replaced by something (the schismatic fifth) which is closer to both a just fifth and to an equal tempered fifth than the fractional-comma fifths that appear between some very frequently-used notes. So that F sharp-D flat interval, which is pretty easy to avoid in practice, is in some sense “better” than D-A.
I guess that’s just the sort of compromise you have to make if you want to prioritize nice frequently used thirds.
Yes, the schismatic fifth is quite acceptable, unlike what it was replacing: the wolf fifth in quarter-comma meantone:
just perfect fifth = 1.5
schismatic fifth ≈ 1.4983081847
equal tempered perfect fifth ≈ 1.4983070769
quarter-comma wolf fifth ≈1.53123715197
The schismatic fifth is indistinguishable to the ear from a modern equal tempered perfect fifth—that’s the amazing coincidence Kirnberger discovered. And the equal tempered fifth is only 1/12th of a Pythagorean comma flat compared to the just perfect fifth. The quarter-comma wolf fifth is about 7/4 commas sharp—so, about 21 times worse!
And your point is an excellent one: Kirnberger has in some sense fixed the wolf fifth too well, at the expense of some other more commonly used fifths.
But my puzzle about patterns was really trying to get people to notice the way in which Kirnberger I, II and III form a systematic progression. If you numbered them in a different order it would be less nice.