The tuning system called ‘quarter-comma meantone’ dominated western keyboard music from about 1550 to roughly 1690. The reason: it has very nice thirds and fifths in many different keys!
But as I keep saying, every tuning system has problems: like lumps in the carpet, the best you can do is move the problems around. Quarter-comma meantone achieves its greatness by completely flattening out the carpet except for one big lump: a single highly dissonant ‘wolf fifth’. Alas, this utterly spoils keys where that fifth is important—or other chords using the note that creates that wolf fifth, which is F♯ in the chart above.
As Baroque musicians became increasingly interested in switching between keys, there was pressure to find tuning systems where the lumps in the carpet were more evenly spread out. But interestingly, they did not embrace equal temperament, where the lumps are spread out as evenly as possible.
It’s not that equal temperament was unknown! Apparently, musicians wanted some keys to have truly beautiful fifths and thirds, and weren’t willing to sacrifice that beauty and purity to make all keys sound equally good—or bad. Thus, they invented compromise systems, called well temperaments, in which each key has its own personality, but all sound reasonably good.
You can see these personalities discussed in Christian Schubart’s Ideen zu einer Aesthetik der Tonkunst, written in 1806:
C Major. Completely pure. Its character is: innocence, simplicity, naïvety, children’s talk.
C Minor. Declaration of love and at the same time the lament of unhappy love. All languishing, longing, sighing of the love-sick soul lies in this key.
D♭ Major. A leering key, degenerating into grief and rapture. It cannot laugh, but it can smile; it cannot howl, but it can at least grimace its crying. Consequently, only unusual characters and feelings can be brought out in this key.
C# Minor. Penitential lamentation, intimate conversation with God, the friend and help-meet of life; sighs of disappointed friendship and love lie in its radius.
D Major. The key of triumph, of Hallejuahs, of war-cries, of victory-rejoicing. Thus, the inviting symphonies, the marches, holiday songs and heaven-rejoicing choruses are set in this key.
It’s with great effort that I resist listing all 24 keys! You’ll just have to visit the link to see which key “tugs at passion as a dog biting a dress”, and which has “pious womanliness and tenderness of character”.
When equal temperament took over in the early 1800s, all this diversity was flattened, although the reputations of the different keys persisted for quite some time. Some consider this flattening a tragedy; others say it opened the doors to Beethoven and jazz. Perhaps both are true.
But what were these well tempered systems, exactly? What were their distinct advantages? Here things get much more complicated and murky. For example:
The biggest advertisement for well-tempered tuning systems was Bach’s The Well-Tempered Clavier. In 1722 and then again in 1742, he wrote a piece in each of the 12 major and 12 minor keys, to illustrate the flexibility of well temperament—and presumably to showcase how different keys had different personalities. But which well tempered system was he actually using?
Nobody knows! We don’t have Bach’s words on this topic, and despite a vast amount of scholarship nobody has been able to pin it down. Serious musicologists have even spent serious time studying a doodle in Bach’s manuscript of The Well-Tempered Clavier, hoping it contains an encoded description of his tuning system! There’s no proof that it does.
For a fun but insightful introduction to the controversy, watch this:
For more, try this:
• Wikipedia, Bach temperament.
This seems to be the best in-depth survey of the subject:
• Sergio Martínez Ruiz, Temperament in Bach’s Well-Tempered Clavier: A Historical Survey and a New Evaluation According to Dissonance Theory, PhD thesis, Departament d’Art i de Musicologia, Universitat Autònoma de Barcelona, 2011.
I haven’t read most of it, but it looks to be a treasure chest of information on well temperaments, their history and their mathematics. And frankly, I find that much more interesting than the futile quest to figure out what Bach was thinking.
There are a lot of interesting well tempered systems. People wrote a lot about them when they were invented, and much more since. So we are not reduced to decoding Bach’s squiggles to understand well temperaments. The hard part, at least for this mathematician, is figuring out their governing principles.
In my efforts, I’ve been helped immensely by this website:
• Carey Beebe Harpsichords, Technical library: temperament FAQ.
He describes about 30 different tuning systems using circular diagrams—a method that I’ve decided to copy in my blog articles here. He is less interested in the math than I am. But he is more more efficient at explaining tuning systems than other sources, and covers other topics: for example, he describes how to tune a harpsichord in all these systems!
Instead of covering well-tempered systems chronologically, I’ll start with the ones I find easiest to explain. I’ll try to cover some of the most important ones, but certainly not all. Many are named after people like Werckmeister, Kirnberger and Vallotti, while some have descriptive names like sixth-comma meantone.
The danger is getting lost in the undergrowth of these tuning systems and not seeing the forest for the trees. So before diving in, I’ll start by surveying some of the mathematical principles that seem to be at work. You could see these between the lines of what I’ve written so far, but I want to be a bit more explicit.
For example, what’s really going on with these weird numbers:
• the Pythagorean comma, p = 531441/524288 ≈ 1.01364
• the lesser diesis, δ = 128/125 = 1.024
• the syntonic comma, σ = 81/80 = 1.0125
and this relationship:
Why have these funny things mattered so much in the history of tuning systems?
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: