Bach’s Well-Tempered Clavier is one of the most beloved works for the keyboard. A clavier can be any keyboard instrument. But what is a “well-tempered” clavier?
Musical intervals – the perceived space between the notes that make up scales and chords – are based on ratios of integers. Pythagoras, who copyrighted the famous theorem that bears his name, also discovered the mathematical relationship between musical tones. If a string of a given length is cut in half, the short string will sound one octave higher than the original. If the long string sounded middle C, the short string will sound the C above middle C. It will vibrate twice as fast.
If the ratio is 3:2 instead of 2:1, we get a “perfect” fifth. If the longer string gave us middle C, the shorter string will give us the G above middle C. It will vibrate 1.5 times faster than the longer string. Pythagorean tuning derives the entire scale of 12 tones using only these two ratios. They get there by continuing to cut the string in a 3:2 ratio. If you cut the string that produced G, you get D, the “perfect” fifth above G. The next fifth will be A, and so on.
If you perform this process twelve times, you will produce twelve distinct tones.[1] You will have completed the “circle of fifths”. However, there is a flaw in the system. The circle of fifths does not quite close. After we stack our 12 perfect fifths, we find that we have not traveled from C to shining C. Instead, we have ended up at a note that is about a quarter of a semi-tone too sharp. Instead of landing back at C, we are one quarter of the distance from C to C#.
Another way to say this is that if you start at C and go up six perfect fifths, you will arrive at F# (C-G-D-A-E-B-F#). If you start at C and go down six perfect fifths, you will arrive at Gb (C-F-Bb-Eb-Ab-Db-Gb). These will be the same note on a keyboard – the black key between F and G. But in Pythagorean tuning based on perfect fifths, they aren’t the same tone. They are off by the tiniest fraction.[2] That nearly quarter of a semitone is called the Pythagorean comma. The difference is audible. If you go to the Wikipedia article titled Pythagorean comma, you can hear it. When the two notes are played at the same time, the result is unpleasant and unharmonious.
What is to be done with the comma? The Pythagorean solution was to dodge the problem by discarding one of the two notes at the ends of the circle of fifths (either F# or Gb if you start the circle at C) and avoiding combinations of notes that highlighted the comma. In the Pythagorean system, all of the fifths except the last one would produce a beautiful sound. The “last” fifth (which one it is depends on which note you start with) produced a jangling dissonance called a “wolf” fifth. The Pythagorean attitude was that there is plenty of beautiful music to be made with the perfect fifths. All you have to do is avoid the wolf fifth.
Other problems accumulate. Major and minor thirds don’t sound quite right under Pythagorean tuning. The Pythagoreans considered thirds to be dissonances. They were not about to adjust their precious fifths to accommodate an interval that they found unpleasant in the first place.
For those who like major and minor thirds just fine, an alternative is to make the thirds sound better in exchange for less perfection in the fifths. “Just” intonation recognizes that there are other musically significant ratios of whole numbers in addition to the 3:2 ratio that produced a perfect fifth. A ratio of 4:3 gives us a “perfect” fourth, C to the F above it. 5:4[3] gives us a major third (C to the E above it) and 6:5 a minor third (C to the Eb above it). The rates of vibration of the strings, measured in cycles per second, will be in the same ratio.
You can use the 5:4 and 6:5 ratios to build a musical scale that will have “pure” tuning. If you tune a piano so that the thirds line up with a given note, let’s say C, you will have a scale and chords that will possess great beauty. You can move up a key or down a key, up to G (one sharp) or down to F (one flat) and all will still be well. But as you move away from the home key, certain chords will become unpleasant to the ear. Just intonation improves the thirds at the expense of the fifths, but doesn’t solve the problem. The Pythagorean comma is still there and will find us out.
Why doesn’t it find us out on the piano that you might have in your parlor or hear at a kid’s school, or on the one that sits in the corner of the derelict bar down the street where you may be reading this? Why do F-sharp and G-flat sound the same on that piano? The answer is that the tuning of each string on that piano has been altered very slightly to spread the dissonance of the Pythagorean comma over the twelve semi-tones that make up each octave. The octaves are perfectly in tune but nothing else is.
Before roughly 1700, musicians simply worked around the problem. If an instrument were tuned to a particular key, composers and performers could not move very far away from that tonal base. They got to know which chords and which changes of key would not work and they stayed away from them. It appears that early musicians could get 10 of the 24 keys to work, some better than others, and there things stood.
Ingenious musicians and technicians continued to play with the mathematics and by 1700 had found various ways to spread the Pythagorean comma over the octave, to “temper” the notes in the scale so that music could be made in each key. The trick is to face up to the hidden disharmonious factors inherent in the apparently rational musical scale and to put everything except the octaves slightly out of perfect mathematical tuning – just enough to mask the comma without creating other distortions. Scholars say that as many as 150 different methods have been used.
There are twelve semi-tones in a chromatic scale. A key can be built on each one. And that key can be either major or minor, so there are 24 possible keys using the traditional western chromatic scale. An instrument is said to be “well-tempered[4]” if it can play harmoniously (but not necessarily identically) in all 24 keys.
Bach published his “Well-Tempered Clavier” in 1722, when he was in his late 30s. On the title page he describes the work as “Preludes and Fugues in all the tones and semitones, both with the major third . . . and with the minor third . . . . For the use and practice of young musicians who desire to learn, as well as for those who are already skilled in this study, by way of amusement . . . .”
This has to be one of the most modest, self-effacing descriptions of a work ever written by its author. It’s as if Shakespeare had said, “I have strung together a series of infinitives to explore the mind of a man at a crisis in his life, for the amusement and entertainment of actors and audiences.” The music that constitutes Bach’s demonstration of the proper tempering of a scale is a staggering achievement, one of the great treasures of western music. And just to show that it wasn’t a fluke, he published a second set, no less beautiful and profound than the first, some twenty years later. Consequently, we have The Well-Tempered Clavier, Book I and The Well-Tempered Clavier, Book II. Each book contains 24 Preludes and Fugues, so the whole thing is sometimes referred to as “the 48”. A nineteenth century critic called the Well-Tempered Clavier the “Old Testament[5]” of piano music.
The emotional and intellectual content of the music is inexhaustible. The technical and interpretive challenges to the performer must be enormous. Each piece provides a lens through which the performer and listener can view the shape and form of the emotions that the composer invites them to explore. Each piece presents its own mood and feeling and seduces the listener, for a few minutes, into the composer’s emotional world. Yet, despite their intensity, the 48 preludes and fugues maintain a clinical distance from their emotional content. My sense is that Bach is not saying, “Listen to this piece to experience happiness, joy, grief, solemnity, or melancholy (as the case may be)”. Rather, he is saying, “This is what happiness, joy, grief, solemnity and the rest look like when I put them under the microscope.” Each performer brings a different point of view. New insights come with each re-hearing.
How many gradations are there of happiness? How finely can the emotion be dissected? The C# major prelude and fugue of Book I present joyful exuberance. That same feeling is on display in Book II’s G major prelude and fugue, but the composer is now twenty years older and the fires of his enthusiasm have been banked by two decades of experience. There are different species of cheerfulness and well-being in Book I’s F# major, G major, and B-flat major preludes and fugues.
Monumental solemnity is on display in the C# minor prelude and fugue of Book I and, combined with a sense of mourning and resignation, in the final prelude and fugue of Book I, in B minor. Equally fascinating are the numbers where the prelude and fugue offer contrasts of form or feeling. One that stands out to this listener is the D major pair in Book II. Listening to the prelude, you can imagine the members of a choir having some musical fun before practice, tossing phrases back and forth from one section of the choir to the other, adding complications and challenges as they go. Then the fun ends, the choirmaster arrives, and everyone settles down to work through the phrasing of a serious, solemn fugue.
The most remarkable piece may be the very first one, the C major prelude of Book I. The prelude consists of a series of broken chords, eight notes to the measure. The second note of each measure is sustained as the remaining six are sounded. There is no melody; neither is there a sense that one is missing. In any other hands, this piece would have been considered an interesting accompaniment in search of a top line melody. Rosalyn Tureck notes that when Beethoven composed something similar – the first movement of the “Moonlight” sonata – he added a simple melody to give the piece the finished sound he was looking for. Yet Bach, who was gently mocked in his own time as old-fashioned and out of date, manages something that the great revolutionary of classical music did not attempt.
The C major prelude also demonstrates in miniature that the keyboard is correctly tempered. Every note in the chromatic scale is touched at least once. There are no wolf tones, no dissonances, only the interplay of light and shade as the composer uses every tone in the scale to introduce the great work that he has prepared for us.
The gothic arch and the flying buttress led to centuries of spectacular achievements in architecture. Did correct temperament lead the way to centuries of great music in the same way?
It’s doubtful. The paradox is that Bach’s solution is so much greater than the problem. We have been listening with pleasure and astonishment to Bach’s music for nearly three centuries. Had he been limited to the ten keys that his Renaissance forebears had to deal with, would the price have been so very high?
Bach does not appear to make use of remote keys in other works written for the keyboard. The Goldberg Variations are mostly in G major, with excursions into G minor and Eb minor. The English Suites, the French Suites, and the Partitas, are written in unexotic keys.
It is true that one of the most delightful pieces from Book I of the Well-Tempered Clavier is the C# major prelude and fugue. C# major is an unusual key. It requires no less than seven sharps – every tone is raised one half step. Would the piece be ruined if it were transposed down a half step to C major, a key in which there are no sharps or flats? I don’t think so.
My guess is that Bach wouldn’t think so either. He often transposed, re-using material originally written in one key in a new piece where he wanted it in a different key. The availability of a “well-tempered” keyboard provided the occasion for the production of 24 masterpieces in the prelude-and-fugue format, followed two decades later by 24 more. Had the pieces all been written in one key or a handful of keys, the work would still be celebrated – under a different name – as one of the great achievements of western culture.
It’s true that a composer may want to modulate into an exotic key to add tension and interest to a piece that starts in an accessible key. Correct temperament makes such modulations possible, but we can question whether that capacity is what accounts for the accumulation of great music over the course of the eighteenth, nineteenth, and part of the twentieth centuries.
The work was written in 24 distinct keys and each piece presents its own emotional world. Some critics think that this means that Bach associated a particular color with each individual key. If each key has its own atmosphere, that would suggest that Bach preferred one of the many forms of unequal temperament. On the other hand, Bach’s habit of transposing freely between keys argues that he did not associate a piece’s key with its emotional content.
Did Bach use equal temperament, then? Equal temperament has been the standard method of keyboard tuning since the middle of the nineteenth century. Every semitone is equally wide. Every key is identical, except for pitch. However, the mathematics of equal temperament are complicated and it is doubtful that it was widely used prior to about 1850[6].
Contemporaries reported that Bach could tune a harpsichord in about 20 minutes. That fact alone argues against his use of equal temperament. The musician and scholar Bradley Lehman thinks the title page of the work gives Bach’s directions for tuning a keyboard. The top of the title page contains a series of curls:
Please see: https:commons/wikipedia.org/wiki/File:Bach-Loops.png
Note the variation in shape and complexity of each curl. Lehman thinks that the set is a schematic of a scale on the keyboard. The different shapes tell the tuner how much to tweak each individual note to produce a correctly tempered scale. The instruction is not rigid and mathematical. Rather, it’s a model for how to feel the way to a correct tuning. The drawing tells the musician or technician where to add, where to subtract, tiny adjustments[8]. Lehman’s insight is intriguing, although controversial among the academics.
I have listened to a lot of recordings of this music, but the catalog is far larger than my limited experience. The first decision a listener has to make is: Piano or Harpsichord. I have commented before that Bach performances can be divided between those who produce a “wall of sound” and those who produce “intersecting planes”. I prefer the latter. For me, that eliminates the harpsichord, even though this was the instrument for which the music was written. Perhaps an ear trained to listen to the harpsichord can keep the threads of the music separated better than I can. I find it much easier to listen to this music performed on a piano.
Robert Levin offers another alternative. He recorded Book I using four instruments: two different harpsichords, a clavichord, and an organ. For Book II he used two harpsichords, a fortepiano, and an organ. He is a wonderful musician. I thought the pieces played on the organ sounded wonderful, but was less taken with the sound of the other instruments.
If you want “wall of sound” try Grigory Sokolov (known as the “world’s greatest living pianist”) or Glenn Gould. Sokolov’s recording on YouTube is accompanied by a photo of him holding the score of the Well-Tempered Clavier. The performance, in my opinion, has less to do with the score and more to do with the demonstration of how the work affects the world’s greatest living pianist. Glenn Gould does not attract any neutral opinions. Listeners are either enthralled or repelled. His recordings are not ones I feel compelled to return to.
Rosalyn Tureck, the “high priestess of Bach”, was a great musician and scholar. Her love for this composer can be heard in each piece. She is the antithesis of the “wall of sound” performers. The problem for me is that she performs the music at a slow, almost ponderous, pace. Most performances of the complete Book I and Book II require a bit more than four hours. Ms. Tureck takes about five hours.
My two favorite recordings are performances by Edward Aldwell (recorded 1989 & 1990) and Tatiana Nikolayeva (released 1984). Aldwell’s career was devoted to scholarship and teaching. He didn’t record much, but his recordings include these works for Nonesuch. His tempos are well judged and he lets the music speak for itself. It’s a performance of great integrity.
Nikolayeva (1924-1993) was somewhat older than Aldwell. She won the first Bach Competition in Leipzig (then East Germany) held in 1950 to commemorate the 200th anniversary of the composer’s death. Each competitor was supposed to prepare a single prelude and fugue from the Well-Tempered Clavier and perform it for the judges. When it was Nikolaeva’s turn, she went to the stage without a score in hand and asked the judges to name the prelude and fugue they wanted to hear. She then played the named piece from memory and won the competition.
Her performance is warm-hearted. Her love of this music comes through every note. Her recording and Mr. Aldwell’s are two that I return to with pleasure.
Aldwell finishes his notes to the recording of Book I with a quotation from a nineteenth century Bach biographer that provides a fitting close to these musings on Bach’s “Old Testament”:
There is a legend which tells us of a city of marvels that lies sunk beneath the sea: the sound of bells comes up from the depths, and when the surface is calm, houses and streets are visible through the clear water, with all the stir and turmoil of busy, eager human life – but it is infinitely far down, and every attempt to clutch the vision only troubles the waters and distorts the picture. We feel the same thing as we listen to this music. All the emotions that stirred the soul of the composer . . . lie deep below the surface: faintly, remotely, we hear their echoes . . ..
[1] Double the length of the string and drop down an octave now and again as you perform this operation. Otherwise, the string will be measured in fractions of an inch as you approach the last few intervals.
[2] Technicians divide the musical scale into “cents”. There are 1200 cents in an octave. Therefore, there are 100 cents in each semitone if you divide the 12-tone scale equally. When all semitones are equal, the distance from C to G, one fifth higher, is an even 700 cents. But Pythagorean tuning does not divide the scale evenly. The mathematics of Pythagorean tuning gives us a fifth that is slightly less than 702 cents. Our hearing cannot detect a difference smaller than 5 cents. But after you repeat the operation twelve times to generate all twelve tones, you have built up nearly 24 cents of differences. That’s clearly audible as you can confirm by playing the Pythagorean comma file available from Wikipedia.
[3] The tones of a major third under Pythagorean tuning are in the ratio of 81:64. Just intonation would use a ratio of 80:64. That tiny adjustment makes a big difference.
[4] Some experts say “correctly tempered” is a better description.
[5] The same critic, Hans von Bülow (1830-1894), labeled Beethoven’s piano sonatas the “New Testament”.
[6]The mathematics of equal temperament involve an exotic number. In equal temperament, the rate of vibration of each semitone is greater than the one below it by the same factor, X. When we stack twelve semi-tones in a row, the last note is an octave above the first note, which means that it is vibrating at a rate twice the original note. Therefore, X to the twelfth power equals 2. X is the twelfth root of 2. This conclusion became obvious to me after I read the explanation three times.
Twelfth root of two is an irrational number. The approximate value of this number had been calculated centuries before Bach. However, producing a scale using this number alone would have been technically challenging in Bach’s time. Note that equal temperament gives up all of the natural ratios except the 2:1 ratio for the octave.
I have read that some Renaissance organs were tuned to equal temperament. Organs don’t have to be tuned from day to day. The builder could set it and forget it. Not so with harpsichords, pianos, and the like.
[7] I regret that I could not figure out how to paste the image, which is in public domain.
[8] See http://www.larips.com/ for his explanation. He can also be found on YouTube, where you can hear the adjustments as he makes them. Daniel Jencka offers a possible refinement of Lehman’s insight. His website is down. A link to a paid site can be found here: https://academic.oup.com/em/article-abstract/33/3/545/2928360/Tempering-Bach-s-temperament?redirectedFrom=fulltext. The paper was available at no charge at one time. I did not think to download it when I read it.
The Passing Scene 