Solving Bach's temperamental puzzle

By Jodi H. Beyeler

Bradley Lehman '86 has solved the centuries-old mystery of what appeared to be an arbitrarily scribbled design on an original copy of one of J.S. Bach's compositions (see the design below). The results have significant ramifications for the world of music history, performance, theory and instrument building.

As with most puzzles, once the key concept is discovered, the design appears obvious and everyone wonders how it could previously have been overlooked – in this case, for 250 years. The very nature of the problem itself often becomes clearer than it had been before; working backwards a decipherer can also see what an excellent solution reveals about the questions it answers.

But as another mystery-solver, Galileo, said, "All truths are easy to understand once they are discovered; the point is to discover them."

The puzzle

Lehman, a Goshen native who now resides in Dayton, Va., was primed to unlock the puzzle of Bach's keyboard tuning – a classic problem in Bach research for at least a century. His love for music, one of his concentrations in a Goshen College double major with mathematics, and his three graduate degrees from the University of Michigan were all essential background to understanding the problem and seeing the solution.

Lehman has taken the liberal arts to heart, appreciating deeply how fields of study can overlap and inform one another. He was Goshen's work-study student in charge of harpsichord maintenance and tuning, a valuable hands-on start for understanding the practical problems. He studied German and French, both of which were essential in his tuning research. He also designed part of Goshen's administrative computer system and became an avid player of the game of bridge. His degree programs at Michigan gave him further specialization in music history and early-keyboard performance.

Always fascinated by games and problem-solving puzzles, Lehman is a full-time business software developer for EPOS Corp., which serves large companies and state governments.

Keyboard tuning is an essentially similar puzzle, setting up a system that responds most appropriately to the music that is to be played. Whether it's a bridge hand, computer program, tuning keyboards or playing a Bach fugue, it's all the same problem, in Lehman's mind.

The puzzle Bach unknowingly left to be solved involves a tuning method, or temperament. Essentially, the practical challenge for keyboard musicians over the past several hundred years in performing particularly Bach's music has been both a mathematical and a musically artistic one. The whole and half notes in a scale, and the thirds and fifths of chords, cannot all be made to fit into an octave in exactly the proportions that physics and geometry would suggest are appropriate. So, tuners have had to make careful adjustments and de-tune some of the notes, coming up with tasteful compromises. The methods used to determine these compromises are called "temperaments" – and there are hundreds of them.

Today, most musicians and tuners use equal temperament, which places all the notes in a scale at exactly the same distance apart in pitch in order to resolve the differences. That is the norm on pianos and most organs, but it creates as many musical problems as it proposes to solve, Lehman said.

Last December, while visiting family in Goshen, Brad Lehman (pictured at left) had an opportunity to play Opus 41 – installed in its permanent home, Rieth Recital Hall – after first seeing the instrument in the workshops of Boody and Taylor, which is located near Lehman's home in Virginia.

"The disadvantage of equal tuning is that it's tricky to set up, and it's relatively boring to listen to. Some of the notes are far out of tune due to this particular compromise, and the overall resonance of the instruments is reduced," Lehman said. "People 300 years ago knew about equal tuning as an option. But most purposefully did not use it because they didn't like the resulting lack of interesting variety" in how their music sounded.

With all other temperaments – those that are unequal – the notes in the scales are shifted by subtly different amounts, that give each temperament a certain character and renders some scales or chords purposefully more in tune than others. Most tuning methods tend to favor the simple keys that do not include many sharps and flats, sounding "sweeter and gentler" in that music. Consequently, they are also sometimes decidedly wrong when many sharps or flats are involved.

Historians and musicians had long concluded that if Bach held a preference for a tuning other than equal temperament, he did not note it or no one knew about it. But historical accounts suggest that Bach could tune a harpsichord in 15 minutes – a quite speedy rate – thus suggesting to Lehman, working as a harpsichord tuner himself, that the composer must have used a temperament that was simpler and quicker to set up than equal tuning. In addition, Bach would have wanted a solid all-purpose solution to leave in place for years, in which all the music to be played, as retuning an organ takes weeks and a lot of money. None of the known temperaments addressed those problems adequately.

So for the past 22 years in which he has studied harpsichord and organ, Lehman has always tuned keyboards based on the pieces he would play, which at times limited the repertoire he could perform within the same concert since the different temperaments are preferred for different pieces.

For Lehman, and thousands of other keyboard musicians, the need has persisted to discover a method of tuning that would work equally well – and yet differently – for all scales and chords. "None of the tunings I had studied and tried on the harpsichord worked right for all of Bach's music. And the reference books said it couldn't be figured out, or else somebody would have done it already," he said. "But music shouldn't sound either boring or randomly out of tune! It doesn't make much sense that Bach would write music that sounds bad, on purpose."

As it turns out, Lehman's mind was perhaps more tuned to Bach's all along than he could have known.

The rediscovery

For 250 years, hand-drawn loops that appeared atop the title page of Bach's 1722 "Well-Tempered Clavier" – seemingly mere decoration – offered a key to the secret of a preferred tuning, but its subtlety disguised its solution to the puzzle.

The flourish was not even considered as a link to Bach's temperament, according to Lehman. "Historiography today is so firmly based on the meaning of words. No other types of evidence are typically allowed to come to the table, not even the sound of the music itself, because that is so difficult to put into adequate words. During my lifetime, this particular piece of evidence has been officially meaningless, according to the Bach research institutes," he said. "It's little wonder, then, that few have paid attention to it."

Then in March 2004, Lehman received an e-mail message from a British harpsichord and clavichord enthusiast, David Hitchin, who reported that two German researchers suggested that the irregular line drawing might be a clue to a temperament. But Lehman felt the temperament that the Germans were proposing didn't make sense; some of its premises were clearly incorrect.

"Then it hit me: Bach's diagram does describe his temperament and merely needed to be interpreted correctly," Lehman said. "Bach wrote down the math of his tuning method here in a simple way, not by a chart of numbers, but by drawing a picture of it. This picture tells us exactly how the tuning should be set up, unequally, so the music in that book and his other books sounds best."

Interestingly, Bach, unlike Lehman, hated math, and was poor at writing and spelling. Being a musical genius, Bach knew what temperament sounded most pleasing, but instead of recording an explanation, he simply jotted down a practical description of his preferred temperament. He wrote this in what was, to him, a very logical place: in a book about tuning that was part of the 37-year-old's set of teaching materials.

Bach turned his book upside-down to draw the diagram so that when viewed right side up, it appeared in reverse. "The different types of shapes in Bach's drawing tell the tuner how to make the specific notes almost in tune together, but slightly out of tune by a little amount, on purpose. This drawing is Bach's recipe to set up all 12 of the notes exactly right. I believe that he was hiding a family secret, by writing it upside down and making it appear unimportant," Lehman said.

As soon as Lehman translated Bach's 1722 drawing and tuned his own harpsichord according to Bach's lesson on paper, "the way it sounded, so beautiful and unexpected, made me cry. I knew I was on to something big."

Lehman then began playing through all of Bach's keyboard music "to see what might pop out from Bach's pattern of intonation," he said. "To understand all this, it is crucial to do extensive listening and experimentation. This specific layout of adjusted intervals does some amazing things not only within the keyboard and organ music, but also in Bach's vocal music."

And according to Lehman, Bach's all-purpose temperament is applicable beyond the composer's own pieces. "This particular unequal tuning aids all tonal music to sound lucid and beautiful, full of natural tensions and resolutions," Lehman said. "By comparison, other harpsichord tunings make the instrument sound tinny and rough. And likewise, other tunings make organs sound occasionally sour, or merely dull."

It is perhaps because the tuning was taken for granted that it was not more consciously noted and preserved, yet after Bach's students died, the secret to Bach's temperament was silent as well. As those with even a trained ear will tell you, the difference this tuning makes can be quite subtle, though they also say that it is at times quite obvious – much like the design Bach drew to describe the temperament. "This tuning, almost like magic, simply sounds very easy to listen to all the time, unproblematic," Lehman said. "The tuning does not draw attention to itself, but rather to the progress of the music."

This tuning influenced Bach's composing as well as his performance of his music, according to Lehman. "The correct tuning according to Bach's expectations reveals a lost layer of his art. It makes the spiritual content of his music more easily perceptible and measurable. There are hidden meanings in the sounds produced. We return to Bach's music so often already because the sound of it moves us – and now even the sound itself is shown to be different from our modern expectations."

Solving Bach's temperament puzzle has unleashed a new chapter of music for Lehman. "I feel as if I have had the best single music lesson of my entire life, and that it's happened at Bach's house. The instruments at my house are now tuned the way they were at his house. And with every piece I work on, whether it's by Bach or not, it's now influenced by his way of hearing music. It affects so many different levels of my playing, improvising and composing."

The Goshen homecoming

Since discovering Bach's preferred tuning method in April 2004, and translating the details necessary to tune keyboards accordingly, Lehman has tested his findings with others in the field. The resulting temperament has already been implemented in music performed in concert tours by Apollo's Fire and The English Concert; broadcasts on the BBC and Swiss radio; and in London's Wigmore Hall and New York's Lincoln Center in October 2004 and in Amsterdam's Concertgebouw in February.

Last summer, Lehman approached Taylor and Boody Organbuilders with an explanation of the temperament he prepared for a journal article, inquiring about the possibility of Bach's temperament being built into Goshen's new organ, Opus 41. Taylor and Boody tested the results for several months during a study of organ building by Bach's own associate, Zacharias Hildebrandt; this brought important validation of Lehman's historical and technical research. In October, the organ builders notified Lehman that they were "very excited about the sound" and committed to using it, both for musical and historical reasons. The Goshen College pipe organ is now the first full-sized organ to be tuned this way since Bach's work in the 18th century.

For Lehman, this is merely Goshen College's investment coming back to itself as deserved, both tangibly and in worldwide exposure. "As a friend pointed out to me, this sort of thing is why colleges give academic scholarships – to invest in the students' abilities. I'm thrilled that Goshen is the first to get back the results of that long-range commitment to excellence both in this beautiful organ itself, and in the academic distinction of supporting cutting-edge research. In my 'senior statement' in a chapel 19 years ago, I played Bach's organ music and had some words about my appreciation for interdisciplinary studies. My principles are still similar, but the organ music now sounds different."

Lehman's research is available in full detail in the February and May 2005 issues of Early Music, a journal published by Oxford University Press, and available to download from its Web site at http://em.oupjournals.org. Lehman has also created a personal Web site with extensive information about his findings at http://www.larips.com. He is making a recording of organ music on Opus 41 for all to hear the "Bach/Lehman 1722" temperament.

Related links:

Goshen College press releases: