Case Study 12.1: Bach's Well-Tempered Clavier — A Scientific Experiment in Tuning

The Most Famous Tuning Document in History

In 1722, Johann Sebastian Bach completed the first book of the Well-Tempered Clavier (WTC) — a set of 24 preludes and fugues, one in each of the twelve major and twelve minor keys. He completed a second book in 1742. The collection is often described as the crowning achievement of Baroque keyboard music, and it is certainly that. But it is also something more unusual: a scientific demonstration, an acoustic experiment conducted in notes.

Bach's title — "Das Wohltemperirte Clavier" — tells us something important. "Wohltemperirt" does not mean "equal temperament." It means "well-tempered" — a German term denoting any temperament in which all 24 keys are usable, as opposed to meantone temperament, in which several keys were acoustically unplayable. The WTC was, in part, a proof of concept: here are 24 pieces in 24 keys, demonstrating that all of them can be played on a single keyboard without retuning.

But it was also an artistic exploration of what those 24 keys sounded like. In well temperament, each key has a slightly different acoustic character — determined by how much of the Pythagorean comma is assigned to the fifths in that key's scale. Keys near C major have smaller, purer thirds and sound bright and open. Keys far from C (like C# minor or F# major) have slightly wider thirds and sound more "tense" or "dark." This variety of key character is one of the things equal temperament destroyed when it eventually replaced well temperament as the standard.

What "Well Temperament" Meant in the 18th Century

In Bach's time, the term "well temperament" did not refer to a single specific tuning system. It referred to a family of temperaments — all of which made all 24 keys usable, but which differed in exactly how they distributed the Pythagorean comma. Several specific well temperaments were described in published treatises during this period:

Andreas Werckmeister's temperaments (1687): Werckmeister published multiple numbered systems (Werckmeister I, II, III). Werckmeister III — the most influential — placed most of the comma in the fifths C-G, G-D, D-A, and B-F#, leaving the remaining fifths pure. This gave keys near C pure or nearly pure thirds, with increasingly tempered thirds as you moved toward the sharp keys.

Johann Philipp Kirnberger's temperaments (1771, 1779): Kirnberger proposed two systems, with Kirnberger III being the most historically important. His approach concentrated the comma in specific fifths, leaving many others pure.

Francesco Antonio Vallotti's temperament (1779): Distributed one-sixth of the syntonic comma across the six fifths from F to B, leaving the remaining six fifths pure.

None of these is equal temperament. All give each key a distinct character. All were known to Bach and his contemporaries.

The Ongoing Musicological Debate

The central question of WTC scholarship is: which specific well temperament did Bach intend? This question matters for several reasons. First, the performance of the WTC sounds noticeably different in different well temperaments — some keys that sound similar in equal temperament have strikingly different characters in Werckmeister III or Vallotti. Second, if Bach intended a specific character for each key (as his musical choices suggest), then reconstructing his tuning is necessary for authentic performance.

The debate has been active since at least the 19th century. Several positions exist:

The equal temperament position: Some scholars have argued that Bach used equal temperament — that the WTC's purpose was simply to demonstrate that all 24 keys could be played without retuning, and that equal temperament achieves this most completely. This position has fallen out of favor among specialists. Equal temperament was known in theory in Bach's time but was not in widespread practical use; more importantly, if all keys were acoustically identical (as they are in equal temperament), there would be less artistic reason to write a separate piece in each key.

The Werckmeister position: Many scholars identify Werckmeister III as Bach's likely system. Bach knew Werckmeister (they were near-contemporaries in central Germany), and Werckmeister's system was widely used and published. The WTC's key sequence (C major, C minor, C# major, C# minor, etc.) corresponds to increasing distance from the "good" keys in Werckmeister III.

The Lehman position: In 2005, scholar Bradley Lehman published a provocative article in the journal Early Music arguing that Bach's own handwritten spiral decoration on the WTC title page encodes his tuning system. Lehman decoded the loops of the spiral as a sequence of tempered and pure fifths, producing a specific well temperament he calls "Bach's original temperament." This temperament gives C major the purest thirds, with a smooth gradient toward sharper thirds in remote keys.

The skeptical position: Other scholars, including Mark Lindley and other tuning specialists, argue that there is insufficient historical evidence to determine Bach's specific system. Bach left no explicit written tuning instructions, and his music is compatible with multiple well temperaments.

What Modern Reconstructions Reveal

The most valuable contribution of this debate to musical practice has been the proliferation of historical tuning reconstructions in performance. Period-instrument performers now regularly record the WTC in Werckmeister III, Vallotti, Kirnberger III, Lehman's reconstruction, and various others. Comparing these recordings reveals something striking:

Key character emerges and matters. In any well temperament, the Prelude in C major sounds noticeably brighter and more open than the Prelude in C# minor. The C# minor key's slightly wider thirds give it a quality that contemporary theorists described as "somewhat hard" or "expressive of night" — a character that disappears entirely in equal temperament.

The technical demands vary by key. Fugues in sharp keys (like the B major fugue) require performers to handle more tense intervals and may even have slightly different optimal tempos than they do in equal temperament. This suggests Bach was composing not just against the backdrop of twenty-four available keys but against the specific acoustic character of each key in a specific (if undetermined) tuning.

The emotional variety is greater. Listeners who have heard the WTC only in equal temperament often report being surprised by the additional emotional range that appears in well-tempered performances. The tonal language of the pieces seems richer — not just harmonically but acoustically.

Broader Implications

Bach's WTC represents the historical moment when Western music committed to the circle of all twenty-four keys. Before Bach, composers routinely worked in a small subset of keys; after Bach, all twenty-four became available. This expansion was made possible by well temperament (not equal temperament, as is often said incorrectly).

The WTC also reveals something about the relationship between constraint and creativity. The need to write successfully in every key — including keys with slightly harsh intervals in any well temperament — forced Bach to develop compositional strategies for working with (and against) the acoustic character of each key. The constraints of well temperament, in other words, were generative.

Finally, the ongoing uncertainty about Bach's specific tuning is itself instructive. It demonstrates that historical documents rarely tell us everything we want to know about musical practice, that the same music can sound significantly different in different acoustic realizations, and that questions at the intersection of music and physics — like tuning — remain alive and debated long after the composer's death.

Discussion Questions

1. The importance of key character. Baroque theorists assigned specific emotional affects to different keys — C major was "joyful," D minor was "melancholy," etc. These characterizations reflected partly the acoustic character of the keys in well temperament. Do you think these emotional associations have survived into the equal temperament era, even though the acoustic basis for them has disappeared? Why might emotional associations persist beyond their physical causes?

2. The authenticity question. Period-instrument performers argue that performing the WTC in a well-temperament close to Bach's likely system produces more "authentic" results. What does "authentic" mean in this context? Is the goal to reproduce what Bach heard, what Bach intended, or what produces the best musical result for modern audiences? Can these three goals conflict?

3. The Lehman debate. Bradley Lehman's argument that Bach encoded his tuning in the spiral decoration of the title page is imaginative but contested. What makes this argument persuasive or unpersuasive? What kind of evidence would definitively prove or disprove it?

4. The equal temperament erasure. If Bach intended well temperament but nearly all modern performances of the WTC use equal temperament, how much of the work's intended meaning has been lost? Is this a tragedy, a neutral fact, or an example of living music adapting to its context?