The Well-Tempered Clavier


The Well-Tempered Clavier, BWV 846–893, consists of two sets of preludes and fugues in all 24 major and minor keys for keyboard by Johann Sebastian Bach. In the composer's time clavier referred to a variety of keyboard instruments, namely the harpsichord, the clavichord and the organ, but not excluding the regal and the then newly-invented pianoforte.
The modern German spelling for the collection is Das wohltemperierte Klavier. Bach gave the title Das Wohltemperirte Clavier to a book of preludes and fugues in all 24 keys, major and minor, dated 1722, composed "for the profit and use of musical youth desirous of learning, and especially for the pastime of those already skilled in this study". 20 years later, Bach compiled a second book of the same kind, which became known as The Well-Tempered Clavier, Part Two.
Modern editions usually refer to both parts as The Well-Tempered Clavier, Book 1 and The Well-Tempered Clavier, Book 2, respectively. The collection is generally regarded as one of the most important works in the history of classical music.

Composition history

Each set contains 24 pairs of prelude and fugue. The first pair is in C major, the second in C minor, the third in C major, the fourth in C minor, and so on. The rising chromatic pattern continues until every key has been represented, finishing with a B minor fugue. The first set was compiled in 1722 during Bach's appointment in Köthen, and the second followed 20 years later in 1742 while he was in Leipzig.
Bach recycled some of the preludes and fugues from earlier sources: the 1720 Klavierbüchlein für Wilhelm Friedemann Bach, for instance, contains versions of eleven of the preludes of the first book of the Well-Tempered Clavier. The C major prelude and fugue in book one was originally in C major – Bach added a key signature of seven sharps and adjusted some accidentals to convert it to the required key.
In Bach's own time just one similar collection was published, by Johann Christian Schickhardt, whose Op. 30 L'alphabet de la musique contained 24 sonatas in all keys for flute or violin and basso continuo, and included a transposition scheme for alto recorder.

Precursors

Although the Well-Tempered Clavier was the first collection of fully worked keyboard pieces in all 24 keys, similar ideas had occurred earlier. Before the advent of modern tonality in the late 17th century, numerous composers produced collections of pieces in all seven modes: Johann Pachelbel's Magnificat fugues, Georg Muffat's Apparatus Musico-organisticus of 1690 and Johann Speth's Ars magna of 1693 for example. Furthermore, some two hundred years before Bach's time, equal temperament was realized on plucked string instruments, such as the lute and the theorbo, resulting in several collections of pieces in all keys :
  • a cycle of 24 passamezzo–saltarello pairs by
  • 24 groups of dances, "clearly related to 12 major and 12 minor keys" by Vincenzo Galilei
  • 30 preludes for 12 course lute or theorbo by John Wilson
One of the earliest keyboard composers to realize a collection of organ pieces in successive keys was Daniel Croner, who compiled one such cycle of preludes in 1682. His contemporary Johann Heinrich Kittel also composed a cycle of 12 organ preludes in successive keys.
J. C. F. Fischer's Ariadne musica neo-organoedum is a set of 20 prelude and fugue pairs in ten major and nine minor keys, and the Phrygian mode, plus five chorale-based ricercars. Bach knew the collection and borrowed some of the themes from Fischer for the Well-Tempered Clavier. Other contemporary works include the treatise Exemplarische Organisten-Probe by Johann Mattheson, which included 48 figured bass exercises in all keys, Partien auf das Clavier by Christoph Graupner with eight suites in successive keys, and Friedrich Suppig's Fantasia from Labyrinthus Musicus, a long and formulaic sectional composition ranging through all 24 keys which was intended for an enharmonic keyboard with both 31 notes per octave and pure major thirds. Finally, a lost collection by Johann Pachelbel, Fugen und Praeambuln über die gewöhnlichsten Tonos figuratos, may have included prelude-fugue pairs in all keys or modes.
It was long believed that Bach had taken the title The Well-Tempered Clavier from a similarly named set of 24 Preludes and Fugues in all the keys, for which a manuscript dated 1689 was found in the library of the Brussels Conservatoire. It was later shown that this was the work of a composer who was not even born in 1689: Bernhard Christian Weber. In fact, it was written in 1745–1750 in imitation of Bach's earlier example.

Intended tuning

Bach's title suggests that he had written for a 12 note tuning system, in which all keys sounded in tune. One of the opposing systems in Bach's day was meantone temperament in which keys with many accidentals sound out of tune on keyboards limited to 12 pitches per octave. Bach would have been familiar with different tuning systems, and in particular as an organist would have played instruments tuned to a meantone system.
During much of the 20th century it was presumed, possibly mistakenly, that Bach intended equal temperament, which after Bach's death became popular as the standard keyboard tuning, and had been described by theorists and musicians for at least a century before Bach's birth. Accounts of Bach's own tuning practice are few and inexact. The three most cited sources are Forkel, Bach's first biographer; Friedrich Wilhelm Marpurg, who received information from Bach's sons and pupils; and Johann Kirnberger, one of those pupils. Despite the presumption of equal temperament, research has continued into various unequal systems contemporary with Bach's career; there is debate whether Bach might have meant a range of similar temperaments, perhaps altered slightly in practice from piece to piece, or possibly some single, specific, "well-tempered" solution for all purposes. Modern scholars suggest some form of unequal well temperament instead of equal temperament.
Forkel reports that Bach tuned his own harpsichords and clavichords and found other people's tunings unsatisfactory, and also that Bach's personal tuning system allowed him to play in all keys, and to modulate into distant keys almost without the listeners noticing. In the course of a heated debate, Marpurg and Kirnberger appear to agree that Bach required all the major thirds to be sharper than pure – which is not very informative, since it is essentially a prerequisite for any temperament to sound tolerable in all keys.
Johann Georg Neidhardt, writing in 1724–1732, described a range of unequal and near-equal temperaments, which can be successfully used to perform some of Bach's music, and were later praised by some of Bach's pupils and associates. J.S. Bach's son Carl Philipp Emanuel Bach himself published a rather vague tuning method which was close to, but still not equal temperament: He wrote that it had only "most of" the fifths tempered, without saying which ones nor by how much.
Since 1950 there have been many other proposals and many performances of the work in different and unequal tunings, some derived from historical sources, some by modern authors. Whatever their provenances, these schemes all promote the existence of subtly different musical characters in different keys, due to the sizes of their intervals. However, they disagree as to which key receives which character:
  • Herbert Anton Kellner argued from the mid-1970s until his death that esoteric considerations such as the pattern of Bach's signet ring, numerology, and more could be used to determine the correct temperament. His result is somewhat similar to Werckmeister's most familiar "correct" temperament. Kellner's temperament was widely adopted worldwide for the tuning pipe organs, and contains seven pure fifths and five comma fifths. It is especially effective as a moderate solution to play 17th century music, if one avoids music that requires more than two flats.
  • John Barnes analyzed the Well-Tempered Claviers major-key preludes statistically, observing that some major thirds are used more often than others. His results were broadly in agreement with Kellner's and Werckmeister's patterns. His own proposed temperament from that study is a comma variant of both Kellner and Werckmeister, with the same general pattern tempering the naturals, and concluding with a tempered fifth B–F.
  • Mark Lindley, a researcher of historical temperaments, has written several surveys of temperament styles in the German Baroque tradition. In his publications he has recommended and devised many patterns close to those of Neidhardt, with subtler gradations of interval size. Since a 1985 article in which he addressed some issues in the Well-Tempered Clavier, Lindley's theories have focused more on Bach's organ music than the harpsichord or clavichord works.

    Title page tuning interpretations

More recently there has been a series of proposals of temperaments derived from the handwritten doodle of loops on the title page of Bach's personal 1722 manuscript.
  • In the course of studying German Baroque organ tunings, Andreas Sparschuh in 1999 assigned mathematical and acoustic meaning to the loops. Each loop, he argued, represents a fifth in the sequence for tuning the keyboard, starting from A. From this Sparschuh devised a recursive tuning algorithm, resembling the Collatz conjecture in mathematics: It subtracts one beat per second each time Bach's diagram has a non-empty loop. In 2006 he retracted his 1998 proposal based on A = 420 Hz, and replaced it with another at A = 410 Hz.
  • Michael Zapf in 2001 reinterpreted the loops as indicating the rate of beating of different fifths in a given range of the keyboard in terms of seconds-per-beat, with the tuning now starting on C.
  • J. C. Francis reported a mathematical analysis of the loops using Mathematica under In 2004, he also distributed several temperaments derived from BWV 924.
  • B. Lehman proposed a and comma layout derived from Bach's loops, which he published in 2005 in articles of three music journals. Reaction to this work has been both vigorous and mixed, with other writers producing further speculative schemes or variants.
  • D. Jencka proposed a variation of Lehman's layout where one of the commas is spread over three fifths, resulting in a comma division. Motivations for Jencka's approach involve an analysis of the possible logic behind the figures themselves, and his belief that a wide fifth found in Lehman's interpretation is unlikely in a well-temperament from the time.
  • Interbartolo, Venturino, & Bof proposed a tuning system deduced from the W.T.C. title page. Their work was published the next year in a book by the same title.
Nevertheless, some musicologists say there is insufficient proof that Bach's looped drawing signifies anything reliable about a tuning method. Bach may have tuned differently per occasion, or per composition, throughout his career.
  • D. Schulenberg allows that Lehman's argument is "ingenious" but counters that it "lacks documentary support " and concludes that the swirls cannot "be unambiguously interpreted as a code for a particular temperament".
  • L. Swich more recently presented an alternative reading from that of Lehman, and others, of Bach's tuning method as derived from the title page calligraphic drawing: It differs in significant details, resulting in a circulating but unequal temperament using Pythagorean-comma fifths that is effective through all 24 keys and, most important, tunable by ear without an electronic tuning device.
Swich's proposal is based on the equal timing of the beats between the fifth F–C and the third F–A and between the fifth C–G and the third C–E. Such a system is reminiscent of Kellner's 1977 temperament and even more closely to the temperament used for the Organ of St. Ludgeri in Norden, built in 1688 by Arp Schnitger, and the temperament later described by Carlo Gervasoni.
A system like Swich's, with all its major thirds more or less sharp, is confirmed by Friedrich Wilhelm Marpurg's description of the way Bach's famous student J.P. Kirnberger was taught to tune in his lessons with Bach: Kirnberger's tuning allows all 24 keys to be played through without changing tuning nor unpleasant intervals, but with varying degrees of difference. The temperament is unequal, and the keys do not all sound the same. Compared to Werckmeister III, the other 24 key-circulating temperaments, Kirnberger's version of Bach's tuning is much more differentiated, with its 8 different kinds of major thirds.
The manuscript Bach P415 in the Berlin State Library is the only known copy of the W.T.C. that shows the doodle. It would be a bit too cryptic for Bach's spirit, but seems to the hopeful to represent the purpose for which the masterpiece was written, and at the same time, a clue to its decipherment. In perspective, this is not surprising, since the document with the doodle is most probably the working copy Johann Sebastian Bach used in classes with his students.