The Well-Tempered Clavier - AbsoluteAstronomy.com

The Well-Tempered Clavier (German: Das Wohltemperierte Klavier), BWV 846–893, is a collection of solo keyboard music composed by Johann Sebastian Bach

Johann Sebastian Bach

Johann Sebastian Bach was a German composer, organist, harpsichordist, violist, and violinist whose sacred and secular works for choir, orchestra, and solo instruments drew together the strands of the Baroque period and brought it to its ultimate maturity...

. He first gave the title to a book of preludes

Prelude (music)

A prelude is a short piece of music, the form of which may vary from piece to piece. The prelude can be thought of as a preface. It may stand on its own or introduce another work...

and fugue

Fugue

In music, a fugue is a compositional technique in two or more voices, built on a subject that is introduced at the beginning in imitation and recurs frequently in the course of the composition....

s in all 24 major and minor keys

Key (music)

In music theory, the term key is used in many different and sometimes contradictory ways. A common use is to speak of music as being "in" a specific key, such as in the key of C major or in the key of F-sharp. Sometimes the terms "major" or "minor" are appended, as in the key of A minor or in the...

, 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." Bach later compiled a second book of the same kind, dated 1742, but titled it only "Twenty-four Preludes and Fugues." The two works are now usually considered to make up a single work, The Well-Tempered Clavier
Clavier
Clavier is a Walloon municipality located in the Belgian province of Liège. On January 1, 2006, Clavier had a total population of 4,172. The total area is 79.12 km² which gives a population density of approximately 53 inhabitants per km²....

, and are referred to respectively as Books I and II. The Well-Tempered Clavier is generally regarded as one of the most influential works in the history of Western classical music.

Composition history

The first book was compiled in the year 1722 during Bach's appointment in Köthen; the second book followed it 20 years later in 1742 while he was in Leipzig

Leipzig

Leipzig Leipzig has always been a trade city, situated during the time of the Holy Roman Empire at the intersection of the Via Regia and Via Imperii, two important trade routes. At one time, Leipzig was one of the major European centres of learning and culture in fields such as music and publishing...

. Both were widely circulated in manuscript, but printed copies were not made until 1801, by three publishers almost simultaneously in Bonn

Bonn

Bonn is the 19th largest city in Germany. Located in the Cologne/Bonn Region, about 25 kilometres south of Cologne on the river Rhine in the State of North Rhine-Westphalia, it was the capital of West Germany from 1949 to 1990 and the official seat of government of united Germany from 1990 to 1999....

, Leipzig

Leipzig

and Zurich

Zürich

Zurich is the largest city in Switzerland and the capital of the canton of Zurich. It is located in central Switzerland at the northwestern tip of Lake Zurich...

. Bach's style went out of favour in the time around his death, and most music in the early Classical period

Classical period (music)

The dates of the Classical Period in Western music are generally accepted as being between about 1750 and 1830. However, the term classical music is used colloquially to describe a variety of Western musical styles from the ninth century to the present, and especially from the sixteenth or...

had neither contrapuntal complexity nor a great variety of keys. But, with the maturing of the Classical style in the 1770s, the Well-Tempered Clavier began to influence the course of musical history, with Haydn

Joseph Haydn

Franz Joseph Haydn , known as Joseph Haydn , was an Austrian composer, one of the most prolific and prominent composers of the Classical period. He is often called the "Father of the Symphony" and "Father of the String Quartet" because of his important contributions to these forms...

and Mozart

Wolfgang Amadeus Mozart

Wolfgang Amadeus Mozart , baptismal name Johannes Chrysostomus Wolfgangus Theophilus Mozart , was a prolific and influential composer of the Classical era. He composed over 600 works, many acknowledged as pinnacles of symphonic, concertante, chamber, piano, operatic, and choral music...

studying the work closely.

Each book contains twenty-four pairs of preludes and fugues. The first pair is in C major

C major

C major is a musical major scale based on C, with pitches C, D, E, F, G, A, and B. Its key signature has no flats/sharps.Its relative minor is A minor, and its parallel minor is C minor....

, the second in C minor

C minor

C minor is a minor scale based on C, consisting of the pitches C, D, E, F, G, A, and B. The harmonic minor raises the B to B. Changes needed for the melodic and harmonic versions of the scale are written in with naturals and accidentals as necessary.Its key signature consists of three flats...

, the third in C-sharp major, the fourth in C-sharp minor, and so on. The rising chromatic

Chromatic scale

The chromatic scale is a musical scale with twelve pitches, each a semitone apart. On a modern piano or other equal-tempered instrument, all the half steps are the same size...

pattern continues until every key has been represented, finishing with a B-minor

B minor

B minor is a minor scale based on B, consisting of the pitches B, C, D, E, F, G, and A. The harmonic minor raises the A to A. Its key signature has two sharps .Its relative major is D major, and its parallel major is B major....

fugue.

Bach recycled some of the preludes and fugues from earlier sources: the 1720 Klavierbüchlein für Wilhelm Friedemann Bach
Klavierbüchlein für Wilhelm Friedemann Bach
Klavierbüchlein für Wilhelm Friedemann Bach is a collection of keyboard music compiled by the German Baroque composer Johann Sebastian Bach for his eldest son Wilhelm Friedemann...

, for instance, contains versions of eleven of the preludes. The C-sharp major prelude and fugue in book one was originally in C major - Bach added a key signature

Key signature

In musical notation, a key signature is a series of sharp or flat symbols placed on the staff, designating notes that are to be consistently played one semitone higher or lower than the equivalent natural notes unless otherwise altered with an accidental...

of seven sharps

Sharp (music)

In music, sharp, dièse , or diesis means higher in pitch and the sharp symbol raises a note by a half tone. Intonation may be flat, sharp, or both, successively or simultaneously...

and adjusted some accidentals

Accidental (music)

In music, an accidental is a note whose pitch is not a member of a scale or mode indicated by the most recently applied key signature. In musical notation, the symbols used to mark such notes, sharps , flats , and naturals , may also be called accidentals...

to convert it to the required key. The far-reaching influence of Bach's music is evident in that the fugue subject in Mozart's

Wolfgang Amadeus Mozart

Prelude and Fugue in C Major K. 394 is isomorphic to that of the A-flat major Fugue in Book II of the Well-Tempered Clavier. This pattern is found also in the C Major fugue subject of Book II. Another similar theme is the third movement fugue subject in the Concerto for Two Harpsichords, BWV 1061

Harpsichord concertos (J. S. Bach)

The harpsichord concertos, BWV 1052-1065, are concertos for harpsichord, strings and continuo by Johann Sebastian Bach. There are seven complete concertos for a single harpsichord, , three concertos for 2 harpsichords , two concertos for 3 harpsichords , and one concerto for 4 harpsichords,...

.

Bach's title suggests that he had written for a (12-note) well-tempered

Well temperament

Well temperament is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier...

tuning system in which all keys sounded in tune (also known as "circular temperament"). The opposing system in Bach's day was meantone temperament

Meantone temperament

Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...

in which keys with many accidentals

Accidental (music)

sound out of tune. (See also musical tuning

Musical tuning

In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...

). It is sometimes assumed that Bach intended equal temperament

Equal temperament

An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

, the standard modern keyboard tuning which became popular after Bach's death, but modern scholars suggest instead a form of well temperament

Well temperament

. There is debate whether Bach meant a range of similar temperaments, perhaps even altered slightly in practice from piece to piece, or a single specific "well-tempered" solution for all purposes.

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 eight modes: Johann Pachelbel

Johann Pachelbel

Johann Pachelbel was a German Baroque composer, organist and teacher, who brought the south German organ tradition to its peak. He composed a large body of sacred and secular music, and his contributions to the development of the chorale prelude and fugue have earned him a place among the most...

's magnificat fugues (composed 1695–1706), Georg Muffat

Georg Muffat

-Life:He was born in Megève, Savoy, , and of Scottish descent. He studied in Paris with Jean Baptiste Lully between 1663 and 1669, then became an organist in Molsheim and Sélestat. Later, he studied law in Ingolstadt, afterwards settling in Vienna...

's Apparatus Musico-organisticus of 1690 and Johann Speth's Ars magna of 1693 are but a few examples. Furthermore, some two hundred years before Bach's time, equal temperament

Equal temperament

was realized on plucked string instruments, such as the lute

Lute

Lute can refer generally to any plucked string instrument with a neck and a deep round back, or more specifically to an instrument from the family of European lutes....

and the theorbo

Theorbo

A theorbo is a plucked string instrument. As a name, theorbo signifies a number of long-necked lutes with second pegboxes, such as the liuto attiorbato, the French théorbe des pièces, the English theorbo, the archlute, the German baroque lute, the angélique or angelica. The etymology of the name...

, resulting in several collections of pieces in all keys (although the music was not yet tonal in the modern sense of the word):

a cycle of 24 passamezzo–saltarello pairs (1567) by Giacomo Gorzanis (c.1520–c.1577)
24 groups of dances, "clearly related to 12 major and 12 minor keys" (1584) by Vincenzo Galilei
Vincenzo Galilei
Vincenzo Galilei was an Italian lutenist, composer, and music theorist, and the father of the famous astronomer and physicist Galileo Galilei and of the lute virtuoso and composer Michelagnolo Galilei...

(c.1528–1591)
30 preludes for 12-course lute or theorbo by John Wilson
John Wilson (composer)
John Wilson , was an English composer, lutenist and teacher. Born in Faversham, Kent, he moved to London by 1614, where he succeeded Robert Johnson as principal composer for the King's Men, and entered the King's Musick in 1635 as a lutenist. He received the degree of D.Mus from Oxford in 1644,...

(1595–1674)

One of the earliest keyboard composers to realize a collection of organ pieces in successive keys was Daniel Croner (1656–1740), who compiled one such cycle of preludes in 1682. His contemporary Johann Heinrich Kittel (1652–1682) also composed a cycle of 12 organ preludes in successive keys.

Ariadne musica neo-organoedum
Ariadne musica
Ariadne musica is a collection of organ music by Johann Caspar Ferdinand Fischer, first published in 1702. The main part of the collection is a cycle of 20 preludes and fugues in different keys, so Ariadne musica is considered an important precursor to Johann Sebastian Bach's Well-Tempered Clavier,...

, by J.C.F. Fischer

Johann Caspar Ferdinand Fischer

Johann Caspar Ferdinand Fischer was a German Baroque composer...

(1656–1746) was published in 1702 and reissued 1715. It is a set of 20 prelude-fugue pairs in ten major and nine minor keys and the Phrygian mode

Phrygian mode

The Phrygian mode can refer to three different musical modes: the ancient Greek tonos or harmonia sometimes called Phrygian, formed on a particular set octave species or scales; the Medieval Phrygian mode, and the modern conception of the Phrygian mode as a diatonic scale, based on the latter...

, plus five chorale

Chorale

A chorale was originally a hymn sung by a Christian congregation. In certain modern usage, this term may also include classical settings of such hymns and works of a similar character....

-based ricercar

Ricercar

A ricercar is a type of late Renaissance and mostly early Baroque instrumental composition. The term means to search out, and many ricercars serve a preludial function to "search out" the key or mode of a following piece...

s. Bach knew the collection and borrowed some of the themes from Fischer for Well-Tempered Clavier. Other contemporary works include the treatise Exemplarische Organisten-Probe (1719) by Johann Mattheson

Johann Mattheson

Johann Mattheson was a German composer, writer, lexicographer, diplomat and music theorist.Mattheson was born and died in Hamburg. He was a close friend of George Frideric Handel, although he nearly killed him in a sudden quarrel, during a performance of Mattheson's opera Cleopatra in 1704...

(1681–1764), which included 48 figured bass

Figured bass

Figured bass, or thoroughbass, is a kind of integer musical notation used to indicate intervals, chords, and non-chord tones, in relation to a bass note...

exercises in all keys, Partien auf das Clavier (1718) by Christoph Graupner

Christoph Graupner

Christoph Graupner was a German harpsichordist and composer of high Baroque music who lived and worked at the same time as Johann Sebastian Bach, Georg Philipp Telemann and George Frideric Handel.-Graupner's life:Born in Hartmannsdorf near Kirchberg in Saxony, Graupner received his first musical...

(1683–1760) with eight suites in successive keys, and Friedrich Suppig

Friedrich Suppig

Friedrich Suppig was an 18th century music theorist and composer. Practically nothing is known about him or his life, or even if he was in fact a professional composer. He is known for two manuscripts; in one of which he discussed theoretical tuning systems:...

's Fantasia from Labyrinthus Musicus (1722), a long and formulaic sectional composition ranging through all 24 keys which was intended for an enharmonic keyboard

Enharmonic keyboard

An enharmonic keyboard is a musical keyboard based on an enharmonic scale. At the very least such keyboards will have 17 keys per octave, and enharmonically equivalent notes will have different pitches. A typical keyboard will have one key for, for instance, C sharp and D flat, but a basic 17 key...

with 31 notes per octave and pure major thirds

Interval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

. Finally, a lost collection by Johann Pachelbel

Johann Pachelbel

(1653–1706), Fugen und Praeambuln über die gewöhnlichsten Tonos figuratos (announced 1704), may have included prelude-fugue pairs in all keys or modes.

Bach's example inspired numerous composers of the 19th century, however, in his own time no similar collections were published, except one by Johann Christian Schickhardt

Johann Christian Schickhardt

Johann Christian Schickhardt was a German composer and woodwind player.-Biography:Schickhardt was born in Braunschweig and received his musical education at the Braunschweig-Wolfenbüttel court under the patronage of Augustus William, third son and heir of Anthony Ulrich, Duke of...

(1681–1762), whose Op. 30 L'alphabet de la musique, contained 24 sonatas for recorder/flute/violin, in all keys.

Musical style and content

Musically, the structural regularities of the Well-Tempered Clavier encompass an extraordinarily wide range of styles, more so than most pieces in the literature.
The Preludes are formally free, although many individual numbers exhibit typical Baroque melodic forms, often coupled to an extended free coda

Coda (music)

Coda is a term used in music in a number of different senses, primarily to designate a passage that brings a piece to an end. Technically, it is an expanded cadence...

(e.g. Book I preludes in C minor

C minor

, D Major

D major

D major is a major scale based on D, consisting of the pitches D, E, F, G, A, B, and C. Its key signature consists of two sharps. Its relative minor is B minor and its parallel minor is D minor....

, and B-flat major).

Each fugue is marked with the number of voices, from two to five. Most are three- and four-voiced fugues. The fugues employ a full range of contrapuntal devices (fugal exposition, thematic inversion, stretto

Stretto

The term stretto comes from the Italian past participle of stringere, and means "narrow", "tight", or "close".In music the Italian term stretto has two distinct meanings:...

, etc.), but are generally more compact than Bach's fugues for organ

Pipe organ

The pipe organ is a musical instrument that produces sound by driving pressurized air through pipes selected via a keyboard. Because each organ pipe produces a single pitch, the pipes are provided in sets called ranks, each of which has a common timbre and volume throughout the keyboard compass...

.

The best-known piece from either book is the first prelude of Book I, a simple progression of arpeggiated

Arpeggio

An arpeggio is a musical technique where notes in a chord are played or sung in sequence, one after the other, rather than ringing out simultaneously...

chords. The technical simplicity of this C Major prelude has made it one of the most commonly studied piano pieces for students completing their introductory training. This prelude also served as the basis for the Ave Maria
Ave Maria (Gounod)
The Bach/Gounod Ave Maria is a popular and much-recorded setting of the Latin text Ave Maria.Written by French Romantic composer Charles Gounod in 1859, his Ave Maria consists of a melody superimposed over the Prelude No. 1 in C major, BWV 846, from Book I of The Well-Tempered Clavier, written by...

of Charles Gounod

Charles Gounod

Charles-François Gounod was a French composer, known for his Ave Maria as well as his operas Faust and Roméo et Juliette.-Biography:...

Later significance and influence

Although the Well-Tempered Clavier was not the first pantonal

Twelve-tone technique

Twelve-tone technique is a method of musical composition devised by Arnold Schoenberg...

(using all keys) composition, it was by far the most influential. The very nature of the piece (as implied by its title page) established a tuning requirement for harmonies which were to become the basis for all Western music developed through the early 20th century. The Well-Tempered Clavier does not include very remote modulation

Modulation (music)

In music, modulation is most commonly the act or process of changing from one key to another. This may or may not be accompanied by a change in key signature. Modulations articulate or create the structure or form of many pieces, as well as add interest...

s, but instead demonstrates the ability of a single instrument in tempered tuning to play in all 24 keys without having to be tuned to new fundamentals. Beethoven

Ludwig van Beethoven

Ludwig van Beethoven was a German composer and pianist. A crucial figure in the transition between the Classical and Romantic eras in Western art music, he remains one of the most famous and influential composers of all time.Born in Bonn, then the capital of the Electorate of Cologne and part of...

, who made remote modulations central to his music, was heavily influenced by the Well-Tempered Clavier, since performing it in concerts in his youth was part of his star attraction and reputation. Further reaching modulations to remote harmonic regions were mostly associated with later Romantic

Romantic music

Romantic music or music in the Romantic Period is a musicological and artistic term referring to a particular period, theory, compositional practice, and canon in Western music history, from 1810 to 1900....

and post-Romantic music, ultimately leading to the functional extension in jazz

Jazz

Jazz is a musical style that originated at the beginning of the 20th century in African American communities in the Southern United States. It was born out of a mix of African and European music traditions. From its early development until the present, jazz has incorporated music from 19th and 20th...

harmony. The atonal system of the 20th century, although still taking the 12-tone chromatic scale (that Bach used) as a foundation, effectively did away with musical keys altogether.

In addition to its use of all keys, the Well-Tempered Clavier was unusual in the very wide range of techniques and modes of expression used by Bach in the fugues

Fugues

Fugue can refer to:* Fugue for the type of musical piece* See :Category:Fugues for individual pieces.* Fugues for the Canadian gay magazine* Fugue for the American literary journal.* Fugue state, a psychological term...

. No other composer had produced such vividly characterised and compelling pieces in the fugal form, which was often regarded as a theoretical exercise. Many later composers studied Bach's work in an effort to improve their own fugal writing: Verdi

Giuseppe Verdi

Giuseppe Fortunino Francesco Verdi was an Italian Romantic composer, mainly of opera. He was one of the most influential composers of the 19th century...

even found it useful for his last work, Falstaff

Falstaff (opera)

Falstaff is an operatic commedia lirica in three acts by Giuseppe Verdi, adapted by Arrigo Boito from Shakespeare's plays The Merry Wives of Windsor and scenes from Henry IV. It was Verdi's last opera, written in the composer's ninth decade, and only the second of his 26 operas to be a comedy...

.

The first complete recording of the Well-Tempered Clavier was made by Edwin Fischer

Edwin Fischer

Edwin Fischer was a Swiss classical pianist and conductor. He is widely regarded as one of the greatest pianists of the 20th century, particularly in the traditional Germanic repertoire of such composers as J. S. Bach, Mozart, Beethoven and Schubert...

between 1933 and 1936. Other notable recordings have been made by Wanda Landowska

Wanda Landowska

Wanda Landowska was a Polish harpsichordist whose performances, teaching, recordings and writings played a large role in reviving the popularity of the harpsichord in the early 20th century...

, Rosalyn Tureck

Rosalyn Tureck

Rosalyn Tureck was an American pianist and harpsichordist who was particularly associated with the music of Johann Sebastian Bach...

, Glenn Gould

Glenn Gould

Glenn Herbert Gould was a Canadian pianist who became one of the best-known and most celebrated classical pianists of the 20th century. He was particularly renowned as an interpreter of the keyboard music of Johann Sebastian Bach...

, Friedrich Gulda

Friedrich Gulda

Friedrich Gulda was an Austrian pianist and composer who worked in both the classical and jazz fields.Born in Vienna as the son of a teacher, Gulda began learning to play the piano from Felix Pazofsky at the Wiener Volkskonservatorium, aged 7...

, Gustav Leonhardt

Gustav Leonhardt

Gustav Leonhardt is a highly renowned Dutch keyboard player, conductor, musicologist, teacher and editor. Leonhardt has been a leading figure in the movement to perform music on period instruments...

, Sviatoslav Richter

Sviatoslav Richter

Sviatoslav Teofilovich Richter was a Soviet pianist well known for the depth of his interpretations, virtuoso technique, and vast repertoire. He is widely considered one of the greatest pianists of the 20th century.-Childhood:...

, Ralph Kirkpatrick

Ralph Kirkpatrick

Ralph Kirkpatrick was an American musician, musicologist and harpsichordist. He is most famous for his chronological catalog of Domenico Scarlatti's keyboard sonatas.-Life and work:...

, Angela Hewitt

Angela Hewitt

Angela Hewitt, OC, OBE is a Canadian classical pianist. She holds British nationality through her father, Godfrey, who was the organist and choirmaster at Christ Church Cathedral in Ottawa, Ontario for almost fifty years.-Career:...

and Andras Schiff

András Schiff

András Schiff is a Hungarian-born British classical pianist, who has won a number of awards including the Grammy and made numerous recordings.- Biography :...

Intended tuning

During much of the 20th century it was assumed that Bach wanted equal temperament

Equal temperament

, which had been described by theorists and musicians for at least a century before Bach's birth. Internal evidence for this may be seen in the fact that in Book 1 Bach paired the E-flat minor prelude (6 flats) with its enharmonic key of D-sharp minor (6 sharps) for the fugue. This represents an equation of the most tonally remote enharmonic

Enharmonic

In modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...

keys where the flat and sharp arms of the circle of fifths

Circle of fifths

In music theory, the circle of fifths shows the relationships among the 12 tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys...

cross each other opposite to C major. Any performance of this pair would have required both of these enharmonic keys to sound identically tuned, thus implying equal temperament in the one pair, as the entire work implies as a whole. However, research has continued into various unequal systems contemporary with Bach's career. Accounts of Bach's own tuning practice are few and inexact. The three most cited sources are Forkel

Johann Nikolaus Forkel

Johann Nikolaus Forkel , was a German musician, musicologist and music theorist.-Biography:...

, Bach's first biographer

Biography

A biography is a detailed description or account of someone's life. More than a list of basic facts , biography also portrays the subject's experience of those events...

, Friedrich Wilhelm Marpurg

Friedrich Wilhelm Marpurg

Friedrich Wilhelm Marpurg was a German music critic, music-theorist and composer. He was friendly and active with many figures of the Enlightenment of the 18th century.-Life:...

, who received information from Bach's sons and pupils, and Johann Kirnberger

Johann Kirnberger

Johann Philipp Kirnberger was a musician, composer , and music theorist. A pupil of Johann Sebastian Bach, he became a violinist at the court of Frederick II of Prussia in 1751. He was the music director to the Prussian Princess Anna Amalia from 1758 until his death. Kirnberger greatly admired J.S...

, one of those pupils.

Forkel reports that Bach tuned his own harpsichord

Harpsichord

A harpsichord is a musical instrument played by means of a keyboard. It produces sound by plucking a string when a key is pressed.In the narrow sense, "harpsichord" designates only the large wing-shaped instruments in which the strings are perpendicular to the keyboard...

s and clavichord

Clavichord

The clavichord is a European stringed keyboard instrument known from the late Medieval, through the Renaissance, Baroque and Classical eras. Historically, it was widely used as a practice instrument and as an aid to composition, not being loud enough for larger performances. The clavichord produces...

s and found other people's tunings unsatisfactory; his own allowed him to play in all keys and to modulate into distant keys almost without the listeners noticing it. Marpurg and Kirnberger, in the course of a heated debate, appear to agree that Bach required all the major thirds to be sharper than pure—which is in any case virtually a prerequisite for any temperament to be good in all keys.

Johann Georg Neidhardt, writing in 1724 and 1732, described a range of unequal and near-equal temperaments (as well as equal temperament itself), 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

Carl Philipp Emanuel Bach

right|250pxCarl Philipp Emanuel Bach was a German Classical period musician and composer, the fifth child and second son of Johann Sebastian Bach and Maria Barbara Bach...

himself published a rather vague tuning method which was close to but still not equal temperament: having only "most of" the fifths

Interval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

tempered, without saying which ones or 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 what key receives what 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
Numerology
Numerology is any study of the purported mystical relationship between a count or measurement and life. It has many systems and traditions and beliefs...

, and more could be used to determine the correct temperament. His result is somewhat similar to Werckmeister's
Werckmeister temperament
Werckmeister temperaments are the tuning systems described by Andreas Werckmeister in his writings . The tuning systems are confusingly numbered in two different ways: the first refers to the order in which they were presented as "good temperaments" in Werckmeister's 1691 treatise, the second to...

most familiar "correct" temperament. Kellner's temperament, with seven pure fifths and five 1/5 comma
Comma (music)
In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. The word "comma" used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning...

fifths, has been widely adopted worldwide for the tuning of organs. It is especially effective as a moderate solution to play 17th century music, shying away from tonalities that have more than two flats.
John Barnes analyzed the Well-Tempered Clavier's 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 1/6 comma variant of both Kellner (1/5) and Werckmeister (1/4), with the same general pattern tempering the naturals, and concluding with a tempered fifth B-F#.
Mark Lindley
Mark Lindley
Mark Lindley is both a noted musicologist and, more recently, an historian of modern India. Born in Washington DC, he studied at Harvard University , Juilliard School of Music and Columbia University...

, a researcher of historical temperaments, has written several surveys of temperament styles in the German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

Baroque
Baroque music
Baroque music describes a style of Western Classical music approximately extending from 1600 to 1760. This era follows the Renaissance and was followed in turn by the Classical era...

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 where 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 pattern of loops on Bach's 1722 title page. These loops (though truncated by a later clipping of the page) can be seen at the top of the title page image at the beginning of the article.

Andreas Sparschuh, in the course of studying German Baroque organ tunings, 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
Collatz conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture , Kakutani's problem , the Thwaites conjecture , Hasse's algorithm The Collatz conjecture is a...

in mathematics, subtracting one beat per second each time Bach's diagram has a non-empty loop. In 2006 he has retracted his 1998 proposal based on A =420 Hz, and replaced it with another at A = 410.
Michael Zapf in 2001 reinterpreted the loops as indicating the rate of beating
Beat (acoustics)
In acoustics, a beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies....

of different fifths in a given range of the keyboard in terms of seconds-per-beat, with the tuning now starting on C.
John Charles Francis in 2004 performed a mathematical analysis of the loops using Mathematica
Mathematica
Mathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...

under the assumption of beats per second. In 2004, he also distributed several temperaments derived from BWV 924.
Bradley Lehman in 2004 proposed a 1/6 and 1/12 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.
Daniel Jencka in 2005 proposed a variation of Lehman's layout where one of the 1/6th commas is spread over three 5ths (G#-D#-A#/Bb), resulting in a 1/18th comma division. Motivations for Jencka's approach involve an analysis of the possible logic behind the figures themselves and his belief that a wide 5th (Bb-F) found in Lehman's interpretation is unlikely in a well-temperament from the time.
Graziano Interbartolo &. others in 2006 proposed http://www.bach1722.com/presentazione.htm a tuning system deduced from the WTK title page. Their work was also published in a book: BACH 1722

Il temperamento di Dio
Le scoperte e i significati del “Wohltemperirte Clavier"
Pag 136 – Edizioni Bolla, Finale Ligure, ISBN 8033064290935

Despite this recent research, however, many musicologists say it is insufficiently proven 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.

Composition history

Precursors

Musical style and content

Later significance and influence

Intended tuning

Title page tuning interpretations

Media

See also

Sheet music

Recordings

Websites

Proposed 'Bach' tunings derived from the title page