Bach's schematic, as it appears on the page
LaripS.com, © Bradley Lehman, 2005-14, all rights reserved.
All musical/historical analysis here on the LaripS.com web site is the personal opinion of the author,
as a researcher of historical temperaments and a performer of Bach's music.

"Temperament-killer" Tests

The primary evidence for a temperament style includes Bach's music. That point should be obvious. If we tune the instrument in "wrong" ways, the direct result is some ugliness within the music, whether it is some sour harmonies, or some individual notes that sound too high or low either melodically or harmonically. Wrong temperaments can make whole sections of the composition sound raw or out-of-tune, because they don't correctly handle the flats and/or the sharps that occur within that section.

If we tune correctly, no notes or harmonies stick out as startlingly out-of-tune. We are not merely avoiding error, but also revealing the music's character. It emerges as beautiful, brilliant, and expressive: powerful in the right spots, and gentle in the right spots. The right temperament to match the music makes the instrument sound at its best.

Which of Johann Sebastian Bach's music most readily causes improperly-balanced temperaments to fail, bringing unprepared harsh sounds into the texture? An especially stringent workout includes playing through the following compositions, on a good harpsichord (with strong upper harmonics, a "bright" tone), at a variety of tempos:

  • Both books of WTC (and especially C#, E, F#, Ab, Bb, B major, and C, C#, Eb/D#, F, F#, G#, Bb, B minor; and the chromaticism of A minor and D minor)
  • The inventions and sinfonias (BWV 772-801), especially Eb, E, A, Bb major and C, F, B minor. Additionally, the A minor, D minor, D major, and E minor pieces need 13 or 14 different pitch classes, even though they have "easy"/simple key signatures. (In the E minor sinfonia, presumably all four of the notes Eb, Bb, D#, and A# should sound decent!) The G major sinfonia, with only one sharp in the signature, needs a good D# and A#. The D major sinfonia needs D#, A#, and both E# and F natural. The book as a whole needs 25 different pitch classes, with plenty of emphasis on Db, Ab, D#, A#, E#, and B#: notes that are problematic in other circulating temperaments (Werckmeister, Vallotti, et al). The D minor sinfonia is analyzed further at the meantone page, and in this video presentation.
  • The B minor Partita (BWV 831, originally in C minor)
  • The Eb (BWV 819) and F minor (BWV 823) Suites
  • The C minor 'French' Suite (BWV 813), especially with attention to the melodic and harmonic occurrences of Eb, Ab, and Db. Similarly, the B minor 'French' Suite (BWV 814) with attention to G#, D#, A#, and especially E#. Both of these suites (along with the G major and D minor suites) also present accented two-voiced occurrences of open 12ths (or 19ths) C-G, G-D, and D-A...which sound obtrusive if these 5ths have been tempered more tightly than about 1/6 comma.
  • The E-flat major 'French' suite (BWV 815) with its many occurrences of Eb, Ab, Db, and Gb...while it also needs F#.
  • The G major 'French' suite (BWV 816) needing G#, D#, A#, Eb, and Bb. The E major suite (BWV 817) needing G#, D#, A#, E#, and B#.
  • The Toccatas (BWV 910-16, especially 910-12), all of which are from 1704-1713 at the latest. Bach was already writing this wildly adventurous music in his 20s, needing all kinds of enharmonic swaps within each composition.
  • The six violin sonatas (BWV 1014-19)
  • The Bb (BWV 992) and E major Capriccii (BWV 993) with their extremes of modulation
  • The E major Suite (BWV 1006a) arranged from its violin counterpart
  • The Chromatic Fantasy and Fugue (BWV 903)
  • The other Fantasien (BWV 904, 906, 917, 918, 919, 922, 944). For example, one passage in BWV 944's fugue (A minor) modulates through 16 different notes in only 12 bars: Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E#. BWV 918 (C minor) needs the 14 notes Gb, Db, Ab, ..., F#, C#.
  • The Sarabande of the G minor 'English' Suite (BWV 808), which has notes changing their enharmonic spellings within short phrases
  • The A major 'English' Suite (BWV 806), E major 'French' Suite (BWV 817), and A major Suite (BWV 832) - lest any of the sharps become too bright to bear
  • The F major 'English' Suite (BWV 809) - already within the first two movements it must handle a wide range of keys and their relatives, both melodically and harmonically: F major, F minor, D minor, A major; all three of C major, E major, Ab major....
  • The E minor 'English' Suite (BWV 810) - brightness of the dominants; handling of A# vs Bb, D# vs Eb, E# vs F; passages of four or five parallel major 10ths in the Prelude (bars 11 and 127) and the Allemande (bar 20); decent melodic smoothness whenever the music goes B-C#-D# or F#-G#-A#
  • The D minor 'English' Suite (BWV 811) - D# vs Eb in the outer two movements; G# vs Ab in the first movement
  • The E minor Partita (BWV 830) - chromaticism, and the same problems as with BWV 810...the character of E minor, and the way Bach used it so adventurously
  • The C minor Partita (BWV 826) - the Sinfonia has some garish moments, melodically, if the Db is too low. The Allemande is full of exposed 10ths and 17ths on strong beats, with Ab or Db in the bass: the two-voiced texture makes these intervals sound remarkably sour if those notes are tuned too low.
  • The C minor Partita for lute or keyboard (BWV 997)
  • Prelude, fugue, and allegro in Eb (BWV 998) - D-flats and A-flats in straightforward harmony as the roots of major chords; enharmonic swaps of E natural to F-flat, and B natural to C-flat at several places; D-flats and A-flats also as minor 7ths of chords; and the whole piece should (arguably) sound rich and resonant, despite the odd key areas and Neapolitan modulations
  • The harpsichord concertos in E major (BWV 1053 and cantatas 49 & 169), and F minor (BWV 1056 and cantata 156)
  • The harpsichord concerto in G minor (BWV 1058) needing notes all the way from Cb, Gb, and Db up to F# and C#
  • The keyboard solo concerto in B minor (BWV 979) - extensive use of D#, A#, and E#
  • The sonata in B minor (BWV 1030) for flute and harpsichord, and its G minor version - extreme chromatic modulations, some at very close range (enharmonic swaps within the same bar)
  • O Mensch, bewein dein' Sünde groß (BWV 622, discussion in supplementary files for part 2)
  • Kleines harmonisches Labyrinth (BWV 591, discussion in supplementary files for part 2)
  • The 'Neumeister' chorales, especially BWV 1093, 714, 742, 1108, 1110, and 1113; on 714 (Ach Gott und Herr as a canonic setting) agreement with Lindley's remark (Michaelstein article 1994/7) "Here are the most straightforward uses of C#-major (or Db-major) triads that I have found in Bach's organ music." [My performance of BWV 714]
  • All of Clavierübung III (discussion in supplementary files for part 2, Early Music May 2005), with an especially good distillation of the issues in the four Duetti (BWV 802-5).
  • The Eb major prelude and fugue, BWV 552 (beginning and end of Clavierübung III). I agree with Lindley (Michaelstein article 1994/7) that the prelude BWV 552 is an important test piece. However, I do not agree with his excerpting to play the staccato-chord portions in isolation and then to claim: "The other two excerpts show how Bach remained cautious in his use of Db-major triads and, though to a rather lesser extent, of Ab-major triads. The staccato marks are due to the composer." If Bach was allegedly so "cautious" about Db-major and Ab-major triads, quitting them with light staccato or whatever, why did he write them in accented positions and sustained/tied at other places in this same piece?! (The excerpting to staccato parts happens to fit an oft-reused Lindley hypothesis, in multiple articles of his, that the Db-F ought to be the widest major 3rd in a Bach temperament, and that Bach somehow finessed around them compositionally....)
  • The G minor Fantasia and Fugue (BWV 542) - enharmonic shifts and adventurous modulations
  • Von Gott will ich nicht lassen (BWV 658) - F minor: Db, Gb frequently in the texture; Ab major cadences
  • Allein Gott in der Höh' sei Ehr' (BWV 664) - A major: D#, A#, E#, F naturals
  • Jesus Christus, unser Heiland (BWV 665) - E minor with intense chromatic lines in contrary motion, from bar 27 forward
  • Das Jesulein soll doch mein Trost (BWV 702) - Bb major with Ab, Db, and Gb coming into it; and the surprisingly unprepared sonority of Bb-Eb-C-F# (bar 7) immediately following a Bb major triad
  • The Praeludium/Toccata in either E or C (BWV 566)
  • The Praeludium and Fugue in C (BWV 547), needing the extraordinary range of 16 different pitch classes: Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, and A#! It also has an accented A-flat major chord preceded and followed by silence. Any temperament with a rough Ab major triad fails blatanly.
  • The C minor fugue (BWV 575), and the C minor fantasia and fugue (BWV 537)
  • The C minor section of the F major Pastorella (BWV 590) - A-flat and D-flat major modulations
  • Chorale partitas "Christ, der du bist der helle Tag" (BWV 766, F minor) and "O Gott, du frommer Gott" (BWV 767, C minor) - listening both melodically and harmonically to the various occurrences of Ab, Db, and Gb
  • The F major toccata and fugue (BWV 540), especially its sequences of deceptive modulations
  • The B minor prelude and fugue (BWV 544), with the way it has resolutions into weak beats on the triads of F# major and C# major
  • Schmücke dich (BWV 654) and O Lamm Gottes, unschuldig (BWV 656) from the Leipzig chorales, where the organ must be able to handle extreme flats and sharps and a cadence into C# major
  • 'Goldberg' variation 25 of Clavierübung IV - enharmonic shifts and deep flats. Variation 25 has 17 different notes from Fb to G#, and the other variations contribute D# and A#, for a total of 19.
  • The fugues in A major (BWV 949 and 950) and B minor (BWV 951 and 951a both on a theme by Albinoni), plus the Praeludium in B minor BWV 923 - handling of E# and B#, and the overall effects of these compositions (how bright is too bright?); and harmonies that have an A# on the top of the texture.
  • The fugue in D minor (BWV 948)--especially bars 53 to the end where all twelve minor keys are traversed in turn
  • Contrapuncti 3, 4, 8, 11, and Fuga a 3 Soggetti in Die Kunst der Fuge (BWV 1080)
  • The two ricercare and the canon per tonos of Das musikalisches Opfer
  • Basso continuo in vocal music in the keys where the original organist read the part, i.e. allowing for the transposing Chorton organ at each venue. For example, the C minor movements in Bach's Leipzig audition cantatas (BWV 22-23) read in B-flat minor, or any B-flat movements wherein the organ was playing A-flat major (likewise E-flat movements in D-flat); such harmonically adventurous cantatas as BWV 12, 21, 27, 48, 56, 78, 82, 89, 97, 98, 116, 134, 140, 143, 159, 166, 176; St Matthew Passion BWV 244 (especially movements 9, 10, 19, 32, 51, 52, 59, 60, 63, 65, 68)
  • Music by Wilhelm Friedemann Bach and Carl Philipp Emanuel Bach
  • Bach four-part chorales, listening carefully to the characteristic sounds of the scale and the harmonies, with the meaning of the text that is being sung or the overall spiritual themes of the cantata. [Note: See also the "Ensemble music" section of part 1 of the article, and this feature page about the vocal music. If the chorale came directly from a cantata written in a transposing-organ situation, either play the chorale by reading everything down a whole step, or transpose the entire temperament to the "D" version as described in the article or here.]
  • L'enharmonique (1728) by Jean-Philippe Rameau, having enharmonic modulations and a second theme spelling out the letters B-A-C-H
  • Harpsichord music by Francois Couperin, especially the harmonically adventurous Ordre 25, 26, 27 (in the 4th book, 1730)
  • Harpsichord and clavichord music by Johann Kuhnau, Bach's predecessor at Leipzig: the Biblical Sonatas (Biblische Historien, published 1700), Fresh Keyboard Fruit (Frische Clavier Früchte, 1696), and several books of New Keyboard Practice (Neuer Clavier-Ubung, 1689, 1692). For example, the 6th sonata in Frische Clavier Früchte is in B-flat major. It requires a good A-flat major triad with the C voiced on top; plus, it needs both C# (used in the A-major dominant triad to D minor) and Db (melodically, and within B-flat minor triads). Where is the C#/Db to be placed? The 5th sonata, immediately before this in the book, is in E minor and needs good a F#, C#, G#, D#, A#, and Bb. Where is the A#/Bb to be placed? Are any notes supposed to be retuned if one chooses to play both these sonatas? Or, did Kuhnau have a good system (especially on fretted clavichord) allowing everything to work decently without retuning?
Further remarks about the testing process are on the Affekt page.

Some of these compositions are available as free samples on the samples page, on Last.fm, and on ilike.com. It is of course still important also to play through them oneself, in various temperaments including this one, to experience the music most directly at harpsichords and organs!

See also the video and streamed programs pages for demonstrations of some of these compositions.


Which notes are required in playing the WTC?

Book 1
Here is a list of all the note names (pitch classes) required by each prelude and fugue of book 1:
  • Book 1, C major: 13 notes: Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#
  • Book 1, C minor: 12 notes: Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#
  • Book 1, C# major: 15 notes: A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, Cx, Gx, Dx, Ax
  • Book 1, C# minor: 14 notes: G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, Cx
  • Book 1, D major: 14 notes: Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E#
  • Book 1, D minor: 13 notes: Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#
  • Book 1, Eb major: 14 notes: Cb, Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#
  • Book 1, Eb minor/D# minor: 25 notes: Bbb, Fb, Cb, Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, Cx, Gx
  • Book 1, E major: 13 notes: C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#
  • Book 1, E minor: 15 notes: Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E#
  • Book 1, F major: 12 notes: Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#
  • Book 1, F minor: 14 notes: Cb, Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#
  • Book 1, F# major: 14 notes: D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, Cx, Gx
  • Book 1, F# minor: 14 notes: G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx
  • Book 1, G major: 14 notes: Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#
  • Book 1, G minor: 12 notes: Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#
  • Book 1, Ab major: 12 notes: Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B
  • Book 1, G# minor: 14 notes: D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, Cx, Gx
  • Book 1, A major: 12 notes: G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#
  • Book 1, A minor: 14 notes: Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#
  • Book 1, Bb major: 14 notes: Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#
  • Book 1, Bb minor: 13 notes: Fb, Cb, Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E
  • Book 1, B major: 12 notes: D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx
  • Book 1, B minor: 17 notes: Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx
See also a video presentation of this material, with musical examples played in each key.

On a standard 12-note keyboard, obviously, the same key lever (and same selected pitch) must be used to play each enharmonic pair of notes, such as Bb/A#, or B#/C. The tuner's task is to find appropriate compromises so the selected pitches are not jarring. Better yet, one must not merely avoid error, but actively provide something sublimely beautiful.

For example, in book 1's E major prelude and fugue (calling for 13 different notes), where should the C/B# be placed so all its occurrences sound musically plausible? And, how does such a compromise affect all the other notes that are doing only single duty?

How should the 17 notes of the B minor prelude and fugue be handled? And the different 14 notes of the F minor music?

These are the problems solved by the present work. The problem is provided by the music itself, the book calling for 27 different note names. The key to Bach's solution is drawn on the title page, showing exactly the pattern in which these compromises should be made.

Look at the way Bach used his scale resources through modulation. This is a general feature of tonal music's behavior, but the example here is Bach's, and it shows why we need more than 12 notes for even a short piece. Whenever a note gets swapped out to a sharp or flat version of itself, we are moving to the resources of a different scale. The A minor prelude of book 1 does this:

  • The first 9 bars require all seven naturals and all five sharps (no flats): the C major and B major diatonic scales combined to provide the resources of those 12 notes
  • Bars 10-16: suddenly, no accidentals at all anymore, but only the seven diatonic notes of the C major scale
  • Bars 17-28 (end): this section of the piece requires all the classic and old-fashioned set of notes, Eb to G#, and no D# or A# anymore
  • Two of the keyboard's 12 notes have to do double duty: Eb/D# and Bb/A# for different sections of the piece
  • A closer look at the scale resources needed: bars 1-3 use only the A minor diatonic scale, plus G# in place of G
  • The G natural in bar 4 replaces G, signaling a move away from A minor
  • The D# at the end of bar 4 replaces D, and signals that we are now in E minor
  • The A# at the end of bar 8 further decorates E minor, as leading tone to its dominant
  • Bars 9, 10, and 11 are the same music repeated three times, but down a step each time: walking down from E minor to C major
  • Bars 11-16 are entirely in C major
  • Bar 17 suddenly gives us F#, Bb, and Eb (replacing F, B, and E), lurching us into G minor
  • Bar 19 adds C# (replacing C) and flips Eb back to E: D minor
  • Bar 21 adds G# (replacing G) to put us back to A minor
  • Bar 26's downbeat could have ended the piece, but this bar gives us two notes foreign to A minor (and borrowed from D minor: C# and Bb)
  • The Bb changes back to B in bar 27, and G to G#: a plagal cadence from D minor back into A minor for the end (with a Picardy third).
Even in such a short piece with an empty key signature, A minor, we need 14 different notes; and the overlapping notes Eb/D# and Bb/A# ought to sound "good" in all situations, although they belong to different scales.

Book 2
Here is the similar analysis of book 2:
  • Book 2, C major: 14 notes: Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#
  • Book 2, C minor: 13 notes: Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#
  • Book 2, C# major: 13 notes: A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, Cx, Gx
  • Book 2, C# minor: 14 notes: C, __, D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, Cx
  • Book 2, D major: 14 notes: Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E#
  • Book 2, D minor: 13 notes: Db, __, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#
  • Book 2, Eb major: 16 notes: Fb, Cb, Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#
  • Book 2, D# minor: 14 notes: A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, Cx, Gx, Dx
  • Book 2, E major: 14 notes: G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, Cx
  • Book 2, E minor: 13 notes: F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E#
  • Book 2, F major: 15 notes: Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#
  • Book 2, F minor: 13 notes: Cb, Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B
  • Book 2, F# major: 15 notes: D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, Cx, Gx, Dx
  • Book 2, F# minor: 14 notes: C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx
  • Book 2, G major: 13 notes: Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#
  • Book 2, G minor: 15 notes: Cb, __, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#
  • Book 2, Ab major: 16 notes: Ebb, Bbb, Fb, Cb, Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B
  • Book 2, G# minor: 17 notes: C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, Cx, Gx, Dx
  • Book 2, A major: 13 notes: C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#
  • Book 2, A minor: 15 notes: Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#
  • Book 2, Bb major: 15 notes: Cb, Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#
  • Book 2, Bb minor: 16 notes: Ebb, Bbb, Fb, Cb, Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B
  • Book 2, B major: 14 notes: D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, Cx, Gx
  • Book 2, B minor: 14 notes: F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#
Both books taken together

Across both books, from the perspective of each individual note, how many of the 48 pieces call for it? (That is, how many of the pieces will sound "wrong" at the occurrences of this note, if it is not tuned into a position that well represents its spelling?)

  • Ebb: 2
  • Bbb: 3
  • Fb: 5
  • Cb: 10
  • Gb: 12
  • Db: 17
  • Ab: 19
  • Eb: 26
  • Bb: 29
  • F: 31
  • C: 36
  • G: 39
  • D: 45
  • A: 48
  • E: 48
  • B: 47
  • F#: 43
  • C#: 39
  • G#: 35
  • D#: 30
  • A#: 28
  • E#: 24
  • B#: 20
  • Fx: 16
  • Cx: 12
  • Gx: 9
  • Dx: 4
  • Ax: 1

How many of the 48 pieces have enharmonic equivalence on each given note (so it has to sound musically good when spelled as either one)?

  • Ebb and D: 2
  • Bbb and A: 3
  • Fb and E: 5
  • Cb and B: 9
  • Gb and F#: 7
  • Db and C#: 8
  • Ab and G#: 6
  • Eb and D#: 8
  • Bb and A#: 9
  • F and E#: 7
  • C and B#: 8
  • G and Fx: 7
  • D and Cx: 9
  • A and Gx: 9
  • E and Dx: 4
  • B and Ax: 1


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Bach's schematic, rotated for use