Bach's schematic, as it appears on the page
LaripS.com, © Bradley Lehman, 2005-22, 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

(2005 section)

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 notes, 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 notes, 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 15 notes Cb, 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 notes: 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 required by each prelude and fugue of book 1 (Bbb to Ax):
  • 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: 15 notes: Cb, 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, which uses a different 27 (adding Ebb, not needing Ax):
  • 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: 11
  • 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: 10
  • 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

See also the 2009 article where I showed why all of this enharmonic perspective is important, along with debunking the statistical methods of John Barnes and his followers.


Bach's other keyboard music outside the Well-Tempered Clavier

(2022 section, the data set for the article "The Notes Tell Us How to Tune")

In Bach's music, we can see that the Well-Tempered Clavier is not an outlier in its requirements with adventurous modulations. A survey of his keyboard music, the more than 400 pieces given here, makes it obvious that it is not rare to go beyond 12 notes within a single piece. He went beyond 12 notes in all stages of his career.

For example, one of the earliest pieces in this list is the Toccata in F# minor, BWV 910, which has been dated to 1707 when Bach was 22. It needs the 15 notes F, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, and Cx. This is 15 years before his compilation of the Well-Tempered Clavier. A yet earlier piece is the Capriccio in Bb, BWV 992, which needs the different 15 notes from Gb to G#. It is from 1704, when Bach was 18 or 19.

I use the standard BWV numbers to distinguish any similarly named pieces, along with stating the home key: a capital letter for a major key, or lowercase for a minor key.

  • Inventions: C (Bb-G#), c (Ab-F#), D (C-A#), d (Eb-G#), Eb (Db-B), E (D-Cx), e (C-A#) F (Eb-C#), f (Gb-F#), G (F-C#), g (Ab-C#), A (G-B#), a (Ab, Bb-D#), Bb (Ab-F#), b (C-E#).
  • Sinfonias: C (Eb-C#), c (Db-F#), D (F-E#), d (Eb-D#), Eb (Db-B), E (G-B#), e (Eb-A#), F (Eb-C#), f (Bbb-B), G (F-A#), g (Ab-G#), A (C-B#), a (Bb-D#), Bb (Ab-C#), b (C-E#).
  • Inventions and sinfonias taken together: 24 notes from Bbb to Cx.
  • French suites: d 812 (Eb-G#), c 813 (Db-C#), b 814 (C-B#), Eb 815 (Gb-F#), G 816 (Eb-A#), E 817 (G-B#).
  • English suites: A 806 (Bb-B#), a 807 (Eb-E#), g 808 (Db-D#), F 809 (Db-D#), e 810 (Ab-E#), d 811 (Ab-D#).
  • Partitas: Bb 825 (Cb-C#), c 826 (Gb-C#), a 827 (Ab-E#), D 828 (Bb-E#), G 829 (Eb-A#), e 830 (Ab-B#).
  • Art of Fugue (Die Kunst der Fuge) 1080: d (Gb-A#)
  • Ricercars from the Musical Offering 1079: c a3 (Bbb-C#), c a6 (Fb-C#).
  • Canon per tonos (modulating canon) from the Musical Offering: If written out for performance by keyboard or ensemble, all 26 notes from (Bbb-Dx), and needing enharmonic exchange somewhere.
  • Little Preludes 924-930: C (Ab-G#), D (Bb, C-G#), d (Eb-G#), F (Bb-E), F (Eb-G#), g 929 (Ab-F#), g 930 (Ab-C#).
  • Little Preludes 933-938: C (Bb-F#), c (Db-F#), d (Ab-C#), D (C-D#), E (D-B#), e (F-A#).
  • Preludes 939-943: C 939 (Bb-F#), d (Bb-G#), e (F-D#), a (Ab, Bb-D#), C 943 (Ab, Bb-C#).
  • Little Preludes taken together: 18 notes from Db to B#. (For example, if a clavichord or harpsichord is to be set up for these together for lessons, it must have a temperament that handles all 18 of those notes.)
  • Minuets in Wilhelm Friedemann's book: G 841 (C-C#), g 842 (Ab-F#), G 843 (C-A#), g 929 (menuet by Stolzel, trio by Bach) (Ab-C#).
  • Overture F 820 (Ab-C#).
  • Miscellaneous suites: a 818 (Bb-D#), a 818a (Ab-A#), Eb 819 (Fb-F#), Bb 821 (Gb-C#), f 823 (Bb-B#), A 832 (Bb-B#).
  • Praeludium and Partita F 833 (Ab-C#).
  • Prelude, fugue, and allegro Eb 998 (Fb-C#).
  • Miscellaneous fugues: c 574 Legrenzi (Db-C#), c 575 (Gb-C#), G 576 (Bb-A#), G 577 (C-A#), g 578 (Ab-C#), b 579 (C-E#), D 580 (F-E#), A 896 (G-E#), a 944 (Ab-E#), e 945 (by Graupner) (C-D#), C 946 (Eb-D#), a 947 (Bb-D#), d 948 (Cb-Cx), d 948 shorter version (Cb, Db-D#), A 949 (C-E#), A 950 Albinoni (C-B#), G 950a (Bb-A#), b 951 Albinoni (C-E#), b 951a Albinoni (F-B#), C 953 (Bb-D#), Bb 954 Reincken (Db-G#), Bb 955 Erselius (Db-C#), e 956 (Bb-A#), a 958 (Bb-D#), a 959 (Bb-A#), e 962 (by Albrechtsberger) (Bb-A#).
  • Miscellaneous preludes and fugues: a 894 (Db-A#) , a 895 (Bb-D#), F 901 (Db-G#). (894 = Later reworked into the Triple Concerto 1044.) (901 = Fugue later transposed and expanded to become the Ab major fugue of WTC book 2.)
  • Miscellaneous sonatas: D 963 (C-E#), d 964 (Gb-D#) , a 965 Reincken (Eb-A#), C 966 Reincken (Eb-A#), a 967 (Eb-D#), G 968 Adagio (Ab-A#). (964 = Arranged from the A minor violin sonata 1003.) (968 = Arranged from the C major violin sonata 1005.)
  • Capriccii: Bb 992 (Gb-G#), E 993 (C-Cx).
  • Goldberg variations: G 988 (Fb-A#).
  • Aria variata: a 989 (Bb-D#).
  • Canonic Variations: C 769 and 769a (Ab-D#, E#).
  • Passacaglia c 582 (Gb-F#).
  • Trio Sonatas: Eb 525 (Gb-F#), c 526 (Gb-C#), d 527 (Db-D#), e 528 (Bb-E#), C 529 (Ab-A#), G 530 (Eb-E#).
  • Trios: d 583 (Ab-G#), c 585 (Db-C#), G 586 (Eb-A#), F aria 587 (Ab-G#), Eb 597 (Gb-B).
  • Toccatas pedaliter: F 540 (Cb-G#), d 538 (Ab-D#), d 565 (Ab-G#).
  • Toccata, adagio, and fugue: C 564 (Ab-A#).
  • Toccatas manualiter: f# 910 (F, G-Cx), c 911 (Gb-C#), D 912 (Bb-Fx), d 913 (Cb-D#), e 914 (F-E#), g 915 (Cb-C#), G 916 (F-A#).
  • Preludes and Fugues pedaliter: C 531 (Eb-D#), D 532 (Eb-B#), e 533 (F-E#), f 534 (Cb-F#), g 535 (Fb-D#), g 535a (Db-C#), A 536 (G-E#), d 539 (Ab-G#), G 541 (Ab-A#), a 543 (Eb-A#), b 544 (F-B#), C 545 (Ab-D#), C 545a (Eb-D#), c 546 (Cb-C#), C 547 (Db-A#), e 548 (Db-B#), c 549 (Db-C#), d 549a (Eb-D#), G 550 (Bb-A#), a 551 (Ab-A#), E 566 (C, D-Cx), C 566a (Ab, Bb-A#).
  • Little Preludes and Fugues: C 553 (Bb-G#), d 554 (Eb-D#), e 555 (F-A#), F 556 (Eb-D#), G 557 (F-D#), g 558 (Ab-C#), a 559 (Bb-D#), Bb 560 (Ab-F#).
  • Prelude and Fughetta: d 899 (Eb-G#), e 900 (Bb-A#), G 902 (Bb-A#).
  • Preludes: G 568 (F-A#), a 569 (Eb-E#), b 923 (F-Fx), C 943 (Ab-C#). C 567 by Krebs (Ab-D#).
  • Canzona d 588 (Eb-D#).
  • Allabreve D 589 (Bb-E#).
  • Pedal Exercise g 598 (Gb-C#).
  • Fantasia and fugue: c 537 (Cb-C#), g 542 (Fb-E#), a 561 (Ab-E#), a 904 (Ab-A#).
  • Fantasia on a rondo: c 918 (Cb-C#).
  • Fantasia con imitatione: b 563 (C-E#).
  • Fantasia duobis subjectis: g 917 (Ab-C#).
  • Chromatic fantasy and fugue: d 903 (Fb-B#, and some sources include a Bbb).
  • Fantasias: g 517 (Ab-C#), c 562 (Gb-C#), C 570 (Bb-G#), G 571 (Eb, F-A#), c 906 (Fb-G#), c 919 (Db-F#), g 920 doubtful (Db-G#), a 922 (Ab-E#, Fx), a 944 (Ab-E#).
  • Pastorale F 590 (Gb-G#).
  • Little Harmonic Labyrinth C 591 (Db-Cx).
  • Piece d'orgue: G 572 (Eb-A#).
  • Concerto arrangements pedaliter: G 592 (F-A#), a 593 (Bb-D#), C 594 (Ebb-A#), C 595 (Ab-D#), d 596 (Db-D#),
  • Concerto arrangements manualiter: G 592a (F-A#), D 972 (Bb-E#), G 973 (Eb-A#), d 974 (Ab-G#), g 975 (Db-D#), C 976 (Ab-G#), C 977 (Db-D#), F 978 (Eb-G#), b 979 (F-E#), G 980 (Bb-E#), c 981 (Db-C#), Bb 982 (Db-G#), g 983 (Ab-C#), C 984 (Db-D#), g 985 (Gb-C#), G 986 (F-A#), d 987 (Eb-G#).
  • Concerto and fugue: c 909 (Fb-G#).
  • Italian concerto: F 971 (Db-G#).
  • French Overture: b 831 (F-Fx), early version c 831a (Gb-G#). (Italian Concerto and French Overture taken together as Clavierübung II, the way Bach published them: 19 notes from Db to Fx. Presumably, the harpsichordist is not expected to retune the instrument from one piece to the other. The single setup shows two contrasting aspects of a harpsichord simulating a full orchestra.)
  • Orgelbüchlein: a 599 (Bb-D#), F 600 (Eb-F#), A 601 (D-D#, E#), F 602 (Eb-B, C#), g 603 (Ab-F#), G 604 (Bb-C#, D#), G 605 (F-C#, D#), D 606 (C-E#), g 607 (Eb-F#), A 608 (G-E#), G 609 (F-D#), c 610 (Db-F#), d 611 (Ab, Bb-D#), g 612 (Eb-C#), b 613 (G-E#), a 614 (Eb-D#), G 615 (Bb-D#), d 616 (Bb-D#), a 617 (Bb-A#), F 618 (Eb-G#), F 619 (Bb-G#), a 620 (Eb-D#), e 621 (Bb-G#), Eb 622 (Fb-F#), G 623 (F-D#), g 624 (Ab-C#), d 625 (Bb-G#), a 626 (Bb-D#), d 627 (Eb-G#), D 628 (C-E#), d 629 (Bb-D#), g 630 (Ab-C#), C 631 (Bb-C#), F 632 (Eb-B, C#), A 633 (D-D#), A 634 (D-D#), C 635 (Ab-F#), d 636 (Eb-G#), a 637 (Ab-D#), D 638 (C-A#), f 639 (Gb-B), e 640 (C-E#), G 641 (Bb, C-C#, D#), a 642 (F-G#), G 643 (C-A#), g 644 (Ab-F#).
  • Orgelbüchlein 599-644, in summary of its 45 chorales:
    • 6 chorales need more than 12 notes within their one or two pages.
    • 19 chorales need D# instead of Eb, 18 need Eb instead of D#, and two need both.
    • 2 chorales need both Ab and G#.
    • 10 chorales need Ab instead of G#, and three of those pieces go beyond it into Db, Gb, Cb, and Fb.
    • 7 chorales need A# instead of Bb, and one needs both.
    • 6 chorales need E# instead of F.
    • The whole book needs Db to E#, 17 notes.
    • Orgelbüchlein is from the same time as the Well-Tempered Clavier and the Inventions and Sinfonias: Bach's first year at Leipzig, using these as his textbooks and teaching students to play suitable music in all the keys.
  • Chorale Partitas: a 743 (Bb-G#), e 745 (Eb, F-E#), f 766 (Gb-B), c 767 (Cb-F#), g 768 (Db-C#), e 770 (Bb-A#), G 771 (F-D#).
  • Schübler Chorales: Eb 645 (Ab-F#), e 646 (C-A#), c 647 (Db-F#), d 648 (Db-D#), Bb 649 (Ab-F#), G 650 (Bb-A#).
  • Leipzig Chorales: F 651 (Db-D#), G 652 (F-A#), G 653 (Bb-D#), G 653a (Eb-D#), Eb 654 (Gb-B), G 655 (F-A#), A 656 (G-B#), A 656a (C-B#), G 657 (F-A#), f 658 (Gb-B), g 659 (Ab-C#), g 660 (Ab-C#), g 661 (Ab-C#), A 662 (C-E#), G 663 (Bb-A#), A 664 (Bb-E#), e 665 (Eb-A#), e 666 (Eb, F-A#), G 667 (Eb-C#), G 668 (Eb, F-E#).
  • Yale "Neumeister" chorales (Edition: Bärenreiter 5181): G 719 (F-G#), g 1090 (Eb-C#), a 1091 (Eb-D#), a 1092 (Bb-A#), g 1093 (Ab-C#), g 1094 (Ab-C#), F 1095 (Eb-C#), a 1096 (Bb-D#), D 1097 (F-D#), d 1098 (Eb-G#), G 1099 (Bb-D#), C or a 1100 (Bb-G#), b 714 (G-E#), f# 742 (C-E#), d 1101 (Bb-D#), Bb 1102 (Ab-F#), g 1103 (Eb-C#), e 1104 (F-A#), d 1105 (Bb-G#), G 1106 (Bb-C#), D 1107 (Bb, C-G#), b 1108 (F-A#), d 1109 (Bb-D#), Bb 1110 (Ab-F#), G 1111 (Bb-D#), F 1112 (Eb-G#), b 1113 (C-E#), f 1114 (Db-B), C 1115 (Eb, F-D#), G 1116 (F-C#, D#), Bb 1117 (Ab-C#), G 957 (F-C#, D#, A#), G 1118 (Bb-G#), d 1119 (Eb-G#), e 1120 (C-D#).
  • Miscellaneous chorale preludes (Edition: Breitkopf 6589 (Lohmann)): a 690 (Bb-D#), a 691 (F-G#), g 694 (Ab-G#), d 695 (Bb-G#), d or a 696 (Eb-G#), G 697 (Bb-D#), G 698 (Eb, F-A#), g 699 (Ab-C#), C 700 (Bb-D#), C 701 (Bb-G#), Bb 702 (Gb-C#), F 703 (Eb-E), d or F 704 (Eb-G#), d 705 (Eb-G#), A 706 (G-D#, E#), a 707 (Eb-A#), a 708 (F-D#), G 709 (F-A#), g 710 (Ab-C#), G 711 (Bb-D#), A 712 (Bb-B#), a 713 (Eb, F-A#), b 714 (G-E#), G 715 (Ab-D#), G 716 (F-D#), G 717 (F-D#), e 718 (F-E#), G 719 (F-C#), D 720 (C-A#), f# 721 (F-E#), G 722 (Eb-D#), G 723 (F-C#), G 724 (F-C#), e 725 (Eb-D#), G 726 (Ab-A#), b 727 (C-E#), C 728 (Bb-D#), A 729 (C-B#), G 730 (Bb-C#, D#), G 731 (F-C#, D#), E 732 (C, D-B#), d or F 733 (Eb-G#), G 734 (F-D#), Bb 735 (Ab-C#), D 736 (Bb, C-E#), d 737 (Bb-G#), D 738 (C-E#), G 739 (F-A#), d or F 740 (Ab-G#), d 741 (Ab-G#), a 744 (F-D#), g 747 (Ab-G#), G 751 [by Johann Michael Bach] (F-C#), Bb 754 (Eb-F#), G 758 (Bb-C#), d 762 (Bb-G#), d 765 (Ab-G#), a 691a (Bb-D#), d 695a (Bb-D#), a 708a (F-G#), G 722a (Eb-D#), A 729a (G-B#), E 732a (C, D-E#), Bb 735a (Ab-C#), D 738a (C-E#), G Anh.55 (F-D#), a Auf meinen lieben Gott (canon) (F-D#), F O Lamm Gottes (Ab-C#). [The version of BWV 714 here is shorter than the one in the Yale manuscript.]
  • Clavierübung III: Eb prelude and fugue 552 (Cb-F#). Chorales: g 669 (Db-F#), c 670 (Gb-F#), g 671 (Fb-F#), e 672 (Bb-D#), e 673 (Bb-D#), e 674 (Bb-D#), F 675 (Eb-G#), G 676 (Bb-A#), A 677 (F, G-B#), G 678 (Bb-D#), G 679 (Eb-A#), d 680 (Db-G#), e 681 (C-A#), e 682 (Ab-E#), d 683 (Eb-G#), g 684 (Db-C#), d 685 (Bb-G#), e 686 (Bb-D#), f# 687 (C-E#), d 688 (Db, Eb-G#), f 689 (Gb-F#). Duetti: e 802 (Db-E#), F 803 (Gb-A#), G 804 (F-A#), a 805 (Ab-D#).
  • Sonatas for flute and harpsichord: b 1030 (Ab-B#), g minor alternative version 1030b (with violin? oboe?) (Fb-G#), A 1032 (Bb-B#), C 1033 (Bb-G#).
  • Sonatas for flute and continuo: e 1034 (F-A#), E 1035 (G-Fx).
  • Sonatas for viola da gamba and harpsichord: G 1027 (Ab-B#), D 1028 (Eb, C-B#), g 1029 (Gb-G#).
  • Sonatas for violin and harpsichord: b 1014 (C-E#), A 1015 (C-B#), E 1016 (G-Fx), c 1017 (Gb-C#), f 1018 (Cb-F#), G 1019 (Bb-E#).
  • Sonatas for violin and harpsichord: g 1020 (Gb-C#), F 1022 (Db-G#).
  • Sonata for violin and continuo: G 1021 (Bb-A#), e 1023 (Eb-A#), g 1026 (Db-C#).
  • Harpsichord concertos: d 1052 (Cb-D#), E 1053 (F-Cx), D 1054 (F-B#), A 1055 (Bb, C-B#), f 1056 (Cb-B), F 1057 (Ab-D#), g 1058 (Cb-G#).
  • 2-harpsichord concertos: c 1060 (Gb-C#), C 1061 (Ab-A#), c 1062 (Fb-C#).
  • 3-harpsichord concertos: d 1063 (Gb-D#), C 1064 (Gb-D#).
  • 4-harpsichord concerto: a 1065 (Eb-A#).
  • Brandenburg concerto 5: D (Eb-E#).
Observations
Notice that Bach frequently requires flats out to Gb or beyond (Cb, Fb, Bbb), or sharps out to A# or beyond (E#, B#, Fx, Cx). Db and E# are common notes, even in very short pieces.

As we saw with the Well-Tempered Clavier, pieces frequently need 13, 14, 15, or 16 notes.

  • The Fantasia 922 is an example of another piece that needs 16.
  • We get occasionally to 17 notes in pieces with especially adventurous modulations (B minor Flute Sonata 1030, Art of Fugue, 3-voiced Ricercar from the Musical Offering, Concerto and Fugue 909).
  • The four Duetti taken together need 18 (Gb-E#) within a book that needs 21 (Clavierübung III, Fb-B#). The "Wedge" Prelude and Fugue in E minor (BWV 548) needs a different 18.
  • The Goldberg Variations need 19, as does the Fantasia and Fugue 542.
  • The Little Harmonic Labyrinth needs 20. (BWV 591, attributed variously to Bach or Heinichen. It includes at least six occasions with direct enharmonic swaps of perception or notation (a held or restruck key gets reinterpreted as a different note). C changes to B#, F to E#, Fx to G, D# to Eb, Db to C#, and G# to Ab. It is approaching a note in one way with diatonic expectation, but exiting that note in a different way, rather like navigating a labyrinth.)
  • Some of the most extravagant examples in this list need 21 (Concerto 594) or 22 (Fugue 948).
  • The Chromatic Fantasy and Fugue needs 22 or 21, which depends on choosing a manuscript source that includes Bbb or not.
  • The whole book of Inventions and Sinfonias together needs 24, on the presumption that we will not retune the harpsichord or clavichord. The question of retuning during the above pieces would be moot, anyway.

The organ pieces do not show simpler enharmonic ranges than the harpsichord or clavichord repertoire. Every keyboard (and player) had to be capable of all the sharps and flats. (Barbour exclaimed about this in his 1951 book, page 196, after he already explored this path of reckoning sharps and flats separately. "The organ works of Bach show as great a range of modulation as his clavier works do. Except for a dozen chorale preludes in the Orgelbüchlein, there are only 3 organ works of 148 examined that do not overstep the compass of the conventionally tuned organ. The compass of individual organ pieces is very frequently 13, 14, and 15 scale degrees, and even 18, 19, and 21 degrees have been observed. The compass of Bach's organ works as a whole is Ebb-Cx, 25 degrees!" Barbour's next sentence then drew a wrong conclusion, asserting that the triads in remote keys "would have been dreadfully dissonant in any sort of tuning except equal temperament". His evidence-based assessment by scale degrees was correct, though.)

The miscellaneous chorales also go beyond 12 notes in single pages. For example, the C major Gelobet seist du, Jesu Christ (BWV 697) is only 14 bars long, but it needs both Bb and A#.

As noted above, with its extra length and the thematic enterprise of playing in all keys, the Well-Tempered Clavier needs 27 differently for each book. Book 1's prelude and fugue in Eb minor and D# minor needs 25 by itself. Book 1's B minor prelude and fugue needs 17, and the fugue subject alone has 13. (There is plenty of musical literature exclaiming that this B minor fugue has a "12 tone subject", but that's obviously being framed incorrectly from a future perspective with equal temperament's assumptions of interchangeability. It needs 13, including both C natural and B#.)

It is easy to recognize the 15-note pieces in the list. Their endpoints by the spiral of fifths have notes of similar names. For example, the first English Suite (Bb-B#) needs the three varieties of B: Bb, B natural, and B#, plus everything between those. The Prelude in A minor for organo pleno (BWV 569) needs Eb to E#.

I find the D major viola da gamba sonata intriguing. It needs the 13 notes of C to B#, but then it suddenly needs also Eb without using Bb or F. Those moments in the piece are remarkably expressive because the Eb is so foreign to the scale context. There are some other pieces like that, as well, where a single note is not connected at one end of the spiral of fifths. One of those has a possibly humorous stroke in it: the G major Fantasia/Concerto 571 spends all its time needing only the range of F to A# (12 notes), but then in the very last bar it reaches out for an Eb.

Where Bach is directly arranging solo keyboard pieces by Reincken, or borrowing themes from Erselius or Albinoni, or arranging other composers' ensemble concertos for a solo keyboardist: it's the same story. He goes beyond 12 notes, or he requires a 12-note compass that is far from the traditional limits of Eb to G#.

The concerto arrangements are somewhat constrained by the tonal ranges of the original orchestral pieces. Even with that restriction, notice that many of them go far beyond any old-fashioned compass of Eb to G#. That is true whether they are for keyboard or for orchestra.

It is exceedingly rare to find Bach keyboard pieces that have no accidentals at all, and therefore using as few as the 7 notes of a single diatonic scale. The only piece in the above list that does this is the extremely short Little Prelude in F 927, using only Bb, F, C, G, D, A, and E.

Consider also the instrumentation, and the feasibility (or not) of adjusting a temperament when going from piece to piece.

  • Organ: Little or no adjustment during a year or more. It is the organ builder's job, not the player's job.
  • Fretted clavichord: With some strings playing more than one different note, some tangents must be bent sideways to strike the string at a different place (in the task of altering the temperament). This is feasible but inadvisable, potentially destructive, and not everyday work.
  • Harpsichord: Possible, and not difficult for a careful listener. Against that, consider the work load of being busy with other musical and administrative tasks, and the amount of time available while tuning the entire harpsichord by ear in 15 minutes. For convenience, a busy person such as Bach might not adjust the temperament at all from piece to piece. If the session is for practice of a piece that is intended for organ or clavichord performance, it might be desirable to match that instrument's fixed temperament to be able to practice similar effects.
  • Harpsichord having different temperaments in different sets of strings: Feasible, but then these cannot be coupled together in performance.
  • Unfretted clavichord: Same observations as harpsichord, but note that it is more difficult to tune clavichords quickly or accurately. The player and tuner must guess at the average key pressure that will be used in the performance. (The pitch is sharper when pressing the keys more firmly.) This is a practical argument to get it working acceptably enough, and then not use many further adjustments from piece to piece, because the time could be spent practicing the music rather than tuning.
  • Lautenwerk: Same observations as harpsichord.
  • Early piano: Same observations as harpsichord, but taking longer to tune if the tuner must use mutes when working on double strings.
  • Double-strung or triple-strung clavichord: Same observations as early piano regarding the use of mutes while tuning.
  • Organ with several isolated ranks in a different temperament: Feasible, if the player can reach those pipes, but then these ranks cannot be coupled with the rest of the instrument.
  • Keyboard instruments with more than 12 levers per octave: Feasible, but consider that they probably will not have distinctions between E# and F, B# and C, or Cb and B. The E# is the most frequent problem in such an approach.

General patterns by keys
Generally, pieces in major keys require Fa, Ri (the diatonic semitone under Mi), and everything in between by the spiral of fifths. Fa-Do-Sol-Re-La-Mi-Ti-Fi-Di-Si-Ri. If the piece also goes into the parallel minor, the notes at Te, Me, and Le are added.

Pieces in minor keys typically require Ra (the diatonic semitone above Do), Ti, and everything in between. Ra-Le-Me-Te-Fa-Do-Sol-Re-La-Mi-Ti. Beyond that, it is common to add Se (the diatonic semitone above Fa) or Fi (the diatonic semitone under Sol).

See also the presentation of Fa, Ti, and modulation.


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Bach's schematic, rotated for use