# Enharmonic key signatures in Bach's Well Tempered Clavier

I have been listening to Bach's Well Tempered Clavier Book 1 and have noticed something intriguing. I have two (related) questions about it. The work consists of 24 preludes and fugues in each major and minor keys. So, for instance, it begins with:

No. 1: Prelude and Fugue in C major, BWV 846; No. 2: Prelude and Fugue in C minor, BWV 847.

But in number 8 the prelude is in E♭ minor and the fugue is in D♯ minor:

No. 8: Prelude in E♭ minor and Fugue in D♯ minor, BWV 853.

Since E♭ and D♯ are enharmonic and are the same key in the keyboard, I wonder what is the harmonic justification for this. Why didn't he just name both E♭ or both D♯?

Later, following his G minor prelude and fugue (no. 16), he writes his A♭ major prelude and fugue and, after it, the prelude and fugue are in G♯ minor, not in A♭ major:

No. 17: Prelude and Fugue in A♭ major, BWV 862; No. 18: Prelude and Fugue in G♯ minor, BWV 863.

Again, why is this so? Why not write both in the same key?

• A little confusion here with enharmonic minors, then enharmonic major/minor. Two different questions/answers.
– Tim
Commented Feb 9, 2020 at 18:23
• IIRC, the D# minor fugue already existed in a d minor version before the entire collection was compiled, and presumably Bach preferred re-copying it into D# rather than Eb because this means only adjusting the accidentals rather than the accidentals and all notes. Commented Feb 10, 2020 at 7:22
• Why would it be easier to transpose to D♯ instead of transposing it to E♭? Commented Feb 13, 2020 at 17:56
• @PedroG.Mattos consider the C sharp diminished seventh chord in D minor: C#, E, G, and Bb. To transpose that to D sharp minor (Cx, E#, G#, B), you change the key signature, copy the noteheads to the same places on the staff, and modify all of the accidentals. To transpose to E flat minor (D, F, Ab, Cb), you change the key signature, move all the noteheads up from a line to the next higher space or from a space to the next higher line, and adjust the accidentals. In practice, the need to of noteheads to a different position on the staff results in a significant increase in copying errors. Commented Jan 17 at 12:35
• Thanks for the example, this is very clear now. Commented Jan 19 at 3:38

There are a few leading factors which have a bearing:

• pedagogical / theoretical

• convenience / historiography

• musical references

The first is well covered by other answers: that there was a didactic element to Book I of Das Wohltemperirte Clavier. By having this key, the most remote of the minor keys from the non-accidental A minor, represented by two different notations, it cemented the full circularity of the 12-tone system, both in terms of understanding the equivalence of both Eb minor and D# minor for the keyboard, but also how the idea of temperament worked to bring both to the same note.

Ledbetter's comment that the equivalence of Eb and D# minors was well-known among theorists and composers stands, although its use of Rameau's Traité de l'Harmonie does not seem to be correct (Rameau only gives Eb minor in his Supplément, at least in the original printing).

The second is because of strong historiographical evidence for the fugue of BWV 853 being originally in D minor. According to Ledbetter, the "most obvious pointer" is the scale in the soprano voice of bar 15 which in D minor would have gone up to the note c''', the top note of the keyboard that Bach was writing for, but now has to jump down to c#'' instead.

Ledbetter also gives the example of some stray accidentals from the "original" notation, B naturals instead of B sharps, E flat for E natural. However, the older convention of "sharp/natural" = go a semitone up, "flat" = go a semitone down still holds in many Bach manuscripts and printed editions from this time in the 18th century.

The factor of musical reference is more arcane but also very interesting. The prelude is in the style of a tombeau, making reference to the French Baroque school, with their highly dotted rhythms and tirata scale passages. Many recordings make use of notes inégales. The abundance of flats in the key signature is also part of this reference: one can think of Froberger's Tombeau in C minor or that of Louis Couperin, or how Bach himself uses the dissonance of the extreme flat keys in the Passions at the death of Christ.

On the other side, the fugue is one of the stile antico fugues and Ledbetter says "it is the only piece in Book I to have a substantial element of the modal tradition". Its subject is very clearly based in Dorian mode, and it has a very strict treatment of dissonance while being rhythmically very loose (so much so that the subject itself mutates quite a bit, again a stile antico feature). Hence it hearkens back to a D dorian modal palette, and by simply transposing up from D minor, it refreshes the "old" into something "new" while still making its roots clear.

Why didn't he just name both E♭ or both D♯?

Why not both in D# major seems clear to me as D# major would need 9 sharps. Eb minor has 6 flats like D# minor has 6 sharps. As usual Bach’s pieces are extending to the dominant and subdominant what would imply additional sharps for the secondary dominants: e.g. F7 in measure 9 would be a E#7, the major 3rd of V/iv, which is G in Eb7 would be a double sharp F => Fx (##), also the secondary vii dim7 chords would need more sharps. So the reason to decide for one or the other key is to find in the intension to avoid additional accidentals, as a natural resolved sign is easier for reading.

• What's the reason for your first sentence? There is no D# major, unless you meant a modulation during the piece itself. Commented Feb 10, 2020 at 7:30
• 1. I didn't understand that OP was asking only about the minor variants. 2. Bach wrote the 3rd prelude in C#-major (7 sharps) instead of Db (5 flats) this was in opposite of my advise and is rather not congruent with my argumentation concerning the secondary dominants! (I hope they won't down vote it now ;) 3. May be Bach wrote the latter prelude in C# that the students learn to read sharp keys? :) One point that we can exclude is that Bach thought the key of C# would be another character than Db. (joke) But I'm sure some interprets will swear they hear it different looking at all the #. Commented Feb 10, 2020 at 8:40
• I just got confused by the mention of D# major, especially since it's the first sentence, since it doesn't come up anywhere else, either in your answer or the question. Commented Feb 10, 2020 at 11:10
• Thanks for the answer! This seems to be the way to understand it, but it is still not clear to me... I have to think about it for a while. Commented Feb 10, 2020 at 11:27
• I think I should add some examples what exactly means secondary dominants ... or do you understand this? Commented Feb 10, 2020 at 15:23

The reason may be that, at least in the first book of the 48, Bach was deliberately pushing the notation of tonal music to its limits instead of taking the easier options.

The E flat minor prelude and D sharp minor fugue are a good example. Because of the differing modulation schemes, the prelude contains some important chords of F flat major, while the fugue has some of E sharp major. In fact writing the prelude in D sharp and the fugue in E flat would have been "easier to read" if Bach had intended the music to be "easy to read". (When I first learned that prelude and fugue, as a teenager, my teacher gave me the task of writing out the music in the "opposite" keys by hand. That was before personal computers and notation software which could do the job with a couple of mouse clicks existed!)

The same may be true of the prelude and fugue in C sharp major, which arguably would be easier to read in D flat.

Some of the other pieces in Book I were original written specifically for educational purposes. For example the first prelude, in C major, was originally an improvisation exercise on the given chord sequence, written for Bach's son Wilhelm Friedemann, and the version in the WTC is Bach's "solution manual" to the exercise.

WTC Book II was compiled about 20 years after Book I, and Bach may have lost interest in such overt educational objectives by that time.

• (When I first learned that prelude and fugue, as a teenager, my teacher gave me the task of writing out the music in the "opposite" keys by hand.) That's what I use to do when practicing the prelude in C#-major, or even ignore all sharps and play it in C, so I can better understand what's happening harmonically) Commented Feb 10, 2020 at 8:37
• Sources, please. Commented Feb 11, 2020 at 0:43

Bach was simply completing his task of writing a movement in every key. It would be unlike Bach to leave any out.

Bach's new keyboard tuning/temperament allowed him for the first time to move through all of the keys pleasingly. The most accurate recreation of this tuning is the Lehman Tuning. Here is a demonstration

. Each key had its own character and would lend itself to open or closed intervals. Bach composed for each key noting its particular characteristics; for example the opening of the first prelude in C, It consists of many open intervals due to the tuning being based on perfect fifths.

Bach would have also taken into consideration the modulations into relevant keys. Without equal temperament, The 8th Fugue would sound like a different key due to its modulations.

• This doesn't make sense as an explanation of choosing to notate something in one key rather than another. Why is there a video in the answer?
– user9480
Commented Feb 10, 2020 at 16:19
• You mention equal temperament, but I thought that well-tempered was meant to refer to some other system such as some Werckmeister system that would also allow all keys to sound reasonable. Indeed, your video says equal temperament was a 20C development. You also speak of "Bach's new tuning" as though he had invented it, which I did not think was the case. Commented Feb 13, 2020 at 23:45

You must see Bach for what he was. He was a pioneer in seeing what could be done. He got to the point where he was a master in counter-point and he wanted to see how far he could take it. These pieces were an exploration in what the art of fugue sounded like in all the different keys, this was his treatise into how keyboard players should determine a suitable key for there musical ideas and how the harmonical possibilities were influenced by that choice, all that and they sounded pretty good too.