I’m referring to Ab the note, not only Ab-C the major third.
Ab must serve as Fa or Do, or other scale degrees, within hundreds of Bach’s keyboard pieces and thousands of his vocal pieces. The keyboards must work in all keys to do the jobs of the keyboard player. My roster of the keyboard pieces is here (and the 2022 article explains all of that):
As I suggested in 2004-5, and wrote more thoroughly in 2021-22, tonality doesn’t behave the same way (symmetrically) for flats and sharps. It matters how the compromises are done, when setting a pitch that is supposed to work either way. If there is some note that is some amount out of tune for the context of the other scale notes around it (sharps that are too sharp, and flats that are too flat), it’s more tolerable to have that out-of-tuneness happen in Ti or Mi of the scale, than to happen where the out-of-tune note is Do or Fa.
That roster of the keyboard pieces is not meant as a pretext for statistics, but it’s observation that all the notes of all the scales must work. We need 24 differently named notes to be able to play the Inventions and Sinfonias, learning how to play keyboards in a singing melodic style in these 15 keys. (We’re not going to bend clavichord tangents or retune the harpsichord for every piece. We’re also not going to go read organ construction books to figure out monochord calculations to tune harpsichords. We’re just going to tune the harpsichord by ear and then play the music…) And, being “in” a key doesn’t mean merely looking at the key signature or playing root-position triads. Every note in the score must make sense in its scales, whether it’s notated as an accidental or in the signature.
In the first piece of the WTC, the prelude in C major demands Ab before the fugue demands G#. Whatever compromise is chosen for the pitch of G#/Ab, it must sound OK as both…within all the scales that ask for Ab, and within all the scales that ask for G#. Bach says on the title page of the book that we must be able to play in all the major scales, and in all the minor scales. (Not triads, and not isolated major thirds, but melodic music based on scales.) Where Kuhnau’s first two books advertised and presented exemplary pieces in 7 of the major scales (Ut Re Mi), and 7 of the minor scales (Re Mi Fa), Bach in taking over Kuhnau’s job is going to make us play music in ALL the Ut Re Mi and ALL the Re Mi Fa scales. We must be able to do everything. Here are Kuhnau’s title pages and pieces:
Once the trained keyboardists can play in every key, and tune the instrument so it works in every key, we can go on to do the jobs of improvising continuo, improvising pieces, composing, arranging, teaching, playing in ensembles, and everything else. We can play pieces with 19 different notes in them, like the Goldberg Variations, or the Art of Fugue with its required 5 flats, 7 naturals, and 5 sharps. We don’t have to worry that any of the music will ever sound bad from the different uses of sharps and flats on a 12-lever keyboard.
To remind myself how some people choose to live, I set up Vallotti this morning.
In the Goldbergs, I played all the G minor variations, all the canons, and 19 and 22. In the G major pieces, the note that screamed at me the most was D# being so high above B, Pythagorean. As a 10th or a 17th above B in the bass, it’s worse. I played through the WTC 1 preludes and fugues in Bb minor, B major, C# minor, and E major. The Pythagorean thirds that occur within those WTC pieces made me want to stop playing harpsichord and go do something else.
I am on board with Sorge’s dislike of Pythagorean thirds, as stated in his 1744 book, page 12. It’s a primer of keyboard tuning. He said he would rather listen to cats and dogs attempting music than to listen to a keyboard set up with any Pythagorean thirds. "Ich lieber die Katzen und Hunde wolte musiciren hoeren, als einen Clavieristen mit seinem auf solche Weise gestimmten Clavier oder Orgel.”