I don't believe Bach "hid" anything there, but rather he drew a straightforward practical diagram: knock certain notes one or two little nudges off spot, or leave them at pure-5th position.
Bach wouldn't have even had to know any of the math about 1/6 or 1/12 comma, necessarily, or any speculative theory; only that one nudges the harpsichord tuning lever by these tasteful amounts from experience, or from trial-and-error. By 1722 Bach had probably already been tuning harpsichords by ear himself for over 20 years. (And his extant music from his early 20s, before 1710, shows that he already had a circulating system in hand, presumably good-sounding, by that time. See, for example, both the capriccii BWV 992 and 993, and all the toccatas BWV 910-916...especially 910 in F# minor with its wild modulations.)
I'm curious: those of you who disagree with my work, have you actually read my papers and watched the videos, or is your reaction mainly to other people's remarks about things they didn't like? (Source vs hearsay....) It's all available for free download from http://www.larips.com
A vital part of the process, I maintain, is to play through all the music I mentioned in the academic papers, and also at
to hear the issues involved. Other temperaments (Kellner's, Werckmeister's, Barnes's, et al) fall on their noses whenever the music gets out too far beyond the usual two or three flats, or four or more sharps.
It's not just about looking at key signatures. It's about playing the whole piece, and observing what notes actually turn up in use. The set of inventions and sinfonias, for example, uses 25 different enharmonic notes. Many of those individual two-page pieces use 13, 14, or even 15 different notes...not only 12. Even if one sets out to retune the harpsichord a little bit between pieces, favoring the correctly-spelled notes for the piece to be played, it falls down whenever we get past 12 notes. And on fretted clavichords, retuning is right out: one needs to bend and reposition tangents. On organs we're even more locked in with a hardware commitment than on fretted clavichords.
Take a quick trip through WTC book 1. Try the first prelude and fugue. Obviously the prelude needs an Ab...but the fugue needs a G# instead of the Ab. What's going on here? Surely one isn't supposed to tweak notes on the harpsichord between playing the prelude and the fugue? No, there has to be an intermediate G#/Ab of some sort that sounds decently good as both.
Flip on to the next piece that looks relatively easy to play (yes, we'll come back to some of the others) and alight on the D major prelude with its formulaic and etude-like texture. Ah yes, we need a D# instead of the Eb, so that's easy...but we need both an A# and Bb in this little prelude, and we need both an E# and an F! 14 different notes in just one piece of a couple of pages!
What's going on here? And surely we're not supposed to move the common note F around whenever we need an E#, are we? What is this Bach guy up to? Maybe some tuning scheme where one is supposed to set up notes once for the whole book (egads!) and not move them? [There's also a report that Bach did play the whole WTC 1 straight through at lessons for a student named Gerber.]
Try the two-voiced fugue in E minor. What notes shall we set up? Ah, it needs 15 different notes! D#/Eb, A#/Bb, E#/F, plus all the usual suspects in the regular cycle between them.
Try the book's first piece that is in a totally funky key signature, seven sharps (!). What notes should be set up to play it? Well, once again the prelude by itself uses 14: A-E-B-F#-C#-G#-D#-A#-E#-B#-Fx-Cx-Gx-Dx. If the tuner favors A and E, the Gx and Dx are going to sound way too high for their contexts. And conversely, if he favors the Gx and Dx (why would he ever do that?), the A and E will be too low. We're supposed to compromise A and E somehow, the downtown notes on violins? Intrepidly we continue into the fugue and notice that it uses a 15th note, Ax.
These are the types of issues to deal with. Again, I emphasize: the evidence is right there in the music, before even bothering to look at the Bach title page drawing or argue about its interpretation. The interpretation of the drawing only puts the final 5% of clarity on top of the layout that is already constrained by the music, within narrow limits.
At my web site there are also CDs with organ (Taylor & Boody #41, Goshen College, Goshen IN) and harpsichord. There's also the newer Taylor & Boody #46 here in Virginia, in the church right across the street from the factory, tuned in this way from my papers.
Other organ builders have been using it, too: I try to keep up a current list at
As for Kellner? Kellner's scheme has nothing to do with that Bach drawing on the WTC title page. I've read most of his papers. They are mostly based on numerology. Yes, he also did a little thing with Bach's signet ring toward the end, but Kellner mainly based his "reasoning" on a couple of odd notions: (1) that Werckmeister somehow had a private 1/5 comma scheme that he chose not to publish, and (2) Bach somehow knew about it and championed it. Kellner's temperament from the mid-1970s is a shaved-down version of Werckmeister III, plain and simple; and everything else by him is after-the-fact rationalization looking for patterns in Bach's work that can be forced to point in that direction.
I have a handy survey of such other "Bach" temperaments here: