While Bradley Lehman's tuning system may or may not have been the one Bach used, it's still a very interesting take and at the very least a happy coincidence. Just for a bit of fun I thought I'd take the time to respond to some of Fred Sturm's responses as I have a lot to say about it from my perspective with my extensive usage of microtuning/xenharmonic experiments... and from the fact baroque is a very tantalizing genre.
I suppose [my] biggest issue with Bradley Lehman's tuning is that it's so close to equal temperament, to the point where it's not spicy enough, but that doesn't mean Bach didn't have the idea for that tuning system. Our good friend Kyle Gann prefers Thomas Young's temperament for enough spiciness, and I can tell you right now without having to compare them that Thomas Young's temperament is most likely more 'unequal' than Bradley Lehman's.
Fred Sturm: "From the start, we need to state very clearly that not a single shred of historical evidence supports the theories of Bradley Lehman. There is a decorative design that appeared with the WTC, and that is all. Everything else is speculation, laid out in a persuasive way by Mr. Lehman. Mr. Lehman has never offered any further proof, resting the majority of his case on the "fact" that his temperament scheme "sounds right," a purely subjective judgment."
That's kind of the *whole point* of the discovery, that there [was] no evidence for it until Bradley Lehman analyzed the drawing. I guess nobody likes it when information is presented like that, but for another analagous comparison: academics denied the existence and importance of the Hittite empire despite there being limited evidence for it at the time... and as it turns out it was in fact larger than the Egyptian empire. It happens. Sometimes the surrounding context doesn't satisfy some people even when it's obvious.
When dealing with alternative tuning systems it's actually very important to use your ears. While I haven't ran all of Bach's works through different master pitches and different tuning systems -- if you run them through 'enough' variables it'll totally be possible to figure out what offers the best listening experience.
And to that point, there's nothing saying that Bach strictly used a single tuning system or reference pitch (I use different reference pitches and tuning systems all the time myself), some pieces using specific unequal temperaments for stronger harmony or stronger dissonance.
Fred Sturm: "Supposing that the diagram is mnemonic device for a tuning scheme that was used with Bach's many students, we might expect that some evidence would survive among their writings and manuscripts, or at least a mention. No such evidence has come to light. However, some of Bach's students did copy the WTC in manuscript. Most of those copies omit the decorative design. One includes a similar decoration, but it is not the same pattern: B. C. Kayser's decoration has fewer loops. (My source: J. S. Bach, a Life in Music (2007: Cambridge) by Peter Williams, pp 336-337). To me this is pretty conclusive evidence that the hypothesis of the students and the design is an untenable one."
You could just as easily argue this the other way around, that all of the surrounding documents don't necessarily lay out the specifics for conventional tuning systems of the period either. Bach clearly didn't use 440 Hz @ equal temperament, but 'what' that was we can only make assumptions. You get a nothing burger for both conventional and unconventional tuning systems, so of course there's probably not going to be much in the way of discussing a completely personalized tuning system.
With that said, the reference pitch is quite important because depending on where it falls, it can dramatically affect the whole scale. For instance, 415 Hz fits into 440 Hz as it just interposes all of the notes down by one. But if you were to use something like 410 Hz it no longer fits in the reference 440 Hz that current modern Western tuning likes to use. Therefore, if Bach did use the Bach-Lehman tuning, it wouldn't even be in 440 Hz, either... which means everyone playing Bach in 440 Hz regardless if it's Werckmeister III, Bach-Lehman, equal temperament... are actually listening to Bach in the wrong key. Ooops!
Fred Sturm: "Bach had many students, some of whom left behind a good deal of writing. Two of them - Bach's son CPE Bach and Johann Philipp Kirnberger - wrote a great deal, indeed, and both wrote about JS Bach and propagated his ideas. CPE Bach took up much of his book (Versuch über die Wahre Art . . .) in explaining his father's keyboard technique and fingering patterns, and in expounding on his father's methods of realizing figured bass. The treatise is long and detailed, and covers a great deal of ground. CPE Bach does write about tuning briefly, in the course of talking about stringed keyboard instruments. He mentions not one word about his father having a special tuning pattern, and his description is vague enough to admit various interpretations. Interesting. Not conclusive, but very suggestive. If JS Bach considered his tuning method to be very important in its details, one would expect his son to mention this fact, and perhaps describe it, at least briefly, in a book devoted largely to the father's legacy."
Again, it could easily just be argued the other way around that 'various interpretations' simply makes room for different tuning systems rather than Bach sticking with one (and again it wouldn't be 440 Hz @ equal temperament). There's nothing saying Bach couldn't use both Werckmeister III and Bach-Lehman. Because Bach was a relatively obscure composer at the time, if he had a custom personal tuning system it's unlikely most people would have cared back then, honestly.
Fred Sturm: "Kirnberger based much of his career on the fact that he was a student of the "Great Bach." He wrote a book explaining Bach's method and principles of composition. He also wrote voluminously about tuning. One would expect him to espouse or at least mention any particular tuning scheme his mentor may have had. Not a word to that effect. Instead, Kirnberger first proposed a tuning pattern consisting of a string of eleven just fifths, with the whole Pythagorean comma contained in one wolf fifth. When this was roundly jeered and shouted down, he retreated to a scheme of ten just fifths, with the comma shared by two fifths. This temperament he claimed to be appropriate for the music of Bach."
This actually works for the narrative that Bach may have experimented with his own tuning system, as Kirnberger himself was heavily involved with *drum roll* alternative tuning systems... perhaps influenced by Bach? And of course Bach would experiment; if you're a genius at music to Bach's calibre it's inevitable to look at any tuning system and see what else is possible. Wendy Carlos did the same with the Alpha/Beta/Gamma.
Fred Sturm: "Marpurg, a proponent of equal temperament, argued vociferously with Kirnberger. At one point he "pulled a trump card," stating that the "Great Bach" had used equal temperament, and appealing to CPE Bach to confirm this fact. Apparently CPE Bach demurred, he seemed to have preferred to avoid taking sides in the argument. But what is most telling is the deafening silence that occurred - a deafening silence IF Bach had a particular tuning that was laid out in plain sight on the title page of his WTC and if his students knew about it. Why would none of them say so at this juncture? Why not CPE? Why not Kirnberger? Or any of the others? This was a very public dispute, one everybody knew about. How easy to show off the fact that you knew the inside secret of JS Bach's tuning. But there was nothing."
This has been a controversy in itself (whether or not Kirnberger's statement was actually implying 12edo), but since Kelletat it's not really taken seriously anymore. If we want to argue from a "historics" perspective, it's literally impossible for Bach to have used 12edo. And even if Bach knew about it (which many will say he did) he wouldn't have cared for it even if they could tune everything perfectly back then, which they couldn't.
You've already lost the plot if you're considering Bach in equal temperament... it's a ridiculous thing to entertain from both a historical and aural perspective.
Fred Sturm: "I am a believer in following the evidence. The evidence in this case is entirely against the speculations of Bradley Lehman, and there is none to support it. He writes in a persuasive way, and has been very successful in converting many, many people to accept his interpretation. His temperament does have a generally authentic shape to it: it is a mild unequal temperament with narrower thirds in the natural keys, wider ones in the flat/sharp keys, as is Vallotti and as are many, many other temperaments. It is mild enough to be innocuous and pleasing to the ear, not enough different from equal temperament to be noticed by most people who have not been told what they are listening to. But in the end, it is pure speculation, fantasy, nothing more. There are no solid grounds to believe Bradley Lehman's story. In the end, his case rests on his musical analysis: "In this piece, this interval only sounds right when it is tuned this way," repeated over and over with example after example. Many other writers have made precisely the same kind of argument, supposedly proving that their own proposed temperament patterns "must have been" the one Bach used. This is a matter of subjective judgment, and one that is so susceptible to suggestion as to be utterly useless."
I do agree that unequal temperaments can start to 'blend together' if the pitch variances are not strong enough... in fact if you look at Werckmeister III and Bach-Lehman III, they're quite similar in practice but it still offers a little bit of zing missing in equal temperament (it's the same reason why VCOs "sound better" due to the minor variances in tuning slop). As I mentioned prior, you also need to figure out the appropriate reference pitch Bach used too, because that can influence how the temperament behaves if it falls on a rather odd value. The infamous 432 Hz is a good example as to how grating it is on equal or unequal temperament, and before anyone asks 432 Hz has nothing to do with 'healing' and it's actually quite a horrible reference pitch to use. Unless you want to add a little bit of uneasiness!
I also found it depends on the piece of music itself... some music can sound exactly like equal temperament even if it's using an unequal temperament, whereas other music all of the sudden it can sound too much out of tune using the same unequal temperament. A few mentioned that with the Bach-Lehman tuning you actually have to find a piece of Bach's that hits the right notes for hearing anything different.
Fred Sturm: "It would be possible to go into much more detail, particularly as to Lehman's interpretation of the various loops and related matters, and to show the historical context in more detail to make clear how unlikely it is that such a "secret" would be kept for over 150 years, waiting for a brilliant code solver to discover it. However, I think what I have written above suffices to explain my point of view."
Well, apparently it's a "secret" now that everyone is playing Bach incorrectly when they reach for a piano (single register) tuned 440 Hz @ equal temperament. Most people don't know any other tuning systems exist, and if they do they generally don't care which could have been the exact sentiment Bach faced. It's beyond me why nobody wants to actually use period-correct baroque tuning for... baroque pieces. You don't necessarily have to use Bach-Lehman, but for all that's holy, do NOT use 440 Hz and do NOT use a piano. At the very minimum you'll want to tune Bach below 440 Hz or transpose the scale a few notes lower if you're going to keep everything referenced to 440 Hz.
I personally think Bach is spoiled in 12edo on an instrument he never used. It's generally meant to be played on an organ or harpsichord because you (can) have two registers, a sudden stark change between registers is more interesting to listen to than trying to manipulate the velocity sensing on a single register. It's also unpleasant how modern pianists over-exaggerate the tempo and velocity on baroque pieces: this is exactly why I prefer Scott Ross over Glenn Gould. Although I wish Scott Ross actually used unequal temperament with an appropriate reference pitch... but at least he's using the correct instrument and a sensical handling of tempo.
If we've regressed this far with mishandling Bach in modern times, it wouldn't be far-fetched to say Bach had his own personal flavour of tuning similar to Werckmeister (and because Bach-Lehman is pretty similar that you could logically say the scenario makes sense).
That's pretty much all I have to say and I'd like to give my personal thanks to Fred Sturm for the great food for thought, and Bradley Lehman for stirring the pot and figuring out a very interesting tuning system.