Bach's schematic, as it appears on the page, © Bradley Lehman, 2005-22, all rights reserved.
All musical/historical analysis here on the web site is the personal opinion of the author,
as a researcher of historical temperaments and a performer of Bach's music.

Frequently Asked Questions, part 2

Continued from part 1...

Would Bach himself have needed to understand any of your mathematical theory or explanations? No. The concept is as simple as nudging certain notes slightly off a pure 5th by some "single" amount, and other specific notes off a pure 5th by a "double" amount. The amounts are easily learned through practice and experience, without needing to know any numbers. The notes are worked on in a typical sequence of setting all the naturals first, and then all the sharps/flats. This is a straightforward process, working hands-on with a tuning lever at a harpsichord, and using musical listening skill. The mathematical explanations in my papers and web pages are only to make the resulting layout clear to modern readers, and to show correspondences with other practical tuning methods in Bach's environment. I believe Bach drew a shape of a practical step-by-step process.

Isn't this temperament too difficult to set up by ear? Some people have used that argument against it. Not at all. And, who's to say that something "too difficult" for modern people (accustomed to electronic tuning devices?) would have been too hard for Bach or his students, doing the work every day or several times a week?

I do it myself at least every week, more than 300 times so far (as of winter 2009), and never once needed any electronic device. It is simple and easy to do by ear, from a single tuning fork, so why should I bother with any electronics (along with the fact that Bach didn't have them, either)? It usually takes me about seven to ten minutes to do a complete "8-foot" set of strings on a harpsichord, tuning them in this manner. It simply requires practice and experience. [Instructions] [Demonstration video on YouTube] [Explanatory video on YouTube]

I've set it on a clavichord, a continuo organ, and several pianos as well: always entirely by ear.

Why do a few individuals generate so much heat and nastiness against this temperament, and against this hypothesis? Most of the controversy that I've seen follows predictable patterns. The interlocutors load up their own counter-arguments and personal complaints with irrelevancies (red herrings, and straw-man mis-representations of my hypothesis -- not necessarily deliberate mis-representations, but errors of understanding, nonetheless). The irrelevancies can usually be found among the points on the following representative list:
  • Numerology
  • Other cabalistic speculations about Bach
  • The alleged spirituality of Bach and other 17th-18th century tuners, as if it affected the ways in which they chose to tune
  • Kellner
  • Werckmeister
  • Over-emphasis on 5ths and major 3rds played in isolation, instead of in musical context
  • Barnes's analytical methods from the 1970s
  • Any temperaments based fundamentally on counting beats per second
  • Any temperaments based fundamentally on ratios
  • Temperaments that put an emphasis on equal-beating comparisons
  • Difficulties of calculating logarithms, and using any irrational numbers
  • Difficulties of approximating this temperament into just-intonation ratios: jamming it into spreadsheets that take only integers or simple ratios as inputs, and then offering value judgments on the way it doesn't fit that expectation neatly
  • Their own unfamiliarity with, or disregard for, regular ("meantone") temperaments as a musical and historical standard
  • Emphasis on any mathematical calculations to be done prior to the tuning process
  • Quarrels about the "small c" serif, as if it's a make-or-break point of the hypothesis
  • Alleged difficulty of anyone living or dead setting up this temperament by ear, finishing the entire instrument in 15 minutes
  • Making value judgments without tuning by ear oneself
  • Making value judgments without playing Bach's music on harpsichords
  • Perceived personalities of anyone involved in this research or debate (i.e., "ad hominem" character assassinations)
  • Citations of supposedly authoritative Internet chatter that never went through any academic review, and that doesn't stand up to any such scrutiny
  • Pianos, synthesizers, or MIDI (i.e., instruments Bach didn't have): used to evaluate any suitability or unsuitability of temperaments
  • Repetitive citations of selected lines lifted from other scholars who disagree with some of the points (to give one's own complaints more veneer of authority)
  • Repetitive citations of remarks from performing musicians who have had celebrated careers to their own or others' satisfaction, without needing any such temperament in their work; meanwhile, pointedly neglecting to cite the musicians who embrace this temperament and use it regularly
  • Mis-quotation (whether deliberately or not): pulling other people's words selectively out of context, alleging that they said things they didn't, or projecting some absurd premises into their work, and then knocking down that "straw man"
  • Confusions about the 1/12th comma diminished 6th, A#-F
  • Confusions about the A#-F being some kind of "wolf" that Bach allegedly never could have tolerated
  • Unhappiness with my writing style, my choice of words, the sound quality of my video camera, the split of my 2005 article into multiple sections, or the organization of this web site
With such a variety of "dead ends", all misunderstandings of the work (or the proper ways to prove or disprove the work), it is no surprise that the debate becomes vigorous: people arguing against the way in which their own expectations or preconceptions are being crossed!

Does one need any timekeeping device (watch, clock, pendulum, metronome) to set up this tuning by ear? None at all; merely a few trained listening skills that musicians should already have: the ability to memorize a tempo, and to sense intonational quality. The only mathematical constant that is necessary to get right in practice (within reasonable tolerance), by beats, is the size of one of the major 3rds on the naturals: either C-E or F-A. That is to start the regular 1/6 comma tempering of naturals properly, on either C-G-D-A-E or F-C-G-D-A: fitting all four of the 5ths/4ths into those endpoints of the major 3rd. At A=440, that's 3 beats per second in the tenor F-A; or it's 2.2 beats per second at tenor C-E, or 4.5 at middle C-E. For A=415, that's 2.8, 2.1, or 4.2. (See endnote #29 and the discussion on page 8 of part 1, February 2005.)

How does one do this without a timekeeping device? Easily and musically! Simply memorize the tempo of some convenient piece of music at the appropriate speed.

  • For the C-E I like to use the C major fugue of WTC book 1, where the quavers (eighth notes) can go at the 2.1 or 2.2 speed, and the semiquavers demonstrate the way middle C-E should be beating, twice as fast as that.
  • For the F-A I like to use the F major prelude of WTC book 1, where the left-hand quavers demonstrate the 3-per-second speed needed at tenor F-A, and the right hand the 6-per-second beating of F-A an octave above that (i.e. in the middle of the treble staff).
Those two examples are easy to remember, since they have the same tonic pitch as the interval they are tuning.

One can start with a fixed pitch source on any of those four notes C, E, F, or A; and then obtain the major 3rd with the memorized speed, fit all the intervening 5ths/4ths into that with even quality on each, and then proceed. There is no other calculation or counting necessary. An entire register of harpsichord strings at 8-foot pitch takes less than 10 minutes, with practice, starting from nothing but a single tuning fork.

Thousands of performances and recordings in the past 30 years have used "Vallotti" as a standard generic temperament for Baroque music. From the perspective of this Bach temperament, how far off is Vallotti? True, it has become omnipresent...and its name is often misspelled as "Valotti".

The good news, from the perspective of this Bach temperament, is that the Vallotti has seven of the 12 notes already on the mark. C, D, E, F, G, A, and Eb are exactly the same as in the Bach. Vallotti has lower B, F#, C#, and G#, and a slightly higher Bb.

What impact does this have on the music? With its lower sharps, Vallotti has less brilliance when music modulates to the scales of D major, A major, and E major. It sounds blander in that regard. And playing in key signatures such as C minor, F minor, E-flat major, or A-flat major, Vallotti often brings up problematic triads such as A-flat major and D-flat major: both having such a wide major 3rd that the music can begin to sound raw and dissonant where it shouldn't be.

More subtly, Vallotti also has trouble with keys as simple as E minor, B minor, D major, and A major. Because of its relatively low placements of the notes B, F#, and C#, the triads of B major, F# major, and C# major end up having very wide major 3rds in them...and this makes a poor effect whenever the music modulates to use those triads as dominants.

Vallotti has had such widespread use over the past several decades that many listeners probably shrug off these problems as "just the things that unequal temperaments do", and live with it. But, should the music really be so ugly in those spots? It needn't be! The Bach and some of the Neidhardt temperaments do better....

If the Vallotti temperament is already installed on a harpsichord, it is a simple matter of less than five minutes to convert the whole thing to the Bach: nudge the B and F#-C#-G# each the appropriate amount higher, and the Bb slightly lower, in every octave.

How about the similarly ubiquitous Werckmeister III? Worse than Vallotti! It was published as an organ conversion temperament...and not for harpsichords at all. With the overtone structure of harpsichord tone, Werckmeister III gives often-raucous results. It has all the faults of Vallotti with respect to the high-accidental keys...and its questionable virtue is that C major and F major are a bit sweeter (but uneven). The 5ths C-G-D-A and B-F# are so tight in Werckmeister III that these open intervals can sound poor by themselves, especially within two-part or three-part counterpoint (i.e. in passing during fugues).

After playing in the Bach temperament for as little as half an hour, and then going back to Werckmeister III, the Werckmeister simply sounds crude and brutal. I spent more than five years myself playing primarily in Werckmeister III, but I have a hard time tolerating its sound anymore. Listeners and players should of course formulate their own judgments in this matter by trying it out, side by side in all kinds of early-18th-century music. Werckmeister III makes a better showing for itself in mid-17th century music, with generally simpler harmonies and less use of the flat keys.

And while talking about these two: Kellner's famous temperament is in character about midway between Vallotti and Werckmeister III. Its recipe is little more than a slightly more moderate version of the latter. All three of these temperaments have the same set of problems, and differ mainly in their degree of smoothness at hiding those problems (or not).

What about the objections that have been raised by various people against turning the Bach title page diagram upside down to read it? Look at the far right side of the diagram, the little cauliflower-like thing that is clearly a separate stroke from the rest of the diagram. In the main article I suggested that it might have something to do with 3 beats per second. More recently (2006), I believe it has a yet simpler meaning: merely a gesture to "spin the book around" on the table (i.e. rotate it left through two 90-degree turns), or "flip it over".

Draw that gesture as pantomime in the air, without explanation or context, to a friend: and then ask for the first several impressions. Is "flip something over" among them? Or, "Turn around so I can look at you?"

What are your own assumptions, guiding the research? I have a series of historical and practical points, as both a musicologist and a practicing musician. These are given at the History page.

Where can I get the cent-value deviations from equal temperament, to try this with an electronic device? The equal temperament deviations (or deltas)--reckoned from either C or A--are the middle rows in the table on page 9 of the article. They are labeled:
  • "d ET (C) c" [Deviation from Equal Temperament in cents, with constant note C]
  • or "d ET (A) c" [likewise, but reckoned from constant note A as 0.0 cents of deviation]
Also, those are available here at the mathematical analysis page.

Where might one begin, meaningfully, to assess the contents of this Oxford article and web supplements? It's all rather dense and involved! To get the point, I believe the following process (at least) is necessary:

  1. Read all of the article, not only part 1!
  2. Set up this temperament on a good harpsichord: and preferably by ear both to understand its structure and because Bach himself had no electronic aids.
  3. Play through all the musical examples of the article, footnotes, and supplementary files--listening carefully to the highlighted features in the sound. The article is about sound and listening, an attempt to hear things inside an aesthetic scheme written down by Bach, testing my hypothesis of its correct interpretation and its place in Bach's milieu.
  4. Set up all eight of the temperaments presented in Table 1 (page 215) of part 2, and play through all the major and minor scales in each. Listen closely to the treatment of enharmonic equivalences, as discussed in that section of the article.
  5. Play Bach's repertoire, and especially the compositions that had been most problematic in other temperaments, listening closely for overall balance of musical elements.
  6. Include at least the B minor Mass, St Matthew Passion, and St John Passion in similar play-throughs, using the "D" version of the temperament as specified in the "Ensemble Music" section of part 1. Note the way the harmonic and melodic tensions align with the words and the dramatic thrust of each movement, within the whole.
  7. Do similar listening tests with at least Werckmeister III, Vallotti, regular 1/6 comma, and equal temperament to hear the differences; and additionally with any other proposed "Bach" temperaments for further comparison. A point here is to locate the Bach expectations within musical and historical context, which presupposes the reader's ability to hear and compare this with other contemporary layouts (including Neidhardt's and Sorge's, for the adventurous and thorough!).
Any attempted assessment of the work, short of that walking-in-the-footsteps process in the musical sound, is not in itself deep enough. There could be plenty of niggling about my words and my writing style, and I have seen a lot of that in response to the article--some of it interesting and helpful, other of it appearing merely petty or confused. But that is ultimately about any failures of my writing to communicate the musical points adequately to the skeptical reader, and about the readers' own expectations as to what I should consider important or plausible.

The content of the paper is musical sound that is there for direct experience and experimentation, fit into the extant historical record around Bach, and especially into the evidence of his music in practice. The sound of this temperament--in musical contexts--cannot be described adequately in words and numerals. Reproduce all these musical experiments to hear the point! And let some of that listening be right-brained openness, not merely left-brained criticism from reading words (mine or anyone else's), or bringing preconceptions in front of the music.

How does this affect the vocal music? This question now has its own page of response.

Does this temperament have anything to do with Herbert Kellner's "Bach" temperament of 1975, or his methods of divination? No. Kellner's conjecture was that Andreas Werckmeister had a secret unpublished method of tuning, and that Bach somehow knew about it (or had anything to do with Werckmeister at all, in the area of tuning). Kellner invented this himself and then marketed it under the name of Bach Wohltemperirt, using arbitrary esotericism and mystical numerology as his manner of "proof". Kellner's demonstrations merely elevate mathematical truisms in the behavior of "proportional beating" (which has no bearing on musical performance) and regular temperament in general, 1/5 comma or otherwise; he assigned spiritualized meaning to this and then assigned it further to Bach. In short, Kellner's argumentation was salesmanship from selective coincidences, and remarkably thin on taking seriously the historical record about Bach's preferences and performances.

Kellner's tuning works reasonably well for 17th century music but not Bach's. Kellner's and the authentic Werckmeister layouts continue the meantone premise that B major, F# major, C# major, and Ab major should all be farther out of tune than the neighboring keys of A major, E major, Eb major, and Bb major. Such layouts run into problems in Bach's music, since the harmonies of B major, F# major, C# major, and Ab major and their relatives are used frequently in the normal musical texture. Nothing in the music argues that these should be vastly out of tune.

Mathematical analysis (no Kellneresque esoteric numerology!) is a focus of the second half of my Early Music article, which see. See also these measurements of the intervals.

How good are John Barnes's "Bach" temperament of 1979, and "Vallotti" of the 18th (17th!) century? These two differ from one another by only the single note B, where Barnes's is slightly higher! From the perspective of Bach's own temperament, the "Vallotti" got seven of the twelve notes correct and Barnes got eight. The errors (measured against Bach's) are that Barnes's F#, C#, and G# are too low, and his Bb is too high; and additionally the "Vallotti" B is too low. Furthermore, Barnes's method of deriving temperaments is nonsensical, and has led people astray. I dealt with that in a book review in 2009.

Is this like the Andreas Sparschuh temperament of the late 1990s, or Michael Zapf's 2001 revision of it? No. The only similarity here is that theirs were derived from the same Bach drawing. In their interpretations they did not use Bach's letter C (where he placed it second from the right) as a meaningful part of the drawing; they assigned C elsewhere. Also, they have read the drawing in the opposite direction, and used a premise that the loops have something to do with seconds and half-seconds of time in the beat rates. In practice their methods yield temperaments that have ten different sizes of fifths, and have their required beat rates (multiples of half a second) at only one particular diapason (starting pitch)! Sparschuh's additionally holds a premise that all the resulting frequencies in Hz must be integers and in superparticular-ratio relationships with one another, yielding integer beat rates in the thirds, fourths, fifths, and sixths. These layouts are interesting and clever. But it has all looked arbitrary to me, and not like something that Bach would have done or expected, as I have explained here and in my broader survey of "Bach" temperaments.

A further analysis of Zapf's resulting layout is provided in my supplementary materials where I have compared more than forty temperaments. Another is at the temperament units explanatory page.

Further detail about the discovery sequence is at the errata/clarifications page.

The 55-step scale of an octave What's "55edo" and "55-division" at this web site and in the article? That is explained fully in the second half of the article. Briefly, it is the 18th century standard of "extended 1/6 comma" or "55 equal divisions of the octave" for string and wind ensembles, and for 18th century vocal pedagogy. It is a close approximation situated between 1/6 PC and 1/6 SC, and indistinguishable from them in practice. The natural notes are in regular 1/6 comma temperament. The accidentals (sharps and flats) are also in regular 1/6 comma, and therefore different from one another: a sharp such as C# is one comma lower than its enharmonic "equivalent" D-flat.

The whole octave, then, is comprised of a total of 55 of these little commas. A tone ("whole step") is 9 commas, a semitone is either 4 (chromatic semitone) or 5 (diatonic semitone) commas, and a major third is 18 commas. This particular major third is "pure" in the sense of agreeing exactly with placement in this standard system, "purely played" in geometrically precise position, not pure as a simple 5:4 ratio of frequencies! This has bearing on CPE Bach's and Quantz's remarks about tuning, as well.

For more information about the 55-division, see especially the articles "Beyond temperament: non-keyboard intonation in the 17th and 18th centuries" by Bruce Haynes (Early Music 19, August 1991), and "Mozart's teaching of intonation" by John Hind Chesnut (Journal of the American Musicological Society 30, summer 1977).

What specific sounds in Bach's music make you now go, "Oh YEAH, baby!"? High on my list has to be the way he handled seventh-chords in the keys of several flats. They are so suave! Examples: bars 124ff of the six-voiced Ricercar of the Musical Offering; bar 37 of the B-flat major fugue of WTC 2; bar 63 to the end of the E-flat major prelude of WTC 1; bar 195 to the end of the "St Anne" prelude in E-flat (BWV 552); and the C minor aria movement in the F major Pastorale (BWV 590). I also like the way the Neapolitan chord C-Eb-Ab comes up so often in Bach's keyboard music in G minor.

But with all that said about the flats, I'm also still bowled over by the brilliance of the sarabande in the A major English Suite, with the vibrato of its sharps. Bars 18-19 of the D major fugue in WTC 2 have a glow to the weak beats, as the linear motion creates complex harmony. The B major prelude and fugue of WTC 1 have a balance of delicacy and noble strength. The G# minor prelude has to be a cure for whatever ails anyone. This stuff in especially high sharps is hardly in the same universe with the flat music, and yet it's all on the same keyboard.

E minor has such a brisk and no-nonsense power to it, for example in the partita in that key. All the incidental sharps sound crisp and tense when playing through music in A minor. F minor has that opposite dichotomy between its tonic and dominant, along with its overall plaintiveness. When F minor goes chromatic, like in the sinfonia or in the fugue of WTC 1, it still sounds completely different from Bach's often-chromatic music in A minor or D minor.

And at the opposite pole from those complexities, the familiar C major prelude of WTC 1 is so simple and perfectly poised, with the way the sharps and flats come into it gently like little spots of color.

In WTC 2 some highlights for me, directly in the powerful sound, have to be the preludes/fugues in F# major, C# major, B major, G# minor, and Ab major. All overwhelming in their strength and crisp tension, without ever veering into harshness....

The sensuousness of playing through all this music, I can't really put it into words. I think it has to be experienced hands-on at the instrument. The bigger point of this all is the enormous range available, and I wouldn't want to be without any of it. And the way Wilhelm Friedemann and Carl Philipp Emanuel used all these contrasts in their music, wow! Friedemann's polonaise in E-flat minor, for starters....

Are the specialists and the general public excited about the Early Music article? There has been a mixture of enthusiastic acceptance and vigorous discussion: especially from those who formulated their own counter-opinions during the three-month wait to read the second printed half, or those whose epistemology requires a more positivistic treatment of the source material.

That is: for some readers and listeners, the evidence is overwhelmingly convincing. It makes sense that Bach was as musically sensitive and brilliant as the history books say he was, and it makes sense that he would express himself in this practical manner vis-a-vis any specifics of tuning, and his music sounds great this way whether the empirical historians are completely satisfied or not. For others (while conceding that the musical results are plausible, marvelous, and pleasing to modern taste), historical veracity has not been proven...and is indeed unprovable without further documentation. The agnosticism that "this problem of Bach's preferences is unsolvable" dies very hard, when forced to rely only on the diplomatic evidence (properly pedigreed physical documents) and the methods of positivistic philology.

To which I ask: what further type of documentation would be sufficient to convince this latter group? Firmer precedent of 1/6 comma tuning? Firmer aesthetic evidence in Bach's own milieu? A willingness to admit that extant compositions are empirical evidence of temperament practice, by practical inferences? Firmer emphasis that Bach was not a theoretical mathematician? Firmer evidence that he (as an expert and practiced harpsichord tuner) really understood accurate comma-splits and didn't get caught up in equal-beating approximations? Additional versions of Bach's drawing, from the 18th century, more clearly labeled as tuning aids with specific reference points? Firmer evidence that 17th century people were not all constrained to use only meantone (because any other common practices are too poorly documented for the comfort of historical empiricists), but that their ears and taste were indeed as sensitive as ours? Just because a proof doesn't look as a positivist might expect, doesn't mean it's an invalid or insufficient proof! Bach was a human artist, he worked by his instincts and practical experience, and he didn't necessarily produce the types of documentation that are the only items allowed onto a positivist's table.

So, yes, there is controversy! Discussion is good and vibrant, at least to the extent that it is conducted fairly and with valid methods of argumentation. (See my letter about this printed in Early Music, February 2010.)

And if the thing sounded rotten in musical practice, none of us would care...but the stake is that it sounds too seductively good for some people's comfort, as to questions of historical truth. Somebody sent me a good joke: "A theorist gets to heaven and interviews Bach, to clear up this matter once and for all. 'Did you really use equal temperament?' Bach shrugs and replies, 'Yes, until I read Early Music.'"

I received many questions or counter-arguments about part 1 that had already been answered in part 2, and I compiled this Frequently Asked Questions list. I have also received correspondence from a number of people who flatly refused to read part 2, or to play the musical examples for themselves, being already self-satisfied that there could not be sufficient evidence in it. I emphasized, and continue to do so, that it is all one article and no portions of it can really be omitted, in fairness to considering the musicological argument. I believe the evidence of Bach's extant music is at least as strong as the evidence in his title page drawing, and both "halves" of the printed article (and web supplements) support one another in that presentation. But, those readers who assume the main research is merely about graphology or positivistic documentation have already dismissed the argument without considering all of it. For such readers, my style of careful practical inferences does not suit their expectations, and I do not have an answer for them.

Meanwhile, practical musicians both expert and amateur have reported great excitement about their trials of this system on their keyboards. I especially like the following comments by a professional harpsichordist who is working on a recording of WTC 1 (now released), after practicing for several months with his instruments tuned this way:

"Forget about mathematical formulas (Bach never used them): the thing works perfectly for every bar in the WTC. So well that (unlike when you use other unequal tunings) you soon forget about the temperament and simply glory in the greatly increased resonance and depth that this tuning imparts to the instrument. The first prelude, for example, truly becomes a voyage of discovery: a foretaste of what is about to explored in greater depth throughout the entire work. That this temperament is 'the' one will be immediately apparent to any who has the ears to hear." (Peter Watchorn, 2006)

Continue with more questions in part 3...


Bach's schematic, rotated for use