How did Johann Sebastian Bach tune his harpsichord?
Direct historical evidence is scarce. Indirect evidence has been sought for, not without results. A review will not be attempted here. My intention is to comment on just one relatively recent result – and especially on the discussion that followed it.
In 2005, Bradley Lehman published his twin “Rosetta Stone” papers1-2 in which a well-tempered tuning (Figure 1) was derived from the ornament that decorates the top of the handwritten front page of Das Wohltemperierte Clavier.
Some debate followed. As is often the case in the humanities and even in some natural sciences, polemics was not altogether avoided. Rather than repeat who said what, I prefer to go directly to the point: After several years of pondering, I find that a number of facts were overlooked in the debate. Let us take the following one first, because it is fun.
Lehman observed that the letter masquerades as a calligraphic serif of a letter below it; and that similar serifs appear elsewhere in Bach’s handwriting and in Altnickol’s title page for the second part of the WTC; and that the serif belongs “also” to the diagram which it touches (Ref. 2, Note 63), keeping a door open for the possibility that it could serve double duty.
Mark Lindley and Ibo Ortgies3 supplied some additional examples of such serifs. They also supplied examples in which a handwritten small letter c is shaped differently, and dismissed the idea that the serif was a letter at all.
In my eyes, the stroke resembles a capital letter C fairly well. The question, then, is: Was the stroke meant by Bach to be a letter, or a serif, or both?
In this Section, I shall demonstrate that the “double duty” hypothesis, which was also mentioned by others who contributed to the debate, has a historical basis in the form of other illustrations in which some details had two simultaneous meanings.
The traditional Vexierbild is an illustration, for instance an ink drawing in a book, in which some of the strokes that outline a tree, a forest, a landscape, etc., are found upon closer inspection to also outline a person, an animal, a face, or whatever. Those strokes are intentionally drawn as to convey two simultaneous meanings, one of which is often meant to be concealed. The eye may recognize the secondary meaning promptly or after some time, depending on the level of difficulty intended by the artist. An easy example from before Bach’s time is shown in Figure 2. The analogy with the C-shaped feature in Bach’s ornament (Figure 3) should be evident.
The concept of Vexierbild is thought to have originated from Germany4 and to have spread from there to other countries. Vexierbilder are still used for entertainment. Many of them are pen drawings, and a pen drawing in place of Figure 2 might have made the analogy more obvious. However, those pen drawings that the author could find originate mostly from the 19th or early 20th century. The artist of Figure 2, Václav or Wenzel or Wenceslaus Hollar, is known to have spent part of his lifetime in southern Germany.
I believe the C-shaped stroke in Bach’s ornament is intentionally ambiguous, and that the ambiguity is intentionally analogous to ambiguities found in the traditional Vexierbild.
(But why did Bach want to conceal it? Here is my guess – and it is nothing more than a guess.
Bach must have been aware that the Well-Tempered Clavier would get to circulate in the form of handwritten copies among experienced colleagues. In such a situation, a too direct “to-be-played-in-my-tuning” message could have been psychologically counterproductive. To stand out as more qualified than one’s colleagues was not considered tactful. Andreas Werckmeister circumvented a similar problem explicitly in his Musicalische Temperatur. A quote or two will be supplied in Section 4 below.)
The secondary motive in many a Vexierbild appears upside-down. Therefore, the conclusion of this Section holds even in case the content of the next Section is someday disproved.
At least two of Lehman’s readers got the impression that Bach wrote his title-page with the ornament turned upside-down. Now, several years later, it seems as if everybody, notably Lehman himself as is clear from material on his website, takes it for granted that Bach wrote his ornament upside-down.
May I – as perhaps the only person in the world – dare to say the following.
Forget about rotating Bach’s diagram. The idea is an unnecessary logical detour. It may once have been a reasonable response to a feeling of discomfort with having to read the ornament in the direction from right to left, but the advantage is small compared with the damage it does by introducing a weak link into the chain of reasoning.
Omit the rotation, and tell people to read the ornament from right to left. That will simplifiy the process of interpretation and leave readers less sceptical. The temperament obtained is unaffected.
Right-to-left is a natural reading direction for a circle-of-fifths that has been folded out into a horizontal line. This ought to be self-evident – but, just for safety, here is the explanation in full detail.
Look carefully at a fifth near the bottom of the circle, for instance, F# – C#. Observe that C# is to the left of F#. In other words, you are actually reading that fifth from right to left. Maybe you are so accustomed to it that you hardly noticed it before.
Now, imagine the circle being cut with a pair of scissors midways between F and C and straightened out into a horizontal line segment. The F will fall near the left end, and the C will fall near the right end. The F# does not move.
The fifth F# – C# is still to be read right-to-left. This is as natural as it was before. In fact, to read the entire horizontal line from right to left, starting at C and ending at F, feels quite normal. It is merely a consequence of reading the original circle clockwise.
The conclusion of this Section is:
Bach’s ornament does not need to be turned upside-down for reading. You may – as Lehman suggested initially – turn it upside-down if you have it on paper and want to copy it by hand.
All of these features appear in temperaments published by Andreas Werckmeister.5 Specifically,
Werckmeister III has been widely used, but it is not representative of Werckmeister’s temperaments. It is the only one of the four correct temperaments published in the Musicalische Temperatur that does not share any of the above characteristics when tuned in the usual modern manner.
And then there is the septenario Werckmeister VI temperament, so called because it is defined by two lists of whole-numbered string lengths with the number 49 = 7 × 7 as a factor of the monochord length and without reference to any comma. It has tempered fifths among the chromatic keys, a sharp fifth, and more than one size of tempered fifths. However, one should perhaps not draw conclusions from that temperament too quickly, since it looks to some extent like an experiment. (On the other hand, Num.VI is, according to Werckmeister’s page 57, “nevertheless in praxi so correct” that one can be satisfied with it. Note that Werckmeister VI is not a 1/7 comma temperament as claimed in some relatively modern literature.)
“Just as it was not my intention in my Musicalischen Wegweiser to prescribe anything to any outstanding Musico, inasmuch as I find myself much too humble for that, and would commit a huge mistake: Similarly, in the present Tractat no experienced Musico will be burdened with how to tune a tempered keyboard instrument.”
I find it difficult to think of Bach as a person who would use someone else’s temperament. It appears more likely that he made his tuning decisions himself. Perhaps one may say that by doing so, Bach followed Werckmeister’s intentions despite not using one of his temperaments.
The conclusion of this Section is: There is no conflict between Werckmeister’s Musicalische Temperatur and Lehman’s Bach temperament.
Lehman’s interpretation of Bach’s ornament does not have a similar problem, since the ornament does not specify the sizes of the major thirds.
There is no indication in the Musicalische Temperatur that mathematical concepts such as the ditonic comma were needed, or that meantone tuners had to be re-educated, in order to tune Werckmeister’s temperaments in practice. Werckmeister assumed that experienced tuners could tune his temperaments using the skills they already had. We are applying mathematics today in our attempts to interprete historical temperaments, not because we think they did so historically, but because attempts to reinvent the old experience-and-skill based subjective tuning techniques would introduce too many uncertainties, and because our more or less accurate results can be communicated reliably between colleagues. Our usage of mathematics does not imply that Bach used a rigorous mathematical scheme when tuning.
My personal experience during several years of pondering was that various potential counterarguments against Lehman’s interpretation fell apart one by one. The conclusion of the whole debate, as I see it, is that Lehman’s interpretation of Bach’s ornament is not easy to disprove.
Perhaps it should be pointed out that there is no conflict with Section 3 above,
in which it is argued that right-to-left is a natural reading direction for a folded-out circle-of-fifths.
O’Donnell assumes that the ornament refers to the tones in monochord string length order
(C, C#, D, etc.); and a string length list is conventionally read from
left to right.
 Bradley Lehman, “Bach’s extraordinary temperament: our Rosetta Stone - 2, ”Early Music Vol. 33, No. 2, May 2005, pp. 211-231.
 Mark Lindley and Ibo Ortgies, “Bach-style keyboard tuning,” Early Music Vol. 34, No. 4, November 2006, pp. 613-624.
 Poul Malmkjær, “Flere tricks, tryllerier og gåder for hele familien,” Forlaget Sesam, 2005, ISBN 87-11-22347-2, page 11. (Note that Vexierbild is pronounced [v-] despite being German and despite being Fikserbillede in Danish.)
 Andreas Werckmeister, “Die Musicalische Temperatur,” Quedlinburg 1691. Facsimile (1996): Guido Bimberg and Rüdiger Pfeiffer (Ed.), Verlag Die Blaue Eule, Annastrasse 74, D-45130 Essen, Germany; ISBN 3-89206-736-8.
 John O’Donnell, “Bach’s temperament, Occam’s razor, and the Neidhardt factor,” Early Music Vol. 34, No. 4, November 2006, pp. 625-634.