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Lehman’s Bach Temperament:  Some New and Old Insight

Bjarne Pagh Byrnak

© 2011 Bjarne Pagh Byrnak (All Rights Reserved)

It is demonstrated that an ambiguity in the ornament that decorates the top of Bach’s hand-written title page of the Well-Tempered Clavier, in which a C-shaped pen stroke may be seen as both as the name of that tone and as a serif of a letter below it, is analogous to a technique used in the traditional hidden-object puzzle picture (Vexierbild) in which the eye may recognize either of two simultaneous motives. It is also pointed out that the ornament needs not be rotated for inter­pre­ta­tion, and that certain features of the tempera­ment derived from the ornament by Bradley Lehman have historical basis in tempera­ments published by Werck­meister in 1691.

1. Introduction

Circle-Of-Fifths showing Lehman’s Bach temperament
Figure 1.  Circle-Of-Fifths for Bradley
Lehman’s Bach Temperament.
Details at Lehman’s site

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.

(Readers who need to see Bach’s ornament in order to understand the logic might want to study Bradley Lehman’s sites www.larips.com and www-personal.umich.edu/~bpl.)

2. Did The “Vexierbild” Escape Attention?

There are eleven loops in Bach’s ornament, one for each of the eleven fifths to tune. A pen stroke shaped as the letter C next to one loop tells – according to Lehman – which loop corresponds to which tone.

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.

Source: Wikimedia.  Public Domain
    Is this a portrait, or is it a landscape on a     peninsula?
Figure 2.  “Landschafts-Kopf,”
Vexierbild  by Wenzel Hollar (1607-1677).
Larger sizes at Wikimedia

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.

Detail from the frontpage of Das Wohltemperierte Clavier
Figure 3.  Detail from the title page of
 Das Wohltemperierte Clavier.
Full title page at Lehman’s site
So, the conclusion of this Section is that Bach indeed wrote down the letter C as part of his ornament, but concealed it by using an ambiguity-based technique that was a well-established and widely known tradition in his geographical region.

(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 TemperaturA 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.

3. Bach’s Ornament Is Not Upside-Down

Lehman, in his first Rosetta paper,1 suggested that one may turn Bach’s ornament upside-down to see and copy the schematic, and provided a figure on his page 7 where the ornament was rotated 180 degrees.

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.

Consider a circle-of-fifths of the usual kind, drawn with C at its top, F# at its bottom, and to be read clock­wise starting from C in the usual manner. In the example shown in Figure 1 near the top of this page, G# has been turned enharmonically into Ab so that a roundtrip passes via F, not via E#.

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.

4. Did Werckmeister’s “Musicalische Temperatur” Escape Attention?

Lehman’s Bach temperament has tempered fifths among the chromatic keys, more than one size of tempered fifths, and a sharp tempered fifth.

All of these features appear in temperaments published by Andreas Werck­meister.5  Specifically,

  1. Werckmeister IV and V have three tempered fifths each among the chromatic keys.
  2. There are two sharp fifths in Werck­meister IV and one in Werck­meister V.
  3. Werckmeister IV has more than one size of tempered fifths when tuned according to the string length table on page 80 in the Musi­calische Temperatur.

I believe, therefore, that the appearance of these features in the temperament published by Lehman cannot be used as an argument against his inter­pre­ta­tion of Bach’s ornament.

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.)

Werckmeister regarded his readers as fully competent, and he took care not to insinuate that he could teach them anything. He said, in the Musicalische Temperatur near the end of the foreword,

“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.”

Gleichwie ich in meinen Musicalischen Wegweiser keinen vornehmen Musico etwas vorzuschreiben gemeinet, sintemal ich mich viel zu gering dazu befinde, und eine grosse Schwachheit begehen würde: Also wird auch in diesem Tractat keinen erfahrenen Musico, wie er ein clavier temperiert stimmen solle, aufgebürdet.
A little later, Werckmeister continues: “In this book, I demonstrate to those who are eager to learn it, how the tempera­ments can be formulated and arranged in various ways. One may place the beatings of the fifths in whichever keys one wants; it is just that the perfect consonants should not be treated too much. It is enough when a keyboard is so tempered that it is usable throughout” (that is, only as many fifths should be tempered as needed for the instrument to be play­able in all keys):
Ich bezeuge hierinnen der Lehrbegierigen, wie man die Temperaturen einrichten, und auf unterschiedliche Arten anstellen könne, es mag einer die Schwebung der quinten hinbringen in welche claves er will, nur daß den perfecten consonantien nicht zu viel gethan werde, genug ist es, wenn ein clavier so temperiret wird, daß es durchaus wohl kan gebraucht werden.
In other words, Werckmeister did not want to impose any particular tuning on his colleagues. Instead, they were granted the freedom to develop well tempera­ments of their own, and not just use the ones that Werck­meister had published.

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.

5. Additional Remarks and Conclusion

Before criticizing Werckmeister III, one should be aware that the usual modern version of that tempera­ment is based solely on one-half of Werck­meister’s specification table (the table on top of page 78 in the facsimile editions of the Musicalische Temperatur), namely the left column in which the sizes of the fifths are specified. The table’s right column gives us the sizes of the major thirds, but that part is cus­tom­arily ignored. While much space in the Musi­calische Temperatur is devoted to discussing the fifths, about equally much space is devoted to the major thirds, suggesting that modern theorists should pay attention to them when inter­preting Werckmeister’s tables. A deeper analysis in which one or more major thirds are taken into account leads to a version of Werck­meister III that should, at least in theory, modulate a little more smoothly than the commonly used version. The two best major triads get structured in a manner inter­mediate between one-fifth and one-quarter syntonic comma meantone tempera­ment. The tempered fifths become unequal-sized. One may speculate that experienced tuners at Werckmeister’s time may have tuned Werck­meister III along such lines, making the two best major triads to sound with a quality as close as possible to meantone temperament.6

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.

6. Postscript

Shortly after Lehman published his twin papers, John O’Donnell7 proposed an interpretation in which he read the ornament from left to right and assigned the tones chromatically.

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.


[1] Bradley Lehman, “Bach’s extraordinary temperament: our Rosetta Stone – 1, ”Early Music Vol. 33, No. 1, February 2005, pp. 3-23. Sup­ple­men­tary material available elec­tron­ically.

[2] Bradley Lehman, “Bach’s extraordinary temperament: our Rosetta Stone - 2, ”Early Music Vol. 33, No. 2, May 2005, pp. 211-231.

[3] Mark Lindley and Ibo Ortgies, “Bach-style keyboard tuning,” Early Music Vol. 34, No. 4, November 2006, pp. 613-624.

[4] 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.)

[5] 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.

[6] Details in this report.

[7] John O’Donnell, “Bach’s temperament, Occam’s razor, and the Neidhardt factor,” Early Music Vol. 34, No. 4, November 2006, pp. 625-634.

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