What Tunings Did Bach Use?

Reflections on Tuning


2011-08-09 — Original posting (on Blogger)
2014-10-28 — Re-posting as is (WordPress)
2016-06-20 — Brushed up for better readability
2020-12-11 — Added proper literature references
2021-11-23 — Expansions and corrections, based on additional literature studies and input by Bradley Lehman


Table of Contents


Introduction

People see Johann Sebastian Bach (1685 – 1750) as the baroque composer. He was a master at writing, even improvising the most complex of fugues. With this preference for traditional polyphonic forms such as fugues and canons, at his time, many must have viewed him as rather traditional composer. On the other hand, he did expand into new, previously (almost) unexplored keys on the keyboard. Thus, he may have helped advancing music towards the pre-classical era. Some of his sons, especially Carl Philipp Emanuel Bach (1714 – 1788), clearly were early classical (pre-classical) composers.


Tuning at Bach’s Time

Similarly, over Bach’s lifespan tuning went through a lot of evolution: in Bach’s time, the Silbermann family of organ makers essentially still tuned their instruments in Meantone temperament. Bach (a well-sought organ expert / tester) disliked the tuning of Silbermann organs. When he played one of these instruments, he on purpose selected keys or modulations that would make the presence of a “wolf fifth” (see my previous blog entry) most obvious, just to exhibit his opinion about such tuning. He must have had fun doing this! To the organ maker’s dismay he also used to pull all the stops and hold the fullest possible chord in order to see whether the instrument “had enough breath”. However, Bach’s opinion on the Silbermann tunings does not automatically imply that he meant to use equal temperament tuning!

In Bach’s time, Andreas Werckmeister (1645 – 1706) developed tuning schemes that permitted using a growing number (ultimately all) of the keys on keyboard instruments, without re-tuning. Evolution in instruments and music / composition style continued. Some of Bach’s sons started writing for the fortepiano (instruments that one didn’t need tune nearly as frequently as a harpsichord). Therefore, they are likely to have used tuning close to equal temperament. Hence the question about the tunings that Bach used for his own instruments. What is the best / desirable tuning for playing Bach’s works on keyboard instruments?


Using All Possible Keys

Over the 19th century, people largely lost the knowledge about baroque and pre-baroque tuning. They “naturally” assumed that Bach was using equal temperament tuning. The fact that Bach has written the Well-tempered Clavier appeared to support that opinion: a series (actually, two series) of compositions (prelude / fugue pairs) that “walk” through all keys of the chromatic circle. He also wrote at least one piece (a canon “per tonos” — through the keys — for 2 voices in the “Musical Offering”. It starts in C minor and modulates up by one (full) tone. It can be repeated until it reaches the original key (C minor) again, after modulating through 6 keys. The player can repeat this for as long as (s)he likes, or until reaching the end of the keyboard.

Mis-Understanding “Well-Tempered”

In German, people sometimes mistakenly use “temperiert” or “temperierte Stimmung” as synonymous to “equal temperament tuning”. Still many artists and listeners are convinced that it was Bach’s intent to have the Well-tempered Clavier played with equal temperament tuning. Some have even gone through lots of contortions to find arguments supporting their opinion. If one looks carefully enough, one sure can find one or the other (weak) argument that appears to support this point-of-view.

One should note, though, that the title is “Well-tempered Clavier” (“Das Wohltemperierte Clavier“), not “Equal-tempered Clavier”. I think that—strictly speaking—equal temperament tuning doesn’t leave room for a qualifier “well(-tempered)”. It’s either a tuning with equal (uniform) distribution of the Pythagorean comma (see my previous blog entry), or it simply isn’t equal temperament tuning.

Bach actually did not invent the concept of a series of compositions going through all keys — there have been precursors (as far back as 1682 for keyboard music), at times when the notion of equal temperament tuning clearly did not exist yet.


Reading A Mysterious Title Page

However, the key point here is: we do have a hint (to say the least) by Bach, describing how he expected the keyboard instrument (harpsichord, Virginal, Clavichord) to be tuned. In fact, countless people have seen this description—without realizing what they were looking at! As Bradley Lehman (*1964) found out (Lehman, 2005 and Lehman, 2005a), Bach’s recipe is right there, on the front page of the manuscript of the first volume and has been reproduced numerous times:

Tunings — WTC

It’s not in the text, though (that is a description of the composition cycle, as well as a dedication), but hidden / encrypted at the very top of the page:

Tunings — WTC_2

I remember having looked at this funny ornament myself, several times — but I did not reach the key enlightenment. The top ornament is a funny garland that follows the circle of fifths (F – C – G – D – etc.) from right to left:

  • There are 5 loops containing a double-squiggle, followed by
  • 3 simple, empty  loops, and finally
  • 3 loops with a single squiggle inside.
  • The “C” of “Clavier” links to the first loop on the right (the “link” is even a repetition of the “C”).

How To Interpret the Curly Ornamentation?

One can summarize Bradley Lehman’s interpretation and the resulting “recipe” as follows:

  • the 5 fifths F – C – G – D – A – E are short by 2/12 of the Pythagorean comma,
  • while the following fifths E – B – F♯ – C♯ are pure, and
  • the 3 fifths C♯ – G♯ (A♭) – E♭ – B♭ are short by 1/12 of the Pythagorean comma.

Bradley Lehman not only lists these interval “ratings”, but he also gives a concrete recipe for interval tuning to achieve these values. In addition, Bradley Lehman has done extensive research and mathematical modeling that led to the above conclusions.

I do not have the means to tell how much Bradley Lehman’s conclusions have been accepted in the “baroque keyboard communities”. As Bernhard Billeter (*1936) notes in his article “Wie hat Bach seine Cembali gestimmt?” (Billeter, 2007), there is little doubt that Lehman’s conclusion is correct, at least in basic terms.

How does Bradley Lehman’s Tuning Sound?

With this tuning, similar to “Werckmeister III” tuning,

  • the keys around C are the “cleanest” ones,
  • there is a slight preference to the “flats” keys (F, B♭, E♭) over the “sharps” keys (G, D, A), and
  • the keys around F♯ are still playable, but sound noticeably sharper, less pure.

For details see Bradley Lehman’s papers (Lehman, 2005 and Lehman, 2005a).

Was Bach Really so Meticulous About Tuning?

Bach had great fun hiding numbers in his compositions. For example, he used the number 14 (B + A + C + H = 2 + 1 + 3 + 8 = 14) by having the theme of a fugue appear 14 times. But he was not a theoretical mathematician. I seriously doubt that he was using extended calculations to set up a tuning method. His (hidden) recipe rather describes the result of practical tuning, i.e., the (approximate) relative purity of the fifths in his tuning scheme. Furthermore, Harpsichord tuning is never 100.0% accurate, see my notes on practical aspects of tuning.


Critical Voices

Needless to say, however, there is also criticism. There have been several follow-up articles proposing modifications. I do not have the resources or the capacity to review Bradley Lehman’s argumentation or that of his critics in detail. That would be a research project on its own. However, let me still try doing justice to Bradley Lehman’s research, while at the same time also presenting some alternatives / proposed modification. All of these still relate to Bach’s “curly header”, but they come to slightly different conclusions.

Billeter’s Notes on the Exact Deviations from Pure Fifths

In his article “Wie hat Bach seine Cembali gestimmt?” (Billeter, 2007), Bernhard Billeter noted that Lehman’s recipe would require tuning the last fifth (F – C) too large by 1/12 of the Pythagorean comma. At Bach’s time, several people warned against using larger-than-pure fifths: this gives that interval the slight flavor of a “wolf fifth”, see my previous blog entry. Bradley Lehman argues that this “residual” fifth (B♭ – F) is only slightly too large—actually “more nearly pure than the F – C – G – D – A – E fifths”.

Billeter also argues that the theory around the numeric value of the Pythagorean comma only gained more widespread around 1724 – 1742, after the time of creation of Bach’s title page (1722). Also, Bach barely was using such theoretical considerations. Billeter considered it sufficient, simply to assume that for the intervals F – C – G – D – A – E the deviation from pure intervals ought to be about twice as big as that of C♯ – G♯ (A♭) – E♭ – B♭.

With this in mind, Billeter proposed a tuning scheme that leaves the residual fifth B♭ – F as a pure interval. As this last interval does not associate with a squiggle in Bach’s “graph”, I see this as still in (possible) agreement with Bach’s hidden instruction.

Balancing the Keys

A few months after Bernhard Billeter’s article, the Swiss organist Pierre-Alain Clerc (*1955) posted an article (Clerc, 2007) that quotes findings by the French harpsichord builder Émile Jobin. Clerc and Jobin argue that Lehman’s scheme favors the “flat” tonalities (F, B♭, E♭, etc.) at the expense of purity in the “sharp” keys (G, D, A, E, etc.). This led to doubts about the relative positioning of the (linearized) circle of fifths in Bach’s graph:

Where To Start Reading Bach’s Header Ornament?

Émile Jobin argued that that the very left end of Bach’s garland (not really interpreted in Bradley Lehman’s articles) matches the letter “F” (e.g., the way baroque organ builders marked “F” pipes). He therefore assumed that the last (single-squiggle) loop refers to the interval F – B♭. If we then follow the graph from left to right (F – B♭ – E♭ – A♭, etc.), the fifths all end up shifted by one interval (now ending in C, not F):

  • F – B♭ – E♭ – A♭ are short by 1/12 of the Pythagorean comma (approximately, see above),
  • G♯ (A♭) – C♯ – F♯ – B are pure
  • B – E – A – D – G – C are short by 2/12 of the Pythagorean comma (approximately, see above).

With this, the last loop (with double-squiggle) on the right (next to the little “C”) is interpreted as fifth G – C, while in Bradley Lehman’s interpretation, that double-squiggle is linked to the interval C – F. One may find this arguable—however, it has the advantage of shifting the balance towards more favorable “sharp” keys.

More “Features” in Bach’s Header Ornament?

Émile Jobin also argued that the long vertical line in the initial “D” of “Das” reaches up to the third single-squiggle loop (from left), which in his view refers to the interval A♭ – E♭ (As – Es in German notation), or G♯ – D. And in the center of the initial capital “C”, there are squiggles resembling E and and a tiny above, again apparently referring to the interval A♭ – E♭. Is that far-fetched?

Jobin further claims that the right edge of the garland reads like an appended “c”-squiggle-“3”. His conclusion is that the third C – E ought to be pure. In Billeter’s view, though, this is somewhat speculative, see also below. On the other hand, with this, the only “features” in Bach’s header ornament that so far remain without interpretation are the two dots in the center of the capital “D”.

The Importance Thirds vs. Fifths

Émile Jobin’s and Pierre-Alain Clerc’s focus actually was on the thirds rather than on the fifths. Consequently, Pierre-Alain Clerc’s article bears the title “J. S. Bach et les tierces justes” (J.S. Bach and the just / pure thirds).

Naturally, Bach’s graph and the bulk of the (interval) tuning works on the basis of fifths. That’s simply because on average, these are easier to tune than thirds. With the exception of one or at most two intervals (especially with early tuning schemes, such as meantone tuning), these are either pure or only slightly off. The frequency of the beats is low in the middle of the keyboard, the common (beating) harmonic easy to hear.

However, in major keys, the human ear is much more sensitive to the purity of the thirds. In particular, Billeter mentions that Lehman’s proposal implies a particularly “bad” third E – G♯, that occurs fairly frequently in Bach’s Well-tempered Clavier. On the other hand, in his comment to Pierre-Alain Clerc’s article (Clerc, 2007), he also expresses reservations about Jobin’s conclusion about the requirement for a pure third C – E. In his proposal, he opts for the thirds C – E and G – B to be a tad large.

Bernhard Billeter’s Conclusions

With the above, Billeter ends up with the following tuning deviations for the fifths (1 cent = 1/100 of a half-tone):

  • C – G: -3.5 cents
  • G – D – A – E: -4.0 cents
  • E – B: -4.5 cents
  • B – F♯ – C♯ – G♯ (A♭): pure
  • A♭ – E♭: -2.0 cents
  • E♭ – B♭ – F: -1.0 cent
  • F – C: pure

Conclusions for Practical Tuning

One Tuning Temperament for All? No!

Needless to say that we will never know how exactly Bach tuned his instruments. Moreover, Bach’s own tuning scheme has certainly evolved over his lifespan. Therefore, even if Bradley Lehman’s findings (with or without the above corrections / amendments by other authors) were accurate and historically correct, this would only reflect a snapshot for the time when Bach published his Well-tempered Clavier.

In addition, one should keep in mind that Bach’s tuning recipe may not have been universal, even for Bach himself. For example, for chamber music, the tuning of string instruments in pure intervals may have required alterations in the tuning of keyboard instruments in the basso continuo. Also: even for Bach himself, organs may have required slightly different temperaments, given that tuning an organ is a major undertaking and is never re-adjusted on a case-by-case basis. And today, organ tuning is almost always a compromise that reflects the breadth of the expected or required repertoire.

Practical Recipes — Interval Tuning

No (interval) tuning recipe works purely based on fifths. Typically, thirds (such as the ones mentioned above) serve as intermediate “check points”, securing the tuner against the accumulation of minor errors into a major discrepancy. Bradley Lehman (Lehman, 2005 and Lehman, 2005a) has produced a recipe for interval tuning for his scheme. Bernhard Billeter (Bernhard Billeter, 1979) has published a booklet with recipes for interval tuning according to various temperaments, and also his original responses to Bradley Lehman’s publication (Billeter, 2007 and Clerc, 2007) include procedures for interval tuning. See also my separate posting “Harpsichord Tuning”.

Using a Tuning Meter

People who don’t tune keyboard instruments on a frequent / regular basis probably find it easier and safer to avoid interval tuning (see my posting “Interval Tuning for Harpsichords”). Very likely, they fill find it easier, safer, and more (time-)efficient to tune with either a digital or analog tuning meter, or—as we do here—with a tuning app on a smartphone, see my posting “Harpsichord Tuning”. To complete this note, let me include my tuning table from the latter posting:

tuning table, in Cents, version 2021 (© Rolf Kyburz)

Explanations

The table indicates deviations from equal temperament tuning, in cents (1 cent = 1/100 half-tone), starting on “A”. The numbers in this table are rounded to half-cent values. Most of these tuning schemes are explained in my earlier postings—in short:

  • Meantone: meantone temperament tuning (see above)
  • Muri—Organ: original tuning of the third organ in the Abbey of Muri / Freiamt, Switzerland
  • Schlick—Organ: organ tuning by Arnolt Schlick (c.1455 – 1521)
  • Werckmeister III: one of the tuning schemes described by Andreas Werckmeister (1645 – 1796)
  • Silbermann 1 / Silbermann 2: alleged tuning schemes by the German organ builder Gottfried Silbermann (1683 – 1753)
  • Billeter — Bach/WTC: Bernhard Billeter’s original proposal for a “Bach harpsichord tuning” (Bernhard Billeter, 1979)
  • Billeter — Organ: Bernhard Billeter’s original proposal for a (more versatile) organ tuning (Bernhard Billeter, 1979)
  • Bach WTC / Lehman: Bradley Lehman’s interpretation of Bach’s header decoration (Lehman, 2005 and Lehman, 2005a)
  • Bach WTC / Lehman, mod. Billeter: Bernhard Billeter’s slightly moderated version of Bradley Lehman’s proposal (Billeter, 2007)
  • Bach WTC / Jobin & Billeter: Bernhard Billeter’s adaptation of Émile Jobin’s reading of Bach’s header decoration, as published by Pierre-Alain Clerc, see above.
  • Equal Temperament: equal temperament tuning, just for reference.

Precautionary Remarks

For some of the historic tuning schemes described above we do have written evidence. For some, we even have physical evidence, e.g., the unaltered lengths of pipes in the organ in the Abbey of Muri. However, it remains unclear to what degree these schemes were generally known, and how widely they were actually used (and how accurately). If Bach’s curly header in the Well-Tempered Clavier is interpreted correctly, it very likely applies to this specific work. However, whether Bach himself would have regarded it appropriate for a wider range (let alone all) of his keyboard works remains speculative at best.

Keep in mind that at Bach’s time, tuning always meant interval tuning, and hence never was 100% accurate or reproducible. Such tuning temperament retain a certain individual & momentary character.


WTC and Tuning Temperaments

To me, there is little or no doubt that tuning temperaments such as Lehman’s or Jobin’s readings of the curly header in Bach’s title page describe what Bach had in mind for the Well-Tempered Clavier. Central points in all this are that all (major and minor) keys have / retain their own and specific characteristics. I have no doubt that Bach wrote or selected Preludes and Fugues for specific keys. He selected keys which produce specific colors / effects with a given piece. Performing such pieces in a different key (let alone with equal temperament tuning, and/or on a modern concert grand) distorts the music and ignores the composer’s intent.

The key “take home message” from all the above is: we may not know exactly what tuning Bach was using, but undeniably, it was not equal temperament tuning.

It is true that a few of the pieces in the Well-Tempered Clavier had a “previous life” in a different context, and even in a different key. But again, I have no doubt that in the Well-Tempered Clavier, Bach either write pieces for specific keys, or he selected keys that produces a specific effect with a given composition. The example cited most often: the prelude in C major in WTC I consists of resting arpeggiando chords that exhibit the (relative) purity of keys around C major. In contrast, the prelude in C♯ major avoids resting chords, as these sound much sharper than those around C.


Personal Recommendations

Since we know about them, we tune our harpsichord following one of the recipes from the “WTC” family of temperaments. My first selection would be “Bach WTC / Jobin & Billeter” from the table above. We are happy with this. More on this in my note on practical aspects of tuning. Additional information on the tuning aspects of Bach’s Well-tempered Clavier are also available in Wikipedia.


Acknowledgements

Many thanks to Bradley Lehman for feedback / input, and for pointers about omissions and incorrect / incomplete statements.


Literature

Bernhard Billeter. (1979). Anweisung zum Stimmen von Tasteninstrumenten in verschiedenen Temperaturen. Verlag Merseburger Berlin GmbH, Kassel.

Lehman, B. (2005). Bach’s extraordinary temperament: our Rosetta Stone—1. Early Music, 33(1), 3–24. https://doi.org/10.1093/em/cah037

Lehman, B. (2005a). Bach’s extraordinary temperament: our Rosetta Stone—2. Early Music33(2), 211–232. https://doi.org/10.1093/em/cah067

Billeter, B. (2007). Wie hat Bach seine Cembali gestimmt? Schweizer Musikzeitung10(4), 13-14.

Clerc, P.-A. (2007). J. S. Bach et les tierces justes: la solution proposée par Émile Jobin. Schweizer Musikzeitung / Revue Musicale Suisse, 10(7/8), 26–27.


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  1. Pingback: Progress in Tuning? | Rolf's Music Blog

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