2011-08-09 — Original posting (on Blogger)
2014-10-28 — Re-posting as is (WordPress)
2016-06-20 — Brushed up for better readability
People probably regard Bach as being 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, and some of his sons (especially Carl Philipp Emanuel, 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 at 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 tuning does not automatically imply that he meant to use equal temperament tuning! In Bach’s time, Andreas Werckmeister 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 tuning 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”, starting in C minor) that modulates up by one (full) tone and can be repeated until it reaches the original key (C minor) again, after modulating through 6 keys. The played can be repeat this for as long as (s)he likes, or until reaching the end of the keyboard.
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 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.
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 found out (“Bach’s extraordinary temperament: our Rosetta Stone”, in “Early Music” 33 (2005), February, pp. 3 – 23, and May, pp. 211 – 231), Bach’s recipe is right there, on the front page of the manuscript of the first volume and has been reproduced numerous times:
It’s not in the text, though (that is a description of the composition cycle, as well as a dedication), but hidden at the very top of the page:
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 one needs read 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 People Interpret the Curly Ornamentation
To cut a long story short: Bradley Lehman concludes that one needs to
- tune the 5 fifths C – G – D – A – E – B short by 2/12 of the Pythagorean comma,
- the fifths B – F# – C# – G# as “pure”, and
- the 3 fifths G# (Ab) – Eb – Bb – F short by 1/12 of the Pythagorean comma.
As Bernhard Billeter notes in his article “Wie hat Bach seine Cembali gestimmt?” in “Schweizer Musikzeitung“, 10(4), pp. 13/14 (2007), there is little doubt that Lehman’s conclusion is correct, at least in basic terms. I concur with Billeter’s minor reservation that the exact numeric conclusion is speculative and likely not entirely correct. It would require tuning the last fifth (F – C) too large by 1/12 of the Pythagorean comma. In 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.
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 (e.g., in using calculations to set up a tuning method): his recipe rather describes the result of practical tuning, i.e., the (approximate) relative purity of the fifths in his tuning scheme. Harpsichord tuning is never 100.0% accurate, see my upcoming note on practical aspects of tuning.
Conclusions for Practical Tuning
In his article, Billeter proposes a slight modification to Lehman’s conclusion. He avoids a fifth that is larger than pure:
- the fifths Eb – Bb – F are short by only about 1 Cent (1/24th Pythagorean comma), and
- the last fifth, F – C, then is pure.
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, Bb, Eb) over the “sharps” keys (G, D, A), and
- the keys around F# are still playable, but sound noticeably sharper, less pure.
Of course all keys retain individual characteristics. No doubt, Bach “adjusted” the 24 preludes and fugues to the characteristics of the various keys, at least to some degree. The example cited most often: the prelude in C consists of resting arpeggiando chords that exhibit the (relative) purity of keys around C major. In contrast, the prelude in C# avoids resting chords, as these sound much sharper than those around C.
The key “take home message” from Lehman’s findings is: we may not know exactly what tuning Bach was using, but clearly, it was not equal temperament tuning.
Since we have this information, we follow Billeter’s recipe for a “modified Lehman tuning” for our own harpsichord, and we are happy with this. More in my upcoming note on practical aspects of tuning. Additional information on the tuning aspects of Bach’s Well-tempered Clavier is present in Wikipedia.