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A G by Any Other Name

This is a follow-up post to Jens's survey review of recordings of the Goldberg Variations.

Other Reviews:

Richard Scheinin, Harpsichordist thrills despite plastic quills (San Jose Mercury News, March 12)

Melinda Bargreen, Recital reinvents beauty of Bach (Seattle Times, March 13)

Anne Midgette, More Variety for 'Goldberg': In 1600's Style, With Quills (New York Times, March 21)
In a recent post (Decoding the Goldberg Variations, March 7), I wrote about Prof. Bradley Lehman's theory about a "squiggle" on the title page of J. S. Bach's The Well-Tempered Clavier. He believes that this mark, long thought to be merely a doodle of the kind found everywhere in musical manuscripts, is the key to the composer's special tuning system. In January, Prof. Lehman released recordings he has made of Bach's music on harpsichords and organs tuned to this special system. Lehman's theory really got my attention because Richard Egarr released a recording last month, in which he plays the Goldberg Variations and the Goldberg Canons using Lehman's tuning system on his harpsichord. He played the work in a few cities in the United States, too, although I was not able to hear him. The Seattle Times critic noted that Egarr's harpischord, now fitted with seagull quills to pluck its strings, "either has remarkable properties of reverberation, or profited from placement in the acoustical 'sweet spot' on the Town Hall stage," noting about the tuning only that "it also was tuned unusually low."

Credit goes traditionally to Pythagoras for determining the ratios that produce pure intervals. As repeated by any number of medieval music theorists, if you sound a string of length X and then divide string X in half and sound it (ratio of 2:1), for example, you will have produced a sound an octave higher. The intervals that make up the tonal scale -- fourths, fifths, thirds, seconds -- can all be explained by such mathematical ratios. This has given rise to misguided defenses of tonality as a "natural" musical system. The fact is that the actual tuning systems used during most of the common practice period and indeed today are as unnatural and artificial as can be. We encounter a serious problem when tuning an instrument with a broad range, like a harpsichord or other keyboard instrument. The fact is that if you tune the notes on a harpsichord all as perfect fifths (3:2 ratio) -- usually starting on F-C, C-G, G-D, D-A, A-E, E-B, B-F#, F#-C#, C#-G#, G#-D#, D# to A#, A# to E#(F) -- the last note of that circle of fifths is not at all the same note as the first note, which it should be. That is what was called the wolf fifth: if you tuned that A# in Pythagorean intervals, the fifth with E-sharp (or sometimes G#-E-flat) would be several cents sharp. Several key signatures on the keyboard are, as a result, unplayable.

There have been several tuning solutions, all of which involve slightly mistuning, or tempering, various intervals flat at the keyboard, so that there are none of those perfect mathematical intervals (except usually for octaves), but the notes all sound more or less good and you can play in any key. Some have been more successful than others. Scholars have argued back and forth over just what tuning system Bach had in mind when he compiled The Well-Tempered Clavier: since he brought together in each volume a prelude and fugue in each of the major and minor keys of the chromatic scale, he clearly had in mind a tempered system that would make all of the keys sound good. But tempered where and by how much? No one knew, and Bach left no indication.

Squiggle by J. S. Bach, photo by Bradley Lehman
Or did he? The squiggle, shown above as captured by Prof. Lehman, may be more than a random series of loops. In fact, there is quite clearly what anyone would agree is a letter C on the first loop. Lehman suggests that we are intended to begin at that loop, turning the squiggle upside down. The first five loops each have two loops inside, meaning to tune the first five fifths (F-C, C-G, G-D, D-A, A-E) flat by 2/12 comma. The next three are empty loops, so you tune those fifths (E-B, B-F#, F#-C#) perfect. The last three have one loop, so you tune those fifths (C#-G#, G#-D#, D#-A#) flat by 1/12 comma. That adds up to 13/12 comma, enough to remove the wolf fifth. Prof. Lehman's Web site is much more detailed.

It's an ingenious explanation, but with no other evidence to support it, it remains at best hypothetical. The only other real criterion for proof that Lehman offers is that Bach's music sounds good when instruments are tuned to this system. In fairness to that point, as Prof. Lehman wrote to me, it is best not to judge only by Egarr's Goldberg Variations recording, since all the pieces are in the same key. In fairness, I have also listened to the music examples Lehman has made available on his site. Particularly mind-blowing is Bach's little harmonic labyrinth, BWV 591, which starts in C major and takes the player through all of the keys.

Presumably, if the point of compiling The Well-Tempered Clavier was to show that his personal tuning system was the best one, then he may have composed some of the pieces to bring out combinations of wolf notes. If you played the piece in some other system, it would sound bad. At the same time, if that were Bach's intention, I am hard pressed to believe that he would not have been more explicit about the system he favored. For me to accept this as anything more than a very interesting theory -- which it certainly is -- I would require some written mention of the system by Bach or a colleague, family member, or student. If he indeed advocated such a tuning system, there would have to be a mention of it somewhere, in a letter or elsewhere on another manuscript piece of music. I am quite sure that Prof. Lehman is poring through libraries around the world in his spare time looking for just that. I hope that he finds it.

[Added remarks: At Prof. Lehman's request, I have gone over his article and supporting Web posts again. I did not read anything that convinced me any further, but I will be more specific about some of the evidence that he supplies. I am convinced by Lehman's argument that J. S. Bach had a special tuning system. His son C. P. E. Bach, one of whose jobs was to tune his father's harpsichord at Leipzig, seemed to know about it. I am also convinced that Bach's system, which he apparently had in mind when he created The Well-Tempered Clavier, is probably not equal tuning. In the treatise Lehman cites, C. P. E. Bach describes a tuning system in which most, but not all, of the 5ths are narrowed slightly. Prof. Lehman may see his "eight or nine of twelve" in C. P. E.'s "most," but it is still not specific evidence of what Lehman sees in the title page squiggle. He is convinced, once again, that the strongest evidence is the C. P. E. Bach's music also "sounds right" when an instrument is tuned to the Lehman system. I stand by my statements above, that there is not sufficient evidentiary information to accept Prof. Lehman's assertion that the squiggle means what he surmises it does. I still think it is an interesting hypothesis, but only a hypothesis.]

As for whether I am interested in hearing Lehman or Egarr play Bach's music using this tuning, I certainly am. I am not as convinced as they are that Bach's music sounds the best in this system, and even if I were, that judgment is so subjective that it cannot be accepted as evidential. I could easily be convinced that the squiggle means something, because it does appear to have a pattern and that added C is not something that looks completely random to me. However, before we can be sure that Bach meant it to mean something and just how to decode it, we would need something else. I do intend to tune my harpsichord to Prof. Lehman's specifications. I may eventually agree that the tuning system makes Bach's music sound good.

J. S. Bach, Goldberg Variations, Céline Frisch (released on August 17, 2004)
Just about anyone can make room on one's CD shelf for more than one recording of the Goldberg Variations. The one I have been listening to the longest is the Glenn Gould 1955 recording, which is still in many ways the best, at least in terms of virtuosity of playing. It is quite simply astounding. (In my opinion, Gould's 1981 recording is inferior. I do not have it in my collection.) Murray Perahia's 2000 recording is also quite good. However, as far as owning a definitive recording, it must be harpsichord. In terms of harpsichord recordings, the choice in my opinion comes down to Richard Egarr and Céline Frisch. Both are on excellent harpsichords, have very good sound, and are encyclopedically complete (with the Goldberg Canons), although only the Frisch has a version of the folksongs used in the Quodlibet. As I wrote above, the tuning theory is interesting but, by itself, not enough to tilt the scale to Egarr's recording. Musically, Egarr's interpretation is leisurely, clocking in at a languorous 90:25, and perhaps a bit dry.

The Frisch recording is also a very musicological CD, planned by Jean-Paul Combet, the French musicologist (to be specific, he did graduate work in musicology, not a doctorate, before going to Sciences-Po) and now the head of the Alpha Productions recording label. (Alpha has a number of of interesting projects and recordings in their catalogue, including a recording of the Bach cello suites in which Bruno Cocset recorded each suite on a different instrument best suited to that piece. I would like to listen to that one of these days.) There is a good interview with Jean-Paul Combet by Hannah Krooz. At a total length of 77:52, Frisch's recording has all of the repeats but is much more vital and exciting listening than Egarr's.

Frisch's playing is the most sparkling in terms of virtuosity, combined with the soundest musical choices, in my opinion. A lot of players take each movement at wildly different tempi, which destroys Bach's carefully planned crescendo of rhythm. By contrast, Frisch's pulse mostly remains the same from movement to movement, and almost every movement struck me as at a near-perfect tempo. In some movements, the rhythmic plan is interrupted for Bach to use a different style, like Variation 7, which is marked in some of the sources as al tempo di Giga. Frisch's Variation 7 is a delightful gigue. That driving pulse is lost, only at moments, in Variation 29, which is still very exciting. Her Quodlibet is a tender reading, in which there is nothing raucous and the notes are caressed. The final statement of the Aria is about 20 seconds longer than the initial one, to my ears pregnant with nostalgia for the cycle's beginning. (By comparison, Masaaki Suzuki's playing is very exciting listening, especially in some of the variations, crisp, enervated -- at a fleet 73:17 -- with flashy embellishments, but not as intellectually organized, I find.) The Frisch CD's version of the canons is arranged for the six string players of Café Zimmermann. The two folksongs used in the Quodlibet are sung, in idiosyncratic fashion, by countertenor Dominique Visse.

I have enjoyed listening to several (but not all, yet) of the recordings covered in Jens's review. The first Pierre Hantaï recording (at 77:26) has very nice embellishments and some really exciting parts, but I still like Frisch better. Two other more recent recordings on piano mentioned by Jens are of interest. Lifschitz's live recording (79:01) -- made at a graduation recital -- has some terrific playing but is too self-indulgent for my taste. The aria is too slow, he distorts some of the rhythmic relationships, and the Variation 7 (gigue) is as close to a dirge as I can imagine. However, at the piano, where the hand-crossing problem is so pronounced in the 2-manual variations, Lifschitz's Variation 8 is quite remarkable.

Keith Jarrett reportedly plays the harpsichord regularly, although the sound of his Goldberg recording, on a harpsichord, does not seem to indicate that. It's the shortest recording, at 61:39, only because Jarrett plays mostly without repeats. The aria is about as boringly slow as one can get, worse even than Gould's pedantic 1981 reading. The harpsichord's sound is not as pretty as Frisch, Hantaï, or Egarr and truly reminds me of Thomas Beecham's famous quip about the harpsichord, that it sounds like "two skeletons copulating on a galvanized tin roof." Good harpsichords recorded well do not sound like that. The tempi are quite reserved, the touch is monochromatic, although when he does take the repeats he adds interesting (if not particularly Baroque, which is OK) embellishments. All the same, I would have expected more freeness from a player with a primarily jazz background. Jarrett's reading is very much wedded to Bach's score in a somewhat disappointing way.


Mark said...

A squiggle by any other name. That is amazing Charles. Never underestimate the complexity of a line.

Bradley Lehman said...

Mr Downey, thanks for this thoughtful and enthusiastic response to my work!

In re this passage especially: "For me to accept this as anything more than a very interesting theory -- which it certainly is -- I would require some written mention of the system by Bach or a colleague, family member, or student. If he indeed advocated such a tuning system, there would have to be a mention of it somewhere, in a letter or elsewhere on another manuscript piece of music."

...Have you read all the portions of my Oxford paper, presenting the corroborative evidence in that regard? Those are all available for download through this page:

...and I have some further/later corroboration (expanding parts of that article) from CPE Bach, at this page:

It seems to me that Bach's own most loquacious and systematic son--trained in law, and a professional musician himself--would supply this information in his roles of colleague, family member, student, and biographer!


Bradley Lehman

Charles T. Downey said...

Prof. Lehman, thanks for the response. I will take another look at the two things you mention and add an update.

Charles T. Downey said...

Acting upon Prof. Lehman's request, I have gone over his arguments again. As a result, I have added a supplementary paragraph to my original post.

Bradley Lehman said...

Looks good. An update for you: two new recordings by other harpsichordists have been released in the first week of June, using this temperament.

Peter Watchorn has recorded/released all of the Well-Tempered Clavier book 1, and is planning to record all the other Bach solo music within the next few years.

Elizabeth Anderson has released a new disc of Anna Magdalena Bach's notebook (1725), tuned this way.

I have links to their info at

...and other concert usage, pipe organs, etc by various people at

The adventure continues!

Charles T. Downey said...

Exciting news for a fascinating project. I wish you good luck in your research.