The “Bach Temperament”

Kenneth Mobbs’ amplification of the letter he and Alexander Mackenzie of Ord published in the

August 2005 edition of Early Music, (pp.546-7)

In the letter I wrote with Sandy Mackenzie to the August 2005 edition of Early Music, (pp.546 –7), we acknowledged the email contact I had had with Michael Zapf in early 2002, when I had spent quite a lot of time considering his interpretation of what Andreas Sparschuh had proposed way back in 2001 – namely that the apparent random “squiggles” at the top of the title page of the first book of J.S. Bach’s “48” could hold the clue to Bach’s method of tuning which would make all keys acceptable enough in sound, i.e. “well-tempered”.  However, Zapf’s interpretation was based on recognising varying speeds of beats, and I found it impossible to be sure in which octave area the individual intervals would have been tuned.  As well as that was the worry that beats apparently were not recognised as an aid to tuning for another thirty years.  I therefore left the matter unresolved.

Bradley Lehman was unaware of this "squiggle" possibility until April 2004, and Sandy and I were amazed therefore that neither Sparschuh nor Zapf were acknowledged in Lehman's two printed articles in Early Music (February and May 2005).

In addition we were worried that he had been misled into wrongly thinking he should invert the diagram, because he had seen a copy of the original Bach m.s. printed in a 1911 edition of Grove.  But this must have been re-drawn for printing ("engraved"? - I don't know what the printing techniques were at that date), for here the decoration, which looks like a little "c" on top of the capital letter "C" of "Clavier", was separated from its body.  The original m.s. of Bach's frontispiece ( see for instance Associated Board edition by Tovey and Samuel, 1924) shows quite clearly the two were joined, and therefore that this is merely decoration, as Carl Sloan also points out in his letter to the August 2005 Early Music, (p.546).

Our third worry was that Lehman had jumped straight away to his preferred solution, and was constantly referring to “normal” experience – whatever that is.  When he attempted to give a practical method for tuning his solution to the riddle, his very first instruction was to tune a chain of 1/6th comma Vths, but unfortunately without providing either any evidence for this specific value or giving any details as to how it should be done.  This is one of the most difficult challenges to do by ear alone, particularly so if one does not make use of reference beatings from other notes already tuned, for instance from those notes in a chain of pure Vths.  As was said earlier, listening for beats was apparently not recognised as an aid to tuning until approximately 30 years later. This point was also made by Carl Sloan.

So Sandy and I decided to start from first principles, and consider what was the essential equipment needed at that period by someone wishing to tune by ear (or, indeed learning to tune by ear – e.g. JSB’s 11-year-old son Wilhelm Friedemann).

Let us now refer in detail to our letter on pp. 546 and 547 of the August 2005 Early Music. Incidentally, we were very conscious that the longer our letter became, the less chance we would have of its being printed.  We therefore were not as detailed in our explanations as we would have wished.

For instance, in our Stage 1, the easiest way of getting the first three equally-flat mean-tone fifths (C : G : D : A) reasonably accurate is  by continuing with one more, namely A : E, in order to try to obtain a pure major third, C : E.  In this way, one can check that there has not been too much mistuning already.

For Stage 2, one needs firstly to retune the E to make it pure to A, and then to continue with also  making E : B and B : F sharp pure.

Stage 3 contains the marginally-flat fifths, namely F sharp : C sharp : G sharp : E flat : B flat : F .

N.B. the one remaining fifth F : C is not specified – oh for a Bachian circular diagram, rather than his wiggly horizontal line!  This also means that, theoretically, another variable has been introduced into the equation, giving many solutions according as to whether the final fifth is flattened, (if so, by how much); pure; or even widened (if so, by how much).

But in practice a tuner-harpsichordist will arrive at the F and then see how it relates to C.  If the interval sounds reasonable enough, he will breathe a sigh of relief, tune the rest of the instrument, and then get on with his practising.  If the interval “howls”, he will have to find out whether F is too high or too low in pitch relative to C, and then back-track and re-adjust some of his earlier intervals.

In our letter, in order to see how our scheme would compare with Lehman’s, we had to make some sort of decision over this interval F : C, and decided, for simplicity’s sake, on the same slight flattening as that given to its five immediate predecessors.  It so happens, that if we had made F : C a marginally wider fifth, we would have been even closer to Lehman’s scheme – or, more to the point, Lehman’s would have been even closer to ours!  In terms of the widths of the major thirds, our temperament peaks at A major, while Lehman’s peaks at E major.  But ours has a smoother descent from this peak, as Lehman’s goes over the same contours twice during his descent. 

It has been suggested that, if one starts a tuning on G and follows the same sequence of three mean-tone fifths followed by three pure fifths, etc., then the peak occurs in E major after all – a sort of “British Summer Time” arrangement!  But when has a tuning scheme ever started on G?

Always bearing in mind the inaccuracy inherent in trying to flatten fifths marginally with a tuning hammer (particularly so if one cannot check tunings by listening for beats), we feel our solution is as simple and straightforward as one will find.  On the other hand, Lehman’s solution involves inverting the page and then assumes a method of tuning which is unnecessarily complicated.  The principle of Occam’s Razor is that, if two rival solutions give identical or virtually identical results, then one should choose the simpler one. QED.



(a) David Ponsford, in his thoughtful letter on p.545 of the August Early Music issue, acknowledges that Lehman’s scheme, amongst others, is “good indeed”.  We trust he would agree that ours is too.  In fact, some of these other tuning schemes give a still wider range of major thirds in all the keys without pushing any of them into the “ouch” category!  But we feel that our tuning scheme, or something very close to it, is what Bach most probably wanted to be implied by his diagram, if indeed it is a tuning code.

(b) However, it is certainly now time to scotch, once and for all, the erroneous belief that Bach wrote the “48” in order to prove the ascendancy of Equal Temperament.

(c) I have been in email contact with both Andreas Sparschuh and Michael Zapf in the last three months, and I trust that the comments in the final paragraph of our letter will help to lead to a worldwide recognition of the initial and fundamental part these two gentlemen played in 2001 in this important piece of Bachian research.  It was disturbing that Bradley Lehman, in his interview with Andrew Manze on the BBC Radio 3 “Early Music Show” today, still continued to skate over this fact.  Neither name was mentioned.

© Kenneth Mobbs,

 28th August, 2005.



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