Copy of Early Music, Vol XXXIII no. 3 (August 2005), Correspondence,  pp. 546 – 547. 

© Oxford Journals, Oxford University Press

Rather than listing our serious fundamental objections to the two Bradley Lehman Bach articles – which could wait for a later occasion, if required – more importantly, may we briefly suggest an obviously much simpler interpretation of Bach’s ‘squiggles’?

   

Let us consider what a tuner of the period, using his ear alone, basically needed to know.  Namely, how to tune pure octaves and fifths, and how to flatten other fifths equally.   In this way, he would be equipped to construct a chain of four ¼-Comma fifths making a pure third, either to complete a Mean-tone tuning in its entirety, or as a basis for starting an Unequal tuning.

So, instead of turning Bach’s apparently coded cipher upside down, as Lehman thought necessary, let us begin from the left-hand side, starting the conventional keyboard tuning from C.  The majority of all unequal temperaments at this time were based upon the tuning of initial intervals as in Mean-tone, leaving the remainder to be tuned unequally.  This is much more likely than Lehman’s interpretation in which he immediately starts with the extremely complicated 1/6th-Comma tempering.

We therefore propose the following scheme:

Stage 1: From C, tune three normal Mean-tone fifths, C : G : D : A.

Stage 2: From A: tune three pure fifths, A : E : B : F sharp.

Stage 3: From F sharp: tune five equally-flat, very slightly tempered fifths,

 F sharp : C sharp : G sharp : E flat : B flat :  F    (i.e. in effect flatten them almost imperceptibly until the remaining fifth  F : C  is also judged to be of the same size as its immediate predecessors, and so closing the tuning circle).

This produces the following scale of values, to the nearest Cent, here shown in both our and Lehman’s interpretations:

    Scale

Mobbs/Mackenzie

Lehman

C

            0

      0  

C sharp

          96

    98

D

        193

  196

E flat

        298

  298

E

        392

  392

F

        499

  502

F sharp

        596

  596

G

        697

  698

G sharp

        797

  798

A

        890

  894

B flat

        999

  998

B

      1094

1094

C

      1200

1200

 

Major Thirds

Mobbs/Mackenzie

Lehman

C : E

         392

   392

G : B

         397

   396

D : F sharp

         402

   400

A : C sharp

         407

   404

E : G sharp

         405

   406

B : D sharp

         404

   404

F sharp : A sharp

         403

   402

C sharp : E sharp

         403

   404

A  flat : C

         403

   402

E  flat : G

         399

   400

B  flat : D

         395

   398

F : A

         390

   392

C : E

         392

   392

Because of these surprisingly similar characteristics as shown above, we trust that Dr. Lehman will appreciate that all his effusive enthusiasm can be transferred equally easily to this much simpler interpretation.  We are sure that almost any inevitably tiny tuning discrepancies would, in practice, remain undetectable by even the most professional ear.

Our grateful thanks go to Andreas Sparschuh and Michael Zapf for having first drawn our attention to the possibility of this Bach coded tuning cipher, when we were in communication in February, 2002.  We also thank David Hitchin for his recent help, particularly in clarifying for us the order of events since 2001 leading up to this, our proposed solution.

 KENNETH MOBBS and ALEXANDER MACKENZIE OF ORD

                                               Mobbs Keyboard Collection, Bristol

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