“Occam’s razor”, history and temperaments

To the Editor, Early Music

As a physicist I often come across “Occam’s razor”, but I confess I do not recognize the form of the principle that John O’Donnell claimed to have been using in his article on Bach and temperaments. Ockham wrote “Entities are not to be multiplied without necessity” – or in plain English, Don’t make up more things than you need. I hope Mr. O’Donnell will not be offended if I say that to me, his article made up several unneeded things: chiefly, yet another highly speculative attempt at manufacturing a near-equal temperament out of the calligraphic adornments of the WTC title page. He also unnecessarily assumes that Bach could only have tuned his harpsichord by steps of 1/12 Pythagorean comma (or multiples thereof). Even worse, he proceeds on the basis that a supposed tuning method may be understood, not by applying it on an actual musical instrument, but by a process of mathematical trial and error involving choosing, without any clear rationale, the sizes of tempered fifths, in the hope that it will result in something sufficiently close to what Mr. O’Donnell thinks historical harpsichord tuning was like.

Like all recent tunings based on the same scroll-like ornament, it is composed almost entirely out of a modern author’s own personal tastes for what he thinks suitable and proper in a “Bach tuning”. A correct use of “Occam’s razor” would shave away all of these modern syntheses, created with the help of extensive mathematical underpinnings, and return the scrolling to its former and entirely plausible status as a slightly eccentric ornament.

Another idea redolent of science is that of testing a theory. We have the theory that Bach wrote down the scrolls for the purpose of communicating some specific keyboard tuning method. How can this idea be meaningfully tested or evaluated? The main method so far put forward is that of listening. This has many drawbacks, the chief one being that the results are always dependent on personal taste (as well as a host of other factors) – and to imagine that one’s own taste in subtle differences of tuning resembles Bach’s is a gross and elementary fallacy. Other difficulties with the “listening method” is that the task is never done: there are so many possible variant squiggle-tunings and so many pieces – and even if you do find a tuning you like very well, who is to say there are not ten or a hundred more that you would like better, were you to try them? As a tool of history (rather than of musical performance), listening can only weed out temperaments that sound grossly cacophonous in Bach’s music.

John Barnes’ method of comparing the quality of thirds to their frequency of use (1979) has a certain amount of historical justification (Werckmeister often says that the more frequently used thirds should be purer), but is subject to so many uncertainties that it cannot be more than an approximate starting-point. Different pieces use thirds in quite different musical contexts: then how can they be objectively compared? (Though I would not go so far as to say that one cannot compare the quality of tuning in, say, the C major prelude to that in the F# major.) It is remarkable, and unlikely to be a random fluctuation, that Barnes’ results show a strong trend for Bach to use diatonic (major) thirds more frequently than chromatic in the WTC. This might though reflect the relative ease of merely playing in different keys; or, as suggested by P. Jones in 1980) reflect Bach's accommodation to how other players were likely to tune.

But there are other ways to test the squiggle-tuning hypothesis: by following through its probable implications and comparing them with the historical record. Let us attempt to take seriously the idea that Bach wanted to communicate a specific and musically important keyboard tuning method via the scrolls. The efficient, practically useful, and unambiguous way to do so would be to have a clearly indicated starting-note; to draw no more and no less than twelve loops; and to devise a notation that clearly and immediately showed the degree of tempering of each fifth without the need to perform any calculations. In order to check the tuning of fifths, the amount by which major thirds are tempered should also be indicated. Bach’s scrolls fulfil none of these desirable conditions. (Even Lehman’s supposed ‘C’ is not a starting-note - he proposes to start on F.) Different modern authors have by now plausibly advocated many different starting-notes, directions of reading and/or tuning up or down by fifths, and substantially different degrees of temperament for the fifths. (Jobin…) And the more of these incompatible modern “interpretations” there are, the weaker is the idea that Bach intended to communicate one specific tuning - because the more indisputable is the incompleteness and ambiguity of the scrolls if read as tuning method. So the theory fails its first test.

If any one special tuning method for a well-tempered keyboard were musically important, would Bach communicate it to his pupils (including his sons)? One would expect so. He must have taught harpsichord tuning to his pupils in any case, if they did not simply pick it up by eavesdropping. Yet no pupil seems to have referred to a specific tuning method with special properties. The closest we have to such evidence is the posthumous tribute mentioning how Bach tuned in a way that made all keys and all modulations possible. If the WTC scrolls were important in communicating one special tuning, one would expect to find somewhere among manuscripts of Bach’s pupils some similar or even identical scrolls – just as one finds many copies, more or less exact, of Bach’s compositions. But nothing has emerged, and the theory fails its second test. Two possibilities remain: that Bach’s pupils agreed to a bond of secrecy, never to reveal or reproduce the invaluable content of the sacred scrolls; or that one or other modern author really does understand Bach’s keyboard tuning better than his own pupils did.

Perhaps some readers will now say: he can’t disprove the idea! To which I can only reply: when a historical theory has no direct evidence, fails the simplest and most direct tests, can only be kept alive by adding complicated and implausible explanations for why those tests do not apply, and draws its support entirely from subjective experience, it has become a conspiracy-theory – and such theories will always have their adherents, regardless of the weight of evidence.

But I do not expect the advocates of “squiggle-tuning” just to give up. I will also address the apparent necessities that motivated them. These are twofold: the apparent inadequacies of currently known historical tunings for the WTC; and the unusual form of the WTC title-page scroll itself, which does not resemble any other known calligraphic ornament used by Bach or his pupils.

On the first point, my response takes two forms. First, many tunings used for the WTC have been inadequate in the ears of many: for example “Werckmeister III” where the fifths on C, G, D and B are tempered, the others being pure. While tolerable as an organ tuning, this is unsuccessful in some people’s judgment on the harpsichord, in remote keys such as B, F#, C# and Ab major and C#, G#, Eb or D# and Bb minor. Even in “central” keys such as G or D major it has the fault of combining ¼-comma flat fifths with noticeably impure thirds. Bradley Lehman has already noted objective reasons why one might dislike Werckmeister III and many related temperament recipes – and correctly said that “historical” temperaments need not sound harsh or odd in remote keys.

Second, many historical guides do exist to perfectly adequate tunings for the WTC, without having in the least to start “interpreting” debatable calligraphic flourishes or tinkering with mathematical fractions.

As first witness I call Werckmeister himself, in his later publications after 1695. As noted by Rasch (1985), he came to advocate either equal temperament, or something quite close to it. In his 1698 tuning instructions, specifically for stringed keyboard instruments, he recommends either ET or an unquantified circular tuning, in which the most used intervals are kept as pure as possible, but most fifths are tempered (narrow) by an amount definitely less than ¼ comma. This would remove some faults of his No.III temperament. He also remarks that temperaments calling for quarter-comma tempered fifths are absolutely intolerable if in practice the tempering turns out slightly uneven. In his last (posthumous) publication Werckmeister recommends an equal temperament, or one in which the most used keys are made a little purer. (This means that the thirds in such keys are somewhat purer than in ET, and the fifths should be tempered a little more narrow than in ET.)

Very similar ideas were propagated in the first half of the 18th century by J.G. Neidhardt, although his works were so mathematically based, and devoid of any hints as to how his irregular temperament schemes might be applied in practice (cf. Norrback), that we may doubt whether these schemes were at all commonly applied, even among experienced musicians. His works are chiefly useful as indicating the probable desire of some musicians to play with good intonation in a wide range of keys, and the opinion (whether derived from direct experience or theoretical grounds) that ET was musically lacking in variety.

That some musicians did use ET is attested by the report of Buemler having already done so for some years, relayed by Mattheson in 1722. One attempted application of ET in the early 18th century is also shown by a manuscript tuning instruction for harpsichords and clavichords found in documents of the organist Ulich of Anhalt-Zerbst (ref. Hochgartz). This gives a complete cycle of fifths from C through to C again, with the remarks that the fifths should be tempered narrow, while the major thirds should be sharp. With a little practice, such instructions might produce a reasonable practical approximation to ET. This source is also remarkably close to J.S. Bach in his Coethen years, though no firm conclusion need follow from this proximity.

Essentially the same instructions were given in 1737 by Rameau, with the refinement that the fifths were to be only the tiniest bit flat, and one could start from any note one liked. The tuning of (something like) ET by simple trial and error round the cycle of fifths was becoming a familiar possibility for professional musicians.

From here our witnesses become strictly post-(J.S.)Bachian, and must be regarded as circumstantial, although quite strongly connected in the case of Marpurg, Kirnberger and C.P.E. Bach. Marpurg, although primarily an advocate of ET, also in 1755-6 allowed merit to a type of unequal harpsichord tuning in which seven fifths from F through F# were narrowed, and the remaining fifths left purer, or less narrow. (Essentially the same tuning recipe is given for organs in a later 18th century document of the Wiegleb organ workshop, in which five fifths from F flatwards through F# are pure, and the remaining ones tempered narrow.) This type of tuning was not explicitly described in previous Baroque writings, but it may be a clue to reading C.P.E. Bach’s somewhat vague tuning advice that something is to be taken away from the absolute purity of “most of the fifths”, to give a tuning in which every key is “purely” tuned. Both criteria are fulfilled by the ‘7-5’ tuning mentioned by Marpurg, if the five “purer” fifths are not all absolutely pure or wide. The interpretation of C.P.E. Bach’s remarks is a puzzle: clearly they do not indicate ET, but the author did not designate any key as better or worse than any other. While Lehman has managed to argue that the remarks do not preclude his favoured Bach tuning, the most straightforward reading of C.P.E. Bach’s lack of verbal precision is that he did not much care about the details of tuning, so long as no key sounded noticeably impure – that is, out-of-tune.

The next two witnesses, while differing in their approaches, introduce an important concept: that of checking temperament through the sharpness of thirds. Sorge’s ET method of 1744 was the first really modern instruction in that it started not from fifths but from a closed cycle of thirds C-E-G#/Ab-C; once one has put these in place and checked that no third is audibly more impure than any other, they provide a framework for fifths to be placed with the right degree of almost imperceptible narrowing. Barthold Fritz’s method of 1756 has a different bearing plan, asking for a check of F-A after four narrow fifths have been tuned, with an approximate beat rate for the sharpness of this third – but Fritz’s beat rate is definitely slower than the correct ET rate of 6.5 to 7 beats per second. His method was likely to result in a tuning similar to the Marpurg unequal tuning, in that the thirds in “central” keys (F-A, C-E, …) would be somewhat purer than in ET, at the cost of those in remote keys (F#-Bb, C#-F, …) which would of necessity be almost Pythagorean. Since C.P.E. Bach wrote a warm recommendation for the second edition of Fritz’s pamphlet, it is unlikely that he was committed to any specific unchanging “Bach family tuning”.

Hence from 1698 through the generation after J.S. Bach there existed in German-speaking musical society a thread of ideas and practices which could give tunings ranging from 1) a truly equal temperament possessing no audible differences between keys, to 2) a practical substitute for equal temperament, probably with the fifths ‘sharpwards’ of F tempered somewhat more than 1/12 comma, and those ‘flatwards’ tempered less, if at all; and with the thirds purer than ET in frequently-used keys; but not as bad as Pythagorean in rare keys.

This context may give us some insight into a remark made in one of Marpurg’s and Kirnberger’s spats over their idol’s legacy, that (necessarily in later life) J.S. Bach had instructed Kirnberger to tune a harpsichord by making “all the thirds sharp”. Whether this was the entire instruction, and what Kirnberger made of it, is unknown. It seems equally plausible that Bach may have wanted an equal temperament (a la Sorge?), by beginning with three thirds equally sharp; or an unequal., graduated tuning (a la Fritz?) judging the sharpness of the thirds by their musical uses (cf. Barnes), then fitting chains of fifths in between. In any case this four-word quotation is of more practical use than any number of squiggles or scrolls, because it tells the tuner how to check whether the chain of fifths is anywhere near correct: namely by listening to thirds.

One possible objection to such “not-quite-equal” tunings is that they are an incompetent attempt at ET. But on the contrary, I believe Bach could have gone beyond a competent ET, at least after he had become aware of the possibility (probably at least as early as 1706), to a form of tuning which produced subtle key differences, without making any key uncomfortably impure; and done so without resorting to any such mathematical formulations as used by Neidhardt. Once one has practiced closing the circle of fifths, it is relatively easy to execute by ear by slightly increasing or decreasing the temperament of one or another fifth, constantly checking chords, or indeed using simple chordal pieces such as the openings of the WTC Preludes as checks on various keys.

I do not want to be more specific about what I believe a “well-tempered” keyboard was for Bach. This is because I believe numerical (not to say electronic!) aids are completely inimical to the spirit of good Baroque keyboard tuning, which required and requires the use of hearing as much as possible. Also as many different good temperaments for the WTC may be tuned by following simple guidelines and personal taste, always informed by extensive practice in closing the circle of fifths correctly. The construction of “Bach-tunings” should and must be given as a task to the ear and hand of keyboard musicians wielding their tuning hammers – rather than to manipulators of occult symbols and mathematical fractions. (I write this as a theoretical physicist!)

If pressed I would suggest the following starting-point. From F tune C-E-B-F#-C#-G# and F-Bb-Eb by pure intervals (thirds or fifths as given). Now flatten C, and raise first B then E, by the smallest amount possible that produces audible beats in F-C and B-F# respectively, and immediately check the chords of F minor, Ab major, C# major, B major, F# major and E major. If desired, E may be raised very slightly to form a narrow fifth with B, then E major and C-E should be checked to make sure that C-E has not become too wide. Finally tune G, D, and A by slightly narrow fifths so that the intervals C-G-D-A-E sound equally out-of-tune, checking each major chord as it becomes available.

There remains the enigma of why the WTC title page looks the way it does. An inspection of other manuscripts of Bach and his pupils shows that this page is unique in its layout and in the nature of the scrolls placed above the title.

It is extremely remarkable that the title pages of Friedrich Suppig’s “Calculus Musicus” and “Labyrinthus Musicus”, that exist only in a single copy dated in Dresden on St. John’s Day 1722, have very close similarities in layout, wording and calligraphic scrolling to Bach’s. It was in fact this similarity that, towards the end of the 1990’s, led Andreas Sparschuh, the mathematican and piano tuner, to conceive a possible connection between these title pages and tuning schemes. He had been examining a reproduction of the Suppig manuscript, then happened upon an old edition of “Musik in Geschichte und Gegenwart” (the German “Grove”) which had a photographic reproduction of the Bach title page.

Anyone who makes a detailed examination of the layout, text and calligraphy of these three pages, and compares them with Bach’s other manuscripts, can scarcely doubt that these are three of a kind – and either Bach or Suppig must have seen the other’s manuscript before writing his own.

As noted by Rasch, Suppig dealt with two tuning schemes: one a just intonation scheme with 19 notes per octave, extended through four octaves (though totally impracticable for musical performance); the other a set of 31 pitches obtained from a chain of pure thirds, and closely approximating a 31-equal division of the octave, or ¼-comma meantone – which was equally impractical, unless one possessed an instrument with 31 keys per octave! (Such instruments were not absolutely unknown, being associated with Vicentino and Trasuntino in 16th century Italy, Valentini in early 17th century Italy and Vienna, and Bulyowsky in the later 17th century Austrian Empire.) To illustrate the use of this “Funfffach Transponir-Clavir” Suppig wrote a pantonal piece ranging slightly haphazardly through the “circle of fifths” (although by rights this circle should have 31 tonalities rather than 12!), and ending with a bizarre microtonal coda in which voice-leading by a diesis (the difference between G# and Ab in ¼ comma meantone) is used in a sequence of abrupt modulations. Apart from this coda, the piece is extremely commonplace and tedious.

I believe the simplest explanation of the peculiarities of Bach’s title page lies in his probable reaction to Suppig’s productions: would he not have been highly amused that another musician had been writing pieces in “all 24 keys” at exactly the same time, but on a vastly lower level of competence and invention – and in addition, promoting not one but two totally impractical keyboard tuning schemes, which required loads of extra keys and strings to do what Bach was able to achieve with only 12? What might the highly adorned and pretentious incipit be, if not an ironic tribute or parody of Suppig’s? (Suppig's calligraphy was itself a laborious attempt to imitate copper-plate engraved title pages.)

One may object that this version of events has no direct evidence in its favour. How could we ever know in what mood Bach created his title-page? But it has the merit of accommodating all the known facts, including the remarkable visual and textual similarities of Bach and Suppig, without having to introduce any new entities, except for Bach’s sense of humour – and I can only suppose that he must have had an extraordinary temperament!

Afterword 1.

Mr. O’Donnell’s claim that Baroque tuning instructions took the form of a chromatically rising sequence (like today’s recipes for pre-programmed electronic tuning devices!) omits or ignores a large number of instructions presented in many different forms. Although chromatic rows existed and were intended for application on a monochord, or for constructing multiply-fretted clavichords, they are totally impractical for tuning harpsichords by ear, which is faster, cheaper and more accurate than tuning by monochord. A chromatic row would require the tuner to hop back and forth along it by fifths and fourths, and at each step calculate whether the next note should be pure, flat, sharp or what not.

Practical harpsichord tuning instructions invariably took the form of a chain or cycle of fifths, supplemented by the use of thirds or triads as checks. In Werckmeister’s “Musicalische Temperatur”, to go no farther, one finds an instruction for “No.IV” laid out both diagrammatically and verbally as a closed cycle of fifths, calling for each to be tuned either pure, narrow or wide in turn. One other popular method for displaying tuning instructions was simply a series of musical notes showing the order of tuning and checking (Stimmweg), as for example in the Ulich instruction, together with a short description of the quality of the respective fifths and thirds. The tuning instruction published by Godfrey or Gottfried Keller, a German immigrant to England, in 1701/7, takes this form, with the simple remark that the fifths should be as flat, and the thirds as sharp, as can be borne. (Since the bearing plan does not encompass a complete cycle of fifths, this particular instruction results in some variety of meantone - Ref. previous EM article)

I am astonished to find that no-one in this debate has yet mentioned that German organists and organbuilders did have their own diagrammatic method for displaying tunings: namely the ^ and v symbols, representing use of the conical tuning tool to flatten or raise the pitch respectively. These appear in publications of the older Trost, Werckmeister, Neidhardt, and in minutes of the debate over tuning the new Trost organ at Altenburg. It is inconceivable that Bach did not know these symbols; if he had wanted to indicate a tuning scheme economically and without mathematical or verbal fuss, he could easily have used them, as Werckmeister already did in the tables of his “Orgelprobe” of 1681.

However I believe it much more likely that Bach would have instructed his pupils in tuning by oral and aural methods – as he found to be most convenient for teaching Anna Magdalena figured bass, rather than trying to reduce a complex and artistic matter to a paper formula.

Afterword 2

It has been suggested that the WTC I titlepage was a rough copy for an engraver to follow. While this hypothesis cannot be excluded, a few things speak against it. First, the extreme length and difficulty of the work, and the fact that many potential customers probably used meantone or similar tunings which would make nonsense of half the movements. Second, the likelihood that the engraver would not need any guidance from Bach to invent and dispose decorative scrolls, spirals etc., or to produce an elegant arrangement of text down the page. (And might well alter the arrangement Bach had already produced!) Third, the fact that no-one did publish the work until many decades after Bach’s death, suggesting that (in contrast to the Art of Fugue) there was no desire for publication within the Bach circle. It was preferable to keep the work within a network of family and pupils who might pass on Bach’s tradition of playing, and of course tuning. By which I do not mean one specific unchanging method, any more than a tradition of playing implies one specific unalterable tempo or articulation.

Thomas Dent: Heidelberg, 2006-2008 email

changed February 6, 2009