Bach's schematic, as it appears on the page, © Bradley Lehman, 2005-22, all rights reserved.
All musical/historical analysis here on the web site is the personal opinion of the author,
as a researcher of historical temperaments and a performer of Bach's music.

"Bach's temperament, Occam's razor, and the Neidhardt factor"

Article by John O'Donnell, November 2006 issue of Early Music.

This is a nine-page article, as a PDF file for download from Oxford's web site. (Subscription or library access are necessary to download this file; I am not at liberty to re-distribute it directly....)

The final version was released here on November 10 2006, and in the printed issue as well; download from the Early Music web site. [2011: A free version is now available here through Stanford's "HighWire" service.]

Here is its abstract, as copied from that web site:

Bradley Lehman's Bach temperament, its predecessors and successors proceed from a single assumption concerning Bach's diagram: that it represents a circle of 5ths. Some have even followed Lehman in inverting the diagram in order to read it, though there appears to be no logical reason for doing so. Temperaments in Bach's day were commonly notated chromatically from C, no doubt following the model of the monochord, but also because of increasing interest in measuring the equality or otherwise of successive semitones in the quest for equal temperament. It also happens to represent the order in which Bach chose to lay out his preludes and fugues. Can it be that Bach's diagram is to be read in this way? The key to the reading of the diagram is surely the ‘Eb/D#’ embedded in the ‘D’ of ‘Das’. The resulting temperament bears strong similarities to Neidhardt's Circulating Temperament no.1 published in 1724, as well as to Sorge's temperament for a Kammerton harpsichord published in 1744, while it is all but identical to The Neidhardt's 3rd-circle no.4 published in 1732. And though we know of no direct association between Neidhardt and Bach, circumstances suggest that they would certainly have been familiar with one another's work. It is proposed, on the principle of Occam's razor, that the linear interpretation given here should supersede all interpretations presented to date.

My remarks, November 2006 and April 2007:

Take Neidhardt's "Third-Circle #4" from 1732, and lower the Bb...

...According to O'Donnell, his temperament also somewhat "bears strong similarities to Neidhardt's Circulating Temperament no.1 published in 1724", which is to say: Neidhardt's 1724 temperament for "Dorf" (Village). Let's take a look. O'Donnell's G#-Eb-Bb-F-C-G-D-A have the same placement as in that Neidhardt temperament, holding either C or A as constant for comparison; but O'Donnell's remaining four notes (E-B-F#-C#) are all higher in pitch than Neidhardt's. This creates a situation where only four of O'Donnell's 12 major 3rds (F-A, Bb-D, Eb-G, and Ab-C) have the same size as Neidhardt's, while the other eight are different. Here is a table of that Neidhardt temperament:

...He also says here that it is similar to "Sorge's temperament for a Kammerton harpsichord published in 1744". Let's see that one. This time all of O'Donnell's naturals F-C-G-D-A-E-B plus (coincidentally) G# are placed the same as Sorge's. The other four notes are different: O'Donnell's F# and C# are higher, and Eb and Bb are lower. That is, it's easy to convert from one to the other in practice by simply moving those four notes. Among the major 3rds, the yield here is that five of O'Donnell's are the same as Sorge's (C-E, F-A, G-B, E-G#, and Ab-C), and the other seven are different. That treatment of the sharps and flats makes O'Donnell's temperament closer to equal temperament than Sorge's was: softening B-D#, F#-A#, and Db-F at the expense of Eb-G, Bb-D, D-F#, and A-C#. Sorge's 1744 temperament for harpsichords:

So, what's the resemblance here among O'Donnell's temperament, and these two by Neidhardt and one by Sorge? Yes, it's easy to convert from his to any of these others, or vice versa, by moving only four notes...or in the case of the "Third-Circle #4", only one note (the Bb). That resemblance, though, with the Sorge 1744 and Neidhardt's "Dorf" is not so much in musical effect but rather the tuner's convenience. Where most of the major 3rds, minor 3rds, semitones, tones, and some 4ths have changed sizes (due to the moving of four notes), the overall sound of the temperament takes on different character.

For this alleged "resemblance" to have any historical thrust: Neidhardt and Sorge -- either or both -- would have had to have known O'Donnell's temperament (in the guise of Bach's practice), and then chosen to move the four notes they did. It's possible.

I made a broadly similar argument myself, in my article, citing a relationship between my own Bach layout and Sorge's 1758 (his last published temperament, where he was promoting only equal temperament or this particular unequal shape, anymore -- which I believe is important). I wrote, p16: "Sorge's 1758 temperament has the same harmonic shape as Bach's (see illus. 3), but is slightly gentler in those contours: more like equal temperament. In summary, Sorge by 1758 had realized that the Bach shape is the only unequal one that really works for tonal music (and especially for Chorton organs), even though Sorge himself preferred equal temperament in theory." I also explained there why the moving of those four notes (the difference) would have made logical sense to an equal-temperament maven such as Sorge, starting from my putative Bach layout. I came to this, myself in 2004, not from examining the placement of notes but rather from playing in Sorge's 1758 temperament, and noticing that the overall sound is the same as "Bach's" as to the movement of harmonic tensions, but just a bit gentler.

OK; so, in reusing this line of presentation, O'Donnell has cherry-picked other Neidhardt and Sorge temperaments from earlier in their careers that happened to match his own within eight or more identical notes...but hasn't shown why those several temperaments would have been especially important.

The alleged "D#" that guides O'Donnell's placement of C...but why not more simply a giant D that's sitting right there in the word "Das"?

Let's take another look at Bach's title page:
Bach's title page, 1722, from Grove Dictionary 1911
O'Donnell claims to see a "D#" in there somewhere, guiding his placement of D# with that particular loop in the drawing. How? Where? First, let's take a look at the way Bach actually wrote the note D# in organ tablature, when he really meant D#.
(This example is explained further in the section "'Hidden letters' in Bach's title-page drawing" at my own errata page.)

It's time for a closer look at this, side by side. O'Donnell wrote, p627: "The detail of Bach's diagram that triggered this realization is one not mentioned by Lehman. The letter 'D' of 'Das' includes a diagonal line that goes directly to the third loop and has 'Eb' on the left side of the line joined to the German keyboard tablature version of 'Dis' (D#) on the other side of the line. How are we to interpret this other than as an indication that this loop represents Eb/D#? It may be that the choice of Eb/D# was made to symbolize the whole art of well-tempered tuning, wherein enharmonic notes are made one, and it is interesting in this regard that in the ensuing volume only the Eb minor Prelude and D# minor Fugue are notated as an enharmonic pair. But I think that it has a greater significance, as we shall see when we come to the tuning process."

Hold on, for a moment! O'Donnell's article asserts but does not demonstrate this! Where in the "Das" area of the drawing is this alleged tablature "Dis" symbol?

If that little jiggle inside the D is anything, under those two little dots with a line running between them: why is it a "Dis" (but missing one of its loops, and the other loop goes in the opposite direction!)? And isn't it already being used as a lowercase "s", to comprise the note-name "Es" (Eb)?

And why does that giant D itself have to turn through prestidigitation into D#/Eb? Why can't it simply be a D? Borrowing a phrase from the first page of O'Donnell's paper: "There is a failure of logic here."

The "small c", again and again, as if it collapses my theory when the C goes away....

Part of O'Donnell's argument in this paper is to assassinate the "small c" as merely a capitalization stroke for the capital C of "Clavier" -- thereby rendering it irrelevant and harmless, supposedly. His words, p625: "The 'C' identified by Lehman as the key to the position of the temperament has effectively been dismissed from the argument by Carl Sloan, who points out that this C 'is actually an ornamental hook on the "C" of "Clavier"'. Yes, it could be doing double duty, but if the diagram is to be read inverted why is this essential element now upside-down?"

Well, then: why does this giant D (or D#!) get to do double duty as a letter and a note, as foundation of O'Donnell's assignment of the notes to the diagram, but the other putative "small c" doesn't in mine? The big D is definitely part of something else, namely the word "Das"; the small C is arguably part of something else, namely a capitalization stroke on the big C of "Clavier". Why choose the D to be so important, instead of the C? The same argument has to cut both ways, if any elements here in the drawing are doing double duty with the diagram and the words on the page.

In any case this is rather moot, to me. I've pointed out in my newer paper and at several places on this web site ("How did this discovery happen?", the FAQ pages, et al) that my argument doesn't stand or fall on this "small c" business anyway! That note C is where it is, in my layout, because that's the obvious place to start if one is going to tune all seven naturals before the five sharps/flats. The F-C-G-D-A-E must be there, if all the naturals are on one side of the diagram and the sharps on the other, and if all those naturals are to have the same (i.e. regular) tempering as one another. That's the place in Bach's drawing where there are five of some "same" thing next to one another. See my explanation in "Bach's Art of Temperament" for details about this start on C, whether the "small c" counts as part of the diagram or not.

And did anyone notice my own endnote #63, way back in the May 2005 printed half of my first paper? "63. The small C of the temperament diagram itself masquerades as merely a calligraphic capitalization stroke on the bigger C of 'Clavier'. The occurrence of such C-like hook strokes elsewhere (for example, above Bach's opening K in the Entwurff, on a C of the title-page of Concerto BWV 1043, and several occurrences on Altnickol's title-page of WTC 2) throws sleuths off the steganographic trail that the C belongs also with the diagram, which it touches."

Unequally-sized 5ths that look the same in Bach's drawing

O'Donnell pointed out the basic problem with his own layout: that it takes a dive in the middle where the major 3rds F#-A# and Db-F are narrower than the B-D# and Ab-C on either side of them.

He said it thus, on p628: "(...) [W]e can already deduce a lot of information from the placement of the pure intervals. For example, regardless of the amount of the narrowing of the 5ths or widening of the 4ths, we can make the following statements about major 3rds: C-E and F-A will always be identical in size; G-B will always be narrower than D-F#; A-C# will always be wider than E-G#; B-D# will always be wider than F#-A#; F#-A# will always be narrower than B-D#; Db-F will always be narrower than Ab-C; and Bb-D will always be wider than F-A." (Emphasis mine, with boldface.)

Take another look at the resulting major 3rds in O'Donnell's temperament:

C-E 5, F-A 5, Bb-D 7, Eb-G 8, Ab-C 9, Db-F 8, F#-A# 7, B-D# 8, E-G# 7, A-C# 8, D-F# 7, G-B 5.

Sure enough, the Db-F and the F#-A# are narrower than the Ab-C on one side and the B-D# on the other side; they're "too good" if we want to preserve the normal behavior of circulating temperaments, which is that C-E and its neighbors should be the smallest major 3rds. We would expect the triads of F# major and Db major to have wider major 3rds than their neighbors, since there are more sharps or flats involved. This is the normal behavior of "well temperaments".

And, it is that problem that then compelled O'Donnell to fudge identical-looking loops (next to one another) in the drawing to be interpreted as two different sizes: to minimize this problem with F#-A# and Db-F! If the 5ths in O'Donnell's F#-C#-G#-D#-A# area are made any narrower than 1/12 comma, he ends up with an F#-A# that too much rivals C-E (or surpasses it) as narrowest.

That problem suggests to me that his assignment of D#, and correspondingly everything else, is mistaken according to his own argument. He has had to make up this arbitrarily assigned two-sizes business, to prevent his own scheme from turning out too well in F# major and its nearby keys! Nothing in the drawing suggests that that the three loops at the left should indicate any differently-sized 5ths among themselves; or similarly for the group of five loops at the right. O'Donnell has had to impose his own rule of varying sizes, to get his layout to work out.

O'Donnell's layout interprets the Bach drawing as follows:

Bach's schematic, as it appears on the page
C-F pure (the single tiny loop at far left);
three tempered loops for C#-F# 1/12 comma, D-G 1/6 comma (why different?), D#-G# 1/12;
three plain loops for pure 5ths E-A, F-Bb, and F#-B;
five tempered loops for G-C 1/6, G#-C# 1/12, A-D 1/6, Bb-Eb 1/12, and B-E 1/6.

The notes in Bach's drawing, to O'Donnell, indicate the top note of a 5th or the bottom note of a 4th. C with a pure 4th above; C# with a tempered 4th above; D with a tempered 4th above; D# with a tempered 4th above; E with a pure 4th above or pure 5th below; F with pure 5th below; F# with pure 5th below; G with a tempered 5th below it (i.e. middle C); etc with more tempered intervals, all being arranged chromatically up the scale. The bearing octave of the instrument is apparently middle C up to the next C, listening either above or below the named note wherever it falls into this octave.

I find that all rather confusing. Why would Bach do such a thing? O'Donnell's explanation from pages 627 to 628 is not very clear, either, as to why all this is being done. To use Bach's diagram, in O'Donnell's manner, one has to skip around and to keep careful track of which loop does what...and with two different sizes! Nor is it explained why the C to F, the tiny loop at far left, is drawn differently from all the others...yet tuned purely like the several in the middle. "There is a failure of logic here." We are to believe that Bach drew his "pure" loops in two different ways, but tuned identically; and drew his "tempered" loops in two other ways that both don't show how much that tempering should be.

It all looks unbearably forced, to me. And "Occam's Razor" is being cited to justify this!

Let's try it with the big D being a D

Let's discard O'Donnell's notion that there are two different qualities of tempered 5ths/4ths in play here, and move the whole line over by one note so the big D really means D (instead of D#/Eb). But, we'll keep most of the rest of O'Donnell's argument about the whole drawing being a chromatic sequence instead of a series of 5ths. What do we get?

Let's take another look at the section I have had on my "practical instructions" page since March 2006.

Bonus 3: an easy Lehman temperament, emphasizing tasteful listening
Posted to HPSCHD-L on 8 March 2006; added here 29 March 2006....

I have formulated this temperament also from elements in the Bach drawing, but in a less obvious way. I still like my first one better, overall, while this one works very well in the same repertoire. Both are equally easy to set up in practice.

  • C from fork, and over to middle C.
  • Set E from C, somewhere near the quality of regular 1/6 comma (i.e. with the familiar sound from Vallotti, or slightly sharper than in Werckmeister 3). If it happens to be a little high or low to taste, that's OK. We just want to stay out of the range where it's so high that C-E turn into a blur, or so low that there's no room left to work with in the following steps.
  • Fit the G, D, and A into this so C-G-D-A-E all have the same quality as 5ths or 4ths. Whatever regular size we're setting here, this is our basic unit.
  • From E, pure B, pure F#. G-B should sound slightly "harder" or brighter than C-E, and then D-F# even more so.
  • From C, pure F. F-A has the same character as G-B. The C major, F major, and G major triads are our three best.
  • From F, temper Bb as a narrow 5th (or wide 4th) of approximately the same quality as the others above. Listen also that Bb-D is similar quality to the F-A already available; and that F#-A# is high but acceptable-sounding.
  • From Bb, pure Eb. Confirm that Eb-G and D-F# have the same quality as one another, both sounding much like they do in equal temperament. Confirm also that B-D# has approximately the same character as F#-A#, and the whole triad B-D#-F# is quite good, owing in part to the pure 5th.
  • From Eb, make Ab a narrow 5th or wide 4th of approximately the same quality again, or perhaps a little bit gentler to taste. Test that E-G# is high and bright, but not quite as wide as Pythagorean. Ab-C should sound very slightly wider in character than Eb-G does, but still a good complete triad Ab-C-Eb.
  • From F#, make C# a narrow 5th or wide 4th similarly. Check that A-C# makes a nice transitional character between D-F# and E-G#. Also check that Db-F has a character resembling that of B-D#. Finally, note that our leftover interval C# to G# is probably slightly wide as a 5th (or narrow as a 4th), but that the character of C#-G# sounds the same as Eb-G#. They simply happen to be tempered in opposite directions, but a similar amount, not that anyone would notice during the playing of music.

Play suitable music in a variety of keys, to test that everything works nicely.

That's what this is: a chromatic layout (humoring some of O'Donnell's argument ahead of its publication, but at the time I was not at liberty to say anything in public about the earlier draft of O'Donnell's paper I had been sent for comments...).

Mine here starts starts on C as the first large loop of Bach's drawing, at the left. The tiny loop to its left is the pure B-F#: no bobble in it. Like this:

As seen here, the third large loop indicates the note D (and the fifth D-A); not D# as O'Donnell's layout has it, with everything shifted over to the right by a semitone.

Play through all those 4ths/5ths in turn, chromatically: the B-F# is pure; the C-G and C#-G# and D-A are all tempered (the same amount/quality as one another, sounding identical in musical practice, even if one of those might be in the opposite direction!); the Eb-Bb, E-B, and F-C are all pure; and the F#-C#, G-D, G#-D#, A-E, and Bb-F are all tempered the same as one another...and the same as the first set. There are eight tempered 5ths, and four pure. Its table works out like this:

Humoring part of O'Donnell's argument, but applying it to this temperament of mine: the difference between the single-bobble and the double-bobble loops is that the ones at the left are tuned as widened 4ths below (such as middle C down a 4th to tenor G), and the ones at the right as narrowed 5ths above. The bearing area of the keyboard works out to be approximately the F below middle C, up to the F# above middle C. Both F#s are made pure to the B, and both Fs are made pure to middle C. That's the switchover point where we take the named note as the bottom of a 5th, instead of as the top of a 4th.

Repeated, for convenience of reading:

Does this scheme -- where the big D really means D! -- not serve the "Occam's Razor" invocation of O'Donnell's paper, better than his own layout does?

This same sequence of tempering instructions given above also works decently, if the regular amount of tempering is relaxed to approximately 1/7 or 1/8 comma...but I find the musical results to be less compelling (too bland) than with 1/6 (and the single "overshoot" at Ab/G#), so I do it that way. Sometimes I set it up in exactly the way shown here, or sometimes with this 1/6 comma base I come back at the end and lower the G#/Ab very slightly (by about 1/12th comma), to taste.

I've played through the WTC with this, and with O'Donnell's temperament, to compare them. In my opinion as a musician playing this repertoire, both of my layouts give a stronger focus to the music than O'Donnell's does. I personally don't like the way his resulting major 3rds of size 5 (C-E, G-B, and F-A) sound. They run into an odd property of harpsichord tone -- a surprising harshness, while some wider major 3rds than that sound better! -- which I've also brought up in my remarks about the Lindley "Bach-style" temperament, at the section "Playing through the WTC in Lindley's temperament". I'd rather hear more resonance in the instrument, with the size 3 major 3rds that my layouts have, whenever the music modulates back to cadences in F or C. The O'Donnell and Lindley temperaments both seem to me to be unsettled-sounding, and frankly dull.

Other players and readers are invited to do the same exercise, and to formulate their own opinions, using good harpsichords (not merely electronic simulations!): going through this hands-on tuning and playing process to hear how it works. Perhaps some will enjoy the O'Donnell and Lindley temperaments more than I do.

This "new" one of mine from 2006 works nicely in music, but I still don't like it as well as my first and main one (2004). Why would Bach make a layout arranged chromatically, by results playing ascending 4ths/5ths on the keyboard to test it? Would he not more simply (as I said the first time, and subsequently) provide a more direct and practical scheme, sequenced by the way one actually tunes a harpsichord: as a series of chained 5ths/4ths in turn? Taking a C or F or A or whatever from some reference source, then doing all the naturals first, and then doing the sharps/flats last to finish off?

Return... Bach's schematic, rotated for use