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.

Errata and clarifications

...for "Bach's extraordinary temperament: Our Rosetta Stone"

Early Music, February and May 2005

Numbering and detail of the 1724 Neidhardt temperaments

This is a corrected version of the 1724 Neidhardt "village" temperament chart, as used at various places in my article and web supplements. The error was in the note G#: being placed 1/12 comma too low. (Correction posted here: September 1 2005)

When preparing the article and its supplements in 2004, I had taken the reading of this temperament from two published sources that have turned out to be mistaken. These were Mark Lindley's article "J. S. Bach's Tuning" in The Musical Times 126 #1714, December 1985, table 2 on page 723; and the same table (this time "table 1") as reproduced in his article "Well-tempered clavier" in New Grove 2nd edition (2001), page 276. (Yes, the printed New Grove has two errors in each of two Neidhardt temperaments, and unfortunately in my trust of its authoritative precedent I have perpetuated those same errors! The errors are still in the online version of New Grove as well, accessed 8/31/05.)

Lindley has corrected the "village" reading tacitly in some of his other articles. The formula is also correct in Dominique Devie's book Le temperament musical. C#-G# is pure and G#-D# is tempered; not vice versa.

Musical Times 1985:

New Grove 2, "Well-Tempered Clavier" 2001:

This same "village" temperament by Neidhardt has a different typographical error at the note Eb (1/12 comma too low), as reported in J Murray Barbour's book Tuning and Temperament: A Historical Survey table 151; but Barbour's cent values there are correct.

The complete set of Neidhardt's original string lengths and the derived modern cent values (to two decimal digits) are available in Johan Norrback's book A Passable and Good Temperament, page 37. He copied the string lengths directly from microfilmed copies of Neidhardt's publications, and then converted these to cents. Norrback also provides an interesting contemporary map of the region, showing which specific cities and towns were generally considered "Dorf" vs "Kleine Stadt" vs "Grosse Stadt"!

In his tables 12 and 13, Norrback addresses all four of the 1724 temperaments (Hof, Grosse Stadt, Kleine Stadt, Dorf), next to their replacements in Neidhardt's publication of 1732. The "Kleine Stadt" of 1724 became 1732 "Grosse Stadt". The "Dorf" of 1724 (the "village" temperament under consideration here) was promoted in 1732 to "Kleine Stadt". There is a new "Dorf" temperament in 1732. "Hof" (court) in both cases is equal temperament.

The 1724 publication had four temperaments, and the 1732 publication had a total of 21; see details in Barbour, page 179, and the facsimiles and transcriptions at

Corrected chart for Neidhardt's 1724 'village' temperament

In my Appendix (i.e. the Oxford web supplements of part 1, and reproduced here) I simply listed and numbered the three 1724 temperaments in the same sequence as Lindley's 1985 article and the New Grove gave them, and where they are not labeled separately. Unfortunately, this might create further confusion next to Barbour! To clear that up, here is a table of concordances:

Neidhardt in Sectio canonis (1724)Barbour in Tuning and TemperamentLindley in New GroveLehman in Rosetta Stone
Kleine Stadt (small city)Neidhardt's Circulating Temperament, No. 2 (table 155) or Fifth-Circle, No. 8Neidhardt (1724-32)Neidhardt #1
Grosse Stadt (large city)Neidhardt's Circulating Temperament, No. 3 (table 156)Neidhardt (1724)Neidhardt #2
Dorf (village)Neidhardt's Circulating Temperament, No. 1 (table 151) or Third-Circle, No. 2Neidhardt (1724-32)Neidhardt #3
Hof (court)equal temperamentequal temperamentequal temperament

Beware also: Barbour's table 155 for "Kleine Stadt" has a typo in the note E's superscript, which should be "-7/12" rather than "-1/12". His table 151 for "Dorf" has a typo in E-flat's superscript: should be "+1/12" rather than "0". And his table 156 for 1724 "Grosse Stadt" also has typos: the note G should have superscript "-1/6" instead of "-1/4" and its cent value should be 698, not 696!

Accordingly, the 1724 "Grosse Stadt" is similarly wrong in New Grove and at various places in my article and web supplements. The error in this one is in the note G: being placed 1/12 comma too low. (Correction posted here: November 5 2005) Like the "Dorf" problem described above, the correct version is in Norrback's and Devie's books, but has not made it into Lindley's New Grove article yet.

I thank Paul Poletti and Johan Norrback for catching the "Dorf" problem (August 2005), and Thomas Dent for the "Grosse Stadt" (November 2005).

Here is the corrected "Grosse Stadt" analysis:

Corrected chart for Neidhardt's 1724 'big city' temperament

The three 1724 temperaments by Neidhardt are also presented in facsimile and transcription at as of 31 December 2005. The 1732 facsimile and transcription were added soon after that (both sets prepared by Gordon Collins, I believe.) In October 2006 I added a set of my usual analysis tables, for Neidhardt's 21 temperaments of 1732.

See also my setup instructions by ear for some of these Neidhardt temperaments.

The corrected version of my PDF file "Major third comparisons" (Appendix 3) is here. The original version, among Oxford's four "Supplementary Data" files (the one about "Sorge matrices"), had the above-noted errors in the two Neidhardt 1724 temperaments "Village" and "Big City".

In my PDF file "Appendix: comparison of Bach's method with other temperaments" (Appendix 4) I remarked that ""Kirnberger 3" is also known variously as another of Sorge's 1744 temperaments (but in a Pythagorean comma version)." That assessment of "Pythagorean comma version" was based on Devie's mistaken analysis at his page 142. A closer look at Sorge's 1744 book reveals that he did intend it as syntonic comma, with a pure C-E, and he simply missed putting a schisma somewhere around the back (among the pure fifths). He says it's a clumsy (ungeschickte) temperament. He says immediately below this table that it sounds bad when playing in the scales of E, Ab, B, C#, or F# majors, or C, Eb, Bb, C#, F, or G# minors....but, it's still better than the old way [meantone].

Sorge, Anweisung zur Stimmung und Temperatur sowohl der Orgelwerke, als auch anderer Instrumente, sonderlich des Claviers 1744, page 27, available at the Munich Digital Library:

So, the temperament well-known today as "Kirnberger 3" was in print by Sorge 35 years before Kirnberger mentioned this temperament pattern without explanation or attribution, in his private letter to Forkel, 1779. Kirnberger's own 1779 formulation of it did not distribute the four tempered fifths C-G-D-A-E quite evenly, because he was using his typical approximations as ratios.

Nomenclature about hexachords

In part 2 (May), left column of page 220: "The older system of hexachords uses the set of six rising notes: ut, re, mi, fa, sol, la. Like a complete major scale, a major hexachord (natural hexachord, hexachordum naturale) can also be broken down into halves: as 'ut re mi' twice in succession. Similarly, a minor hexachord (soft hexachord, hexachordum molle) has 're mi fa' twice in succession."

Delete the words "(soft hexachord, hexachordum molle)" from that sentence. The technical point about that group of six notes is a valid one, about the layout of the intervals in it (the sequence of tones and semitones). However, the uses of the specialized terms "soft hexachord" and "hexachordum molle" were in error within that context. The hexachordum molle, historically, was the set of notes F-G-A-Bb-C-D, as contrasted against the hexachordum naturale of C-D-E-F-G-A and the hexachordum durum of G-A-B-C-D-E. All three of these classic hexachords are ut, re, mi, fa, sol, la. The ut, re, mi of one overlaps the fa, sol, la of another; and that is the classic principle of "mutation" (akin to later tonal modulation).

Re, mi, fa, re, mi, fa is simply one way of describing the intervals in a "minor hexachord", and that is the only point I was trying to make in that sentence.

The "wide 5th Bb-F" that many readers reject or misunderstand

This is not an error in the article or at! It becomes an error of comprehension, however, when readers misunderstand and misquote it, or try to "improve" the temperament by eliminating it.

A wide "5th" is apparently an anomaly to some readers who cannot believe that any "well temperament" (a 20th century English term, and ungrammatical, and based on theoretical expectations invented long after Bach's death!) could ever have even the most slightly wide 5th in it.

This problem has already generated a number of confused, angry, or "helpful" letters to "correct" this point on which I have been accused (some in public) as bewilderingly ignorant. The presence of a wide 5th evidently upsets some readers so much--at least in their conceptual understanding of the temperament--that they must try to "correct" it in practice, or in argument against the printed material...even though this particular interval is so nearly pure that it's not noticeable from even two meters away from a harpsichord!

The A#-F diminished 6th in this temperament is exactly that: a diminished 6th, not a 5th. (Every 12-note keyboard temperament must have an enharmonic diminished 6th in it somewhere!) The A#-F interval in this proposed Bach temperament is incidentally almost a pure Bb-F 5th, but it happens not to be one; and this is based directly on Bach's diagram as source. I have explained this point fully at the "Frequently Asked Questions" page 3, which see.

Sparschuh and Zapf

Several people have suggested that Andreas Sparschuh and Michael Zapf "should have been" credited much more prominently in the Oxford materials, and especially in the printed body of the article. I have explained this unfortunate omission in responsive letters to several internet discussion groups, in personal letters, in two interviews for radio broadcast and three others for print, and further here at

I continue to receive personal letters about this, and to observe other discussion about it on the internet (and some of that looks to me like people repeating one another's cynicism and hearsay, with rumors rolling). Through summer 2005 in the internet discussions, where several people lecture both me and Oxford University Press in public (and quite presumptuously!) on the topic of ethics, some of this feels to me like harassment/defamation against me and against OUP, given that both I and OUP have already responded to it to clear up misunderstandings.

Let me try once again to set the record straight on this matter, putting everything here for reference.

  • Dr Andreas Sparschuh, a mathematics professor in Germany, was (to my knowledge) the first person in modern times to suggest that the drawing at the top of Bach's autograph title page of the WTC means something about tuning. I fully acknowledged and credited this in my original manuscript submitted to Oxford, summer 2004.
  • Sparschuh continued to present his ideas in various academic lectures, not known to me. As I have learned in e-mail dialogue with him during 2005, his main direction of research in this area is the depiction of musical intervals by ratios, as several of the writers around Bach did. Sparschuh reportedly also ties it into a research component about measuring brain-waves electronically, as music is imagined silently while viewing a page of score. All this is quite interesting, and he and several others continue to develop it; but I personally believe that any practical connection with Bach's daily musical work is tenuous. According to the historical record, Bach was not a mathematician; and even if he were, I do not see any compelling reason why he would encode any preferred method as an arcane mathematical algorithm dealing with numbers and ratios. (That is, if Bach had worked along those lines of thought, he probably would not have chosen to represent it in a line drawing without any figures. And furthermore, it makes no sense to me that Bach as an expert practical musician would constrain himself to start at only a single pitch level--Sparschuh's 420 Hz as derived from Friedrich Suppig's 1722 manuscript Calculus musicus--and then hide that somehow into an unstated mathematical algorithm in the form of a drawing.) I believe that the Bach drawing's significance--and therefore a more reasonable interpretation of it--is much simpler and more direct than that.
  • Michael Zapf, early in 2001, met with Sparschuh and soon thereafter he posted onto several internet discussion groups his own suggested revision of Sparschuh's layout. He reorganized the beat-rates and the intervals giving a layout that sounds nothing like Sparschuh's, in practice. (Sparschuh's remains rather close to the familiar "Werckmeister III"; Zapf's features especially high sharps.) Apparently a number of people discussed Zapf's informal suggestion at that time, according to archives; but I was unaware of all this.
  • In March 2004 I received an e-mail question, privately, from David Hitchin. He had downloaded my temperament-analysis spreadsheet from my web site and input Zapf's layout into it, for my perusal. This was the first I had ever heard about Sparschuh, Zapf, or anything on Bach's border decoration being perhaps meaningful. Hitchin and I both copied our ensuing e-mail discussion on this to Michael Zapf, although Zapf never responded to me during those several weeks of dialogue.
  • The e-mail discussion with Hitchin sparked me to take a closer and more direct look at Bach's page, as reproduced in The New Bach Reader and subsequently in other facsimiles. And that closer look, myself, plus my research into the tuning literature and my background as a harpsichord tuner/player, is the basis of my work. The only thing in common here is that Sparschuh, Zapf, and I (among others) have looked at the same diagram drawn by Bach... and we have come to different theoretical conclusions about it.
  • I wrote the narrative of this discovery trail explicitly as part of my Oxford manuscript, and submitted it as a section entitled "How did this 2004 discovery happen?" crediting both Sparschuh and Zapf. I explained how this material came to my attention, and explained why I disagree with the premise of both Sparschuh and Zapf, that Bach's loops mean something about constant numbers of beats per second. Also, I included and credited the Zapf layout among the others in the Appendix charts, surveying the context of competing temperaments; and its fuller citation was that section in the body of the article.
  • The manuscript was copy-edited by Oxford's staff for publication, shortening the presentation and splitting the very long prose into two printed halves plus five web supplements. They also moved most of the mathematical material to the web supplements, rather than the printed pages, in keeping with normal style expectations for their broader readership (i.e. music scholars and enthusiasts, non-mathematicians). In this reorganization of the article, this section explaining the Sparschuh/Zapf/Hitchin trail unfortunately did not survive the cuts. [Outline of the published version]
  • Accordingly, I salvaged and recycled that section immediately (the same week as the February issue's publication) onto a page here at, "How did this discovery happen?" That was one of the first pages available in the initial release of this web site, February 2005, to accompany and clarify the article. The manuscript's Appendix was issued by Oxford as two of the supplementary files of part 1, also immediately in February: with credit to both Sparschuh and Zapf.
  • Several people, not bothering to look at those various web resources (or not knowing about them? or perhaps feeling that those mentions were inadequate? or perhaps simply being uncharitable against me?), immediately protested the February half of the printed article, insisting that Sparschuh and Zapf "should have" been credited more fully in the printed pages of the journal. I have continued to receive personal notes and letters to this same effect, through the spring and summer of 2005, pressing this same concern with varying degrees of vehemence.
  • Sparschuh's algorithm from the drawing was published 1999 as a single-page abstract for a conference lecture. This page of abstract was not available to me during the writing of my article, in several searches and interlibrary loan requests during 2004; therefore I could not go into any detail about it. The abstract is "Stimm-Arithmetik des wohltemperierten Klaviers", in Deutsche Mathematiker Vereinigung Jahrestagung 1999, Mainz: page 154-155. Dr Sparschuh kindly made this available to me in June 2005, after both halves of my Early Music article were already in print, so that there should not be any public confusion between his theory and mine. He posted it in full onto his own web site 25/Jun/2005; and with his permission I posted his scanned copy of the same here at, reviewing it as part of my survey of other "Bach" temperaments.
  • My standard mathematical summary of Sparschuh's resulting layout is here. As should be obvious by direct comparison with my proposed layout, we have taken the Bach drawing entirely differently.
  • Is Andreas Sparschuh's one-page abstract the first printed source to make something of the Bach diagram? Yes, of course...unless he himself has some earlier one he has not mentioned yet!
  • And Michael Zapf has not attempted to publish anything formally on his own layout; he presented it simply for internet discussion and for anyone who would utilize its musical resources, and to spark any discussion about Sparschuh's work. Whether by design or accident, a layout very close to Zapf's is suggested by Mobbs and MacKenzie as published in the August 2005 issue of Early Music. (Compare my standard charts analyzing both: [Zapf] [Mobbs/MacKenzie]) Furthermore, in July 2005, Thomas Dent showed me his own suggested formulation of a "theoretical Zapf" which also gives similar musical effects, pointing out that it somewhat resembles my proposed Bach temperament except for a rotation of the whole temperament by the interval of a 5th. Clearly, there is some ongoing interest in the way that particular harmonic shape interacts with Bach's music.
  • On 25 June 2005 Michael Zapf posted an "open letter" to several public internet newsgroups, including "Yahoo! Tuning" and HPSCHD-L, among some other groups that do not have public archives. Much of it was an ad hominem attack against my character, which drew (among others) a defense by another Tuning-list member (Aaron Krister Johnson), and my own open public letter of response, on HPSCHD-L where I answered Zapf's allegations. It was a pre-emptive strike by him against my rights to do further work in this field! Zapf issued this public apology in "Yahoo! Tuning" on the 28th.
  • Another person, in particular, has launched what I feel has been a more deplorable smear campaign against me, a personal and sustained attack to try to damage my professional reputation. His name is Charles Francis. (And, a posting of his here, October 3rd 2005 expresses his general attitude about disgruntled consumers properly using the internet to damage people's professional reputations.) His personal campaign against me and this particular article started within the first three days that part 1 was available at Oxford, in February 2005: before the print edition was available. Mr Francis presented a "review" of my work that was little but red-herring accusations against me, distributing it in several internet discussion groups, alleging that OUP and I had done something unethical. Charles Francis's cynicism, and his refusal to listen to my explanations subsequently, has set at least some of the acrimonious tone for subsequent discussion of this issue on the internet. I have explained myself, answered his allegations, and yet his apparent grudge continues. He is still at this same campaign to make me look bad, as of mid-October in various forums both in English and German; he simply takes it to new audiences each time. This all looks to me like an attempt by him to bias any unwary readers against my material and my ethics before they have a chance to read my work, or to read my public statements in my defense in this controversy.
  • More recently, Kenneth Mobbs has issued several public letters on his web site, reproducing his own Early Music letter for the August issue and then amplifying it on August 28th, immediately after hearing my BBC broadcast, and writing in obvious disgust/condemnation of things he believed I "should" have said in that program! I do not know the extent (if any) to which Mobbs has been influenced by the allegations by Francis and/or Zapf. My perception has been that this is just another person offering bitterly gratuitous advice and condemnation, but without having all the facts or listening to my explanations.
  • Charles Francis, likewise, took several occasions during the week of the BBC broadcast (it was available on the internet for seven days) to make his similar allegations about the things he believed I "should" have said in that program: again spreading his opinion to several internet groups. (And once again, I perceived this as his personal attempt to bias people against me before they could listen to the program!) In response to this round of accusations from him, I pointed out that I did explain the Sparschuh/Zapf connections more than adequately in my hour-long interview that was recorded in July for that BBC program. I spent more than five minutes talking about it, clarifying all these controversial points, and giving due credit to those equal-beating theories by Sparschuh and Zapf. (I explained the same material that is available here.) The production staff, editing that recorded interview for broadcast, evidently decided not to use that portion of it: as they mixed one hour of our conversation and more than two hours of available musical examples down to a half-hour segment. These accusations by Mobbs and Francis, presenting their opinions as to what I "should" have said on public radio, reflect their own cynicisms and expectations. Should the production decisions for a public radio broadcast be determined only by the satisfaction of several listeners who had already decided the featured guest is wrong?
  • During 2005 I have continued to clarify the Sparschuh and Zapf credits here at, in its additional sub-pages. My intention, always, is respectful and fair treatment of their material, and meaningful musical/mathematical comparison of the results of their keyboard layouts. My most extensive page about their work is the survey of other "Bach" temperaments; additional occurrences can be located using the "Search this site" box on my introduction page.
  • Likewise I believe that Oxford University Press's personnel have done nothing deliberately to downplay any Sparschuh/Zapf connection here. This was all just an unfortunate misunderstanding, and has been pressed to a fervor by several who apparently feel their own expectations about printed material must be satisfied (as to what format counts as "publication", etc...). The editor of Early Music has issued a printed apology on behalf of OUP and me, on page 547 in the August 2005 issue: in the extensive section of Correspondence about the article.

Additional details of many analytical points

There is available a continuing digest of postings to HPSCHD-L clarifying research details and answering questions.

"Hidden letters" in Bach's title-page drawing

November 15, 2006

If anyone is going to search for handwritten letters in Bach's drawing (as I suggested in my Early Music May 2005 portion, page 222; as O'Donnell has done, page 627ff in November 2006; as Lindley/Ortgies have remarked on pages 614, 622, and in their endnote #33, November 2006) is helpful to have at hand a picture of Johann Sebastian Bach's handwritten letters as he used them in organ tablature. These are examples of his letters that were definitely intended to be read as musical notes, within that shorthand method of tablature notation.

The following illustration is drawn from the recently-discovered (2006) tablature copy that Bach made of a composition by Reincken. This was done in 1700 while he was studying with Georg Böhm, according to Bach's notation on the manuscript. It is now the earliest known music manuscript by JS Bach.

The exhibition of this manuscript opened September 2006 at the Bach-Archiv Leipzig, and small photographs are given in the commemorative book published with that exhibit, Expedition Bach (see the "Shop" section there). From that book this is an enlargement of illustration 14, page 10, extracting the letters from the Reincken/Bach manuscript:

"Johann Sebastian Bachs Tabulaturalphabet (hier der Reincken-Abschrift entnommen)."

click to display horizontally
click to display horizontally

This is only an example, of course, and we must also allow for changes in his handwriting between 1700 and 1722/3....

February 25, 2007

Additional discussion of Bach's keyboard tablature is in Robert Lewis Marshall's book The Compositional Process of J. S. Bach: A Study of the Autograph Scores of the Vocal Works, volume 2 (Princeton University Press, 1972). See especially pages 6-7 for a table of his symbols, plus the sketch transcriptions from BWV 2, 26, 27, 30a, 39, 51, 57, 65, 72, 81, 82, 88, 91, 110, 116, 133, 134, 135, 138, 144, 151, 169, 170, 174, 180, 198, 201, 205, 206, 243a, and 1053. In all of these instances Bach sketched ideas in keyboard tablature (for vocal lines, or vocal-instrumental score!), before preparing his final versions in score; not merely the more familiar examples for organ/keyboard music (Orgelbüchlein et al.). He was still using tablature for himself, at least as late as the 1730s: in the Prelude, Fugue, and Allegro for lute or keyboard, BWV 998 (from 1735 or later), the last part of the Allegro is written that way.

Wir Christenleut (BWV 612) from Orgelbüchlein, where the tablature saves space :
BWV 612 showing the tablature completion

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