Temperament / Key Character / Tuning
Discussions - Part 6

Continue from Part 5

Tunings and Bach

Ludwig wrote (June 18, 2004):
Yes Bach had certain tunings in mind when he wrote music--that is if he had anything to do with it. He is attributed to being the inventor of the well tempered scale.. the tunings which Bach seemed to be in favor was in the neighborhood of A=440 based on the Organs which he was a jurist and consultant in their building.

Some of the Organs Bach played (and had no say so in their construction etc) were not so well tempered and in certain reconstructed HIP instruments one can easily hear the Wolf at the door if certain music is played. In such instuments; it was not uncommon to find certain notes of the chomatic series missing. Aside from Organs; one often finds this is surviving Carillons from the period. As a Carillonneur I find
this building practice despicable because when playing each different instrument; one has to try to find out what how the bells are tuned and one has to transpose at sight and just hope that the music comes our correctly as one is often caught between the devil and the deep blue sea---running out of batons/keys to play. This also limits the use of Carillon in combinations with other instruments.

Nowhere is this more evident than in the French Baroque School of such composers as Lully, Couperin etc. Their music played in the tunings that they used sounds very different from the same music played in the well tempered scale. For me in a HIP scale (for such composers other than Bach); it has a more agreeable tenure but that is simply a matter of taste as others might not so agree.

My personal preference given the differences between well tempered and no so well tempered scales is that of Bach---as it seems to be the diplomat of all scales and tunings.

Bradley Lehman wrote (June 18, 2004):
Ludwig wrote:
< Yes Bach had certain tunings in mind when he wrote music--that is if he had anything to do with it. He is attributed to being the inventor of the well tempered scale.. the tunings which Bach seemed to be in favor was in the neighborhood of A=440 based on the Organs which he was a jurist and consultant in their building. (...)
My personal preference given the differences between well tempered and no so well tempered scales is that of Bach---as it seems to be the diplomat of all scales and tunings. >
Ludwig, you very likely haven't heard Bach's "well tempered scale" yet. It was only discovered in April 2004 (by me), and it's only been used in two public performances so far: a broadcast of fortepiano music on Swiss radio and a live performance of the St John Passion in the USA. As for private performances, I've played through almost all of Bach's harpsichord music, and the violin sonatas, and about half the organ music in it here in my house; and a few others assisting me in this research project have done likewise.

But, please, hold your assertions about its characteristics until you've heard it, and until you've read my paper (to be published later this year). Its sound does NOT resemble equal temperament, most of the time; and it sounds even less like Werckmeister's or Kellner's.

The remarks above about "A=440" and the presumption of modern equal temperament are both superficial and mistaken; the history books have been wrong, and those legends about Bach supposedly inventing equal temperament have been copied and recopied ad infinitum while still not being true.

John Pike wrote (June 18, 2004):
[To Bradley Lehman] Does this mean you have a definite journal publishing it, Brad? Please tell us as soon as you are able about publication and, if possible, post us a copy of the article at the time of publication. I get more intrigued every day.

Bradley Lehman wrote (June 18, 2004):
[To John Pike] I'm trying to finish the proofreading of all my footnotes this weekend, and send it off. The basic paper's been done already for about a month, other than the small fixes and restructuring I've been doing (from the private suggestions and questions of several informal referees who have been reading my drafts: professional musicians, instrument builders, and university professors...peers qualified to assess portions of it).

There might be an appendix or two explaining the basic and normal techniques of tuning keyboard instruments (necessary background to understand where all this comes from), and the comparison with about 25 other currently used temperaments for Bach's music...if there's space in the journal. Bach's temperament arises absolutely from normal 17th century practices, not from walking out of a cornfield, while it's also an exquisitely different blend of several other methods.

I'll of course post a link to the copyrighted results when publication happens. Obviously I can't say more about those details until the paper is formally accepted for publication.

John Reese wrote (June 18, 2004):
[To Bradley Lehman] Does anyone know of a software utility that can filter MIDI data so that it produces different tunings?

The reason I ask is that it sounds like it would make for a good programming project (although I would have to find a way to account for MIDI's treatment of enharmonic spellings).

F# and Gb being different

Bradley Lehman wrote (June 22, 2004):
Juozas Rimas wrote:
<< I wonder how that "piquancy" sounds. It could be I have heard it already in a recording but I can't say - perhaps I didn't spot it, without knowing that it was unequal temperament and that I should have excepted the piquancy. >>
Ivea Mark wrote:
< Apparently, and of course, I don't really know much about this, but I have heard that in unequal tuning, the F# and Gb is supposed to be different in sound (is that technically? 'Cos they're the same physical key, how do you get two sounds?).
Can someone elaborate? >
F# and Gb, in Bach's time, were approximately 22 cents apart: 1/5 or 1/4 of a semitone (there being two different sizes of semitones) in common usage among musicians. In the century before that, the discrepancy was even wider than that...F# being even more than 22 cents flat from Gb.

That's true of all the enharmonic pairs of notes that (today) are crammed into a single key on the piano keyboard. That's why some of the 16th/17th/18th century keyboards were built with split keys, so the player may select either one as appropriate. Or, lacking that, musicians and theorists had to come up with tuning compromises so the sound isn't violently awful when the wrong one is played on keyboards that don't have both. Basically it involves putting the sharp/flat notes of the keyboard somewhere between the points where they really should be, according to acoustical properties, such that they are useful (but differently useful) either way. The tasteful ways to do that are legion, and everybody had an opinion, and the field is still a minefield in that regard.

That's the basic historical problem of temperament, in a nutshell. For a good historical overview, I recommend Mark Lindley's article "Temperaments" in New Grove (2001) as the place to start understanding the complexities; and its bibliography as well.

As of last night, my paper about Bach's tuning method is submitted: it will be in the hands of the peer reviewers shortly, and (I hope) published later this year yet. And then, New Grove and quite a few other reference books will eventually need revisions, in light of my findings. An exciting project...while I'm now also relieved to have turned it in so I can catch back up with other things in my life that have been on hold! That's after I get all the library books turned back in, and all the articles and drafts filed appropriately.

If I stop answering tuning questions here for a while, it's because (1) I have to be careful about revealing too much pre-publication information, and (2) I simply need a rest from it...like recuperation after giving birth, or something (according to my wife's report of that experience).

To answer Juozas' question above: MOST harpsichord recordings of Bach from the past 20 years or so are in various unequal temperaments, with various levof piquancy or subtlety...listen to as many as you can find! Ditto for many (but I wouldn't say "most") organ recordings of the past 20 years. But none of those on either instrument are tuned Bach's way, yet: for the simple reason that his way hadn't been discovered yet until April 2004.

Fumitaka Sato wrote (June 23, 2004):
In history there have been many professionals who thought that a new and right temperament had been found out or created by themselves. And in my case, I found out the Sacred Temperament. In this Sacred Temperament all the things will be forgiven. Any will not be discriminated out. Of course the coming new temperament is included in the Sacred Temperament.

Bradley Lehman wrote (June 23, 2004):
[To Fumitaka Sato] Again, that's true. Herbert Kellner comes especially to mind: in the 1970s he invented a temperament that has some pretty good musical qualities, but no direct historical tie to Bach (and is indeed contradicted by historical information). Relentlessly Kellner spent the next 30 years explicating his invention through numerology and semantics, and the marketing of truisms about 1/5 comma temperaments in general: all to make it appear that his invention had more historical validity for Bach (as a secret reading of Werckmeister!) than it really did. The musical results have convinced a lot of people, as his temperament sounds especially good for 17^th century music; but, that's not proof that Bach used it or even anything similar. In Kellner's published articles, he said explicitly that he got it from an assumption that Werckmeister published less than his own best, reserving this similar solution as a secret offshoot of it. That's a forcing of the evidence by Herbert Kellner, reading his own goals and expectations into the historical record, along with an assumption that Werckmeister was dishonest! Not an especially good scholarly thing to do. The bylines of his articles also explain that his own (Kellner's) training was in engineering, not music or music history; it shows up additionally in the thinness of his musical/historical arguments as he presented them. From a modern scientific standpoint, he's invented something nice and useful; and he spent his remaining years spiritualizing it, in print. But his ties of it back to Bach were too tenuous, for the simple reason that Bach's tuning is entirely different from his. Bach's way has only two intervals in common, anywhere, with Kellner's!

This Sacred Temperament of which you speak: what are its musical details, and does it sound in any way different from equal temperament in practice? Or, is it really a repackaging of equal temperament itself...or, maybe even something removed from mere sound altogether?

Bach's way sounds as "sacred" to me, now, as your method undoubtedly sounds to you. And, Bach's deployment of his as an expressive device in his sacred music is (I believe) both overwhelming and rich; he used it to make theological points directly in the sound of the music, through the subtle inequalities. The evidence here is the music itself, the way it sounds tuned his way, according to his explicitly handwritten instructions; the sound of it affected the way he composed his pieces, the way he deployed specific sounds to make expressive points. That, to me, is sacred: Bach using the sound itself to express his soul and edify his listeners.

Charles Francis wrote (June 24, 2004):
[To Bradley Lehman] Your real quarrel appears to be with the poor quality of the peer-review process in musicology: Herr Dr. Kellner was, after all, the author of 52 publications on the tuning system of J. S. Bach! Given the alleged thinness of Kellner's musical/historical arguments, it would seem to speak volumes about the peer-review process in this "soft" science. If you are to be believed, good marketing (smoke and mirrors, as you once put it) is all that is needed to be become widely respected.

For myself, I will avoid passing judgement on Kellner. Having corresponded with him both publicly and privately, I was saddened to learn of his passing. His last posting to the tuning group, was to express dismay at the 9/11 attacks, after that silence.

Bradley Lehman wrote (June 24, 2004):
Charles Francis wrote:
< Your real quarrel appears to be with the poor quality of the peer-review process in musicology: Herr Dr. Kellner was, after all, the author of 52 publications on the tuning system of J. S. Bach! Given the alleged thinness of Kellner's musical/historical arguments, it would seem to speak volumes about the peer-review process in this "soft" science. If you are to be believed, good marketing (smoke and mirrors, as you once put it) is all that is needed to be become widely respected. >
Translation: all academia is bogus, Kellner and Isacoff were as smart as Lindley and Barbour, Lehman's got nothing, pass the beer nuts.

Anyone who thinks this is a "soft" science is advised to read Lindley's book Lutes, Viols, and Temperaments, and Rita Steblin's dissertation (2^nd edition), and Easley Blackwood's Structure of Recognizable Diatonic Tunings.

And as for becoming widely respected, there's a difference between impressing people who have no technical background to make value judgments about the material, and impressing people who really know the musical and historical problems of tuning. "Wide respect" in terms of numbers of papers, and wowing the crowd, really means nothing in a truly scientific field as this is; accuracy to the material does, and that most of all is listening to the sound.

< For myself, I will avoid passing judgement on Kellner. Having corresponded with him both publicly and privately, I was saddened to learn of his passing. His last posting to the tuning group, was to express dismay at the 9/11 attacks, after that silence. >
I was saddened to learn of his departure, too, told to me by a longtime colleague and friend of his who has been an excellent sounding board for drafts of my paper. I don't pass judgment on Kellner either, personally, but his work was smoke and mirrors to promote a good system of his own invention (with heavy reliance on Werckmeister) that just happened to have nothing to do with Bach. Somebody can say the same thing 52 times and still be wrong, because of ignoring inconvenient evidence; that's what happened here with Herr Dr K. He found several mathematical truisms and coincidences in his temperament and promoted them about 7000% beyond their real significance (making them appear mystical), eventually falling back on numerology to keep the ball rolling: numerology where anything can be "proven" through enough coincidence. That's not a science. I read about 6 or 7 of Kellner's papers (variously in German, French, and English) over the past month and found them amusing, but probably not for a reason he intended. They're at least good role models of how not to write a convincing paper: about 2% content (the temperament he made up himself) and 98% fluff....

Which temperament to use?

Fredrik Sandström wrote (July 16, 2004):
As a complete amateur I'd like to ask a simple and practical question regarding temperaments:

On my digital piano (that also has a remarkably pleasant harpsichord sound for an electronic instrument, I think) it is possible to use Equal temperament, Pythagorean, Mean tone, Werckmeister and Kirnberger. Which one of these would you suggest that I use for playing Bach's music? (It also offers pure major and pure minor tunings, but these are of course not very usable...)

Bradley Lehman wrote (July 16, 2004):
[To Fredrik Sandström] From my experience playing Bach's music in all of those, the Kirnberger is the one that sounds best (assuming it's the 1/4 comma variety ascribed rather sloppily to him, and not the 1/2 comma one he published). But, that is a very limited set of choices there and not all that attractive to begin with. Quite a few people will prefer equal temperament over the Kirnberger; others might prefer the Werckmeister. All of those choices run into some serious problems of one kind or another.

That "Kirnberger" solution is not thought of very highly in some scholarly circles, due to some different problemwith his work as a theorist; but musically it brings some decent effects to the repertoire. And I suspect that--because it's so extraordinarily easy to set up by ear, by beginners--that same solution or very similar may have been in use already 100 years or more, informally, by musicians before Kirnberger. It is a very easy, and elegant, layout that sounds decent.

To do it from scratch, by ear, simply tune C-E pure and then fudge the other fifths C-G-D-A-E within it so they're all equally tempered (as compared with one another) or close enough not to bother you; and then finish up by tuning pure fifths C-F-Bb-Eb-Ab-Db and from E-B-F#. The schisma discrepancy shows up in F#-C# but is so slight that the impurity of that fifth is almost negligible. (Technically, it's not a fifth; F#-Db is a diminished sixth here at this place where the circle is forced to close, or almost to close.)

Hope that helps,

Chorton / Cammerton & Bach's Tuning

Charles Francis wrote (February 1, 2005):
I noticed the recent discussion on Chorton / Cammerton (Dale Gedcke, Thomas Braatz et al.) in relation to BWV 51, and thought some of you might be interested in an article of mine that touches on this topic. It's an abridged version of a longer document that I shortened for technical reasons. But, even so, the PDF-file is still
over 2 Megabytes in size and will take a while to download. The title is "The Esoteric Keyboard Temperaments of J. S. Bach" and the document can be found in the files section of the group: http://launch.groups.yahoo.com/group/BachCantatas/files/

Bradley Lehman wrote (February 1, 2005):
[To Charles Francis]
Quite entertaining.

Some salient remarks and questions. (Well, I'll keep it to twelve, in honor of twelve semitones on a keyboard) :

1. Dr Rudolf Rasch has debunked John Barnes's method as presented in that article, already some 20 years ago. The premise that Bach would pick key signatures with a statistically valid correlation to interval (im)purity is the most flawed part of it; it's only ameliorated slightly by enlarging the data set as seen here (in the brute-force search through key signatures in the BWV). That is, the Barnes premise itself doesn't really do much to convince people who weren't predisposed to take it as a foregone conclusion.

2. Information about the "Texas Sharpshooter Fallacy" is available here, and elsewhere:
http://www.fallacyfiles.org/texsharp.html
http://skepdic.com/texas.html
The Texas-sharpshooting method is of course made a whole lot easier by scooping various unacknowledged postings of mine to BachRecordings and BachCantatas, and then drawing the target around the results (in the permutations discussed in this article) that I presumably would find most appealing. For example, I recall a conversation with Dale Gedcke and others here where I made some remarks about the sound of Bach's D-major tonic. Sure enough, the coincidence here in this article seems rather strong. So is the coincidence that this is released on the first day of the month when the author had heard that my research is being published. But, let us move along to more interesting questions and remarks.

3. The fundamental flaw in the Sparschuhe-Zapf methodology is the premise that the loops in that drawing have anything to do with seconds of time, whatsoever.

4. The thoroughness in trying all the permutations is admirable, celebrating the author's virtuosity with computers and other electronic equipment. Nevertheless, it is a truism that all temperaments sound pretty good (and pretty close to equal temperament) if all the beat rates in the fifths are kept slow, like the rates that are common to all the temperaments investigated here. So what? Where's the proof that any one in particular, among the scattershot possibilities presented here, would have any pride of place? I see there are some "Euclidean distance to other temperaments" proposing to offer such a correlation, but it is not adequately explained. How would a man like Bach come to decide on any one of these possibilities over any of the others; what musical reasons might motivate him? What features of this particular "glyph" might swing such a balance? These things are not explained, and it doesn't appear to matter to the author that they are missing.

5. Pendulums were in use several thousands of years ago. See, for example, the discussion in the book Uriel's Machine by Knight and Lomas, where it is demonstrated that pendulum length was a unit of measurement that united time and distance, way back among megalithic-age astronomers. The claim here that Christiaan Huygens's 1656 findings validate the Sparschuhe-Zapf premise is tenuous at best.

6. To see which keys people in the 18^th century really expected to sound the most harmonious, why not consult and cite standard reference works such as Owen Jorgensen's gigantic red tome Tuning and Rita Steblin's dissertation, and the many standard articles by Mark Lindley? That would seem to me useful to improve Tables 35ff, markedly. Likewise, the definition of the new term "goodness" in footnote #2 strikes me as rather arbitrary, and in need of explication.

7. Whatever happened to the esoteric temperaments allegedly derived from BWV 924, and presented in June? http://www.bach-cantatas.com/Articles/Keyboard-Temperament[Francis].htm
That temperament (or should I say "set of four temperaments" to amuse any fans of Carl Nielsen), too, would seem to deserve some attention here: at least to explain why its own author has abandoned it in favor of this new presentation. It gives me an impression once again as an example of Texas sharpshooting.

8. Have any classically trained musicians, and preferably specialists in historically-informed performance practices, tried any of these dozens of temperaments in practice to inform the musical decisions offered in the present paper? If not, why not? Or if so, why aren't they credited?

9. Why would Bach have any interest in the material presented as Tables 4 to 25, or any stake in that manner of calculation by long fractions? By his own son's admission in the historical record, Bach disliked "dry, mathematical stuff".

10. As the present paper is so long on raw undigested data, and so parsimonious on explication, it's not clear (at least to me) what it is expected that historically-informed musicians should do with these results. Are we supposed to plug these 144 different temperaments into electronic tuning devices, or what?

11. The paper doesn't really present much proof of anything, does it? It just looks to me pretty much like a warmed-over version of alt.music.j-s-bach discussions seen here from January 2001: http://tinyurl.com/3uks4 In short, the present paper gives me the distinct impression of other people's work (most notably Sparschuhe's and Zapf's), but dressed up to look impressive and new with a bunch of colors and charts. I didn't happen upon that 2001 discussion thread myself in any archives until July or August 2004, at which time I read it with amusement at the author's remarks about children scribbling. He dismissed this particular drawing (which he's now calling a "glyph") of Bach's as non-evidence, at: http://tinyurl.com/6vxlr . I'm wondering what has compelled the author to change his mind about its relevance.

12. Has Dr Sparschuhe himself been contacted about this enthusiastic further running with his work? Or, for that matter, Michael Zapf from whom that discussion thread mentioned in my point #11 originated? It would seem that they should be very excited about this.

Well, one more practical question to make it a baker's dozen:

13. When tuning or doing other maintenance on a harpsichord that has flat-topped and unthreaded (i.e. smooth) pins, when one taps a loose tuning pin more firmly into the instrument's pinbwith the top of the tuning hammer, does the string's pitch go up or down?

Charles Francis wrote (February 1, 2005):
[To Bradley Lehman] My belated response to: http://launch.groups.yahoo.com/group/BachCantatas/message/12311

Im glad you found my article entertaining - I enjoyed yours as well and my review will follow shortly.

1. On Barnes, I have read his paper and concluded it doesn't deserve to be referenced, given his obvious contrivance to fit the historical record. Early Music should perhaps confine itself to writing entertaining pieces for musicians, rather than dabbling in
technical matters?

2. Was this ad hominem really needed?

3. See my review of your article.

4. The method in the paper is based on a mathematical analysis of the glyph and does not rely at all on the historical record or musical decisions. The historical record (e.g. Haynes book on pitch) and musical factors (e.g. correlation with historic temperaments, Bach's key usage) are only used to verify the predictions. The mathematics yields two unique Cammerton-Cornet-ton solutions corresponding to reading the glyph in both directions. Both solutions accord with the historical record. We both agree that Bach marked a specific point on the glyph indicating a particular solution for the Well Tempered Clavier.

5. The only point here is to establish that seconds were a known quantity to Bach and for the purposes of tuning sufficiently close to today's unit. Is this point in dispute?

6. There is no question of improving any tables by considering musical factors. The results derive exclusively and uniquely from mathematical analysis without any historical or musical decisions. Get a pen and paper and work out the fractions for yourself if you don't believe this!

7. You appear hung up on the idea that Bach confined himself to one temperament. Anything in the historical record to support this?

8. There are no musical decisions informing the present article - the results derive exclusively from mathematical analysis of the glyph.

9. Bach had no need to reverse-engineer the temperament he used, as he knew the reference pitch of his day. The method in my paper avoids any dependency on the historical record, however, so this information must be derived. See my review of your article for more specific comments.

10. Providing raw data to allow results to be reproduced is the norm in most scientific papers. I believe that in any field, the reader should take the trouble to understand the facts and be allowed to draw their own conclusions. Evangelism and hype have no place in the scientific process, and nothing more than fractions are needed to reproduce the fundamental results here. Bach did mark a specific solution for the Well Tempered Clavier, and we both agree on the markings. That solution corresponds to one of the Cammerton-Cornet-ton pairs derived mathematically in the paper. The tuning method is actually very simple.

11. The method of Sparschuh yields a tuning solution starting on `A', and I have analytic solutions for alternative tuningmethods which include a solution starting from A. Zapf's method,described in his spreadsheet on the Clavichord group, is based on an assumption of seconds-per-beat, not beats-per-second. I also have analytic derivations of Zapf's results. The suggestion that these the glyph was the scribbles of Bach's children came from my wife. Female intuition is not incompatible with scientific enquiry, after all. And, no I haven't "changedmy mind" about the glyph as you incorrectly suggest. Indeed, I pointed you to these finding concerning Bach glyph in April last year: http://launch.groups.yahoo.com/group/BachCantatas/message/7956

12. Not yet, although a copy of my paper has been placed on Zapf's Clavichord group for some time (I note you are a long-standing member, there). I will soon inform both Sparschuh and Zapf of my findings and point them to your paper, of course.

13. Once again, my solutions are derived exclusively from a mathematical analysis of Bach's glyph.

From Charles Francis (February 12, 2005):
New file uploaded to BachCantatas

This email message is a notification to let you know that a file has been uploaded to the Files area of the BachCantatas group.

File : /ReviewLehman.pdf
Uploaded by : bachjohann_sebastian <Francis@datacomm.ch>
Description : Review of Bach tuning article in Early Music

You can access this file at the URL: http://groups.yahoo.com/group/BachCantatas/files/ReviewLehman.pdf

To learn more about file sharing for your group, please visit: http://help.yahoo.com/help/us/groups/files

My article about Bach's tuning

Bradley Lehman wrote (February 13, 2005):
< Uploaded by: Charles Francis
Description : Review of Bach tuning article in Early Music
You can access this file at the URL:
http://groups.yahoo.com/group/BachCantatas/files/ReviewLehman.pdf >

As of yesterday morning, the first half of my article "Bach's Extraordinary Temperament: Our Rosetta Stone" is now available in the February issue of Early Music (Oxford University Press), and for download as PDF from their web site: http://em.oupjournals.org/current.dtl

The article describes JS Bach's specific keyboard tuning, its basic sound, and the historical background (in both the 18^th and 20^th centuries).

The second half (May 2005 issue) gets more deeply into the musical and mathematical analysis, showing how the method that Bach wrote down interacts with his music, and showing how it solves the practical "problems" that other temperaments have created.

Additionally, there are 50 pages of supplementary mathematical material to accompany the article, scheduled for publication on Oxford's web site (and promised in the footnotes); not out there yet as of today. Those supplements present analytical comparison of more than forty documented temperaments that are either contemporary with Bach's or are modern published speculations as to his methods.

My own new supplementary web site at http://www.larips.com , separately, presents various additional introductions to this research project. The practical issues here have occupied me for more than the past 15 years as a professional harpsichordist and organist, tuning these instruments for my own and other people's gigs.

Enjoy!

=====

As for a response to the review and allegations offered by "bachjohann_sebastian" (Charles Francis), I will post something at: http://www-personal.umich.edu/~bpl/freview.html when I finish drafting an appropriate response. Obviously Mr Francis was very eager to review the first half of the article within its first two days of publication, and get the word out about its existence before my own announcement of it here.

Dr Bradley Lehman, A.Mus.D. (harpsichord), University of Michigan, 1994

John Reese wrote (February 13, 2005):
I haven't had a chance to fully analyze Charles' critique, nor even Brad's original paper, so I can only offer an observation at this point.

People are always talking about the mathematical quality of music, but in actuality a purely mathematical analysis of music leads to some counter-intuitive results. Trying to explain the rules of tonal harmony, for instance, in mathematical terms, might come up with confusing and disapointing results for someone expecting to find the sort of mathematical symmetry that many non-musicians believe is there. This is because music is a balance of science and art, and, just as the ancients had to include imperfections in the great stone columns in temples, so they would look "right" to the eye, composers and other musicians make compromises betweenthe mathematical ideal and the expectations of the ear.

To achieve mathematical symmetry in music would mean not only adopting even-temperament, but throwing out the diatonic scale and using either 12-tone, whole-tone, or octatonic scales. Any mathematical study of traditional tonal music would have to use the more general approach of discrete mathematics -- set theory, graphs and dependencies, etc.

This is why I find the graphs showing lack of symmetry unconvincing, at first glance. Although the art of tuning is somewhat more precise than the art of composition, the same principles of theory vs. practice would seem to apply.

Without more information, that's all I have to say...

Bach/Lehman tuning [was: Bach's Birthday]

Ed Myskowski wrote (April 1, 2011):
Ed Myskowski wrote: (Mar 22, 2011)
< (1) Brian McCreath interviewed Richard Egarr on the radio Bach Hour, with extensive reference to (and endorsement of) Bach/Lehman tuning. Available for a week, I believe, at: http://www.99.5allclassical.org
and with a side reference to Brads site: http://www.larips.com
Not much new to astute BCML readers, but nice to confirm that we are on (either side of) the >cutting edge.
For a sample of that other blade of the scissors: http://temper.braybaroque.ie/lehman.htm
[end Mar. 22 post] >
For a bit more on Richard Egarr, see: http://www.bach-cantatas.com/NVD/BWV988-Gen8.htm

with this quote from notes to::

Richard Egarr, harpsichord, Goldberg Variations plus 14 Canons, Harmonia Mundi, 2006

<A timely coincidence, musicological discovery allowed me to use what may be Bach's own tuning system for the recording. There have been many claims and counter-claims since Bachs death as to his own preferred method of tuning. Most of them have little actual connection to Bach himself. Incredibly to me, there is still even a camp in favor of equal temperament. [see BCW archives, for example] In a brilliant and refreshingly musical piece of thinking and musicology, Bradley Lehman seems to have discovered the Truth, or at least a Truth. ... I am sure that Lehmans idea will not receive universal acceptance, but I find it utterly convincing.>

Consider also this humorous comment by Egarr, re the tuning system, on radio KPCK:

<KPCK: <You obviously endorse it [Bach/Lehman tuning]. Is it universally accepted?>

Egarr: <Of course not! Its more like a religion. You believe it or you dont. There is plenty of evidence in Early Music, and you can read it any way you want.> (end quote)

I usually make reference to Bach/Lehman tuning when it is relevant to recordings or other topics of current discussion. One motive is to provide some balance to the BCW archives. When I first joined BCML in early 2006, there were mainly negative posts, full of faulty logic and dubious *facts*, discrediting Brads contribution. A quick example of the worst, written by the same person in the same (multiple) posts:

(1) It is absurd to treat the squiggle on the WTC title page as code.
(2) But if it is code, I thought of it first.

In fact Brad has accurately credited the contribution of others in suggesting that the squiggle might be coded information; his claim to originality lies in cracking the code.

I expect that the topic of unequal temperaments is outside the interests of many (perhaps most) BCML readers, and it is certainly a topic on which I have no specific expertise nor prior experience. Nevertheless, I enjoy the intellectual challenge of grappling with the concepts, and the ease of listening to the practical applications. I am happy to add thoughts and encouragement to any ongoing discussions, and I will continue to mention Bach/Lehman tuning when appropriate, with the understanding that it is not universally accepted (as Richard Egarr acknowledges) as Bachs personal choice among a number of competing unequal temperaments, or the emerging (now dominant) equal temperament. I do find the quality of the musicians who have adopted Bach/Lehman tuning to be impressive.

There are many related posts available in the BCW archives under the General Topic of Tuning, for the truly dedicated reader.

I hope the thread is clear, apologies for the transmission error for the earlier post.

George Bromley wrote (April 1, 2011):
[To Ed Myskowski] For the benefit of us lesser mortals, what is Bach Lehman tuning?

Evan Cortens wrote (April 1, 2011):
[To George Bromely] The Bach/Lehman temperament is a method of tuning keyboard instruments derived by Brad Lehman, a list member, from the spiral on the title page for the well-tempered clavier. He described it first in the journal Early Music (Vol 33, Nos. 1 and 2, 2005), but the information is also on his website: http://larips.com (spiral backwards).

George Bromley wrote (April 1, 2011):
[To Evan Cortens] Thankyou Eric, so equal temperament is not the same, does this blow a hole in the theory why he wrote the '48'

George Bromley wrote (April 1, 2011):
[To Evan Cortens] sorry Evan was jumping the gun a little and have just read the article now (although a yes/no) [or where I come from ja/nee] would be nice

Ed Myskowski wrote (April 1, 2011):
George Bromley wrote:
< sorry Evan was jumping the gun a little and have just read the article now (although a yes/no) [or where I come from ja/nee] would be nice >
I believe it is a fair statement that all aspects of Bach/Lehman tuning, even the name, remain matters of scholarly discussion. Much of that discussion is not easy of access nor understanding for us lesser mortals. That does not mean it cannot be fun, from time to time, to try.

Richard Egarr addressed the issue of *the 48* (WTC) and equal temperament in his Mar. 21 radio interview with Brian McCreath, which was the original impetus for my raising the topic. Independent of Bach/Lehman tuning, Egarr pointed out there are different German words for <well-temperament> and <equal-temperament>. In his interpretation, equal-temperament is not a reasonable consideration for WTC. However, even the validity of that language distinction, for Bach in 1722, has been contentious in previous BCML discussions.

Yes/no? Depends who you ask, still, it appears. It is fair to say that Bach/Lehman (or any other) unequal temperament is either/or, compared with equal temperament.

Bruce Simonson wrote (April 2, 2011):
I first heard about this temperament from discussions on the list around 2006. I had my piano tuned to it, and I love it. I may never go back.

While temperaments are a bit esoteric for those who don't have experience with them, Brad presents a clear explanation (and in-depth analyses) of this temperament on his website, and also demonstrates how simple it is to implement. I'm a believer.

PS: Repeat after me: (3/2)^12 <> 2^7

Ed Myskowski wrote (April 3, 2011):
Bruce Simonson wrote:
< I'm a believer.
PS: Repeat after me: (3/2)^12 <> 2^7 >
One of the beauties of math expressions is that they can be clear and concise, independent of language. Compare this eqivalent statement, from the Temperament entry in the Norton Grove Concise Encyclopedia of Music:
<the circle of pure 5ths does not quite cumulate in a perfect unison>

Any temperament, unequal (including Bach/Lehman), or equal, is a compromise to resolve the incompatibility between the pure fifth (3/2 frequency ratio) and the pure octave (2 ratio). Equal temperament makes all fifths slightly and equally impure. Detailed comparisons among temperaments can run to hundreds of pages, and math that is even less friendly to the non-specialist reader than the inequality Bruce cited. I am in that group of non-specialist readers. I did not mean to imply otherwise with the links I provided recently, nor to imply that there are any easy or quick answers therein. I continue to be impressed with the musical results, and the reputations, of professional performers who have adopted Bach/lehman tuning.

George Bromley wrote (April 3, 2011):
[To Ed Myskowski] All this chat re tuning BUT flisteners point of view does it sound different?

Ed Myskowski wrote (April 4, 2011):
George Bromley wrote:
< All this chat re tuning BUT from listeners point of view does it sound different? >
The short answer is yes. Note two categories of comparison:

(1) Bach/Lehman, or other unequal temperament, compared to equal temperament. This is the point which Richard Egarr is surprised to see ongoing, at all.

(2) Bach/Lehman compared to other unequal temperaments. Ongoing scholarly discussion, with more subtle distinctions.

George Bromley wrote (April 4, 2011):
[To Ed Myskowski] No thats fine, but can you suggest some examples, sorry to go on about this but I am only a tenor form a Bach Choir Tot Siens

Claudio Di Veroli wrote (April 4, 2011):
[To George Bromely] There are many examples in YouTube searching for the temperament name. Some need a very sharp hearing to discern.

This is a didactic introduction, showing how in "remote" tonalities meantone sounds horribel while circular temperaments are fine.
http://www.youtube.com/watch?v=teVlrYJGKAE
< http://www.youtube.com/watch?v=teVlrYJGKAE&feature=related&feature=related >

Search YouTube for Dr. Pitches there are a few other recordings.

Hearing is not all: every musician interested in pre-classical music needs to understand the "how" and the "why" of these different tuning system. You really need to READ about unequal temperaments: search the internet for
material.

Bradley Lehman wrote (April 4, 2011):
George Bromely wrote:
<< All this chat re tuning BUT from listeners point of view does it sound different?
The short answer is yes. Note two categories of comparison: (...)
< No thats fine, but can you suggest some examples, sorry to go on about this but I am only a tenor form a Bach Choir >
Some direct comparisons of Bach/Lehman vs Equal temperament....

Part 1: http://www.youtube.com/watch?v=iMy-oTPD5mI
Part 2: http://www.youtube.com/watch?v=dw0GXkymEhg

More: http://www.youtube.com/profile?user=thebpl

Ed Myskowski wrote (April 5, 2011):
Claudio Di Veroli wrote:
< Hearing is not all: every musician interested in pre-classical music needs to understand the "how" and the "why" of these different tuning system. You really need to READ about unequal temperaments: search the internet for material. >
I have been hinting that this is true for general listeners as well, not only for performing musicians and/or scholars. There is a connection between intellect and ear which is impossible to describe, but which is certainly real for me, adding to enjoyment of performances and recodings.

Tuning & Fermatas - "Du Friedefürst, Herr Jesu Christ"

Charles Francis wrote (October 2, 2012):
BACH uses the chorale "Du Friedefürst, Herr Jesu Christ" in the Leipzig cantatas BWV 67 and BWV 116, and in the earlier BWV 143: http://www.bach-cantatas.com/CM/Du-Friedefurst.htm

The BWV 67 and BWV 116 chorale harmonisations both have the same key signature, namely one sharp at Corntet-Ton pitch - the Leipzig organ providing the immutable tuning reference for the other instruments and voices. Accordingly, if we assume for arguments sake some irregular, well-tempered system in place on Bach's organ, and thereby diverse thirds, fifths etc., then the peaceful/pure affekt that might be expected from the
subject matter of this chorale would be enhanced by the anticipated slow beating of relevant intervals, for example, the prominent G-B Cornet-Ton third [this remark is especially relevant to slow tempo or expansive
performances, where tempo and fermata interpretation don't conceal the beat rates]. Note that if we assume everything is kept in tune, then any optimised thirds on the fixed intonation Leipzig organ, e.g. the diatonic
intervals of the commonly used C major / A minor scales, would be transposed when viewed from the Cammerton perspective, to D major / B minor with the aforementioned example. Of relevance here is CPE Bach's remark concerning his father: "Das reine Stimmen seiner Instrumente sowohl als des ganzen Orchestres war sein vornehmstes Augenmerk".

In the case of BWV 67, three continuo parts have survived (see Bach Digital), one with figured bass at Cornet-Ton pitch, a second with figured bass at Cammerton pitch and a further Cammerton part without figured bass. From this we may suppose that the supporting basso continuo consisted of an organ, a second chordal instrument (e.g. theorbo, harpsichord) and a wind or string instrument. In the case of BWV 116, only one Cammerton continuo part survives and is without figures.

With regard to fermatas, it's instructive to compare the older 17^th century hymnal version of the chorale melody given at the link above, with Bach's various settings - note the rest corresponding to the first fermata. Also
compare the first movement of BWV 116, which uses the same choral melody and has extended rests corresponding to each fermata elsewhere. Likewise, the second and seventh movements of BWV 143, both of which feature extended rests corresponding to these fermatas.

By way of audio illustration, I have realised the BWV 67 and BWV 116 closing chorales on a well-tempered virtual copy of the 1721 Gottfried Silbermann organ from the St. Georgenkirche - Rötha (as with most existent organs, the actual Rötha instrument was modernised in the centuries following its construction and is today tuned to 1/12 Pythagorean comma meantone). My pedagogic video is here: http://www.youtube.com/watch?v=oBcaFQrTkL4

Douglas Cowling wrote (October 2, 2012):
Charles Francis wrote:
< In the case of BWV 67, three continuo parts have survived (see Bach Digital), one with figured bass at Cornet-Ton pitch, a second with figured bass at Cammerton pitch and a further Cammerton part without figured bass. >
One of the speakers at the ABS was discussing organ specifications and noted that a particular organ had one 8' stop which was tuned to Kammerton even though the rest of the organ was in chorton. It was obviously used for
concerted music.

Resolving problems of pitch and temperament

Charles Francis wrote (December 16, 2013):
David Stancliffe wrote (December 16, 2013);
< What interests me in particular is resolving the problems of pitch and temperament in early cantatas like 131, when the oboe is playing a part notated in a at 392, where strings (and organ) may be using 440 and playing in g: and a dulzian for fagotto as we didn't have a 392 fagotto available. Where do you pitch eg 182, where the basic notation in both the Weimar and Leipzig versions (and it looks as though it may have been performed 5 or 6 times) is in G, but where you have the flauto notated in Bb originally, and then in G? I last played it in the same programme as 131, so kept the strings in G at 440, and had the flauto play in F at 392. But what about reconciling the temperaments? Sometimes the organ part is notated in a bizarre number of flats! >
IMO, there is the possibility that one or more of Bach’s organs was tuned to a transposed temperament to eliminate the intonation errors that would necessarily have arisen from the simultaneous use of an unequal temperament, given divergent pitch standards. Minimally, that would have required appropriate tuning of, say, the Gedacht stop for continuo use. However, one might well argue that the 1739 puof Bach’s Clavier-Übung III, given its tonality, suggests a more widespread practice of favouring the flat side, and also for the Plenum. Recently, on another discussion group, the tuning expert Andreas Sparschuh pointed me to a 1743 article by the Swedish organ builder Daniel P. Stråhle containing a ‘Five Limit’ intonation that is notably so skewed: http://upload.wikimedia.org/wikipedia/commons/b/b9/1743_strahle_nyttpafund_firstpage.png

At Bach’s time Equal Temperament was discussed by some theorists, especially as its use would have resolved the intonation issue when using divergent pitch standards; but it seems to have failed to gain traction, being perceived as lacking variety (a criticism that presumably likewise applies to the various regular temperaments sometimes assumed to have been in common use at the turn of the 18th century). In the past, I’ve suggested that ‘decoration’ on the cover page of Bach’s 1722 Well-Tempered Clavier may actually have been a way to tune a transposed unequal temperament at divergent pitch standards by synchronising beats: http://www.eunomios.org/contrib/francis2/francis2.pdf

As the theory was not well understood, such acoustic techniques were largely out of bounds for tuning theorists at the time, although there are indeed various written indications of proprietary heuristic methods being used by German-speaking organ builders.

Claudio Di Veroli wrote (December 16, 2013):
Charles Francis wrote:
< IMO, there is the possibility that one or more of Bach’s organs was tuned to a transposed temperament to eliminate the intonation errors that would necessarily have arisen from the simultaneous use of an unequal temperament, given divergent pitch standards. >
Sure enough, Charles. Good observation. And there is evidence (in different 18th C. French publications) of tuning directions specifically allowing to start from different notes thus obtaining transposed temperaments.

< Minimally, that would have required appropriate tuning of, say, the Gedacht stop for continuo use. However, one might well argue that the 1739 publication of Bach’s Clavier-Übung III, given its tonality, suggests a more widespread practice of favouring the flat side ... >
On this there is coincidence among modern publications (for example by Lyndley, Lehman and myself, with all our differences in other details).

Claudio Di Veroli wrote (December 16, 2013):
Charles Francis wrote:
< the 1739 publication of Bach’s Clavier-Übung III, given its tonality, suggests a more widespread practice of favouring the flat side, and also for the Plenum. Recently, on another discussion group, the tuning expert Andreas Sparschuh pointed me to a 1743 article by the Swedish organ builder Daniel P. Stråhle containing a ‘Five Limit’ intonation that is notably so skewed: http://upload.wikimedia.org/wikipedia/commons/b/b9/1743_strahle_nyttpafund_firstpage.png >
Dear Charles. It is obvious that you have not scrutinised the above tuning.
I just have fully analysied it on the computer. This is NOT a circular temperament, but a classical Renaissance Just Intonation with 2 reduced fifths: it sports 9 pure fifths, plus 2 reduced ones (unplayable being 21.51 C. wide) and a wolf (unplayable being 19.55 Cents large). As per the Major thirds, it has the classical 8 pure ones and 4 wolves at 41 C. each.

Clearly, none of J.S. Bach's works (and scarcely any from the Renaissance and Baroque repertoire) are playable on such a tuning, which is why organ tuners preferred meantone ever since the beginning of the 16th century, a whopping 2 centuries before J.S. Bach's era.

Charles Francis wrote (December 16, 2013):
[To Claudio Di Veroli] Naturally no one is suggesting that this is a preferable tuning for the Well Tempered Clavier, the cantatas, or what have you; please note my original characterisation of it as ‘Five Limit’:
http://en.wikipedia.org/wiki/Five-limit_tuning

Although it’s purely a subjective matter from which little can meaningfully be deduced, I actually do find that Bach’s Art of Fugue, for example, which reaches back to the traditions of earlier ages, sounds rather excellent when performed on an organ in this old-fashioned tuning. My ears are perhaps more attuned to such sounds than most, however.

The relevant point for me was the skewing to the flat-side of Stråhle:

-1

0

1

2

1

A

E

B

F#

0

F

C

G

D

-1

Db

Ab

Eb

Bb

versus, for example, Werckmeister (Musicae Mathematicae Hodegus Curiosus):

-1

0

1

2

2

C#

G#

D#

1

A

E

B

F#

0

F

C

G

D

-1

Bb

Both agree on the ‘just’ diatonic notes, as do Robert Smith and much later Kirnberger and, needless to say, such notionally ‘correct’ diatonic positioning (constructed using F, C and G triads) is but the starting point for a discussion of temperament (Meantone and ‘Good’); albeit, that may influence the shape of the resulting temperament. Stråhle, an organ builder and neither a mathematician nor man of learning, came up with a heuristic geometric method for approximating equal temperament to within a few cents, but unfortunately his construction was incorrectly analysed and so was lost for centuries:
http://en.wikipedia.org/wiki/Str%C3%A4hle_construction

Anthony Kozar wrote (December 17, 2013):
Charles Francis wrote:
< Although it’s purely a subjective matter from which little can meaningfully be deduced, I actually do find that Bach’s Art of Fugue, for example, which reaches back to the traditions of earlier ages, sounds rather excellent when performed on an organ in this old-fashioned tuning. My ears are perhaps more attuned to such sounds than most, however. >
The D minor triad is very noticably out-of-tune in this tuning though, isn't it? (though I would not call it "unplayable")

[the rest of this message is off-topic]

For music that is almost purely diatonic (eg. some organ works of Andrea Gabrieli), I have used a 12-note tuning that repurposes most of the black keys as comma-shifted versions of the diatonic tones:

D-1 E-1 F#-1 A-1 B-1

C D E F G A B

The white keys are tuned per Pythagorean intonation. The D-1, E-1, etc. pitches are all one syntonic comma flat of the corresponding Pythagorean pitch so that C to E-1, D to F#-1 are pure (5/4), etc.

Using this tuning, I can play Gabrieli or Cabezon and almost always keep the fifths and thirds pure (at least when they matter most). The shifting commas keep things interesting and (in C major) after hearing D-1/F/A-1 as the ii chord for awhile, the appearance of a V/V chord as D/F#-1/A is startlingly beautiful -- like suddenly being in a new sound world. If I immediately play the same piece again with equal temperament, the endless diatonic sequences sound lifeless and dull.

Unfortunately, I don't see any way to play Bach in a similar manner -- but at least Bach is never lifeless or dull

(I did recently tune my keyboard to 31-tone equal with the intention of experimenting with playing some chorales or the WTC that way but the mapping of the keyboard is so much harder to grok -- and difficult to reach -- that I haven't yet attempted to play any real music in this tuning ...)

Charles Francis wrote (December 18, 2013):
[To Anthony Kozar] Indeed, if you sequentially tune three pure triads F-A-C, C-E-G and G-B-D, then D-A will be dissonant. But is this a bug or a feature? For example, all the pieces in the Art of Fugue are in d-minor, which suggests to me that Bach wanted the opening note of each fugal entry to be dissonant to highlight it and propel the music forward. A few snippets to illustrate:
https://dl.dropboxusercontent.com/u/35465264/AoF-Examples-JohnCharlesFrancis.mp3

Note Kirnin his temperament 'II' also adopts this diatonic construction, although he then tempers the 'A' to 'improve' D-A and obtain a 'circular' temperament (however, his Galant contemporaries no doubt considered the near-equal proposals of Neidhardt and Marpurg more suited to their pretty, simplistic, music). One supposes that Kirnberger II is somewhat indicative of Bach's manner of tuning in his later years, or at least as he taught young Kirnberger. Bach no longer had to produce functional music for church or court and could therefore indulge his interest in past music and tunings.

Claudio Di Veroli wrote (December 19, 2013):
[To Charles Francis] your comments are certainly interesting.

That said, and for the benefit of the "non-temperati", I have to report that you are on your own with your conclusions, which run straight against the understanding and research results of every single leading temperament scholar known to me. This is for two reasons:

1) The D-A, being a fifth out of tune by 1/5th of an ET semitone, is not just dissonant (like a Pythagorean Major Third), but unplayable. It cannot play anything like a recognisable major or minor chord.

2) You wrote: "One supposes that Kirnberger II is somewhat indicative of Bach's manner of tuning in his later years, or at least as he taught young Kirnberger. " Indeed, "one supposes": that one is yourself. Nobody else does, because, as is quite easy to deduce (see my UT book for the details), this is impossible, given what we know about this matter via his polemics with Marpurg. Summarizing, it does not make sense that, as his proposals were (rightly) attacked, not even once Kirnberger suggested that JSBach's teachings were at the origin of his temperaments. The only thing Kirnberger tells us that JSBach told him was to tune all the major thirds really wide. This is something Kirnberger, decades later, did not do in any of his II or III temperament. And certainly far away from any Just Intonation, by the way.

The inevitable conclusion - agreed by current musicology - is that Kirnberger's temperaments are as useless today as they were in their time when, let me recall, all the first-rate German musicians known to us had been using Equal Temperament for at least two decdes.

Anthony Kozar wrote (December 19, 2013):
[To Charles Francis] Thanks for posting your examples which push these musings out of the theoretical tones in my head and into the realm of easily ponderable! ^_^

I will say that I found the sound of the AoF theme by itself in this tuning enchanting. But as soon as the theme is answered by the second voice (D-A), in my humble opinion, it was much uglier. I was tempted to rescind my

earlier statement that this fifth is "not unplayable", but later decided it sounds positively fine following the putrid C#-A (really Db-A) at the entrance of the answer in your third example.

Have you tried centering your tuning on D or at least tuning C# and G# instead of Db and Ab?

(For the record, I am something of a just intonation enthusiast and have also experimented with many stranger tunings over the years. A few years ago, I made some demos of Old 100th being played in 12TET, 12T-just, and extended just. Using two different pitches for the second scale degree is essential, IMHO, to applying just intonation succesfully to triadic music. I could post them online if there is any interest).

Charles Francis wrote (December 19, 2013):
[To Claudio Di Veroli] The assumption of certain musicologists seems to be that Bach always tuned in the same manner. Any evidence for that, as it seems unlikely to me given the absence of a clear legacy from his students? What better pedagogic way to illustrate the incongruity of 2, 3 and 5, and the need for temperament in remote keys, than to tune in just manner?

Obviously, one of the attractions of a group such as ours is that we can discuss aesthetic reactions, although we should be aware that subjective impressions (e.g. what is ‘unplayable’) are primarily a matter of habit and conditioning, and not set in stone. One can acclimatise to almost any tuning by using it often, at least that is my own experience; Marpurg, as I recall, even advised his readers to tune in Pythagorean manner (for a week?) to prepare their ears for equal temperament, a remark that does indeed tell us something about tuning norms at that time.

Marpurg’s anecdote concerning Bach is on page 213 of his ‘Versuch über die musikalische Temperatur’ and you may also, I believe, find it in Bach-Dokumente 3, Item 815, p. 304. For the record, he doesn’t say ‘really wide’, merely ‘wide’. It is at best anecdotal evidence: a report of a report written a quarter century after Bach’s death by someone with an axe to grind. And even if it were proven correct, and I’m certainly not stating otherwise, it would only establish that one of the tunings Bach taught to Kirnberger had wide thirds, but would not tell us which were wide nor how wide they were (Lindley has suggested Equal Temperament here as I recall).

As to ‘all the first-rate German musicians known to us had been using Equal Temperament...’, bear in mind CPE Bach’s characterisation of Equal Temperament as one where most of the fifths are slightly narrowed (a statement misquoted by Lindley, incidentally), which suggests to me that CPE’s father either didn’t teach him such a tuning or that he forgot the details because it was never used.

Claudio Di Veroli wrote (December 19, 2013):
Charles Francis wrote:
< The assumption of certain musicologists seems to be that Bach always tuned in the same manner. Any evidence for that, as it seems unlikely to me given the absence of a clear legacy from his students? >
Indeed strong evidence is found in published research by John Barnes and myself. In Barnes's statistics we find that JSBach's peculiar distribution of the use of major thirds is almost identical between the two books of the WTC, composed about two decades apart. It has also been shown that his major thirds were all wide between approx. 7 Cents and 21 Cents. This allows for only for a few variants of Good temperament, and absolutely excludes any of the tunings proposed by Kirnberger. (This matter has been fully scrutinised in several pages in my Unequal Temperament book, which I cannot obviously copy here).

< ... Marpurg, as I recall, even advised his readers to tune in Pythagorean manner (for a week?) to prepare their ears for equal temperament, a remark that does indeed tell us something about tuning norms at that time. >
Marpurg was correct even by modern norms: Pythagorean tuning will yield beautiful pure fifths (except a single wolf) and Pythagorean major thirds, dissonant but perfectly playable. It will NOT produce the unplayable fifths of Kirnberger.

I share some of your other arguments, Charles, but they do not carry any weight against the above about Kirnberger which is our main discussion point.

Charles Francis wrote (December 20, 2013):
[To Claudio Di Veroli] It wouldn’t be surprising if a similar tuning was expected for Book 1 and 2 of the WTC, as the requirements are identical. But a Barnes-type analysis applied to the Art of Fugue or Goldberg’s, for example, would presumably produce different results. Likewise, the continuo parts for the cantatas, which are set in orchestra-friendly keys, and are often offset by transposition.

I have an old photocopy of Barnes EM paper somewhere that I read many years back, but would need to dig it out as I don’t specifically remember the statistical correlation between WTC Books 1 and 2 being discussed. My overriding impression at the time if I’m honest (and I don’t wish to provoke or cast doubts on the veracity of the late John Barnes), was of an apparent forcing of results by what appeared to me to be arbitrary quantising of his data (I think this was done with thresholds pulled out of thin air, but may be wrong on this detail as much time has passed).

One could add that the philosophic assumption that Bach would tend to seek out purer thirds rather than wider ones, can be doubted, as an overriding motif of his music is dissonance. If, as some have speculated, Bach wrote thWTC while in prison, then would aural memory allow discernment of one type of third over another? – same question if he worked at a desk. And if he fell back on a habitual way of doing things, would that not be influenced by the various organs he had encountered in his youth (tuned in Meantone or who knows what). The issue of transposition should also be mentioned, as many pieces existed in earlier versions in different keys. Last but not least, there are physical constraints such as hand size and fingering to consider that could have influenced his choice of thirds.

Claudio Di Veroli wrote (December 20, 2013):
Charles Francis wrote:
< The issue of transposition should also be mentioned, as many pieces existed in earlier versions in different keys. >
Of course. (Yet another matter fully covered in my UT book, specifically in the case of the French Ouverture in c minor, later in b minor.)

< One could add that the philosophic assumption that Bach would tend to seek out purer thirds rather than wider ones, can be doubted, as an overriding motif of his music is dissonance. ... >
Interesting observation Charles! Indeed, pieces with many dissonances, such as the Fantasy in the "great" organ Fantasy and Fugue in g minor, can be played in many different temperaments (even in Rameau's very unequal ordinaire) with no problems at all. While "simpler" yet involved harmonies with a few flats or sharps where you expect lots of consonances as you are playing with a solo harpsichord stop, for example the sonata for violin and harpsichord in f minor, first movement which is largely a harpsichord solo, really requires a fully circular temperament.

Continue on Part 7

The Keyboard Temperament of J. S. Bach: Article | Music Examples | Feedback: Part 1 | Part 2 | Part 3
Discussions of Temperament / Key Character / Tuning: Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6 | Part 7 | Meantone