Continue from Part 4

Sacred Temperament

Mocfujita wrote (May 13, 2004):
Which temperament system is sacred?
The Sacred Temperament exists in your mind.
http://www.geocities.jp/imyfujita/index.html

Bradley Lehman wrote (May 14, 2004):
[To Mocfujita] Yes, I remember that site from a few years ago. Looking at it again tonight, I see the following representative quote that sums up the author's message:
http://www.geocities.jp/imyfujita/wtcpage004.html

"Once upon a time, the equal temperament and the pure temperament should have been the same. There was only one temperament. But since a certain old time, the confusion of temperament system became one of the sins of human being. LORD made a fifth, a fourth and a third never be under one temperament. I think that we have been sufferring for our arrogance, like saying that we, people, tried to unify these all fifth, fourth and third into a heavenly harmony only by our own knowledge. Our predecessors were scared of roars of Wolf which we could not get rid of. We should humbly admit that actual pure fifth, fourth and third are inevitably out of tune from the equal temperament which was given to us by LORD. Those of us who can not help adhering to the thought that only the pure temperament is the trueth and the equal temperament is nothing but a compromise might be staying in a situation without an awareness of this sinfulness of ours."

According to that sermon, I'm sinful whenever I tune my harpsichord! I guess that means I'm also sinful for offering knowledge about those other wicked temperaments, and a tool to analyze them, here: http://how.to/tune

Mocfujita wrote (May 20, 2004):
I don't intend to accuse anyone who loves music especially tuning professionals who do very complicated tuning works for many keyboards.

Bradley Lehman wrote (May 21, 2004):
[To Mocfujita] I agree that we need Bach's "Sacred Temperament" to hear his music to best advantage, but that temperament (according to Bach!) is not equal temperament. The derivation of temperament has to be based on something much more solid than merely measuring Walcha's harpsichord recording with an oscilloscope. Walcha was only reflecting 20^th century tastes.

The observations on: http://www.geocities.jp/imyfujita/wtcpage1241m.html
are sort of on the right track, as to the symmetry of the Dorian mode, but that's a truism of any piece in a Dorian mode, nothing special about the prelude 24 of WTC 1.

The prelude and fugue 24 actually illustrate something much more interesting than that, when Bach's specified temperament is put onto the instrument. The Dorian scale on B (B-C#-D-E-F#-G#-A-B), taken half by half as tetrachords (B-C#-D-E and F#-G#-A-B: the way Bach has laid it out as the bass line of the first bar), is the only Dorian scale in Bach's temperament in which all of the steps have sizes different from one another: four different sizes of tones (whole steps) there, and two different sizes of semitones. In all the other minor scales, tones of the same size get used more than once. Similarly, in the fugue, the oddly sinuous subject illustrates the distinctive sounds of five different sizes of semitones.

That's not a proof of Bach's temperament (as in, being able to derive the temperament sufficiently backward from this by assuming this goal), but a remarkable consequence of it. I believe he was deliberately illustrating these expressive distinctions (subtly different sizes of tones and semitones), and similar musical quirks that arise from the way his temperament sounds on the keyboard, in many other compositions as well. His compositions, remarkably, illustrate the various fields of symmetries, asymmetries, and balances of his own preferred temperament that is, showing off its depths and uniquenesses to best advantage. This is all a consequence of putting his tuning onto the instrument, playing the music, and carefully noting the resultant sounds (with some help from mathematical analysis of his temperament, also, to confirm and explain why it sounds as it does).

The important concept here is: this isn't just a bunch of numerological counting, which can be forced to say anything the seeker wishes it to say, and look like magic on paper; but the different sizes of the steps are perceptible in the sound of the music, and that's what Bach is illustrating. Bach was working in his own field of specialty: the sound.

Numerology (at least of the sort practiced by Kellner in the articles of his I've been reading: a bunch of circumstantial coincidences and truisms exalted to the level of looking significant...thereby proving nothing) is the "dry, mathematical stuff" that Bach was not interested in. Bach's temperament is explicitly not Kellner's solution, and that's why all of Kellner's rationalization to support his own temperament with a circumstantial Blitzkrieg doesn't really mean anything. It's merely a distraction by irrelevancies, a very large boat full of red herrings. The only people Kellner's argument convinces are those who don't fully understand what they're looking at or listening to, who are willing to put up with grossly out-of-tune major thirds (since nothing better appeared to be available), and who are bamboozled by his spectacular presentation of smoke and mirrors. In vehemence and persistence, through about 50 publications, he tried to make up for a lack of really solid evidence by showing that his own patented creation "must have" been what Bach had in mind, due to all the coincidental alignments (mostly truisms) he could demonstrate. By his own admission, his own temperament is based on an assumption that Werckmeister had a secret method that he did not write down, but reserved only for his own use. Kellner has simply changed and "corrected" a few of Werckmeister's premises, reconstituted Werckmeister's alleged secret, secured a patent for himself, and then baldly asserted that all the outcomes from that point forward must have been Bach's own. BAAAAP, that's not good science; the logical steps don't follow from one another, the tie to Bach is a tenuous modern assertion, and there's no reason to assume that Werckmeister even had such a secret, or that Bach would have cared about it. Kellner's reasoning collapses at all these points, and his "proofs" are repeated citations back to his own papers, a circular method that goes nowhere. One good thing he did accomplish, anyway, was to get people to start thinking more charitably and practically about possibilities of unequal temperaments; and that was a major achievement (not to be underestimated).

Another book about symmetries in Bach's music is J S Bach's Critique of Pure Music by C L van Panthaleon van Eck. The author tries to show that certain pieces by Bach "prove" a use of equal temperament: the author inserts his own premise that the intervals themselves should be symmetrical when subjects are transposed or inverted, and then he reasons all this backward to "prove" that equal temperament (and indeed any regular meantone) must be present. That is a fallacy (i.e. his assumption that his own goal, complete symmetry of intervals, was necessarily Bach's goal), and it spoils a book that otherwise has some interesting observations in it. It's a circular argument, where his "conclusion" is nothing more than his own premise, plus some truisms about the behavior of regular meantone (including equal temperament). Furthermore, the author shows no awareness of other strains of meantone except 1/4 comma, or many irregular temperaments except Kirnberger's, and this hurts some of the other points he's trying to make....

Bach's temperament really comes into its own realm, I believe, in Clavieruebung III. The prelude and fugue, the chorale preludes, and the duets all illustrate different expressive aspects of this same tuning, givit a complete workout. And, remarkably, this is the one of his four published books of "Clavieruebung" that has a different intended audience, according to its title page. It's not only for the refreshment of spirits of the Liebhaber, but especially for the appreciation of the Kenner...which, I believe, may be those who understand the subtleties of his ideal method of tuning (which was never any secret: he taught it directly to his students), and the extraordinary things he makes of it in that book, theologically and otherwise. Also remarkably, that book does not work well in any tuning other than his; one runs into barbarisms of one type or another, in any other tuning but his.

The tuning is not only "sacred", whatever that means, but practical and necessary if Bach's music is to be heard with best clarity, with sensitivity to the subtleties of the music's content.

John Pike wrote (May 21, 2004):
[To Bradley Lehman] Have you tried any of the unaccompanied violin or cello pieces using Bach's temperament, especially the 3 violin fugues and the chaconne. I'd be most interested to know whether they sound different from recordings you know on the violin (probably perfect 5ths and perfectly tuned major/minor 3rds)?

Bradley Lehman wrote (May 21, 2004):
[To John Pike] Good idea, John. I have already played through the keyboard-or-lute version (BWV 1006a) of the E major partita, since I was especially curious about the way that key would come across, but haven't played through 964 or 995 yet, or done my usual improvisations straight from the violin or cello works.

The E-major capriccio 993, that normally cast-off piece of weirdness (among keyboard players today, anyway), was one of the first things I checked out. It comes out beautifully and is finally starting to make some sense to me. The special properties of E major in Bach's tuning make a good showing here; and this is another of those pieces that really don't hold together in anything but Bach's tuning.

I see with some surprise that the Adagio 968 (G major harpsichord arrangement of the first movement of C major violin sonata 1005) has been moved in the BWV catalog to the group of questionable works.

To answer your speculation about the tuning of E major: no, there aren't any major or minor 3rds anywhere close to pure in that key, but the 5ths are very good. E major is one of the brightest and "hardest" (in sound) keys available, like a brilliant and sometimes violent yellow (referring to 18th century reports about key characteristics, as presented in Rita Steblin's dissertation). Those 18th century reporters and theorists were right, along that line of the keys having specific Affekts. Steblin also does a good job, by the way, of explaining how the physical and psychological elements of key character are tied not only to keyboards in unequal temperaments, but also to the other instruments: it's a big picture. [And, quite apart from any 18th century meanings of keys, and at the risk of sounding irrelevant: the tonal organization of Wagner's "Die Meistersinger" is brilliant, in the way he reserves meaning for each key and puts them together into the drama. Ditto for the Mahler symphonies, and Bernard Herrmann's soundtrack scores: the ways in which specific chords and keys can cause specific physiological responses in listeners, when organized carefully.]

As I play through all these Bach pieces in various keys, I've been finding that I have different physiological reactions to them and to the music, from the sound. A major and D major are the keys that I personally find least comfortable in Bach's music, to my own personality, but everybody is of course different, and different people like different things. Bach's relationships with the keys of several flats are more to my preferences; but the keys of four or more sharps also sound magical to me, the way Bach used them.

The point is, there's a whole range there, and all of it is usable to express different things. All the keys are equally usable, but they are absolutely not interchangeable: when the music modulates so the same ideas are heard in a different key, or when the whole piece is transposed for some other use, the character changes: sometimes radically. This isn't a consequence only of Bach's tuning, as it happens in some others also, but his is exceptionally well organized both for maximum contrast and for smooth transitions, with never any harshness anywhere.

That's what is so remarkable about it. Some other temperaments differentiate some or most of the keys; I have the data to lay all this out. But in Bach's tuning every major scale and every minor scale has an absolutely distinctive sound, due to the different sizes of the intervals in the first four notes. That is, the lower half and the upper half of each scale (lower tetrachord and upper tetrachord) are different from one another, and there is a smooth metamorphosis from one scale to the next--by the circle of fifths--by taking half of the existing scale and adding four more notes on either end to make the new scale. F-G-A-Bb, C-D-E-F, G-A-B-C, D-E-F#-G, etc. Put any two neighboring groups together and you have a complete scale. Ditto for the minors.

That is, whenever the music modulates to a new tonal area there is OBJECTIVELY a different overall "color" (for lack of a better word, for now; it doesn't have to be a visual thing) where the tonality is immediately recognizable; and this isn't tied to absolute pitch, either. It's all about relationships among the notes, not absolute frequencies. Whenever the music modulates we're moved to a different world that retains half the features of the previous one. It's this whole bunch of dynamic dimensions, flipping all over the place all the time, but so well-organized that it all seems smooth and natural at the same time. That's why this stuff is so nifty. It demonstrates some astonishing things about tonal music. And if one ever skips from one world directly into another one, as JSB does for special effect and his son CPE does even more egregiously and dramatically, leaping over the usual transitions, it's really powerful stuff: like a midnight ride of Don Carlo Gesualdo, but with 18^th century balance.

Kellner

Bradley Lehman wrote (May 2, 2004):
< Purcell Quartet, without conductor? (2000) (...) Kellner temperament is an interesting and worthwhile feature of this CD, but unfortunately the Kyrie performance is seriously impaired by others factors. >
Whatever.

With regard to the title of Kellner's most famous piece, and despite both his vehemence and tenacity defending his preference, here's a closing sentence I'm considering for my paper.

"Wanda Landowska's spiritual heirs might now have a new mantra: 'you tune the keyboard your way, as you please, but I'll tune it Bach's way.'"

Granted, out of context that sentence might appear astoundingly arrogant. But, coming as it does at the end of a paper where I've proven my discovery of Bach's single preferred temperament (as documented directly by him), and following a thorough musical and historical analysis of its implications, I think it has some merit. The music really does sound better in Bach's tuning, not too surprisingly. I have it set on my harpsichord and clavichord right now, and have been playing through almost all of his keyboard works, in awe of Bach's musicality.

Charles Francis wrote (May 2, 2004):
[To Bradley Lehman] In the past I've experimented with several alternative temperaments for Bach (Meantone, Werkmeister III, Kirnberger, Barnes, Kellner). For the WTC, I find Kellner and Barnes generally give satisfactory results (as does ET). Somewhere I recall a paper Kellner sent me where he finds his temperament coded in the WTC. Are you proposing a different temperament from him?

Jef Lowell wrote (May 2, 2004):
[To Bradley Lehman] Can't wait to learn more about Bach's tuning system!

Bradley Lehman wrote (May 2, 2004):
Charles Francis wrote: < In the past I've experimented with several alternative temperaments for Bach (Meantone, Werkmeister III, Kirnberger, Barnes, Kellner). For the WTC, I find Kellner and Barnes generally give satisfactory results(as does ET). Somewhere I recall a paper Kellner sent me where he finds his temperament coded in the WTC. Are you proposing a different temperament from him? >
Yes. Barnes and Kellner had some reasonable results, but the evidence I have for a different one is overwhelming.

Charles Francis wrote (May 2, 2004):
[To Bradley Lehman] If you post the 12 deltas from ET in cents, I'll try it out (I presume no octave stretching is required).

Bradley Lehman wrote (May 2, 2004):
[To Charles Francis] What, I should just give all eleven of the cents values away?

The request itself here is disturbing. Members of the general public should know the crucial details of new findings, given in a public posting, BEFORE expert colleagues can finish confirming them??!!! Really?!

Isn't this like asking a chef to hand out his recipes for free, letting people decide for themselves willy-nilly if the dish is worth anything, or if the chef knows how to cook? And to let anyone invent whatever unreasonable proof and/or refutation they wish: based on nothing more than the respondent's creativity and preconceptions, and the selective mining of reference books, and personal reactions to the results...before the theory is even put out there to stand on its own merit, with the full presentation of evidence?

Challenge by casual consumers (i.e. the public) is not science. I can understand the enthusiasm of wanting to try the results, like picking up a sandwich from a drive-through restaurant, but now is not the time for that.

The recipe will be given eventually. I already have a group of experts examining my preliminary data, helping me as colleagues to test the practical and theoretical sides of my findings. Release outside that circle comes later.

Besides, the use of deltas from ET is absolutely a wrong way to conceptualize this temperament, and any "findings" of people's reactions from that (whether expert or amateur) would be irrelevant. (Murray Barbour and his followers have been down a wrong path.) Bach never heard of a cent or an electronic tuning device, and never heard a modern piano (re the remark above about stretching octaves), so all that is irrelevant.

The truth in this particular issue comes not from anybody today liking the results on any modern instrument, but the proof of a procedure Bach used on his own keyboards. This is a fundamental point of historically-informed performance practice. It is not about what we like but about what those people back then did and liked. In this case, Bach's own discovery was phenomenal, and I'm pretty sure we will like it too (I certainly do, already)...but my enjoyment of it must not cause the scientific presentation to be slipshod.

It's historically important because Bach did it. It's musically important because it makes the music and the instruments sound wonderful, inexhaustibly rich.

Ask again in six months, maybe, after (I hope) publication of my paper somewhere.

Charles Francis wrote (May 3, 2004):
Bradley Lehman wrote: < What, I should just give all eleven of the cents values away? >
The request itself here is disturbing. Members of the general public should know the crucial details of new findings, given in a public posting, BEFORE expert colleagues can finish confirming them??!!! Really?! >
I'm surprised at this comment, given you chose to mention your current research paper to the Bach Cantatas group. What was the motivation, then, if not for feedback from members?

Bradley Lehman wrote (May 3, 2004):
[Contiune of his previous message] Simply sharing information that some exciting new findings exist. These findings could eventually have direct effect upon the performance of the cantatas and many other pieces by Bach: especially among those of us who value doing things Bach's way as far as can be determined, and not merely playing into people's modern preferences. The keyboards should be tuned differently from the ways they are now. That affects the sound of everybody else in the ensemble, also. Is that not interesting?

But, feedback from list members as to the factual correctness of the findings, just putting the information out there and letting anyone chomp on it? That would be awfully premature, and anti-scientific.

And doesn't that presuppose that everybody's opinion is equally valid, as to ability to judge correctness? Doesn't expertise mean anything to you, in that experts might be better equipped to evaluate one another's work than ordinary consumers (who don't do the work, and who are blissfully unaware of the problems to be solved, and who judge things mainly by their own likes and dislikes)?

What if, for example, some expert in astronomical observation discovered that a comet is likely to strike the Earth in a couple of months, according to her calculations? Wouldn't you want that to be confirmed first by other experts studying the evidence, or would you rather just have that rumor put out there for the general public to panic and decide for themselves (with no evidence whatsoever, or maybe with a cheap telescope or some general reference books) if it's true or not?

=====

Besides, the criticism above (the alleged "surprise" at my comment) is especially hollow coming from a person who has belittled my expertise directly already: http://launch.groups.yahoo.com/group/BachCantatas/message/7808
...and who dismisses scholarly responsibility to the material as nothing more than "political correctness": http://launch.groups.yahoo.com/group/BachCantatas/message/8064

If we like something, it's true, and if we dislike it, it's false? That's not a scientific approach no matter how loudly one might protest anyone else's methods or abilities.

Just look at the subject line of this thread: Kellner. Some have advocated his keyboard temperament as it works pretty well in some music...but that still doesn't make his temperament true to Bach's own preference. What we like today, or are accustomed to, is irrelevant to the issue of discovering historical truth. Factually, Kellner's solution was mostly reasonable yet it's still wrong, as it conveniently ignores some inconvenient bits of evidence, or runs over them as if they're not important. That's not good science, no matter how many papers are written about it or how strong the writer's reputation is. Truth in a scientific field such as this isn't about reputations, it's about evidence and facts. People who don't know that evidence, or know how it all fits together, are not in a position to judge if a model is correct or not!

Charles Francis wrote (May 4, 2004):
Bradley Lehman wrote: < Just look at the subject line of this thread: Kellner. Some have advocated his keyboard temperament as it works pretty well in some music...but that still doesn't make his temperament true to Bach's own preference. What we like today, or are accustomed to, is irrelevant to the issue of discovering historical truth. Factually, Kellner's solution was mostly reasonable yet it's still wrong, as it conveniently ignores some inconvenient bits of evidence, or runs over them as if they're not important. That's not good science, no matter how many papers are written about it or how strong the writer's reputation is. Truth in a scientific field such as this isn't about reputations, it's about evidence and facts. People who don't know that evidence, or know how it all fits together, are not in a position to judge if a model is correct or not! >
You will be aware that Barnes statistical results are compatable with Kellner's proposed temperament (allowing for experimental error). Is this also true of your proposal? Perhaps you might wish to post a pre-publication copy to Kellner's tuning group to get some expert feedback? You may be aware that Kellner acted as reviewer for Barnes seminal paper. Having the world's leading expert on Bach's temperament on your side might add essential credibility to your results.

Bradley Lehman wrote (May 4, 2004):
[To Charles Francis] Any idea at all how insulting these suggestions are, as to the nof the work and the value of evidence? They also reduce the value of real peer-reviewed publication to zero, or even less (especially the part about posting a pre-publication copy to a public group): as if scientific inquiry is merely a popularity contest, and as if experts get to be experts merely by deciding for themselves that they are sufficiently qualified to judge others' work.

But scientific points are made with solid evidence, and careful and thorough observation of it, not the reputation of authority.

Perhaps we will continue this conversation someday, AFTER it is realized that scientific truth (a mathematical proof) does not depend on personalities or popularity, or on vehement recent evangelism (Werckmeister, to Bach; Kellner et al, to me). Knowledge and mastery of a topic come from doing the work, not from consuming other people's packaged ideas.

I'm not interested in the regularly argumentative nonsense here that belittles the meaning of scientific knowledge itself, wittering it down as if nothing can be true or acceptable unless cherished authority figures say so. Mathematical facts are true!

The things I have to say in the paper are true whether they're said by me (whom you obviously do not respect), or by someone who has a longer list of previous publications to his/her credit (whom you might respect); so, get used to it. The truth of the findings does not depend who says them. At the same time, I do deserve at least some credit for the 20 years of necessary background work (on many fronts) I've put into this before making the discovery, and there is some value in the direct musical illustrations I will give, from practical experience playing the music...this isn't empty theory here, it's practical information. Bach was no lover of "dry, mathematical stuff" anyway!

Charles Francis wrote (May 4, 2004):
Bradley Lehman wrote: < But scientific points are made with solid evidence, and careful and thorough observation of it, not the reputation of authority. >
So are your results, like Kellner's, within the error bound of Barnes statistical analysis of Das Wohltemperierte Klavier? This is a scientific criteria which any serious scientific reviewer would use to eliminate a bogus proposal for a new Bach temperament.

Ref: John Barnes, Early Music, v.7, no.2 (1979).

Dale Gedcke wrote (May 4, 2004):
A lot of the discussion group members are eager to find out what Brad Lehman has discovered about Bach's preferred tuning temperament. Thus, all the requests arise to just tell us something more about it.

However, I think Brad is choosing the right course to reveal the information by submitting the research paper to a reputable, peer-reviewed journal. That's the best way to make sure the evidence and conclusions are confidentially fully vetted by knowledgeable experts.

Having a lot of scientific publications and two books under my name, I can shed some light on the process. I cannot speak about a specific journal, because I don't have adequate experience in musicology. However here are the principles that are generally applicable for most peer-reviewed journals.

1) Upon receipt of the submitted research paper, the editor of the journal sends the paper to several acknowledged experts in the field for their critique. Usually the author is not given the names of the reviewers, but he is presented with their critique.

2) Frequently, the editor of the journal rejects the paper as "unacceptable for publication" or "acceptable if the improvements suggested by the reviewers are made". In the latter case, the author has the opportunity to make the revisions and re-submit.

3) Upon receipt of acceptable revisions, the editor schedules the paper for inclusion in a future issue of the journal.

4) The time from initial submission to distribution in the printed issue can be at least a year. That's a long time. Some journals have shortened the time interval by publishing an electronic copy on the Internet. But, the process still takes about 9 months from start to finish.

5) There is a danger that in that long time period between submission and publication that someone else will independently discover the same information and publish first. The more that snippets of the discovery are talked about, the more likely it is that someone else will jump on the fast track to publishing the same content first. Back in the late 1960s, I published a paper on a new instrumentation technique for timing the arrival of nuclear particles. It revolutionized the industry and became the standard that is still used today. In the following 5 years, it talked to at least 4 people who said they were on the threshold of publishing a similar concept, but they lagged my time-to-market by about 6 months.

6) There is a method for beating the long time-to-publication in order to benchmark your work as the leading edge. When I was at the University of Alberta, we would submit the paper to a journal, and the next week we would issue a "pre-print" of the paper to all the researchers who mattered. The pre-print was published under the name of the Nuclear Research Center at the University of Alberta, and it mentioned that the original paper had been submitted to the specific journal. That pre-print put the earliest possible time stamp on the publication of the research, and helped to establish a leadership position.

Brad, you can try option 6), but you need to make sure it does not become cause for rejection by the selected journal. Some journals don't want pre-prints going out, because it detracts from the journal's audience.

Dale Gedcke, B.Eng., M.Sc., Ph.D.
(But, .... AMATEUR trumpeter)

Bradley Lehman wrote (May 4, 2004):
[To Charles Francis] I know where that stuff is, having read it at least 15 years ago. Are you suggesting that I'm not a serious scientific reviewer of the material, and/or lecturing me how to do my job as a researcher or musical analyst?

Anyway, what is this, the preparation of an ad hominem offensive attack against my work, with the assumption that my proposal will be in any way "bogus" just because it's me doing it? Why?! And why the offensive choice of that word "bogus" with its implications of deliberate deception?

This tuning I've discovered, coming directly to us from Bach, is the harmony of the musical cosmos in a perceptible and provable sense, which I'm now proving, and showing how Bach used its unique properties in musical situations.

"Error bounds" will have a different meaning after my findings are released, as any other "wanna-be" temperament can be simply measured against this saddle-point solution--as I have been measuring already. What reason would you have to assume otherwise, that I'm not capable of this (or that Bach wasn't!), and/or that such information is unknowable, just because the late Kellner and Barnes happened not to know it? Kellner did pretty well but he still failed to leave the baggage of some wrong premises behind, and that's why he didn't get there....

Bradley Lehman wrote (May 5, 2004):
Bradley Lehman wrote: < "Error bounds" will have a different meaning after my findings are released, as any other "wanna-be" temperament can be simply measured against this saddle-point solution--as I have been measuring already. What reason would you have to assume otherwise, that I'm not capable of this (or that Bach wasn't!), and/or that such information is unknowable, just because the late Kellner and Barnes happened not to know it? Kellner did pretty well but he still failed to leave the baggage of some wrong premises behind, and that's why he didn't get there.... >
For example, according to my preliminary analysis, the worst of Kellner's notes are off by more than 9 cents from their proper positions in Bach's temperament. In the famous "Barnes-Bach" temperament derived from Barnes' statistical research (i.e. from Bach's use of specific intervals in the music), the notes are never off by more than 5 cents, suggesting that Barnes as an analyst and Bach as a composer were both onto something. But anyway, Bach's intentions are even clearer on paper than this 5-cent error would suggest; much clearer. A 5-cent "error boun" derived merely from statistical analysis is not unreasonable, for any sufficiently large composition.

Translation: the "Barnes-Bach" temperament sounds better for Bach's music than Kellner's does: less garishly wrong. That is according to BACH'S written intentions for the way it should sound, not my or anyone else's modern preferences.

Not that my paper goes into statistics very much, as it's not necessary beyond a brief mention. The point is not to prove that those guys and a dozen others have been wrong, but that Bach was right. The fact that approximations are wrong is simply a truism about approximation. So what?

I seriously doubt that monkeys and typewriters would have delivered this in any reasonable amount of time.

Jef Lowell wrote (May 5 2004):
[To Bradley Lehman] I'm intrigued by your suggestion that Bach left some kind of theoretical record of his method of tuning. I've never come across such a thing, although I try to read everything about him I can get my hands on. What am I missing here? If it's not letting the cat out of the bag (before you publish), could you please tell us what the source of this important information is?
Best Regards,

Smoovus wrote (May 5, 2004):
< I seriously doubt that monkeys and typewriters would have delivered this in any reasonable amount of time. >
using the cents method there are 1.76146947602 x 10 to the power of 26 ways of tuning (thats 1199 nCr 11), as long as theirs no octave streching. then if we give a hundred mokeys with a hundred harpsichords about an hour to tune their harpsichords and play a french suites or something to try it out (i dont think a monkey could play anything much harder). we can only make them work 16 hours per day since we have to give them 8 hours to sleep and throw their feces around, because of the damn monkey unions. that means it will take the monkeys 3.01167620568 x 10 to the power of 20 years to try all the tunings. (11 nCr 1199) / (100 x 16 x 365.25)

Congratulations Dr Lehman you are smarter than a monkey by several orders of magnitude.

Back on topic: Bach's music in his tuning

Bradley Lehman wrote (May 20, 2004):
Mats Winther wrote: < However, it takes time to understand very deep and sophisticated matters. Such matters are by nature quite different from the down-to-earth subject of tuning instruments and the subject of "which fugue was historically connected to which toccata". I have tried to discuss metaphorically about the "serpent that lies coiled up inside the fugue and which must be magnified". But how on earth will I be able to elaborate on this symbol if you go on like this? >
By understanding that Bach's own tuning method bites its own tail (like our friend Ourobouros) in a most peculiar and compelling way, giving extraordinary musical results to all his compositions. But, such understanding is not going to come from dismissing those who know more than oneself about it, and refusing to learn about its features as if it didn't matter. (And I'm not citing Ourobouros in my paper, because then the paper would get dismissed by some scientists as ridiculous and insubstantial mumbo-jumbo.)

And, any assertion that tuning is merely a "down to earth" mundane matter is grossly mistaken; Bach's ideal tuning is (to venture an equally bold assertion) a representation of the harmony of the spheres. That's why his music sounds so good in it.

It does take time to understand very deep and sophisticated matters: such as correct tuning methods. The intonation affects the expressive meanings of every interval, every moment of the music, the behavior of the entire dynamic field (i.e. the music's texture). Those who wish to understand the music could perhaps give it a try sometime to understand how tuning fits into all this, and to know scientifically how and why it works, as yet another (but certainly not the only) aspect of knowing it.

Bradley Lehman wrote (May 20, 2004):
I wrote: < And, any assertion that tuning is merely a "down to earth" mundane matter is grossly mistaken; Bach's ideal tuning is (to venture an equally bold assertion) a representation of the harmony of the spheres. That's why his music sounds so good in it.
It does take time to understand very deep and sophisticated matters: such as correct tuning methods. The intonation affects the expressive meanings of every interval, every moment of the music, the behavior of the entire dynamic field (i.e. the music's texture). Those who wish to understand the music could perhaps give it a try sometime to understand how tuning fits into all this, and to know scientifically how and why it works, as yet another (but certainly not the only) aspect of knowing it. >

To explain that a bit further, here's something I drafted already about a week ago in response to the following provocation:
>> I was under the strong impression that you've made a major discovery in how to the tune the instrument and that this will resolve many problems and produce the authentic sound, not only of C.P.E. Bach. This is revolutionary. But now you suddenly describe this as some trivial excercise in everyday tuning. You are clearly tottering. If you've said A, you must say B. You ought to make up your mind whether you are going to present this as a new way of tuning or whether it's a mere adjustment "to what is most appropriate to the music to be played on the occasion."<<

There is no "tottering" here, at least by me; and those who dismiss it as "trivial" merely reveal their own ignorance of both the material and its importance. Herewith, let's cross some T's and dot some I's.

(1) Equal temperament is inappropriate for JS Bach's music (even though many people today like it just fine, and put it forth as a current preference, and probably 95% of music listeners don't know any better anyway). That incorrect status quo today is our own problem, not Bach's.

(2) All the known unequal temperaments are also inappropriate for Bach's music, although some of them have been "forced" to fit (uncomfortably) by picking different ones for different pieces by him, or by fooling around with adjustments to individual notes ad hoc, or by deciding simply to live with things that sound odd, and convincing ourselves that maybe he intended it to sound odd.

(3) The solution, if it exists, therefore must be some previously unknown unequal temperament. I have discovered which one it is, specifically as documented by Bach, and have been learning from it that (when it's taken seriously as it stands) it works for all of Bach's keyboard music, and perhaps also his ensemble music. All the messing-about of (2) will no longer be necessary! One can simply put Bach's temperament onto the instrument and then play anything by him. It's not surprising at all that Bach would know how to make his own music sound at its best.

It is not my task to prove (1) or (2) to the unprepared. It is my task to show what the solution is, show how it fits Bach's music, and (by a thorough analysis of its properties) suggest with some general remarks why it works as well as it does.

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That is, I incidentally open up some huge questions in music theory, music history, reception history, performance practice, and the meaning of music. Some of the implications are staggering:

- Relationships of consonance, dissonance, and the degree of dissonance are all shifted around somewhat, from anything we have known before. As a set, the available sounds are organized differently from some familiar patterns.

- If all the intervals in Bach's music have specific and distinct sounds, a complex web of inter-relationships as a complete package, there may be levels of meaning in his music (within pieces, and within groups of pieces) that no one has suspected or heard since his lifetime. This is a rare new window into the ways he may have thought, as a composer...at least as far as he chose to write some things down and not others.

- The differences of intonation also affect phrasing, timing, articulation, and other musical elements: at least for performers who are sensitive to the music's expressivity and to any of these previously hidden levels of meaning. That is, those of us who care adoing things Bach's way have some re-learning to do! (It's already changed my own harpsichord-playing and clavichord-playing, substantially, over the past six weeks of work with the "new" sounds.)

- Having the correct solution in hand, it is now simple to measure all other temperaments against it to see how wrong (quantifiably) and in what ways they sounded odd to Bach, and to see why (also documented) he preferred some of those over others. Incidentally, I was also able to prove (yesterday!) the usefulness of another theorist's analytical work from 1979, a specific statistical analysis as a model of understanding the music and its tuning; and I confirmed that his method yields both this solution I've found and another temperament (by somebody else, Bach's contemporary) that Bach is on record (by circumstantial evidence only) as enjoying, a solution different in several important details. That is, the other theorist's work of the 1970s was on the right track, although he didn't take it this far: it is able to provide a small set of "best fit" solutions, as further corroborative evidence that these solutions are plausible for Bach's music. (i.e. Scientific proof why they sound good, measurable data to add to the already overwhelming evidence that they do sound good in practice.) At the same time, that theorist's methods themselves are not sufficient to suggest this solution, but only to measure its effectiveness. And, as I noticed yesterday, a slightly different interpretation of his own results (using a sesquialtera relationship rather than doubling) yields an even stronger set of "best fit" solutions, where his own suggested solution makes an even better showing than it did already. Again, this merely confirms that he was mostly on the right track, but didn't take his own theory as far in print as he might have. [I've also found a small error of tabulation in his work as he presented it, having reproduced his numbers exactly but then getting a different linear regression mapped from them, although it doesn't really harm his own results. It merely shows that he was a bit more enthusiastic than careful, and that he didn't have Microsoft Excel to do his own least-squares analysis of a best-fit line. To get his line, one has to stick an additional constant into the equation to force the result, and there's no compelling reason to do so.]

- Instrument builders and restorers will have to grapple with this set of "best fit" temperaments as proven, in instruments that are designed to have Bach's music played on them and to stay in the same temperament for years (e.g., pipe organs....). It's not enough simply to put on temperaments that we're accustomed to, or that sound good to us, if we're interested in the way the music sounded to Bach in our quest to understand it.

- Bach's written solution, beyond its usefulness for his own and his students' music, may be a remarkably elegant and useful one for the complex balances in tonal music long after his death. That will have to be explored, both in theory and practice. That is, his musical legacy in the melodic/harmonic functions of tonality might have been shaped by the distinctive properties of THIS temperament (because it in turn shaped the way he thought and the way he developed musical techniques), propagating into Western music. In some sense, this goes back to the old searches for the music of the spheres, and perfection.

- Open-minded individuals may finally be able to let go of their incorrect conclusions that equal temperament was ever the only, or best, option for Bach's music. With Bach's own tuning method in hand, sounding better in his music, there's no solid reason to keep that misleading hagiography about equal temperament alive anymore...except perhaps by those self-serving musicians or critics who would overrule Bach's documented preference/intentions with their own priorities, not caring so much what Bach's own way was. The negative "proof" of equal temperament (an invalid one: the observation that all the known unequal temperaments sound wrong, and conjecture that therefore Bach couldn't have intended one, or had any specific choice in mind at all) no longer matters, due to the discovery of the correct tuning. The incorrectness of equal temperament, and the incorrectness of the other unequal temperaments, is now measurable. With the perfect platonic object in hand, the conformance (or not) of everything else to its shape is plain to see, by direct comparison.

I simply bring those up as issues that others will have to deal with. It's huge.

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Nor is this any trivial exercise in everyday tuning. People who have done "everyday tuning" for their entire careers (including myself, and many who are more experienced than I am) have not found this one, due to some misleading and blinding assumptions that informed the habits and expectations, and discouraged entire directions of investigation. I know this from review of the scientific literature on this topic, and practical understanding of the issues at the instruments. If anybody before me has found this solution, he or she has not written it down in any place that has become public knowledge. That's a void my paper is to fill.

Meanwhile, I believe Bach knew what he had (as an outstanding musician, knowing what sounds good, and knowing why it sounds good).

And I'm convinced, he did not want this to be a mystery. He spelled it out as clearly as he could do as practical instructions and in his musical examples, and he taught it to his students, and all his children grew up hearing it at home every day. This is not difficult stuff at all...provided that one comes to it with a background of knowing what's normal to begin with, and seeing where it could have come from, and knowing the practical things that players/tuners do in our everyday work, and knowing how theory and practice overlap or disagree. That is, thinking like Bach (through practical knowledge of the same things he did) enough to grasp what he wrote down, and to fill in the several details he didn't need to state directly, and to recognize the importance of the results. A recipe for Kirschtorte does no good unless one already knows how to cook pretty well, and has all the correct ingredients available, and has the commitment and patience to get it done; the master chef doesn't have to write down all that rudimentary stuff, and indeed does not waste his time doing so. (And a slip of the fingers gives an even more interesting word to chew on: Kirchtorte.)

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To understand this, it's just like musical interpretation, taking Bach's notation seriously (everywhere): Bach wrote down only the essence, the parts that are absolutely necessary to distinguish his ideas from other ideas; and we still need to know the other 75% (or whatever) filling in the everyday normal stuff he didn't bother to write down, because everybody already knew it or could get it from his directly spoken instructions.

Nobody on record has found this solution before, because this particular set of scattered clues hasn't been connected adequately before, and because others have been focusing on the wrong types of clues altogether, expecting answers to be where they are not. And many have forced coincidental or circumstantial information to seem more important than it really is, for lack of anything stronger to go on; but that doesn't work. Brute force tuning theory (as typically practiced both back then and now) and/or lexicography and/or historiography simply do not yield this solution, by themselves. Nor will those be successful who would try to knock it down with those same tools, coming to it from incorrect yet unquestioned assumptions; Bach created his own reality that doesn't submit to those expectations. According to all the available records of Bach's reality, taken collectively as pieces of the same huge puzzle, this is the solution he favored; one that is not automatically the same as anyone else's.

To find this I had to recognize and discard all those typical assumptions that had kept me from perceiving what's there; and then figure out what to do with what's left over. At that po, after the necessary flash of insight free of the distractions, everything fell into place: the work after that has been merely crossing a long series of T's and dotting a long series of I's, and marveling at how many long-standing problems are solved at this single sweep of the table. My paper lays out all the premises, forthrightly, and then shows why this solution satisfies them all. My sense of wonder at all this is magnified every day: and, isn't that the way a scientific discovery should be, along with any intuitive satisfaction?

In the case of these tuning rules, Bach's instructions are complete enough and perfectly clear, coming down to two similar and similarly reasonable readings of detail, both of which are stunningly elegant aspects of the same solution: but only to a reader who already knows all the required background (something which is rare, but which comes to those of us who really have done the work). One can't throw away wrong assumptions without first knowing what they are, and understanding why they're in the way.

My paper is an attempt to fill in all that middle layer that has been invisible: clairvoyance, clear-seeing, especially into the several most important clues. All I'm doing is helping Bach's written work to be more obvious and clear, revealing what's there. This is nothing odd; that's the job of any historically-informed performer who takes seriously the work of delivering the music on its own (and the composer's own) terms.

Yes, I understand "higher cognizance". And when it's backed up with scientifically demonstrable evidence, one really has something. Intuitions can be much more wrong than their owners would ever care to admit, even to themselves, and that's why careful and sensible scientific inquiry (done properly, else it doesn't contribute anything of value!) brings in the other side of the balance. (I've already had some of my intuitions
corrected at least twice during this project; and if I hadn't been willing to let them go, but had dug in defensive heels, I'd still be stuck at incorrect conclusions.) If there isn't such balance, the findings merely come across as made-up garbage, and indeed it's more likely that they are garbage, if no process of refinement is allowed to burn off the errors.



Continue on Part 6

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 | Meantone