Piano World Home Page Forums Our Most Popular Forums Piano Tuner-Technicians Forum I've made up my mind about Bach's WTC and ET

http://imslp.org/wiki/Das_wohltemperierte_Klavier_I,_BWV_846-869_(Bach,_Johann_Sebastian)

(laughing out loud at how a simple "I agree with Phil" elaboration turns into another Bill Bremmer Special Essay that inevitably won't be read!)
You don't skimp on your verbiage do you, Bill. I bet you're great fun over a beer or two

For many of us here, there could not be a clearer description of what you have taken on in this thread! It is the zeal which you try and get others to agree with your ET quest that keeps this thing going - most wouldn't care a bit if you just said that you don't like listening to the WTC in anything but equal...

Ron Koval

No, of course it won't be read or if it is, it will be disregarded. My arguments are "weak". I "misquote" him. Jeff obviously knows I am an advocate of Historical Temperaments. Everybody who knows me, knows that.

It is a tough position to take and I have stood my ground on it fro 23 years, more than half of my career as a piano technician. I was once a skeptic myself but it was not what anyone said about them, it was what I heard that convinced me. I had been told the same thing once by an eminent Jazz artist, "You don't need to explain anything to me, I heard it!"

20 years ago, I knew of just a few people who tuned Historical Temperaments. They were quiet, timid and secretive about it because if they said or wrote much at all, they would get blasted and ridiculed.

It seems that the very idea of an unequal temperament is disturbing to some people. It goes against the grain of everything they always knew and believed in. To continue to write about them publicly rubbed many people the wrong way. There was a musical once that had the line, "There are certain things you just don't do!" To tune a piano in a way that is different from what another person thinks is the one and only right way is one of those things that you just don't do!

So, here we have a person trying to prove the unprovable and indefensible. Practically no one has supported his efforts but plenty of people have said that he is just plain wrong about the premise. I am afraid the no amount of verbiage, no amount of statistics will convince him. There will always be that doubt and the lingering notion that if Bach could have written the WTC music for ET, then he did!

I really do hope that Doel (or whoever it was) will put up some comparative samples. Not for whichever "sounds" better to whomever; that is not the point as Jeff so aptly points out. One may still like the ET version better as the WT version sounds strange to those who prefer ET. To show, however that the music works in either a simple or remote key, whichever case that may be and is adversely affected when transposed to an inappropriate key. We already know that transposition in ET will have no effect one way or the other.

Kees,

Do you mean that you could obtain BC bud in Amsterdam in the 70's? I am curious, because I am also trying to understand the "Triangle Trade" of Bach's time, and am trying to bring this idea back to the "Topic At Hand" for the value of the historical significance. Context, you know...

--Andy

Can someone please provide a link?

It was Phil who brought up the BC bud, not me. In Amsterdamned there was mainly Lebanese and Nepalese hasj available.

Coffee was Bach's drug of choice as it is for me.

Kees

Kees

Emphasis added. This is critically important. Jeff, if you consider only two paragraphs of Bill's post, consider these.

I am all for it, Jeff. Go for it! There is much to be discovered in your effort! But as you do, please address Phil's questions. They are important, and will strengthen your study. Also, consider your edition. Not all minor pieces end in major chords, depending on which edition you consult...

Yes. Yes. Is see That That is so. Is that why Gould played so much Bach so fast?

Yes I know. It seems that when I say "I think..." it is read as "I think YOU should think..."

It is probably going to get worse after a few more posts.

Folks:

I did not go into a whole lot of detail when I described how and why I chose the type of analysis that I did. I was not expecting much interest. Much was for a simple, repeatable test case that does not require musical evaluation, something that a computer would do. The clearest way that I can explain is that these are the note numbers of the highest tonic notes whose fifth partial has a nearly coincidental partial in another note. I used the Alfred Masterwork Edition. All 48 pieces end in major chords in this edition, unless I made an error.

C 28, 40, 28, 40
C# 29, 41, 29, 29
D 30, 18, 42, 30
D# 31, 19, 31, 19
E 20, 20, 32, 20
F 21, 45, 33, 21
F# 46, 22, 22, 22
G 23, 35, 23, 23
G# 24, 24, 24, 24
A 25, 25, 37, 47
A# 50, 26, 26, 38
B 27, 51, 27, 39

So the average note number for the remote keys is 42, or D4 28 or C3.

And the average note number for the close keys is 47.25 or about G4 31.5 or between D#3 and E3.

Is this significant? Maybe yes, maybe no. In ET this would mean the remote keys were used more harmoniously. In WT, well I think it would depend on which WT. That is why I chose to use note numbers rather than beat rates. For beat rates comparison I think an approximation can be made by taking the average beatrates of the three M3s above and the three M3s below the average note number for each group. Then various WTs could be compared.

But before doing so the question should be raised as to what would it mean if the average beatrates were the same or if one was much greater than the other. If you wait until after they are known it can be just rationalization.

Thinking along these lines is why I am up so early. Also I may not get back to the keyboard much before Monday. (I am going to be getting some experience on function block programming on a particular PLC.) Here is the question that is going through my mind: During Bach’s time, was WT used for tonal color or was WT a compromise between MT and ET with the tonal color only an artifact? If Bach did compose WTC in WT, it may have only been because ET could not yet be tolerated with all the busy RBIs. But MT prohibited the convoluted passages that Bach desired. And tonal color was a by-product, not the intent. Later composers might have utilized the tonal shading more.

Now I am not saying that this very simplistic analysis would answer such a question. But trying to decide what the result might mean does. For instance if the average beatrates are the same, that could mean that the desire was not for tonal shading but for harmonious equality, a tendency towards ET. (hoohoo, I bet that’s what the beatrates will show. The interval from D4 to G4+ is about halfway around the circle…) But if tonal variation was the goal, wouldn’t the average note number be about the same so the remote keys DID beat more?

Jeff:

I think the answer to your question is well put in the piece I quoted before:

The Revolution of Bach’s ‘Well-Tempered Clavier’
http://www.larouchepub.com/eiw/public/2010/2010_20-29/2010-24/2010-24/pdf/44_3724.pdf

In 1691, the German organist and mathematician Andreas Werckmeister published a treatise entitled, “Musical Temperament or . . . mathematical instruction how to produce . . . a well-tempered intonation on the clavier.” Bach, Werckmeister, and others who supported the well-tempered system, rejected the previously held idea that musical intervals in the physical universe, had to conform to abstract mathematical proportions. This idea had put a straitjacket on the musical universe, limiting it to only those keys in which “pure” intervals could be played.

The new movement, of which Bach was a leader, created systems in which it would be possible to play music in all keys. The “comma” (the part of the octave that is left over if only mathematically “pure” musical intervals are used) was distributed unequally throughout all of the keys. (Different keys had different-sized intervals, giving each key its own nuance or “color,” creating a “musical palette,” which is lost in the modern practice of “equal-tempering,” where all half-notes have the same value.) It were then possible both to write music in every key, and to modulate—to move from one key to any another—within the same piece of music, in a way not possible before.

From the structural point of view you mentioned my guess is that Bach would not have been too worried about the distinction between WT and ET (which is why concert pianists are happy to play WTC today). But that would be only half the story.

PS If JI and MT were a straitjacket of abstract mathematical proportions then so, arguably, is ET!

An instrument is never perfectly in tune. A "slightly-off" unison is, mathematically-speaking, much more dischordant than a perfectly-tuned major 7th. But human perception is a wonderful thing. We hear what we expect to hear, what we WANT to hear.

thank you for your enlightenment.

Two things I don't understand, what coincident partials do you use to tune your perfect major 7th. And how does a piano technician tune and tone regulate a piano if they can only hear what they expect to hear or want to hear?

If anyone is following what I am doing, here are the beatrates and their averages.

For the remote M3s, M10s or M17s centered around C3:

E3 7.3
D#3 7.1
C#3 7.9
B2 5.8
G#2 5.5
F#2 4.7
Average 6.4 bps

For the close M3s, M10s, or M17s centered around D#3-E3:

A3 7.8
G3 5.2
F3 4.7
D3 4.6
C3 2.9
A#2 4.3
Average 4.9 bps

Fwiw the distance between the ET M3s with these beatrates is a P4.

I still don’t know what it might mean. And when it comes down to it, this is examining evidence for WT, not for ET, which I think needs to come from the convoluted passages.

I remember a better example. Doel mentioned this in another Topic. I can't remember if it was the same one that discussed the difference between "texture" and "form" as it relates to pieces affected by WT.

If I remember right, Doel said that Bach wrote the C# Prelude and Fugue in C and then transposed it for C#. I have have every reason to believe him. And I am thinking more about what he said in regards to what I call "texture" vs. "form."

I can see how it all might sound like double talk.

Phil, Andy:

You encouraged me to do the analysis. Any comments, or is it time to let this Topic die?

Sure Jeff, I read your analysis before the weekend but didn't get round to replying. I must say I do like your method now I understand it.

The difference in the averages is a quarter of an octave, so I think it is significant. Especially with so many inconsistencies between the different pieces in the same key throughout the works. But there are many anomalous results. The most glaring is the first D major prelude. That's a major 10th from the bottom D #18... but as you say, the beat rate is the same as if it was #30, so it should be the same result in the texture of the chord. Is there something I'm missing here? Would it not be better to treat it as if it was a M3rd on #30, and the same with any other M10th? I've just found a M17th as well... I guess it's not quite as clear cut as that, a M10th clearly has a thinner texture than a M3rd regardless of the beatrates.

Looking at the score, something else comes up about that chord - the presence of a minor third directly above it. It would be interesting to also look at whether these crop up more often in close or remote keys, and where in the compass.

I also see you don't make provision for pieces where there is just a unison or octave at the end without the 3rd in it.

Criticisms aside, it's certainly interesting. And I think further investigation would throw up some more results. I'd enjoy doing it, if I get round to it... so I might add to this topic in the future!

I also wondered whether the trend is different between the different sets of 12 - the first preludes, first fugues, second preludes and second fugues. So I split up your results, and took the averages again.

For the remote keys, the averages are, for the different sets in that order:
29.5 29.5 27.5 25.5
And for the close keys
29.5 31.5 31.5 33.16666667

So there's no difference in the preludes overall in the first book, and a tone difference in the fugues.
The second book has much bigger differences. A major third in the preludes, and just over a major 6th in the fugues.

So maybe Bach got more out of the WT over time. You could interpret that to mean he was being more expressive with the temperament the second time round, or he was just mitigating better against the unevenness of it. So I don't think it supports the idea that it was written for ET, but it could indicate a desire for it - he avoided the fast beats more over time, so if he'd had ET maybe he'd have liked it as he wouldn't have to work around the bad intervals.

So a little bit of objective data... but still totally subjective interpretation

Some inner voice tells me that it will be futile to become too analytical with WTC intervals, beat rates, modulations and keys etc. It might be intellectual fun to try but there will always be inconsistencies and exceptions as in so much of art and nature.

If I could summarise all that I have ascertained from my studies and through this thread: All Bach was trying to demonstrate was that a keyboard could be tuned in a manner that reduced dissonance in remote keys to an acceptable level and yet did not unacceptably increase dissonance in close keys. A keyboard instrument is therefore capable of making acceptable music in any key. It is that simple. As a consequence I am not convinced that for WTC Bach was too concerned about exact tuning to any particular scheme and that a range of tempering could be acceptable which could be exact equal temperament if that were possible. Isn't it obvious that if Bach had a specific tuning in mind that he would have provided more instruction than a cryptic squiggle?