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Temperament / Key Character / Tuning

Part 1



Bach's understanding of key character
Key character

Bradley Lehman wrote (May 4, 2003):
< Alex Riedlmayer wrote:
<< Bradley Lehman wrote:
Bach is doing a clever thing to illustrate the tensions in the text. >>
But how can you be so sure of Bach's cleverness, when I have denied it? Easily: you have put Bach's mind over your own, and, by extension, over mine. The immortality of the former is confirmed by your use of the present tense.

<< (And one would assume he's doing it deliberately.) >>
The proposition "Bach knew what he was doing" is credulous and can only be taken on faith. It implies that Bach imbued His music with an exacting comprehensive meaning, one which mortals can never fully discover >
."An exacting comprehensive meaning, one which mortals can never fully discover"? Huh?

All I'm saying here is: Bach knew very well what the various keys sound like in meantone temperaments, and in well temperaments. Organs he played were in various versions of meantone, as a standard. (And some still are.) Other wind instruments were also designed to use those specific twelve notes: A-B-C-D-E-F-G, F#-C#-G#, Bb-Eb. The other notes were by design not as well in tune in the temperaments in practical use. Composers understood this. This was practical musical knowledge.

As for putting one mind over another: I hear meantone temperaments and modified meantone temperaments EVERY DAY in my harpsichord practice, and this is my professional field. I know from experience how these temperaments behave, as a language of practical expression. Anybody who has plenty of keyboard experience in meantone knows these practical expressive quirks of the temperaments, knows what works well in improvisation and in composition, and knows what stands out as special
effects. And, when I look at the music Bach wrote, it is inconceivable to me that he did not know what he was doing. This is a very easy issue of faith for me, because I speak that same musical language (playing keyboards tuned in those same manners) and I recognize what he's done with it. He was playing with the musical resources that exist. They existed then, and they exist now.

How much practical experience do you have working in meantone? Have you, personally, ever played through the Chromatic Fantasy and Fugue (BWV 903) and the G minor fantasia (BWV 542) in any of the meantone, modified meantone, or well temperaments? If you have, if you've been there with your hands on the keyboard savoring those different characters as the music modulates, you KNOW that Bach knew what he was doing. (A clincher for me is the C#-major part of the Chromatic Fantasy.)

So, what's this mumbo-jumbo about comprehensive meanings that mortals can't discover?! Bach himself said that anybody who works hard enough at it can come to the same skills he had. If you haven't done so already, get your hands on a keyboard tuned in 1/6 comma meantone, and PLAY.

Hugo Saldias wrote (May 4, 2003):
[To Bradley Lehman] Even Bach knew what he was doing, Richter, Harnoncourt, Rilling and others ALSO KNOW what they are doing.

The problem is that many of us do not like it, not all the time but sometimes...

Well nobody is perfect and mistakes are always made because we are humans and these conductors are too... But the only think we ALL AGREE is that we like BACH.

Alex Riedlmayer wrote (May 5, 2003):
[To Bradley Lehman] I do not dispute your having played keyboards tuned in mean-tone and other temperaments. I have not worked with mean-tone keyboards, and I have no immediate plans to practice with them. That is why I do not wish to contest your understanding of practical applications of mean-tone temperaments, except for your presumption that it is universal among people who have played upon mean-tone keyboards, with J.S. Bach as a particular example. I would rather hear about what you might make of Bach's music than what you would have Bach make of his.

Bradley Lehman wrote (May 5, 2003):
For those of you who are not acquainted with the sound of 1/6 comma meantone temperament (which Bach knew from Silbermann organs, &c), I've prepared a little "Magical Mystery Tour." I have uploaded four mp3 samples to:

These are played by Otto Winter on a Silbermann organ; they are from the ultra-budget CD "Johann Sebastian Bach: Famous Organ Works" on the Pilz/Vienna Master Series label...a two-disc set that also includes a fine performance of the Goldberg Variations by Jaccottet. I wish that Winter were more flexible with his tempos and phrasing, but the point here is to hear the temperament and what Bach did with it.

- BWV 614, "Das alte Jahr vergangen ist" - a good example of the chromatic melodic lines I mentioned, where the semitones are two different sizes. Goodbye to the troubles of the old year (here they are), and the darkness of the winter solstice, happy new year!

- BWV 564, the Adagio of the "Toccata, Adagio, and Fugue" - the toccata explores C major, one of the best in-tune keys (because it has fewest sharps or flats in the signature), until its last few phrases (which I have included at the beginning of this sample)...bringing in the A-flat (really tuned as a G#). That A-flat is like a slap in the face that something new is coming. And it's this adagio section: a calm interlude for the most part, other than the mild surprises of the Neapolitan harmony (B-flat major, in first inversion). And then at the end of the Adagio there are nine bars of extremely chromatic music: a huge contrast with the rest of the Adagio, and with the unproblematic C major fugue that is to follow. Drama!

- BWV 542, Fantasia of the Fantasia & Fugue in G minor - a rollercoaster ride through wild modulations and into the weirdest keys (including E-flat minor, A-flat minor, and F minor)...and the extraordinary enharmonic shift at bar 38.

- BWV 622, "O Mensch, bewein' dein' Sünde gross" - an exploration of the extreme flat keys. Bach especially used the melodic poignancy of D-flat and A-flat here: notes that are really tuned as C# and G#. And then the C-flat major surprise of the penultimate bar (but spaced cleverly so the out-of-tune major third of that harmony is scarcely noticeable), in that final chromatic descent.... Mmm, mmm, mmm!

This 1/6 comma is the temperament that I've suggested is also the most appropriate one for the cantatas: both empirically from the effects Bach made with it (as noted in last week's cantata), and historically. This is also the style of temperament on oboes and the other wind instruments, by design: as I noted below, and in other postings here, the five accidentals used are F#-C#-G#, Bb-Eb.


And did you ever wonder why so much music from the 16th-18th centuries is in D minor? It's because that is the best all-around key that sticks to the twelve available notes, without misspellings!

Here's a fun fact: in the Toccata & Fugue in D minor BWV 565 (maybe by Bach, maybe not), all the notes are spelled correctly all the way through the piece...except for a few A-flats. Where do these occur? Take the total number of bars (143) and multiply by the Golden Proportion (0.6180339+) 88. And there it is: from bars 87 to 90 there is a pair of trills on G, with the note A-flat...wildly out of tune, as the
instrument is tuned with a G# instead of an A-flat! Then after one more A-flat in bar 96, everything through the end of the piece is spelled correctly, and very well in tune, even when it's chromatic. Is it coincidence that this dramatic use of a "wrong" accidental happens at the point of the Golden Proportion? I don't think so.... That is the most obvious place to put the climax of a piece, if the composer is acquainted with that number. (Bartok was another composer who got quite a bit of use from this principle.)

Here are some other places to explore the Golden Proportion:


Back to the Bach cantatas....

For those of you who perpetually whine about Harnoncourt's and Leonhardt's oboes being "out of tune": get over it. They are designed to play THESE notes, and they are doing it accurately according to this temperament which Bach knew. With our modern ears accustomed to equal temperament, this sound world may seem quite foreign and "wrong" and "out of tune": but again, get over it. (I mean, you're entitled to enjoy what you enjoy, and hate what you hate; but it's simply not accurate to say that they are "out of tune"! They are only "out of tune" against modern expectations.) The expressivity of the temperament is part of the music! It is easy to notice (and complain about) the notes that are "out of tune" to our ears, at the places where the composer has used the wrong (misspelled) accidentals; but also it should be noticed that all the other passages are much BETTER in tune than they are in equal temperament! It is a much "sweeter" sound, with more repose, more of a harmonic "center of gravity" when we are in the best keys; and more dramatic dissonance when we are not. (For every time that somebody whines about the Teldec series' oboes, I could just as strongly whine that all the Rilling/Richter/Ramin oboes are wildly out of tune, because they are playing equal temperament...and using vibrato...they are NEVER in tune, because there ARE no in-tune intervals in equal temperament except the octave!)

Although some of you may feel this is an overstatement: I suggest that performances of the cantatas and the organ music in equal temperament (while they may sound lovely) miss the boat, having none of this intonational expressivity. Equal-tempered performances may seem much more comfortable to us; but comfort is not necessarily the point of the texts, Bach's "sermons in music" in the cantatas. (And, speaking of comfort, when the music gets to those best keys, the comfort is also MORE comfortable.)

In short, when the temperament is a strongly tonal one as heard here, the music emerges with a much wider range of expression, more intensity all around...and when that's all missing (as in equal temperament, the atonal temperament) the music is washed out, bland, merely pretty. [In my opinion, of course. Everyone's mileage may vary.] Performers in equal temperament can of course do some other things to be "expressive"...but they've turned their backs on one of the most obvious resources available.

And this also might seem like an overstatement, but: "if you haven't heard Bach's music in these unequal temperaments, you don't know his music." There. I said it. Now I will duck and run for cover.

Enjoy. It's a different world.

Thomas Braatz wrote (May 5, 2003):
Brad Lehman stated: >>And this also might seem like an overstatement, but: "if you haven't heard Bach's music in these unequal temperaments, you don't know his music." There. I said it. Now I will duck and run for cover.<<
No need to duck and run for cover. Almost everything that you have stated makes sense and is very informative for listeners who wish to educate themselves about many characteristics of Bach's music. As you have demonstrated in your excellent discussion of Mvt. 2 (BWV 116) and now regarding the finer points of temperament illustrated on a keyboard instrument, all of us can obtain a deeper insight into Bach's musical world by heeding details that we (and many other performers of Bach's music) have overlooked in the past.

My objection to your defense of Jürg Schaeflein (?) playing on a Paul Hailperin (or did the instrument maker play his own instrument) 1975 reconstruction of an oboe d'amore (based on which model?) is that even if the he had been playing an instrument in 1/6 comma meantone temperament (the fact that the original model upon which the reconstruction is based is omitted here is very suspicious indeed), the extreme reedy sound coupled with an unsteady, wide (and fairly slow vibrato) makes it nearly impossible to hear the difference between the two temperaments and to attribute good musicianship to the rendition of mvt. 2 (BWV 116) in the Harnoncourt recording. Your point regarding the vibratos on the modern instruments playing the same aria are well taken: the fact is that there is a greater probability that reed instruments were played without vibrato in Bach's time than assuming that constant vibrato would have been present. It is the unmusical sounds created by this uncertain (in more ways than one) oboist that I find objectionable. It would be doing listeners a disservice to describe this kind of playing as 'authentic' because a few holes were drilled in different places or at a slightly different angles than those in modern instruments and then expect even a good ear to distinguish whether somewhere in the midpoint of this rather wide vibrato a note is being sounded at 1/6 comma meantone temperament vs. a note at equal temperament and then derive significance from this regarding Bach's understanding of key character.

Again, Brad, your description of what Bach is doing in mvt. 2 of BWV 116 is meaningful and worthy of consideration, but your insistence that Harnoncourt's oboist presents an 'authentic' sound with redeemable qualities which the listener must learn to accept in order to understand (and 'enjoy' if that is the correct word to use in the context of this aria) is one with which I can not identify.

Bradley Lehman wrote (May 5, 2003):
[To Thomas Braatz] I can already hear the howls of indignation that are on the way on this one. Before they get here, let me clarify the logic of my statement.

First, I am NOT saying that performances in meantone temperaments are the only correct (or appropriate or whatever) way to hear Bach's music. We can play and sing his music in many ways; he is gone, and we do as we please (what pleases and moves US, now).

Second, I am also NOT saying that meantone performances automatically reveal the music. Performers still have to do their (our) jobs putting the music across, using all the resources available!

I AM saying: these tonal and intonational aspects of Bach's music remain hidden until one has heard the pieces (at least once) in these stronger temperaments. We may think we know Bach's music from hearing it 100 times in various performances, but we haven't experienced these intonational subtleties (the ones I suggest he knew very well, and used as deliberate effects as vital parts of the compositions) until we've heard them in a strongly tonal temperament with the notes A-B-C-D-E-F-G, F#-C#-G#, Bb-Eb.

We may or may not like the strange sounds he came up with, especially in the enharmonic spice of notes outside that set, but it's part of the music and deserves a fair and serious hearing. That expressive stretching is (I suggest) a vital and important part of Bach's thinking, coming up as evidence in the music, and it should be taken seriously. And that's something that has not been done strongly enough yet, even among "Historically Informed" performers and listeners. That intonational aspect of the music is still being played down, swept under the carpet.

To me, this reduction of intonational content (because people don't like it, or whatever) is just as bad as the bowdlerization of a piece's words, changing the passages that make us uncomfortable. It's censorship. It softens Bach's expressivity to our own comfort levels, rather than taking his techniques seriously at face value.

Is that clear enough?

Neil Halliday wrote (May 6, 2003):
[To Bradley Lehman] Thanks for these extremely interesting examples , Brad.

But they confirm what I have suspected, which is that, for the listener, the temperament of the instrument is not all that important.

Unless you had told me this Silbermann organ was tuned to mean temperament, it would not have entered into my consciousness - indeed the only place where I really noticed 'strange goings-on' was in the bars beginning 31 to 34 of BWV 542, but again, unless I knew, I probably would have been unconcerned. I first heard this music played byFernando Germani on the (then) new Royal Festival Hall organ in London, and that performance was every bit as exciting as this example; presumably that instrument was tuned to equal temperament.

Indeed, in that fabulous section leading up to bar 38 ( the one with the last pedal quarter note being written as two 8th notes - E sharp and F natural (!) but presumably the listener is only to hear one note), Germani achieved greater tension and majesty than Winter, because he takes these passages at a more measured pace, rather than the slight increase in tempo shown by Winter. I can remember greater clarity in bar 38, despite the fact that Germani was probably using more stops (!) and I suppose it's an even larger instrument than the Silbermann.

As for that 'off the planet' section at the end of the Adagio in BWV 564, again, the discords here in this 7 part writing are extremely 'wild' and engaging, no matter what the temperament, and I would certainly not have guessed that this was in mean temperament.

I get back to my point about the timbre of the instruments, and the realisation of the continuo's figured bass, being more important than matters of temperament.

Do you prefer the Taylor/Haynes very expressive HIP example of the 1st aria of BWV 116 (with the soft organ part that makes only one or two lines audible), or would you like to hear the addition of some pungent diminished 7th chords etc., on the piano, punctuating the three music lines written on the staves, in this very chromatic and emotional music?

This is the question. I don't know if the Taylor/Haynes example is mean or equal temperament, and I don't really think it's important.

Neil Halliday wrote (May 6, 2003):
Thomas Braatz wrote: "....even if he had been playing an instrument in 1/6 comma meantone temperament (the fact that the original model upon which the reconstruction is based is omitted here is very suspicious indeed), the extreme reedy sound coupled with an unsteady, wide (and fairly slow vibrato) makes it nearly impossible to hear the difference between the two temperaments...."
Agreed 100%, Tom.

Matthew Neugebauer wrote (May 6, 2003):
Bradley Lehman wrote: < Performers in equal temperament can of course do some other things to be "expressive"...but they've turned their backs on one of the most obvious resources available. >
I wonder how much the word "composer" can be replaced with "performer".

There's another career goal for Matt Neugebauer!

Matthew Neugebauer wrote (May 6, 2003):
[To Neil Halliday] I think the problem is that you, me and others possibly can't directly recognize the differences in temperament, while Bradley can. This makes sense especially because Brad has spent many years playing in unequal temperaments, while we may not have. It really takes experience, and if I'm carefully listening to it, I might be able to hear it. Just a few minutes ago I listened to Gerhard Gnann playing BWV 614 and then the example Brad put up. While I didn't really hear a difference between interval x in equal temp and interval x in meantone, I actually sensed a darker mood in the meantone performance. Does this make sense?

As well, this probably is a huge reason why I prefer HIP performances over modern-it's just so subtle that I don't specifically recognize it, but can sense something different on the whole.

p.s. When I learn more about temperament, I'll try to implement it in my own work. There's just a few things I'm not sure about-by enharmonic differences/changes does it mean that a G# and an Ab are supposed to sound like different notes?

Bradley Lehman wrote (May 6, 2003):
< Matthew Neugebauer asked: There's just a few things I'm not sure about-by enharmonic differences/changes does it mean that a G# and an Ab are supposed
to sound like different notes? >
Matthew, yes. G# and Ab ARE different notes, and always have been, functionally (until atonal music came around). They are different notes, and they sound like different notes, in every temperament other than equal temperament.

Some numbers for you:

1/4 comma meantone: diatonic semitones are 117 cents, and chromatic semitones are 76 cents. (A huge difference!) That is, if G# and Ab were both available, they would be about 41 cents apart from one another: almost a quarter-tone!

1/6 comma meantone, a milder temperament: diatonic semitones are 108.5 cents, and chromatic semitones are 88.5 cents. (Still a perceptible difference!) That is, if G# and Ab were both available, they would be about 20 cents apart from one another.

Equal temperament: all semitones are 100 cents. Enharmonic notes sound the same as one another. [BOR-ing.]

As I explained earlier, "diatonic" semitones are those where the note name changes, such as F# moving to G. "Chromatic" semitones are those where the note name stays the same, such as G to G#.

< Neil Halliday wrote: "I think the problem is that you, me and others possibly can't directly recognize the differences in temperament, while Bradley can."

I fail to see that as a problem. If I have experienced the entire gamut of emotions from unbridled joy to blackest grief in Bach's music played on modern instruments tuned to equal temperament, why would I want to concern myself with matters of tuning, (or period instruments, for that matter)?
In conclusion, why bring up the matter of temperament - Brad can take his pick of temperaments (tuning), but the success or failure of the music will depend on a host of other factors. >
Neil, if you say you've "experienced the entire gamut of emotions from unbridled joy to blackest grief in Bach's music played on modern instruments tuned to equal temperament", great! But (I suggest) those joys and griefs are even more intense when the music is played in other temperaments.

This may sound like an advert for bleach detergent (or High Definition TV, or Monosodium Glutamate), but the brights really are brighter and the whites are whiter and the darks are more colorful, and the sweet parts are sweeter and the vinegary parts are more pungent. I suggest there's a gamut of emotions out there that resembles those same ones, but more intense all around. You haven't heard the "blackest grief" until you've heard what F minor and F# minor and some of the other far-out keys do in the more extreme temperaments. :)

If you haven't heard Bach's music in temperaments other than equal, you've heard only "Bach Lite." (Even if it may have been intense in other ways.) I agree with you, the success or failure of the music depends on many other factors as well. But this factor of intonation is also one of them.

Bradley Lehman wrote (May 6, 2003):
< Neil Halliday wrote: Bach wrote the Well Tempered Clavier for an equal-tempered keyboard; isn't it likely that he played BWV 542 on an organ with this tuning, thus allowing him to show of the daring modulations in this piece, without the unexpected 'beatings' or tremolo-like effects that occur with other tunings? >
No, he wrote the Well-Tempered Clavier for a well temperament. In the well temperaments, every key is different. Equal temperament is not well temperament. Well temperaments have some of the fifths pure (and none wider than pure). Equal temperament has no pure fifths: they are all narrow.

(Notwithstanding the many music-history books from the first half of the 20th century, where it is blithely stated that "Bach invented equal temperament" or some such nonsense, handed down from generations of people who did not do their homework on this.)

Bart Stolzel wrote (May 6, 2003):
< Bradley Lehman wrote: The different types of meantone, and modified meantone, and well temperaments, all give similar effects but vary by degree. Some are more colorful (or have stronger flavors) than others. It's a continuum by the amount of character.

And to make a comparison: with equal temperament (at the end of that same continuum), unless one has absolute pitch, everything is gray and everything tastes like chicken. Any effects that were composed into the music are washed out. All keys are the same, except for being higher or lower. The performers have to work much harder to keep the 's attention to a similar degree. >
OK Brad, you convinced me. Now whose CDs do I go to for characterful, colourful, flavourful non-equal temperament? Harnoncourt/Leonhardt? Who else?

Bob Henderson wrote (May 6, 2003):
Convinced me as well. How about a fistful of examples.

Bradley Lehman wrote (May 7, 2003):
[To Bob Henderson]
- The organ samples....

- My analysis of last week's cantata 116; read that while looking at a score and listening to the Harnoncourt recording....

- Edward Parmentier's CD "17th Century French Harpsichord Music" on a harpsichord tuned in 1/3 comma meantone (pure minor thirds!)....

- Blandine Verlet's recording of Bach's Chromatic Fantasy and Fugue on a harpsichord tuned in 1/4 comma meantone (pure major thirds)...a temperament that I think is too extreme for this music, but is interesting to hear anyway

- The set of Handel wind sonatas played by Bruggen, Haynes, et al on Sony/SEON....

- Lou Harrison's piano concerto (recorded by Keith Jarrett and a Japanese orchestra): it's in a well temperament invented by Bach's student especially strange well temperament as there is only one tempered note, but it's A (and the dicey fifths are D-A and A-E)....

Neil Halliday wrote (May 7, 2003):
Bradley Lehman wrote: "(Notwithstanding the many music-history books from the first half of the 20th century, where it is blithely stated that "Bach invented equal temperament" or some such nonsense, handed down from generations of people who did not do their homework on this.)"
Thanks for clearing this up, Brad. I suspected I was out on a limb with this.

On the matter of timbre and tuning, I notice in a later post you draw the contrast between the "brightness and tension" of equal temperament, and the "resonance" of meantone. This would be a positive for equal temperament, would it not?

Taking our specific example, the Silbermann organ would still 'lift the roof off' (which is what I meant by "timbre", ie, the excitement and drama of the sound) with the opening chord of BWV 542, even if it were tuned in equal temperament? (The equal-tempered instrument, with Germani at the console in London's Royal Festival Hall, certainly did).

Christian Panse wrote (Maay 7, 2003):
< Bradley Lehman wrote: You haven't heard the "blackest grief" until you've heard what F minor and F# minor and some of the other far-out keys do in the more extreme temperaments. >
A striking example for this is the Sacred Concert "Vater Abraham, erbarme Dich mein" by Heinrich Schütz, SWV 477. When the text comes to the words "an diesen Ort der Qual [to this place of torment]", the music modulates to F# major, which in non-equal temperament is really painful. In a performance using equal temperament, the singer can of course sing in some credible way about being tormented, but the use of the ill-sounding chord as direct demonstration of the text content would be completely missed.

Just my two cents,

Bart Stolzel wrote (May 7, 2003):
[To Bradley Lehman] Thanks for the examples Brad. Can I check that understand?

Just sticking to CDs of Bach cantatas: You are saying that:

All Leonhardt/Harnoncourt performances are well temperament and No performances by any of the other prominent figures in this field are well temperament

Bradley Lehman wrote (May 7, 2003):
[To Bart Stolzel] No, I haven't said that at all.

All I've said is: in Cantata BWV 116 Harnoncourt used a temperament that sounds like regular 1/6 comma meantone, or perhaps slightly modified (lowered E-flat); and Leusink in that same cantata uses a well temperament (i.e. something more moderate than that meantone, but also not equal temperament).

The big bad wolf

Bradley Lehman wrote (May 6, 2003):
< Neil Halliday wrote: On further listening, perhaps I should say I would have noticed something different about the tuning - I see that what I took to be the use of a tremolo stop in BWV 542 (unexpected in this piece) only occurs in certain places! Is this the "wolf' often referred to? >
A "wolf" comes up anywhere that a composer uses an interval that is spelled incorrectly. For example, if C#-F is played as a major third (but it's really a diminished fourth) it's a wolf and will be noticeably out of tune. Indeed, in some situations it sounds like a vibrato or a tremolo, because the beats are so fast and loud between the upper partials of the tone. (And that's how we tuners set the temperaments on the instruments: by counting the rates of beats in some of the intervals, specifically the correctly-spelled fifths and the correctly-spelled thirds.)

The even more serious wolf is the diminished sixth G#-Eb, when played as if it were a fifth. The ear tolerates less leeway in fifths than in major or minor thirds. That particular interval G#-Eb is the most important wolf.

Although, in a moderate temperament such as 1/6 comma, it's more like a border collie than a wolf. Nice doggie. It still bites, but it doesn't chew your arm off. In the tighter temperaments such as 1/4 or 1/3, that guy is nasty.

Alex Riedlmayer wrote (May 6, 2003):
< Bradley Lehman wrote: That particular interval G#-Eb is the most important wolf. >
But why this particular fifth?

Bradley Lehman wrote (May 7, 2003):
[To Alex Riedlmayer] Ooooo, you're not going to like this chicken-and-egg answer, but : because the music most commonly uses those notes Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#. If one is going to choose only 12, those are the most useful ones. The next two to add (in terms of usefulness) would be D# and A# because they are in the dominant harmonies of E minor and B minor; but composers got along quite well without those notes, for centuries.

Some other practical schemes that have historical precedent:

- Crank the G# up so it is halfway between where it belongs and where an A-flat would be; and it is only demi-awful in this rough service as either one. Optionally, do the same thing to E-flat (taking it down halfway to D#) and fudging as many others as you feel like doing. This was Grammateus' strategy.

- Retune several notes on the instrument if the music to be played uses different accidentals...most often bringing in a D#, thereby moving the wolf to be D#-Bb.

- Split the keys so the front half plays the sharp and the back half plays the flat (or vice versa)...I'm not joking, this scheme of 13 or more notes in an octave has been done. I've played a harpsichord on which the G# and the E-flat were split keys.

- Just "lie back and think of England" -- play the music on the given 12 notes and don't worry about it; get used to it; might even be enjoyable.

- Move the G#, C#, and F# up only part of the way (not halfway) and bring the Eb and Bb down only part of the way; preserves their identity but makes the opposite one more usable. In some temperaments of this style, F is also lowered so it becomes pure to C, or sometimes wide.

- Write music that cleverly puts the wilder intervals into spacings where they are less noticeable (chord inversions, making sure the wolf notes aren't struck simultaneously, disguise them with ornamentation, etc.)...there is a typical set of tricks.

- Enjoy the augmented seconds Bb-C#, Eb-F#, F-G#...they are so far away from normal 6:5 minor thirds that they sound instead like 7:6 minor thirds, a different pure interval...and are quite lovely as such; an interesting harmonic resource. (That's what makes the key of F minor so special: the way the G# can fake its way in there to make a startlingly in-tune tonic triad with the exquisitely dark 7:6 minor third in it.)

Timbre and temperament

Bradley Lehman wrote (May 6, 2003):
< Neil Halliday wrote: In any case, it's the magnificent timbre of this Silbermann organ which grabs my attention, and the meantone tuning I am inclined to regard as a minor drawback. >
Neil, the "magnificent timbre" of an organ and its temperament are inter-related! Here is the physical explanation:

1. In any organ that has tierces and/or mixtures, they are tuned pure to the fundamental. These are major thirds (plus some intervening octaves). The point of having mixtures on an organ is to reinforce the upper partials of the fundamentatone: hitting those overtones exactly. That's what makes the tone more resonant. Pure major thirds are much smaller (386 cents) than equal-tempered major thirds (400 cents). (**)

Now consider two notes a major third apart on the keyboard, such as C and E. The mixtures on the note C are reinforcing pure C's and E's and G's high in the series. The mixtures on the note E are reinforcing E's and G#'s and B's. If the fundamental notes C and E are tuned purely with one another (as in 1/4 comma meantone) or nearly pure (as in 1/5 or 1/6), those upper partial E's from both notes will also align. Big tone. But if the C and E are nowhere near a true major third (as in equal temperament) those upper E's will squeal against one another and never line up.

The effect of all this is: in the meantone temperaments, the overall sound of the organ (when playing chords) is more resonant because the tierces and mixtures line up and reinforce one another. In equal temperament, those high overtones are always fighting it out and beating against one another. More brightness, more tension, but less overall resonance.

(**) This can be demonstrated easily, even on an equal-tempered organ. Pull only a tierce stop on one manual, and only an 8-foot or 4-foot or 2-foot stop on another manual. Find the equivalent pitches on both manuals, and play them in succession (for example: playing an F [sounding A] on the tierce manual, and the equivalent A on the other manual). The pitches of the tierce stop will be noticeably lower than the notes on the other manual. That is because they are tuned to be pure to the pitch they are designed to reinforce.

2. Pipes on the same chest "draw" together...intervals that are already almost in tune pull themselves better into tune than they really are (which is a fortunate circumstance for the listener, and a headache for the tuner who is trying to set a temperament accurately). That is, if a major third is nearly in tune (such as in 1/5 or 1/6 comma) it will improve itself in practice, as the wind is shared between them: the valves of both pipes are open, and to some extent they resonate through the chest into one another. The beat rate (sounding like a vibrato) will slow down soon after both pipes are sounding. But if the major third is already far out, as in equal temperament, less of this "drawing" occurs; the interval simply beats with a fast vibrato. In equal temperament, the fifths are almost in tune, so those draw together; but the thirds don't.

3. Some overall effects:

- 1/4 comma meantone: major thirds are pure(!)--no vibrato--and fifths have mild beats.

- 1/5 comma meantone: major thirds and fifths both have the SAME beat rates, a slow vibrato.

- 1/6 comma meantone: major thirds have a mild vibrato and fifths have less.

- (...)

- equal temperament (which is 1/12 comma): all major thirds have a fast vibrato and fifths have almost none.

This beat-rate business is from the harmonic series in each individual tone. Those upper partials line up exactly (if the interval is pure), or are slightly out (slow beats/vibrato, like in 1/5 or 1/6 comma), or are far out. This is similar to point #1 above, and is true even if the mixtures aren't on; every pipe (or every string of a stringed instrument) is resonating with some combination of these upper partials, in various strengths. [That's what causes differences of timbre between different instruments: the proportions of that blend. It's like on a Hammond electronic "organ" where the player can sculpt tone colors by moving the sliders of the harmonics, rather like playing with an equalizer in a stereo system. The tone is assembled from those components, the overtones of the fundamental.] When those upper partials line up exactly, or almost line up, the whole instrument gives a more resonant effect as those upper partials reinforce one another.

Too much to go into here, but that's the gist of it. This isn't the time and place for a university course in acoustics.... :)

Continue on Part 2

The Keyboard Temperament of J. S. Bach: Article | Music Examples | Feedback: Part 1 | Part 2 | Part 3
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Last update: July 17, 2004 19:00:19