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On “The Well-Tempered Clavier” on a Well-Tempered Clavier
Recently WGUC in Cincinnati played a recording of a piece from The Well-Tempered Clavier played on a well-tempered clavier. Or a well-tempered something that sounded like a clavier, whatever a clavier sounds like, which I wouldn’t know from a harpsichord. Know from the sound of a harpsichord, I mean. (I’m confessing here about my ignorance of classical music, not bragging about that ignorance, which ardly hever appens.)
According to the DJ, the lady who made the recording had enthused over the glorious, rich experience of hearing The Well-Tempered Clavier played on a well-tempered clavier for the first time in her life, and was eager to share the excitement with the world, which is why she made the recording. She pointed out that she was finally hearing the music the way Bach intended for it to sound when he wrote it.
I admit I couldn’t tell the difference, but I am utterly fascinated by the quirky science and math involved in all this arcane stuff about tuning instruments and singing*, how it has changed over the millennia, and how it differs from instrument to instrument even today.
I feel the temptation to share what I’ve learned over the years, and especially over the recent months, and especially in the last 48 hours. But it is a pretty quirky subject, as I said, and I hear someone moving about upstairs sounding like she needs coffee, so I won’t.
Thanks for reading this.
* * *
*Like Pythagorean tuning, the infinite variety of just tunings, all the different mean-tone, well, and equal temperaments, scales (pentatonic, diatonic, 12-tone, and other), the tuning perfectionism of Barbershop Quartet singing, stretched octaves on acoustic pianos (but not electronic ones!), etc.
Published in General
I said I won’t share, but now that coffee has been served, etc., etc., and so on and so forth, I will take questions and educational comments and corrections from the audience, at least one of whom is an actual credentialed expert on music theory and its history.
I am not a musicologist, I just play one on Ricochet.
(But if your chorus’s second bass pulled up lame due to a Little League coaching injury and you need a sub, let me know.)
Fifty years ago, there was a loft in New York City, Zuckerman’s, where carpenters made assemble-it-yourself kits of harpsichords and clavichords. Quite a few Greenwich Village residents bought them and tried to re-learn what little they remembered from piano lessons in Akron or Shaker Heights. It was a little known, brief lived fad.
But until then I was unfamiliar with the clavichord. It is a quiet instrument.
Nice post, Mark!
I see now on Wikipedia…
Well, I could talk about this stuff for hours… pick a starting point. You’ve certainly hit a number of related issues right there.
The “Well-Tempered Clavier” is a collection of preludes and fugues, in each of the 12 chromatic major keys, and minor keys.
Some argue that it was intended to promote even temperament, as it hit all the keys. And some argue that it was intended for specific temperaments, but those were never specified. So I’m thinking it was more of a workout for trying out various temperaments. As well an incentive to avoid personal cliches by working in atypical keys.
I’m basically of the mind that, while temperament is mathematically fascinating, and you can nerd-out on it forever if you want, in practice it doesn’t matter all that much. Less in composition, and more of a performance attribute.
Fascinating story:
https://en.wikipedia.org/wiki/Wolfgang_Zuckermann
And still in production:
https://zhi.net
I agree!
(It must tell us something about God’s nature and his relationship to us, and about the relationship of science to the human mind and to pure math and logic. But I have no idea what and don’t expect to know until I get to heaven. I think it’s possible that no one else understands it, either.)
The gal who made the recording (and some other folks) think it was promoting well temperament, and the reason for using every key was to showcase how well it solved the problem, all the while demonstrating the variety of characters of the different keys.
But as you point out it is very controversial.
And which well temperament did Bach intend the pieces to be played on? Nobody knows for sure, though people have their pet theories. There is no clear historical evidence of what his organ and other keyboards were tuned to. I did notice just now on Wikipedia that he could not stand the way anyone tuned his instruments so he tuned them himself!
So I guess it was important to him, at least, if not to you and me.
Which is consistent with my theory that the WTC served as a workout for trying out different temperaments.
The Fisk-Nanney pipe organ at Stanford’s Memorial Church was built with 19 pipes per octave, and a big-ass lever (“großer Arschhebel”) that switches the black keys between two different tunings.
Quote:
The magnificent Charles Fisk organ is an eclectic 4-manual Baroque-type instrument of 73 ranks and 4,422 pipes. In addition to the usual Baroque organ features of mechanical action, pipes speaking directly into the room, bright and high upperwork, and short, straight and flat pedalboard, this organ is equipped with both French and German reeds and choruses, a Brust-positiv division with mean-tone tuning for early 17th-century music, and a special lever to switch the other three manuals from well-tempered to mean-tone tuning. This change in tuning is made possible by having five extra pipes (two for each “black” key) in each octave. The Brust-positiv which is fixed in mean-tone has two split-keys per octave (namely D sharp-E flat and G sharp-A flat).
Well, I didn’t know what the heck you were writing about, Mark, but I came back and found these delightful comments! So I’m in!
This was JS Bach’s view. He wrote music as a tribute to God, and thus sought perfect mathematical symmetry in his music, as such perfection would surely please our perfect God. The more I read about the way he designed his music, the more fascinated I became. Until I tried to listen to it.
I lack the understanding and the background to appreciate it. So I view Bach’s compositions as one of mankind’s greatest achievements, but I don’t enjoy them myself.
Bach was one smart dude. Amazing stuff. But I enjoyed reading about his music more than listening to his music.
Nothing fosters an appreciation for Bach like trying to play it.
I can only imagine.
“Wait, let me get my slide rule…“
For those who aren’t music physics nerds…
Musical pitches are ratios of frequencies. A major chord has pitches in the ratios 4:5:6. Simple and beautiful.
If you have a bugle nearby, you’ll see that it has no valves, and can only play multiples of a single pitch. And by varying lip pressure you can hit frequencies 3, 4, 5, and 6 times a lower frequency. And with that you can play all your favorite bugle calls (“Reveille”, “Taps”, “Charge”, “Never Gonna Give You Up”, etc.).
There are 12 notes to each octave, 7 white keys and 5 black keys. And we can assign musical pitches to each of these notes that are related by ratios of small integers. BUT, if you want to transpose, or play chords around a new key, everything is changed and it sounds very different. Often bad.
The ability to start on a different note requires the ratio of each note to the next to be the same. And for 12 notes per octave, that has to be the 12th root of 2, or 1.059-something. We call this “even temperament”.
The beauty of the system is that each of the 12 even-tempered notes are really, really, really close to simple and beautiful ratios. So it’s a question of appreciating this helpful mathematical coincidence as a gift, or ranting about tiny discrepancies.
Meanwhile, small offsets in pitch are often a good thing. The slight pitch offset between multiple instruments or voices creates a lovely chorus effect. On a piano, there are three strings under each note, and they’re tuned ever so slightly off so they don’t sound like a single string. So I would argue that this is a good thing.
The barbershop quartet is especially interesting because close voices naturally tune in exact ratios. And to tighten the harmony they’ll sing a “minor 7th” interval as a ratio of lower integers than normal (7/4 = 1.750 vs. 16/9 = 1.778).
Thx, this is fascinating.
I felt bad about starting.an article and then ending it without having made any sense.
But if you knew Kate without coffee in the morning you would understand that this was an emergency.
Just keep at it. You’ll get used to it. Trust me.
“Wagner’s music is better than it sounds.” – Mark Twain
Here’s an intro to the Fisk organ at Stanford Memorial Church with Robert “Organ” Morgan:
I know next to nothing about music but found this book fascinating:
Temperament https://a.co/d/3un1tMk
So for instance, let’s say that you and your ensemble were going to play “Louie Louie”.
“Louie Louie” is in a Mixolydian mode, the seventh is flatted. And that’s an important part of the song. Should that seventh be two fourths, 4/3 * 4/3 => 16/9 or 1.778 times the key? Or a minor third on a fifth, 3/2 * 6/5 => 9 / 5 or 1.800 times the key? Or the barbershop 7/4 or 1.750 times the key? Or equal tempered at 2^(10/12) => 1.782 times the key?
It’s a tough decision, and the math isn’t really helping.
I’m sure that questions like that kept the “Louie Louie” composer up at night…
Can’t say much about Wagner, but the trick with Bach is to understand that there isn’t one kind of “Bach music”. There are a hundred kinds. When you say you don’t like it, you really mean you simply haven’t run into the pieces that you would like.
An example off the top of my head, he wrote cantatas about the death of Christ, as well about a girl addicted to coffee (considered somewhat disreputable at the time). Some of his instrumental works are light and sparkling; others suitable for funerals. (Not to mention horror movies!)
Bach left us over a thousand works, so there’s something for everybody.
There’s parts of the Goldberg Variations (I think that’s what it’s called) that I like. But you’re right – the variety and volume of his output is amazing.
Indeed. For those who aren’t up on this…
Plucked string instruments have this issue where the ends of the string are a little more rigid than the rest of the string due to being held down at the bridge or fret. (‘Makes sense…) And this causes the higher harmonics to run a little sharp.
(If you connect an electric guitar to an oscilloscope and pluck a string, you can see the waveform has ripples on it and they’re moving to the right when the rest of the waveform is standing still.)
So when tuning a piano, they use “stretch tuning” where the lower notes are tuned slightly lower and the higher notes are tuned slightly higher, so that the higher harmonics of the lower notes are in tune with the higher notes.
I take this into account when I tune my guitar. I tune the first two strings normally, I use the 12-fret 2nd harmonic to tune the middle strings, and the 6th fret 4th harmonic to tune the bottom strings.
‘Just noting that this tuning stuff can get endlessly complicated.
Wikipedia gives a competing explanation of stretched octaves. This is a good example of the complexity.
This: https://en.wikipedia.org/wiki/Stretched_tuning ?
Wow.. it’s very wrong, and very badly written.
I meant this one…
https://en.wikipedia.org/wiki/Pseudo-octave#Stretched_octave
But now that you mention it, I think I saw a third competing theory a few days ago. If I can find it I will let you know.
My vague recollection is that it explained the inharmonicity of a vibrating string simply by the fact that it had non-zero stiffness (resistance to bending.) Presumably the model that predicts that overtones will all be harmonics assumes no bending resistance. Actually, that is consistent with the theory I was taught! The only restorative force considered was due to the tensile force.
Oh dear… that whole article is bogus. (Check the “talk” section.)
String stiffness, by itself, will just cause the vibration to damp out quicker. And it wouldn’t explain sharpening the harmonics.
Here, let’s take a look… [Google up an image…]
A vibrating spring will sustain a note ringing with lots of harmonics.
It’s clear that the higher harmonics require more flexibility at the endpoints. But the endpoints have less flexibility as they are fixed. So the length of the string is effectively shortened for the higher harmonics. Which sharpens them.
Another example; a guitar bridge needs to be adjusted for “compensation” to play in tune, which involves moving the position of the bridge back a little bit.
[Google up an image…]
Here you can see the high E string is forward while the low E string is set back aways, maybe 3/16 of an inch.
Strings are less flexible at their endpoints, and the lower strings have more mass and thus less flexibility at their endpoints. So the effective length of the string is shorter. And playing notes higher up the neck will run a little sharp. And the bridge can compensate for this by being positioned back from the theoretical location, more for the lower strings.
Nylon strings are way more flexible than steel strings, so nylon string guitars don’t bother with this.
When I was in high school, a member of our church bought and assembled one of those harpsicord kits. I attended the inaugural concert, and so heard a couple of the preludes and fugues from the Well-Tempered Clavier (don’t remember which book) on that “original” instrument. As I already liked a lot of Bach, it was great. But I enjoy them on piano even more. Given that “clavier” meant whatever keyboard instrument was handy, I think these are also as “Bach meant people to hear them.”
(I may have used the wrong terminology above. When I said “even” temperament I should have said “equal”temperament.)
Here’s a diagram I drew up for an article I was writing about temperament but never got around to finishing.
The vertical lines are the pitches of the 12 notes (plus octave) of the equal tempered chromatic scale, in a log format so they’re equally spaced.
And the orange circles are pitches of the notes if they were ratios of small integers.
The tan beams link ratios that reflect the octave; that is one times the other is 2. (4/3 * 3/2 => 2)
The diameter of the circles is 50 cents, half the distance between semitones.
So you can see that the fourth, F, is almost dead on, while the major and minor thirds (E, Eb) are off by a bit.
And you can see which notes to complain about.