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Posts Tagged ‘scale’

Well Tempered Clavier Title Page
(Wikimedia Commons)

Well-Tempered Clavier. What kind of a title is that, anyway? If you have a vague idea that it has something to do with how good the clavier (whatever that is) sounds, you’re cruising round the right neighborhood.

The musical octave – from C to C on the piano keyboard, for example – is a basic building block of music. Within the octave, there are certain intervals – the difference in pitch between one note and another – that have given us the fundamental sounds of Western music since the Middle Ages. These include the fifth and the third.

But here’s the problem: these intervals don’t quite come out even with the octave. To put it another way, the intervals that make an octave sound good and true and right to our ears aren’t compatible with the pitch intervals that make for a velvet-smooth third or a sweet, consonant, glorious fifth.

Suppose you have an instrument with the 7+ octave compass (range of pitches) of a modern piano, but where the pitch of each note is completely under your control. (One candidate that comes immediately to mind is the Theremin, one of the earliest electronic instruments. You can hear a Theremin in, of all things, the Beach Boys’ Good Vibrations.)

Play your instrument’s lowest note. Now, play 12 ascending perfect fifths. When you get to the last note, it should be exactly 7 octaves above the first note. (Count ‘em for yourself.)

But it isn’t! It turns out that when you play 12 consonant perfect fifths, the last note overshoots 7 octaves by just a little. Not a lot – not even a half step – but enough to be noticeable.

Herein lies the reason for temperament. Temperament systems cram fifths and thirds into a mathematically perfect octave by main force – by making the thirds and fifths something less than perfect. It’s kind of like the calendar, where some months have 30 days, some have 31, and one has 28 – or, some years, 29. The tweaked intervals won’t sound quite as rich, quite as right, but the octaves will come out even.

The obvious way to do this is to simply make all 12 notes of the octave evenly spaced. This is called equal temperament. It was probably the first temperament system invented, and it’s still used today. In fact, that’s the way the modern piano is tuned, which is why you couldn’t do this experiment on a piano.

Equal temperament may have been the first solution to the problem, but it was far from the last. Quite a few musicians just didn’t care for the way it sounded, so they made up their own temperaments, different ways of distributing the error round the octave, adding a little here, subtracting a little there. Usually, they managed to make thirds and fifths sound close to perfect in some (not all) of the possible key signatures. The tuning systems they devised are generically called mean-tone systems.

In mean-tone systems, the varying distances between notes of the scale meant that different keys had different musical characteristics. C major might be (and was) described as the key of joy and sunlight. D major was called the key of triumph. G minor was the key of darkness and despair. For example, Mozart’s powerful 40th and 25th symphonies are written in G minor.

Into this minefield of different tunings steps Bach (if it isn’t too much of a nonsequitur for me to bring him in after Mozart).

I’m no Bach scholar, but everything I’ve read about him suggests a man almost obsessed with numbers and mathematics. For Bach, numbers had deep spiritual meanings. He attached significance to the numeric intervals in a fugue’s subject, its length in number of notes, the number of measures between entrances, and much more. Some musicologists have built their entire careers (or at least their master’s theses) on unearthing and divining the meaning of these arcane relationships.

Now, Bach’s life was music. For him, this flaw in his world must have been an endless source of frustration. But his answer wasn’t equal temperament; that’s not what the Well-Tempered Clavier was about. Nor was Bach showing off some new system of temperament he’d invented.

Rather, the Well-Tempered Clavier was Bach’s argument for a tuning system – someone else’s invention – that he called "well temperament."

Remember what I said above: mean-tone systems make different key signatures sound different. They make some keys sound better – more in tune, with those nearly-perfect thirds and fifths – and some worse. Most keyboard players and composers dealt with this by simply avoiding the keys that didn’t sound good to them.

Bach threw that practice back in their faces. The Well-Tempered Clavier comprises two books of preludes and fugues in all 24 major and minor keys. The keyboard player who bought a copy, but insisted on sticking with mean-tone tuning, would have only two choices. He could either put up with certain intervals being out of tune in the more remote keys (the ones with lots of sharps and flats), or else just not play those pieces.

The question is, how did Bach tune his harpsichords? Exactly what is this tuning system he promoted, the one he called "well-tempered," giving it the name we still use today?

The sad fact is, we don’t really know for sure. (We do know, though, that no one else could tune a harpsichord to his satisfaction.)

However, we can guess at a few candidates. The one most often suggested is a system invented in 1691 by organist Andreas Werckmeister, which Werckmeister said was for the "chromatic genius." You could say that Werckmeister’s system is a compromise between mean-tone and equal temperament. It preserves much of the distinctive character of the different keys, but makes all 24 major and minor keys – and all of Bach’s WTC preludes and fugues – playable.

Bach made his point. In the end, though, he lost – not to the mean-tone mavens, but to equal temperament. Today, few harpsichordists and pianists routinely tune their instruments in any system Bach would say was "well-tempered." Ironically, the simplest and most direct answer to the problem won out. For today’s keyboard instruments, equal temperament is nearly universal, even among musicians who otherwise embrace the principles of historically informed performance.

Before I close, one last thought about the Well-Tempered Clavier. What’s a "clavier"? Is that a clavichord, as in Well-Tempered Clavichord, the title you used to see on recordings many years ago? Well, it can be, but it’s not just that. Clavier means keyboard – that is, the part of the instrument your fingers actually play. Bach simply intended the WTC for any instrument that has a keyboard. You can find modern recordings of the WTC played on the the harpsichord, the organ, the piano, and – yes – even the clavichord.

Further reading:

Introduction to Historical Tunings by musicologist and educator Kyle Gann

The Wolf At Our Heels by Jan Swafford in Slate

This article was first published in WKSU Classical on 25 April 2010.


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