Equal Temperment Tuning; The Wolf at Our Heels

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The Wolf at Our Heels: The centuries-old struggle to play in tune.
By Jan SwaffordPosted Tuesday, April 20, 2010, at 10:08 AM ET

You are about to enter the Twilight Zone. I submit for your consideration an oddly named book lying on an ordinary desk: How Equal Temperament Ruined Harmony (and Why You Should Care), by professor Ross W. Duffin. This book was written by a madman. Or is he? You should understand: If Duffin is mad, he's not alone. And the spaces between the lines of his book are filled with the silent laughter of the gods.

The gods are laughing at their little joke on musicians. When it comes to the tuning of instruments, especially keyboards and fretted instruments, nature drops a giant hairball in our path. Here's a short course on the arcana of tuning. It will take us to the meaning of a celebrated collection of keyboard pieces: J. S. Bach's The Well-Tempered Clavier, humankind's greatest musical riposte to the laughter of the gods.

In dealing with tuning, there are two main terms to know. One is interval. It means the distance between notes. The basic science of intervals was laid out in ancient Greece, perhaps first by the mathematician Pythagoras. The first notes of the C major scale are C, D, E, F, and G. The note E is the third note up from C, so the interval C-E is a third. The note G is five notes up, so C-G is a fifth. So musical intervals run second, third, fourth, fifth, and so on. (Some intervals can be major, like F to A, or minor, like F to A flat.)

OK? Now, as Pythagoras discovered, intervals are also mathematical ratios. If you take an open guitar string sounding E, stop it with your finger in the middle and pluck, you get E an octave above. The octave ratio, then, is 2:1. If you stop the string in the ratio 3:2, you get a fifth higher than the open string, the note B. The other intervals have progressive ratios; 4:3 is a fourth, and so on.

So far, all very tidy. But this is where things get hilarious. As Pythagoras also realized in mathematical terms, if you start with a C at the bottom of a piano keyboard and tune a series of 12 perfect 3:2 fifths up to the top, you discover that where you expect to have returned to a perfect high C, that C is overshot, intolerably out of tune. In other words, nature's math doesn't add up. A series of perfect intervals doesn't end at a perfect interval from where you started. If you tune three perfect 5:4 major thirds, it should logically add up to an octave, but it doesn't; the result is egregiously flat. It is this innate irreconcilability of pitch that, through the centuries, has driven men mad. Professor Duffin is a living representative of a long line of obsessives. Personal and institutional battles have been fought over the issue of tuning, fame won and lost. It was ever thus, wrestling with the gods.

What all this means in practice is that in tuning keyboards and fretted instruments, you have to screw around with the intervals in order to fit the necessary notes into an octave. In other words, as we say, you have to temper pure intervals, nudge them up or down a hair in some systematic way. Otherwise, you get chaos. So there's the second word you need to remember: The business of adapting tuning to nature's messy math is called temperament. And now we're halfway to understanding The Well-Tempered Clavier: It has to do with the art and science of keyboard tuning. We'll get to the wellness in a minute.

There have been some 150 tuning systems put forth over the centuries, none of them pure. There is no perfection, only varying tastes in corruption. If you want your fifths nicely in tune, the thirds can't be; if you want pure thirds, you have to put up with impure fifths. And no scale on a keyboard, not even good old C major, can be perfectly in tune. Medieval tunings voted for pure fifths. By the late Renaissance the tuning systems favored better thirds. The latter were various kinds of meantone temperament. In meantone, most of the accumulated fudges were dumped onto two notes, usually G# (aka A flat) and E flat. The shivery effect of those two notes played together in meantone temperaments earned it the name "wolf," which, like its namesake, was regarded with a certain holy fear.

By and large, in composing music for meantone keyboards you avoided the wolf, so never, for example, wrote in the key of A flat. In fact, those temperaments left only a few keys that were well-enough in tune to be usable: the keys between two flats and three sharps. Between the 16th and 18th centuries a lot of splendid music was written in meantone tuning, within that range of a dozen major and minor keys. But the inability to write in all 24 possible keys ate at composers' guts. More and more, there was a demand for a tuning system that would render all keys usable—and escape the wolf.

One of those tunings was already known to the ancients: equal temperament. Here the poison is distributed equally through the system: The distance between each interval is mathematically the same, so each interval is equally in, and slightly out of, tune. Nothing is perfect; nothing is terrible. So now it's all fixed, yes? The laughter of the gods has been stilled, right? Are you kidding? You fools: The gods never lose.

For centuries, equal temperament didn't catch on because musicians tended not to like it. Even when fretted instruments were invented and lutes and guitars were mostly tuned in equal temperament, they still didn't like it. Most especially, musicians didn't like the fat major thirds of equal temperament, which are way out of tune with nature. They preferred the sweet thirds of meantone temperaments, with all their limitations. For another thing, in meantone each key had an audible personality, from, say, the almost-pure and upstanding C major, suitable to moods of equanimity and celebration, to shadowy C minor, suitable for doubt and despair. Equal temperament leaves every key with exactly the same personality, which was widely felt to be boring. Musicians still preferred, then, the old varieties of what is generically called unequal temperament.

In the late 17th century, tuning geeks came up with a new idea: Let's hair-split all over the keyboard, tweaking this and that in minuscule ways, letting, say, a third be a bit larger in one spot and a bit smaller in another. These kinds of flexible temperaments accomplished several things at once: 1) They made all keys usable; 2) yet they preserved the individual character of keys, because each still had its distinctive collection of intervals; 3) and they tamed the big bad wolf.

Hey, said adherents of this more sophisticated unequal system, this really works well! So they called it well-temperament. One of those adherents was J. S. Bach. He wanted, he said somewhat testily, to write in any damn key he felt like, and he tuned his harpsichord himself to make that possible. When a famous organ tuner who did meantone tuning showed up, Bach would play an A flat major chord on one of his organs with its howling wolf, just to torture the old man.

Bach wrote the preludes and fugues of The Well-Tempered Clavier (clavier meaning any kind of keyboard instrument) not only to show off this improved system but to help make well-temperament mandatory by writing irreplaceable pieces in every key. Anybody who wanted to play from the WTC was pressured to use well-temperament, because many of the pieces sounded sour in any other tuning. (However, heh-heh, there's no precise record of which well-tempered system Bach used.)
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The various kinds of meantone and well-temperament help explain why, in the 18th into 19th centuries, keys had particular emotional associations. Key descriptions of the time sound outlandish, and indeed some were on the loony side, but they were founded on the reality that in unequal temperaments each key had its distinctive color and personality. "Is something gay, brilliant, or martial needed?" wrote one theorist. "Take C, D, E [majors]." Another: "D major … the key of triumph, of Hallelujahs, of war-cries, of victory-rejoicing." All those keys were relatively well in tune on the keyboard. Minor keys were innately less in tune, so darker in sound and import: G minor, for example, is "suited to frenzy, despair, agitation. ... The lament of a noble matron who no longer has her youthful beauty." You want a pretty pastoral piece? You want a relaxing key like F major—the key of Beethoven's Pastoral Symphony:

Two of Beethoven's favorite keys tell us a lot about him. The most famous is C minor, described by one writer of the time as "a tragic key … fit to express grand misadventures, deaths of heroes, and grand but mournful, ominous, and lugubrious actions."
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On the other hand, in the prevailing unequal temperaments there was still the presence, or at least the ghost, of the old wolf. Thus, croaked one theorist concerning that key, "Death, grave, putrefaction, judgment, eternity lie in its radius." Beethoven studied the theorists carefully, then did what he wanted. As for the putrefaction of A flat major: baloney. For Beethoven, that key, with its complex and distinctive coloration, suggested feelings in the direction of nobility, devotion, and resignation, as in the second movement of the Pathètique, again by Andras Schiff:

When composers stretched for more harmonic variety and tension in the first decades of the 19th century, as a practical matter the once-despised equal temperament won out over unequal tunings, which withered away during the century. But as professor Duffin exemplifies in How Equal Temperament Ruined Harmony, many tuning geeks today still find that temperament loathsome. Actually, Duffin's book is less rabid than its title sounds, amounting to a plea for playing keyboard music through the early 19th century in period tunings, especially in one called "sixth-comma meantone," which Duffin believes is the tuning Bach had in mind for the Well-Tempered Clavier. His reasoning is of Glenn Beckian deviousness. Some claim to find a cabalistic clue to Bach's intended tuning, close to sixth-comma, in the curlicues at the top of his title page for the WTC:

That idea may be nuts, or not. Bach was into puzzles, numerology, and all kinds of musical cabala, so the nuttiest idea of all about his tuning might well be right. It would figure. Listen to Watchorn in the curlicue tuning—or rather a theory about it—in the C# Major Prelude, one of the world's most happy-making pieces. This is another key out of the pale in older temperaments:

How do the travails of keyboard temperament apply to instruments without fixed tuning, like violins, trombones, flugelhorns, and the human voice? They don't apply at all. Most of the time violinists, et al., tune by ear, on the fly, note by note, and chord by chord. That's why a string quartet or an a cappella choir can be better in tune with nature than a guitar or a piano can. As a high-school trombonist playing with a piano for the first time, I found adjusting to keyboard tuning a pain in the neck—without knowing why. String recitalists know that pain intimately. Meanwhile, an orchestra is made of a bunch of instruments, some of which tune naturally by ear—strings, woodwinds, brass—but also instruments in fixed, equal temperament: harp, marimbas and xylophones, harpsichord and piano, etc. What do orchestras do to harmonize all those conflicting demands? They do the best they can and try not to think about it too much. It can make you crazy.

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Read the full article and its accompanying sound file examples HERE.