Second Monday Music: A terrible temperament

By Chuck Smythe

Many years ago, I attended a seminar by Owen Jorgensen, a famous piano tuner. He discussed historical musical temperaments, the system of spacings between the notes of a musical scale. He performed on four large Bösendorfers, each tuned to a different temperament. The seminar was held in the showroom of Chris Finger Pianos, a very high end dealer—the only way such a collection of instruments could be found in a single room!

    Like most people with a casual musical education, I had a vague notion that temperaments used before Bach could play only in a few keys, and sounded terrible in all the others. Bach showed the light in the “Well-Tempered Clavier,” a collection of short pieces written in each of the twenty-four possible keys, demonstrating that “well temperament” made them all performable. A couple of generations later, people got the great idea of spacing the twelve notes of the western musical scale equally, a method called “equal temperament” and that system has been used ever since. So the standard urban legend tells it. And, of course, that myth is altogether wrong.

I had assumed that the differences between one temperament and another would be rather subtle, and was astonished to learn how dramatic the changes actually were. One of the pianos was tuned in Just Temperament, a system common before the Renaissance. In this system, the important intervals, the thirds, fourths, and fifths, are tuned as accurately as possible...for the most commonly used keys. When I played it in these keys, the harmony was the most beautiful I’d ever heard in a keyboard instrument, far better tuned than standard equal temperament. As advertised, though, when I tried to play in a really remote key, say B Major, it was unbearably out of tune.

Frédéric Chopin

    The second instrument was tuned to a system popular in the time of Beethoven and Chopin, I believe Well Temperament or one of its close relatives. Jorgensen performed several Chopin pieces on this instrument, and the result was fascinating. I had, of course, heard the claim that each key has its own emotional color, and was baffled by this. Surely all keys sound the same except for pitch? This is indeed so in Equal Temperament, but in this system some keys were sweet, some were dark, some were harsh and jangly—and it was obvious that Chopin had picked his keys to take advantage of this. There was an entire emotional dimension to the music that was completely inaccessible to modern equal temperament.
    The third piano was in equal temperament, the fourth in a strange experimental system Jorgensen had invented for himself, and we won’t go there.

The theory behind all this is much too complicated for this space. I’ll try to just sketch what is going on. The basic point is that every musical tone has a basic lowest pitch , the fundamental, plus a series of overtones rising above it, with frequencies in simple ratios to the fundamental: 2:1 (the octave), 3:2 (the fifth), 4:3 (the third), and so on to infinity.
    These intervals are those used in building Western music; for instance the fifth is the interval between the first two notes of “Twinkle, twinkle, little star.” Almost all melody in common musical practice is built on chords produced by stacking these first few simple intervals, and melodies made by moving among them. So, when you sing the first note of “Twinkle,” the sound includes the overtone at a ratio of 3:2, the fifth. Then when you move to the second note, its fundamental is the same pitch as that first overtone, so the two notes “harmonize,” i.e., sound well together. If you instead sing a note not in the harmonic series, the waves will “beat” against the first note, with a painful “wow wow wow” sound. The two notes are said to be out of tune.

    The first few notes of the harmonic series were discovered by Pythagoras, he of high school geometry fame. He was also a famous religious crackpot: he got the idea that all of reality was constructed of perfectly harmonious vibrations such as this. He acquired many followers. But there was a Beast in the garden. The idea was that if you walk up the musical scale in octaves (2:1 frequency ratio) for seven octaves, the last frequency will be 2 x 2 x 2 x 2 x 2 x 2 x 2 = 128 times the starting frequency (and, of course, if you start on C, each note visited is also a C). According to the scheme of Pythagoras, you must reach the same frequency by walking up the scale in fifths (3:2 frequency ratio) 12 times, and each of these multiplications will visit one note of the chromatic scale (white plus black keys). Thus the “circle of fifths” starting on C will go to G, then to D, then to A, etc., defining the musical scale. The problem is that when you calculate 3/2 x 3/2 x 3/2.... = (3/2)¹², it equals 129.746. And 129.746/128 = 1.036, so the circle of fifths doesn’t quite close. The ending C is about 3.6 percent sharp, so the attempt to build a perfect musical scale this way fails. Irrational numbers (not expressible by a simple ratio) have crept in. This was Pythagoras’s religious scandal, and he solved it in the usual fashion of religion—by making it heresy to talk about the problem.

For centuries after, musicians, mathematicians, and philosophers quarreled about what to do about this fundamental problem. At least several dozen systems were invented for “tempering” the basic scale; that is, for adjusting the intervals between notes in some way to make harmony work as well as possible despite this scandal. Contrary to the urban legend I mentioned above, equal temperament—that is, simply dividing the musical scale into twelve equal parts—was thought of very early, and mostly rejected. A minor problem was that calculating these intervals was hard until logarithms were invented; a far more serious one is that in this system the interval of the major third, the ratio 5:4 (the first two notes of Kumbaya, for instance) is badly out of tune. After a century and a half of brainwashing by pianos in equal temperament, we Westerners have more or less gotten used to it, but you only need to listen to properly tuned thirds to realize that it is actually quite awful.
    All this is a problem only for instruments with fixed pitches, such as keyboards; singers and string players can and do make small adjustments to get perfectly tuned intervals. The conductor of my Baroque chamber choir, in fact, insists that we do so. The (amateur) string players I know say this is far beyond them, but I’m told that professionals are routinely expected to be able to do this.
    But we keyboardists are left with the problem. Systems such as Well Temperament were designed to make the necessary compromises in various ways less glaring than those awful wide thirds. Many of them do so by making all keys at least playable, some in better tune than others—thus my experience with Chopin. This came to be seen as a virtue, offering another dimension to musical composition. I’m forced to more or less agree, and am eager to play around with it. I’m looking for a keyboard with adjustable temperament. Does anyone have one?

In writing this, I revisited three books on the subject, which I recommend for those who would like to learn more.

• Tuning and Temperament: a Historical Survey, by J.M. Barbour, is an older work, a thesis expanded into a book. It is very thorough and very technical. Barbour admits that he hadn’t actually heard the older systems, and comes down heavily on the side of Equal Temperament as The March of Science.
• Temperament, by Stuart Isacoff, is as non-technical as possible, and is a grand tour of the cultural, philosophical, and artistic wars that have accompanied the argument over temperament through the centuries. He also comes down in favor of Equal Temperament, though in an afterward added for the current edition—replying to his critics—he shows a much more open mind.

• How Equal Temperament Ruined Harmony (And Why You Should Care), by Ross Duffin, takes a very different tack, as the title warns—he thinks that much of the music composed before, say, 1850 would be better performed in other temperaments, as might some modern music. This book is, much more than the others, a guide for working musicians. It assumes a basic knowledge of music, but beyond that offers much the clearest explanation of the technicalities.

    What I am eager to find is recordings made in the older temperaments. I have one CD on order from Amazon; I’ll be grateful to any reader who can suggest other sources.
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Copyright © 2014 by Chuck Smythe

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