Temperament: The Idea That Solved Music's Greatest Riddle

Good Vibrations

Our world today is discrete, compartmentalized. The arts and the sciences bump up against each other mainly in the acronyms of graduate schools. Music, for most of us, takes place in a concert hall, or a jazz club, or a car radio; our encounters with the world of cells and atoms are limited to the doctor's office and the garden. Our universe is filled with black holes and dark matter and other unknowables; our home is a speck that orbits another speck in an ever-expanding cosmos. But there was a time when there existed a glorious synthesis of music and mathematics, and in the imaginations of scientists and philosophers and musicians it wove the entire universe into a grand design. The planets were imagined as enormous spheres, each moving in concert with the others. This motion produced "seven tones, this number being, one might almost say, the key to the universe," Cicero wrote. "The ears of mortals are filled with this sound, but they are unable to hear it...you might as well try to stare directly at the sun, whose rays are much too strong for your eyes."

The very idea of a "key to the universe" today seems as quaint as the belief that the Earth is flat. We are more familiar with concepts such as Heisenberg's uncertainty principle, or chaos theory, or irrational numbers that can be calculated to an infinite and patternless number of decimal places. Even if a key to the universe could be discovered, the lock that it fits long ago disappeared. But for thousands of years, from the ancient Greeks to the Church fathers to the Enlightenment, the existence of such a key was not a fantasy but a premise of intellectual life, and the key was situated at the intersection of music, science, and religion. The proportions that govern musical harmony, causing certain tones that vibrate together to produce a beautiful sound, were believed to regulate also the positions and the motions of the celestial bodies. These proportions — simple ratios built on the integers 1, 2, 3, and 4 — were proof of the divine organization of the cosmos. As Stuart Isacoff succinctly puts it: "The hand of God tunes the world."

Isacoff's book describes the history of musical harmony and its most important offshoot: the evolution of equal temperament, a tuning system that had an enormous impact on every aspect of the history of music, from the development of the sonata and the symphony to the design of instruments. Equal temperament divides the octave into twelve equal parts, or the notes of the scale: C, C-sharp, D, D-sharp, and so on. In previous tuning systems, the precise placement of these pitches varied; depending on which system was used, they could be higher or lower, in tune with certain harmonies and out of tune with others. But when instruments are tuned in equal temperament, there is no variation. Each note must be exactly a half step away from the note above or below it.

Over the last four hundred years or so, equal temperament, once a radical heresy, has become an almost universally accepted tenet. It is synonymous with the modern piano, and it defines the musical scale as we know it. Except for music theorists, people are now surprised to learn that Western instruments could be tuned in any other way. But for thousands of years there was another way of tuning, and it was inseparable from the theories about the structure of the cosmos. When equal temperament was first proposed, it was seen as "akin to taking a file to the pieces of a particularly frustrating jigsaw puzzle to force their irregular shapes into submission," Isacoff writes. "The result...was regarded by many as repugnant, even catastrophic: a violation of nature." It was like throwing away the key to the universe.

Legend has it that it was Pythagoras who first discovered the basic principle of musical harmony. Passing by a forge one day, he noticed that as the blacksmiths' hammers struck different anvils, they produced various tones. When the anvils were struck simultaneously, some combinations of tones were pleasing to the ear, while others were discordant. Pythagoras is said to have realized that the hammers that produced the most pleasant sounds were of certain proportions in relation to one another. Two hammers whose weights were in the ratio 2:1 produced the most harmonious sound, an octave; those in the relation 3:2 produced the perfect fifth. (This interval, logically, spans five notes of the musical scale, say from C to G.) The other harmonious ratio was 4:3, which resulted in the perfect fourth: the interval from C to F.

Pythagoras, of course, was not primarily a music theorist. His theory of the ratios for octaves and fifths was a small subset of his general conception of the universe, which he perceived as defined by number. The proportions of the musical intervals, like the laws governing geometric shapes and the laws directing the movements of the planets, were all part of the natural order of the world, which had as its basis certain simple mathematical formulas. The musical proportions reflected the vibrations of man's nature, which Pythagoreans such as Boethius called musica humana: the continuous soundless music produced by each human being, particularly the resonance between the soul and the body, which could be harmonious or disharmonious. Musica humana mirrored musica mundana, the music of the heavenly spheres as they moved in their orbits. The concept of the music of the spheres and its relation to the terrestrial world formed a backdrop to much of classical literature. In Plato's Republic, the myth of Er includes a detailed description of the spheres; they also make an appearance in Aristotle's On the Heavens and in Scipio's dream in Cicero's Republic.

In the phenomenon known as sympathetic vibration, when a string on any instrument is plucked, other strings tuned to the same pitch will vibrate at the same time, though almost silently. Similarly, in the vastly thrumming universe of the ancients, the vibrations of man's nature and the harmony of the celestial spheres mirrored each other. Thus music literally had the power to heal, as well as to damage: it could correct discord in the soul, but the myth was also told of a piper who incited men to violence by accidentally playing a tune in the wrong mode.

The church fathers adopted a very similar idea of the relationship between music and man. The earliest Church music was plainchant, of which Gregorian chant is the best-known type. The primary characteristic of this music is its simplicity; unlike the multi-part cantatas and chorales that would later become popular, it consists of a single melodic line, which commonly travels up or down gradually, one step at a time. This music has an austere beauty, to which the recent surge in the popularity of chant recordings attests. But its long strands of unison melody can bore modern ears, which are accustomed to greater complexity.

Soon the singers of plainchant learned to embellish a simple melody with the addition of another melodic line paralleling the first but located a fifth or a fourth above it. By the thirteenth century, these pieces had evolved into motets, which introduced the greater variety of polyphony, or counterpoint: the use of two, three, four, or more melodic lines simultaneously, as in Bach's fugues. But this music had not yet achieved anything like the Baroque era's sophisticated counterpoint: the fundamental harmonies continued to be based on the octave, the fifth, and the fourth — the "perfect" intervals, which were believed to emulate the choirs of heaven.

The Renaissance brought the extension of the ideas underlying musical harmony to the visual arts. Leon Battista Alberti wrote in On Painting that "the same numbers that please the ears also fill the eyes and the soul with pleasure." When Guillaume Dufay was called upon to write a musical tribute to the successful completion of the Duomo in Florence, the huge dome of which posed the engineering problem famously solved by Filippo Brunelleschi, he based his motet upon the ancient proportions for sacred buildings laid out in 1 Kings — 6:4:2:3, the dimensions of Solomon's Temple. (Augustine believed that churches should be built according to the Pythagorean ratios, but Abelard found in Solomon's Temple further proof of God's intermingling of music and architecture.)

But as composers grew more inventive in their use of harmony, a problem with the entire musical system became more and more apparent. Theoretically speaking, if you were to take any stringed instrument and play a series of perfect fifths, each building on the last — C to G, G to D, D to A, and so on — after cycling through twelve fifths, you would end up on C again, eight octaves above where you started. But if the fifths are "pure" — that is, tuned exactly according to the Pythagorean ratio — the C on which you end up is slightly out of tune with the C on which you began. (The mathematical explanation for this phenomenon is that fifths vibrate in the 3:2 ratio, and octaves in 2:1; 2 and 3 are prime numbers, and the powers of different prime numbers can never coincide.) The difference between the two C's, known as the "Pythagorean comma," is infinitesimal, but even an untrained ear will find it disconcerting.

The problem of making the two C's match up — the "riddle" of Isacoff's title — occupied composers and musicians for more than two thousand years. It is less significant for stringed instruments such as the violin; since the musician adjusts the pitch of each note as he plays, these instruments are not tied to any particular tuning, or temperament. But with the growth in popularity of the harpsichord and other early keyboard instruments that did demand a fixed temperament, musicians and composers began to experiment with various ways of adjusting the tuning systems.

The simplest way to solve the conundrum is to shorten, or "temper," one link in the chain of fifths. But tempering creates a whole new problem: when just one or two notes in the scale are adjusted, the result is a "wolf" interval, an altered fifth with a hideously dissonant sound. On a piano tuned in this way, some chords — those based on the "pure," or untempered, intervals — would be exquisitely in tune; but approximately one-third of the keys — any that involved the tempered notes in the "wolf" — were dissonant to the point of being unusable. Composers had to take great pains to avoid the objectionable harmonies, which sharply limited the effects that they were able to achieve.

The effort to preserve as many pure harmonies as possible while simultaneously rendering the maximum number of keys playable led to the development of two competing methods of tuning. The fifteenth-century music theorist Bartolomeo Ramos de Pareja advocated a system called "just intonation," in which fifths and thirds were kept pure, but the individual steps of the scale were not of a uniform size. The harmonies based on the pure intervals were exquisite, while the others, as Isacoff puts it, were "like a splash of vinegar in the ears." In the other system, known as "mean-tone temperament," selected fifths were tempered by a fraction of the Pythagorean comma. Writing at around the same time as Ramos, Franchinus Gaffurius noted its appearance in his treatise Practica musicae, commenting that it was common practice for organists to abandon "perfect" tuning in favor of more utilitarian methods, including tempering fifths "by a very small and hidden and somewhat uncertain quantity." Several decades later, Giovanni Maria Lanfranco proposed an early approximation of equal temperament, suggesting that fifths be tuned flat enough so that "the ear is not well pleased with them," and thirds as sharp as possible.

In the sixteenth century, the concept of equal temperament began to gain momentum. The court composer Adrian Willaert demonstrated the need for it in 1530 with his choral piece Quid non ebrietas, which is impossible to perform in any tuning system other than equal temperament: it modulates through each key, ending with an octave leap. But the first serious challenge to the old system came from Vincenzo Galilei, the father of Galileo. Galilei, who studied for a time under the music theorist Gioseffo Zarlino, published in 1589 a treatise attacking his teacher, who had sought to expand the Pythagorean system to include sixths and thirds as "natural" intervals. Since (almost) all of these could be derived from the division of a string into six segments, Zarlino argued that six, not four, must be Pythagoras's "perfect number." Such contortions are nonsense, Galilei countered. All scales are man-made, with no basis in nature whatsoever. Indeed, just intonation is itself only an ideal; in practice singers automatically temper their intervals for the sake of overall harmony. Zarlino responded by calling Galilei's treatise "an assault on God's plan."

In retrospect, equal temperament seems an obvious solution. Rather than fiddling with shrinking thirds here and fifths there, why not spread out the tempering over the entire keyboard, thus altering each interval as little as possible? But this brought yet another mathematical conundrum: it was still not known how to divide the octave into twelve equal parts. The whole step is formed by the ratio 9:8, and an attempt to divide it (and thus to produce uniform half steps) yields an irrational number. More importantly, musicians were not yet prepared to abandon entirely the principles behind Pythagorean tuning. The theory of equal temperament was proposed by as early a thinker as Aristoxenus, a pupil of Aristotle, but it was considered an attack on Pythagoras and quickly dismissed. The Pythagorean proportions literally defined music: how could they be put aside?

There was another reason for musicians' and composers' resistance to equal temperament. Though unequal temperaments rendered certain keys unusable, the harmonies that were in tune retained an aesthetic quality that was utterly lost in the new system. Pure intervals, particularly thirds, simply sound more harmonious than tempered intervals. "In a triad everything sounds bad enough; but if the major thirds alone, or minor thirds alone, are played, the former sound much too high, the latter much too low," Johann Georg Neidhardt wrote in 1732. "Thus equal temperament brings with it its comfort and discomfort, like blessed matrimony."

Moreover, the musical keys, when tuned properly, were believed to possess inherent characteristics that do not come across in equal temperament, with its even spacing of intervals. "Every fear, every hesitation of the shuddering heart, breathes out of horrible E-flat minor," wrote the eighteenth-century composer Christian Schubart, one of the more exuberant chroniclers of this phenomenon. "Preparations for suicide begin in this key," he noted of B-flat minor. Similarly, some argue that Bach wrote the preludes and fugues of The Well-Tempered Clavier, often taken as a demonstration of the wonders of equal temperament, to showcase instead an instrument tuned in "well temperament," a somewhat irregular tuning somewhere in between mean-tone and equal temperament that allows the use of all the keys but still preserves their distinctive attributes. Theoretically, a piece of music intended to be played in equal temperament can be transposed into any other key with the same effect; but these theorists argue that Bach's preludes and fugues take advantage of the wider or narrower intervals of certain well-temperament keys, and when played in the proper temperament will sound wrong when transposed.

In the early seventeenth century, the Dutch mathematician Simon Stevin tried to strike a compromise of a sort: the Greeks may have believed 3:2 to be a real ratio for the perfect fifth, he said, but obviously it was just an approximation of the true interval, the perfect interval, which was itself as unattainable as the Platonic forms or the Pythagorean triangle. He went on to argue that only equal temperament could be natural, because the other methods are simply unworkable. Around the same time Marin Mersenne, a French Jesuit, took the opposite approach, advocating that keyboards be constructed with anywhere from nineteen to thirty-two keys to the octave, thus allowing for every possible permutation. Meanwhile, far from the din of Europe, Prince Chu Tsai-yü of China was quietly inventing equal temperament on his own, calculating the tempered fifth at the ratio 749:500 — just a smidgen away from 3:2.

The concept that natural intervals somehow had precedence over others got a boost in the late seventeenth century when the acousticians William Noble and Thomas Pigot discovered what are now known as "overtones." On stringed instruments, when a note is sounded, other notes can be discerned in the vibrations — most noticeably the fifth and the fourth. But in a famous dispute with Rousseau, Rameau invoked the phenomenon of overtones as an argument for equal temperament. Rousseau, in his "Letter on French Music," which appeared in 1753, had argued that since music was rooted in the first cries of a baby or an animal, music based on melody was natural, while music based on harmony was artificial and cultivated, further removed from natural sound. Rameau replied that the close study of nature revealed that harmony was in fact the real basis of musical expression: since vibrating objects emit overtones in addition to their primary tone, nature favors harmony. He concluded that equal temperament could thus be the only natural system, since only it allowed for harmonic movement between all of the keys.

Meanwhile, the first prototypes for the pianoforte were created by Bartolomeo Cristofori around 1700. The sonorous tone produced by the instrument's characteristic action — rather than being plucked, as with a harpsichord, the piano's strings are struck by hammers — made the slight dissonance of tempered intervals easier to bear. And as composers began to write for the piano more and more, they came to develop the sonata form, which relies on often-complicated modulations from one key to the next. Once composers had experienced the freedom of modulation that equal temperament allows, there was no turning back. At least that is how Isacoff would have it.

Isacoff's book makes admirable sense of a topic that is normally broached only in turgid texts of music theory. He writes engagingly about abstract and often counterintuitive matters, and he lends his subject a bit of drama by placing it, rightly, in the context of the history of ideas. The revolution of equal temperament took place primarily in the musical arena, but Isacoff shows that the concepts that brought it about were inseparable from contemporaneous ideas about astronomy or man's place in the universe or God. Like every other important idea, equal temperament did not develop in a vacuum.

Sometimes the connections that Isacoff draws seem a bit forced, as when he argues that the new polyphony of the motet was the musical counterpart to Giotto's paintings, which Petrarch described as "images bursting from their frames, and with the lineaments of breathing faces....It is here that the danger lies, for great minds are greatly taken with this." Surely similar warnings were given about polyphonic music, but it seems difficult to call music, which is non-representational, a reflection of nature in the same way that painting can be said to be. And Isacoff's rundown of the history of music, astronomy, art, and physics sometimes has a sort of flip-book quality, as figures such as Kepler or Newton pop up for a few paragraphs and then disappear.

But the deeper problem with Isacoff's linkage of historical temperaments with the history of science is that it implies that these ideas from deep within the history of music are no more applicable today than are Ptolemy's epicycles or Copernicus's models of circular planetary orbits. Though at the end of his book Isacoff raptly recounts an experience of listening to a piano adapted to play untempered tones, his forward-thrusting approach to equal temperament's development clearly indicates that he regards the positions against it as better off left behind. Indeed, most of the theorists and the philosophers who favored just intonation or mean-tone temperaments are presented as kindly old eccentrics situated comfortably in the distant past.

By essentially ending his book with Rameau, Isacoff ignores some of the more controversial temperament research of the past few decades. A strong camp of contemporary music theorists continue to argue against what they see as the "homogeneity" of equal temperament. It is required for the performance of modern atonal music, which exists outside traditional harmonic systems. But these scholars argue that perfect equal temperament was itself merely a theoretical ideal until the introduction of scientific tuning methods in the early twentieth century. Until then — that is, throughout the Classical and Romantic periods — pianos were tuned to various approximations of equal temperament, with some intervals tempered more, some less. "Nineteenth-century tuning by ear was a highly developed art based on aesthetic judgments for every tone," the theorist Owen Jorgensen observes in Tuning, which appeared in 1991. "By contrast, twentieth-century tuning is a mathematical skill," with little art left.

These scholars also contend that the complete dominance of equal temperament in modern times has resulted in contemporary musicians' loss of the ability to distinguish between pure intervals and their tempered forms. "We have become so accustomed to hearing the tempered scale that we have become, in effect, `earwashed': having heard tempered fifths all our lives, they sound right, or right enough, to us," writes Jamie James in The Music of the Spheres. But the intervals are not supposed to sound "right." They are supposed to sound sublime. It is worth recalling that "temper" comes from the Latin temperare, which has among its meanings "to restrain oneself." It is the beauty of pure intervals that equal temperament restrains. Galileo described the pure perfect fifth as "a tickling of the eardrum such that its softness is modified with sprightliness, giving at the same moment the impression of a gentle kiss and of a bite." You can strike C and G together on the piano, but that is not what you will hear.

In a logical offshoot of the "authenticity" movement that has brought increasing interest in period instruments, some musicians have begun to experiment with reconstructing historical temperaments. Jorgensen's book gives detailed explanations for tuning more than one hundred different temperaments, and the pianist Enid Katahn has put them to use on two recent recordings, "Six Degrees of Tonality: A Well Tempered Piano" and "Beethoven in the Temperaments: Historical Tunings on the Modern Concert Grand," both on the Gasparo label. On the first recording, Katahn performs a number of repertoire staples in various temperaments. The temperament that she employs for Mozart, for instance, was common during the mid-eighteenth century: a version of well temperament, all the keys are usable, though some intervals are tempered more than others, making it an irregular temperament.

Katahn's recordings carry a mock "warning label" that reads: "This CD contains pure intervals which may be habit forming!" And Katahn writes in the notes to "Beethoven in the Temperaments":

Not only do certain keys have an overall aura or character, but within each key, every chord seems to have its own color. One is given the impression of a constant shift between light and shade as one moves through the changing chordal landscape....Listen to the tension created by the diminished seventh chords in the opening of the Pathetique sonata or by the strange progressions of the second movement of the Waldstein. Notice the contrast in the Moonlight sonata as it moves from the "dark" key of C# minor to the "lighter" A and D major chords of the third bar....These are exciting passages in any tuning, but in a Well Temperament they become musical events.

After comparing Katahn's interpretation of the Appassionata Sonata to a recording by Richard Goode, however, I could not be certain that I actually heard any of these effects in Katahn's version. Despite the supposed drawbacks of equal temperament, Goode's opening chords are just as rich as Katahn's. The trill at the end of the chord progression leads more inevitably to the introduction of the theme, which sings more brightly under his hands, and the arpeggios that follow are more bubbly. It is Katahn's version, in fact, that lacks drama, because her approach to the music simply feels less strong. Not surprisingly, the skill of the pianist clearly has far more to do with the success of an interpretation than does the temperament of the instrument.

But I listened to Goode again after hearing Katahn, and the quality of the pitches did seem a little flat. And Katahn's recording of Mozart's Fantasy in D minor provides further evidence for the benefits of well temperament. With its abundance of diminished chords and seventh chords, the Fantasy is perfectly suited for temperament comparison, since it is these notes that are most likely to be affected by the irregularities of unequal temperaments. The piece begins with deep, low arpeggiated chords that cycle from one hand to the other, progressing from D minor to a diminished second and a seventh chord, then through F major and E-flat major to another diminished chord before resolving to A major and then D minor through a series of treble half-steps in the right hand. When Katahn plays the piece in eighteenth-century temperament, the accidentals — particularly the G sharps of the last diminished chord — are noticeably unstable. This phenomenon repeats in the series of half-steps: the space between G sharp and A, as well as between D sharp and E, is audibly larger than the norm. Katahn hovers on the strange notes, so that the resolution to D minor is oddly refreshing, despite the darkness of the minor key. When she gives the piece again in equal temperament, the instability is gone, along with the strangeness.

Did Mozart intend this strangeness? For years musicologists have been debating whether it is possible satisfactorily to answer questions about the way composers intended their music to be heard. One thinks of Christopher Hogwood's attempt — to cite only one of the many once-notorious experiments of the "authenticity" movement in performance practice — to recreate the original performance conditions of the Eroica Symphony, up to and including the use of an amateur orchestra. But aside from such obvious absurdities, problems such as inconsistencies in scores and lack of dynamics notations are a clear stumbling block to establishing definitive versions of many compositions. Just as reading literature written in the past, even in one's native language, requires an act of mental translation, a certain kind of translation may also be necessary to interpret music written under conditions that we can never fully know.

It seems clear that true equal temperament was not common on pianos until at least the middle of the nineteenth century, if not later. Thus we should recognize that, particularly in the Classical period, composers likely worked with different temperaments in mind than the one to which we are accustomed. But in order truly to appreciate how these temperaments sounded, it would seem necessary to hear them on a period instrument. The very concept of "historical tunings on the modern concert grand" is a bit of an absurdity, since the instrument is built to be tuned in equal temperament. Indeed, modern pianos have to be "taught" to accept historical temperaments, with repeated tunings required before the soundboard will hold the new shape. (Katahn's liner notes reveal that her piano actually had to be re-strung after it was used for the most radical mean-tone temperament.)

When "authenticity" is for so many reasons unavailable to us, it is the performer who represents our best avenue of approach to the music. Katahn has also recorded Chopin's Fantaisie-Impromptu in C-sharp minor, and I again compared her version with another recording that I happened to have, this one by Horowitz. I have been listening to this piece for years, and there was a time when I could play it myself. Yet as I concentrated on Horowitz's recording I heard something in it that I had never before noticed: a delicate and beautiful ornamentation that sounded like double triplets on one of the repetitions of the sweet second theme. Eagerly I turned to Katahn's recording to hear how this beautiful moment would sound in irregular temperament, and was surprised to discover that it was entirely missing. I checked my edition of the score: it wasn't there, either. Was Horowitz using a different version? Or did he improvise the ornamentation himself? I do not know the answer, but in sheer beauty that one measure in Horowitz far surpassed all the untempered intervals in Katahn.

Leibniz famously wrote that "music is the hidden arithmetic exercise of a soul unconscious that it is calculating." Nowhere is the connection between math and music more visible than in the meticulous calculations required to determine temperament. But music is not only an arithmetic exercise for the soul; it is also an aesthetic exercise, even a spiritual exercise. All the tuning adjustments in the world can only be of secondary importance. The temperament of an instrument is finally far less interesting than the temperament of a musician.

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