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Around the start of the seventeenth century, a new sense of the word “harmony” emerged. To that point, harmony in music had been produced by the pleasing opposition of two melodies according to the principles of counterpoint. In the 1600s, “harmony” began to denote the non-melodic accompaniment to a line of melody. This sense remains colloquial today. If I say, “Here is a melody, let’s add some harmony,” I mean that there is a tune and it is getting some additional, subordinate music to accompany it. It is basically synonymous with chords. We would describe Taylor Swift as the melody, and her back-up band—its guitars and piano and bass—as the harmony.

But what is “harmony” in the older sense of the word, captured particularly in Bach’s counterpoint? Counterpoint is the accumulation of multiple melodies. It is like Louis Armstrong playing an improvised tune on his trumpet at the same time as Ella ­Fitzgerald sings “La Vie en Rose”two different melodies simultaneously. Neither is subordinate to the other, or, if there is subordination (perhaps we listen a little more to Ella’s voice than the trumpet), they are both melodies, a status that the piano, plunking out chords in the background, does not share. In true counterpoint, all the sound created is produced by people singing or playing melodies. If we lived several hundred years ago, we would say that “harmony” is what joins and holds together those melodies, their counterpoint, in a pleasing fashion.

And though this older sense of “harmony” is no longer dominant, idiomatic English retains it alongside the modern sense. If I were to say, for instance, “In marriage, the husband plays the melody and the wife plays the harmony,” I would be expressing both a bad view of marriage and a modern view of harmony. Harmony here is a discrete entity supporting the melody, existing beneath and for it. But if I said, “The husband and wife, though they disagreed, finished the conversation in harmony,” I am using “harmony” in a different, but recognizable, sense. It is no longer a distinct line, no longer a role one of the two parties plays; it is instead a relation between the two, both of whom are human subjects existing in the same plane.

Very often, “harmony” serves as a synonym for perfect agreement: A harmo­nious marriage or society is one in which all members are in perfect accord. But the contrapuntal idea of harmony implies a different vision of social concord, one in which the various parts retain autonomy but find their fullness in relation to each other and to a certain order that arises from their life in common. “Implicit in the term contrapuntal,” says Walter Piston, “is the idea of disagreement. The interplay of agreement and disagreement between the various factors of the musical texture constitutes the contrapuntal element in music.” This account mirrors the almost mystical formulation of Franchino Gaffurio, the fifteenth-­century music theorist, for whom harmony could be defined as discordia concors, “agreeing disagreement” or “concordant discord.” Contrapuntal harmony is an almost miraculous occurrence, a sonic solution to the problem of the one and the many.

The science of composing contrapuntal music has been, since the late eighteenth century, portrayed as dry and excessively difficult. But counterpoint is simply a playbook for the rules of social conduct between two melodies. There are, in essence, two principles.

When two melodies coexist, the glory is their coexistence. But there is no harmony among things that are too dissimilar. The melodies must have an awareness of and reliance on each other in order to live in concord. That is the first principle: Melodies in counterpoint must not be too different.

But if the two melodies resemble each other too closely, they lose their identity. The glory of harmony, of concord, is that the elements are different. That is the second principle: The melodies must not be too similar.

Rules about how to execute dissonances, then, are simply rules applying the first principle. Too much dissonance, and the melodies have ceased to notice each other, and they live in worlds of their own. It may be egalitarian, but it is not social. They are not sacrificing their own desires and ends for the sake of the other. So we must carefully prepare the dissonances. They must not last too long. “Mi against fa,” as the medieval jingle has it, “is the devil in musica.”

The prohibition of parallel fifths and parallel octaves has long befuddled beginning music students: For some reason, counterpoint insists that two melodies may not move in the same direction in subsequent octaves or fifths. But it is simply an application of the second rule: Fifths and octaves, from the perspective of a medieval, are the “perfect consonances.” The two notes of a fifth or an octave are as harmonious with one another as any two notes can be. When two voices begin moving in perfect, parallel consonance, their identities become blurred, because so many of their overtones are shared so prominently. They have become too similar. That medieval phrase, discordia concors, is gently urging the composer: “Do not make these melodies agree so much. They need to fight each other more.” In Yvain, Chrétien de Troyes offers an extreme vision of this concordance of discord:

I saw so many birds gathered in the pine tree (if any one will believe my words) that not a branch or twig was to be seen which was not entirely covered with birds. . . . All the birds sang in harmony, yet the note of each was different, so that I never heard one singing another’s note. . . . I have never heard such happy song . . .

In the real world, if thousands of birds were singing in a tree, each a different note, they would sound like the score of The Shining. I mean that literally, because in composing music for The Shining, György Ligeti attempted what he called “micropolyphony,” a kind of counterpoint that ignores the first rule and makes all melodies dissonant. The result is an orchestra of hundreds of people playing different notes. It works quite well, for horror films. So, there are limits to counterpoint’s ability to incorporate disagreement. But the classical vision recognized that harmony—social as well as musical—required variety.

There is a way to make two melodies sound both alike and at variance, as though they were engaged in a spirited discussion. This method was developed in the late Middle Ages, and it is perhaps the most marvelous invention of Western music. It is called fuga or, nowadays, a canon. If we are prohibited from moving two melodies in parallel simultaneously, why not move them in parallel asynchronously? Medieval composers first experimented with asynchrony as a game between two melodies. In the game, whatever one melody did, the other did afterward. The first was called dux or leader, and the second was called comes or companion. This staggering of gestural similarity means that, if I am the comes, my identity is determined by what you, the dux, decide to do. But in another way, you, the dux, have no will apart from me.

This paradox can be illustrated if I think of myself as the composer. If I were to write a canon, I could not write just any old melody for my dux. I must think carefully about each note. For every dux note I write will be heard, eventually, against previous notes as the comes sings them, and so the dux can never be written simply with its own ends in mind. Every note is beholden to its own actions in the future. To speak of the dux as the “independent” or “­autocratic” melody would be ridiculous. In a canon, neither the first melody nor the second can exercise its will with complete independence; both must sacrifice for the sake of harmony.

Metaphors are powerful, and one of the most potent metaphors for marriage—encapsulated in the word “complementarianism”—is geometry. To complement is to complete, to fill up what is lacking. Geometrically, an angle and its complement can never add up to more than 180 degrees. And if we are dealing in three dimensions, two solid objects cannot exist in the same place. Something similar is going on in our modern conception of harmony: As one autocratic virtuoso begins to monopolize our attention, the chord-playing aspects of the orchestra (the “rhythm section”) become more marginal. Yet in the older vision of “harmony,” harmony understood as counterpoint, one person singing is no threat at all to another person singing. Sounds are not quantities or physical objects; for one to exist in the same space as one another is not only possible but desirable. The challenge is to get them to sound good together. This requires some chronological hierarchy—one party needs to lead and the other follow—but this, as we discovered above, does not mean that one party will sacrifice more autonomy than another. Both must sacrifice independence for the sake of symmetry.

If marriage or social order is conceived geometrically, the whole will always be the sum of its parts, and an increase of one party will entail the subjugation of another. This is not the case if our metaphor is musical harmony in the classical sense. Long before the invention of counterpoint, ­Tertullian seemed to be aware of something like this when he encouraged husband and wife to sing to each other: “Psalms and hymns sound between the two of them, and they challenge each other to see who better sings to the Lord. Seeing and hearing this, Christ rejoices. He sends them his peace.” This reminds me, in turn, of C. S. Lewis’s description of Arthur and Camilla Denniston’s marriage in That Hideous Strength—a book with a parochial vision of marriage even by conservative standards—yet here nothing could be more egalitarian:

Mother Dimble always remembered Denniston and her husband as they had stood, one on each side of the fireplace, in a gay intellectual duel, each capping the other, each rising above the other, up and up, like birds or aeroplanes in combat. . . . Never in her life had she heard such talk—such eloquence, such melody (song could have added nothing to it), such toppling structures of ­double meaning, such skyrockets of metaphor and ­allusion.

The comparison of music and marriage—of a counterpoint between two melodies that are constantly “capping” one another—reminds me of Bach’s ­Widerstehedoch der Sünde. Lewis has a love of metaphor and double meaning. Whatever could possibly be metaphysical about a conceit or profound about a pun is probably true of the frivolity of counterpoint. Most curious of all about this passage, almost strange enough to go unnoticed or be dismissed as a mistake: “Denniston and her husband.”

In Classical and Christian Ideas of World Harmony, Leo Spitzer argues that the old idea of harmony was adopted socially before it found musical expression in the art of counterpoint. Whether or not that is the case, certainly this idea of harmony had long been understood by Christian thinkers. “The concord of different sounds,” says Augustine, “controlled in due proportion, suggests the unity of a well-ordered city, welded together in harmonious variety.” Spitzer also recalls Chrysostom’s teaching on friendship, “A musical performance which consists of the tuning of two souls.” And it is in the hymns of Ambrose, of course, that Spitzer sees a “performance” and “incarnation” of the ancient Greek notion of harmony. The Church “was represented in his hymns as echoing the music of the universe” by “transforming pantheistic fullness into Catholic polyphony.”

As counterpoint developed, it seemed to embody the ideals set forth for the social order in the word “harmony” as the Greeks and the Fathers used it. Anyone who orders his loves and subordinates his desires to others’ needs can be expected to appreciate this sort of music. Shakespeare expressed the classical vision of harmony in the Merchant of Venice:

The man that hath no music in himself,
Nor is not moved with concord of sweet sounds,
Is fit for treasons, stratagems and spoils;
The motions of his spirit are dull as night
And his affections dark as Erebus:
Let no such man be trusted. Mark the music. (V.1)

For Shakespeare, “music in himself” means the harmony that reconciled a human’s bodily form to his spiritual essence. This discordia concors produced a “human music.” More than that, “concord of sweet sounds” means peace within a social order. If the unharmonious man is unmoved by the delicate balance of wills and desires which counterpoint enforces on its players, then he can hardly be expected to achieve that balance in his dealings with others. It was Shakespeare’s Elizabethan England whose Book of Common Prayer invoked God as “the author of peace and the lover of concord.”

Dorothy Sayers chose a performance of Bach’s double violin concerto as the backdrop for the resolution to her three novels describing Lord Peter ­Wimsey’s unrequited love for Harriet Vane. At the end of Gaudy Night, Harriet says yes to his proposal. What decides her in Wimsey’s favor is the gentleman detective’s analysis of Bach’s counterpoint. This incident convinces Harriet that Wimsey’s view of marriage does not involve her subjugation.

Peter and Harriet are both strong-willed. They are bound to disagree. As a friend tells Harriet, “A marriage of two independent and equally irritable intelligences seems to me reckless to the point of insanity.” It is this consideration that has prevented Harriet from accepting the proposal, until the moment she observes Peter listening to Bach. “Peter, she felt sure, could hear the whole intricate pattern, every part separately and simultaneously, each independent and equal, separate but inseparable, moving over and under and through, ravishing heart and mind together.”

From the outset, Harriet has believed that any woman with a brain must choose between marriage and the cultivation of her intellect. She sees the withered intellects of her old classmates at an Oxford ­reunion and notes that all have chosen husband over intellect, “heart” over “brain.” Counterpoint provides an answer to her worries. “Bach isn’t a matter of an autocratic virtuoso and a meek accompanist,” says Peter. Harriet comes to believe that a marriage between two intellectually independent people can be fulfilling—that heart and mind are not at odds, that they can coexist.

Most music today is divided between, as Sayers put it, “autocratic virtuoso” and “meek accompanist.” Because modern harmony is all we know, when we listen to the contrapuntal music of Josquin or Bach, it is almost impossible for us not to hear it as a hierarchy. We assume that whatever leads dominates what follows. Very few of us have Peter Wimsey’s contrapuntal intelligence. This makes it harder for us to imagine a sociality neither too homogeneous nor too individualistic, but balanced by the rules of counterpoint. It is this vision of harmony—in music, in marriage, in friendship, in society—that we so desperately need to recover.

John Ahern is a doctoral candidate in musicology at Princeton University.